Recent Applications in Nonlinear Systems

**Chapter 8**

**Abstract**

Nonlinear Resonances in 3D

Nonlinear resonators can have advantages over linear designs including increased sensitivity towards changes in their physical properties and environment, and high quality factors which make them attractive in applications such as mass/ chemical sensors or signal filters. Designing nonlinear structures, however, requires much understanding of nonlinear behavior characteristics of structures. Similarly, the proliferation of 3D or additive manufacturing/printing capabilities has opened the doors to deploying nonlinear resonators on scales not possible earlier. However, to obtain consistent nonlinear dynamic performance the designer must perform a careful analysis to explore the existence and repeatability of desired nonlinear behavior. Also, the use of 3D printing with the associated substrate material properties poses its own challenges in regards to device simulation in view of the fact that most of the traditional literature on nonlinear resonators assumes linear material stiffness. In this chapter, the authors discuss computational design methods for structural design, and specifically study the case of 1:2 internal resonances in resonators made of nonlinear (hyperelastic) materials. The design methods allow for development of large number of candidate resonator designs without a required significant nonlinear structural design experience, and the study of the dynamic response of the resonators provides a glimpse in to the 1:2 nonlinear internal reso-

**Keywords:** nonlinear dynamics, internal resonances, hyperelastic materials, 3D

The development of the micro- and nano-electronics industry coupled with the advances in semiconductor manufacturing techniques led to an interest in developing and applying micro- and nano-electromechanical systems (MEMS and NEMS) [1, 2]. Due to the small length scales of such devices and the popular modes of actuation employed by designers for MEMS or NEMS, such as electrostatic actuation, it led to the observations that nonlinear effects in many of these devices were the norm rather than the exception. This realization led to the study of effects of nonlinearities on the dynamic response of MEMS and NEMS in their various modes of operation, the effects in some cases being detrimental to linear design performance, and in some cases being beneficial to performance [3]. The study of nonlinear dynamics is an area of research with long history in structural and mechanical systems [4]. Several attempts have been made to incorporate the

Printed Structures

*Astitva Tripathi and Anil K. Bajaj*

nance exhibited by the candidate resonators.

printing, topology optimization

**1. Introduction**

**149**

#### **Chapter 8**

## Nonlinear Resonances in 3D Printed Structures

*Astitva Tripathi and Anil K. Bajaj*

#### **Abstract**

Nonlinear resonators can have advantages over linear designs including increased sensitivity towards changes in their physical properties and environment, and high quality factors which make them attractive in applications such as mass/ chemical sensors or signal filters. Designing nonlinear structures, however, requires much understanding of nonlinear behavior characteristics of structures. Similarly, the proliferation of 3D or additive manufacturing/printing capabilities has opened the doors to deploying nonlinear resonators on scales not possible earlier. However, to obtain consistent nonlinear dynamic performance the designer must perform a careful analysis to explore the existence and repeatability of desired nonlinear behavior. Also, the use of 3D printing with the associated substrate material properties poses its own challenges in regards to device simulation in view of the fact that most of the traditional literature on nonlinear resonators assumes linear material stiffness. In this chapter, the authors discuss computational design methods for structural design, and specifically study the case of 1:2 internal resonances in resonators made of nonlinear (hyperelastic) materials. The design methods allow for development of large number of candidate resonator designs without a required significant nonlinear structural design experience, and the study of the dynamic response of the resonators provides a glimpse in to the 1:2 nonlinear internal resonance exhibited by the candidate resonators.

**Keywords:** nonlinear dynamics, internal resonances, hyperelastic materials, 3D printing, topology optimization

#### **1. Introduction**

The development of the micro- and nano-electronics industry coupled with the advances in semiconductor manufacturing techniques led to an interest in developing and applying micro- and nano-electromechanical systems (MEMS and NEMS) [1, 2]. Due to the small length scales of such devices and the popular modes of actuation employed by designers for MEMS or NEMS, such as electrostatic actuation, it led to the observations that nonlinear effects in many of these devices were the norm rather than the exception. This realization led to the study of effects of nonlinearities on the dynamic response of MEMS and NEMS in their various modes of operation, the effects in some cases being detrimental to linear design performance, and in some cases being beneficial to performance [3]. The study of nonlinear dynamics is an area of research with long history in structural and mechanical systems [4]. Several attempts have been made to incorporate the

nonlinear dynamic effects and use the plethora of associated phenomena into operating mechanisms of MEMS devices particularly as mass and/or chemical sensors and filters [3]. While such nonlinear devices have several advantages over linear ones with the same functionality in terms of measurement resolution, there have been challenges in making sure that the designed topologies are "in-tune" with the semiconductor manufacturing processes. At a conceptual level, researchers in nonlinear dynamics often work with lumped-parameter models [3] and many interesting applications have been considered [5, 6] for systems characterized by a single degree of freedom. For systems with more than one degree of freedom, one of the most compact representations is a system exhibiting a nonlinear 1:2 internal resonance, the spring-pendulum system [4, 7]. While this system lends itself very conveniently to a systematic analytical study [8], it is a relatively hard task to replicate it physically at micro- or nano-scales. This is even more so when the structural components fabricated involve two- or three-dimensional elastic structures.

optimization of geometry and material distribution to affect frequency distributions as well as internal resonances [19, 20]. A systematic approach is based on the concepts in "topology optimization" [21]. Applications of topology or shape optimization have now appeared in the literation on nonlinear dynamics as well, with the works in [22–24] focusing on general one-dimensional elastic systems whereas the works in [12, 13] focusing on plate structures with internal resonances. The overall goal is to tailor the system's dynamic response to some desired form for

In the present study, particular classes of resonator designs consisting of rectan-

This work has two following sections: Section 2 describes the design and optimization process which leads to a desired candidate structure. It discusses the aspects of the hyperelastic material model as well as the use of mode shapes to construct the reduced order model. Section 3 describes the development of the structure's Lagrangian, the extraction of the nonlinear equations of motion for the modal amplitudes, and the steady state dynamic response of the system under harmonic excitation of the higher frequency mode. Section 4 contains some con-

The principal objective of the structural synthesis proves is to obtain a resonator design with commensurable natural frequencies. As it is difficult to come up with such a structure by just relying on the researcher's experience, a computational optimization method is proposed to design the candidate resonators. For 1:2 internal resonances, the desired frequency relation between the two modes taking part in

> *ω*1 *ω*2 ¼ 1

example optimization problem for such a task can be represented by

*minimize, c*ð Þ¼ *<sup>ω</sup>* <sup>1</sup>

where *ω*<sup>1</sup> and *ω*<sup>2</sup> are the natural frequencies of the lower and the higher mode, respectively. To obtain such a candidate resonator, the natural frequency requirement represented by Eq. (1) can be formulated as an optimization problem. An

> <sup>2</sup> � *<sup>ω</sup>*<sup>1</sup> *ω*2

<sup>2</sup> (1)

(2)

gular plates with cutouts which can be easily fabricated using 3D printing are analyzed for their nonlinear dynamic response. To obtain a suitable resonator design with commensurable (1:2) natural frequencies, a parametric optimization process which varies the sizes of the cutouts is employed. The natural frequencies themselves are computed using linear finite element analysis (FEA). The resonators are assumed to be made of a Mooney-Rivlin hyperelastic material [25] which is anticipated to provide the material nonlinearity necessary to produce 1:2 internal resonances. Once the optimization process is able to provide a candidate structure, the mode shapes obtained by the finite element analysis are used to build a reduced order model of the resonator displacements. This displacement field can then be used to derive the kinetic and strain energies of the structure which can provide the system Lagrangian. This Lagrangian is then averaged and subjected to the Euler-Lagrange conditions to derive the slow-amplitude equations of motion of the struc-

appropriate external excitations.

*Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

cluding remarks for this work.

**2. Candidate structure synthesis**

the energy transfer can be expressed as:

**151**

ture that provide the dynamic steady state response.

3D printing or additive manufacturing offers several appealing advantages in terms of building devices based on nonlinear dynamic principles. The fabrication processes and dimensions are such that there is better repeatability with complex mechanical designs, as well as initial prototyping and low volume production costs come without the need for significant capital expenditure [9, 10]. Recently studies have been reported for prototyping nonlinear vibratory components made with 3D printers having a feature size of less than 1 mm [11]. While the dimensions of polymeric resonators created by 3D printers may limit their measurement resolution and frequency range of operations, the manufacturing process itself is more repeatable for certain kinds of resonators. However, using 3D printing for producing nonlinear resonators also comes with its own challenges. While micro- and nano-resonators operate in an environment with many kinds of nonlinearities, 3D printed structures have to largely rely on only two sources of nonlinearities, namely, geometric and material nonlinearities unless controlled material variations or composite structures are explicitly introduced in fabrication. Fortunately, as was demonstrated by Tripathi and Bajaj [12, 13], both geometric nonlinearities due to finite deformations and material nonlinearities due to nonlinear hyperelastic properties of the 3D printed material can produce nonlinear dynamic effects such as 1:2 internal resonance.

A 1:2 internal resonance is a popular mechanism exhibited and employed by many nonlinear dynamics based resonators [14]. Internal resonance in a structure refers to the energy transfer that occurs between two modes of the structure when their natural frequencies are almost commensurable and the structure has some appropriate nonlinearity. For example, for 1:2 internal resonance, if the two modes of a structure have their natural frequencies close to the ratio 1:2 and harmonic excitation of the higher mode is above a certain threshold, energy can be transferred from the resonant response of the higher mode to the lower frequency mode in the presence of quadratic nonlinearities. The mathematical description of a 1:2 internal resonance and the dynamics is well established [4, 7, 8]. For the purposes of evaluating the suitability of using 3D printing to produce resonators exhibiting 1:2 internal resonances, it is important to demonstrate that the dynamic response equations for the resonators exhibit the same mathematical characteristics as the cardinal examples.

In the context of elastic structures exhibiting various internal resonances, the present work focuses on elastic plate-type structures [4, 15]. A few representative works on different aspects of nonlinear vibrations of rectangular plates with internal resonances are [16–18]. In general, isotropic plates with different simple boundary conditions do not exhibit any commensurate frequencies unless there exists some type of symmetry of the structure. Some works have considered

#### *Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

nonlinear dynamic effects and use the plethora of associated phenomena into operating mechanisms of MEMS devices particularly as mass and/or chemical sensors and filters [3]. While such nonlinear devices have several advantages over linear ones with the same functionality in terms of measurement resolution, there have been challenges in making sure that the designed topologies are "in-tune" with the semiconductor manufacturing processes. At a conceptual level, researchers in nonlinear dynamics often work with lumped-parameter models [3] and many interesting applications have been considered [5, 6] for systems characterized by a single degree of freedom. For systems with more than one degree of freedom, one of the most compact representations is a system exhibiting a nonlinear 1:2 internal resonance, the spring-pendulum system [4, 7]. While this system lends itself very conveniently to a systematic analytical study [8], it is a relatively hard task to replicate it physically at micro- or nano-scales. This is even more so when the structural compo-

3D printing or additive manufacturing offers several appealing advantages in terms of building devices based on nonlinear dynamic principles. The fabrication processes and dimensions are such that there is better repeatability with complex mechanical designs, as well as initial prototyping and low volume production costs come without the need for significant capital expenditure [9, 10]. Recently studies have been reported for prototyping nonlinear vibratory components made with 3D printers having a feature size of less than 1 mm [11]. While the dimensions of polymeric resonators created by 3D printers may limit their measurement resolution and frequency range of operations, the manufacturing process itself is more repeatable for certain kinds of resonators. However, using 3D printing for producing nonlinear resonators also comes with its own challenges. While micro- and nano-resonators operate in an environment with many kinds of nonlinearities, 3D printed structures have to largely rely on only two sources of nonlinearities, namely, geometric and material nonlinearities unless controlled material variations or composite structures are explicitly introduced in fabrication. Fortunately, as was demonstrated by Tripathi and Bajaj [12, 13], both geometric nonlinearities due to finite deformations and material nonlinearities due to nonlinear hyperelastic properties of the 3D printed material can produce nonlinear dynamic effects such as 1:2

A 1:2 internal resonance is a popular mechanism exhibited and employed by many nonlinear dynamics based resonators [14]. Internal resonance in a structure refers to the energy transfer that occurs between two modes of the structure when their natural frequencies are almost commensurable and the structure has some appropriate nonlinearity. For example, for 1:2 internal resonance, if the two modes of a structure have their natural frequencies close to the ratio 1:2 and harmonic excitation of the higher mode is above a certain threshold, energy can be transferred from the resonant response of the higher mode to the lower frequency mode in the presence of quadratic nonlinearities. The mathematical description of a 1:2 internal resonance and the dynamics is well established [4, 7, 8]. For the purposes of evaluating the suitability of using 3D printing to produce resonators exhibiting 1:2 internal resonances, it is important to demonstrate that the dynamic response equations for the resonators exhibit the same mathematical characteristics as the

In the context of elastic structures exhibiting various internal resonances, the present work focuses on elastic plate-type structures [4, 15]. A few representative works on different aspects of nonlinear vibrations of rectangular plates with internal resonances are [16–18]. In general, isotropic plates with different simple boundary conditions do not exhibit any commensurate frequencies unless there exists some type of symmetry of the structure. Some works have considered

nents fabricated involve two- or three-dimensional elastic structures.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

internal resonance.

cardinal examples.

**150**

optimization of geometry and material distribution to affect frequency distributions as well as internal resonances [19, 20]. A systematic approach is based on the concepts in "topology optimization" [21]. Applications of topology or shape optimization have now appeared in the literation on nonlinear dynamics as well, with the works in [22–24] focusing on general one-dimensional elastic systems whereas the works in [12, 13] focusing on plate structures with internal resonances. The overall goal is to tailor the system's dynamic response to some desired form for appropriate external excitations.

In the present study, particular classes of resonator designs consisting of rectangular plates with cutouts which can be easily fabricated using 3D printing are analyzed for their nonlinear dynamic response. To obtain a suitable resonator design with commensurable (1:2) natural frequencies, a parametric optimization process which varies the sizes of the cutouts is employed. The natural frequencies themselves are computed using linear finite element analysis (FEA). The resonators are assumed to be made of a Mooney-Rivlin hyperelastic material [25] which is anticipated to provide the material nonlinearity necessary to produce 1:2 internal resonances. Once the optimization process is able to provide a candidate structure, the mode shapes obtained by the finite element analysis are used to build a reduced order model of the resonator displacements. This displacement field can then be used to derive the kinetic and strain energies of the structure which can provide the system Lagrangian. This Lagrangian is then averaged and subjected to the Euler-Lagrange conditions to derive the slow-amplitude equations of motion of the structure that provide the dynamic steady state response.

This work has two following sections: Section 2 describes the design and optimization process which leads to a desired candidate structure. It discusses the aspects of the hyperelastic material model as well as the use of mode shapes to construct the reduced order model. Section 3 describes the development of the structure's Lagrangian, the extraction of the nonlinear equations of motion for the modal amplitudes, and the steady state dynamic response of the system under harmonic excitation of the higher frequency mode. Section 4 contains some concluding remarks for this work.

#### **2. Candidate structure synthesis**

The principal objective of the structural synthesis proves is to obtain a resonator design with commensurable natural frequencies. As it is difficult to come up with such a structure by just relying on the researcher's experience, a computational optimization method is proposed to design the candidate resonators. For 1:2 internal resonances, the desired frequency relation between the two modes taking part in the energy transfer can be expressed as:

$$\frac{\alpha\_1}{\alpha\_2} = \frac{1}{2} \tag{1}$$

where *ω*<sup>1</sup> and *ω*<sup>2</sup> are the natural frequencies of the lower and the higher mode, respectively. To obtain such a candidate resonator, the natural frequency requirement represented by Eq. (1) can be formulated as an optimization problem. An example optimization problem for such a task can be represented by

$$\text{minimize}, c(\alpha) = \left(\frac{1}{2} - \frac{\alpha\_1}{\alpha\_2}\right) \tag{2}$$

Thus, the optimization process attempts to minimize the deviation of the two natural frequencies from the perfect 1:2 natural frequency ratio. Solving the optimization problem posed by Eq. (2) would lead to a structure with two of its natural frequencies close to the ratio of 1:2 which is a major requirement for resonators exhibiting 1:2 internal resonance. In this study, two methods for solving the optimization problem are discussed. The first method is a topology optimization method based on simple isotropic material with penalization (SIMP) model and the method of moving asymptotes (MMA) [21]. The second method is a parametric optimization method in which a starting parameterized base structure is chosen whose topology is similar to the final desired candidate structure. Then this base structure is optimized by a nonlinear quadratic programming process to produce a viable candidate structure.

#### **2.1 Topology optimization with SIMP**

Topology optimization techniques have been widely used to solve a broad range of structural optimization problems. While quite versatile, an occasional drawback against topology optimization generated optimal topologies has been the difficulty of their reproduction using conventional manufacturing processes. In this regard 3D printing is eminently suited to produce topologically optimized design as both techniques are adept at producing extruded structures with complex planar patterns. Topology optimization methods are based on finite element discretization of the design spaces. In the context of designing candidate hyperelastic resonators for 1:2 internal resonance, the design space can be discretized with finite elements and the density and material stiffness of the *i th* element in the design space can be written using the SIMP formulation as

$$
\rho\_i = \rho\_{\min} + \mathbf{x}\_i^{\text{n1}} \rho\_0 \tag{3}
$$

$$E\_i = E\_{min} + \varkappa\_i^{n1} E\_0 \tag{4}$$

The ratio of the first two planar natural frequencies of the optimized structure shown in **Figure 2** was 1.99. Thus, the topology optimization process was successful in bringing the natural frequencies of interest close to the ratio of 1:2. As the optimization process uses finite elements to compute the natural frequencies of the structure, the mode shapes of the optimized structure also become available and are

*The optimized structure obtained after applying the topology optimization process to the base structure shown in*

*The base structure used as a starting point for the topology optimization process. The red line indicates the edge*

shown in **Figure 3**.

**Figure 2.**

*Figure 1.*

**153**

**Figure 1.**

*that is fixed.*

*Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

where, *ρ***<sup>0</sup>** is the material density, *E***<sup>0</sup>** is the material stiffness and *xi* is the design variable which varies between 0 and 1. *ρmin* and *Emin* are infinitesimal constants to prevent numerical singularities in case *xi* becomes equal to zero. The exponents, *n1* and *n2* are usually chosen larger than three so as to penalize any intermediate values of the design variable. As can be observed from Eqs. (3) and (4), a value of *xi* ¼ **1**, implies that material is present and *xi* ¼ **0** implies presence of a void. Any intermediate value of *xi* would produce non-physical results which is why a high value of exponents *n1* and *n2* are chosen to initially penalize intermediate values followed by filtering process at the end of optimization in which intermediate properties are taken to one extreme or the other depending on the filter design.

As an example of the topology optimization based resonator generation process, consider the structure shown in **Figure 1**. This structure is a rectangular plate which is constrained at its bottom edge. This plate is assigned Mooney-Rivlin material properties and is meshed with four node planar elements as it is assumed that the plate is undergoing vibrations in its plane.

The ratio of the first two planar natural frequencies of the base structure shown in **Figure 1** was 3.3. The aim of the topology optimization process is to fill the central cavity of the starting structure so that its first two planar natural frequencies are in the ratio close to 1:2. Thus, the design space is discretized with finite elements and the optimization problem posed by Eq. (2) is solved using the method moving asymptotes (MMA) which yields the optimized structure shown in **Figure 2**.

*Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

**Figure 1.**

Thus, the optimization process attempts to minimize the deviation of the two natural frequencies from the perfect 1:2 natural frequency ratio. Solving the optimization problem posed by Eq. (2) would lead to a structure with two of its natural frequencies close to the ratio of 1:2 which is a major requirement for resonators exhibiting 1:2 internal resonance. In this study, two methods for solving the optimization problem are discussed. The first method is a topology optimization method based on simple isotropic material with penalization (SIMP) model and the method of moving asymptotes (MMA) [21]. The second method is a parametric optimization method in which a starting parameterized base structure is chosen whose topology is similar to the final desired candidate structure. Then this base structure is optimized by a nonlinear quadratic programming process to produce a

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

Topology optimization techniques have been widely used to solve a broad range of structural optimization problems. While quite versatile, an occasional drawback against topology optimization generated optimal topologies has been the difficulty of their reproduction using conventional manufacturing processes. In this regard 3D printing is eminently suited to produce topologically optimized design as both techniques are adept at producing extruded structures with complex planar patterns. Topology optimization methods are based on finite element discretization of the design spaces. In the context of designing candidate hyperelastic resonators for 1:2 internal resonance, the design space can be discretized with finite elements and

*<sup>ρ</sup><sup>i</sup>* <sup>¼</sup> *<sup>ρ</sup>min* <sup>þ</sup> *xn***<sup>1</sup>**

*Ei* <sup>¼</sup> *Emin* <sup>þ</sup> *<sup>x</sup>n***<sup>1</sup>**

where, *ρ***<sup>0</sup>** is the material density, *E***<sup>0</sup>** is the material stiffness and *xi* is the design variable which varies between 0 and 1. *ρmin* and *Emin* are infinitesimal constants to prevent numerical singularities in case *xi* becomes equal to zero. The exponents, *n1* and *n2* are usually chosen larger than three so as to penalize any intermediate values of the design variable. As can be observed from Eqs. (3) and (4), a value of *xi* ¼ **1**, implies that material is present and *xi* ¼ **0** implies presence of a void. Any intermediate value of *xi* would produce non-physical results which is why a high value of exponents *n1* and *n2* are chosen to initially penalize intermediate values followed by filtering process at the end of optimization in which intermediate properties are taken to one extreme or the other depending on the filter

As an example of the topology optimization based resonator generation process, consider the structure shown in **Figure 1**. This structure is a rectangular plate which is constrained at its bottom edge. This plate is assigned Mooney-Rivlin material properties and is meshed with four node planar elements as it is assumed that the

The ratio of the first two planar natural frequencies of the base structure shown

in **Figure 1** was 3.3. The aim of the topology optimization process is to fill the central cavity of the starting structure so that its first two planar natural frequencies are in the ratio close to 1:2. Thus, the design space is discretized with finite elements and the optimization problem posed by Eq. (2) is solved using the method moving asymptotes (MMA) which yields the optimized structure shown in **Figure 2**.

*th* element in the design space can be

*<sup>i</sup> ρ***<sup>0</sup>** (3)

*<sup>i</sup> E***<sup>0</sup>** (4)

viable candidate structure.

**2.1 Topology optimization with SIMP**

the density and material stiffness of the *i*

plate is undergoing vibrations in its plane.

written using the SIMP formulation as

design.

**152**

*The base structure used as a starting point for the topology optimization process. The red line indicates the edge that is fixed.*

**Figure 2.**

*The optimized structure obtained after applying the topology optimization process to the base structure shown in Figure 1.*

The ratio of the first two planar natural frequencies of the optimized structure shown in **Figure 2** was 1.99. Thus, the topology optimization process was successful in bringing the natural frequencies of interest close to the ratio of 1:2. As the optimization process uses finite elements to compute the natural frequencies of the structure, the mode shapes of the optimized structure also become available and are shown in **Figure 3**.

function being described by Eq. (2). The design parameters for this optimization were the cut-out size and positions on the cantilever plate and the optimization was performed by a sequential quadratic programming method. The optimized structure obtained by applying the optimization process on the base structure is shown in

The ratio between the third and second natural frequencies of the optimized structure shown in **Figure 5** was 2.0. Thus, the optimization method was able to successfully bring the natural frequencies of the structure close to the desired ratio of 1:2. The mode shapes of the optimized structure also become available from the finite element model and are shown in **Figure 6**. These mode shapes can then be used to construct a reduced order model for the system which will be used to

The method of parametric optimization allows for development a wide range of topologies which can each potentially exhibit 1:2 internal response. The optimal topology obtained depends on the starting structure, reflecting the local optimal nature of the solution. For example, consider the starting structure shown in **Figure 7**, the ratio between the natural frequencies of the higher and lower mode of interest (third and second natural frequencies, respectively) was computed as 1.6. Also note that the boundary conditions in this case involve fixing the resonator

*The optimized structure obtained after applying the parametric optimization process to the base structure shown*

*Mode shapes of the optimized structure shown in Figure 5. (a) Lower Mode (Mode 2 of the structure) (b)*

develop the nonlinear dynamic response for the candidate structure.

**Figure 5**.

*Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

**Figure 5.**

**Figure 6.**

**155**

*Upper Mode (Mode 3 of the structure).*

*in Figure 4.*

**Figure 3.**

*Mode shapes of the optimized structure shown in Figure 2. (a) Lower Mode (Mode 1 of the structure) (b) Upper Mode (Mode 2 of the structure).*

#### **2.2 Parametric optimization**

Parametric optimization process is a simple but powerful tool which can also be used to generate various candidate structures for 1:2 internal resonance. As an example to illustrate the essential aspects of this procedure, consider the base structure shown in **Figure 4**. This base structure consists of a rectangular cantilever plate with two cutouts.

This plate can be assigned Mooney-Rivlin material properties and meshed with four node shell elements. In this study, Abaqus software is used to compute the natural frequencies of the base structure with the frequencies of interest being the second and third natural frequencies respectively. For the base structure shown in **Figure 4**, the ratio between the natural frequencies of the higher and lower mode of interest (third and second natural frequencies, respectively) was computed as 2.4. This base structure was then subjected to an optimization process with the objective

#### **Figure 4.**

*The base structure used as a starting point for the optimization process in parametric optimization. The red line indicates the edge that is fixed.*

*Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

function being described by Eq. (2). The design parameters for this optimization were the cut-out size and positions on the cantilever plate and the optimization was performed by a sequential quadratic programming method. The optimized structure obtained by applying the optimization process on the base structure is shown in **Figure 5**.

The ratio between the third and second natural frequencies of the optimized structure shown in **Figure 5** was 2.0. Thus, the optimization method was able to successfully bring the natural frequencies of the structure close to the desired ratio of 1:2. The mode shapes of the optimized structure also become available from the finite element model and are shown in **Figure 6**. These mode shapes can then be used to construct a reduced order model for the system which will be used to develop the nonlinear dynamic response for the candidate structure.

The method of parametric optimization allows for development a wide range of topologies which can each potentially exhibit 1:2 internal response. The optimal topology obtained depends on the starting structure, reflecting the local optimal nature of the solution. For example, consider the starting structure shown in **Figure 7**, the ratio between the natural frequencies of the higher and lower mode of interest (third and second natural frequencies, respectively) was computed as 1.6. Also note that the boundary conditions in this case involve fixing the resonator

**Figure 5.**

**2.2 Parametric optimization**

*Upper Mode (Mode 2 of the structure).*

**Figure 3.**

**Figure 4.**

**154**

*indicates the edge that is fixed.*

plate with two cutouts.

Parametric optimization process is a simple but powerful tool which can also be

This plate can be assigned Mooney-Rivlin material properties and meshed with four node shell elements. In this study, Abaqus software is used to compute the natural frequencies of the base structure with the frequencies of interest being the second and third natural frequencies respectively. For the base structure shown in **Figure 4**, the ratio between the natural frequencies of the higher and lower mode of interest (third and second natural frequencies, respectively) was computed as 2.4. This base structure was then subjected to an optimization process with the objective

*The base structure used as a starting point for the optimization process in parametric optimization. The red line*

used to generate various candidate structures for 1:2 internal resonance. As an example to illustrate the essential aspects of this procedure, consider the base structure shown in **Figure 4**. This base structure consists of a rectangular cantilever

*Mode shapes of the optimized structure shown in Figure 2. (a) Lower Mode (Mode 1 of the structure) (b)*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*The optimized structure obtained after applying the parametric optimization process to the base structure shown in Figure 4.*

**Figure 6.**

*Mode shapes of the optimized structure shown in Figure 5. (a) Lower Mode (Mode 2 of the structure) (b) Upper Mode (Mode 3 of the structure).*

**3. Nonlinear dynamic response**

*Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

*Upper Mode (Mode 3 of the structure).*

**Figure 9.**

plate can we written as:

**Figure 10.**

*excitation.*

**157**

For the development of the nonlinear dynamic response of a 3D printed structure, consider the structure shown in **Figure 10**. This structure was designed using the simple iterative optimization procedure detailed in Section 2, and the resulting structure's modes 2 and 3 are in near internal resonance of 1:2. Thus, the frequency ratio achieved was 2:0005. The resonator was then fabricated using 3D printing machine Stratsys Dimension 1200es. This structure has the ratio of its second and third natural frequencies as 2.0. The two transverse modes of interest for this candidate structure are shown in **Figure 11**. Using the mode shapes shown in **Figure 11**, assuming that the structure is subjected to a base excitation, and using the Kirchhoff plate theory [15], the displacement at any point on this rectangular

*The candidate structure for which the nonlinear dynamic transverse response is to be developed subject to a base*

*Mode shapes of the optimized structure shown in Figure 8. (a) Lower Mode (Mode 2 of the structure) (b)*

#### **Figure 7.**

*The base structure used as a starting point for the parameter optimization process. The red lines indicate the edges that are fixed.*

#### **Figure 8.**

*The optimal structure obtained after implementing the parametric optimization process on the base structure shown in Figure 7.*

along both of its vertical sides. In this particular case, the optimization parameters were the size and location of the three circular cutouts. After performing the shape optimization process again using the sequential quadratic programming method, the optimized structure obtained is shown in **Figure 8**. The ratio between the third and second natural frequencies of the optimized structure shown in **Figure 8** was 2.0. Thus, the examples of **Figures 5** and **8** illustrate the possibilities of generating a large number of examples with different topologies as candidate resonators for 1:2 internal resonance. The mode shapes of the optimized structure shown in **Figure 8** are shown in **Figure 9**.

*Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

**Figure 9.**

*Mode shapes of the optimized structure shown in Figure 8. (a) Lower Mode (Mode 2 of the structure) (b) Upper Mode (Mode 3 of the structure).*

#### **3. Nonlinear dynamic response**

For the development of the nonlinear dynamic response of a 3D printed structure, consider the structure shown in **Figure 10**. This structure was designed using the simple iterative optimization procedure detailed in Section 2, and the resulting structure's modes 2 and 3 are in near internal resonance of 1:2. Thus, the frequency ratio achieved was 2:0005. The resonator was then fabricated using 3D printing machine Stratsys Dimension 1200es. This structure has the ratio of its second and third natural frequencies as 2.0. The two transverse modes of interest for this candidate structure are shown in **Figure 11**. Using the mode shapes shown in **Figure 11**, assuming that the structure is subjected to a base excitation, and using the Kirchhoff plate theory [15], the displacement at any point on this rectangular plate can we written as:

**Figure 10.**

*The candidate structure for which the nonlinear dynamic transverse response is to be developed subject to a base excitation.*

along both of its vertical sides. In this particular case, the optimization parameters were the size and location of the three circular cutouts. After performing the shape optimization process again using the sequential quadratic programming method, the optimized structure obtained is shown in **Figure 8**. The ratio between the third and second natural frequencies of the optimized structure shown in **Figure 8** was 2.0. Thus, the examples of **Figures 5** and **8** illustrate the possibilities of generating a large number of examples with different topologies as candidate resonators for 1:2 internal resonance. The mode shapes of the optimized structure shown in **Figure 8**

*The optimal structure obtained after implementing the parametric optimization process on the base structure*

*The base structure used as a starting point for the parameter optimization process. The red lines indicate the*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

are shown in **Figure 9**.

**Figure 7.**

**Figure 8.**

**156**

*shown in Figure 7.*

*edges that are fixed.*

can be treated as constants as they do not depend on time. In a similar manner, for a

where, *I*<sup>1</sup> and *I*<sup>2</sup> are the first and second deviatoric invariants, respectively, of the Left Cauchy Green deformation tensor *B*, *J* is the determinant of the deformation gradient, *F*, and *C*10, *C*<sup>01</sup> and *d* are the material constitutive parameters. The deformation gradient *F* is derived from the displacement field of the structure. The relationship between the original coordinates of a point on the plate and the

1 *<sup>d</sup>* ð Þ *<sup>J</sup>* � <sup>1</sup> <sup>2</sup> � �*dXdYdZ* (9)

> *x* ¼ *X* þ *u* (10) *y* ¼ *Y* þ *v* (11) *z* ¼ *Z* þ *w* (12)

*<sup>B</sup>* <sup>¼</sup> *FF<sup>T</sup>* (14)

<sup>3</sup>*I*<sup>1</sup> (15)

<sup>3</sup> *I*<sup>2</sup> (16)

*I*<sup>1</sup> ¼ *tr B*ð Þ (17)

<sup>2</sup> *tr B*ð Þ<sup>2</sup> � *tr B*<sup>2</sup> � � � � (18)

*L* ¼ *T* � *U* (19)

(13)

Mooney-Rivlin material, the strain energy, U can be written as,

Then the deformation gradient *F*, can we written as,

1 þ *∂u ∂X*

*∂v ∂X*

*∂w ∂X* *∂u ∂Y*

*∂w <sup>∂</sup><sup>Y</sup>* <sup>1</sup>

1 þ *∂v ∂Y*

The deviatoric strain invariants of the left Cauchy Green deformation tensor *B*

where *J* is the determinant of the deformation tensor given by Eq. (13), and the

where *tr(B)* refers to the trace of the matrix *B*. Note that using Eqs. (5)–(7) and

(9)–(14), the strain energy of the structure can also be computed. Once the expressions of both the kinetic energy and the strain energy are available, the

This Lagrangian from Eq. (19) will be a nonlinear function of the modal amplitudes owing to the nonlinear nature of the strain energy potential given in Eq. (9). The base excitation of the structure is now assumed to be of the form

*I*<sup>1</sup> ¼ *J* �2

*I*<sup>2</sup> ¼ *J* �4

*<sup>I</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>

*∂u ∂Z*

*∂v ∂Z*

*F* ¼

The left Cauchy Green deformation tensor is given by

*<sup>C</sup>*<sup>10</sup> *<sup>I</sup>*<sup>1</sup> � <sup>3</sup> � � <sup>þ</sup> *<sup>C</sup>*<sup>01</sup> *<sup>I</sup>*<sup>2</sup> � <sup>3</sup> � � <sup>þ</sup>

*U* ¼ ð*V* 0

are written as

**159**

strain invariants *I1* and *I2* are given by

Lagrangian of the resonator can be written as

deformed coordinates can be written as,

*Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

**Figure 11.**

*The mode shapes of the candidate structure shown in Figure 10. The modes are (a) mode 2 and (b) mode 3 respectively of the candidate resonator.*

$$u(X,Y,Z,t) = \varepsilon \left( A\_1(t) \left( u\_{01}(X,Y) - Z \frac{\partial w\_1(X,Y)}{\partial X} \right) + A\_2(t) \left( u\_{02}(X,Y) - Z \frac{\partial w\_2(X,Y)}{\partial X} \right) \right) \tag{5}$$

$$v(X,Y,Z,t) = \varepsilon \left( A\_1(t) \left( v\_{01}(X,Y) - Z \frac{\partial w\_1(X,Y)}{\partial Y} \right) + A\_2(t) \left( v\_{02}(X,Y) - Z \frac{\partial w\_2(X,Y)}{\partial Y} \right) \right) \tag{6}$$

$$w(\mathbf{X}, \mathbf{Y}, \mathbf{Z}, t) = \varepsilon(A\_1(t)w\_1(\mathbf{X}, \mathbf{Y}) + A\_2(t)w\_2(\mathbf{X}, \mathbf{Y})) + w\_f(t) \tag{7}$$

where, *u*, *v* and *w* are the displacements in the *X*-, *Y*- and *Z*-directions, respectively, *A*<sup>1</sup> and *A*<sup>2</sup> are the modal amplitudes, *u01*; *u02* and *v01*; *v02* are the independent in-plane modal displacements in the *X* and *Y* directions, and *w1* and *w2* are the corresponding modal displacements (the mode shapes) in the *Z*- or the transverse direction. The base excitation is applied in the transverse direction (*Z*-direction) and is denoted by *wf* which only depends on time, and *ε* is a small dimensionless parameter to keep track of the significant terms in the system response. It is assumed that the displacement field can be written with linear superposition of the two modes because in the canonical 1:2 internal resonance form, the higher mode is directly excited by external means and the system nonlinearities can cause an energy transfer between the higher mode and the lower mode. All other modes, if present, will see their contribution to the displacement field decay over time in the presence of damping as they are neither directly excited, nor excited by internal energy transfer.

The displacement field given in Eqs. (5)–(7) can now be used to write the kinetic and strain energy of the candidate structure. The kinetic energy,*T* is given by

$$T = \int\_0^V \frac{1}{2} \rho \left(\dot{u}^2 + \dot{v}^2 + \dot{w}^2\right) d\mathbf{X} d\mathbf{Y} d\mathbf{Z} \tag{8}$$

where the dot (. ) represents the derivative of the displacements with respect to time and *ρ* is the material density. For the time derivatives of the displacement, it must be noted that the modal amplitudes will be differentiated and the mode shapes *Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

can be treated as constants as they do not depend on time. In a similar manner, for a Mooney-Rivlin material, the strain energy, U can be written as,

$$U = \int\_{0}^{V} \left\{ \mathbf{C}\_{10}(\tilde{I}\_{1} - \mathbf{3}) + \mathbf{C}\_{01}(\tilde{I}\_{2} - \mathbf{3}) + \frac{\mathbf{1}}{d}(J - \mathbf{1})^{2} \right\} d\mathbf{X} d\mathbf{Y} d\mathbf{Z} \tag{9}$$

where, *I*<sup>1</sup> and *I*<sup>2</sup> are the first and second deviatoric invariants, respectively, of the Left Cauchy Green deformation tensor *B*, *J* is the determinant of the deformation gradient, *F*, and *C*10, *C*<sup>01</sup> and *d* are the material constitutive parameters. The deformation gradient *F* is derived from the displacement field of the structure. The relationship between the original coordinates of a point on the plate and the deformed coordinates can be written as,

$$\mathbf{x} = \mathbf{X} + \mathbf{u} \tag{10}$$

$$\mathbf{y} = \mathbf{Y} + \boldsymbol{\nu} \tag{11}$$

$$z = Z + w \tag{12}$$

Then the deformation gradient *F*, can we written as,

$$F = \begin{bmatrix} \mathbf{1} + \frac{\partial u}{\partial X} & \frac{\partial u}{\partial Y} & \frac{\partial u}{\partial Z} \\ \frac{\partial v}{\partial X} & \mathbf{1} + \frac{\partial v}{\partial Y} & \frac{\partial v}{\partial Z} \\ \frac{\partial w}{\partial X} & \frac{\partial w}{\partial Y} & \mathbf{1} \end{bmatrix} \tag{13}$$

The left Cauchy Green deformation tensor is given by

$$B = F\mathcal{F}^T\tag{14}$$

The deviatoric strain invariants of the left Cauchy Green deformation tensor *B* are written as

$$
\overline{I}\_1 = J^{-\frac{2}{3}} I\_1 \tag{15}
$$

$$
\overline{I}\_2 = J^{-\frac{4}{3}} I\_2 \tag{16}
$$

where *J* is the determinant of the deformation tensor given by Eq. (13), and the strain invariants *I1* and *I2* are given by

$$I\_1 = \text{tr}(B) \tag{17}$$

$$I\_2 = \frac{1}{2} \left( tr(\mathcal{B})^2 - tr(\mathcal{B}^2) \right) \tag{18}$$

where *tr(B)* refers to the trace of the matrix *B*. Note that using Eqs. (5)–(7) and (9)–(14), the strain energy of the structure can also be computed. Once the expressions of both the kinetic energy and the strain energy are available, the Lagrangian of the resonator can be written as

$$L = T - U\tag{19}$$

This Lagrangian from Eq. (19) will be a nonlinear function of the modal amplitudes owing to the nonlinear nature of the strain energy potential given in Eq. (9). The base excitation of the structure is now assumed to be of the form

*u X,Y, Z, t* ð Þ¼ *<sup>ε</sup> <sup>A</sup>*1ð Þ*<sup>t</sup> <sup>u</sup>*01ð Þ� *X,Y <sup>Z</sup> <sup>∂</sup>w*1ð Þ *X,Y*

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*respectively of the candidate resonator.*

*v X,Y, Z, t* ð Þ¼ *<sup>ε</sup> <sup>A</sup>*1ð Þ*<sup>t</sup> <sup>v</sup>*01ð Þ� *X,Y <sup>Z</sup> <sup>∂</sup>w*1ð Þ *X,Y*

energy transfer.

**Figure 11.**

where the dot (.

**158**

*∂X*

*The mode shapes of the candidate structure shown in Figure 10. The modes are (a) mode 2 and (b) mode 3*

*∂Y*

where, *u*, *v* and *w* are the displacements in the *X*-, *Y*- and *Z*-directions, respectively, *A*<sup>1</sup> and *A*<sup>2</sup> are the modal amplitudes, *u01*; *u02* and *v01*; *v02* are the independent in-plane modal displacements in the *X* and *Y* directions, and *w1* and *w2* are the corresponding modal displacements (the mode shapes) in the *Z*- or the transverse direction. The base excitation is applied in the transverse direction (*Z*-direction) and is denoted by *wf* which only depends on time, and *ε* is a small dimensionless parameter to keep track of the significant terms in the system response. It is assumed that the displacement field can be written with linear superposition of the two modes because in the canonical 1:2 internal resonance form, the higher mode is directly excited by external means and the system nonlinearities can cause an energy transfer between the higher mode and the lower mode. All other modes, if present, will see their contribution to the displacement field decay over time in the presence of damping as they are neither directly excited, nor excited by internal

The displacement field given in Eqs. (5)–(7) can now be used to write the kinetic

<sup>2</sup> <sup>þ</sup> *<sup>w</sup>*\_ <sup>2</sup> � �*dXdYdZ* (8)

) represents the derivative of the displacements with respect to

and strain energy of the candidate structure. The kinetic energy,*T* is given by

time and *ρ* is the material density. For the time derivatives of the displacement, it must be noted that the modal amplitudes will be differentiated and the mode shapes

*T* ¼ ð*V* 0 1 2 *ρ u*\_ <sup>2</sup> <sup>þ</sup> *<sup>v</sup>*\_

� � � �

� � � �

*w X, Y, Z, t* ð Þ¼ *ε*ð*A*1ð Þ*t w*1ð Þþ *X, Y A*2ð Þ*t w*2ð Þ *X,Y* Þ þ *wf*ð Þ*t* (7)

<sup>þ</sup> *<sup>A</sup>*2ð Þ*<sup>t</sup> <sup>u</sup>*02ð Þ� *X, Y <sup>Z</sup> <sup>∂</sup>w*2ð Þ *X,Y*

<sup>þ</sup> *<sup>A</sup>*2ð Þ*<sup>t</sup> <sup>v</sup>*02ð Þ� *X,Y <sup>Z</sup> <sup>∂</sup>w*2ð Þ *X, Y*

*∂X*

*∂Y*

(5)

(6)

� �

� �

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

$$
\omega \dot{w}\_f = \varepsilon^2 V\_B \sin\left(\Omega t\right) \tag{20}
$$

*p*0

*Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

*q*0

where a prime (0

transformations are defined

tudes become

<sup>1</sup> <sup>þ</sup> *<sup>ζ</sup>*1*p*<sup>1</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>2</sup>

<sup>1</sup> <sup>þ</sup> *<sup>ζ</sup>*1*q*<sup>1</sup> � *<sup>σ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>2</sup>

*p*0

*a*0

<sup>1</sup> <sup>¼</sup> *<sup>σ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>2</sup> 2

<sup>2</sup> ¼ �*ζ*2*a*<sup>2</sup> � *<sup>Λ</sup>*2*ω*2*a*<sup>2</sup>

<sup>2</sup> <sup>¼</sup> *<sup>σ</sup>*2*a*<sup>2</sup> � *<sup>Λ</sup>*2*ω*2*a*<sup>2</sup>

*a*1*β*<sup>0</sup>

*a*0

*a*2*β*<sup>0</sup>

(37) can be obtained by setting *a*<sup>0</sup>

results is provided in [13].

**161**

*q*0

2

2

<sup>2</sup> <sup>þ</sup> *<sup>ζ</sup>*1*q*<sup>2</sup> � ð Þ *<sup>σ</sup>*<sup>2</sup> *<sup>p</sup>*<sup>2</sup> <sup>þ</sup> *<sup>Λ</sup>*2*ω*<sup>2</sup> *<sup>p</sup>*<sup>2</sup>

represent damping in the system to prevent the solutions from becoming

*<sup>q</sup>*<sup>1</sup> <sup>þ</sup> *<sup>Λ</sup>*1*ω*<sup>1</sup> *<sup>p</sup>*2*q*<sup>1</sup> � *<sup>p</sup>*1*q*<sup>2</sup>

*<sup>p</sup>*<sup>1</sup> <sup>þ</sup> *<sup>Λ</sup>*1*ω*<sup>1</sup> *<sup>p</sup>*1*p*<sup>2</sup> <sup>þ</sup> *<sup>q</sup>*1*q*<sup>2</sup>

*Λi*, *i* = 1, 2, 3, are constants which come from the averaged Lagrangian of the structure and depend on the material constitutive parameters (Eq. (9)) and mode shapes. The modal damping terms *ζ*<sup>1</sup> and *ζ*<sup>2</sup> were introduced in Eqs. (28)–(31) to

unbounded. The expressions in Eqs. (28)–(31) are the same as the expressions in a standard 1:2 internal resonance system [4, 8], thus demonstrating that 3D printed rectangular plates with cutouts are able to exhibit nonlinear dynamic phenomena such as 1:2 internal resonance provided the constants *Λi*, *i* = 1, 2, 3, are non-zero. To make the analysis of Eqs. (28)–(31) a little more tractable, the following variable

where *ai*'s are the amplitudes and *β<sup>i</sup>* are the phase angles. With the transformations introduced in Eqs. (32) and (33), the equations of motion for modal ampli-

The steady-state solutions for the system of equations described by Eqs. (34)–

steady-state solutions to give single-mode (only second modal amplitude *a2* is nonzero) and coupled-mode solutions (both first and second mode amplitudes are nonzero, that is, *a1* 6¼ 0 and *a2* 6¼ 0). The coupled-mode solution is the main nonlinear response as it implies energy transfer from the higher to lower mode when the higher mode is directly excited in the presence of quadratic nonlinearities. Plots in **Figure 12** give a representative set of steady-state solutions for the single and coupled-mode response for the structure shown in **Figure 10** with zero damping. Note that the coupled-mode response is slightly asymmetric about *σ*<sup>2</sup> ¼ 0 axis for the structure as some minimal internal mistuning exists (*σ*<sup>1</sup> 6¼ <sup>0</sup><sup>Þ</sup> due to the fact that *<sup>ω</sup>*<sup>2</sup> *<sup>ω</sup>*<sup>1</sup> ¼ 2:0005. For perfect internal resonance, the coupled-mode solutions will be completely symmetric around *σ*<sup>2</sup> ¼ 0. The non-zero first mode arises as a result of the subcritical pitchfork bifurcations (at *P*<sup>1</sup> and *P*2) from the single mode solution consisting of only the second mode (The lower mode amplitude is zero). A more detailed study of the stability of the solutions for the single and coupled mode

*<sup>i</sup>* ¼ 0 and *β*<sup>0</sup>

<sup>2</sup> þ *ζ*1*p*<sup>2</sup> þ ð Þ *σ*<sup>2</sup> *q*<sup>2</sup> � 2*Λ*2*ω*2*p*1*q*<sup>1</sup> ¼ 0 (30)

*p*<sup>1</sup> ¼ *a*<sup>1</sup> cos *β*<sup>1</sup> ð Þ*, q*<sup>1</sup> ¼ *a*1sin *β*<sup>1</sup> ð Þ (32) *p*<sup>2</sup> ¼ *a*<sup>2</sup> cos *β*<sup>2</sup> ð Þ*, q*<sup>2</sup> ¼ *a*2sin *β*<sup>2</sup> ð Þ (33)

<sup>1</sup> ¼ �*ζ*1*a*<sup>1</sup> � *Λ*1*ω*1*a*1*a*<sup>2</sup> *sin* 2*β*<sup>1</sup> � *β*<sup>2</sup> ð Þ (34)

*<sup>a</sup>*<sup>1</sup> � *<sup>Λ</sup>*1*ω*1*a*1*a*<sup>2</sup> *cos* <sup>2</sup>*β*<sup>1</sup> � *<sup>β</sup>*<sup>2</sup> ð Þ (35)

<sup>1</sup> *sin β*<sup>2</sup> � 2*β*<sup>1</sup> ð Þþ *Λ*3*VB* sin *β*<sup>2</sup> ð Þ (36)

<sup>1</sup> *cos* 2*β*<sup>1</sup> � *β*<sup>2</sup> ð Þþ *Λ*3*VB* cos *β*<sup>2</sup> ð Þ (37)

*<sup>i</sup>* ¼ 0. These equations can be solved for

<sup>1</sup> � *<sup>q</sup>*<sup>2</sup> 1

) denotes a derivative with respect to the slow time *τ*, and.

<sup>¼</sup> 0 (28)

<sup>¼</sup> 0 (29)

<sup>¼</sup> *<sup>Λ</sup>*3*VB* (31)

where *VB* is the amplitude of the base excitation velocity and *Ω* is the excitation frequency. For 1:2 internal resonance, the excitation frequency can be near the lower or the upper natural frequency. In case of subharmonic external resonance, the external frequency is assumed to be close to the upper natural frequency so that the higher mode is resonantly excited [4]. The difference between the excitation frequency and the second natural frequency is known as the external mistuning *σ<sup>2</sup>* defined as

$$
\Delta \mathcal{Q} = a \sigma\_2 + \varepsilon \sigma\_2 \tag{21}
$$

Similarly, another mistuning parameter, the internal mistuning *σ*1, is introduced to take into account the deviation of the two interacting natural frequencies from the perfect 1:2 ratio, that is,

$$
\omega\_2 = 2a\_1 + \varepsilon \sigma\_1 \tag{22}
$$

To further study the nonlinear dynamic response of the structure for small nonlinear motions, and to formulate the application of the method of averaging [4], the modal amplitudes (for Eqs. (5)–(7)) can be written as

$$A\_1(t) = p\_1(et) \cos\left(\frac{\Omega}{2}t\right) + q\_1(et) \sin\left(\frac{\Omega}{2}t\right) \tag{23}$$

$$A\_2(\mathbf{t}) = p\_2(\mathbf{e}t)\cos\left(\Omega \mathbf{t}\right) + q\_2(\mathbf{e}t)\sin\left(\Omega \mathbf{t}\right) \tag{24}$$

where *pi* and *qi* are amplitude components which vary on a slow time scale *τ = εt*, as defined in [4, 8]. Using the expressions for amplitudes in Eqs. (23) and (24), the time derivatives of the amplitudes can be written as

$$\begin{aligned} \dot{A}\_1 &= e \left( p\_1'(\epsilon t) \cos \left( \frac{\Omega}{2} \mathbf{t} \right) + q\_1'(\epsilon t) \sin \left( \frac{\Omega}{2} \mathbf{t} \right) \right) \\ &+ \frac{\Omega}{2} \left( -p\_1(\epsilon t) \sin \left( \frac{\Omega}{2} \mathbf{t} \right) + q\_1(\epsilon t) \cos \left( \frac{\Omega}{2} \mathbf{t} \right) \right) \end{aligned} \tag{25}$$
 
$$\dot{A}\_2 = e \left( p\_2'(\epsilon t) \cos \left( \Omega \mathbf{t} \right) + q\_2'(\epsilon t) \sin \left( \Omega \mathbf{t} \right) \right) + \frac{\Omega}{2} \left( -p\_2(\epsilon t) \sin \left( \Omega \mathbf{t} \right) + q\_2(\epsilon t) \cos \left( \Omega \mathbf{t} \right) \right) \tag{26}$$

where a prime (<sup>0</sup> ) denotes a derivative with respect to the slow time *τ.* Now the Lagrangian given in Eq. (19) is averaged over the period of oscillation<sup>¼</sup> <sup>4</sup>*<sup>π</sup> <sup>Ω</sup>* . The slow time amplitudes are treated as constants for this averaging operation and also only terms till O(*ε*<sup>3</sup><sup>Þ</sup> are retained in the Lagrangian as the cubic nonlinear terms are sufficient to capture the 1:2 internal resonance of the structure. The effects of internal resonance are essentially captured by quadratic nonlinear terms in the equations of motion. The averaged Lagrangian is defined by

$$
\langle L \rangle = \int\_0^{\frac{4\pi}{\alpha}} (T - U)dt \tag{27}
$$

Subjecting the averaged Lagrangian shown in Eq. (27) to the Euler-Lagrange conditions ( *<sup>d</sup> dτ ∂*〈*L*〉 *∂p*0 *i* � *<sup>∂</sup>*〈*L*〉 *∂pi* <sup>¼</sup> 0, *<sup>d</sup> dτ ∂*〈*L*〉 *∂q*0 *i* � *<sup>∂</sup>*〈*L*〉 *∂qi* ¼ 0, *i* = 1, 2) provides the following equations of motion for the slow time amplitudes

*Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

*<sup>w</sup>*\_ *<sup>f</sup>* <sup>¼</sup> *<sup>ε</sup>*<sup>2</sup>

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

defined as

*<sup>A</sup>*\_ <sup>2</sup> <sup>¼</sup> *<sup>ε</sup> <sup>p</sup>*<sup>0</sup>

conditions ( *<sup>d</sup>*

**160**

*dτ ∂*〈*L*〉 *∂p*0 *i* � *<sup>∂</sup>*〈*L*〉 *∂pi*

where a prime (<sup>0</sup>

the perfect 1:2 ratio, that is,

where *VB* is the amplitude of the base excitation velocity and *Ω* is the excitation

Similarly, another mistuning parameter, the internal mistuning *σ*1, is introduced to take into account the deviation of the two interacting natural frequencies from

To further study the nonlinear dynamic response of the structure for small nonlinear motions, and to formulate the application of the method of averaging [4],

> **Ω 2 t** � �

where *pi* and *qi* are amplitude components which vary on a slow time scale *τ = εt*, as defined in [4, 8]. Using the expressions for amplitudes in Eqs. (23) and (24), the

þ *q*<sup>0</sup>

� � � �

Ω

2 t � �

� � � �

the modal amplitudes (for Eqs. (5)–(7)) can be written as

*A***1**ð Þ¼ *t p***1**ð Þ *εt* **cos**

time derivatives of the amplitudes can be written as

<sup>1</sup>ð Þ *εt* cos

equations of motion. The averaged Lagrangian is defined by

<sup>¼</sup> 0, *<sup>d</sup> dτ ∂*〈*L*〉 *∂q*0 *i* � *<sup>∂</sup>*〈*L*〉 *∂qi*

tions of motion for the slow time amplitudes

Ω 2 t � �

Lagrangian given in Eq. (19) is averaged over the period of oscillation<sup>¼</sup> <sup>4</sup>*<sup>π</sup>*

〈*L*〉 ¼

time amplitudes are treated as constants for this averaging operation and also only terms till O(*ε*<sup>3</sup><sup>Þ</sup> are retained in the Lagrangian as the cubic nonlinear terms are sufficient to capture the 1:2 internal resonance of the structure. The effects of internal resonance are essentially captured by quadratic nonlinear terms in the

> ð4*π Ω* 0

Subjecting the averaged Lagrangian shown in Eq. (27) to the Euler-Lagrange

<sup>2</sup> �*p*1ð Þ *<sup>ε</sup><sup>t</sup>* sin <sup>Ω</sup>

*<sup>A</sup>*\_ <sup>1</sup> <sup>¼</sup> *<sup>ε</sup> <sup>p</sup>*<sup>0</sup>

<sup>2</sup>ð Þ *εt* cosð Þþ Ωt *q*<sup>0</sup>

þ Ω

<sup>2</sup>ð Þ *<sup>ε</sup><sup>t</sup>* sin ð Þ <sup>Ω</sup><sup>t</sup> � � <sup>þ</sup>

frequency. For 1:2 internal resonance, the excitation frequency can be near the lower or the upper natural frequency. In case of subharmonic external resonance, the external frequency is assumed to be close to the upper natural frequency so that the higher mode is resonantly excited [4]. The difference between the excitation frequency and the second natural frequency is known as the external mistuning *σ<sup>2</sup>*

*VB* sin ð Þ *Ωt* (20)

*Ω* ¼ *ω*<sup>2</sup> þ *εσ*<sup>2</sup> (21)

*ω*<sup>2</sup> ¼ 2*ω*<sup>1</sup> þ *εσ*<sup>1</sup> (22)

**2 t** � �

> Ω 2 t

<sup>2</sup> �*p*2ð Þ *<sup>ε</sup><sup>t</sup>* sin ð Þþ <sup>Ω</sup><sup>t</sup> *<sup>q</sup>*2ð Þ *<sup>ε</sup><sup>t</sup>* cosð Þ <sup>Ω</sup><sup>t</sup> � �

ð Þ *T* � *U dt* (27)

¼ 0, *i* = 1, 2) provides the following equa-

(23)

(25)

(26)

*<sup>Ω</sup>* . The slow

<sup>þ</sup> *<sup>q</sup>***1**ð Þ *<sup>ε</sup><sup>t</sup>* **sin <sup>Ω</sup>**

*A***2**ð Þ¼ *t p***2**ð Þ *εt* **cos** ð Þþ **Ωt** *q***2**ð Þ *εt* **sin** ð Þ **Ωt** (24)

<sup>1</sup>ð Þ *<sup>ε</sup><sup>t</sup>* sin <sup>Ω</sup>

þ *q*1ð Þ *εt* cos

) denotes a derivative with respect to the slow time *τ.* Now the

2 t

$$(p\_1' + \zeta\_1 p\_1 + \left(\frac{\sigma\_1 + \sigma\_2}{2}\right) q\_1 + \Lambda\_1 o\_1 (p\_2 q\_1 - p\_1 q\_2) = 0\tag{28}$$

$$q\_1' + \zeta\_1 q\_1 - \left(\frac{\sigma\_1 + \sigma\_2}{2}\right) p\_1 + \Lambda\_1 o\_1(p\_1 p\_2 + q\_1 q\_2) = 0\tag{29}$$

$$p\_2' + \zeta\_1 p\_2 + (\sigma\_2) q\_2 - 2\Lambda\_2 a \rho\_2 p\_1 q\_1 = 0 \tag{30}$$

$$\Lambda q\_2' + \zeta\_1 q\_2 - (\sigma\_2) p\_2 + \Lambda\_2 \alpha\_2 \left(p\_1^2 - q\_1^2\right) = \Lambda\_3 V\_B \tag{31}$$

where a prime (0 ) denotes a derivative with respect to the slow time *τ*, and.

*Λi*, *i* = 1, 2, 3, are constants which come from the averaged Lagrangian of the structure and depend on the material constitutive parameters (Eq. (9)) and mode shapes. The modal damping terms *ζ*<sup>1</sup> and *ζ*<sup>2</sup> were introduced in Eqs. (28)–(31) to represent damping in the system to prevent the solutions from becoming unbounded. The expressions in Eqs. (28)–(31) are the same as the expressions in a standard 1:2 internal resonance system [4, 8], thus demonstrating that 3D printed rectangular plates with cutouts are able to exhibit nonlinear dynamic phenomena such as 1:2 internal resonance provided the constants *Λi*, *i* = 1, 2, 3, are non-zero. To make the analysis of Eqs. (28)–(31) a little more tractable, the following variable transformations are defined

$$p\_1 = a\_1 \cos\left(\beta\_1\right), q\_1 = a\_1 \sin\left(\beta\_1\right) \tag{32}$$

$$p\_2 = a\_2 \cos(\beta\_2), \\ q\_2 = a\_2 \sin(\beta\_2) \tag{33}$$

where *ai*'s are the amplitudes and *β<sup>i</sup>* are the phase angles. With the transformations introduced in Eqs. (32) and (33), the equations of motion for modal amplitudes become

$$a\_1' = -\zeta\_1 a\_1 - \Lambda\_1 a\_1 a\_1 a\_2 \sin\left(2\beta\_1 - \beta\_2\right) \tag{34}$$

$$a\_1 \beta\_1' = \left(\frac{\sigma\_1 + \sigma\_2}{2}\right) a\_1 - \Lambda\_1 a\_1 a\_1 a\_2 \cos\left(2\beta\_1 - \beta\_2\right) \tag{35}$$

$$a\_2' = -\zeta\_2 a\_2 - \Lambda\_2 a\_2 a\_1^2 \sin\left(\beta\_2 - 2\beta\_1\right) + \Lambda\_3 V\_B \sin\left(\beta\_2\right) \tag{36}$$

$$
\sigma\_2 \beta\_2' = \sigma\_2 a\_2 - \Lambda\_2 a\_2 a\_1^2 \cos\left(2\beta\_1 - \beta\_2\right) + \Lambda\_3 V\_B \cos\left(\beta\_2\right) \tag{37}
$$

The steady-state solutions for the system of equations described by Eqs. (34)– (37) can be obtained by setting *a*<sup>0</sup> *<sup>i</sup>* ¼ 0 and *β*<sup>0</sup> *<sup>i</sup>* ¼ 0. These equations can be solved for steady-state solutions to give single-mode (only second modal amplitude *a2* is nonzero) and coupled-mode solutions (both first and second mode amplitudes are nonzero, that is, *a1* 6¼ 0 and *a2* 6¼ 0). The coupled-mode solution is the main nonlinear response as it implies energy transfer from the higher to lower mode when the higher mode is directly excited in the presence of quadratic nonlinearities. Plots in **Figure 12** give a representative set of steady-state solutions for the single and coupled-mode response for the structure shown in **Figure 10** with zero damping. Note that the coupled-mode response is slightly asymmetric about *σ*<sup>2</sup> ¼ 0 axis for the structure as some minimal internal mistuning exists (*σ*<sup>1</sup> 6¼ <sup>0</sup><sup>Þ</sup> due to the fact that *<sup>ω</sup>*<sup>2</sup> *<sup>ω</sup>*<sup>1</sup> ¼ 2:0005. For perfect internal resonance, the coupled-mode solutions will be completely symmetric around *σ*<sup>2</sup> ¼ 0. The non-zero first mode arises as a result of the subcritical pitchfork bifurcations (at *P*<sup>1</sup> and *P*2) from the single mode solution consisting of only the second mode (The lower mode amplitude is zero). A more detailed study of the stability of the solutions for the single and coupled mode results is provided in [13].

#### **Figure 12.**

*Non-linear response of the 3D printed structure shown in Figure 10 to a transverse harmonic base excitation. The plots are for the amplitudes of the two interacting modes for both the single-mode and coupled-mode response. Note that σ*<sup>1</sup> 6¼ *0 though very small and the modal damping is low (*ζ*<sup>1</sup> and* ζ*<sup>2</sup> equal to 0.05).*

**Figure 14a**, increasing damping leads to a reduction in the frequency range in

*Non-linear response curves of the hyperelastic structure in Figure 10 for representative low (red) and higher (blue) damping coefficients. The two figures are for (a) mode 1 amplitudes (b) mode 2 amplitudes. The points*

This work explored the possibility of synthesizing 3D printed hyperelastic plate structure exhibiting 1:2 internal resonances. 3D printing occupies a potential sweet spot in terms of dimensional capabilities and repeatability to produce nonlinear resonators which can be used as vibration absorbers, sensors, or for signal

processing applications. The synthesis methodology allows for designing a large set of designs meeting the desired internal resonance conditions resulting in complex

While the nonlinear dynamical response studied here was focused on 3D printed cantilever plates that exhibited 1:2 internal resonances on account of material nonlinearities, the methodology can be easily applied to other boundary conditions and internal resonances, as well as for structures with geometric nonlinearities caused

modal coupling and energy transfer between modes of the structure.

which the nonlinear coupled-mode response is observed.

**4. Summary and conclusions**

*T1 and T2 are turning point bifurcations.*

*Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

**Figure 14.**

by finite deformations of plates.

The authors declare no conflict of interest.

**Conflict of interest**

**163**

As is clear from Eqs. (34)–(37), the modal amplitudes depend upon many parameters. Some of the more interesting of these are the internal mistuning *σ*<sup>1</sup> and the modal damping terms *ζ*<sup>1</sup> and *ζ*2. The effect of change in internal mistuning is shown in **Figure 13**. As can be observed from **Figure 13**, changes in internal mistuning can result in the coupled-mode motion to lose existence and disappear, that is, the modal interaction is lost for large internal mistuning. Thus, in actual physical systems deviation of natural frequencies of participating modes from the perfect 1:2 ratio can cause the non-trivial coupled mode response to not manifest itself.

**Figure 14** shows the effect of damping coefficients on the nonlinear response curves obtained using Eqs. (34)–(37). Increasing damping coefficients *ζ*<sup>1</sup> and *ζ*<sup>2</sup> has interesting effects on the overall nonlinear response. As can be observed from

#### **Figure 13.**

*Non-linear response curves of the 3D printed (hyperelastic) structure shown in Figure 10 for (a) large negative internal mistuning, and (b) large positive internal mistuning.*

*Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

**Figure 14.**

As is clear from Eqs. (34)–(37), the modal amplitudes depend upon many parameters. Some of the more interesting of these are the internal mistuning *σ*<sup>1</sup> and the modal damping terms *ζ*<sup>1</sup> and *ζ*2. The effect of change in internal mistuning is shown in **Figure 13**. As can be observed from **Figure 13**, changes in internal mistuning can result in the coupled-mode motion to lose existence and disappear, that is, the modal interaction is lost for large internal mistuning. Thus, in actual physical systems deviation of natural frequencies of participating modes from the perfect 1:2 ratio can cause the non-trivial coupled mode response to not manifest

*Non-linear response of the 3D printed structure shown in Figure 10 to a transverse harmonic base excitation. The plots are for the amplitudes of the two interacting modes for both the single-mode and coupled-mode response. Note that σ*<sup>1</sup> 6¼ *0 though very small and the modal damping is low (*ζ*<sup>1</sup> and* ζ*<sup>2</sup> equal to 0.05).*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

**Figure 14** shows the effect of damping coefficients on the nonlinear response curves obtained using Eqs. (34)–(37). Increasing damping coefficients *ζ*<sup>1</sup> and *ζ*<sup>2</sup> has interesting effects on the overall nonlinear response. As can be observed from

*Non-linear response curves of the 3D printed (hyperelastic) structure shown in Figure 10 for (a) large negative*

*internal mistuning, and (b) large positive internal mistuning.*

itself.

**Figure 12.**

**Figure 13.**

**162**

*Non-linear response curves of the hyperelastic structure in Figure 10 for representative low (red) and higher (blue) damping coefficients. The two figures are for (a) mode 1 amplitudes (b) mode 2 amplitudes. The points T1 and T2 are turning point bifurcations.*

**Figure 14a**, increasing damping leads to a reduction in the frequency range in which the nonlinear coupled-mode response is observed.

#### **4. Summary and conclusions**

This work explored the possibility of synthesizing 3D printed hyperelastic plate structure exhibiting 1:2 internal resonances. 3D printing occupies a potential sweet spot in terms of dimensional capabilities and repeatability to produce nonlinear resonators which can be used as vibration absorbers, sensors, or for signal processing applications. The synthesis methodology allows for designing a large set of designs meeting the desired internal resonance conditions resulting in complex modal coupling and energy transfer between modes of the structure.

While the nonlinear dynamical response studied here was focused on 3D printed cantilever plates that exhibited 1:2 internal resonances on account of material nonlinearities, the methodology can be easily applied to other boundary conditions and internal resonances, as well as for structures with geometric nonlinearities caused by finite deformations of plates.

#### **Conflict of interest**

The authors declare no conflict of interest.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

**References**

[1] Santuria SD. Microsystem Design.

*Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

[11] Grappasonni C, Habib G,

[Accessed: May 25, 2019]

10.1115/1.4028268

2008;**54**:31-52

263-278

internal resonances in in-plane vibrations of plates with hyperelastic materials. Journal of Vibration and Acoustics. 2014;**136**:061005. DOI:

[13] Tripathi A, Bajaj AK. Topology optimization and internal resonances in transverse vibrations of hyperelastic plates. International Journal of Solids and Structures. 2016;**81**:311-328

[14] Vyas A, Peroulis D, Bajaj AK. Dynamics of a nonlinear microresonator based on resonantly interacting flexuraltorsional modes. Nonlinear Dynamics.

[15] Amabili M. Nonlinear Vibrations and Stability of Shells and Plates. New York: Cambridge University Press; 2008

[16] Ribeiro P, Petyt M. Non-linear free vibration of isotropic plates with

internal resonance. International Journal of Nonlinear Mechanics. 2000;**35**(2):

[17] Amabili M. Nonlinear vibrations of rectangular plates with different boundary conditions: Theory and experiments. Computers & Structures.

[18] Chang SI, Bajaj AK, Krousgrill CM. Non-linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance. Nonlinear Dynamics. 1993;**4**:433-460

2004;**82**(31–32):2587-2605

Detroux T, Kerschen G. Experimental Demonstration of a 3D-Printed Nonlinear Tuned Vibration Absorber [Internet]. Available from: http://citesee rx.ist.psu.edu/viewdoc/download?doi= 10.1.1.1024.23&rep=rep1&type=pdf

[12] Tripathi A, Bajaj AK. Design for 1:2

Microelectromechanical Systems. New

New York: Springer; 2001

York: Springer; 2007

York: Springer; 2011

10.1115/1.4001333

Sons; 2008

[2] Lobontiu N. Dynamics of

[3] Younis MI. MEMS Linear and Nonlinear Statics and Dynamics. New

[4] Nayfeh AH, Mook DT. Nonlinear Oscillations. New York: John Wiley &

[5] Rhoads JF, Shaw SW, Turner KL. Nonlinear dynamics and its applications in micro- and nanoresonators. Journal of Dynamic Systems, Measurement, and Control. 2010;**132**:034001. DOI:

[6] Daqaq MF, Masana R, Erturk A, Quinn DD. On the role of nonlinearities in vibratory energy harvesting: A critical

[7] Bajaj AK, Chang SI, Johnson JM. Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system. Nonlinear

[8] Nayfeh AH. Nonlinear Interactions: Analytical, Computational, and Experimental Methods. 1st ed. New

[9] Galeta T, Raos P, Stojšić J, Pakši I. Influence of structure on mechanical properties of 3D printed objects.

Procedia Engineering. 2016;**149**:100-104

[10] Kotlinski J. Mechanical properties of commercial rapid prototyping materials. Rapid Prototyping Journal. 2014;**20**: 499-510. DOI: 10.1108/RPJ-06-2012-

Dynamics. 1994;**5**:433-457

York: Wiley; 2000

0052

**165**

review and discussion. Applied Mechanics Reviews. 2014;**66**(4): 040801. DOI: 10.1115/1.4026278

#### **Author details**

Astitva Tripathi<sup>1</sup> and Anil K. Bajaj2 \*

1 Caelynx LLC, Ann Arbor, MI, USA

2 School of Mechanical Engineering, West Lafayette, IN, USA

\*Address all correspondence to: bajaj@ecn.purdue.edu

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Nonlinear Resonances in 3D Printed Structures DOI: http://dx.doi.org/10.5772/intechopen.88934*

#### **References**

[1] Santuria SD. Microsystem Design. New York: Springer; 2001

[2] Lobontiu N. Dynamics of Microelectromechanical Systems. New York: Springer; 2007

[3] Younis MI. MEMS Linear and Nonlinear Statics and Dynamics. New York: Springer; 2011

[4] Nayfeh AH, Mook DT. Nonlinear Oscillations. New York: John Wiley & Sons; 2008

[5] Rhoads JF, Shaw SW, Turner KL. Nonlinear dynamics and its applications in micro- and nanoresonators. Journal of Dynamic Systems, Measurement, and Control. 2010;**132**:034001. DOI: 10.1115/1.4001333

[6] Daqaq MF, Masana R, Erturk A, Quinn DD. On the role of nonlinearities in vibratory energy harvesting: A critical review and discussion. Applied Mechanics Reviews. 2014;**66**(4): 040801. DOI: 10.1115/1.4026278

[7] Bajaj AK, Chang SI, Johnson JM. Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system. Nonlinear Dynamics. 1994;**5**:433-457

[8] Nayfeh AH. Nonlinear Interactions: Analytical, Computational, and Experimental Methods. 1st ed. New York: Wiley; 2000

[9] Galeta T, Raos P, Stojšić J, Pakši I. Influence of structure on mechanical properties of 3D printed objects. Procedia Engineering. 2016;**149**:100-104

[10] Kotlinski J. Mechanical properties of commercial rapid prototyping materials. Rapid Prototyping Journal. 2014;**20**: 499-510. DOI: 10.1108/RPJ-06-2012- 0052

[11] Grappasonni C, Habib G, Detroux T, Kerschen G. Experimental Demonstration of a 3D-Printed Nonlinear Tuned Vibration Absorber [Internet]. Available from: http://citesee rx.ist.psu.edu/viewdoc/download?doi= 10.1.1.1024.23&rep=rep1&type=pdf [Accessed: May 25, 2019]

[12] Tripathi A, Bajaj AK. Design for 1:2 internal resonances in in-plane vibrations of plates with hyperelastic materials. Journal of Vibration and Acoustics. 2014;**136**:061005. DOI: 10.1115/1.4028268

[13] Tripathi A, Bajaj AK. Topology optimization and internal resonances in transverse vibrations of hyperelastic plates. International Journal of Solids and Structures. 2016;**81**:311-328

[14] Vyas A, Peroulis D, Bajaj AK. Dynamics of a nonlinear microresonator based on resonantly interacting flexuraltorsional modes. Nonlinear Dynamics. 2008;**54**:31-52

[15] Amabili M. Nonlinear Vibrations and Stability of Shells and Plates. New York: Cambridge University Press; 2008

[16] Ribeiro P, Petyt M. Non-linear free vibration of isotropic plates with internal resonance. International Journal of Nonlinear Mechanics. 2000;**35**(2): 263-278

[17] Amabili M. Nonlinear vibrations of rectangular plates with different boundary conditions: Theory and experiments. Computers & Structures. 2004;**82**(31–32):2587-2605

[18] Chang SI, Bajaj AK, Krousgrill CM. Non-linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance. Nonlinear Dynamics. 1993;**4**:433-460

**Author details**

**164**

Astitva Tripathi<sup>1</sup> and Anil K. Bajaj2

1 Caelynx LLC, Ann Arbor, MI, USA

provided the original work is properly cited.

\*

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

2 School of Mechanical Engineering, West Lafayette, IN, USA

\*Address all correspondence to: bajaj@ecn.purdue.edu

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

[19] Pedersen NL. Optimization of holes in plates for control of eigenfrequencies. Structural and Multidisciplinary Optimization. 2004;**28**:1-10. DOI: 10.1007/s00158-004-0426-8

[20] Pedersen NL. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization. 2005;**30**:297-307. DOI: 10.1007/s00158-005-0529-x

[21] Bendsoe M, Sigmund O. Topology Optimization: Theory, Methods and Applications. New York: Springer; 2003

[22] Dai X, Miao X, Sui L, Zhou H, Zhao X, Ding G. Tuning of nonlinear vibration via topology variation and its application in energy harvesting. Applied Physics Letters. 2012;**100**: 031902. DOI: 10.1063/1.3676661

[23] Dou S, Strachan BS, Shaw SW, Jensen JS. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society, A. 2015;**373**:20140408

[24] Lily LL, Polunin PM, Dou S, Shoshani O, Strachan BS, Jensen JS, et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters. 2017;**110**:081902

[25] Breslavsky IV, Amabili M, Legrand M. Nonlinear vibrations of thin hyperelastic plates. Journal of Sound and Vibration. 2015;**333**:4668-4681

**167**

**Chapter 9**

**Abstract**

Prediction

*and Ricardo L. Armentano*

Nonlinear Systems in Healthcare

Healthcare is one of the key fields that works quite strongly with advanced analytical techniques for prediction of diseases and risks. Data being the most important asset in recent times, a huge amount of health data is being collected, thanks to the recent advancements of IoT, smart healthcare, etc. But the focal objective lies in making sense of that data and to obtain knowledge, using intelligent analytics. Nonlinear systems find use specifically in this field, working closely with health data. Using advanced methods of machine learning and computational intelligence, nonlinear analysis performs a key role in analyzing the enormous amount of data, aimed at finding important patterns and predicting diseases. Especially in the field of smart healthcare, this chapter explores some aspects of nonlinear systems in predictive analytics, providing a holistic view of the field as well as some examples

towards Intelligent Disease

*Parag Chatterjee, Leandro J. Cymberknop* 

to illustrate such intelligent systems toward disease prediction.

 *—Desiderius Erasmus*

cardiometabolic disease, Parkinson's disease

*"Prevention is better than cure"*

health and in disease states [4].

**1. Introduction**

**Keywords:** nonlinear systems, healthcare, artificial intelligence, computational intelligence, machine learning, predictive analytics, chronic disease, cancer,

A nonlinear system is a system in which the change of the output is not proportional to the change of the input [1–3]. Especially in the field of healthcare, most of the health systems being inherently nonlinear in nature, nonlinear systems are of special interest to researchers hailing from multidisciplinary areas. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Nonlinear modeling still has not been able to explain all of the complexity present in human systems, and further models still need to be refined and developed. However, nonlinear modeling is helping to explain some system behaviors that linear systems cannot and thus will augment our understanding of the nature of complex dynamic systems within the human body in

#### **Chapter 9**

[19] Pedersen NL. Optimization of holes in plates for control of eigenfrequencies.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

[20] Pedersen NL. Designing plates for

[21] Bendsoe M, Sigmund O. Topology Optimization: Theory, Methods and Applications. New York: Springer; 2003

[22] Dai X, Miao X, Sui L, Zhou H, Zhao X, Ding G. Tuning of nonlinear vibration via topology variation and its application in energy harvesting. Applied Physics Letters. 2012;**100**: 031902. DOI: 10.1063/1.3676661

[23] Dou S, Strachan BS, Shaw SW, Jensen JS. Structural optimization for

Philosophical Transactions of the Royal

Shoshani O, Strachan BS, Jensen JS, et al. Tailoring the nonlinear response of MEMS resonators using shape

optimization. Applied Physics Letters.

Legrand M. Nonlinear vibrations of thin hyperelastic plates. Journal of Sound and Vibration. 2015;**333**:4668-4681

nonlinear dynamic response.

Society, A. 2015;**373**:20140408

[24] Lily LL, Polunin PM, Dou S,

[25] Breslavsky IV, Amabili M,

2017;**110**:081902

**166**

Structural and Multidisciplinary Optimization. 2004;**28**:1-10. DOI: 10.1007/s00158-004-0426-8

minimum internal resonances. Structural and Multidisciplinary Optimization. 2005;**30**:297-307. DOI:

10.1007/s00158-005-0529-x

## Nonlinear Systems in Healthcare towards Intelligent Disease Prediction

*Parag Chatterjee, Leandro J. Cymberknop and Ricardo L. Armentano*

#### **Abstract**

Healthcare is one of the key fields that works quite strongly with advanced analytical techniques for prediction of diseases and risks. Data being the most important asset in recent times, a huge amount of health data is being collected, thanks to the recent advancements of IoT, smart healthcare, etc. But the focal objective lies in making sense of that data and to obtain knowledge, using intelligent analytics. Nonlinear systems find use specifically in this field, working closely with health data. Using advanced methods of machine learning and computational intelligence, nonlinear analysis performs a key role in analyzing the enormous amount of data, aimed at finding important patterns and predicting diseases. Especially in the field of smart healthcare, this chapter explores some aspects of nonlinear systems in predictive analytics, providing a holistic view of the field as well as some examples to illustrate such intelligent systems toward disease prediction.

**Keywords:** nonlinear systems, healthcare, artificial intelligence, computational intelligence, machine learning, predictive analytics, chronic disease, cancer, cardiometabolic disease, Parkinson's disease

#### **1. Introduction**

*"Prevention is better than cure"*

 *—Desiderius Erasmus*

A nonlinear system is a system in which the change of the output is not proportional to the change of the input [1–3]. Especially in the field of healthcare, most of the health systems being inherently nonlinear in nature, nonlinear systems are of special interest to researchers hailing from multidisciplinary areas. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Nonlinear modeling still has not been able to explain all of the complexity present in human systems, and further models still need to be refined and developed. However, nonlinear modeling is helping to explain some system behaviors that linear systems cannot and thus will augment our understanding of the nature of complex dynamic systems within the human body in health and in disease states [4].

The delivery of healthcare is a complex endeavor at both individual and population levels. At the clinical level, the tailored provision of care to individuals is guided, in part, by medical history, examination, vital signs and evidence. In the twenty-first century these traditional tenets have been supplemented by a focus on learning, metrics and quality improvement. The collection and analysis of data of good quality are critical to improvements in the effectiveness and efficiency of health care delivery [5]. This is also catalyzed by the boost in the field of eHealth across the world. eHealth is emerging as a promising vehicle to address the limited capacity of the health care system to provide health behavior change and chronic disease management interventions. The field of eHealth holds promise for supporting and enabling health behavior change and the prevention and management of chronic disease [6].

With a global increase in the adoption of Electronic Health Records (EHRs) [7–12], the volume and complexity of the data generated increases in all dimensions. In addition to the EHR-sourced patient data, the additional data available from other sources like the data about medical conditions, underlying genetics, medications, and treatment approaches is humongous. But human cognition to learn, understand, and process the data being finite [13], the traditional medical methods of analysis does not stand always to be the most efficient. Thus, computerassisted methods to organize, interpret, and recognize patterns from these data are needed [14].

In the recent years, the underlying value of data is unfolding like never before and newer systems are being developed concentrating on the data analysis to make sense of the data. Especially in the field of healthcare, the aspect of intelligent data analytics is one of the most trending topics worldwide. One of the focal areas where such analyses have been applied is in the field of chronic diseases. By 2020, chronic diseases are expected to contribute to 73% of all deaths worldwide and 60% of the global burden of disease. Moreover, 79% of the deaths attributed to these diseases occur in the developing countries. Four of the most prominent chronic diseases—cardiovascular diseases (CVD), cancer, chronic obstructive pulmonary disease and type 2 diabetes are linked by common and preventable biological risk factors, notably high blood pressure, high blood cholesterol and overweight, and by related major behavioral risk factors. Action to prevent these major chronic diseases should focus on controlling these and other key risk factors in a well-integrated manner [15]. Apart from the chronic diseases, a key area where nonlinear models are applied from the perspective of intelligent prediction is human movement and locomotion. This leads to topics like fall detection, abnormal gait detection and diseases like Parkinson's. The outreach of intelligent prediction is spread to wider domains like transplantations [16], for example, to predict the success of a liver transplant by analyzing all the relevant health parameters.

Moreover, with the recent trends of smart sensors and eHealth devices powered by the Internet of Things (IoT), the data acquired is more comprehensive and detailed. Both prediction and prevention systems in this case usually use some fundamental steps in common, like collection of data from sensors and its analysis, followed by computing the risk and other possibilities [17]. The entire pool of data originating from this field is mostly nonlinear, invoking the need for the development of nonlinear analysis and predictive models.

Exploring the possible actions toward prevention of the chronic diseases, the key challenge lies in early detection of the diseases. Most of these diseases do not exhibit clearly identifiable signs at the early stage. This leads to harvesting the possibility of early detection of these diseases using artificial intelligence (AI). From the perspective of data science, the fundamental and most valuable resource in this aspect is the health data.

**169**

*Nonlinear Systems in Healthcare towards Intelligent Disease Prediction*

**2. Data modeling in healthcare toward predictive analysis**

*"Data is the new oil. It's valuable, but if unrefined it cannot really be used." —Clive Humby*

The backbone of the intelligent prediction systems in healthcare is the data. Thanks to the skyrocketing advancements in data collection strategies and tools, it estimates a yearly growth of 48% [18] with projected growth rate to more than 2000 exabytes by 2020 [19]. On one hand, this poses an enormous challenge to handle this data. But on the other hand, this also keeps the potential in performing detailed analysis toward interesting insights. To understand the health data from a holistic view, the attributes like volume, variety, velocity, and veracity need to be considered, to decipher its value. This implies that on a larger scale, handling the health data as big data is inevitable. The first part is making this data suitable for analysis. About 80% of the world's healthcare data being unstructured [20], it poses a huge challenge for the data preprocessing. Health data obtained from multiple sources often lack seriously in the aspect of interoperability or uniformity to model and process together. Even with the rise of EHRs, a major challenge lies in normalizing the data and making it suitable for modeling. With a lot of heterogeneous smart eHealth devices and components of IoT, tying the data together stands extremely difficult. For this reason, several EHR management system are being designed recently considering the data models and the aspect of preprocessing; thus, the EHRs are collected in such a way that fits the data models or facilitates the same. An inclusive data preprocessing system holds immense potential to support the aspect of health data modeling from a comprehensive perspective. Especially in the case of nonlinear systems, the aspect of data modeling is critical for processing enormous health data. Badly constructed data models not only skew the results but may also

The traditional way modeling the health data includes the field of data mining, with the aspects of knowledge discovery in databases (KDD) and exploratory data analysis (EDA). Most of these old KDD techniques visualize data from a perspective of database, looking for interesting knowledge in the data from a high-level point of view. These techniques provide better view to the data and especially in the field of healthcare, is often quite useful in analyzing patterns in large health datasets. However, specifically in the area of prediction of diseases and risks, recent systems are focused toward predictive models to perform a more formal, scalable and more efficient analysis and prediction. Comparing

The health data holds immense potential for detailed analyses towards the early detection and prediction of diseases. The prediction of diseases using computational intelligence is multilevel. Most of the health systems being inherently nonlinear in nature, it provides an enormous opportunity to analyze those intricate details of the health systems while searching for the traits or early signals of diseases. On the other hand, given that the importance of health data (mostly the superficially and non-invasively obtained behavioral, physiological and metabolic health data) is quite crucial, a huge opportunity lies on the aspect of analysis of this health data toward the prediction of diseases. The computational aspect of disease prediction is also multifold, including aspects of data analysis, signal and image processing and other fields. However, this chapter is focused to the aspect of data analytics and computational intelligence, highlighting the key aspects of the health data of people pertaining to nonlinear systems, discussing the field of machine learning and intel-

*DOI: http://dx.doi.org/10.5772/intechopen.88163*

ligence toward the prediction of diseases.

produce erroneous insights appearing to be correct.

*Nonlinear Systems in Healthcare towards Intelligent Disease Prediction DOI: http://dx.doi.org/10.5772/intechopen.88163*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

chronic disease [6].

needed [14].

The delivery of healthcare is a complex endeavor at both individual and population levels. At the clinical level, the tailored provision of care to individuals is guided, in part, by medical history, examination, vital signs and evidence. In the twenty-first century these traditional tenets have been supplemented by a focus on learning, metrics and quality improvement. The collection and analysis of data of good quality are critical to improvements in the effectiveness and efficiency of health care delivery [5]. This is also catalyzed by the boost in the field of eHealth across the world. eHealth is emerging as a promising vehicle to address the limited capacity of the health care system to provide health behavior change and chronic disease management interventions. The field of eHealth holds promise for supporting and enabling health behavior change and the prevention and management of

With a global increase in the adoption of Electronic Health Records (EHRs) [7–12], the volume and complexity of the data generated increases in all dimensions. In addition to the EHR-sourced patient data, the additional data available from other sources like the data about medical conditions, underlying genetics, medications, and treatment approaches is humongous. But human cognition to learn, understand, and process the data being finite [13], the traditional medical methods of analysis does not stand always to be the most efficient. Thus, computerassisted methods to organize, interpret, and recognize patterns from these data are

In the recent years, the underlying value of data is unfolding like never before and newer systems are being developed concentrating on the data analysis to make sense of the data. Especially in the field of healthcare, the aspect of intelligent data analytics is one of the most trending topics worldwide. One of the focal areas where such analyses have been applied is in the field of chronic diseases. By 2020, chronic diseases are expected to contribute to 73% of all deaths worldwide and 60% of the global burden of disease. Moreover, 79% of the deaths attributed to these diseases occur in the developing countries. Four of the most prominent chronic diseases—cardiovascular diseases (CVD), cancer, chronic obstructive pulmonary disease and type 2 diabetes are linked by common and preventable biological risk factors, notably high blood pressure, high blood cholesterol and overweight, and by related major behavioral risk factors. Action to prevent these major chronic diseases should focus on controlling these and other key risk factors in a well-integrated manner [15]. Apart from the chronic diseases, a key area where nonlinear models are applied from the perspective of intelligent prediction is human movement and locomotion. This leads to topics like fall detection, abnormal gait detection and diseases like Parkinson's. The outreach of intelligent prediction is spread to wider domains like transplantations [16], for example, to predict the success of a liver

Moreover, with the recent trends of smart sensors and eHealth devices powered

Exploring the possible actions toward prevention of the chronic diseases, the key challenge lies in early detection of the diseases. Most of these diseases do not exhibit clearly identifiable signs at the early stage. This leads to harvesting the possibility of early detection of these diseases using artificial intelligence (AI). From the perspective of data science, the fundamental and most valuable resource in this aspect is

by the Internet of Things (IoT), the data acquired is more comprehensive and detailed. Both prediction and prevention systems in this case usually use some fundamental steps in common, like collection of data from sensors and its analysis, followed by computing the risk and other possibilities [17]. The entire pool of data originating from this field is mostly nonlinear, invoking the need for the develop-

transplant by analyzing all the relevant health parameters.

ment of nonlinear analysis and predictive models.

**168**

the health data.

The health data holds immense potential for detailed analyses towards the early detection and prediction of diseases. The prediction of diseases using computational intelligence is multilevel. Most of the health systems being inherently nonlinear in nature, it provides an enormous opportunity to analyze those intricate details of the health systems while searching for the traits or early signals of diseases. On the other hand, given that the importance of health data (mostly the superficially and non-invasively obtained behavioral, physiological and metabolic health data) is quite crucial, a huge opportunity lies on the aspect of analysis of this health data toward the prediction of diseases. The computational aspect of disease prediction is also multifold, including aspects of data analysis, signal and image processing and other fields. However, this chapter is focused to the aspect of data analytics and computational intelligence, highlighting the key aspects of the health data of people pertaining to nonlinear systems, discussing the field of machine learning and intelligence toward the prediction of diseases.

#### **2. Data modeling in healthcare toward predictive analysis**

*"Data is the new oil. It's valuable, but if unrefined it cannot really be used." —Clive Humby*

The backbone of the intelligent prediction systems in healthcare is the data. Thanks to the skyrocketing advancements in data collection strategies and tools, it estimates a yearly growth of 48% [18] with projected growth rate to more than 2000 exabytes by 2020 [19]. On one hand, this poses an enormous challenge to handle this data. But on the other hand, this also keeps the potential in performing detailed analysis toward interesting insights. To understand the health data from a holistic view, the attributes like volume, variety, velocity, and veracity need to be considered, to decipher its value. This implies that on a larger scale, handling the health data as big data is inevitable. The first part is making this data suitable for analysis. About 80% of the world's healthcare data being unstructured [20], it poses a huge challenge for the data preprocessing. Health data obtained from multiple sources often lack seriously in the aspect of interoperability or uniformity to model and process together. Even with the rise of EHRs, a major challenge lies in normalizing the data and making it suitable for modeling. With a lot of heterogeneous smart eHealth devices and components of IoT, tying the data together stands extremely difficult. For this reason, several EHR management system are being designed recently considering the data models and the aspect of preprocessing; thus, the EHRs are collected in such a way that fits the data models or facilitates the same. An inclusive data preprocessing system holds immense potential to support the aspect of health data modeling from a comprehensive perspective. Especially in the case of nonlinear systems, the aspect of data modeling is critical for processing enormous health data. Badly constructed data models not only skew the results but may also produce erroneous insights appearing to be correct.

The traditional way modeling the health data includes the field of data mining, with the aspects of knowledge discovery in databases (KDD) and exploratory data analysis (EDA). Most of these old KDD techniques visualize data from a perspective of database, looking for interesting knowledge in the data from a high-level point of view. These techniques provide better view to the data and especially in the field of healthcare, is often quite useful in analyzing patterns in large health datasets. However, specifically in the area of prediction of diseases and risks, recent systems are focused toward predictive models to perform a more formal, scalable and more efficient analysis and prediction. Comparing

with the traditional data mining models, these predictive models are made in a tailored-approach, mostly dedicated to a specific goal (for example, prediction of cardiovascular diseases). Often, such models also involve machine learning and computational intelligence for the aspect of analysis and prediction.

One of the most crucial tasks in this respect is to identify the mathematical model of a system from measurements of the system inputs and outputs. Especially in the field of disease and risk prediction, the data handled is mostly multidimensional. Keeping the focus toward nonlinear modeling, the first check is of course the identification of the data model, if it is a linear model or a nonlinear one. This is usually done using the superposition principle (properties of additivity and homogeneity). However, in some cases for finding the pattern in the health data, linear models do count useful. A more crude but common approach is to start with a linear model. After the initial tests to check the suitability of the linear model, which if turns out not good enough, leads to the replacement by a nonlinear approach to model the system.

Among several nonlinear models used in analyzing health data especially aimed at intelligent disease prediction, methods like nonlinear regression, clustering, decision trees, nonlinear support vector machines (SVMs), and artificial neural networks (ANNs) are quite profuse in recent times within the field of predictive medical analytics. For example, in nonlinear regression, the observational health data is modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables. The data is fitted by a method of successive approximations. If the health data available is labeled, supervised learning models like SVMs are used often, to analyze the data using classification and regression analysis. Given a set of training examples, each marked as belonging to one or the other of two categories, a SVM training algorithm builds a model that assigns new examples to one category or the other. Training data is usually divided into a training data (70%) and test data (30%). To map nonlinear functions, kernels can be used in SVMs. A kernel is a function that maps the data into a higher dimensional space where the linear mapping is possible. One of the main advantages of SVM with respect to modeling nonlinear systems is the possibility to use kernels, making it possible to represent very complex functions. However, when compared to linear regression, the main drawback is the need of more training and prediction times [21].

Neural networks on the other hand encompass a large class of models and learning methods and are nonlinear statistical models [22]. A recent survey of AI applications in healthcare reported uses in major disease areas such as cancer or cardiology and artificial neural networks as a common machine learning technique [23]. Such networks are organized in layers made of a number of interconnected nodes which contain an activation function. Data is provided to the network via the input layer, following which, the processing is performed in one or more hidden layers using a system of weighted connections. The last hidden layer is linked to the output layer where the result is given [21]. The healthcare domain of intelligent risk prediction is largely governed by the aspect of pattern recognition or finding relationships among several health and behavioral parameters and to study their impacts. One of the principal advantages of ANN (**Figure 1**) [24] is that it can model different types of relationships; systems which otherwise may have been very difficult to represent correctly could be modeled quickly and relatively easily using ANNs. However, compared to other types of networks, ANNs tend to be slower in training. Despite being a system of parallel computation, the slowness of the training step is due to the fact that individual artificial neurons are usually processed sequentially [21].

Ensemble classifiers are constructed from a given training data set and predict the class of a previously unseen object by combining the predictions obtained from

**171**

**Figure 2.**

*Working principle of random forest.*

**Figure 1.**

*Artificial neural networks.*

*Nonlinear Systems in Healthcare towards Intelligent Disease Prediction*

these basic classifiers. The importance of different ensemble classifiers has also been at rise attributing to the possibility of determining the risk groups among patient population. For example, the family of simple probabilistic classifiers like naive Bayes classifiers discover application in programmed medicinal analysis [25]. Performing regular analysis of healthcare for a large population makes it possible to act early in the case of health hazards and risks [26]. Clinical decision support systems often count useful in this aspect to assist the medical personnel in designing treatment strategies [27]. Such systems are mostly constructed using decision trees. Decision trees are flowchart structures in which each internal node denotes a test on a characteristic, each branch signifies the result of the test, and each leaf node denotes a class label. The paths from the root to leaf denotes classification rules [25]. Random forest algorithms (**Figure 2**) [21] are specifically suited for decision tree classifiers. In this technique, the basic classifiers are decision trees obtained by manipulating the input features. Basically, random forest builds multiple decision

*DOI: http://dx.doi.org/10.5772/intechopen.88163*

#### *Nonlinear Systems in Healthcare towards Intelligent Disease Prediction DOI: http://dx.doi.org/10.5772/intechopen.88163*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

model the system.

ing and prediction times [21].

with the traditional data mining models, these predictive models are made in a tailored-approach, mostly dedicated to a specific goal (for example, prediction of cardiovascular diseases). Often, such models also involve machine learning and

One of the most crucial tasks in this respect is to identify the mathematical model of a system from measurements of the system inputs and outputs. Especially in the field of disease and risk prediction, the data handled is mostly multidimensional. Keeping the focus toward nonlinear modeling, the first check is of course the identification of the data model, if it is a linear model or a nonlinear one. This is usually done using the superposition principle (properties of additivity and homogeneity). However, in some cases for finding the pattern in the health data, linear models do count useful. A more crude but common approach is to start with a linear model. After the initial tests to check the suitability of the linear model, which if turns out not good enough, leads to the replacement by a nonlinear approach to

Among several nonlinear models used in analyzing health data especially aimed

Neural networks on the other hand encompass a large class of models and learning methods and are nonlinear statistical models [22]. A recent survey of AI applications in healthcare reported uses in major disease areas such as cancer or cardiology and artificial neural networks as a common machine learning technique [23]. Such networks are organized in layers made of a number of interconnected nodes which contain an activation function. Data is provided to the network via the input layer, following which, the processing is performed in one or more hidden layers using a system of weighted connections. The last hidden layer is linked to the output layer where the result is given [21]. The healthcare domain of intelligent risk prediction is largely governed by the aspect of pattern recognition or finding relationships among several health and behavioral parameters and to study their impacts. One of the principal advantages of ANN (**Figure 1**) [24] is that it can model different types of relationships; systems which otherwise may have been very difficult to represent correctly could be modeled quickly and relatively easily using ANNs. However, compared to other types of networks, ANNs tend to be slower in training. Despite being a system of parallel computation, the slowness of the training step is due to the

fact that individual artificial neurons are usually processed sequentially [21].

Ensemble classifiers are constructed from a given training data set and predict the class of a previously unseen object by combining the predictions obtained from

at intelligent disease prediction, methods like nonlinear regression, clustering, decision trees, nonlinear support vector machines (SVMs), and artificial neural networks (ANNs) are quite profuse in recent times within the field of predictive medical analytics. For example, in nonlinear regression, the observational health data is modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables. The data is fitted by a method of successive approximations. If the health data available is labeled, supervised learning models like SVMs are used often, to analyze the data using classification and regression analysis. Given a set of training examples, each marked as belonging to one or the other of two categories, a SVM training algorithm builds a model that assigns new examples to one category or the other. Training data is usually divided into a training data (70%) and test data (30%). To map nonlinear functions, kernels can be used in SVMs. A kernel is a function that maps the data into a higher dimensional space where the linear mapping is possible. One of the main advantages of SVM with respect to modeling nonlinear systems is the possibility to use kernels, making it possible to represent very complex functions. However, when compared to linear regression, the main drawback is the need of more train-

computational intelligence for the aspect of analysis and prediction.

**170**

these basic classifiers. The importance of different ensemble classifiers has also been at rise attributing to the possibility of determining the risk groups among patient population. For example, the family of simple probabilistic classifiers like naive Bayes classifiers discover application in programmed medicinal analysis [25]. Performing regular analysis of healthcare for a large population makes it possible to act early in the case of health hazards and risks [26]. Clinical decision support systems often count useful in this aspect to assist the medical personnel in designing treatment strategies [27]. Such systems are mostly constructed using decision trees. Decision trees are flowchart structures in which each internal node denotes a test on a characteristic, each branch signifies the result of the test, and each leaf node denotes a class label. The paths from the root to leaf denotes classification rules [25]. Random forest algorithms (**Figure 2**) [21] are specifically suited for decision tree classifiers. In this technique, the basic classifiers are decision trees obtained by manipulating the input features. Basically, random forest builds multiple decision

**Figure 1.** *Artificial neural networks.*

**Figure 2.** *Working principle of random forest.*

trees and merges them together to get a more accurate and stable prediction. A key advantage of random forest is, that it can be used for both classification and regression problems; a huge part of current machine learning systems in healthcare is related to such problems. However, the major limitation of random forest algorithms is that a large number of trees can make the algorithm slow down and ineffective for real-time predictions. In general, these algorithms are fast to train, but quite slow to create predictions once they are trained. A more accurate prediction requires more trees, which results in a slower model [28].

Unsupervised learning techniques are also profuse in health analytics especially in the field of analyzing health risks. Since a considerable part of health-data often arrive unlabeled, unsupervised learning methods help in finding patterns in the data or to analyze the health scenario over a big population. In this aspect, clustering techniques like k-means, gaussian distribution models, and mean-shift clustering often stand very useful in separating a group of patients into different clusters and then to analyze in detail the salient features and distinct characteristics. Among all the unsupervised learning algorithms, clustering via k-means might be one of the simplest and most widely used algorithms. Briefly, k-means clustering aims to find the set of k clusters such that every data point is assigned to the closest center, and the sum of the distances of all such assignments is minimized [29]. Especially when the relationships among different health parameters and their respective impacts are not known well, clustering techniques are used to separate a patient population in order to study the distinguishing features and influencing factors. In this aspect, nonlinear techniques of clustering also count useful.

Another crucial aspect with respect to machine learning algorithms is the aspect of bias and variance. The bias is an error from erroneous assumptions in the learning algorithm. High bias can cause an algorithm to miss the relevant relations between features and target outputs (underfitting). The variance is an error from sensitivity to small fluctuations in the training set. To visualize the degree to which a machine learning algorithm is suffering from bias or variance with respect to a data problem, learning curves are important tools. Learning curves are displays in which the performance of the machine learning algorithms are plotted with respect to the quantity of data used for training where the plotted values are the prediction error measurements [30].

Nevertheless, the choice of algorithm is dependent on multiple factors, the most important being the type of the dataset. Apart from that the aspect of prior knowledge about the data, computational complexity and expected results are also deciding factors, and the correct use of the model is extremely crucial in this regard [31]. Recent research has delved into uniting different techniques to provide hybrid machine learning algorithms [32]. Nevertheless, it is clear that the use of machine learning and computational intelligence takes an active role in predicting the health risks and the probability of diseases using the intelligence hidden in the health data.

#### **3. Applications of computational intelligence toward prediction of diseases**

The use of computational intelligence with an objective to predict the health risks and diseases is an extensive and multi-step process. In this section, different scenarios are explained with respect to predictive analytics, aimed at disease and risk prediction. The first case is based on cardiometabolic diseases, and therefore, the most important component of the predictive system stands to be the detailed physiological and health data.

For example, in the field of cardiometabolic diseases, several parameters (age, gender, systolic and diastolic blood pressure, cholesterol, diabetes and smoking

**173**

in time.

*Nonlinear Systems in Healthcare towards Intelligent Disease Prediction*

habits) are considered for patients registered in the database and their risk scores was calculated. The risk scores give a general idea to classify the entire population into high and low risk groups [33]. Nevertheless, alternative analyses are performed to identify the underlying risk groups for each health parameter in the entire population. But this analysis is scalable for a further detailed technique for prediction of risks and influence of different health parameters on the cardiometabolic disease of a population. For example, Framingham Risk Score is used to predict the hardcoronary heart diseases (myocardial infarction or coronary death) and is calculated based on predictors like age, total cholesterol, high-density lipoproteins (HDL), systolic blood pressure, treatment for hypertension and smoking status [34]. In this case, the Framingham Risk can be expressed as a linear equation considering all the parameters. However, considering sample sets of people which are smaller and more specific, there exists the possibility of nonlinear relationships of several other parameters pertaining to cardiovascular risks, not usually considered in classical

The traditional approach based on the identification and treatment of risk factors has proven to be insufficient and ignores that the detection of its subclinical stage is valid to define cardiovascular risk strategies. Taking into account that the artery is the main protagonist in this disease, it is necessary to evaluate it directly through a morpho-structural and functional analysis with non-invasive, reliable, reproducible procedures that are applicable in the youngest population. The detection of subclinical disease and the precocity with which it is done defines a safe framework to derive the real individual cardiovascular risk. Because coronary calcifications are an early marker of atherosclerosis detectable non-invasively; a model of cardiovascular risk that incorporates them along with the classic risk factors could have a remarkable interest in clinical practice, having the potential to change the field of preventive cardiology. The traditional approach guides the prevention and treatment of arterial disease, atherosclerotic in particular, based on the detection of cardiovascular risk factors (e.g., hypertension, smoking, dyslipidemia etc.). This approach quantifies the probability (risk) that the subject has a cardiovascular event (accident) in the next 10 years of life. Thus, based on information from large populations and global cardiovascular risk tables, and information obtained regarding the risk factors of each subject (for example, blood pressure, blood lipids (LDL, HDL), etc.), this can be classified in one of three possible categories: low risk, intermediate risk or high risk. However, this method has limitations since it does not take into account the individual cardiovascular risk and to detect early atherosclerosis and other alterations of the arterial wall. It has been demonstrated that a significant number of people considered to be at intermediate risk with the traditional approach, in fact, have a high risk of presenting a cardiovascular event (for example, they have significant atheroma plaques in the coronary arteries). Moreover, quantitatively most deaths due to cardiovascular causes occur in subjects who present low or intermediate risk, evaluated by the traditional approach. This underestimation of individual risk, which shows the traditional risk quantification approach, determines that millions of people do not receive adequate medical treatment every day to reduce their cardiovascular risk. In other words, asymptomatic subjects, but vulnerable to having a cardiovascular or cerebrovascular accident in the short term, are not offered the benefits of available prophylactic therapies, because they have underestimated their real cardiovascular risk. For example, hypertension is considered an asymptotic disease and is easy to detect; however, it has serious and lethal complications if it is not treated

To demonstrate how coronary artery calcium (CAC) can be incorporated into the risk of traditional was calculated in 618 male patients, the Framingham model

*DOI: http://dx.doi.org/10.5772/intechopen.88163*

risk prediction models.

#### *Nonlinear Systems in Healthcare towards Intelligent Disease Prediction DOI: http://dx.doi.org/10.5772/intechopen.88163*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

tion requires more trees, which results in a slower model [28].

this aspect, nonlinear techniques of clustering also count useful.

trees and merges them together to get a more accurate and stable prediction. A key advantage of random forest is, that it can be used for both classification and regression problems; a huge part of current machine learning systems in healthcare is related to such problems. However, the major limitation of random forest algorithms is that a large number of trees can make the algorithm slow down and ineffective for real-time predictions. In general, these algorithms are fast to train, but quite slow to create predictions once they are trained. A more accurate predic-

Unsupervised learning techniques are also profuse in health analytics especially in the field of analyzing health risks. Since a considerable part of health-data often arrive unlabeled, unsupervised learning methods help in finding patterns in the data or to analyze the health scenario over a big population. In this aspect, clustering techniques like k-means, gaussian distribution models, and mean-shift clustering often stand very useful in separating a group of patients into different clusters and then to analyze in detail the salient features and distinct characteristics. Among all the unsupervised learning algorithms, clustering via k-means might be one of the simplest and most widely used algorithms. Briefly, k-means clustering aims to find the set of k clusters such that every data point is assigned to the closest center, and the sum of the distances of all such assignments is minimized [29]. Especially when the relationships among different health parameters and their respective impacts are not known well, clustering techniques are used to separate a patient population in order to study the distinguishing features and influencing factors. In

Another crucial aspect with respect to machine learning algorithms is the aspect of bias and variance. The bias is an error from erroneous assumptions in the learning algorithm. High bias can cause an algorithm to miss the relevant relations between features and target outputs (underfitting). The variance is an error from sensitivity to small fluctuations in the training set. To visualize the degree to which a machine learning algorithm is suffering from bias or variance with respect to a data problem, learning curves are important tools. Learning curves are displays in which the performance of the machine learning algorithms are plotted with respect to the quantity of data used for

training where the plotted values are the prediction error measurements [30].

**3. Applications of computational intelligence toward prediction of** 

The use of computational intelligence with an objective to predict the health risks and diseases is an extensive and multi-step process. In this section, different scenarios are explained with respect to predictive analytics, aimed at disease and risk prediction. The first case is based on cardiometabolic diseases, and therefore, the most important component of the predictive system stands to be the detailed

For example, in the field of cardiometabolic diseases, several parameters (age, gender, systolic and diastolic blood pressure, cholesterol, diabetes and smoking

Nevertheless, the choice of algorithm is dependent on multiple factors, the most important being the type of the dataset. Apart from that the aspect of prior knowledge about the data, computational complexity and expected results are also deciding factors, and the correct use of the model is extremely crucial in this regard [31]. Recent research has delved into uniting different techniques to provide hybrid machine learning algorithms [32]. Nevertheless, it is clear that the use of machine learning and computational intelligence takes an active role in predicting the health risks and the probability of diseases using the intelligence hidden in the health data.

**172**

**diseases**

physiological and health data.

habits) are considered for patients registered in the database and their risk scores was calculated. The risk scores give a general idea to classify the entire population into high and low risk groups [33]. Nevertheless, alternative analyses are performed to identify the underlying risk groups for each health parameter in the entire population. But this analysis is scalable for a further detailed technique for prediction of risks and influence of different health parameters on the cardiometabolic disease of a population. For example, Framingham Risk Score is used to predict the hardcoronary heart diseases (myocardial infarction or coronary death) and is calculated based on predictors like age, total cholesterol, high-density lipoproteins (HDL), systolic blood pressure, treatment for hypertension and smoking status [34]. In this case, the Framingham Risk can be expressed as a linear equation considering all the parameters. However, considering sample sets of people which are smaller and more specific, there exists the possibility of nonlinear relationships of several other parameters pertaining to cardiovascular risks, not usually considered in classical risk prediction models.

The traditional approach based on the identification and treatment of risk factors has proven to be insufficient and ignores that the detection of its subclinical stage is valid to define cardiovascular risk strategies. Taking into account that the artery is the main protagonist in this disease, it is necessary to evaluate it directly through a morpho-structural and functional analysis with non-invasive, reliable, reproducible procedures that are applicable in the youngest population. The detection of subclinical disease and the precocity with which it is done defines a safe framework to derive the real individual cardiovascular risk. Because coronary calcifications are an early marker of atherosclerosis detectable non-invasively; a model of cardiovascular risk that incorporates them along with the classic risk factors could have a remarkable interest in clinical practice, having the potential to change the field of preventive cardiology. The traditional approach guides the prevention and treatment of arterial disease, atherosclerotic in particular, based on the detection of cardiovascular risk factors (e.g., hypertension, smoking, dyslipidemia etc.). This approach quantifies the probability (risk) that the subject has a cardiovascular event (accident) in the next 10 years of life. Thus, based on information from large populations and global cardiovascular risk tables, and information obtained regarding the risk factors of each subject (for example, blood pressure, blood lipids (LDL, HDL), etc.), this can be classified in one of three possible categories: low risk, intermediate risk or high risk. However, this method has limitations since it does not take into account the individual cardiovascular risk and to detect early atherosclerosis and other alterations of the arterial wall. It has been demonstrated that a significant number of people considered to be at intermediate risk with the traditional approach, in fact, have a high risk of presenting a cardiovascular event (for example, they have significant atheroma plaques in the coronary arteries). Moreover, quantitatively most deaths due to cardiovascular causes occur in subjects who present low or intermediate risk, evaluated by the traditional approach. This underestimation of individual risk, which shows the traditional risk quantification approach, determines that millions of people do not receive adequate medical treatment every day to reduce their cardiovascular risk. In other words, asymptomatic subjects, but vulnerable to having a cardiovascular or cerebrovascular accident in the short term, are not offered the benefits of available prophylactic therapies, because they have underestimated their real cardiovascular risk. For example, hypertension is considered an asymptotic disease and is easy to detect; however, it has serious and lethal complications if it is not treated in time.

To demonstrate how coronary artery calcium (CAC) can be incorporated into the risk of traditional was calculated in 618 male patients, the Framingham model and the probability that the CAC of each patient falls in every four categories of CAC (0, 1–100, 101–400 and >400) using linear and nonlinear regression models. Then they were adjusted based on a relative risk (RR) that weighted the risk of coronary heart disease in individuals and that are RR = 1.7 (for a CAC of 1–100), RR = 3.0 (for CAC 101–400), and RR = 4.3 (for CAC > 400) obtained from a meta-analysis published by Fletcher. The predictive power was evaluated using ROC curves (receiver operating characteristic). The model included in the CAC has a remarkable predictive value of atherosclerosis of 0.74, which is the area of the ROC curve as a function of the number of sites with extracoronal plates including carotid, femoral and abdominal aorta (coded as 0–1 sites = 0; 2–3 sites = 1). The predictive scale indicated 0.90–1 = excellent, 0.80–0.90 = good, 0.70–0.80 = median, 0.60–0.70 = weak, 0.50–0.60 = zero. The calcium score is a numerical information that allows quantifying the magnitude of coronary atherosclerotic lesions and provides independent predictive information of risk factors in general mortality. The combination of modeling of the CAC with the modeling of conventional risk factors leads to a remarkable improvement in the predictive value of the overall risk assessment of Framingham through the reclassification of the risk of atherosclerosis to a degree that may be clinically important. Adding to this approach, the other indices of subclinical atherosclerosis such as arterial rigidity, intima media thickness, endothelial function, and the presence of plaques will generate an integrative risk that will determine and classify the subjects in relation to short-term risk of suffering a cardiovascular or cerebrovascular accident. It will allow to know in a more precise way the cardiovascular risk of a particular individual. It allows early detection (subclinical stage) of vascular alterations and offer the best current prophylactic therapies available [35–37].

All standard risk assessment models to predict cardiovascular diseases make an implicit assumption that each risk factor is related in a linear fashion to CVD outcomes [38]. Such models may thus oversimplify complex relationships which include large numbers of risk factors with nonlinear interactions [39]. The aspect of computational intelligence comes into play specifically in this situation to decipher the inherent patterns and relationships among the parameters apart from the known and formally specified set, thereby determining more nuanced relationships between risk factors and outcomes. Current approaches to predict cardiovascular risk fail to identify many people who would benefit from preventive treatment, while others receive unnecessary intervention. Machine learning offers the opportunity to improve accuracy by exploiting complex interactions between the risk factors [39]. The established ACC/AHA 10-year risk prediction model used eight core baseline variables (gender, age, smoking status, systolic blood pressure, blood pressure treatment, total cholesterol, HDL cholesterol, and diabetes). However, in [39], additional 22 variables (like Body Mass Index, Triglycerides, C-reactive protein, Serum fibrinogen, etc.) were included in the machine learning algorithms, aimed at finding the influence on cardiovascular diseases, thereby designing the predictive algorithm for the same. Among other machine learning algorithms, neural networks had a 3.6% improved prediction than the existing algorithm. The system of cardiovascular risk prediction varies widely based on geographical factors. Therefore, several risk scores have been developed in different parts of the world like the SCORE by the European Society of Cardiology or the HellenicSCORE in Greece to address more accurately a specific group of population for calculating the cardiovascular risks. The majority of these scores use a common set of the 'classical' CVD risk factors, e.g., age, sex, smoking, blood pressure and lipids levels, whereas others have also incorporated more advanced markers of CVDs.

Most of these risk-prediction tools are based on stochastic models, incorporating variables, based on cohort studies [40]. However, the alternative approaches of

**175**

accuracy.

**4. Conclusion**

*Nonlinear Systems in Healthcare towards Intelligent Disease Prediction*

machine learning like k-nearest neighbors, random forests and decision tress also generate results quite comparable to the classical risk prediction scores [41], thus demonstrating its possibility as alternative methods of CVD risk prediction along

With the rise in the prevalence of hypertension globally and its associativity with other parameters of cardiovascular risks, computational techniques like feedforward ANNs are used to model systolic blood pressure, diastolic blood pressure and pulse pressure variations with biological parameters like age, pulse rate, alcohol addiction, and physical activity level. In this aspect, ANN approaches provided more flexible and nonlinear models for prognosis and prediction of the blood pressure parameters than classical statistical algorithms [42]. Even with the increase of complex cardiovascular diseases, using machine learning models like random forest, ANNs, SVMs and Bayesian Networks to predict the in-hospital length of stay provides a positive impact on healthcare metrics [43]. Nonlinear models of unsupervised learning like clustering are commonly used in stratification of patient population and knowledge extraction from different groups. This is highly relevant in the prediction of risks because individuals with similar characteristics often pres-

The aspect of computational intelligence through nonlinear machine learning model even applies to other fields like survival prediction in transplantations and early detection of chronic diseases like cancer. Another interesting example of using computational intelligence and predictive analysis is the prediction of neurodegenerative diseases like Parkinson's disease. Disorders like Parkinson's disease and essential tremor which affect the normal movements of a person share some symptoms or manifestations that make the process of discrimination between them a difficult task. Clinical experience of the medical doctor is crucial at the moment of giving an accurate diagnosis. And still in such a case, that diagnosis is subjective and could be contaminated by several factors beyond the usual capacity of a medical personnel to analyze efficiently [45]. Especially with the use of wearable IoT-based sensors, data obtained about a patient's movement is extensive and complex. Nevertheless, it provides huge scope for using computational intelligence toward the prediction or early detection of such diseases. A major challenge in this aspect is the early detection of such disorders based on the patient's data obtained over a period of time, tracking its changes or finding patterns exhibiting similar trends of having the disease. In this case also, linear models do not count useful always since the parameters are quite dynamic and it needs the provision to continuously analyze other non-formalized parameters to find interesting traits leading to prediction. Thus, the aspect of computational intelligence is not only helpful in designing a better model for analysis but is also useful in prediction of diseases with higher

Nonlinear systems constitute an important part of the area of predictive analytics aimed at diseases and risks for people. In the new age of data and eHealth, the inherent knowledge of data has turned out to be of immense importance, which needs specific methods with computational intelligence. Especially for chronic diseases, long-term behavioral data stands quite crucial. Data modeling and predictive analytics open a huge avenue toward clinical decision support systems, which is a fundamental tool now-a-days for preventive and personalized healthcare and supports healthcare providers to have deeper insights into patients' data [46] and take clinical decisions [33]. Therefore, the use of intelligent prediction is primarily based

*DOI: http://dx.doi.org/10.5772/intechopen.88163*

with its added advantages.

ent a similar risk profile [44].

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

the best current prophylactic therapies available [35–37].

All standard risk assessment models to predict cardiovascular diseases make an implicit assumption that each risk factor is related in a linear fashion to CVD outcomes [38]. Such models may thus oversimplify complex relationships which include large numbers of risk factors with nonlinear interactions [39]. The aspect of computational intelligence comes into play specifically in this situation to decipher the inherent patterns and relationships among the parameters apart from the known and formally specified set, thereby determining more nuanced relationships between risk factors and outcomes. Current approaches to predict cardiovascular risk fail to identify many people who would benefit from preventive treatment, while others receive unnecessary intervention. Machine learning offers the opportunity to improve accuracy by exploiting complex interactions between the risk factors [39]. The established ACC/AHA 10-year risk prediction model used eight core baseline variables (gender, age, smoking status, systolic blood pressure, blood pressure treatment, total cholesterol, HDL cholesterol, and diabetes). However, in [39], additional 22 variables (like Body Mass Index, Triglycerides, C-reactive protein, Serum fibrinogen, etc.) were included in the machine learning algorithms, aimed at finding the influence on cardiovascular diseases, thereby designing the predictive algorithm for the same. Among other machine learning algorithms, neural networks had a 3.6% improved prediction than the existing algorithm. The system of cardiovascular risk prediction varies widely based on geographical factors. Therefore, several risk scores have been developed in different parts of the world like the SCORE by the European Society of Cardiology or the HellenicSCORE in Greece to address more accurately a specific group of population for calculating the cardiovascular risks. The majority of these scores use a common set of the 'classical' CVD risk factors, e.g., age, sex, smoking, blood pressure and lipids levels,

whereas others have also incorporated more advanced markers of CVDs.

Most of these risk-prediction tools are based on stochastic models, incorporating variables, based on cohort studies [40]. However, the alternative approaches of

and the probability that the CAC of each patient falls in every four categories of CAC (0, 1–100, 101–400 and >400) using linear and nonlinear regression models. Then they were adjusted based on a relative risk (RR) that weighted the risk of coronary heart disease in individuals and that are RR = 1.7 (for a CAC of 1–100), RR = 3.0 (for CAC 101–400), and RR = 4.3 (for CAC > 400) obtained from a meta-analysis published by Fletcher. The predictive power was evaluated using ROC curves (receiver operating characteristic). The model included in the CAC has a remarkable predictive value of atherosclerosis of 0.74, which is the area of the ROC curve as a function of the number of sites with extracoronal plates including carotid, femoral and abdominal aorta (coded as 0–1 sites = 0; 2–3 sites = 1). The predictive scale indicated 0.90–1 = excellent, 0.80–0.90 = good, 0.70–0.80 = median, 0.60–0.70 = weak, 0.50–0.60 = zero. The calcium score is a numerical information that allows quantifying the magnitude of coronary atherosclerotic lesions and provides independent predictive information of risk factors in general mortality. The combination of modeling of the CAC with the modeling of conventional risk factors leads to a remarkable improvement in the predictive value of the overall risk assessment of Framingham through the reclassification of the risk of atherosclerosis to a degree that may be clinically important. Adding to this approach, the other indices of subclinical atherosclerosis such as arterial rigidity, intima media thickness, endothelial function, and the presence of plaques will generate an integrative risk that will determine and classify the subjects in relation to short-term risk of suffering a cardiovascular or cerebrovascular accident. It will allow to know in a more precise way the cardiovascular risk of a particular individual. It allows early detection (subclinical stage) of vascular alterations and offer

**174**

machine learning like k-nearest neighbors, random forests and decision tress also generate results quite comparable to the classical risk prediction scores [41], thus demonstrating its possibility as alternative methods of CVD risk prediction along with its added advantages.

With the rise in the prevalence of hypertension globally and its associativity with other parameters of cardiovascular risks, computational techniques like feedforward ANNs are used to model systolic blood pressure, diastolic blood pressure and pulse pressure variations with biological parameters like age, pulse rate, alcohol addiction, and physical activity level. In this aspect, ANN approaches provided more flexible and nonlinear models for prognosis and prediction of the blood pressure parameters than classical statistical algorithms [42]. Even with the increase of complex cardiovascular diseases, using machine learning models like random forest, ANNs, SVMs and Bayesian Networks to predict the in-hospital length of stay provides a positive impact on healthcare metrics [43]. Nonlinear models of unsupervised learning like clustering are commonly used in stratification of patient population and knowledge extraction from different groups. This is highly relevant in the prediction of risks because individuals with similar characteristics often present a similar risk profile [44].

The aspect of computational intelligence through nonlinear machine learning model even applies to other fields like survival prediction in transplantations and early detection of chronic diseases like cancer. Another interesting example of using computational intelligence and predictive analysis is the prediction of neurodegenerative diseases like Parkinson's disease. Disorders like Parkinson's disease and essential tremor which affect the normal movements of a person share some symptoms or manifestations that make the process of discrimination between them a difficult task. Clinical experience of the medical doctor is crucial at the moment of giving an accurate diagnosis. And still in such a case, that diagnosis is subjective and could be contaminated by several factors beyond the usual capacity of a medical personnel to analyze efficiently [45]. Especially with the use of wearable IoT-based sensors, data obtained about a patient's movement is extensive and complex. Nevertheless, it provides huge scope for using computational intelligence toward the prediction or early detection of such diseases. A major challenge in this aspect is the early detection of such disorders based on the patient's data obtained over a period of time, tracking its changes or finding patterns exhibiting similar trends of having the disease. In this case also, linear models do not count useful always since the parameters are quite dynamic and it needs the provision to continuously analyze other non-formalized parameters to find interesting traits leading to prediction. Thus, the aspect of computational intelligence is not only helpful in designing a better model for analysis but is also useful in prediction of diseases with higher accuracy.

#### **4. Conclusion**

Nonlinear systems constitute an important part of the area of predictive analytics aimed at diseases and risks for people. In the new age of data and eHealth, the inherent knowledge of data has turned out to be of immense importance, which needs specific methods with computational intelligence. Especially for chronic diseases, long-term behavioral data stands quite crucial. Data modeling and predictive analytics open a huge avenue toward clinical decision support systems, which is a fundamental tool now-a-days for preventive and personalized healthcare and supports healthcare providers to have deeper insights into patients' data [46] and take clinical decisions [33]. Therefore, the use of intelligent prediction is primarily based in two parts—modeling of the health data and analysis of the knowledge obtained. Computational intelligence and predictive analytics not only help in predicting the risks of diseases, but also supports largely in visualizing the holistic picture of health in a large population, aimed at designing more efficient and robust healthcare strategies across the world. Of course, it deals with growing challenges like the complexity of health data obtained, lack of interoperable systems for extended and unified analysis, intrinsic bias of some machine learning algorithms, and the implementational difficulties. Nonetheless, the application of machine learning and nonlinear methods using computation intelligence have already demonstrated its potential in predicting health risks and diseases, and is expected to reshape the field of health analytics, early detection and prediction of diseases in a global perspective.

#### **Acknowledgements**

The authors acknowledge the research facilities and technical assistance of the National Technological University (*Universidad Tecnológica Nacional*), Buenos Aires, Argentina.

The work is supported partly by the National Agency for Research and Innovation (*Agencia Nacional de Investigación e Innovación*) of Uruguay through its grant FSDA\_1\_2017\_1\_143653 (*Inteligencia Computacional en Salud. Creando Herramientas para Predecir la Sobrevida en Trasplante Hepático*).

Also it is supported partly by the National Interuniversity Council (CIN) of Argentina through the program '*Programa Estratégico de Formación de Recursos Humanos para la Investigación y Desarrollo (PERHID)*'.

The authors sincerely acknowledge the scientific view and insights of Dr. Josemaría Menéndez Dutra, Dr. Ofelia María Noceti Penza, and Dr. Solange Gerona Sangiovanni, from the Central Hospital of the Armed Forces (*Dirección Nacional de Sanidad de las Fuerzas Armadas*) in Montevideo, Uruguay.

#### **Author details**

Parag Chatterjee1,2\*, Leandro J. Cymberknop1 and Ricardo L. Armentano1,2

1 Universidad Tecnológica Nacional, Buenos Aires, Argentina

2 Universidad de la República, Uruguay

\*Address all correspondence to: paragc@ieee.org

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**177**

20 April 2014]

2012;**7**(1):130-134

[8] Shah NH. Translational bioinformatics embraces big data. Yearbook of Medical Informatics.

[9] Heinze O, Birkle M, Koster L, Bergh B. Architecture of a consent management suite and integration into

*Nonlinear Systems in Healthcare towards Intelligent Disease Prediction*

IHE-based regional health information networks. BMC Medical Informatics and Decision Making. 2011;**11**:58

[10] Tejero A, de la Torre I. Advances and current state of the security and privacy in electronic health records: Survey from a social perspective. Journal of Medical Systems. 2012;**36**(5):3019-3027

[11] Mense A, Hoheiser-Pfortner F, Schmid M, Wahl H. Concepts for a standard based cross-organisational information security management system in the context of a nationwide EHR. Studies in Health Technology and

Informatics. 2013;**192**:548-552

[12] Faxvaag A, Johansen TS, Heimly V, Melby L, Grimsmo A. Healthcare professionals' experiences with EHRsystem access control mechanisms. Studies in Health Technology and Informatics. 2011;**169**:601-605

[13] Ross MK, Wei W, Ohno-Machado L. "Big data" and the electronic health record. Yearbook of Medical Informatics. 2014;**9**(1):97-104. DOI:

[14] Wagholikar KB, Sundararajan V, Deshpande AW. Modeling paradigms for medical diagnostic decision support: A survey and future directions. Journal of Medical Systems.

[15] WHO. Integrated Chronic Disease Prevention and Control [Internet]. 2019. Available from: https://www.who.int/

[16] VanWagner LB, Ning H, Whitsett M, Levitsky J, Uttal S, Wilkins JT, et al. A point-based prediction model for cardiovascular risk in orthotopic liver transplantation: The CAR-OLT score. Hepatology. 2017;**66**:1968-1979. DOI:

10.15265/IY-2014-0003

2012;**36**(5):3029-3049

10.1002/hep.29329

chp/about/integrated\_cd/en/

*DOI: http://dx.doi.org/10.5772/intechopen.88163*

[1] Boeing G. Visual analysis of nonlinear dynamical systems: Chaos, fractals, self-similarity and the limits of prediction. Systems. 2016;**4**(4):37. DOI:

[2] Explained: Linear and Nonlinear Systems. MIT News [Retrieved:

[3] Nonlinear Systems, Applied Mathematics. University of

Birmingham. Available from: www. birmingham.ac.uk [Retrieved:

[4] Higgins JP. Nonlinear systems in medicine. The Yale Journal of Biology and Medicine. 2002;**75**(5-6):247-260

[5] Wyber R, Vaillancourt S, Perry W, Mannava P, Folaranmi T, Celi LA. Big data in global health: Improving health in low- and middle-income countries. Bulletin of the World Health Organization. 2015;**93**:203-208. DOI:

[6] Ahern DK, Kreslake JM, Phalen JM. What is ehealth (6): Perspectives on the evolution of ehealth research. Journal of Medical Internet Research. 2006;**8**(1):e4. DOI: 10.2196/jmir.8.1.e4

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*Nonlinear Systems in Healthcare towards Intelligent Disease Prediction DOI: http://dx.doi.org/10.5772/intechopen.88163*

#### **References**

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

perspective.

Argentina.

**Author details**

**Acknowledgements**

in two parts—modeling of the health data and analysis of the knowledge obtained. Computational intelligence and predictive analytics not only help in predicting the risks of diseases, but also supports largely in visualizing the holistic picture of health in a large population, aimed at designing more efficient and robust healthcare strategies across the world. Of course, it deals with growing challenges like the complexity of health data obtained, lack of interoperable systems for extended and unified analysis, intrinsic bias of some machine learning algorithms, and the implementational difficulties. Nonetheless, the application of machine learning and nonlinear methods using computation intelligence have already demonstrated its potential in predicting health risks and diseases, and is expected to reshape the field of health analytics, early detection and prediction of diseases in a global

The authors acknowledge the research facilities and technical assistance of the National Technological University (*Universidad Tecnológica Nacional*), Buenos Aires,

The work is supported partly by the National Agency for Research and Innovation (*Agencia Nacional de Investigación e Innovación*) of Uruguay through its grant FSDA\_1\_2017\_1\_143653 (*Inteligencia Computacional en Salud. Creando* 

Also it is supported partly by the National Interuniversity Council (CIN) of Argentina through the program '*Programa Estratégico de Formación de Recursos* 

The authors sincerely acknowledge the scientific view and insights of Dr. Josemaría Menéndez Dutra, Dr. Ofelia María Noceti Penza, and Dr. Solange Gerona Sangiovanni, from the Central Hospital of the Armed Forces (*Dirección Nacional de* 

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

and Ricardo L. Armentano1,2

*Herramientas para Predecir la Sobrevida en Trasplante Hepático*).

*Humanos para la Investigación y Desarrollo (PERHID)*'.

*Sanidad de las Fuerzas Armadas*) in Montevideo, Uruguay.

1 Universidad Tecnológica Nacional, Buenos Aires, Argentina

Parag Chatterjee1,2\*, Leandro J. Cymberknop1

\*Address all correspondence to: paragc@ieee.org

2 Universidad de la República, Uruguay

provided the original work is properly cited.

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[34] Framingham Heart Study. The Adult Treatment Panel III, JAMA. 2001 [Internet]. 2019. Available from: https:// www.framinghamheartstudy.org/fhsrisk-functions/hard-coronary-heart-

[35] Chironi G, Simon A, Megnien JL, Sirieix ME, Mousseaux E, Pessana F, et al. Impact of coronary artery calcium on cardiovascular risk categorization and lipid-lowering drug eligibility in asymptomatic hypercholesterolemic men. International Journal of

Cardiology. 2011;**151**(2):200-204. DOI:

[36] Pessana F, Armentano R, Chironi G, Megnien JL, Mousseaux E, Simon A. Subclinical atherosclerosis modeling: Integration of coronary artery calcium score to Framingham equation. In: Annual International Conference of the IEEE Engineering in Medicine and Biology Society. 2009. DOI: 10.1109/

[37] Bucci CM, Legnani WE, Armentano

machine learning, and clinical medicine. The New England Journal of Medicine. 2016;**375**(13):1216-1219. DOI: 10.1056/

RL. Clustering of cardiovascular risk factors highlighted the coronary artery calcium as a strong clinical discriminator. Health and Technology. 2016;**6**(3):159-165. DOI: 10.1007/

[38] Obermeyer Z, Emanuel EJ. Predicting the future—Big data,

10.1016/j.ijcard.2010.05.024

iembs.2009.5334049

s12553-016-0139-1

NEJMp1606181

disease-10-year-risk/

*Nonlinear Systems in Healthcare towards Intelligent Disease Prediction DOI: http://dx.doi.org/10.5772/intechopen.88163*

Converging Technologies (ICCTCT). 2018. DOI: 10.1109/icctct.2018.8550857

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

networks in health care organizational decision-making: A scoping review. PLoS One. 2019;**14**(2):e0212356. DOI:

[25] Deepa et al. Health care analysis using random Forest algorithm. Journal of Chemical and Pharmaceutical Sciences. 2017;**10**(3):1359-1361. Available from: www.jchps.com/ issues/Volume%2010\_Issue%20 3/20171025\_075052\_0180417.pdf

[26] Chatterjee P, Cymberknop L, Armentano R. IoT-based ehealth toward decision support system for CBRNE events. In: Malizia A, D'Arienzo M, editors. Enhancing CBRNE Safety & Security: Proceedings of the SICC 2017 Conference. Cham: Springer; 2018. DOI:

10.1007/978-3-319-91791-7\_21

2017

d457d499ffcd

[27] Chatterjee P, Cymberknop L, Armentano R. IoT-Based Decision Support System Towards Cardiovascular Diseases. Córdoba, Argentina: SABI;

[28] Donges N. The Random Forest Algorithm. Towards Data Science. 2018. Available from: towardsdatascience. com/the-random-forest-algorithm-

[29] Healthcare.ai. Step by Step to K-Means Clustering. Data Science Blog. 2017. Available from: healthcare.ai/ step-step-k-means-clustering/

[30] Mueller J, Massaron L. Machine Learning for Dummies. Hoboken: John

[31] Raschka S. Model evaluation, model selection, and algorithm selection in machine learning. ArXiv. 2018. Available

Santhosh KD, Mareeswari V. Prediction of cardiovascular disease using machine learning algorithms. In: International Conference on Current Trends towards

from: arxiv.org/abs/1811.12808v2

[32] Dinesh KG, Arumugaraj K,

Wiley & Sons, Inc.; 2016

10.1371/journal.pone.0212356

[17] Hemmatpour M, Ferrero R, Gandino F, Montrucchio B, Rebaudengo M. Nonlinear predictive threshold model for real-time abnormal gait detection. Journal of Healthcare Engineering. 2018;**2018**. Article ID 4750104. 9 p. DOI:

10.1155/2018/4750104

tsunami.html

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Springer; 2009

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[18] Stanford Medicine. Stanford Medicine 2017 Health Trends Report Harnessing the Power of Data in Health [Internet]. 2017. Available from: med.stanford.edu/content/ dam/sm/sm-news/documents/ StanfordMedicineHealth TrendsWhitePaper2017.pdf

[19] Corbin, Kenneth. How CIOs Can Prepare for Healthcare 'Data Tsunami'. CIO [Internet]. 2014. Available from: www.cio.com/article/2860072/howcios-can-prepare-for-healthcare-data-

[20] Health Data Archiver. Health Data Volumes Skyrocket, Legacy Data Archives On The Rise [Internet]. 2017. Available from: https://www. healthdataarchiver.com/health-datavolumes-skyrocket-legacy-data-

[21] Hippolyte T, Adamou M, Blaise N, Pierre C, Olivier M. Linear vs non-linear learning methods—A comparative study for forest above ground biomass, estimation from texture analysis of satellite images. ARIMA Journal.

[22] Hastie T et al. The Elements of Statistical Learning. Vol. 2. Heidelberg:

[23] Jiang F, Jiang Y, Zhi H, Dong Y, Li H, Ma S, et al. Artificial intelligence in healthcare: Past, present and future. Stroke and Vascular Neurology.

[24] Shahid N, Rappon T, Berta W. Applications of artificial neural

**178**

[33] Chatterjee P, Armentano RL, Cymberknop LJ. Internet of things and decision support system for eHealth-applied to cardiometabolic diseases. In: International Conference on Machine Learning and Data Science (MLDS); Noida. Piscataway: IEEE; 2017. pp. 75-79. DOI: 10.1109/MLDS.2017.22

[34] Framingham Heart Study. The Adult Treatment Panel III, JAMA. 2001 [Internet]. 2019. Available from: https:// www.framinghamheartstudy.org/fhsrisk-functions/hard-coronary-heartdisease-10-year-risk/

[35] Chironi G, Simon A, Megnien JL, Sirieix ME, Mousseaux E, Pessana F, et al. Impact of coronary artery calcium on cardiovascular risk categorization and lipid-lowering drug eligibility in asymptomatic hypercholesterolemic men. International Journal of Cardiology. 2011;**151**(2):200-204. DOI: 10.1016/j.ijcard.2010.05.024

[36] Pessana F, Armentano R, Chironi G, Megnien JL, Mousseaux E, Simon A. Subclinical atherosclerosis modeling: Integration of coronary artery calcium score to Framingham equation. In: Annual International Conference of the IEEE Engineering in Medicine and Biology Society. 2009. DOI: 10.1109/ iembs.2009.5334049

[37] Bucci CM, Legnani WE, Armentano RL. Clustering of cardiovascular risk factors highlighted the coronary artery calcium as a strong clinical discriminator. Health and Technology. 2016;**6**(3):159-165. DOI: 10.1007/ s12553-016-0139-1

[38] Obermeyer Z, Emanuel EJ. Predicting the future—Big data, machine learning, and clinical medicine. The New England Journal of Medicine. 2016;**375**(13):1216-1219. DOI: 10.1056/ NEJMp1606181

[39] Weng SF, Reps J, Kai J, Garibaldi JM, Qureshi N. Can machine-learning improve cardiovascular risk prediction using routine clinical data? PLoS One. 2017;**12**(4):e0174944. DOI: 10.1371/ journal.pone.0174944

[40] Panagiotakos D. Health measurement scales: Methodological issues. Open Cardiovascular Medicine Journal. 2009;**3**:160

[41] Dimopoulos AC et al. Machine learning methodologies versus cardiovascular risk scores, in predicting disease risk. BMC Medical Research Methodology. 2018;**18**(1):1-11. DOI: 10.1186/s12874-018-0644-1

[42] Bhaduri A, Bhaduri A, Bhaduri A, Mohapatra PK. Blood pressure modeling using statistical and computational intelligence approaches. In: IEEE International Advance Computing Conference. 2009. DOI: 10.1109/ iadcc.2009.4809156

[43] Daghistani TA, Elshawi R, Sakr S, Ahmed AM, Al-Thwayee A, Al-Mallah MH. Predictors of in-hospital length of stay among cardiac patients: A machine learning approach. International Journal of Cardiology. 2019;**288**:140-147. DOI: 10.1016/j.ijcard.2019.01.046

[44] Paredes S, Henriques J, Rochar T, Mendes D, Carvalho P, Moraisl J, et al. A clinical interpretable approach applied to cardiovascular risk assessment. In: 40th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC). 2018. DOI: 10.1109/ embc.2018.8512956

[45] Romero L, Chatterjee P, Armentano R. An IoT approach for integration of computational intelligence and wearable sensors for Parkinson's disease diagnosis and monitoring. Health and Technology. 2016;**6**(3):167-172. DOI: 10.1007/ s12553-016-0148-0

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

**Chapter 10**

**Abstract**

initial lesion behavior.

**1. Introduction**

**181**

Mathematical Modeling

and Well-Posedness of

in Disorders of Human

*Vishakha Jadaun and Nitin Raja Singh*

Aortic dissection is the most common aortic emergency requiring surgical intervention. Whether the elective endovascular repair of abdominal aortic aneurysm reduces long-term morbidity and mortality, as compared with traditional open repair, remains uncertain. The foundation of shell element based on the Reissner-Mindlin kinematics assumption is widely applicable, but this cannot model applications of shell surface stresses as needed in analysis of shell in human vascular system. The analysis is designed to assess progression of initial lesion in aortic dissection. Using general shell element analysis and tensor calculus, a higher order differential geometry-based model is proposed. Since the shell is thin, a variational formulation for initial lesion is proposed. The variational formulation for initial lesion is well posed. The weak convergence of the solution to initial lesion model is mathematically substantiated. Asymptotic analysis shows that initial lesion is membrane-dominated and bending-dominated when pure bending is inhibited and noninhibited, respectively. At least two observations are to be noted. First, the mathematical analysis of the initial lesion model is distinct from classical shell models. Second, the asymptotic analysis of the initial lesion model is based on degenerating three-dimensional continuum to bending strains in order to assess

**Keywords:** aortic dissection, higher order kinematical assumptions, initial lesion

The shell structure is generally a three-dimensional structure that is elongated in two directions and thinned out in other direction. The shell structures in nature are profusely impressive such as seashells and eggshells. In various industries including aeronautics, naval architecture, and automotive engineering milieu, many engineering designs are analyzed to design shells as thin as possible and optimize the amount of material [1]. Human anatomy develops cyst-related diseases with progressive severity. These disease states involve single to multiple cyst formations

model, variational formulation, asymptotic analysis

Vascular System

Three-Dimensional Shell

[46] Chatterjee P, Cymberknop LJ, Armentano RL. IoT-based decision support system for intelligent healthcare—Applied to cardiovascular diseases. In: 7th International Conference on Communication Systems and Network Technologies (CSNT); Nagpur. 2017. pp. 362-366. DOI: 10.1109/CSNT.2017.8418567

#### **Chapter 10**

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

[46] Chatterjee P, Cymberknop LJ, Armentano RL. IoT-based decision support system for intelligent

diseases. In: 7th International

healthcare—Applied to cardiovascular

Conference on Communication Systems and Network Technologies (CSNT); Nagpur. 2017. pp. 362-366. DOI: 10.1109/CSNT.2017.8418567

**180**

## Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human Vascular System

*Vishakha Jadaun and Nitin Raja Singh*

### **Abstract**

Aortic dissection is the most common aortic emergency requiring surgical intervention. Whether the elective endovascular repair of abdominal aortic aneurysm reduces long-term morbidity and mortality, as compared with traditional open repair, remains uncertain. The foundation of shell element based on the Reissner-Mindlin kinematics assumption is widely applicable, but this cannot model applications of shell surface stresses as needed in analysis of shell in human vascular system. The analysis is designed to assess progression of initial lesion in aortic dissection. Using general shell element analysis and tensor calculus, a higher order differential geometry-based model is proposed. Since the shell is thin, a variational formulation for initial lesion is proposed. The variational formulation for initial lesion is well posed. The weak convergence of the solution to initial lesion model is mathematically substantiated. Asymptotic analysis shows that initial lesion is membrane-dominated and bending-dominated when pure bending is inhibited and noninhibited, respectively. At least two observations are to be noted. First, the mathematical analysis of the initial lesion model is distinct from classical shell models. Second, the asymptotic analysis of the initial lesion model is based on degenerating three-dimensional continuum to bending strains in order to assess initial lesion behavior.

**Keywords:** aortic dissection, higher order kinematical assumptions, initial lesion model, variational formulation, asymptotic analysis

#### **1. Introduction**

The shell structure is generally a three-dimensional structure that is elongated in two directions and thinned out in other direction. The shell structures in nature are profusely impressive such as seashells and eggshells. In various industries including aeronautics, naval architecture, and automotive engineering milieu, many engineering designs are analyzed to design shells as thin as possible and optimize the amount of material [1]. Human anatomy develops cyst-related diseases with progressive severity. These disease states involve single to multiple cyst formations

in distinct organ systems including the lung, liver, kidney, brain, bone etc. Pathophysiologically, these cysts emanate from either the underlying genetic anomaly or infections such as helminths and mycobacterial, among others. Interestingly, these cysts can be modeled as shells, albeit in higher dimensions.

hypertension-related treatment, patients may develop a significant aortic enlargement that necessitates operative intervention. These chronic patients will benefit in

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human…*

Currently, there is no consensus on the management of uncomplicated type B dissection that may be liable for rapid progression. Thus, seeking multiple high-risk attributes/features responsible for rapid progression might help to decide when to treat and how to treat. There is a subgroup of patients who progress very rapidly to terminal dilatation liable for rupture and torrential bleed leading to death. Offering early transthoracic endovascular repair to this subgroup seems to be a life-saving proposition. Finding these patients is a challenge. It is not known that a patient at risk for catastrophic events is following a personal trajectory of disease progression. It is also not known that a threshold for disease progression that can predict a high risk of mortality for a specific patient. By modeling the initial lesion of AD, we can potentially avoid rupture by crossing over to transthoracic endovascular repair at a time that minimizes procedural risks. On asymptotic analysis, we evaluate the point of follow-up; we lose the ability to achieve the same desirable aortic remodeling observed with transthoracic endovascular repair in the more acute setting. Therefore, reliable predictors are needed in the early stage of disease. It aids identification of patients at risk of

In early stages of AD, subintimal intramural hemorrhage occurs due to tunica media degeneration. In certain situations, when strains are known on a plane, the low degree of expansion of the transverse displacement is to be recovered. It is to be noted that by dispensing away the assumption of plane stress, an arbitrary threedimensional material law is applicable in three-dimensional formulation of continuum mechanics. The objective of this chapter is to identify higher-order shell model for initial-stage primary tunica intimal lesion of AD by the general shell element

This chapter is organized in the following manner. In Section 2, we give certain

definitions, conventions, and notations relevant to the shell geometry and its corresponding deformation. Next in Section 3, we derive initial lesions of AD as the higher-order shell model perusing general shell analysis approach. Then in Section 4, we do mathematical analyses of the initial lesions described in the previous section. In Section 5, we assess asymptotic behavior of the model. Finally, in Section 6, we present our conclusions regarding mathematical modeling of shells in human

**2. Conventions and notations in higher-order shell geometry**

We are interested in modeling early stages of AD; the initial subintimal intramural hemorrhage caused by tunica media degeneration undergoes solidification due to clot formation. Thus, this initial lesion closely follows the principles of continuum mechanics. We consider the initial lesion as a solid medium. It is geometrically defined by a mid-surface immersed in the human vascular compartment ϵ (dimensionless thickness parameter) and a parameter representing the thickness

In order to understand the initial lesion of AD, we model the initial lesion using

!, which is a one-one map from the closure of a bounded open

general shell element theory. A shell is defined as a collection of charts. Let us consider the mid-surface of a shell as a collection of two-dimensional charts. These charts are smooth ono-one maps from domains of <sup>2</sup> into Euclidean (physical) space ℰ. We consider an initial lesion with a mid-surface S defined by a two-

the long-term from prophylactic intervention.

*DOI: http://dx.doi.org/10.5772/intechopen.89866*

approach and to perform mathematical analysis.

vascular tissues and future scope.

of the medium around this surface.

dimensional chart *φ*

**183**

aortic enlargement.

Different approaches have been formulated for shell elements discretization. One of the approaches [2] evaluated the shell behavior as the superimposition of membrane bending action as well as plate bending action. The discrete construction of shell elements requires a combination of plane stress matrices as well as plate bending stiffness matrices. However, the resultant shell elements are less accurate since curvature effects are not duly incorporated and the membrane behavior and plate bending behavior are coupled at nodal points only. Another approach [3] is based on variational formulation and perusing relevant shell theory wherein a specific shell theory is constituted of higher-order derivatives and required concomitant nodal point variables beyond the conventional nodal point rotations and displacements. Such an approach is applicable and relevant to certain shell geometries and associated pertinent analysis conditions. Thus, it is difficult to model more complex shell structures. Yet another approach [4] is aimed for very general formulation related to threedimensional continuum degeneration. In this approach, the mid-surface of the shell element that belongs to the three-dimensional continuum is clearly defined and identified. The first assumption is that the fibers are straight and normal to mid-surface prior to the deformation which continues to remain straight during the course of deformation. The second assumption is that the stress normal to the shell mid-surface is zero throughout the shell motion [5, 6]. The shell models based on the aforementioned kinematical assumptions can be interpreted as a truncation of the expansion of displacements in different directions across the thickness of the shell structure. It is to be noted that such truncated expansion contains terms up to degree one and degree zero for the tangential displacements and transverse displacements, respectively. The physiological and pathological states in the human body undergo dynamic transformations. In cardiovascular dynamics, the interaction of blood to the internal vessel lining is associated with large through-the-thickness displacement of local vessel wall surface owing to distension by propulsion of blood and elastic recoil thereafter. Thus, the aforementioned assumptions might not be applicable to shells in human anatomy. In order to make better estimate, higher-order kinematical assumptions are effective. Yet the detailed analysis of biological shell structures frequently presents challenging problems. One of the difficulties is that the shell structure resists applied loads largely along its curvature such that, in case, curvature is changed and the load bearing capacity of shell is transformed. Therefore, analysis of boundary conditions of a shell structure plays a vital role in shell behavior and its response to stress.

The aorta is the largest diameter blood vessel, emerging from the left ventricle to supply oxygenated blood to the human body. Whenever nonlinear degeneration of the tunica media (middle layer of the vessel wall) occurs, the aorta undergoes dynamic dilatation and marginal elongation. Generally, this degeneration is caused by genetic anomaly and prolonged untreated hypertension in young and senile patients, respectively. It is termed as *aortic aneurysm* [7]. Whenever there is structural discontinuity in nonconformal internal vessel wall, the blood surges through the tear causing the inner and middle layers of the aorta to separate. It is termed as *aortic dissection* (AD) [8]. AD is a life-threatening condition [9]. If the blood-filled channel ruptures through the outside aortic wall, AD is often fatal [10]. It is the most common aortic emergency requiring surgical intervention. AD is classified according to the regional involvement of the segment of the aorta with the Stanford type A dissection and the Stanford type B dissection involving the ascending aorta and occurring distal to the left subclavian artery, respectively. According to the international guidelines on clinical therapeutics, uncomplicated type B dissection should receive optimal medical treatment (OMT). However, in spite of adequate

#### *Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human… DOI: http://dx.doi.org/10.5772/intechopen.89866*

hypertension-related treatment, patients may develop a significant aortic enlargement that necessitates operative intervention. These chronic patients will benefit in the long-term from prophylactic intervention.

Currently, there is no consensus on the management of uncomplicated type B dissection that may be liable for rapid progression. Thus, seeking multiple high-risk attributes/features responsible for rapid progression might help to decide when to treat and how to treat. There is a subgroup of patients who progress very rapidly to terminal dilatation liable for rupture and torrential bleed leading to death. Offering early transthoracic endovascular repair to this subgroup seems to be a life-saving proposition. Finding these patients is a challenge. It is not known that a patient at risk for catastrophic events is following a personal trajectory of disease progression. It is also not known that a threshold for disease progression that can predict a high risk of mortality for a specific patient. By modeling the initial lesion of AD, we can potentially avoid rupture by crossing over to transthoracic endovascular repair at a time that minimizes procedural risks. On asymptotic analysis, we evaluate the point of follow-up; we lose the ability to achieve the same desirable aortic remodeling observed with transthoracic endovascular repair in the more acute setting. Therefore, reliable predictors are needed in the early stage of disease. It aids identification of patients at risk of aortic enlargement.

In early stages of AD, subintimal intramural hemorrhage occurs due to tunica media degeneration. In certain situations, when strains are known on a plane, the low degree of expansion of the transverse displacement is to be recovered. It is to be noted that by dispensing away the assumption of plane stress, an arbitrary threedimensional material law is applicable in three-dimensional formulation of continuum mechanics. The objective of this chapter is to identify higher-order shell model for initial-stage primary tunica intimal lesion of AD by the general shell element approach and to perform mathematical analysis.

This chapter is organized in the following manner. In Section 2, we give certain definitions, conventions, and notations relevant to the shell geometry and its corresponding deformation. Next in Section 3, we derive initial lesions of AD as the higher-order shell model perusing general shell analysis approach. Then in Section 4, we do mathematical analyses of the initial lesions described in the previous section. In Section 5, we assess asymptotic behavior of the model. Finally, in Section 6, we present our conclusions regarding mathematical modeling of shells in human vascular tissues and future scope.

#### **2. Conventions and notations in higher-order shell geometry**

We are interested in modeling early stages of AD; the initial subintimal intramural hemorrhage caused by tunica media degeneration undergoes solidification due to clot formation. Thus, this initial lesion closely follows the principles of continuum mechanics. We consider the initial lesion as a solid medium. It is geometrically defined by a mid-surface immersed in the human vascular compartment ϵ (dimensionless thickness parameter) and a parameter representing the thickness of the medium around this surface.

In order to understand the initial lesion of AD, we model the initial lesion using general shell element theory. A shell is defined as a collection of charts. Let us consider the mid-surface of a shell as a collection of two-dimensional charts. These charts are smooth ono-one maps from domains of <sup>2</sup> into Euclidean (physical) space ℰ. We consider an initial lesion with a mid-surface S defined by a twodimensional chart *φ* !, which is a one-one map from the closure of a bounded open

in distinct organ systems including the lung, liver, kidney, brain, bone etc. Pathophysiologically, these cysts emanate from either the underlying genetic anomaly or infections such as helminths and mycobacterial, among others. Interestingly, these cysts can be modeled as shells, albeit in higher dimensions.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

structure plays a vital role in shell behavior and its response to stress.

**182**

The aorta is the largest diameter blood vessel, emerging from the left ventricle to supply oxygenated blood to the human body. Whenever nonlinear degeneration of the tunica media (middle layer of the vessel wall) occurs, the aorta undergoes dynamic dilatation and marginal elongation. Generally, this degeneration is caused by genetic anomaly and prolonged untreated hypertension in young and senile patients, respectively. It is termed as *aortic aneurysm* [7]. Whenever there is structural discontinuity in nonconformal internal vessel wall, the blood surges through the tear causing the inner and middle layers of the aorta to separate. It is termed as *aortic dissection* (AD) [8]. AD is a life-threatening condition [9]. If the blood-filled channel ruptures through the outside aortic wall, AD is often fatal [10]. It is the most common aortic emergency requiring surgical intervention. AD is classified according to the regional involvement of the segment of the aorta with the Stanford type A dissection and the Stanford type B dissection involving the ascending aorta and occurring distal to the left subclavian artery, respectively. According to the international guidelines on clinical therapeutics, uncomplicated type B dissection should receive optimal medical treatment (OMT). However, in spite of adequate

Different approaches have been formulated for shell elements discretization. One of the approaches [2] evaluated the shell behavior as the superimposition of membrane bending action as well as plate bending action. The discrete construction of shell elements requires a combination of plane stress matrices as well as plate bending stiffness matrices. However, the resultant shell elements are less accurate since curvature effects are not duly incorporated and the membrane behavior and plate bending behavior are coupled at nodal points only. Another approach [3] is based on variational formulation and perusing relevant shell theory wherein a specific shell theory is constituted of higher-order derivatives and required concomitant nodal point variables beyond the conventional nodal point rotations and displacements. Such an approach is applicable and relevant to certain shell geometries and associated pertinent analysis conditions. Thus, it is difficult to model more complex shell structures. Yet another approach [4] is aimed for very general formulation related to threedimensional continuum degeneration. In this approach, the mid-surface of the shell element that belongs to the three-dimensional continuum is clearly defined and identified. The first assumption is that the fibers are straight and normal to mid-surface prior to the deformation which continues to remain straight during the course of deformation. The second assumption is that the stress normal to the shell mid-surface is zero throughout the shell motion [5, 6]. The shell models based on the aforementioned kinematical assumptions can be interpreted as a truncation of the expansion of displacements in different directions across the thickness of the shell structure. It is to be noted that such truncated expansion contains terms up to degree one and degree zero for the tangential displacements and transverse displacements, respectively. The physiological and pathological states in the human body undergo dynamic transformations. In cardiovascular dynamics, the interaction of blood to the internal vessel lining is associated with large through-the-thickness displacement of local vessel wall surface owing to distension by propulsion of blood and elastic recoil thereafter. Thus, the aforementioned assumptions might not be applicable to shells in human anatomy. In order to make better estimate, higher-order kinematical assumptions are effective. Yet the detailed analysis of biological shell structures frequently presents challenging problems. One of the difficulties is that the shell structure resists applied loads largely along its curvature such that, in case, curvature is changed and the load bearing capacity of shell is transformed. Therefore, analysis of boundary conditions of a shell

subset of <sup>2</sup> , denoted by *ω*, into ℰ, hence S ¼ *φ* ! *ω* ! . At each point of the midsurface, the vector **z** ! *<sup>α</sup>* is assumed as partial derivative of *φ* ! with respect to *ξ<sup>α</sup>* such that

$$\overrightarrow{\mathbf{z}}\_a = \frac{\partial \rho(\xi\_1, \xi\_2)}{\partial \xi^a}. \tag{1}$$

Interestingly, covariant basis is useful in the modeling of higher-order initial

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human…*

*<sup>i</sup>* � **z***<sup>k</sup>*

**Definition 3**. (Covariant metric tensor). *The covariant metric tensor is the pairwise*

Suppose two vectors **A** and **B** are located at the same point and their components

The length of a vector **B** can be expressed in terms of covariant metric tensor as

q

Interestingly, covariant tensors are useful in modeling of higher-order initial

**Definition 4**. (Contravariant metric tensor *zij*). *The contravariant metric tensor zij*

*zij* � *<sup>z</sup>jk* <sup>¼</sup> *zij* � *<sup>z</sup>kj* <sup>¼</sup> *<sup>δ</sup><sup>i</sup>*

**<sup>z</sup>***<sup>i</sup>* � **<sup>z</sup>***<sup>j</sup>* <sup>¼</sup> *<sup>δ</sup><sup>i</sup>*

**Definition 6**. (Christoffel symbol). *In affine and curvilinear coordinate systems, the covariant basis z<sup>i</sup> is the same at all points and varies from one point to another,*

*j*

ffiffiffiffiffiffiffiffiffiffiffiffiffi *zijBi Bj*

**z***<sup>j</sup>* ¼ **z***<sup>i</sup>* � **z***<sup>j</sup>*

∣**B**∣ ¼

*cij*

*z*1*:z*<sup>1</sup> *z*1*:z*<sup>2</sup> *z*1*:z*<sup>3</sup> *z*2*:z*<sup>1</sup> *z*2*:z*<sup>2</sup> *z*2*:z*<sup>3</sup> *z*3*:z*<sup>1</sup> *z*3*:z*<sup>2</sup> *z*3*:z*<sup>3</sup>

� �*Ai* � *Bj* <sup>¼</sup> *zijAi*

!, where **<sup>z</sup>** 3,*<sup>i</sup>* �! <sup>¼</sup> *<sup>∂</sup>***<sup>z</sup>**

3 7

*Bj*

*<sup>k</sup>*, (11)

*:* (13)

). *The contravariant basis z<sup>i</sup> is defined as*

*<sup>z</sup><sup>i</sup>* <sup>¼</sup> *<sup>z</sup>ijz<sup>j</sup>* <sup>¼</sup> *<sup>z</sup>jiz<sup>j</sup>* (12)

3 ! *<sup>∂</sup>ξ<sup>i</sup>* ,

<sup>5</sup>, (8)

*:* (9)

(10)

(7)

*i* ! � *<sup>ξ</sup>*<sup>3</sup> *<sup>b</sup><sup>k</sup>*

2 6 4

lesions in human vascular system given by

*DOI: http://dx.doi.org/10.5772/intechopen.89866*

*<sup>i</sup>* � *<sup>ξ</sup>*<sup>3</sup> *<sup>b</sup><sup>k</sup> i* � �**z***<sup>k</sup>*

*<sup>∂</sup>ξ*<sup>3</sup> <sup>¼</sup> **<sup>z</sup>**<sup>3</sup> !*:*

*dot product of the covariant basis vectors:*

.

lesions in human vascular system given by

*<sup>j</sup>* <sup>¼</sup> **<sup>z</sup>***ij* � <sup>2</sup>*ξ*<sup>3</sup>

*is the matrix inverse of the covariant metric tensor zij:*

*<sup>k</sup>* is the Kronecker symbol. **Definition 5**. (Contravariant basis **z***<sup>i</sup>*

The bases **z***<sup>i</sup>* and **z***<sup>i</sup>* are mutually orthonormal:

*<sup>∂</sup>ξ<sup>i</sup>* <sup>¼</sup> *zi* <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> **<sup>z</sup>** 3,*<sup>i</sup>* �! <sup>¼</sup> **<sup>z</sup>**

!,

*zij* ¼ *zi:z<sup>j</sup>* ¼

, then the dot product **A***:***B** is given by

**z***i:Bj*

*bij* <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> � �<sup>2</sup>

**<sup>A</sup>***:***<sup>B</sup>** <sup>¼</sup> *<sup>A</sup><sup>i</sup>*

*g* ! *<sup>i</sup>* <sup>¼</sup> *<sup>∂</sup><sup>φ</sup>*

8 >>>>>><

*g* ! *<sup>i</sup>* <sup>¼</sup> *<sup>δ</sup><sup>k</sup>*

*g* ! <sup>3</sup> <sup>¼</sup> *<sup>∂</sup><sup>φ</sup>*

where *z<sup>i</sup>* is in <sup>3</sup>

are *Ai* and *Bi*

1.*gij* ¼ *g* ! *<sup>i</sup>* � *g* !

2. *gi*<sup>3</sup> ¼ *g* ! *<sup>i</sup>* � *g* ! <sup>3</sup> ¼ 0

3. *g*<sup>33</sup> ¼ *g* ! <sup>3</sup> � *g* ! <sup>3</sup> ¼ 1

where *δ<sup>i</sup>*

**185**

>>>>>>:

These vectors are linearly independent from each other, so that they form a basis of the plane tangent to the mid-surface at this point. The unit normal vector is given by

$$
\overrightarrow{\mathbf{z}\_3} = \frac{\overrightarrow{\mathbf{z}\_1} \times \overrightarrow{\mathbf{z}\_2}}{||\overrightarrow{\mathbf{z}\_1} \times \overrightarrow{\mathbf{z}\_2}||}.
$$

**Definition 1**. (Geometric definition of initial lesion). *An initial lesion is a solid medium whose domain* Ω *can be defined by a mid-surface whose map is given by*

$$\rho: \alpha \subseteq \mathbb{R}^2 \to \mathbb{R}^3, \mathfrak{s}.t. \ \rho\left(\xi^1, \xi^2\right) = \left(\xi^1, \xi^2, \xi^3\right) \in \mathbb{R}^3 \tag{2}$$

*The three-dimensional medium corresponding to the initial lesion is then defined by three-dimensional chart given by*

$$
\rho(\xi^1, \xi^2, \xi^3) = \rho(\xi^1, \xi^2) + \xi^3 \overrightarrow{\mathfrak{a}}\_3(\xi^1, \xi^2), \tag{3}
$$

$$\begin{split} \text{where } \left(\xi^{1},\xi^{2},\xi^{3}\right) \in \Omega = \left\{ \left(\xi^{1},\xi^{2},\xi^{3}\right) \in \mathbb{R}^{3} \, \middle|\, \left(\xi^{1},\xi^{2}\right) \in \alpha, \xi^{3} \in \left(-\frac{t\left(\xi^{1},\xi^{2}\right)}{2}, \frac{t\left(\xi^{1},\xi^{2}\right)}{2}\right) \right\}, \\ \text{and } t\left(\xi^{1},\xi^{2}\right) \text{ is the thickness of the initial lesion element at } \left(\xi^{1},\xi^{2}\right). \end{split}$$

In Eq. (1), we have defined tangent vector to a point on the mid-surface of the initial lesion (2) which lies in the region of the Euclidean space. Since we are interested in higher-order parameterization of the initial lesion of AD, the threedimensional chart (3) of this lesion can be very helpful. Thus, transition from the Euclidean space to curvilinear coordinate system will aid to model higher-order initial lesion. It is relevant to grasp few basic notions of surface differential geometry.

#### **2.1 Definitions related to surface differential geometry**

**Definition 2**. (Covariant vector). *Let r z*ð Þ *be a position vector; the differentiation of r z*ð Þ *with respect to each of the coordinate is called covariant basis:*

$$\mathbf{z}\_i = \frac{\partial r(z)}{\partial z^i} \tag{4}$$

If Eq. (4) defines three vectors **z**1, **z**2, and **z**<sup>3</sup>

$$\mathbf{z}\_1 = \frac{\partial \, r(z^1, z^2, z^3)}{\partial z^1}, \quad \mathbf{z}\_2 = \frac{\partial \, r(z^1, z^2, z^3)}{\partial z^2}, \quad \mathbf{z}\_3 = \frac{\partial \, r(z^1, z^2, z^3)}{\partial z^3}.\tag{5}$$

Let **V** be a vector in <sup>3</sup> , and then its expansion n terms of basis is

$$\mathbf{V} = V^i \mathbf{z}\_i = V^1 \mathbf{z}\_1 + V^2 \mathbf{z}\_2 + V^3 \mathbf{z}\_3 \tag{6}$$

The values *V<sup>i</sup>* are called *contravariant components* of vector **V**.

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human… DOI: http://dx.doi.org/10.5772/intechopen.89866*

Interestingly, covariant basis is useful in the modeling of higher-order initial lesions in human vascular system given by

$$\begin{cases} \overrightarrow{\text{g}}\_{i} = \frac{\partial \rho}{\partial \xi^{i}} = \mathbf{z}\_{i} + \xi^{3} \ \mathbf{z}\_{\overline{3},i} = \mathbf{z}\_{\overline{i}} - \xi^{3} \ b\_{i}^{k} \cdot \overrightarrow{\mathbf{z}\_{k}^{i}}, \quad \text{where} \quad \mathbf{z}\_{\overline{3},i} = \frac{\partial \mathbf{z}\_{\overline{3}}}{\partial \xi^{i}},\\ \overrightarrow{\text{g}}\_{i} = \left(\delta\_{i}^{k} - \xi^{3} \ b\_{i}^{k}\right) \overrightarrow{\mathbf{z}\_{k}^{i}},\\ \overrightarrow{\text{g}}\_{3} = \frac{\partial \rho}{\partial \xi^{3}} = \overrightarrow{\mathbf{z}\_{3}}. \end{cases} \tag{7}$$

**Definition 3**. (Covariant metric tensor). *The covariant metric tensor is the pairwise dot product of the covariant basis vectors:*

$$\mathbf{z}\_{i\circ} = \mathbf{z}\_i.\mathbf{z}\_{\circ} = \begin{bmatrix} \mathbf{z}\_1.\mathbf{z}\_1 & \mathbf{z}\_1.\mathbf{z}\_2 & \mathbf{z}\_1.\mathbf{z}\_3 \\\\ \mathbf{z}\_2.\mathbf{z}\_1 & \mathbf{z}\_2.\mathbf{z}\_2 & \mathbf{z}\_2.\mathbf{z}\_3 \\\\ \mathbf{z}\_3.\mathbf{z}\_1 & \mathbf{z}\_3.\mathbf{z}\_2 & \mathbf{z}\_3.\mathbf{z}\_3 \end{bmatrix},\tag{8}$$

where *z<sup>i</sup>* is in <sup>3</sup> .

subset of <sup>2</sup>

such that

surface, the vector **z**

vector is given by

*where ξ*<sup>1</sup>

*and t ξ*<sup>1</sup>

!

, denoted by *ω*, into ℰ, hence S ¼ *φ*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*<sup>α</sup>* is assumed as partial derivative of *φ*

*<sup>α</sup>* <sup>¼</sup> *<sup>∂</sup>φ ξ*1, *<sup>ξ</sup>*<sup>2</sup> ð Þ

! � **z**<sup>2</sup> !

, *<sup>ξ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ξ</sup>*<sup>1</sup>

, *ξ*<sup>2</sup>

, *<sup>ξ</sup>*<sup>2</sup> <sup>∈</sup>*ω*, *<sup>ξ</sup>*<sup>3</sup> <sup>∈</sup> � *<sup>t</sup> <sup>ξ</sup>*1, *<sup>ξ</sup>*<sup>2</sup> ð Þ

, *ξ*<sup>2</sup> *.*

*<sup>∂</sup>zi* (4)

*<sup>∂</sup>z*<sup>3</sup> *:* (5)

**z**<sup>3</sup> (6)

*<sup>∂</sup>z*<sup>2</sup> , **<sup>z</sup>**<sup>3</sup> <sup>¼</sup> *<sup>∂</sup> r z*1, *<sup>z</sup>*2, *<sup>z</sup>*<sup>3</sup> ð Þ

These vectors are linearly independent from each other, so that they form a basis of the plane tangent to the mid-surface at this point. The unit normal

> k**z**<sup>1</sup> ! � **z**<sup>2</sup> !k *:*

*medium whose domain* Ω *can be defined by a mid-surface whose map is given by*

, *s:t: φ ξ*<sup>1</sup>

**Definition 1**. (Geometric definition of initial lesion). *An initial lesion is a solid*

*The three-dimensional medium corresponding to the initial lesion is then defined by*

, *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> *<sup>z</sup>*

j *ξ*1

In Eq. (1), we have defined tangent vector to a point on the mid-surface of the initial lesion (2) which lies in the region of the Euclidean space. Since we are interested in higher-order parameterization of the initial lesion of AD, the threedimensional chart (3) of this lesion can be very helpful. Thus, transition from the Euclidean space to curvilinear coordinate system will aid to model higher-order initial lesion. It is relevant to grasp few basic notions of surface differential geometry.

**Definition 2**. (Covariant vector). *Let r z*ð Þ *be a position vector; the differentiation of*

*<sup>z</sup><sup>i</sup>* <sup>¼</sup> *<sup>∂</sup>r z*ð Þ

, and then its expansion n terms of basis is

**<sup>z</sup>**<sup>2</sup> <sup>þ</sup> *<sup>V</sup>*<sup>3</sup>

**<sup>z</sup>**<sup>1</sup> <sup>þ</sup> *<sup>V</sup>*<sup>2</sup>

! <sup>3</sup> *ξ*<sup>1</sup>

**z** !

**z**3 ! <sup>¼</sup> **<sup>z</sup>**<sup>1</sup>

, *ξ*<sup>2</sup> , *ξ*<sup>3</sup> ∈ <sup>3</sup>

, *ξ*<sup>2</sup> *is the thickness of the initial lesion element at ξ*<sup>1</sup>

**2.1 Definitions related to surface differential geometry**

If Eq. (4) defines three vectors **z**1, **z**2, and **z**<sup>3</sup>

**<sup>V</sup>** <sup>¼</sup> *<sup>V</sup><sup>i</sup>*

**<sup>z</sup>**<sup>1</sup> <sup>¼</sup> *<sup>∂</sup> r z*1, *<sup>z</sup>*2, *<sup>z</sup>*<sup>3</sup> ð Þ

Let **V** be a vector in <sup>3</sup>

**184**

*r z*ð Þ *with respect to each of the coordinate is called covariant basis:*

*<sup>∂</sup>z*<sup>1</sup> , **<sup>z</sup>**<sup>2</sup> <sup>¼</sup> *<sup>∂</sup> r z*1, *<sup>z</sup>*2, *<sup>z</sup>*<sup>3</sup> ð Þ

**<sup>z</sup>***<sup>i</sup>* <sup>¼</sup> *<sup>V</sup>*<sup>1</sup>

The values *V<sup>i</sup>* are called *contravariant components* of vector **V**.

*<sup>φ</sup>* : *<sup>ω</sup>*<sup>⊆</sup> <sup>2</sup> ! <sup>3</sup>

*φ ξ*<sup>1</sup> , *ξ*<sup>2</sup> , *<sup>ξ</sup>*<sup>3</sup> <sup>¼</sup> *φ ξ*<sup>1</sup>

, *<sup>ξ</sup>*<sup>3</sup> <sup>∈</sup> <sup>Ω</sup> <sup>¼</sup> *<sup>ξ</sup>*<sup>1</sup>

*three-dimensional chart given by*

, *ξ*<sup>2</sup>

! *ω* !

. At each point of the mid-

! with respect to *ξ<sup>α</sup>*

, *ξ*<sup>3</sup> ∈ <sup>3</sup> (2)

, *ξ*<sup>2</sup> , (3)

<sup>2</sup> , *t ξ*<sup>1</sup> , *<sup>ξ</sup>*<sup>2</sup> ð Þ 2

*<sup>∂</sup>ξα :* (1)

Suppose two vectors **A** and **B** are located at the same point and their components are *Ai* and *Bi* , then the dot product **A***:***B** is given by

$$\mathbf{A.B} = A^i \mathbf{z}\_i.B^j \mathbf{z}\_j = \left(\mathbf{z}\_i \cdot \mathbf{z}\_j\right) A^i \cdot B^j = \mathbf{z}\_{i\dagger} A^i B^j. \tag{9}$$

The length of a vector **B** can be expressed in terms of covariant metric tensor as

$$|\mathbf{B}| = \sqrt{z\_{\vec{\eta}} \mathbf{B}^{\dot{i}} \mathbf{B}^{\dot{j}}} \tag{10}$$

Interestingly, covariant tensors are useful in modeling of higher-order initial lesions in human vascular system given by

$$\begin{aligned} \mathbf{1}. \mathbf{g}\_{i\boldsymbol{j}} &= \overrightarrow{\mathbf{g}}\_{i} \cdot \overrightarrow{\mathbf{g}}\_{\boldsymbol{j}} = \mathbf{z}\_{i\boldsymbol{j}} - 2\xi^{3}b\_{i\boldsymbol{j}} + \left(\xi^{3}\right)^{2}c\_{i\boldsymbol{j}}, \\\\ \mathbf{2}. \mathbf{g}\_{i\boldsymbol{3}} &= \overrightarrow{\mathbf{g}}\_{i} \cdot \overrightarrow{\mathbf{g}}\_{\boldsymbol{3}} = \mathbf{0} \\\\ \mathbf{3}. \mathbf{g}\_{3\boldsymbol{3}} &= \overrightarrow{\mathbf{g}}\_{\boldsymbol{3}} \cdot \overrightarrow{\mathbf{g}}\_{\boldsymbol{3}} = \mathbf{1} \end{aligned}$$

**Definition 4**. (Contravariant metric tensor *zij*). *The contravariant metric tensor zij is the matrix inverse of the covariant metric tensor zij:*

$$\mathbf{z}\_{\vec{\text{ij}}} \cdot \mathbf{z}^{jk} = \mathbf{z}\_{\vec{\text{ij}}} \cdot \mathbf{z}^{k\vec{\text{j}}} = \delta^i\_k,\tag{11}$$

where *δ<sup>i</sup> <sup>k</sup>* is the Kronecker symbol.

**Definition 5**. (Contravariant basis **z***<sup>i</sup>* ). *The contravariant basis z<sup>i</sup> is defined as*

$$\mathbf{z}^{i} = \mathbf{z}^{i\circ} \mathbf{z}\_{\circ} = \mathbf{z}^{i\circ} \mathbf{z}\_{\circ} \tag{12}$$

The bases **z***<sup>i</sup>* and **z***<sup>i</sup>* are mutually orthonormal:

$$\mathbf{z}\_{i} \cdot \mathbf{z}^{j} = \delta^{i}\_{j}. \tag{13}$$

**Definition 6**. (Christoffel symbol). *In affine and curvilinear coordinate systems, the covariant basis z<sup>i</sup> is the same at all points and varies from one point to another,*

*respectively. This variation can be described by the partial derivatives ∂zi=∂z<sup>j</sup> . Using decomposition of partial derivatives ∂zi=∂z<sup>j</sup> with respect to the covariant basis zk, the Christoffel symbol* Γ*<sup>k</sup> ij is given by*

$$\frac{\partial \mathfrak{x}\_i}{\partial \mathbf{z}^j} = \Gamma^k\_{ij} \mathfrak{x}\_k. \tag{14}$$

directions are called the *principal curvatures*. The product and the half-sum of the principal curvatures are classically known as the *Gaussian* curvature and *mean*

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human…*

*cij* <sup>¼</sup> *bk*

*z* 3,*<sup>i</sup>* ! ¼ *z* 3,*<sup>i</sup>* ! � *z<sup>k</sup>*

The initial tunica intimal lesion in AD is heterogenous in terms of various attributes such as shape, size, and conjugality among others. These notions of surface differential geometry are helpful to model these lesions as higher-order initial lesions. To illustrate, the surface of lesion modeled as initial lesion can be *elliptic*, *parabolic*, or *hyperbolic* according to whether its Gaussian curvature is positive, zero, or negative, respectively. Note that Gaussian curvature is derived from the second fundamental form. From now onwards, we simply use initial lesion

Normally, the aorta is composed of three layers, tunica adventitia, tunica media,

To simplify, it is assumed that collagen fibers are straight and resist deformation

caused by hemodynamic stresses. In addition, hemodynamic stress, normal to mid-surface of tunica media, is zero throughout the cardiac cycle. The modeling of initial lesion of AD based on the aforementioned kinematical assumptions can be interpreted as a truncation of the expansion of displacements across the thickness of the normal human aorta. The kinematical assumptions pertain to the displacements of points located on tunica intima layer of the aorta through the lesion thickness. Such points are orthogonal to mid-surface in the earlier pre-deformed configuration. Note that the kinematical assumptions connect the displacements of points located on the tunica intima layer that is orthogonal to the mid-surface of the tunica media layer in undeformed configuration. The displacement is expressed by the

and tunica intima (from outside to inside in cross section). Tunica adventitia is composed of linear palisades of collagen fibers as an envelope over tunica media that is a smooth muscle layer, capable of elastic recoil for propelling blood forward.

Tunica intima is quite a thin innermost layer comprised of linear array of

*z* 3,*<sup>i</sup>* ! ¼ �*bikz*

*<sup>i</sup> bkj*

! ¼ 0 *that is z* 3,*<sup>i</sup>* ! *which lies in the tangent plane.*

!*k*, (20)

*<sup>i</sup> z*!*<sup>k</sup> :* (21)

It is a derivative along a curve lying on the surface. Note that the expressions of surface Christoffel symbols and surface covariant derivative are inferred from the

*<sup>z</sup>*

!*<sup>k</sup>* ¼ �*b<sup>k</sup>*

curvature, respectively.

third fundamental form. **Remark 1**. *z* <sup>3</sup>

*Hence, we have*

and thus

collagen fibers.

following equation:

**187**

*The third fundamental form*

*DOI: http://dx.doi.org/10.5772/intechopen.89866*

! � *z* <sup>3</sup>

! ¼ 1 ) *z* 3,*<sup>i</sup>* ! � *z* <sup>3</sup>

model to describe initial lesion of aortic dissection.

**3. Modeling of initial lesion**

**3.1 A simplistic view of initial lesions**

Note that the Christoffel symbol is symmetric in lower indices:

$$
\Gamma^k\_{\vec{\eta}} = \Gamma^k\_{\vec{\mu}} = \mathbf{z}\_k \cdot \frac{\partial \mathbf{z}\_i}{\partial \mathbf{z}^j}. \tag{15}
$$

#### **2.2 Fundamental forms**

The *first fundamental form* of the surface is also known as the restriction of the metric tensor to the tangent plane. It is given by its components

$$\mathbf{z}\_{ij} = \overline{\mathbf{z}}\_i^{\rightarrow} \cdot \overline{\mathbf{z}}\_j^{\rightarrow}.$$

Alternatively, its contravariant form is given by

$$\mathbf{z}^{\overrightarrow{ij}} = \mathbf{z}^{\overrightarrow{i}} \cdot \mathbf{z}^{\overrightarrow{j}}.$$

Note that the first fundamental form can be used for the conversion of covariant components into contravariant components, such as

$$\mathbf{v}^i = \mathbf{z}^{ik} \mathbf{v}\_k$$

The Euclidean norm of the two-dimensional tensors is denoted by k k� *<sup>ε</sup>* and the corresponding inner product by < � , � >*ε*. Note that the first fundamental form can be used for the evaluation of such norm quantities:

$$<\underline{u}, \underline{v}>\_{\varepsilon} = u\_i \mathbf{z}^{\ddagger} v\_{\restriction},\tag{16}$$

$$\left\|\underline{v}\right\|\_{\mathfrak{e}}^{2} = \nu\_{i}\mathbf{z}^{\sharp}\boldsymbol{v}\_{j},\tag{17}$$

$$<\underline{T}, \underline{U}\ge = T\_{\vec{\eta}} \mathbf{z}^{ik} \mathbf{z}^{\vec{\eta}} U\_{kl},\tag{18}$$

$$\left\|\frac{T}{\Xi}\right\|\_{\mathfrak{r}}^2 = T\_{\vec{\eta}} \mathbf{z}^{ik} \mathbf{z}^{jl} T\_{kl}.\tag{19}$$

*The second fundamental form*

$$b\_{\vec{\eta}} = \mathbf{z}\_{\vec{\cdot}} \cdot \mathbf{z}\_{\overrightarrow{i\_{\vec{y}}}},$$

where

$$\mathbf{z}\_{\overrightarrow{i,j}} = \frac{\partial^2 \varrho}{\partial \xi^i \partial \xi^j} = \mathbf{z}\_{\overrightarrow{j,i}}$$

is the fundamental form of symmetry.

The second fundamental form is yet another important second-order tensor of the surface. It is also known as the *curvature tensor* since it provides information about the curvature of the surface. The values of these curvatures along the

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human… DOI: http://dx.doi.org/10.5772/intechopen.89866*

directions are called the *principal curvatures*. The product and the half-sum of the principal curvatures are classically known as the *Gaussian* curvature and *mean* curvature, respectively.

*The third fundamental form*

$$c\_{\vec{\eta}} = b\_i^k \ b\_{kj}$$

It is a derivative along a curve lying on the surface. Note that the expressions of surface Christoffel symbols and surface covariant derivative are inferred from the third fundamental form.

**Remark 1**. *z* <sup>3</sup> ! � *z* <sup>3</sup> ! ¼ 1 ) *z* 3,*<sup>i</sup>* ! � *z* <sup>3</sup> ! ¼ 0 *that is z* 3,*<sup>i</sup>* ! *which lies in the tangent plane. Hence, we have*

$$\mathbf{z}\_{\overline{\mathbf{3},i}'} = \left(\mathbf{z}\_{\overline{\mathbf{3},i}'} \cdot \mathbf{z}\_k\right) \overrightarrow{\mathbf{z}'},\tag{20}$$

and thus

*respectively. This variation can be described by the partial derivatives ∂zi=∂z<sup>j</sup>*

*∂zi <sup>∂</sup>zj* <sup>¼</sup> <sup>Γ</sup>*<sup>k</sup>*

Note that the Christoffel symbol is symmetric in lower indices:

Γ*k ij* <sup>¼</sup> <sup>Γ</sup>*<sup>k</sup>*

metric tensor to the tangent plane. It is given by its components

Alternatively, its contravariant form is given by

components into contravariant components, such as

be used for the evaluation of such norm quantities:

*The second fundamental form*

is the fundamental form of symmetry.

where

**186**

*ij is given by*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*Christoffel symbol* Γ*<sup>k</sup>*

**2.2 Fundamental forms**

*decomposition of partial derivatives ∂zi=∂z<sup>j</sup> with respect to the covariant basis zk, the*

*ji* ¼ **z***<sup>k</sup>* �

The *first fundamental form* of the surface is also known as the restriction of the

**z***ij* ¼ **z***<sup>i</sup>* ! � **z***<sup>j</sup>* !*:*

Note that the first fundamental form can be used for the conversion of covariant

**<sup>v</sup>***<sup>i</sup>* <sup>¼</sup> **<sup>z</sup>***ik***v***<sup>k</sup>*

The Euclidean norm of the two-dimensional tensors is denoted by k k� *<sup>ε</sup>* and the corresponding inner product by < � , � >*ε*. Note that the first fundamental form can

> k k*v* 2

*bij* ¼ **z**<sup>3</sup>

! � **<sup>z</sup>***i*,*<sup>j</sup>* !,

*φ ∂ξi*

The second fundamental form is yet another important second-order tensor of the surface. It is also known as the *curvature tensor* since it provides information about the curvature of the surface. The values of these curvatures along the

*<sup>∂</sup>ξ<sup>j</sup>* <sup>¼</sup> **<sup>z</sup>***j*,*<sup>i</sup>* !

*T* 2 *ε*

> **z***i*,*j* ! <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup>

<sup>&</sup>lt; *<sup>u</sup>*, *<sup>v</sup>*>*<sup>ε</sup>* <sup>¼</sup> *ui***z***ijvj*, (16)

<sup>&</sup>lt; *<sup>T</sup>*, *<sup>U</sup>*<sup>&</sup>gt; <sup>¼</sup> *Tij***z***ik***z***jlUkl*, (18)

*<sup>ε</sup>* <sup>¼</sup> *vi***z***ijvj*, (17)

<sup>¼</sup> *Tij***z***ik***z***jlTkl:* (19)

**<sup>z</sup>***ij* <sup>¼</sup> **<sup>z</sup>** *<sup>i</sup>* ! � **z** *j* ! *:*

*∂***z***i*

*. Using*

*ijzk:* (14)

*<sup>∂</sup>zj :* (15)

$$\mathbf{z}\_{\overrightarrow{3,i}} = -b\_{ik}\overrightarrow{\mathbf{z}^k} = -b\_i^k \mathbf{z}\_{\overrightarrow{k}}.\tag{21}$$

The initial tunica intimal lesion in AD is heterogenous in terms of various attributes such as shape, size, and conjugality among others. These notions of surface differential geometry are helpful to model these lesions as higher-order initial lesions. To illustrate, the surface of lesion modeled as initial lesion can be *elliptic*, *parabolic*, or *hyperbolic* according to whether its Gaussian curvature is positive, zero, or negative, respectively. Note that Gaussian curvature is derived from the second fundamental form. From now onwards, we simply use initial lesion model to describe initial lesion of aortic dissection.

#### **3. Modeling of initial lesion**

Normally, the aorta is composed of three layers, tunica adventitia, tunica media, and tunica intima (from outside to inside in cross section). Tunica adventitia is composed of linear palisades of collagen fibers as an envelope over tunica media that is a smooth muscle layer, capable of elastic recoil for propelling blood forward. Tunica intima is quite a thin innermost layer comprised of linear array of collagen fibers.

#### **3.1 A simplistic view of initial lesions**

To simplify, it is assumed that collagen fibers are straight and resist deformation caused by hemodynamic stresses. In addition, hemodynamic stress, normal to mid-surface of tunica media, is zero throughout the cardiac cycle. The modeling of initial lesion of AD based on the aforementioned kinematical assumptions can be interpreted as a truncation of the expansion of displacements across the thickness of the normal human aorta. The kinematical assumptions pertain to the displacements of points located on tunica intima layer of the aorta through the lesion thickness. Such points are orthogonal to mid-surface in the earlier pre-deformed configuration. Note that the kinematical assumptions connect the displacements of points located on the tunica intima layer that is orthogonal to the mid-surface of the tunica media layer in undeformed configuration. The displacement is expressed by the following equation:

$$
\overrightarrow{\mathfrak{D}}\left(\xi^1,\xi^2,\xi^3\right) = \overrightarrow{d}\left(\xi^1,\xi^2\right) + \xi^3 \overrightarrow{\theta}\_k\left(\xi^1,\xi^2\right) \overrightarrow{\mathbf{z}}^k\left(\xi^1,\xi^2\right),
\tag{22}
$$

Moreover,

8 >>><

>>>:

where

*∂***D**

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human…*

� �, *<sup>i</sup>* <sup>¼</sup> 1, 2

� � � *bij***z**<sup>3</sup>

*<sup>θ</sup><sup>i</sup>*∣*<sup>j</sup>* <sup>þ</sup> *<sup>θ</sup><sup>j</sup>*∣*<sup>i</sup>* � *<sup>b</sup><sup>k</sup>*

� �

� �

In the framework of the kinematical assumptions, the second-order tensors, *γ* and *χ*, and the first-order tensor *ζ* are called the membrane strain, bending strain,

In pathological conditions and even in physiological conditions strained to its limits, fluid-structure interaction in the aorta does not follow the kinematical assumptions because the arrangement of collagen fibers in the aortic wall is not straight. Tunica intima is comprised of a single layer of endothelial cells with a subendothelial layer of varying thickness. Tunica intimal surface is nonconformal depending upon the amount of subendothelial ground matrix, contrary to the conventional perspective of the conformal tunica intimal surface. The tunica media is a complex three-dimensional network of smooth muscle cells, elastin, and bundles of collagen fibrils. These well-defined concentrically oriented fibers are mutually reinforcing in radial direction. Tunica adventitia is comprised of fibroblasts,

<sup>2</sup> *<sup>θ</sup><sup>i</sup>* <sup>þ</sup> *<sup>d</sup>*3,*<sup>i</sup>* <sup>þ</sup> *<sup>b</sup><sup>k</sup>*

Substituting Eqs. (28), (29), and (7) into (23)

<sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup>

*χij d* ! , *θ* � � � *<sup>ξ</sup>*<sup>3</sup> � �<sup>2</sup>

¼ 1

2

2 *bk <sup>j</sup> <sup>θ</sup><sup>k</sup>*∣*<sup>i</sup>* <sup>þ</sup> *bk*

fibrocytes, collagen fibers (helically arranged), and ground matrix.

The constituents of the aortic wall including collagen fibers, elastin fibers, smooth muscle fibers, and ground matrix can stretch to deformation and recoil. Histologically and functionally, these constituents are viscoelastic; hence, aortic tissues resist deformation, albeit partially. Note that hemodynamic strain normal to

The assumptions in the simplistic case (22) does not hold true in clinical settings. Thus, an initial lesion model is required to incorporate these attributes. An initial lesion model that is asymptotically consistent with three-dimensional solid mechanics without resorting to any independent kinematical assumptions on the strains requires correction for rotation inaccuracies, while only linearized strain tensor is perused for displacement equation (22). For initial lesion model, the

<sup>2</sup> *di*∣*<sup>j</sup>* <sup>þ</sup> *dj*∣*<sup>i</sup>*

*eij* ¼ *γij d*

*ei*<sup>3</sup> ¼ *ζ<sup>i</sup> d* ! , *θ*

8

>>>>>>>>>>><

>>>>>>>>>>>:

and shear strain, respectively.

*e*<sup>33</sup> ¼ 0,

� �!

*DOI: http://dx.doi.org/10.5772/intechopen.89866*

*γij d* � �!

*χij d* ! , *θ* � � <sup>¼</sup> <sup>1</sup>

*ζ<sup>i</sup> d* ! , *θ* � � <sup>¼</sup> <sup>1</sup>

**3.2 Higher-order model for initial lesion**

mid-surface of tunica intima will not be zero.

**189**

*<sup>κ</sup>ij*ðÞ ¼ *<sup>θ</sup>* <sup>1</sup>

*<sup>∂</sup>ξ*<sup>3</sup> <sup>¼</sup> *<sup>θ</sup>k***z***<sup>k</sup>* (29)

(30)

(31)

*κij*ð Þ*θ* , *i*, *j* ¼ 1, 2

*<sup>j</sup> dk*∣*<sup>i</sup>* � *<sup>b</sup><sup>k</sup>*

� � <sup>þ</sup> *cij***z**<sup>3</sup>

*<sup>i</sup> θ<sup>k</sup>*∣*<sup>j</sup>*

*<sup>i</sup> dk*

*<sup>i</sup> dk*∣*<sup>j</sup>*

In Eq. (22), we consider the tunica intima layer in the direction of **z** ! <sup>3</sup> at the coordinate *ξ*<sup>1</sup> , *ξ*<sup>2</sup> . The displacement *d* ! *ξ*1 , *ξ*<sup>2</sup> represents a global infinitesimal displacement of the linearly arranged endothelial cells of the tunica intima on the line displacing by the similar amount. The displacement *ξ*<sup>3</sup> *θ* ! *<sup>k</sup> ξ*<sup>1</sup> , *ξ*<sup>2</sup> **z** !*<sup>k</sup> ξ*<sup>1</sup> , *ξ*<sup>2</sup> is due to the rotation of the line measured by *θ*<sup>1</sup> and *θ*2.

Hemodynamic flows can cause both linear and rotational strain. The linear strain is caused by laminar flow, while the rotational strain is caused by either turbulent flow and/or concomitant nonlinear geometry of the vessel. Thus, the measure of linear strain is not sufficient, rather inaccuracies emanate from the increments in rotation. We choose the principle of deformation gradient to calculate both the strains. The combined linear and nonlinear strains can be characterized by stretch tensor called Green-Lagrange strain tensor. The 3D-Lagrange-Green tensor, for which the components *eαβ* for general displacement **D** ! *ξ*1 , *ξ*<sup>2</sup> , *ξ*<sup>3</sup> are

$$\boldsymbol{e}\_{a\boldsymbol{\beta}} = \frac{1}{2} \left( \overrightarrow{\mathbf{g}}\_{a} \cdot \overrightarrow{\mathbf{\mathcal{D}}}\_{,\boldsymbol{\beta}} + \overrightarrow{\mathbf{g}}\_{,\boldsymbol{\beta}} \cdot \overrightarrow{\mathbf{\mathcal{D}}}\_{,a} \right), \quad a,\boldsymbol{\beta} = \mathbf{1},2,3. \tag{23}$$

To calculate the components of Green-Lagrange strain tensor, we need to evaluate **<sup>D</sup>**,*<sup>α</sup>* <sup>¼</sup> *<sup>∂</sup>***D***=∂ξ<sup>α</sup>* (displacement of endothelial cells in a line on the tunica intima in *ξ<sup>α</sup>* direction). For the specific displacement in (22), we compute the covariant components of the linearized strain tensor. We have

$$\frac{\partial d}{\partial \xi^i} = \frac{\partial}{\partial \xi^i} \left( d\_k \mathbf{z}^k + d\_3 \mathbf{z}\_3 \right) \tag{24}$$

We peruse the fundamental forms to obtain

$$\frac{\partial}{\partial \xi^i} \left( d\_k \mathbf{z}^k \right) = \mathbf{z}^k \frac{\partial d\_k}{\partial \xi^i} + d\_k \frac{\partial \mathbf{z}^k}{\partial \xi^i} = \mathbf{z}^k \frac{\partial d\_k}{\partial \xi^i} + b\_i^k d\_k a\_3. \tag{25}$$

Hence,

$$\begin{split} \frac{\partial d}{\partial \xi^i} &= d\_{k|i} \mathbf{z}^k + b\_i^k d\_k \mathbf{z}\_3 + d\_{3,i} \mathbf{z}\_3 + d\_3 \mathbf{z}\_{3,i} \\ &= \left( d\_{k|i} - b\_{ki} d\_3 \right) \mathbf{z}^k + \left( d\_{3,i} + b\_i^k d\_k \right) \mathbf{z}\_3, \end{split} \tag{26}$$

where *dk*∣*<sup>i</sup>* <sup>¼</sup> *<sup>∂</sup>dk=∂ξ<sup>i</sup>* . As we have calculated the derivative for linearized strain, we calculate the derivative for rotational strain. From (21)

$$\frac{\partial}{\partial \xi^i} (\theta\_k \mathbf{z}\_k) = \theta\_{k|i} \mathbf{z}\_k + b\_i^k \theta\_k \mathbf{z}\_3. \tag{27}$$

The overall displacement in Eq. (22) is composed of linear displacement and rotational displacement. Therefore,

$$\frac{\partial \mathfrak{D}}{\partial \xi^i} = \frac{\partial d}{\partial \xi^i} + \frac{\partial}{\partial \xi^i} \left( \xi^3 \theta\_k \mathbf{z}\_k \right) = \left( d\_{k|i} - b\_{ki} \mathbf{z}\_3 + \xi^3 \theta\_{k|i} \right) \mathbf{z}\_k + \left( d\_{3,i} + b\_i^k d\_k + \xi^3 b\_i^k \theta\_k \right) \mathbf{z}\_3. \tag{28}$$

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human… DOI: http://dx.doi.org/10.5772/intechopen.89866*

Moreover,

D ! *ξ*1 , *ξ*<sup>2</sup> , *<sup>ξ</sup>*<sup>3</sup> <sup>¼</sup> *<sup>d</sup>*

, *ξ*<sup>2</sup> . The displacement *d*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

to the rotation of the line measured by *θ*<sup>1</sup> and *θ*2.

which the components *eαβ* for general displacement **D**

components of the linearized strain tensor. We have

We peruse the fundamental forms to obtain

*∂d*

*<sup>∂</sup>ξ<sup>i</sup> dk***z***<sup>k</sup>* <sup>¼</sup> **<sup>z</sup>***<sup>k</sup> <sup>∂</sup>dk*

*∂*

Hence,

*∂***D** *<sup>∂</sup>ξ<sup>i</sup>* <sup>¼</sup> *<sup>∂</sup><sup>d</sup> ∂ξ<sup>i</sup>* þ

**188**

where *dk*∣*<sup>i</sup>* <sup>¼</sup> *<sup>∂</sup>dk=∂ξ<sup>i</sup>*

rotational displacement. Therefore,

*∂ <sup>∂</sup>ξ<sup>i</sup> <sup>ξ</sup>*<sup>3</sup>

*θk***z***<sup>k</sup>*

*∂d <sup>∂</sup>ξ<sup>i</sup>* <sup>¼</sup> *<sup>∂</sup>*

*<sup>∂</sup>ξ<sup>i</sup>* <sup>¼</sup> *dk*∣*<sup>i</sup>***z***<sup>k</sup>* <sup>þ</sup> *bk*

we calculate the derivative for rotational strain. From (21) *∂*

<sup>¼</sup> *dk*∣*<sup>i</sup>* � *bki***z**<sup>3</sup> <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup>

¼ *dk*∣*<sup>i</sup>* � *bkid*<sup>3</sup>

*<sup>e</sup>αβ* <sup>¼</sup> <sup>1</sup> <sup>2</sup> *<sup>g</sup>* ! *<sup>α</sup>* � **D** ! ,*<sup>β</sup>* þ *g* ! *<sup>β</sup>* � **D** ! ,*α*

line displacing by the similar amount. The displacement *ξ*<sup>3</sup> *θ*

coordinate *ξ*<sup>1</sup>

! *ξ*1 , *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> *<sup>θ</sup>*

In Eq. (22), we consider the tunica intima layer in the direction of **z**

*<sup>∂</sup>ξ<sup>i</sup>* <sup>þ</sup> *dk*

To calculate the components of Green-Lagrange strain tensor, we need to evaluate **<sup>D</sup>**,*<sup>α</sup>* <sup>¼</sup> *<sup>∂</sup>***D***=∂ξ<sup>α</sup>* (displacement of endothelial cells in a line on the tunica intima in *ξ<sup>α</sup>* direction). For the specific displacement in (22), we compute the covariant

*<sup>∂</sup>ξ<sup>i</sup> dk***z***<sup>k</sup>* <sup>þ</sup> *<sup>d</sup>*3**z**<sup>3</sup>

*∂***z***<sup>k</sup>*

**<sup>z</sup>***<sup>k</sup>* <sup>þ</sup> *<sup>d</sup>*3,*<sup>i</sup>* <sup>þ</sup> *bk*

*<sup>∂</sup>ξ<sup>i</sup>* ð Þ¼ *<sup>θ</sup>k***z***<sup>k</sup> <sup>θ</sup><sup>k</sup>*∣*<sup>i</sup>***z***<sup>k</sup>* <sup>þ</sup> *bk*

The overall displacement in Eq. (22) is composed of linear displacement and

*θ<sup>k</sup>*∣*<sup>i</sup>* **<sup>z</sup>***<sup>k</sup>* <sup>þ</sup> *<sup>d</sup>*3,*<sup>i</sup>* <sup>þ</sup> *bk*

*<sup>∂</sup>ξ<sup>i</sup>* <sup>¼</sup> **<sup>z</sup>***<sup>k</sup> <sup>∂</sup>dk*

*<sup>i</sup> dk***z**<sup>3</sup> þ *d*3,*<sup>i</sup>***z**<sup>3</sup> þ *d*3**z**3,*<sup>i</sup>*

*<sup>∂</sup>ξ<sup>i</sup>* <sup>þ</sup> *bk*

*<sup>i</sup> dk* 

. As we have calculated the derivative for linearized strain,

**z**3,

*<sup>i</sup> θk***z**3*:* (27)

*<sup>i</sup> dk* <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup>

*bk <sup>i</sup> θ<sup>k</sup>*

**z**3*:*

(28)

! *ξ*1

displacement of the linearly arranged endothelial cells of the tunica intima on the

Hemodynamic flows can cause both linear and rotational strain. The linear strain is caused by laminar flow, while the rotational strain is caused by either turbulent flow and/or concomitant nonlinear geometry of the vessel. Thus, the measure of linear strain is not sufficient, rather inaccuracies emanate from the increments in rotation. We choose the principle of deformation gradient to calculate both the strains. The combined linear and nonlinear strains can be characterized by stretch tensor called Green-Lagrange strain tensor. The 3D-Lagrange-Green tensor, for

! *<sup>k</sup> ξ*<sup>1</sup> , *ξ*<sup>2</sup> **z**

!*<sup>k</sup> ξ*<sup>1</sup>

, *ξ*<sup>2</sup> represents a global infinitesimal

! *<sup>k</sup> ξ*<sup>1</sup> , *ξ*<sup>2</sup> **z**

, *α*, *β* ¼ 1, 2, 3*:* (23)

*<sup>i</sup> dka*3*:* (25)

(26)

(24)

! *ξ*1 , *ξ*<sup>2</sup> , *ξ*<sup>3</sup> are

, *ξ*<sup>2</sup> , (22)

! <sup>3</sup> at the

!*<sup>k</sup> ξ*<sup>1</sup>

, *ξ*<sup>2</sup> is due

$$\frac{\partial \mathfrak{D}}{\partial \boldsymbol{\xi}^{\overline{\boldsymbol{\xi}}^{\overline{\boldsymbol{\xi}}}}} = \theta\_k \mathbf{z}^k \tag{29}$$

Substituting Eqs. (28), (29), and (7) into (23)

$$\begin{cases} \boldsymbol{\varepsilon}\_{\vec{\boldsymbol{\eta}}} = \boldsymbol{\chi}\_{\vec{\boldsymbol{\eta}}} \left( \overrightarrow{\boldsymbol{d}} \right) + \boldsymbol{\xi}^{3} \boldsymbol{\chi}\_{\vec{\boldsymbol{\eta}}} \left( \overrightarrow{\boldsymbol{d}}, \boldsymbol{\varrho} \right) - \left( \boldsymbol{\xi}^{3} \right)^{2} \boldsymbol{\kappa}\_{\vec{\boldsymbol{\eta}}}(\boldsymbol{\varrho}), & \boldsymbol{i}, \boldsymbol{j} = \mathbf{1}, 2 \\\\ \boldsymbol{\varepsilon}\_{\vec{\boldsymbol{\eta}}} = \boldsymbol{\zeta}\_{\vec{\boldsymbol{\eta}}} \left( \overrightarrow{\boldsymbol{d}}, \boldsymbol{\varrho} \right), & \boldsymbol{i} = \mathbf{1}, 2 \\\\ \boldsymbol{\varepsilon}\_{33} = \mathbf{0}, \end{cases} \tag{30}$$

where

$$\begin{cases} \begin{aligned} \chi\_{\vec{\eta}} \left( \overrightarrow{d} \right) &= \frac{1}{2} \left( d\_{l\vec{\eta}} + d\_{j\vec{\mu}} \right) - b\_{\vec{\eta}} \mathbf{z}\_3 \\ \varkappa\_{\vec{\eta}} \left( \overrightarrow{d}, \underline{\theta} \right) &= \frac{1}{2} \left( \theta\_{l\vec{\eta}} + \theta\_{j\vec{\mu}} - b\_j^k d\_{k\vec{\mu}} - b\_i^k d\_{k\vec{\eta}} \right) + c\_{\vec{\eta}} \mathbf{z}\_3 \\ \kappa\_{\vec{\eta}}(\underline{\theta}) &= \frac{1}{2} \left( b\_j^k \theta\_{k\vec{\mu}} + b\_i^k \theta\_{k\vec{\eta}} \right) \\ \zeta\_i \left( \overrightarrow{d}, \underline{\theta} \right) &= \frac{1}{2} \left( \theta\_i + d\_{3,i} + b\_i^k d\_k \right) \end{aligned} \tag{31}$$

In the framework of the kinematical assumptions, the second-order tensors, *γ* and *χ*, and the first-order tensor *ζ* are called the membrane strain, bending strain, and shear strain, respectively.

#### **3.2 Higher-order model for initial lesion**

In pathological conditions and even in physiological conditions strained to its limits, fluid-structure interaction in the aorta does not follow the kinematical assumptions because the arrangement of collagen fibers in the aortic wall is not straight. Tunica intima is comprised of a single layer of endothelial cells with a subendothelial layer of varying thickness. Tunica intimal surface is nonconformal depending upon the amount of subendothelial ground matrix, contrary to the conventional perspective of the conformal tunica intimal surface. The tunica media is a complex three-dimensional network of smooth muscle cells, elastin, and bundles of collagen fibrils. These well-defined concentrically oriented fibers are mutually reinforcing in radial direction. Tunica adventitia is comprised of fibroblasts, fibrocytes, collagen fibers (helically arranged), and ground matrix.

The constituents of the aortic wall including collagen fibers, elastin fibers, smooth muscle fibers, and ground matrix can stretch to deformation and recoil. Histologically and functionally, these constituents are viscoelastic; hence, aortic tissues resist deformation, albeit partially. Note that hemodynamic strain normal to mid-surface of tunica intima will not be zero.

The assumptions in the simplistic case (22) does not hold true in clinical settings. Thus, an initial lesion model is required to incorporate these attributes. An initial lesion model that is asymptotically consistent with three-dimensional solid mechanics without resorting to any independent kinematical assumptions on the strains requires correction for rotation inaccuracies, while only linearized strain tensor is perused for displacement equation (22). For initial lesion model, the

displacement vector **D** ! *ξ*1 , *ξ*<sup>2</sup> , *ξ*<sup>3</sup> � � contains at least all terms up to degree two, namely,

$$
\overrightarrow{\mathfrak{D}}\left(\xi^{1},\xi^{2},\xi^{3}\right) = \overrightarrow{d}\left(\xi^{1},\xi^{2}\right) + \xi^{3}\overrightarrow{\theta}\left(\xi^{1},\xi^{2}\right) + \left(\xi^{3}\right)^{2}\overrightarrow{\mathfrak{q}}\left(\xi^{1},\xi^{2}\right).\tag{32}
$$

where the function, Δ

Δ ! *ξ*1 , *ξ*<sup>2</sup> , *<sup>ξ</sup>*<sup>3</sup> <sup>¼</sup> *<sup>δ</sup>*

! *ξ*1 , *ξ*<sup>2</sup>

higher-order displacement equation, setting *<sup>τ</sup>* <sup>¼</sup> <sup>2</sup>*ξ*<sup>3</sup>

<sup>¼</sup> *τ τ*ð Þ � <sup>1</sup> 2

**4. Mathematical analysis of initial lesion**

upper bound and contributing to the severity of disease.

For a particular bound and coercivity for a test function Δ

*d*

**D** !

form of *τ τ*ð Þ � <sup>1</sup> *<sup>=</sup>*2, 1 � ð Þ*<sup>τ</sup>* <sup>2</sup>

disease.

**191**

a unique **D**

! *d* ! , *θ* ! , ϱ ! *ξ*1 , *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> *<sup>η</sup>*

for initial lesion) there exists unique **D**

*DOI: http://dx.doi.org/10.5772/intechopen.89866*

, *ξ*<sup>3</sup> , is called test function; for each Δ

! *ξ*<sup>1</sup>

It obviously comes to mind: what are kinematical assumptions in initial lesion model? Keeping the histological and functional perspective of the vessel wall from the biomechanical point of view, it is known that internal surface of the vessel wall is not smooth. It becomes obvious that it will not follow banal kinematical assumptions as mentioned earlier. Note that the initial lesion of AD might be evolving on the tunica intima due to medial degeneration. The lesion presence is spatially nonlinear. It seems plausible that it is governed by quadratic equation as higherorder tensor has quadratic components. The equation for kinematical assumption in

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human…*

!*bot* <sup>þ</sup> <sup>1</sup> � ð Þ*<sup>τ</sup>* <sup>2</sup> *<sup>d</sup>*

Since the internal lining is not smooth, a gestalt view of affected tunica intima has initial lesions at differing heights. Lesions are on the tunica intima surface. To localize spatial dimension of lesions across tunica intima surface, correction terms are to be introduced to tunica intima surface levels, viz. top, mid, and bottom in

, and *τ τ*ð Þ þ 1 *=*2, respectively.

We did weak formulation to estimate strain tensors. Now, we assess net displacement. In order to do so, well-posedness of variational form (35) is the key. To understand the evolution of AD, the initial lesion from its inception to the advanced stage wherein the lesion contributes to nonlinear radial dilatation and marginal elongation of diseased aortic tissue needs to be evaluated. There are bounds to emergence of lesion, the lower bound is the status of primal lesion first noticed, and the upper bound is advanced stage of lesion that contributes to rupture of the aorta. Within these bounds, the blood flow acts on the lining of the aorta, adversely impacting the primal lesion that is susceptible to progression from lower bound to

The inherent nature of normal aortic tissue is to retain its earlier state despite varying interplay of tensors in higher-order. But this resistive tendency, called *coercivity*, weakens as primal lesion progresses towards upper bound. This transition from lower bound to upper bound depends on the complex interplay of various tensors in higher-order. Interestingly, the interplay between tensors in higher-order and coercivity is responsible for worsening of the disease. Intuitively, coercivity is inversely proportional to the progression towards upper bound. Thus, gaining information about the progression towards upper bound and concomitant decline in coercivity is vital to understand net displacement of initial lesion and progression of

! , exemplifying a particular state of disease. Furthermore, there

are various states of displacement of initial lesion due to progression of the disease.

∈V such that Eq. (35) holds:

*ς* ! *ξ*<sup>1</sup>

*=t*, is given by

! *δ* ! , *η* !, *ς*

! , there exists

*d*

!*mid* <sup>þ</sup> *τ τ*ð Þ <sup>þ</sup> <sup>1</sup> 2

, *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> <sup>2</sup>

!

!

, *ξ*<sup>2</sup> (36)

!*top* (37)

∈V (domain

In the simplistic view, the strain normal to the tunica intima is zero since the vessel wall does not deform. In higher-order model, the vector *θ* ! is arbitrary in the Euclidean space and not constrained to lie in the tangential plane. The modified expression for strain components is as follows:

$$\begin{cases} \begin{aligned} \varepsilon\_{\vec{\eta}} \left( \overrightarrow{\mathfrak{D}} \right) &= \chi\_{\vec{\eta}} \left( \overrightarrow{d} \right) + \xi^{3} \chi\_{\vec{\eta}} \left( \overrightarrow{d}, \overrightarrow{\theta} \right) + \left( \xi^{3} \right)^{2} \kappa\_{\vec{\eta}} \left( \overrightarrow{\theta}, \overrightarrow{\mathbf{q}} \right) + \left( \xi^{3} \right)^{3} l\_{\vec{\eta}} \left( \overrightarrow{\mathbf{q}} \right) \\\ \varepsilon\_{\vec{\eta}} \left( \overrightarrow{\mathfrak{D}} \right) &= \zeta\_{i} \left( \overrightarrow{d}, \overrightarrow{\theta} \right) + \xi^{3} m\_{i} \left( \overrightarrow{\theta}, \overrightarrow{\mathbf{q}} \right) + \left( \xi^{3} \right)^{2} n\_{i} \left( \overrightarrow{\mathbf{q}} \right) \\\ \varepsilon\_{33} \left( \overrightarrow{\mathfrak{D}} \right) &= \varpi \left( \overrightarrow{\theta} \right) + \xi^{3} p \left( \overrightarrow{\mathsf{q}} \right) \end{aligned} \tag{33}$$

where

*γij d* � �! ¼ 1 2 *di*∣*<sup>j</sup>* þ *dj*∣*<sup>i</sup>* � � � *bijd*<sup>3</sup> *χij d* ! , *θ* � �! ¼ 1 <sup>2</sup> *<sup>θ</sup><sup>i</sup>*∣*<sup>j</sup>* <sup>þ</sup> *<sup>θ</sup><sup>j</sup>*∣*<sup>i</sup>* � *bk <sup>j</sup> dk*∣*<sup>i</sup>* � *<sup>b</sup><sup>k</sup> <sup>i</sup> dk*∣*<sup>j</sup>* � � � *bijθ*<sup>3</sup> <sup>þ</sup> *cijd*<sup>3</sup> *κij θ* ! , ϱ ! � � <sup>¼</sup> <sup>1</sup> <sup>2</sup> <sup>ϱ</sup>*<sup>i</sup>*∣*<sup>j</sup>* <sup>þ</sup> <sup>ϱ</sup>*<sup>j</sup>*∣*<sup>i</sup>* � *bk <sup>j</sup> <sup>θ</sup><sup>k</sup>*∣*<sup>i</sup>* � *bk <sup>i</sup> θ<sup>k</sup>*∣*<sup>j</sup>* � � � *bij*ϱ<sup>3</sup> <sup>þ</sup> *cijθ*<sup>3</sup> *lij* ϱ !� � ¼ � <sup>1</sup> <sup>2</sup> *<sup>b</sup><sup>k</sup> <sup>j</sup>* <sup>ϱ</sup>*<sup>k</sup>*∣*<sup>i</sup>* <sup>þ</sup> *<sup>b</sup><sup>k</sup> <sup>i</sup>* ϱ*<sup>k</sup>*∣*<sup>j</sup>* � � <sup>þ</sup> *cij*ϱ<sup>3</sup> *ζ<sup>i</sup> d* ! , *θ* � �! ¼ 1 2 *<sup>θ</sup><sup>i</sup>* <sup>þ</sup> *<sup>d</sup>*3,*<sup>i</sup>* <sup>þ</sup> *<sup>b</sup><sup>k</sup> <sup>i</sup> dk* � � *mi θ* ! , ϱ ! � � <sup>¼</sup> <sup>1</sup> 2 ð Þ 2ϱ*<sup>i</sup>* þ *θ*3,*<sup>i</sup> ni* ϱ !� � <sup>¼</sup> <sup>1</sup> <sup>2</sup> �*b<sup>k</sup> <sup>i</sup>* ϱ*<sup>k</sup>* þ ϱ3,*<sup>i</sup>* � � *ϖ θ* � �! ¼ *θ*<sup>3</sup> *P* ϱ !� � <sup>¼</sup> <sup>2</sup>ϱ<sup>3</sup> 8 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: (34)

Here, the tensors, *γ* and *ζ*, are called the membrane and shear strain tensors as defined in Eq. (31). The tensor *χ* is a generalization of the bending strain tensor, and *k* is a generalization of �*κ* in Eq. (31), since *θ*<sup>3</sup> appears in the expressions of *χ* and *k* in Eq. (34). Because of different orders in *ξ*<sup>3</sup> in higher-order displacement vector, the newer tensors including *l*, *m*, *n*, *ϖ*, and *p* are obtained. In initial lesion model, the different orders in *ξ*<sup>3</sup> introduces complex interplay of various tensors. The continuous interplay among tensors of different orders makes it difficult to calculate resultant displacement, comprised of linear and rotational displacements. It becomes necessary to peruse algebra for weak formulation of this complex interplay of tensors. The variational formulation using a test function on displacement which aids to evaluate displacement equation in higher-order is

$$\int\_{\mathfrak{Q}} H^{a\beta\lambda\mu} e\_{a\beta} \left(\overrightarrow{\mathfrak{D}}\right) e\_{\lambda\mu} \left(\overrightarrow{\Delta}\right) dV = \int\_{\mathfrak{Q}} \overrightarrow{F}. \overrightarrow{\Delta}dV,\tag{35}$$

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human… DOI: http://dx.doi.org/10.5772/intechopen.89866*

where the function, Δ ! *ξ*1 , *ξ*<sup>2</sup> , *ξ*<sup>3</sup> , is called test function; for each Δ ! ∈V (domain for initial lesion) there exists unique **D** ! ∈V such that Eq. (35) holds:

$$
\overrightarrow{\Delta} \left( \xi^1, \xi^2, \xi^3 \right) = \overrightarrow{\delta} \left( \xi^1, \xi^2 \right) + \xi^3 \overrightarrow{\eta} \left( \xi^1, \xi^2 \right) + \left( \xi^3 \right)^2 \overrightarrow{\xi} \left( \xi^1, \xi^2 \right) \tag{36}
$$

It obviously comes to mind: what are kinematical assumptions in initial lesion model? Keeping the histological and functional perspective of the vessel wall from the biomechanical point of view, it is known that internal surface of the vessel wall is not smooth. It becomes obvious that it will not follow banal kinematical assumptions as mentioned earlier. Note that the initial lesion of AD might be evolving on the tunica intima due to medial degeneration. The lesion presence is spatially nonlinear. It seems plausible that it is governed by quadratic equation as higherorder tensor has quadratic components. The equation for kinematical assumption in higher-order displacement equation, setting *<sup>τ</sup>* <sup>¼</sup> <sup>2</sup>*ξ*<sup>3</sup> *=t*, is given by

$$
\overrightarrow{\mathfrak{D}} = \frac{\mathfrak{r}(\mathfrak{r} - \mathfrak{1})}{2} \overrightarrow{d}^{bot} + \left(\mathfrak{1} - \left(\mathfrak{r}\right)^{2}\right) \overrightarrow{d}^{mid} + \frac{\mathfrak{r}(\mathfrak{r} + \mathfrak{1})}{2} \overrightarrow{d}^{top} \tag{37}
$$

Since the internal lining is not smooth, a gestalt view of affected tunica intima has initial lesions at differing heights. Lesions are on the tunica intima surface. To localize spatial dimension of lesions across tunica intima surface, correction terms are to be introduced to tunica intima surface levels, viz. top, mid, and bottom in form of *τ τ*ð Þ � <sup>1</sup> *<sup>=</sup>*2, 1 � ð Þ*<sup>τ</sup>* <sup>2</sup> , and *τ τ*ð Þ þ 1 *=*2, respectively.

#### **4. Mathematical analysis of initial lesion**

We did weak formulation to estimate strain tensors. Now, we assess net displacement. In order to do so, well-posedness of variational form (35) is the key. To understand the evolution of AD, the initial lesion from its inception to the advanced stage wherein the lesion contributes to nonlinear radial dilatation and marginal elongation of diseased aortic tissue needs to be evaluated. There are bounds to emergence of lesion, the lower bound is the status of primal lesion first noticed, and the upper bound is advanced stage of lesion that contributes to rupture of the aorta. Within these bounds, the blood flow acts on the lining of the aorta, adversely impacting the primal lesion that is susceptible to progression from lower bound to upper bound and contributing to the severity of disease.

The inherent nature of normal aortic tissue is to retain its earlier state despite varying interplay of tensors in higher-order. But this resistive tendency, called *coercivity*, weakens as primal lesion progresses towards upper bound. This transition from lower bound to upper bound depends on the complex interplay of various tensors in higher-order. Interestingly, the interplay between tensors in higher-order and coercivity is responsible for worsening of the disease. Intuitively, coercivity is inversely proportional to the progression towards upper bound. Thus, gaining information about the progression towards upper bound and concomitant decline in coercivity is vital to understand net displacement of initial lesion and progression of disease.

For a particular bound and coercivity for a test function Δ ! *δ* ! , *η* !, *ς* ! , there exists a unique **D** ! *d* ! , *θ* ! , ϱ ! , exemplifying a particular state of disease. Furthermore, there are various states of displacement of initial lesion due to progression of the disease.

displacement vector **D**

*eij* **D** � � !

8 >>>>><

>>>>>:

where

8

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

*ei*<sup>3</sup> **D** � � !

*e*<sup>33</sup> **D** � � !

> *γij d* � �!

*χij d* ! , *θ* � �!

*κij θ* ! , ϱ ! � �

*lij* ϱ !� �

*ζ<sup>i</sup> d* ! , *θ* � �!

*mi θ* ! , ϱ ! � �

*ni* ϱ !� �

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

**190**

*ϖ θ* � �!

*P* ϱ !� �

**D** ! *ξ*1 , *ξ*<sup>2</sup> , *<sup>ξ</sup>*<sup>3</sup> � � <sup>¼</sup> *<sup>d</sup>*

namely,

! *ξ*1 , *ξ*<sup>2</sup>

expression for strain components is as follows:

<sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup>

<sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> *p* ϱ !� �

¼ 1 2

¼ 1

¼ 1

¼ � <sup>1</sup> <sup>2</sup> *<sup>b</sup><sup>k</sup>*

¼ 1 2

¼ 1 2

¼ 1 <sup>2</sup> �*b<sup>k</sup>*

¼ *θ*<sup>3</sup>

¼ 2ϱ<sup>3</sup>

aids to evaluate displacement equation in higher-order is

*Hαβλμeαβ* **D**

� � !

*eλμ* Δ � � !

*dV* ¼ ð Ω *F* ! *:*Δ !

*dV*, (35)

ð Ω <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup>

*di*∣*<sup>j</sup>* þ *dj*∣*<sup>i</sup>* � � � *bijd*<sup>3</sup>

<sup>2</sup> *<sup>θ</sup><sup>i</sup>*∣*<sup>j</sup>* <sup>þ</sup> *<sup>θ</sup><sup>j</sup>*∣*<sup>i</sup>* � *bk*

<sup>2</sup> <sup>ϱ</sup>*<sup>i</sup>*∣*<sup>j</sup>* <sup>þ</sup> <sup>ϱ</sup>*<sup>j</sup>*∣*<sup>i</sup>* � *bk*

*<sup>j</sup>* <sup>ϱ</sup>*<sup>k</sup>*∣*<sup>i</sup>* <sup>þ</sup> *<sup>b</sup><sup>k</sup>*

� �

*<sup>i</sup>* ϱ*<sup>k</sup>* þ ϱ3,*<sup>i</sup>* � �

*<sup>θ</sup><sup>i</sup>* <sup>þ</sup> *<sup>d</sup>*3,*<sup>i</sup>* <sup>þ</sup> *<sup>b</sup><sup>k</sup>*

ð Þ 2ϱ*<sup>i</sup>* þ *θ*3,*<sup>i</sup>*

� �

¼ *γij d* � �!

¼ *ζ<sup>i</sup> d* ! , *θ* � �!

¼ *ϖ θ* � �! ! *ξ*1 , *<sup>ξ</sup>*<sup>2</sup> � � <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> *<sup>θ</sup>*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

vessel wall does not deform. In higher-order model, the vector *θ*

*χij d* ! , *θ* � �!

> *mi θ* ! , ϱ ! � �

, *ξ*<sup>3</sup> � � contains at least all terms up to degree two,

, *<sup>ξ</sup>*<sup>2</sup> � � <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> � �<sup>2</sup>

ϱ ! *ξ*<sup>1</sup>

<sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> � �<sup>3</sup>

� *bijθ*<sup>3</sup> þ *cijd*<sup>3</sup>

� *bij*ϱ<sup>3</sup> þ *cijθ*<sup>3</sup>

!

*lij* ϱ !� �

, *ξ*<sup>2</sup> � �*:* (32)

is arbitrary in the

(33)

(34)

! *ξ*1

In the simplistic view, the strain normal to the tunica intima is zero since the

<sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> � �<sup>2</sup>

<sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> � �<sup>2</sup>

*<sup>j</sup> dk*∣*<sup>i</sup>* � *<sup>b</sup><sup>k</sup>*

*<sup>j</sup> <sup>θ</sup><sup>k</sup>*∣*<sup>i</sup>* � *bk*

� �

� �

*<sup>i</sup>* ϱ*<sup>k</sup>*∣*<sup>j</sup>*

*<sup>i</sup> dk*

Here, the tensors, *γ* and *ζ*, are called the membrane and shear strain tensors as defined in Eq. (31). The tensor *χ* is a generalization of the bending strain tensor, and *k* is a generalization of �*κ* in Eq. (31), since *θ*<sup>3</sup> appears in the expressions of *χ* and *k* in Eq. (34). Because of different orders in *ξ*<sup>3</sup> in higher-order displacement vector, the newer tensors including *l*, *m*, *n*, *ϖ*, and *p* are obtained. In initial lesion model, the different orders in *ξ*<sup>3</sup> introduces complex interplay of various tensors. The continuous interplay among tensors of different orders makes it difficult to calculate resultant displacement, comprised of linear and rotational displacements. It becomes necessary to peruse algebra for weak formulation of this complex interplay of tensors. The variational formulation using a test function on displacement which

*<sup>i</sup> dk*∣*<sup>j</sup>*

*<sup>i</sup> θ<sup>k</sup>*∣*<sup>j</sup>*

þ *cij*ϱ<sup>3</sup>

*κij θ* ! , ϱ ! � �

> *ni* ϱ !� �

Euclidean space and not constrained to lie in the tangential plane. The modified

Such a compendium is used to characterize a particular displacement state. A higher-dimensional space, *Sobolev space* which is comprised of all such possible combinations, comes handy. It is constituted of functions with sufficiently many derivative including partial differential equations of fluid-structure interaction and equipped with the norm that measures size and regularity of these functions. The test function Δ ! *δ* ! , *η* !, *ς* ! � � is a replica of **<sup>D</sup>** ! *d* ! , *θ* ! , ϱ ! � � in higher-dimensional metric space, V. Since test function is an idealized version of net displacement vector in continuum mechanics, evaluating the interaction between test function Δ ! *δ* ! , *η* !, *ς* ! � � and force *F* ! yields insights about **D** ! *d* ! , *θ* ! , ϱ ! � �. <sup>Δ</sup> ! *δ* ! , *η* !, *ς* ! � � is a test function present in Sobolev space. Gaining information about the characteristics of Δ ! *δ* ! , *η* !, *ς* ! � �, we can infer about **D** ! *d* ! , *θ* ! , ϱ ! � �. It yields insight about the disease process wherein with progression the initial lesion is contributed by the blood flow. Interestingly, proving well-posedness of Eq. (35) gives insights about **D** ! *d* ! , *θ* ! , ϱ ! � � present in bilinear function *A d*! , *θ* ! , ϱ !; *δ* ! , *η* !, *ς* ! � �, which is given by

c*zik ξ*<sup>1</sup>

*surface tensor r*∈ *H*<sup>1</sup>

j j*r <sup>H</sup>*<sup>1</sup>

ment vector.

<sup>≤</sup> <sup>C</sup>*zik <sup>ξ</sup>*<sup>1</sup>

ð Þ <sup>S</sup> <sup>≤</sup>*δ<sup>k</sup>* <sup>ϵ</sup>ð Þ*<sup>r</sup>* � � �

**Theorem 1**. *Assume F*!

*Eq.* (42) *for some T*!

for any *δ*

*A d*! , *η* !, *ς* !; *δ* ! , *η* !, *ς* ! � �≥*<sup>γ</sup> <sup>γ</sup>*

**193**

! , *η* !, *ς*

equations more compact. (i) First, we prove

> *A d*! , *η* !, *ς* !; *δ* ! , *η* !, *ς* ! � �<sup>≥</sup> *<sup>γ</sup>*

, *ξ*<sup>2</sup> � �*zjl ξ*<sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.89866*

, *ξ*<sup>2</sup> � �*zjl ξ*<sup>1</sup>

ð Þ S *,*

� � � *<sup>L</sup>*2ð Þ <sup>S</sup>

where ϵ is symmetrized gradient tensor.

∈*L*<sup>2</sup>

*is* 0 ! , 0 ! , 0 � �!

*A d*! , *θ* ! , ϱ !; *δ* ! , *η* !, *ς* ! � � <sup>¼</sup> *<sup>F</sup> <sup>δ</sup>*

! � �∈V, and we have

� � � � � � 2 0 þ *χ* � � � � � � 2 0 þ *k* � � � � � � 2 0 þ *l* � � � � � � 2 0 þ *ζ* � � � �2 <sup>0</sup> <sup>þ</sup> k k *<sup>m</sup>* <sup>2</sup>

, *R* � �!

Then there exists a unique *d*

*such that no rigid-body motion is possible, i.e., the only element δ*

! , *θ* ! , ϱ

*d* ! , *θ* ! , ϱ ! � � �

From Eqs. (44) and (45), using *<sup>g</sup>αβgλμeαβeλμ* <sup>¼</sup> *<sup>g</sup>αβeαβ* � �<sup>2</sup>

≥*γ* ð Ω

≥*γ* ð Ω

ð Ω *.*

� � � 1 ≤C *F* !� � � � � � *<sup>L</sup>*2ð Þ <sup>ℬ</sup>

is explained in three steps. We shall write *f* instead of function *f*ð Þ, to make

*Proof.* We prove coercivity of *A* and continuity of *A* and *F*. Coercivity argument

*gαβgλμeαβeλμdV*

! � � in <sup>V</sup> that satisfies

! , *η* !, *ς*

, *ξ*<sup>2</sup> � �*YijYkl* ≤*gik ξ*<sup>1</sup>

þ k k*r <sup>L</sup>*2ð Þ <sup>S</sup> � �, *for* <sup>ϵ</sup>ð Þ¼ *<sup>r</sup>*

, *ξ*<sup>2</sup> , *ξ*<sup>3</sup> � �*gjl ξ*<sup>1</sup>

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human…*

**Lemma 3**. *The gradient of a vector field is on average not distant from the space of*

It is inferred from Lemma 2 that mapping of initial lesion is well-defined in curvilinear coordinate system wherein quantity *g* is volume measure. Also, Lemma 2 suggests that this function is well-defined and continuous. Because the initial lesion is defined over upper bound (C) and lower bound (c), the set of bounds is a compact set. The mid-surface of initial lesion definitely lies within the bounds. Thus, the characterization of initial lesion is well-defined. In order to comment on net displacement of the initial lesion during the progression of disease, we prove the following theorem to establish well-posedness of weak formulation for displace-

*skew-symmetric matrices, the gradient must not be a far from a particular skewsymmetric matrices. Thus, there exists a constant δk*>0 *such that for any first order*

, *ξ*<sup>2</sup> , *ξ*<sup>3</sup> � �*YijYkl*

, *<sup>ξ</sup>*<sup>2</sup> � �*YijYkl*, <sup>∀</sup>ð Þ *<sup>Y</sup>*11, *<sup>Y</sup>*12, *<sup>Y</sup>*21, *<sup>Y</sup>*<sup>22</sup> <sup>∈</sup> <sup>4</sup>*:* (45)

1

<sup>2</sup> <sup>∇</sup> *<sup>r</sup>* <sup>þ</sup> <sup>∇</sup> *<sup>r</sup>*

ð Þ ℬ ; *the essential boundary conditions enforced in* V *are*

<sup>0</sup> <sup>þ</sup> k k*<sup>n</sup>* <sup>2</sup>

� �*:* (49)

*<sup>g</sup>ikgjleijekl* <sup>þ</sup> *<sup>g</sup>ijei*3*ej*<sup>3</sup> <sup>þ</sup> ð Þ *<sup>e</sup>*<sup>33</sup> <sup>2</sup> h i*dV*

**<sup>z</sup>***ik***z***jleijekl* <sup>þ</sup> **<sup>z</sup>***ijei*3*ej*<sup>3</sup> <sup>þ</sup> ð Þ *<sup>e</sup>*<sup>33</sup> <sup>2</sup> h i*dV*

<sup>0</sup> <sup>þ</sup> k k *<sup>ϖ</sup>* <sup>2</sup>

≥ 0, we have

<sup>0</sup> <sup>þ</sup> k k*<sup>p</sup>* <sup>2</sup> 0

! , *η* !, *ς*

! � � (47)

! � � *in* <sup>V</sup> *satisfies*

(48)

(50)

� �*<sup>T</sup>* � �, (46)

$$A\left(\overrightarrow{d},\overrightarrow{\theta},\overrightarrow{\varphi},\overrightarrow{\delta},\overrightarrow{\eta},\overrightarrow{\varphi}\right) = \int\_{\Omega} H^{a\\$\mu} e\_{a\theta} \left(\overrightarrow{d} + \xi^{3}\overrightarrow{\theta} + \left(\xi^{3}\right)^{2}\overrightarrow{\mathsf{Q}}\right) e\_{\lambda\mu} \left(\overrightarrow{\delta} + \xi^{3}\overrightarrow{\eta} + \left(\xi^{3}\right)^{2}\overrightarrow{\xi}\right) dV.\tag{38}$$

The linear function is given by

$$F(\stackrel{\rightarrow}{\delta}, \stackrel{\rightarrow}{\eta}, \stackrel{\rightarrow}{\zeta}) = \int\_{\Omega} \stackrel{\rightarrow}{F} \left( \stackrel{\rightarrow}{\delta} + \xi^{3} \stackrel{\rightarrow}{\eta} + \left( \xi^{3} \right)^{2} \stackrel{\rightarrow}{\zeta} \right) dV \tag{39}$$

The specification of the displacement space is given by

$$\mathcal{V} = \left\{ \left( \overrightarrow{\delta}, \overrightarrow{\eta}, \overrightarrow{\xi} \right) \in H^1(\mathcal{S}) \times H^1(\mathcal{S}) \times H^1(\mathcal{S}) \right\} \cap \mathcal{R} \mathcal{C}, \tag{40}$$

where *<sup>H</sup>*<sup>1</sup> is the Sobolev space of order 1, <sup>ℬ</sup><sup>C</sup> is space for boundary conditions. **Lemma 1**. *Let us consider δ* ! , *η*∈ *H*<sup>1</sup> ð Þ S *and*

$$\left(\underline{\chi}\left(\stackrel{\cdot}{\delta}\right),\underline{\chi}\left(\stackrel{\cdot}{\delta},\underline{\eta}\right),\underline{\zeta}\left(\stackrel{\cdot}{\delta},\underline{\eta}\right)=\left(\underline{\underline{0}},\underline{\underline{0}},\underline{\mathbb{0}}\right)\quad on\quad\mathcal{S}.\tag{41}$$

Then, the displacement (36) in ℬ (higher-dimensional initial lesion body) corresponds to an infinitesimal rigid-body motion, i.e., there exists *T* ! and *R* ! a global translation vector and an infinitesimal rotation vector, respectively, such that

$$
\overrightarrow{\delta}\left(\xi^1,\xi^2\right) = \overrightarrow{T} + \overrightarrow{R} \wedge \overrightarrow{\rho}\left(\xi^1,\xi^2\right);
\qquad \underline{\eta}\left(\xi^1,\xi^2\right) = \overrightarrow{R} \wedge \underline{z}\_3\left(\xi^1,\xi^2\right) \tag{42}
$$

**Lemma 2**. *For any ξ*<sup>1</sup> , *ξ*<sup>2</sup> , *<sup>ξ</sup>*<sup>3</sup> � �<sup>∈</sup> <sup>Ω</sup>*, there exist two constants* <sup>c</sup>, <sup>C</sup>><sup>0</sup> *such that the following inequalities hold*

$$\mathfrak{c}\sqrt{\mathfrak{z}\left(\xi^{1},\xi^{2}\right)}\leq\sqrt{\mathfrak{g}\left(\xi^{1},\xi^{2},\xi^{3}\right)}\leq\mathcal{C}\sqrt{\mathfrak{z}\left(\xi^{1},\xi^{2}\right)}\tag{43}$$

$$\mathbf{c}\mathbf{c}^{\circ j}(\xi^1, \xi^2)\mathbf{Y}\_i\mathbf{Y}\_j \le \mathbf{g}^{\circ j}(\xi^1, \xi^2, \xi^3)\mathbf{Y}\_i\mathbf{Y}\_j \le \mathcal{C}\mathbf{z}^{\circ j}(\xi^1, \xi^2)\mathbf{Y}\_i\mathbf{Y}\_j, \ \forall (\mathbf{Y}\_1, \mathbf{Y}\_2) \in \mathbb{R}^2. \tag{44}$$

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human… DOI: http://dx.doi.org/10.5772/intechopen.89866*

$$\begin{split} & \mathsf{c} \mathsf{c}^{ik} \left( \xi^{1}, \xi^{2} \right) \mathsf{z}^{jl} \left( \xi^{1}, \xi^{2} \right) Y\_{\vec{\eta}} Y\_{kl} \leq \mathsf{g}^{ik} \left( \xi^{1}, \xi^{2}, \xi^{3} \right) \mathsf{g}^{jl} \left( \xi^{1}, \xi^{2}, \xi^{3} \right) Y\_{\vec{\eta}} Y\_{kl} \\ & \leq \mathsf{C} \mathsf{z}^{ik} \left( \xi^{1}, \xi^{2} \right) \mathsf{z}^{jl} \left( \xi^{1}, \xi^{2} \right) Y\_{\vec{\eta}} Y\_{kl}, \quad \forall \left( Y\_{11}, Y\_{12}, Y\_{21}, Y\_{22} \right) \in \mathbb{R}^{4} . \end{split} \tag{45}$$

**Lemma 3**. *The gradient of a vector field is on average not distant from the space of skew-symmetric matrices, the gradient must not be a far from a particular skewsymmetric matrices. Thus, there exists a constant δk*>0 *such that for any first order surface tensor r*∈ *H*<sup>1</sup> ð Þ S *,*

$$\|\underline{r}\|\_{H^1(S)} \le \delta\_k \left( \left\|\underline{\underline{\boldsymbol{e}}}(r)\right\|\_{L^2(S)} + \|\underline{\boldsymbol{r}}\|\_{L^2(S)} \right), \text{ for } \underline{\boldsymbol{e}}(r) = \frac{1}{2} \left( \underline{\underline{\boldsymbol{v}}} \cdot \underline{\boldsymbol{r}} + \left( \underline{\underline{\boldsymbol{v}}} \cdot \underline{\underline{\boldsymbol{v}}} \right)^T \right), \tag{46}$$

where ϵ is symmetrized gradient tensor.

Such a compendium is used to characterize a particular displacement state. A higher-dimensional space, *Sobolev space* which is comprised of all such possible combinations, comes handy. It is constituted of functions with sufficiently many derivative including partial differential equations of fluid-structure interaction and equipped with the norm that measures size and regularity of these functions. The

> ! *d* ! , *θ* ! , ϱ ! � �

space, V. Since test function is an idealized version of net displacement vector in

progression the initial lesion is contributed by the blood flow. Interestingly, proving

! � �

ð Þ� <sup>S</sup> *<sup>H</sup>*<sup>1</sup>

where *<sup>H</sup>*<sup>1</sup> is the Sobolev space of order 1, <sup>ℬ</sup><sup>C</sup> is space for boundary conditions.

ð Þ S *and*

Then, the displacement (36) in ℬ (higher-dimensional initial lesion body)

translation vector and an infinitesimal rotation vector, respectively, such that

, *ξ*<sup>2</sup> � �; *η ξ*<sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *g ξ*<sup>1</sup> , *ξ*<sup>2</sup> , *ξ*<sup>3</sup> <sup>q</sup> � �<sup>≤</sup> <sup>C</sup>

n o

, which is given by

. Δ ! *δ* ! , *η* !, *ς* ! � �

> ! *d* ! , *θ* ! , ϱ ! � �

. It yields insight about the disease process wherein with

*eλμ δ* ! <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> *<sup>η</sup>*

! <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> � �<sup>2</sup>

ð Þ� <sup>S</sup> *<sup>H</sup>*<sup>1</sup>

¼ 0, 0, 0 � �

, *<sup>ξ</sup>*<sup>2</sup> � � <sup>¼</sup> *<sup>R</sup>*

, *<sup>ξ</sup>*<sup>3</sup> � �<sup>∈</sup> <sup>Ω</sup>*, there exist two constants* <sup>c</sup>, <sup>C</sup>><sup>0</sup> *such that the*

! ∧ *z*<sup>3</sup> *ξ*<sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *z ξ*<sup>1</sup> , *ξ*<sup>2</sup>

, *<sup>ξ</sup>*<sup>2</sup> � �*YiYj*, <sup>∀</sup>ð Þ *<sup>Y</sup>*1, *<sup>Y</sup>*<sup>2</sup> <sup>∈</sup> <sup>2</sup>

! � �

*ς*

ð Þ S

*on* S*:*

continuum mechanics, evaluating the interaction between test function Δ

! *d* ! , *θ* ! , ϱ ! � �

in Sobolev space. Gaining information about the characteristics of Δ

in higher-dimensional metric

! *δ* ! , *η* !, *ς* ! � �

present in bilinear

*ς*

*dV* (39)

∩ℬC, (40)

! and *R* !

<sup>q</sup> � � (43)

, *ξ*<sup>2</sup> � � (42)

*dV:* (38)

(41)

a global

*:* (44)

! <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> � �<sup>2</sup>

! � �

! *δ* ! , *η* !, *ς* ! � �

, we

is a test function present

is a replica of **D**

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

yields insights about **D**

well-posedness of Eq. (35) gives insights about **D**

*Hαβλμeαβ d* ! <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> *<sup>θ</sup>* ! <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> � �<sup>2</sup> ϱ

> ¼ ð Ω *F* ! *: δ* ! <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> *<sup>η</sup>*

∈ *H*<sup>1</sup>

, *η*∈ *H*<sup>1</sup>

corresponds to an infinitesimal rigid-body motion, i.e., there exists *T*

, *<sup>ξ</sup>*<sup>3</sup> � �*YiYj* <sup>≤</sup>C*zij <sup>ξ</sup>*<sup>1</sup>

, *ζ δ*! , *η* � �

The specification of the displacement space is given by

!

, *χ δ*! , *η* � �

test function Δ

and force *F*

!

can infer about **D**

function *A d*!

*A d*! , *θ* ! , ϱ !; *δ* ! , *η* !, *ς*

! *δ* ! , *η* !, *ς* ! � �

> ! *d* ! , *θ* ! , ϱ ! � �

, *θ* ! , ϱ !; *δ* ! , *η* !, *ς*

! � �

! � �

¼ ð Ω

> *F δ* ! , *η* !, *ς* ! � �

V ¼ *δ* ! , *η* !, *ς* ! � �

> *γ δ* � �!

> > ! þ *R* ! ∧*φ* ! *ξ*<sup>1</sup>

c

, *ξ*<sup>2</sup> � �*YiYj* ≤*gij ξ*<sup>1</sup>

, *ξ*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *z ξ*<sup>1</sup> , *ξ*<sup>2</sup> <sup>q</sup> � �<sup>≤</sup>

, *ξ*<sup>2</sup>

�

**Lemma 1**. *Let us consider δ*

*δ* ! *ξ*1 , *<sup>ξ</sup>*<sup>2</sup> � � <sup>¼</sup> *<sup>T</sup>*

**Lemma 2**. *For any ξ*<sup>1</sup>

*following inequalities hold*

c*zij ξ*<sup>1</sup>

**192**

The linear function is given by

It is inferred from Lemma 2 that mapping of initial lesion is well-defined in curvilinear coordinate system wherein quantity *g* is volume measure. Also, Lemma 2 suggests that this function is well-defined and continuous. Because the initial lesion is defined over upper bound (C) and lower bound (c), the set of bounds is a compact set. The mid-surface of initial lesion definitely lies within the bounds. Thus, the characterization of initial lesion is well-defined. In order to comment on net displacement of the initial lesion during the progression of disease, we prove the following theorem to establish well-posedness of weak formulation for displacement vector.

**Theorem 1**. *Assume F*! ∈*L*<sup>2</sup> ð Þ ℬ ; *the essential boundary conditions enforced in* V *are such that no rigid-body motion is possible, i.e., the only element δ* ! , *η* !, *ς* ! � � *in* <sup>V</sup> *satisfies*

*Eq.* (42) *for some T*! , *R* � �! *is* 0 ! , 0 ! , 0 � �! *.*

Then there exists a unique *d* ! , *θ* ! , ϱ ! � � in <sup>V</sup> that satisfies

$$A\left(\overrightarrow{d},\overrightarrow{\theta},\overrightarrow{\q};\,\overrightarrow{\delta},\,\overrightarrow{\eta},\,\overrightarrow{\zeta}\right) = F\left(\overrightarrow{\delta},\,\overrightarrow{\eta},\,\overrightarrow{\zeta}\right) \tag{47}$$

for any *δ* ! , *η* !, *ς* ! � �∈V, and we have

$$\left\| \left\| \overrightarrow{d}, \overrightarrow{\theta}, \overrightarrow{\mathbf{q}} \right\| \right\|\_{\mathbf{1}} \leq \mathcal{C} \left\| \overrightarrow{F} \right\|\_{L^{2}(\partial \mathbb{B})} \tag{48}$$

*Proof.* We prove coercivity of *A* and continuity of *A* and *F*. Coercivity argument is explained in three steps. We shall write *f* instead of function *f*ð Þ, to make equations more compact.

(i) First, we prove

$$A\left(\overrightarrow{d},\overrightarrow{\eta},\overrightarrow{\zeta};\overrightarrow{\delta},\overrightarrow{\eta},\overrightarrow{\zeta}\right) \ge \mathbf{y}\left(\left\|\underline{\underline{\boldsymbol{\zeta}}}\right\|\_{0}^{2} + \left\|\underline{\underline{\boldsymbol{\zeta}}}\right\|\_{0}^{2} + \left\|\underline{\underline{\boldsymbol{\zeta}}}\right\|\_{0}^{2} + \left\|\underline{\underline{\boldsymbol{\zeta}}}\right\|\_{0}^{2} + \left\|\underline{\underline{\boldsymbol{\zeta}}}\right\|\_{0}^{2} + \left\|\underline{\underline{m}}\right\|\_{0}^{2} + \left\|\underline{\underline{m}}\right\|\_{0}^{2} + \left\|\boldsymbol{w}\right\|\_{0}^{2} + \left\|p\right\|\_{0}^{2}\right). \tag{49}$$

From Eqs. (44) and (45), using *<sup>g</sup>αβgλμeαβeλμ* <sup>¼</sup> *<sup>g</sup>αβeαβ* � �<sup>2</sup> ≥ 0, we have

$$\begin{split} A\left(\overrightarrow{\boldsymbol{d}}, \overrightarrow{\boldsymbol{\eta}}, \overrightarrow{\boldsymbol{\varsigma}}; \overrightarrow{\boldsymbol{\delta}}, \overrightarrow{\boldsymbol{\eta}}, \overrightarrow{\boldsymbol{\varsigma}}\right) &\geq \chi \int\_{\Omega} \boldsymbol{g}^{a\boldsymbol{\theta}} \boldsymbol{g}^{\boldsymbol{\lambda}\boldsymbol{\mu}} \boldsymbol{e}\_{a\boldsymbol{\theta}} \boldsymbol{e}\_{\boldsymbol{\lambda}\boldsymbol{\mu}} \boldsymbol{d} \boldsymbol{V} \\ &\geq \chi \int\_{\Omega} \left[ \boldsymbol{g}^{\boldsymbol{k}\boldsymbol{g}} \boldsymbol{g}^{\boldsymbol{\ell}} \boldsymbol{e}\_{\boldsymbol{j}\boldsymbol{\ell}} \boldsymbol{e}\_{\boldsymbol{k}\boldsymbol{l}} + \boldsymbol{g}^{\boldsymbol{j}\boldsymbol{\ell}} \boldsymbol{e}\_{i3} \boldsymbol{e}\_{\boldsymbol{j}\boldsymbol{3}} + \left( \boldsymbol{e}\_{33} \right)^{2} \right] \boldsymbol{d} \boldsymbol{V} \\ &\geq \chi \int\_{\Omega} \left[ \boldsymbol{\varpi}^{\boldsymbol{k}\boldsymbol{k}} \boldsymbol{\varpi}^{\boldsymbol{\ell}} \boldsymbol{e}\_{\boldsymbol{j}\boldsymbol{\ell}} \boldsymbol{e}\_{\boldsymbol{k}\boldsymbol{l}} + \boldsymbol{\varpi}^{\boldsymbol{j}} \boldsymbol{e}\_{i3} \boldsymbol{e}\_{\boldsymbol{j}\boldsymbol{3}} + \left( \boldsymbol{e}\_{33} \right)^{2} \right] \boldsymbol{d} \boldsymbol{V} \end{split} \tag{50}$$

Now using Eqs. (43) and (33) and integrating through the thickness, we obtain

$$\begin{split} A\left(\overrightarrow{d},\,\overrightarrow{\eta},\,\overrightarrow{\varsigma};\,\overrightarrow{\delta},\,\overrightarrow{\eta},\,\overrightarrow{\epsilon}\right) &\geq \chi \int\_{\rm{ov}} t \,\textbf{z}^{\rm{i}k} \mathbf{z}^{\rm{i}l} [\chi\_{\overrightarrow{\eta}}\chi\_{\textit{kl}} + \frac{t^{2}}{12}\chi\_{\overrightarrow{\eta}}\chi\_{\textit{kl}} + \frac{t^{2}}{6}\chi\_{\textit{j}}k\_{\textit{kl}} \\ &\quad + \frac{t^{4}}{80}k\_{\textit{ij}}k\_{\textit{kl}} + \frac{t^{4}}{40}l\_{\textit{ij}}\chi\_{\textit{kl}} + \frac{t^{6}}{448}l\_{\textit{ij}}l\_{\textit{kl}}] + \left[\varpi^{2} + \frac{t^{2}}{12}p^{2}\right] \\ &\quad + \mathbf{z}^{\rm{j}l} \left[\zeta\_{\textit{i}}\zeta\_{\textit{j}} + \frac{t^{2}}{12}m\_{\textit{i}}m\_{\textit{j}} + \frac{t^{2}}{6}\zeta\_{\textit{i}}n\_{\textit{j}} + \frac{t^{4}}{80}n\_{\textit{i}}n\_{\textit{j}}\right])d\mathcal{S} \end{split} \tag{51}$$

To simplify the above expression, we use the following inequality:

$$|ab| \le \frac{1}{2} \left( \eta a^2 + \frac{1}{\eta} b^2 \right), \quad \forall \eta > 0. \tag{52}$$

*<sup>η</sup>*3, *<sup>ς</sup>* !

*<sup>η</sup>*3, *<sup>ς</sup>* !

!Þ

!Þ 

In addition, from the definition of *n* and *m* in Eq. (34), we have

<sup>1</sup> ≤C *n ς* ! 

<sup>1</sup> ≤C *m η*

<sup>1</sup> ≤ Cð *m η*

þ *n ς* ! 

<sup>≤</sup><sup>C</sup> *<sup>η</sup>*3, *<sup>ς</sup>* ! 2

!, *ς* !

> 2 0 þ k k *ς*<sup>3</sup> 2 <sup>0</sup> þ k k *η*<sup>3</sup>

 2 0

 2 0 þ k k *ς*<sup>3</sup> 2 <sup>0</sup> þ k k *η*<sup>3</sup>

 2 0 þ *ς* 2 0

≤C *k*ð 0, *η*<sup>3</sup> ð Þ, *ς*

≤ C *k* 0, *η*<sup>3</sup> ð Þ, *ς*

*<sup>ς</sup>*<sup>3</sup> j j<sup>2</sup>

*<sup>η</sup>*<sup>3</sup> j j<sup>2</sup>

From Eqs. (60)–(62), we obtain

*<sup>η</sup>*3, *<sup>ς</sup>* 2

the same limit. Thus, the subsequence in *L*<sup>2</sup>

**195**

Using Lemma 3 and Eq. (34), we have

*DOI: http://dx.doi.org/10.5772/intechopen.89866*

<sup>1</sup> ≤ C *ε ς* 

> 

 

*ς* 2

and we get

<sup>∗</sup> <sup>≤</sup> <sup>C</sup> *<sup>η</sup>*3, *<sup>ς</sup>* ! 

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human…*

<sup>∗</sup> <sup>≤</sup>*γ η*3, *<sup>ς</sup>* ! 

> þ *bς*<sup>3</sup>

 2 0 þ *ς* 2 0

 2 0 þ *ς* 2 0 (62)

þ *k*ð0, *η*Þ, *ς*

2 <sup>0</sup> þ *ς* 2 0Þ

 

!, *ς* !

> 2 0

<sup>∗</sup> <sup>þ</sup> *<sup>η</sup>*3, *<sup>ς</sup>* ! 2 0

*:*

Perusing the norm of the gradient of vector fields, the setting of lower and upper bounds is tantamount to estimating attributes of lesion at the initial and advanced stages, respectively. The sequence of all lower bounds corresponds to the initial stage of disease prevalent in affected population. Similarly, the sequence of all upper bounds corresponds to the advanced stage of disease prevalent in terminally ill patients. Note that each of these sequences is uniformly bounded in the *H*<sup>1</sup>

There exist a subsequence that converges to some limit for each of these sequences. The weak convergence in *H*<sup>1</sup> implies strong convergence in *L*<sup>2</sup> for the same norm to

stronger result about the sequences of upper and lower bounds. Clinically, it indicates various patients might report, at different stages of disease owing to different reasons, their disease initiation be an element, which is a limit point of the subsequence of lower bound sequence. Note that primal lesion presence in any patient whatsoever can be traced back by the convergence of subsequence of lower bound sequence. Its corollary equivalently applies to the advanced stage of the disease.

(iii) Coercivity bound: Coercivity is the measure of the ability of the initial lesion to withstand an external fluid-structure interaction without undergoing deformation. It is obviously dependent on the intensity of hemodynamic forces applied to the lesion. Thus, coercivity bound is the limit point of the ability of initial lesion to withstand deformation. Note that in due course of the progression of the disease, the evolution of the lesion at each stage is dependent on the increment of coercivity bound. In normal circumstances, it

 2 0 þ *cη*<sup>3</sup> 

 2 0 þ *ς* 2 0

(61)

!Þ

 2 0


2 <sup>0</sup> þ *ς* 2 0

<sup>1</sup> (58)

<sup>1</sup> (59)

(60)

(63)


Now we have

$$\begin{split} |\frac{t^2}{6} \mathbf{z}^{ik} \mathbf{z}^{jl} \boldsymbol{\gamma}\_{ij} k\_{kl}| &= \frac{1}{6} |\langle \boldsymbol{\gamma}, t^2 k \rangle| \\ &\leq \frac{1}{12} \left( a\_1 ||\boldsymbol{\gamma}||\_\varepsilon^2 + \frac{t^4}{a\_1} ||k||\_\varepsilon^2 \right) \\ &\leq \frac{1}{12} \mathbf{z}^{ik} \mathbf{z}^{jl} \left( a\_1 \boldsymbol{\gamma}\_{ij} k\_{kl} + \frac{t^4}{a\_1} k\_{ij} k\_{kl} \right), \end{split} \tag{53}$$

and similarly

$$|\frac{t^4}{4\mathcal{O}}\mathbf{z}^{ik}\mathbf{z}^{jl}l\_{ij}\chi\_{kl}| \le \frac{1}{8\mathcal{O}}\mathbf{z}^{ik}\mathbf{z}^{jl}\left(a\_2t^6l\_{ij}l\_{kl} + \frac{t^2}{a\_2}\chi\_{ij}\chi\_{kl}\right). \tag{54}$$

$$|\frac{t^2}{6}\mathbf{z}^{\circ j}\zeta\_i n\_j| \le \frac{1}{12}\mathbf{z}^{\circ j}\left(a\_3\zeta\_i\zeta\_j + \frac{t^4}{a\_3}n\_in\_j\right),\tag{55}$$

where *a*1, *a*2, *a*3>0. Using suitable values of the constants, *a*<sup>1</sup> ¼ *a*<sup>3</sup> ¼ 10, *a*<sup>2</sup> ¼ 6*=*35, and *t*>0, Eq. (51) becomes

$$\begin{split} A\left(\overrightarrow{\boldsymbol{d}}, \overrightarrow{\boldsymbol{\eta}}, \overrightarrow{\boldsymbol{\varsigma}}; \overrightarrow{\boldsymbol{\delta}}, \overrightarrow{\boldsymbol{\eta}}, \overrightarrow{\boldsymbol{\varsigma}}\right) &\geq \ \boldsymbol{\chi} \int\_{\boldsymbol{\alpha}} \{\mathbf{z}^{\text{ik}} \mathbf{z}^{\text{il}} \Big[\boldsymbol{\gamma}\_{\overrightarrow{\boldsymbol{\eta}}} \boldsymbol{\gamma}\_{\textit{kl}} + \boldsymbol{\chi}\_{\overrightarrow{\boldsymbol{\eta}}} \boldsymbol{\chi}\_{\textit{kl}} + \boldsymbol{k}\_{\overrightarrow{\boldsymbol{\eta}}} \boldsymbol{k}\_{\textit{kl}} + \boldsymbol{l}\_{\overrightarrow{\boldsymbol{\eta}}} \boldsymbol{l}\_{\textit{kl}} \Big] \\ &+ \mathbf{z}^{\overrightarrow{\boldsymbol{\eta}}} \Big[\boldsymbol{\zeta}\_{i} \boldsymbol{\zeta}\_{j} + \boldsymbol{m}\_{i} \boldsymbol{m}\_{j} + \boldsymbol{n}\_{i} \boldsymbol{n}\_{j}\Big] + \left[\boldsymbol{\varpi}^{2} + \boldsymbol{p}^{2}\right] \big) d\mathbf{S} \end{split} \tag{56}$$

Hence, Eq. (49) is proved. The bilinear function is bounded below by the sum of norm of strain tensors. This function for mid-surface is integrated through the thickness of the entire lesion giving semblance of the whole lesion.

(ii) Denoting

$$\left\|\left(\eta\_{3},\overrightarrow{\boldsymbol{\varsigma}}\right)\right\|\_{\ast} = \left(\left\|\underline{\boldsymbol{m}}\left(\overrightarrow{\boldsymbol{\eta}},\overrightarrow{\boldsymbol{\varsigma}}\right)\right\|\_{0}^{2} + \left\|\underline{\boldsymbol{n}}\left(\overrightarrow{\boldsymbol{\varsigma}}\right)\right\|\_{0}^{2} + \left\|\underline{\boldsymbol{k}}\left(\left(\boldsymbol{\Omega},\eta\_{3}\right),\overrightarrow{\boldsymbol{\varsigma}}\right)\right\|\_{0}^{2} + \left\|\boldsymbol{\varpi}\left(\overrightarrow{\boldsymbol{\eta}}\right)\right\|\_{0}^{2} + \left\|\boldsymbol{p}\left(\overrightarrow{\boldsymbol{\varsigma}}\right)\right\|\_{0}^{2}\right)^{1/2}\tag{57}$$

We now show that k k� <sup>∗</sup> provides a norm equivalent to the *<sup>H</sup>*<sup>1</sup> -norm over certain subspace of the Sobolev space. Note that *ϖ η*!� � <sup>¼</sup> *<sup>p</sup> <sup>ς</sup>* !� � <sup>¼</sup> 0 gives *<sup>η</sup>*<sup>3</sup> <sup>¼</sup> *<sup>ς</sup>*<sup>3</sup> <sup>¼</sup> 0 and *m η* !, *ς* ! � � <sup>¼</sup> *<sup>η</sup>*<sup>3</sup> <sup>¼</sup> 0 gives *<sup>ς</sup>* <sup>¼</sup> 0. Bounding the norm from above, we get

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human… DOI: http://dx.doi.org/10.5772/intechopen.89866*

$$||\eta\_3, \overrightarrow{\boldsymbol{\varsigma}}||\_\* \le \mathcal{C} ||\eta\_3, \overrightarrow{\boldsymbol{\varsigma}}||\_1 \tag{58}$$

and we get

Now using Eqs. (43) and (33) and integrating through the thickness, we obtain

*lijχkl* þ

1 *η b*2 � �

> *a*1k k*γ* 2 *<sup>ε</sup>* þ *t* 4 *a*1 k k*<sup>k</sup>* <sup>2</sup> *ε*

**z***ik***z***jl a*2*t*

**<sup>z</sup>***ij <sup>a</sup>*3*ζiζ<sup>j</sup>* <sup>þ</sup>

where *a*1, *a*2, *a*3>0. Using suitable values of the constants, *a*<sup>1</sup> ¼ *a*<sup>3</sup> ¼ 10, *a*<sup>2</sup> ¼

<sup>þ</sup> **<sup>z</sup>***ij <sup>ζ</sup>iζ<sup>j</sup>* <sup>þ</sup> *mimj* <sup>þ</sup> *ninj* h i

norm of strain tensors. This function for mid-surface is integrated through the

� � � �

¼ *η*<sup>3</sup> ¼ 0 gives *ς* ¼ 0. Bounding the norm from above, we get

thickness of the entire lesion giving semblance of the whole lesion.

� � � 2 0

We now show that k k� <sup>∗</sup> provides a norm equivalent to the *<sup>H</sup>*<sup>1</sup>

þ *n ς* ! � � � � �

Hence, Eq. (49) is proved. The bilinear function is bounded below by the sum of

þ *k* 0, *η*3Þ, *ς*

� �

*t* 4 *a*1 *kijkkl*

> *t* 2 *a*2 *χijχkl*

� �

� �

<sup>6</sup>*lijlkl* <sup>þ</sup>

*t* 4 *a*3 *ninj*

<sup>f</sup>**z***ik***z***jl <sup>γ</sup>ijγkl* <sup>þ</sup> *<sup>χ</sup>ijχkl* <sup>þ</sup> *kijkkl* <sup>þ</sup> *lijlkl* h i

> !Þ � �

> > ¼ *p ς* !� �

� 2 0

� �

**<sup>z</sup>***ik***z***jl <sup>a</sup>*1*γijkkl* <sup>þ</sup>

*t* 2 <sup>6</sup> *<sup>ζ</sup>inj* <sup>þ</sup>

� �

*mimj* þ

*t* 2 <sup>12</sup> *<sup>χ</sup>ijχkl* <sup>þ</sup>

> *t* 6 448

*t* 2 <sup>6</sup> *<sup>γ</sup>ijkkl*

*t* 4 80 *ninj*

*lijlkl*� þ *<sup>ϖ</sup>*<sup>2</sup> <sup>þ</sup>

*t* 2 <sup>12</sup> *<sup>p</sup>*<sup>2</sup> � �

(51)

(53)

(56)

*:* (54)

, (55)

g*d*S

, ∀*η*>0*:* (52)

,

<sup>þ</sup> *<sup>ϖ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>p</sup>*<sup>2</sup> � �g*dS*

<sup>þ</sup> *ϖ η*! � � � � �

� �1*=*<sup>2</sup> �

� � � 2 0 þ *p ς* ! � � � � �

¼ 0 gives *η*<sup>3</sup> ¼ *ς*<sup>3</sup> ¼ 0 and


� � � 2 0

(57)

*<sup>t</sup>*f**z***ik***z***jl*½*γijγkl* <sup>þ</sup>

*t* 2 12

To simplify the above expression, we use the following inequality:

<sup>2</sup> *<sup>η</sup>a*<sup>2</sup> <sup>þ</sup>

<sup>6</sup> <sup>∣</sup> *<sup>γ</sup>*, *<sup>t</sup>* 2 *k* � �∣

≤ 1 12

≤ 1 12

80

12

*t* 4 40

*kijkkl* þ

<sup>þ</sup>**z***ij <sup>ζ</sup>iζ<sup>j</sup>* <sup>þ</sup>

<sup>∣</sup>*ab*∣ ≤ <sup>1</sup>

<sup>6</sup> **<sup>z</sup>***ik***z***jlγijkkl*<sup>∣</sup> <sup>¼</sup> <sup>1</sup>

**<sup>z</sup>***ik***z***jllijχkl*∣ ≤ <sup>1</sup>

<sup>6</sup> **<sup>z</sup>***ijζinj*∣ ≤ <sup>1</sup>

≥ *γ* ð *ω*

*A d*! , *η* !, *ς* !; *δ* ! , *η* !, *ς*

Now we have

and similarly

*A d*! , *η* !, *ς* !; *δ* ! , *η* !, *ς*

(ii) Denoting

�<sup>∗</sup> ¼ *m η*

�

!, *ς* ! � � � �

� � � 2 0

subspace of the Sobolev space. Note that *ϖ η*!� �

*<sup>η</sup>*3, *<sup>ς</sup>* � � !�

*m η* !, *ς* ! � �

**194**

∣ *t* 2

∣ *t* 4 40

6*=*35, and *t*>0, Eq. (51) becomes

! � �

∣ *t* 2

! � �

≥*γ* ð *ω*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

þ *t* 4 80

$$\left\|\left|\eta\_{3},\overrightarrow{\boldsymbol{\varsigma}}\right\|\right\|\_{\*} \leq \mathcal{Y} \left\|\eta\_{3},\overrightarrow{\boldsymbol{\varsigma}}\right\|\_{1} \tag{59}$$

Using Lemma 3 and Eq. (34), we have

$$\begin{split} \left| \underline{\mathcal{L}} \right|\_{1}^{2} &\leq \mathcal{C} \left( \left\| \underline{\underline{\mathcal{L}}} (\underline{\underline{\mathcal{L}}}) \right\|\_{0}^{2} + \left\| \underline{\underline{\mathcal{L}}} \right\|\_{0}^{2} \right) \\ &\leq \mathcal{C} \left( \left\| \underline{\underline{\mathcal{L}}} (\underline{\mathbf{0}}, \eta\_{3}), \overline{\underline{\mathcal{L}}} \right\|\_{0}^{2} + \left\| \underline{\underline{\mathcal{L}}} \underline{\underline{\mathcal{L}}} \right\|\_{0}^{2} + \left\| \underline{\underline{\mathcal{L}}} \eta\_{3} \right\|\_{0}^{2} + \left\| \underline{\underline{\mathcal{L}}} \right\|\_{0}^{2} \right) \\ &\leq \mathcal{C} \left( \left\| \underline{\underline{\mathcal{L}}} (\underline{\mathbf{0}}, \eta\_{3}), \overline{\underline{\mathcal{L}}} \right\|\_{0}^{2} + \left\| \underline{\mathcal{L}}\_{3} \right\|\_{0}^{2} + \left\| \eta\_{3} \right\|\_{0}^{2} + \left\| \underline{\underline{\mathcal{L}}} \right\|\_{0}^{2} \right) \end{split} \tag{60} \right) $$

In addition, from the definition of *n* and *m* in Eq. (34), we have

$$\left\|\zeta\_{3}\right\|\_{1}^{2} \leq \mathcal{C}\left(\left\|\underline{\mathfrak{u}}\left(\overrightarrow{\boldsymbol{\varsigma}}\right)\right\|\_{0}^{2} + \left\|\underline{\mathfrak{s}}\right\|\_{0}^{2}\right) \tag{61}$$

$$\left\|\eta\_{3}\right\|\_{1}^{2} \leq \mathcal{C}\left(\left\|\underline{m}\left(\overrightarrow{\eta},\overrightarrow{\zeta}\right)\right\|\_{0}^{2} + \left\|\underline{\xi}\right\|\_{0}^{2}\right) \tag{62}$$

From Eqs. (60)–(62), we obtain

$$\begin{split} \left\lVert \left\lVert \eta\_{3}, \underline{\boldsymbol{\varepsilon}} \right\rVert\_{1}^{2} &\leq \mathcal{C} \left( \left\lVert \underline{\boldsymbol{m}} \left( \overrightarrow{\boldsymbol{\eta}}, \overrightarrow{\boldsymbol{\varepsilon}} \right) \right\rVert\_{0}^{2} + \left\lVert \underline{\boldsymbol{k}} (\underline{\boldsymbol{\Omega}}, \boldsymbol{\eta}), \overrightarrow{\boldsymbol{\varepsilon}} \right\rVert \right\rVert\_{0}^{2} \\ &+ \left\lVert \underline{\boldsymbol{n}} \left( \overrightarrow{\boldsymbol{\varepsilon}} \right) \right\rVert\_{0}^{2} + \left\lVert \boldsymbol{\varepsilon}\_{3} \right\rVert\_{0}^{2} + \left\lVert \eta\_{3} \right\rVert\_{0}^{2} + \left\lVert \underline{\boldsymbol{\varepsilon}} \right\rVert\_{0}^{2} \right) \\ &\leq \mathcal{C} \left( \left\lVert \eta\_{3}, \overrightarrow{\boldsymbol{\varepsilon}} \right\rVert\_{\*}^{2} + \left\lVert \eta\_{3}, \overrightarrow{\boldsymbol{\varepsilon}} \right\rVert\_{0}^{2} \right). \end{split} \tag{63}$$

Perusing the norm of the gradient of vector fields, the setting of lower and upper bounds is tantamount to estimating attributes of lesion at the initial and advanced stages, respectively. The sequence of all lower bounds corresponds to the initial stage of disease prevalent in affected population. Similarly, the sequence of all upper bounds corresponds to the advanced stage of disease prevalent in terminally ill patients. Note that each of these sequences is uniformly bounded in the *H*<sup>1</sup> -norm. There exist a subsequence that converges to some limit for each of these sequences. The weak convergence in *H*<sup>1</sup> implies strong convergence in *L*<sup>2</sup> for the same norm to the same limit. Thus, the subsequence in *L*<sup>2</sup> -norm converges strongly. This gives a stronger result about the sequences of upper and lower bounds. Clinically, it indicates various patients might report, at different stages of disease owing to different reasons, their disease initiation be an element, which is a limit point of the subsequence of lower bound sequence. Note that primal lesion presence in any patient whatsoever can be traced back by the convergence of subsequence of lower bound sequence. Its corollary equivalently applies to the advanced stage of the disease.

(iii) Coercivity bound: Coercivity is the measure of the ability of the initial lesion to withstand an external fluid-structure interaction without undergoing deformation. It is obviously dependent on the intensity of hemodynamic forces applied to the lesion. Thus, coercivity bound is the limit point of the ability of initial lesion to withstand deformation. Note that in due course of the progression of the disease, the evolution of the lesion at each stage is dependent on the increment of coercivity bound. In normal circumstances, it

$$\left\|\vec{v}\_1 + a\vec{v}\_2\right\|^2 + \left\|\vec{v}\_2\right\|^2 \ge \gamma \left(\left\|\vec{v}\_1\right\|^2 + \left\|\vec{v}\_2\right\|^2\right),\tag{64}$$

$$\begin{split} \left\| \underline{\underline{\chi}} \left( \overrightarrow{\delta}, \overrightarrow{\eta} \right) \right\|\_{0}^{2} + \left\| \underline{\boldsymbol{w}} \left( \overrightarrow{\eta} \right) \right\|\_{0}^{2} &= \left\| \underline{\underline{\chi}} \left( \overrightarrow{\delta}, (\boldsymbol{\eta}, \mathbf{0}) - \underline{\underline{b}} \eta\_{3} \right\|\_{0}^{2} + \left\| \eta\_{3} \right\|\_{0}^{2} \\ &\geq \underline{\underline{\chi}} \left( \underline{\underline{\chi}} \left( \overrightarrow{\delta}, (\boldsymbol{\eta}, \mathbf{0}) \right) \right) \right\|\_{0}^{2} + \left\| \eta\_{3} \right\|\_{0}^{2}, \end{split} \tag{65}$$

$$\begin{split} \left\lVert \underline{\chi} \left( \overrightarrow{\delta} \right) \right\rVert\_{0}^{2} + \left\lVert \underline{\chi} (\overrightarrow{\delta}, \overrightarrow{\eta}) \right\rVert\_{0}^{2} + \left\lVert \underline{\zeta} (\overrightarrow{\delta}, \overrightarrow{\eta}) \right\rVert\_{0}^{2} + \left\lVert \underline{\sigma} \left( \overrightarrow{\eta} \right) \right\rVert\_{0}^{2} &\geq \gamma \left( \left\lVert \underline{\zeta} \left( \overrightarrow{\delta} \right) \right\rVert\_{0}^{2} + \left\lVert \underline{\zeta} (\overrightarrow{\delta}, \overrightarrow{\eta}) \right\rVert\_{0}^{2} \\ &+ \left\lVert \underline{\chi} (\overrightarrow{\delta}, (\eta, \mathbf{0})) \right\rVert\_{0}^{2} + \left\lVert \eta\_{3} \right\rVert\_{0}^{2} \geq \gamma \left( \left\lVert \overrightarrow{\delta}, \eta \right\rVert\_{1}^{2} + \left\lVert \eta\_{3} \right\rVert\_{0}^{2} \right) \end{split} \tag{66}$$

$$\left\|\underline{\underline{k}}\left(\overrightarrow{\eta},\overrightarrow{\varepsilon}\right)\right\|\_{0}^{2} + \left|\underline{\eta}\right|\_{1}^{2} \geq \gamma \left(\left\|\underline{\underline{k}}\left(\left(\underline{\underline{0}},\eta\_{3}\right),\overrightarrow{\varepsilon}\right)\right\|\_{0}^{2} + \underline{\eta}\right\|\_{1}^{2}\right) \tag{67}$$

$$\begin{split} \left\lVert \underline{\mathbb{k}} \left( \overline{\boldsymbol{\eta}}, \overline{\boldsymbol{\varsigma}} \right) \right\rVert\_{0}^{2} &+ \left\lVert \underline{\boldsymbol{\eta}} \right\rVert\_{1}^{2} + \left\lVert \underline{\boldsymbol{m}} (\overline{\boldsymbol{\eta}}, \overline{\boldsymbol{\varsigma}}) \right\rVert\_{0}^{2} + \left\lVert \underline{\boldsymbol{m}} \left( \overline{\boldsymbol{\eta}} \right) \right\rVert\_{0}^{2} + \left\lVert \boldsymbol{p} \left( \overline{\boldsymbol{\varsigma}} \right) \right\rVert\_{0}^{2} \\ &\geq \gamma \left( \left\lVert \underline{\boldsymbol{\varrho}} (\underline{\boldsymbol{\varrho}}, \boldsymbol{\eta}\_{3}), \overline{\boldsymbol{\varsigma}} \right\rVert\_{0}^{2} + \left\lVert \underline{\boldsymbol{m}} (\overline{\boldsymbol{\eta}}, \overline{\boldsymbol{\varsigma}}) \right\rVert\_{0}^{2} + \left\lVert \underline{\boldsymbol{m}} \left( \overline{\boldsymbol{\varsigma}} \right) \right\rVert\_{0}^{2} + \left\lVert \boldsymbol{p} \left( \overline{\boldsymbol{\varsigma}} \right) \right\rVert\_{0}^{2} + \left\lVert \underline{p} \left( \overline{\boldsymbol{\varsigma}} \right) \right\rVert\_{0}^{2} + \left\lVert \underline{p} \left( \overline{\boldsymbol{\varsigma}} \right) \right\rVert\_{0}^{2} \\ &= \gamma \left( \left\lVert \boldsymbol{\eta}\_{3}, \overline{\boldsymbol{\varsigma}} \right\rVert\_{\*}^{2} + \left\lVert \underline{\boldsymbol{\varrho}} \right\rVert\_{\*}^{2} \right) \\ &\geq \gamma \left( \left\lVert \boldsymbol{\eta}\_{3}, \overline{\boldsymbol{\varsigma}} \right\rVert\_{1}^{2} + \left\lVert \underline{\boldsymbol{\varrho}} \right\rVert\_{1}^{2} \right). \end{split} \tag{68}$$

$$\begin{split} A\left(\overrightarrow{d},\overrightarrow{\eta},\overrightarrow{\varphi},\overrightarrow{\delta},\overrightarrow{\eta},\overrightarrow{\varphi}\right) \geq & \gamma \left( \|\boldsymbol{\nu}\|\_{0}^{2} + \|\boldsymbol{\chi}\|\_{0}^{2} + \|\boldsymbol{k}\|\_{0}^{2} + \|\boldsymbol{\zeta}\|\_{0}^{2} + \|\boldsymbol{\zeta}\|\_{0}^{2} + \|\boldsymbol{m}\|\_{0}^{2} + \|\boldsymbol{m}\|\_{0}^{2} + \|\boldsymbol{p}\|\_{0}^{2} + \|\boldsymbol{p}\|\_{0}^{2} \right) \\ \geq & \gamma \left( \left\|\overrightarrow{\boldsymbol{\delta}},\underline{\boldsymbol{\nu}}\right\|\_{1}^{2} + \|\boldsymbol{\eta}\_{3}\|\_{0}^{2} \left\|\underline{\boldsymbol{\varepsilon}}\right\|\_{0}^{2} + \|\boldsymbol{m}\|\_{0}^{2} + \|\boldsymbol{n}\|\_{0}^{2} + + \|\boldsymbol{p}\|\_{0}^{2} \right) \\ \geq & \gamma \left( \left\|\overrightarrow{\boldsymbol{\delta}},\underline{\boldsymbol{\eta}}\right\|\_{1}^{2} + \|\boldsymbol{\eta}\_{3},\overrightarrow{\boldsymbol{\varepsilon}}\|\_{1}^{2} \right) = \gamma \left\|\left.\overrightarrow{\boldsymbol{\delta}},\overrightarrow{\eta},\overrightarrow{\boldsymbol{\varepsilon}}\right\|\_{1}^{2}. \end{split} \tag{69}$$

$$\begin{split} \left| \int\_{\Omega} \overrightarrow{F} \cdot \left( \overrightarrow{\delta} + \xi^{3} \overrightarrow{\eta} + \left( \xi^{3} \right)^{2} \overrightarrow{\xi} \right) dV \right| &\leq \left\| \overrightarrow{F} \right\|\_{L^{2}\left(\mathcal{\mathcal{S}}\right)} \left\| \overrightarrow{\delta} + \xi^{3} \overrightarrow{\eta} + \left( \xi^{3} \right)^{2} \overrightarrow{\xi} \right\|\_{L^{2}\left(\mathcal{\mathcal{S}}\right)} \\ &\leq \left\| \overrightarrow{F} \right\|\_{L^{2}\left(\mathcal{\mathcal{S}}\right)} \left\| \overrightarrow{\delta} \right\| \left\| \overrightarrow{\delta} \right\| \left\| \overrightarrow{\eta} \right\|\_{\mathfrak{U}} . \end{split} \tag{70}$$

$$\left\|\mathcal{Y}\left\|\left.\vec{d},\vec{\theta},\vec{\varphi}\right\|\right\|\_{1}^{2}\leq A\left(\vec{d},\vec{\theta},\vec{\varphi};\vec{d},\vec{\theta},\vec{\varphi}\right) = F\left(\vec{d},\vec{\theta},\vec{\varphi}\right) \leq \mathcal{C}\left\|\left.\vec{F}\right\|\_{L^{2}(\mathcal{A})}\left\|\left.\vec{d},\vec{\theta},\vec{\varphi}\right\|\right\|\_{0}.\tag{71}$$

$$\mathcal{V}\_0 = \left\{ \left( \overrightarrow{\delta}, \overrightarrow{\eta}, \overrightarrow{\zeta} \right) \in \mathcal{V}, \mid \: \gamma\_{\overrightarrow{\eta}} \left( \overrightarrow{\delta} \right) = \mathbf{0}, \; \zeta\_i \left( \overrightarrow{\delta}, \overrightarrow{\eta} \right) = \mathbf{0} \quad \forall \; \ i, j = \mathbf{1}, \mathbf{2} \right\}. \tag{72}$$

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

$$\mathcal{V}\_0 \cap \left\{ \left( \overrightarrow{\delta}, \overrightarrow{\eta}, \overrightarrow{\varepsilon} \right) \in \mathcal{V} \right\} = \{ (0, 0, 0) \},\tag{73}$$

and situation 2, when pure bending is non-inhibited

$$\mathcal{V}\_0 \cap \left\{ \left( \overrightarrow{\delta}, \overrightarrow{\eta}, \overrightarrow{\varepsilon} \right) \in \mathcal{V} \right\} \neq \{ (0, 0, 0) \}, \tag{74}$$

Let us define higher-dimensional body force as

$$
\overrightarrow{F} = \varepsilon^{(\rho - 1)} \overrightarrow{\mathbf{G}},
\tag{75}
$$

We now discuss the cases of non-inhibited pure bending versus inhibited pure

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human…*

Assume that V displacement space for the initial lesion contains few nonzero elements. The terms of order zero in *ξ*<sup>3</sup> in the strain Eq. (33) vanishes by a penalization mechanism, and the appropriate scaling factor is then *ρ* ¼ 3. We define the

bounded in the norm k k� *<sup>b</sup>*, we extract a subsequence weakly converging in V to a

vessel wall, tunica intima, is smooth, we can expand the constitutive tensor:

! � �. Since in the early stage of the disease, the internal lining of the

, *<sup>ξ</sup>*<sup>2</sup> � � <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup>

where *Am* is the bilinear form to assess net displacement caused by the mem-

þ C*ε*<sup>2</sup> *<sup>δ</sup>* ! , *η* !, *ς* ! � � �

� �!

, *ξ*<sup>3</sup> � � is bounded over initial lesion body ℬ. Using the uniform

!<sup>1</sup>

1 2 ∇*η*<sup>3</sup> � � � �

*; ρε*! � � is uniformly

2

2

, *ξ*<sup>3</sup> � �, (80)

(79)

0

� � � �

*; θε*!

*Hαβλμ ξ*<sup>1</sup>

!*w* , *θ* !*<sup>w</sup>*; *δ* ! , *η* ! � �, (81)

! <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> � �<sup>2</sup>

! � �*dV*

� � � 0 *:*

! , *η* !, *ς*

, ∀ *δ* ! , *η* !, *ς*

*ς*

� � � �

! � �<sup>∈</sup> <sup>V</sup>*:* (83)

! � �∈V0*:* (84)

(82)

, *ξ*<sup>2</sup>

**5.1 The impact of non-inhibited pure bending on the initial lesion**

bending.

norm

limit *dw* ! *; θ<sup>w</sup>* ! *;* ϱ*<sup>w</sup>*

*δ* ! , *η* !, *ς* ! � � �

where *Hαβλμ ξ*<sup>1</sup>

boundedness of *<sup>ε</sup> <sup>d</sup><sup>ε</sup>* !

1 *ε A d*!*<sup>ε</sup>* , *θ* !*ε* , ϱ !*<sup>ε</sup>*; *δ* ! , *η* !, *ς*

� � �

When *δ* ! , *η* !, *ς*

*d* !*w* , *θ* !*<sup>w</sup>*, ϱ !*w*

**199**

� � �

*<sup>b</sup>* <sup>¼</sup> *<sup>δ</sup>* !� � � � � � 2 1 þ *η* � � � � � � 2 1 þ k k *η*<sup>3</sup> 2 <sup>0</sup> þ k k *ς*<sup>3</sup> 2 <sup>0</sup> þ *ς* þ

*DOI: http://dx.doi.org/10.5772/intechopen.89866*

*Hαβλμ ξ*<sup>1</sup>

, *ξ*<sup>2</sup>

� �

lim*ε*!0 1 *ε A d*!*<sup>ε</sup>* , *θ* !*ε* ; *ρ* !*<sup>ε</sup>*, *δ* ! , *η* !, *ς* ! � � <sup>¼</sup> *Am <sup>d</sup>*

! � � �

brane strain. This is equivalent to

*Ab d* !*w* , *θ* !*<sup>w</sup>*, ϱ !*<sup>w</sup>*; *δ* ! , *η* !, *ς* ! � � <sup>¼</sup> *<sup>G</sup> <sup>δ</sup>*

*; θε*! *;* ϱ*ε*! k1

, *ξ*<sup>2</sup>

� , we get

<sup>≤</sup> <sup>C</sup>*ε*<sup>2</sup> *<sup>δ</sup>* ! , *η* !, *ς* ! � � �

! � � is fixed in <sup>V</sup>, we get

tensor predominates whose bilinear form is given by

*Am d* !*w* , *θ* !*<sup>w</sup>*; *δ* ! , *η* ! � � <sup>¼</sup> <sup>0</sup> <sup>∀</sup> *<sup>δ</sup>*

, *<sup>ξ</sup>*<sup>3</sup> � �¼<sup>0</sup>*Hαβλμ <sup>ξ</sup>*<sup>1</sup>

� � � � ¼ 1 *ε* ð Ω *F* ! *: δ* ! <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup> *<sup>η</sup>*

� � � �

> � � � *b*

Using equivalence relations among norms and semi-norms, infer that

� �<sup>∈</sup> <sup>V</sup>. This result (83) shows that bilinear form for the membrane strain tensor vanishes. In this case, non-inhibited pure bending, bending strain

for which the convergence is anticipated. Since *<sup>d</sup><sup>ε</sup>* !

where the exponent (*ρ* � 1) is used for consistency when the external work involves an integration over the thickness which is relevant for general asymptotic analysis; *G* ! represents a force field:

$$
\overrightarrow{G}\left(\xi^1,\xi^2,\xi^3\right) = \overrightarrow{G}\_0\left(\xi^1,\xi^2\right) + \xi^3\overrightarrow{B}\left(\xi^1,\xi^2,\xi^3\right),
\tag{76}
$$

where *G* ! <sup>0</sup> is in *L*<sup>2</sup> ð Þ S and *B* ! is a uniformly bounded function over ℬ in *t*. Since it is improbable to obtain strong convergence result in context of asymptotic analysis, we make weaker assumption about *G* ! . We also forgo regularity assumption in context of weak convergence to introduce abstract bilinear forms. Depending upon boundary conditions, nonzero pure-bending displacements of initial lesion are assessed. The displacement is in response to inhibited and non-inhibited purebending lesion as we have already argued that only bending strain matters in asymptotic analysis. In the current framework of asymptotic analysis for initial lesion of a given thickness, specific membrane-dominated bilinear form is given by

$$\begin{split} A\_{m}(\overrightarrow{\boldsymbol{d}},\overrightarrow{\boldsymbol{\theta}};\overrightarrow{\boldsymbol{\delta}},\overrightarrow{\boldsymbol{\eta}}) &= \int\_{\boldsymbol{\alpha}} \boldsymbol{l} [^{0}H^{\vec{\eta}\boldsymbol{l}\boldsymbol{d}}\boldsymbol{\gamma}\_{\overrightarrow{\boldsymbol{\eta}}}(\overrightarrow{\boldsymbol{d}})\boldsymbol{\gamma}\_{\boldsymbol{\omega}\boldsymbol{l}}(\overrightarrow{\boldsymbol{\delta}}) + {^{0}H^{\vec{\eta}\boldsymbol{\beta}3}} \left( \boldsymbol{\gamma}\_{\overrightarrow{\boldsymbol{\eta}}}(\overrightarrow{\boldsymbol{d}}) \boldsymbol{\varpi} \left( \overrightarrow{\boldsymbol{\eta}} \right) + \boldsymbol{\gamma}\_{\overrightarrow{\boldsymbol{\eta}}}(\overrightarrow{\boldsymbol{\delta}}) \boldsymbol{\varpi} \left( \overrightarrow{\boldsymbol{\theta}} \right) \right) \\ &+ {^{0}H^{\vec{\eta}\boldsymbol{\beta}3}} \zeta\_{i} \left( \overrightarrow{\boldsymbol{d}},\overrightarrow{\boldsymbol{\theta}} \right) \zeta\_{j} \left( \overrightarrow{\boldsymbol{d}},\overrightarrow{\boldsymbol{\eta}} \right) + {^{0}H^{333}} \boldsymbol{\varpi} \left( \overrightarrow{\boldsymbol{\theta}} \right) \boldsymbol{\varpi} \left( \overrightarrow{\boldsymbol{\eta}} \right) \boldsymbol{l} \boldsymbol{d} \boldsymbol{\}, \end{split} \tag{77}$$

bending-dominated bilinear form is given by

*Ab d* ! , *θ* ! , ϱ !; *δ* ! , *η* !, *ς* ! � � <sup>¼</sup> ð *ω l* 3 12 0 *Hijklχij d* ! , *θ* � �! *χkl δ* ! , *η* ! � �þ0*Hij*<sup>33</sup> *<sup>χ</sup>ij <sup>d</sup>* ! , *θ* � �! *p ς* !� � <sup>þ</sup> *<sup>χ</sup>ij <sup>δ</sup>* ! , *η* ! � �*<sup>p</sup>* <sup>ϱ</sup> ! � � � � <sup>þ</sup>40*Hi*3*j*<sup>3</sup> *mi θ* ! , ϱ ! � �*mj <sup>η</sup>* !, *ς* ! � �þ0*H*3333*<sup>p</sup>* <sup>ϱ</sup> !� �*<sup>p</sup> <sup>ς</sup>* !� ��*dS*, (78)

where the tensor <sup>0</sup>*H* is defined by

$$\prescript{0}{}{H}^{a\beta\dot{\lambda}\mu} = H^{a\beta\dot{\lambda}\mu}|\_{\xi^3 = 0},$$

and linear form is given by

$$G\left(\overrightarrow{\delta}\right) = \int\_{\alpha} \overrightarrow{lG}\_0 \cdot \overrightarrow{\delta}dS.$$

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human… DOI: http://dx.doi.org/10.5772/intechopen.89866*

We now discuss the cases of non-inhibited pure bending versus inhibited pure bending.

#### **5.1 The impact of non-inhibited pure bending on the initial lesion**

Assume that V displacement space for the initial lesion contains few nonzero elements. The terms of order zero in *ξ*<sup>3</sup> in the strain Eq. (33) vanishes by a penalization mechanism, and the appropriate scaling factor is then *ρ* ¼ 3. We define the norm

$$\left\| \left\| \vec{\delta}, \vec{\eta}, \vec{\xi} \right\| \right\|\_{\dot{\mathfrak{b}}} = \left( \left\| \vec{\delta} \right\|\_{\mathbf{1}}^2 + \left\| \underline{\eta} \right\|\_{\mathbf{1}}^2 + \left\| \eta\_3 \right\|\_{\mathbf{0}}^2 + \left\| \underline{\sigma}\_3 \right\|\_{\mathbf{0}}^2 + \left\| \underline{\sigma} + \frac{\mathbf{1}}{2} \underline{\nabla} \eta\_3 \right\|\_{\mathbf{0}}^2 \right)^{\frac{1}{2}} \tag{79}$$

for which the convergence is anticipated. Since *<sup>d</sup><sup>ε</sup>* ! *; θε*! *; ρε*! � � is uniformly bounded in the norm k k� *<sup>b</sup>*, we extract a subsequence weakly converging in V to a limit *dw* ! *; θ<sup>w</sup>* ! *;* ϱ*<sup>w</sup>* ! � �. Since in the early stage of the disease, the internal lining of the vessel wall, tunica intima, is smooth, we can expand the constitutive tensor:

$$H^{a\beta\lambda\mu}\left(\xi^1,\xi^2,\xi^3\right) = {}^0H^{a\beta\lambda\mu}\left(\xi^1,\xi^2\right) + \xi^3\overline{H}^{a\beta\lambda\mu}\left(\xi^1,\xi^2,\xi^3\right),\tag{80}$$

where *Hαβλμ ξ*<sup>1</sup> , *ξ*<sup>2</sup> , *ξ*<sup>3</sup> � � is bounded over initial lesion body ℬ. Using the uniform boundedness of *<sup>ε</sup> <sup>d</sup><sup>ε</sup>* ! *; θε*! *;* ϱ*ε*! k1 � � � , we get

$$\lim\_{\varepsilon \to 0} \frac{1}{\varepsilon} A\left(\overrightarrow{d}^{\varepsilon}, \overrightarrow{\theta}^{\varepsilon}; \overrightarrow{\rho}^{\varepsilon}, \overrightarrow{\delta}, \overrightarrow{\eta}, \overrightarrow{\zeta}\right) = A\_m\left(\overrightarrow{d}^w, \overrightarrow{\theta}^w; \overrightarrow{\delta}, \overrightarrow{\eta}\right), \tag{81}$$

where *Am* is the bilinear form to assess net displacement caused by the membrane strain. This is equivalent to

$$\begin{split} \left| \frac{1}{\varepsilon} \mathcal{A} \left( \overrightarrow{d}^{\varepsilon}, \overrightarrow{\theta}^{\varepsilon}, \overrightarrow{\varphi}^{\varepsilon}; \overrightarrow{\delta}, \overrightarrow{\eta}, \overrightarrow{\xi} \right) \right| &= \left| \frac{1}{\varepsilon} \int\_{\Omega} \overrightarrow{F} \left( \overrightarrow{\delta} + \xi^{3} \overrightarrow{\eta} + \left( \xi^{3} \right)^{2} \overrightarrow{\xi} \right) dV \right| \\ &\leq \mathcal{C} \varepsilon^{2} \left\| \left| \overrightarrow{\delta}, \overrightarrow{\eta}, \overrightarrow{\xi} \right| \right\|\_{b} + \mathcal{C} \varepsilon^{2} \left\| \left| \overrightarrow{\delta}, \overrightarrow{\eta}, \overrightarrow{\xi} \right| \right\|\_{0}. \end{split} \tag{82}$$

When *δ* ! , *η* !, *ς* ! � � is fixed in <sup>V</sup>, we get

$$A\_m\left(\stackrel{\rightarrow}{d}^w, \stackrel{\rightarrow}{\theta}^w; \stackrel{\rightarrow}{\delta}, \stackrel{\rightarrow}{\eta}\right) = \mathbf{0} \qquad \forall \left(\stackrel{\rightarrow}{\delta}, \stackrel{\rightarrow}{\eta}, \stackrel{\rightarrow}{\xi}\right) \in \mathcal{V}.\tag{83}$$

Using equivalence relations among norms and semi-norms, infer that *d* !*w* , *θ* !*<sup>w</sup>*, ϱ !*w* � �<sup>∈</sup> <sup>V</sup>. This result (83) shows that bilinear form for the membrane strain tensor vanishes. In this case, non-inhibited pure bending, bending strain tensor predominates whose bilinear form is given by

$$A\_b\left(\stackrel{\dashv}{d}^w, \stackrel{\dashv}{\theta}^w, \stackrel{\dashv}{\varphi}^w; \stackrel{\dashv}{\delta}, \stackrel{\dashv}{\eta}, \stackrel{\dashv}{\zeta}\right) = G\left(\stackrel{\dashv}{\delta}\right), \qquad \forall\left(\stackrel{\dashv}{\delta}, \stackrel{\dashv}{\eta}, \stackrel{\dashv}{\zeta}\right) \in \mathcal{V}\_0. \tag{84}$$

V0∩ *δ* ! , *η* !, *ς* ! � �

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

and situation 2, when pure bending is non-inhibited

V0∩ *δ* ! , *η* !, *ς* ! � �

Let us define higher-dimensional body force as

represents a force field:

ð Þ S and *B* !

*G* ! *ξ*1 , *ξ*<sup>2</sup> , *<sup>ξ</sup>*<sup>3</sup> � � <sup>¼</sup> *<sup>G</sup>*

<sup>0</sup> is in *L*<sup>2</sup>

we make weaker assumption about *G*

¼ ð *ω l*½

¼ ð *ω l* 3 12 0

where the tensor <sup>0</sup>*H* is defined by

and linear form is given by

<sup>þ</sup>4<sup>0</sup>*H<sup>i</sup>*3*j*<sup>3</sup>

bending-dominated bilinear form is given by

<sup>þ</sup>40*Hi*3*j*<sup>3</sup>

<sup>0</sup>*Hijklγij d* � �!

> *ζ<sup>i</sup> d* ! , *θ* � �!

*Hijklχij d* ! , *θ* � �!

> *mi θ* ! , ϱ ! � �

> > *G δ* � �! ¼ ð *ω lG* ! <sup>0</sup> � *δ* ! *dS:*

analysis; *G*

where *G* !

is given by

*Am d* ! , *θ* ! ; *δ* ! , *η* ! � �

*Ab d* ! , *θ* ! , ϱ !; *δ* ! , *η* !, *ς*

**198**

! � �

!

∈V

∈V

<sup>¼</sup> *<sup>ε</sup>*ð Þ *<sup>ρ</sup>*�<sup>1</sup> *<sup>G</sup>* !

, *<sup>ξ</sup>*<sup>2</sup> � � <sup>þ</sup> *<sup>ξ</sup>*<sup>3</sup>*<sup>B</sup>*

is improbable to obtain strong convergence result in context of asymptotic analysis,

context of weak convergence to introduce abstract bilinear forms. Depending upon boundary conditions, nonzero pure-bending displacements of initial lesion are assessed. The displacement is in response to inhibited and non-inhibited purebending lesion as we have already argued that only bending strain matters in asymptotic analysis. In the current framework of asymptotic analysis for initial lesion of a given thickness, specific membrane-dominated bilinear form

! *ξ*1 , *ξ*<sup>2</sup>

<sup>þ</sup><sup>0</sup>*Hij*<sup>33</sup> *<sup>γ</sup>ij <sup>d</sup>*

<sup>þ</sup><sup>0</sup>*H*<sup>3333</sup>*ϖ θ*

<sup>þ</sup>0*Hij*<sup>33</sup> *<sup>χ</sup>ij <sup>d</sup>*

<sup>þ</sup>0*H*3333*<sup>p</sup>* <sup>ϱ</sup>

� *<sup>ξ</sup>*3¼<sup>0</sup>, � �!

� �!

! , *θ* � �!

!� � *p ς* !� � �*dS*,

*ϖ η*!� �

þ *γij δ* � �!

�*dS*,

þ *χij δ* ! , *η* ! � �

! � � � �

� � � �!

*ϖ η*!� �

*p ς* !� � *ϖ θ*

(77)

*p* ϱ

(78)

is a uniformly bounded function over ℬ in *t*. Since it

. We also forgo regularity assumption in

where the exponent (*ρ* � 1) is used for consistency when the external work involves an integration over the thickness which is relevant for general asymptotic

¼ f g ð Þ 0, 0, 0 , (73)

6¼ f g ð Þ 0, 0, 0 , (74)

, (75)

, *ξ*<sup>3</sup> � �, (76)

n o

n o

*F* !

> ! <sup>0</sup> *ξ*<sup>1</sup>

!

*γkl δ* � �!

> *ζ<sup>j</sup> d* ! , *η* ! � �

*χkl δ* ! , *η* ! � �

*mj η* !, *ς* ! � �

<sup>0</sup>*Hαβλμ* <sup>¼</sup> *<sup>H</sup>αβλμ*�

Eq. (84) equivalently holds for any *δ* ! , *η* !, *ς* ! � �∈V<sup>0</sup> (pure-bending subspace of initial lesion). The uniqueness of solution implies that *d* !*w* , *θ* !*w*, ϱ !*w* � � <sup>¼</sup> *<sup>d</sup>* !0 , *θ* !0 , ϱ !0 � �. If Eq. (83) equivalently holds for any weakly converging subsequence *d* !*w* , *θ* !*w*, ϱ !*w* � �, we affirmatively conclude that the whole sequence converges weakly to *d* !0 , *θ* !0 , ϱ !0 � �.

subsequence *d*

*d* !*<sup>ε</sup>* <sup>þ</sup> *<sup>t</sup>* 2 <sup>12</sup> *ς* !, *θ* !*w* , *θ* !*w*, ϱ !*w*

**6. Concluding remarks**

**6.1 Conclusion**

**6.2 Future scope**

bounds.

**201**

!*<sup>ε</sup>* converges weakly to *<sup>d</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.89866*

, we affirmatively conclude that the whole sequence

!*m*, *θ* !*<sup>m</sup>* in <sup>V</sup>*m*. Finally, asymptotic analysis, both types of initial lesion problems, including case of non-inhibited pure bending and case of inhibited pure bending, has weak convergence. Asymptotic analysis revealed that initial lesion is bending-dominated when pure bending is non-inhibited and that initial lesion is membrane-dominated when pure bending is inhibited. Clinically, the primal lesion undergoes transformations under the influence of membrane, bending, and shear tensors. In advanced stages, the transition towards upper bound occurs due to change in coercivity bounds. During the advanced stages of disease, the bending is responsible for introducing progressive disarray of collagen fibers, smooth muscle cells, and ground matrix and thus contributes to rapid progression. Asymptotic analysis suggests that bending strain is relevant for the progression of disease in advanced stages. Hence, asymptotic analysis is a valuable technique for theoretical supplementation to

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human…*

model building and provide insights into the behavior of initial lesion.

this chapter can be used in the convergence studies.

We constructed the model by using higher-order kinematical assumptions relevant to human cardiovascular system. We called this model the initial lesion model. The weak convergence of the solution to initial lesion model was mathematically substantiated. In the analysis of the initial lesion, we concentrated to seek biological and mathematical insights in order to understand early stages of AD. A general understanding of evolution of initial lesion in aortic dissection is presented. The results presented in this chapter are relevant for the assessment of shell-type lesion in biological systems including human physiology and pathology. At least two observations are to be noted. First, the mathematical analysis of the initial lesion model is distinct from classical shell models. Second, the asymptotic analysis of the initial lesion model is based on degenerating three-dimensional continuum to bending strains to initial lesion behavior. For very thin shells as seen in human vessels' internal lining, the analytical perspective to the initial lesion model given in

Clinically complex situations such as the formation of false lumen either blind or patent in advanced stage of AD merit mathematical analysis perusing coercivity

#### **5.2 The impact of inhibited pure bending on the initial lesion**

We define pure-bending subspace <sup>V</sup>#, of displacement space <sup>V</sup> for initial lesion such that

$$\mathcal{V}'' = \left\{ \left( \overrightarrow{\delta}, \overrightarrow{\eta} \right) \mid \left( \overrightarrow{\delta}, \overrightarrow{\eta}, \overrightarrow{0} \right) \in \mathcal{V} \right\}.$$

In this case, pure bending is inhibited; k k� *<sup>m</sup>* gives a norm in pure-bending subspace <sup>V</sup># such that

$$\left\| \left| \overrightarrow{\delta}, \overrightarrow{\eta} \right\rangle \right\|\_{\mathfrak{m}} = \left\| \underline{\chi} \left( \overrightarrow{\delta} \right) \right\|\_{\mathfrak{0}} + \left\| \underline{\zeta} \left( \overrightarrow{\delta}, \overrightarrow{\eta} \right) \right\|\_{\mathfrak{0}} + \left\| \left| \underline{\sigma} \left( \overrightarrow{\eta} \right) \right\|\_{\mathfrak{0}} \right\| $$

Since *d* !*ε* , *θ* !*<sup>ε</sup>* � � is uniformly bounded in pure membrane subspace of displacement space for initial lesion, *ε*<sup>2</sup> *δ* ! , *η* !, *ς* ! � � is uniformly bounded in *<sup>H</sup>*<sup>1</sup> ð Þ S ; we infer that the sequence *d* !*<sup>ε</sup>* <sup>þ</sup> *<sup>t</sup>* 2 <sup>12</sup> *ς* !, *θ* !*<sup>ε</sup>* � � is also uniformly bounded in <sup>V</sup>. Due to the weak convergence in pure membrane subspace V*m*,

$$
\left(\underline{\underline{\chi}}\left(\overrightarrow{\underline{d}}^{\varepsilon} + \frac{t^2}{12}\overrightarrow{\underline{\mathbf{q}}^{\varepsilon}}\right), \underline{\underline{\zeta}}\left(\overrightarrow{\underline{d}}^{\varepsilon} + \frac{t^2}{12}\overrightarrow{\underline{\mathbf{q}}^{\varepsilon}}, \overrightarrow{\underline{\boldsymbol{\theta}}^{\varepsilon}}\right), \underline{\boldsymbol{\sigma}}\left(\overrightarrow{\underline{\boldsymbol{\theta}}^{\varepsilon}}\right)\right) \overset{\varepsilon \to 0}{\rightarrow} \left(\underline{\underline{\chi}}\left(\overrightarrow{\underline{\mathbf{d}}}^{\boldsymbol{w}}\right), \underline{\underline{\zeta}}\left(\overrightarrow{\underline{\mathbf{d}}}^{\boldsymbol{w}}, \overrightarrow{\underline{\boldsymbol{\theta}}^{\boldsymbol{w}}}\right), \underline{\boldsymbol{\sigma}}\left(\overrightarrow{\underline{\boldsymbol{\theta}}^{\boldsymbol{w}}}\right)\right), \underline{\underline{\chi}}
$$

converges weakly in *L*<sup>2</sup> ð Þ S . Hence, for any fixed *δ* ! , *η* !, *ς* ! � � in displacement space V, we infer

$$\lim\_{\varepsilon \to 0} \frac{1}{\varepsilon} A \left( \overrightarrow{d}^{\varepsilon}, \overrightarrow{\theta}^{\varepsilon}, \overrightarrow{\varphi}^{\varepsilon}; \overrightarrow{\delta}, \overrightarrow{\eta}, \overrightarrow{\zeta} \right) = A\_m \left( \overrightarrow{d}^w, \overrightarrow{\theta}^w; \overrightarrow{\delta}, \overrightarrow{\eta} \right). \tag{85}$$

We have

$$\frac{1}{\varepsilon} \mathbf{A} \left( \overrightarrow{\overline{d}}^{\varepsilon}, \overrightarrow{\theta}^{\varepsilon}, \overrightarrow{\overline{\mathbf{q}}}^{\varepsilon}; \overrightarrow{\delta}, \overrightarrow{\eta}, \overrightarrow{\xi} \right) = \mathbf{G} \left( \overrightarrow{\delta} \right) + \frac{R}{\varepsilon}. \tag{86}$$

Here, *<sup>R</sup> <sup>ε</sup>* ! 0 when *ε* ! 0. As *δ* ! , *η* !, *ς* ! � � is fixed, we infer

$$A\_m\left(\overrightarrow{d}^w, \overrightarrow{\theta}^w; \overrightarrow{\delta}, \overrightarrow{\eta}\right) = G\left(\overrightarrow{\delta}\right) \qquad \forall \left(\overrightarrow{\delta}, \overrightarrow{\eta}\right) \in \mathcal{V}.\tag{87}$$

Eq. (87) equivalently holds for any *δ* ! , *η* ! � �∈V*<sup>m</sup>* (membrane subspace of initial lesion). From the uniqueness of the weak convergence result, it follows that *d* !*w* , *θ* !*<sup>w</sup>* � � <sup>¼</sup> *<sup>d</sup>* !*<sup>m</sup>*, *θ* !*<sup>m</sup>* � �. If this equivalently holds for any weakly converging

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human… DOI: http://dx.doi.org/10.5772/intechopen.89866*

subsequence *d* !*w* , *θ* !*w*, ϱ !*w* , we affirmatively conclude that the whole sequence *d* !*<sup>ε</sup>* <sup>þ</sup> *<sup>t</sup>* 2 <sup>12</sup> *ς* !, *θ* !*<sup>ε</sup>* converges weakly to *<sup>d</sup>* !*m*, *θ* !*<sup>m</sup>* in <sup>V</sup>*m*.

Finally, asymptotic analysis, both types of initial lesion problems, including case of non-inhibited pure bending and case of inhibited pure bending, has weak convergence. Asymptotic analysis revealed that initial lesion is bending-dominated when pure bending is non-inhibited and that initial lesion is membrane-dominated when pure bending is inhibited. Clinically, the primal lesion undergoes transformations under the influence of membrane, bending, and shear tensors. In advanced stages, the transition towards upper bound occurs due to change in coercivity bounds. During the advanced stages of disease, the bending is responsible for introducing progressive disarray of collagen fibers, smooth muscle cells, and ground matrix and thus contributes to rapid progression. Asymptotic analysis suggests that bending strain is relevant for the progression of disease in advanced stages. Hence, asymptotic analysis is a valuable technique for theoretical supplementation to model building and provide insights into the behavior of initial lesion.

#### **6. Concluding remarks**

#### **6.1 Conclusion**

Eq. (84) equivalently holds for any *δ*

such that

subspace <sup>V</sup># such that

Since *d* !*ε* , *θ* !*<sup>ε</sup>* � �

*γ d* !*<sup>ε</sup>* <sup>þ</sup> *t* 2 <sup>12</sup> <sup>ϱ</sup> !*ε* � �

V, we infer

We have

Here, *<sup>R</sup>*

¼ *d* !*<sup>m</sup>*, *θ* !*<sup>m</sup>* � �

*d* !*w* , *θ* !*<sup>w</sup>* � �

**200**

that the sequence *d*

*δ* ! , *η* ! � � �

ment space for initial lesion, *ε*<sup>2</sup> *δ*

� � �

!*<sup>ε</sup>* <sup>þ</sup> *<sup>t</sup>* 2 <sup>12</sup> *ς* !, *θ* !*<sup>ε</sup>* � �

convergence in pure membrane subspace V*m*,

� � � �

, *ζ d* !*<sup>ε</sup>* <sup>þ</sup> *t* 2 <sup>12</sup> <sup>ϱ</sup> !*<sup>ε</sup>*, *θ* !*ε*

lim*ε*!0 1 *ε A d*!*<sup>ε</sup>* , *θ* !*ε* , ϱ !*<sup>ε</sup>*; *δ* ! , *η* !, *ς*

> 1 *ε A d*!*<sup>ε</sup>* , *θ* !*ε* , ϱ !*<sup>ε</sup>*; *δ* ! , *η* !, *ς*

*<sup>ε</sup>* ! 0 when *ε* ! 0. As *δ*

*Am d* !*w* , *θ* !*<sup>w</sup>*; *δ* ! , *η*

Eq. (87) equivalently holds for any *δ*

converges weakly in *L*<sup>2</sup>

initial lesion). The uniqueness of solution implies that *d*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

! , *η* !, *ς* ! � �

We define pure-bending subspace <sup>V</sup>#, of displacement space <sup>V</sup> for initial lesion

j *δ* ! , *η* !, 0 � �!

<sup>þ</sup> *ζ δ*!

� � �

, *ϖ θ*!*<sup>ε</sup>*

ð Þ S . Hence, for any fixed *δ*

! � �

! � �

! , *η* !, *ς* ! � �

> ¼ *G δ* � �!

! , *η* ! � �

lesion). From the uniqueness of the weak convergence result, it follows that

! � �

, *η* !Þ � � � � 0

is uniformly bounded in pure membrane subspace of displace-

! *ε*!0

¼ *Am d* !*w* , *θ* !*<sup>w</sup>*; *δ* ! , *η*

¼ *G δ* � �! þ *R*

is fixed, we infer

. If this equivalently holds for any weakly converging

∀ *δ* ! , *η* ! � �

In this case, pure bending is inhibited; k k� *<sup>m</sup>* gives a norm in pure-bending

� � � 0

n o

If Eq. (83) equivalently holds for any weakly converging subsequence *d*

we affirmatively conclude that the whole sequence converges weakly to *d*

**5.2 The impact of inhibited pure bending on the initial lesion**

<sup>V</sup># <sup>¼</sup> *<sup>δ</sup>* ! , *η* ! � �

*<sup>m</sup>* <sup>¼</sup> *γ δ* � �! � � �

� �

! , *η* !, *ς* ! � � ∈V<sup>0</sup> (pure-bending subspace of

¼ *d* !0 , *θ* !0 , ϱ !0 � �

> !*w* , *θ* !*w*, ϱ !*w* � �

!0 , *θ* !0 , ϱ !0 � �

ð Þ S ; we infer

in displacement space

*:* (85)

*<sup>ε</sup> :* (86)

∈ V*:* (87)

∈V*<sup>m</sup>* (membrane subspace of initial

,

.

,

.

!*w* , *θ* !*w*, ϱ !*w* � �

∈V

*:*

<sup>þ</sup> *ϖ η*! � � � � �

is also uniformly bounded in V. Due to the weak

! � �

, *ζ d* !*w* , *θ* !*<sup>w</sup>* � � , *ϖ θ*!*<sup>w</sup>* � � � �

is uniformly bounded in *H*<sup>1</sup>

*γ d* !*<sup>w</sup>* � �

! , *η* !, *ς* ! � � � � � 0 *:*

> We constructed the model by using higher-order kinematical assumptions relevant to human cardiovascular system. We called this model the initial lesion model. The weak convergence of the solution to initial lesion model was mathematically substantiated. In the analysis of the initial lesion, we concentrated to seek biological and mathematical insights in order to understand early stages of AD. A general understanding of evolution of initial lesion in aortic dissection is presented. The results presented in this chapter are relevant for the assessment of shell-type lesion in biological systems including human physiology and pathology. At least two observations are to be noted. First, the mathematical analysis of the initial lesion model is distinct from classical shell models. Second, the asymptotic analysis of the initial lesion model is based on degenerating three-dimensional continuum to bending strains to initial lesion behavior. For very thin shells as seen in human vessels' internal lining, the analytical perspective to the initial lesion model given in this chapter can be used in the convergence studies.

#### **6.2 Future scope**

Clinically complex situations such as the formation of false lumen either blind or patent in advanced stage of AD merit mathematical analysis perusing coercivity bounds.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

**References**

2006

1960

1956;**23**:9

419-451

[1] Bathe KJ. Finite Element Procedures.

*DOI: http://dx.doi.org/10.5772/intechopen.89866*

[10] Sakalihasan N, Limet R, Defawe OD. Abdominal aortic

1577-1589

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human…*

aneurysm. The Lancet. 2005;**365**(9470):

[2] Argyris JH, Kelsey S. Energy Theorems and Structural Analysis. A Generalised Discourse with Applications on Energy Principles of Structural Analysis Including the Effects of Temperature and Non-linear Stress-Strain Relations. London: Butterworths;

[3] Clough RW, Martin HC, Topp LJ, Turner MJ. Stiffness and deflection analysis of complex structures. Journal of the Aeronautical Sciences.

[4] Ahmad S, Irons BM, Zienkiewicz OC.

[5] Reissner E. The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics. 1945;**12**:A-69-A-77

[6] Mindlin RD. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics. 1951;**18**:31-38

treatment for ruptured abdominal aortic aneurysm. Cochrane Database of Systematic Reviews. 2017;**5**

[8] Zankl AR et al. Pathology, natural history and treatment of abdominal aortic aneurysms. Clinical Research in Cardiology. 2007;**96**(3):140-151

[9] Legarreta JH et al. Hybrid decision support system for endovascular aortic

aneurysm repair follow-up. In: International Conference on Hybrid Artificial Intelligence Systems. Berlin, Heidelberg: Springer; 2010. pp. 500-507

**203**

[7] Badger S et al. Endovascular

Analysis of thick and thin shell structures by curved finite elements. International Journal for Numerical Methods in Engineering. 1970;**2**(3):

### **Author details**

Vishakha Jadaun and Nitin Raja Singh\* Indian Institute of Technology, Delhi, New Delhi, India

\*Address all correspondence to: nitinrsingh22@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human… DOI: http://dx.doi.org/10.5772/intechopen.89866*

### **References**

[1] Bathe KJ. Finite Element Procedures. 2006

[2] Argyris JH, Kelsey S. Energy Theorems and Structural Analysis. A Generalised Discourse with Applications on Energy Principles of Structural Analysis Including the Effects of Temperature and Non-linear Stress-Strain Relations. London: Butterworths; 1960

[3] Clough RW, Martin HC, Topp LJ, Turner MJ. Stiffness and deflection analysis of complex structures. Journal of the Aeronautical Sciences. 1956;**23**:9

[4] Ahmad S, Irons BM, Zienkiewicz OC. Analysis of thick and thin shell structures by curved finite elements. International Journal for Numerical Methods in Engineering. 1970;**2**(3): 419-451

[5] Reissner E. The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics. 1945;**12**:A-69-A-77

[6] Mindlin RD. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics. 1951;**18**:31-38

[7] Badger S et al. Endovascular treatment for ruptured abdominal aortic aneurysm. Cochrane Database of Systematic Reviews. 2017;**5**

[8] Zankl AR et al. Pathology, natural history and treatment of abdominal aortic aneurysms. Clinical Research in Cardiology. 2007;**96**(3):140-151

[9] Legarreta JH et al. Hybrid decision support system for endovascular aortic aneurysm repair follow-up. In: International Conference on Hybrid Artificial Intelligence Systems. Berlin, Heidelberg: Springer; 2010. pp. 500-507 [10] Sakalihasan N, Limet R, Defawe OD. Abdominal aortic aneurysm. The Lancet. 2005;**365**(9470): 1577-1589

**Author details**

**202**

Vishakha Jadaun and Nitin Raja Singh\*

provided the original work is properly cited.

Indian Institute of Technology, Delhi, New Delhi, India

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

\*Address all correspondence to: nitinrsingh22@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**Chapter 11**

**Abstract**

**1. Introduction**

**205**

**1.1 Solar sail history**

Friedrich Zander (1887–1933).

Problems of Control Motion of

*Vladimir Stepanovich Korolev, Elena Nikolaevna Polyakhova*

The problems of spacecrafts with a solar sail-controlled motion lead to the study of mathematical models for translational orbital motion and for the spaceship rotation about the mass center in photogravitational fields. There are opportunities to choose the optimal maneuvering conditions to realize orbital motion or to move to a given orbit point. The realization of the given optimal sail orientation about the sunlight flow allows to obtain motions in the vicinity of a possible relative equilibrium or stationary state. This realization also takes into account the stability change according to the process models with perturbations. For the motion control, we can change the properties, dimensions, or location of sail system elements. The spacecraft flights using light pressure are already a reality. Such space sailing ships may soon be used to fly to the big and small planets, for asteroids and comets meeting, to form special motion conditions in the vicinity of the Sun or the Earth. New technologies will bring visible benefits for solving complex problems, based on the

The principle of movement in space by solar sail is based on the light pressure

In 1920, F.A. Zander and K.E. Tsiolkovsky (1857–1935) suggested that a very thin flat sheet, illuminated by sunlight, is able to achieve high speeds in space. As for the question of whether this property of photons can be used for space motion, they answered positively. The idea to use this effect for space flight was advanced by scientist and inventor F.A. Zander in 1924 [2]. He proposed the construction of solar sails and developed the foundation of the spacecraft motion theory. He was the first person to realize the potential of large specularly reflecting surfaces for space flight, proposed to build solar sails and developed the basis of the theory of motion of spacecraft. It can be considered the founding father of the innovative concept of

effect on all the bodies, which experimentally are detected and measured by Russian scientist P.N. Lebedev in 1899 [1]. The development of the first engineering project of a space flight under a solar sail belongs to a Russian scientist and engineer

Solar Sail Spacecraft in the

Photogravitational Fields

direct use of practically unlimited source of solar energy.

**Keywords:** spaceflight, solar sail, control, stability

*and Irina Yurievna Pototskaya*

#### **Chapter 11**

## Problems of Control Motion of Solar Sail Spacecraft in the Photogravitational Fields

*Vladimir Stepanovich Korolev, Elena Nikolaevna Polyakhova and Irina Yurievna Pototskaya*

#### **Abstract**

The problems of spacecrafts with a solar sail-controlled motion lead to the study of mathematical models for translational orbital motion and for the spaceship rotation about the mass center in photogravitational fields. There are opportunities to choose the optimal maneuvering conditions to realize orbital motion or to move to a given orbit point. The realization of the given optimal sail orientation about the sunlight flow allows to obtain motions in the vicinity of a possible relative equilibrium or stationary state. This realization also takes into account the stability change according to the process models with perturbations. For the motion control, we can change the properties, dimensions, or location of sail system elements. The spacecraft flights using light pressure are already a reality. Such space sailing ships may soon be used to fly to the big and small planets, for asteroids and comets meeting, to form special motion conditions in the vicinity of the Sun or the Earth. New technologies will bring visible benefits for solving complex problems, based on the direct use of practically unlimited source of solar energy.

**Keywords:** spaceflight, solar sail, control, stability

#### **1. Introduction**

#### **1.1 Solar sail history**

The principle of movement in space by solar sail is based on the light pressure effect on all the bodies, which experimentally are detected and measured by Russian scientist P.N. Lebedev in 1899 [1]. The development of the first engineering project of a space flight under a solar sail belongs to a Russian scientist and engineer Friedrich Zander (1887–1933).

In 1920, F.A. Zander and K.E. Tsiolkovsky (1857–1935) suggested that a very thin flat sheet, illuminated by sunlight, is able to achieve high speeds in space. As for the question of whether this property of photons can be used for space motion, they answered positively. The idea to use this effect for space flight was advanced by scientist and inventor F.A. Zander in 1924 [2]. He proposed the construction of solar sails and developed the foundation of the spacecraft motion theory. He was the first person to realize the potential of large specularly reflecting surfaces for space flight, proposed to build solar sails and developed the basis of the theory of motion of spacecraft. It can be considered the founding father of the innovative concept of

the solar sail, which was developed in two manuscripts but was not published until 1947. Flight spacecraft using light pressure energy is no longer a fantasy but the reality of the near future [3, 4]. The first attempt of the project implementation and deployment of a solar sail in space was made in 1993.

• The formation of special frame elements for control and support of the sail.

Only by resolving all problems can we talk about space travel and maneuvering in reality. It should provide a sufficiently sophisticated control sail itself, as desired by changing its size, shape, and position relative to the main body. We can use the sail elements that can change the reflection coefficient of the surface of a given program. Successful construction has been recognized as the slit-like sails helicopter rotor, each blade which is rolled out from the container and can be rotated radially relative to the axis of fixing at a predetermined angle. In some projects, the spacecraft with the sail offers spinning relative to the main axis for the stability of the sail

The most successful may be the design of the sail system, which provides the installation of the desired orientation and control over its preservation. After creating and placing such mirror elements with certain proportions in orbit, we obtain orientation stability relative to the Sun for coplanar trajectories of the transition to a new orbit or preservation of a given final orbit. More complex options and models make it possible to program sequential control of the orbital or rotational motions of

The efficiency of using solar sails is primarily associated with the angle of their orientation relative to the beam of rays. In a complex system of mirror surfaces, the beam path can be adjusted in such a way that the direction of the incident and reflected light beam is independent, creating new opportunities for a given direc-

Only having solved all the problems, we can talk about space travel. This can be ensured by a difficult choice of controlling the elements of the sailing system, which will allow you to change the size or shape and position relative to the main body and the flow of sunlight. We can use sail elements that can change the reflection

Of interest is the unique possibility of functioning in special zones in the vicinity of the Sun, even near the solar corona, where the sail can simultaneously play the role of a reliable heat-resistant screen that protects the main instrument compartment from overheating. This design will be indispensable for studying solar space and observing sunspots from close range. The disclosure of the sail creates a force in the direction that compensates for the force of gravity and, therefore, will change

Many options relate to near-Earth space maneuvers. The use of the solar sail is

the sail elements with a parameter (1), which is determined by a reflection coefficient *qi* and inversely proportional to the square of the distance *r* from the Sun (**Figure 1a**). It also depends on the direction force (**Figure 1b**) of the vector normal *n*

surface of the respective element *b*ð Þ *ϑ<sup>i</sup>* relative to the radial direction with angle *ϑ*. Stability may be ensured by the torques of pressure forces about the center of mass, which can modify the value at change the settings. The main vector of the

! is proportional to the surface *Si* area of

! to the

possible to put the spacecraft into a given orbit and to support further stable

operation of satellite systems without additional fuel consumption.

• Providing required initial orientation of the elements of the sail.

*Problems of Control Motion of Solar Sail Spacecraft in the Photogravitational Fields*

*DOI: http://dx.doi.org/10.5772/intechopen.89452*

• Motion control and stability of the given position in flight.

shape under the action of light pressure and disturbing forces.

coefficient of a part of the surface according to a given program.

a spacecraft with a solar sail.

tion of the thrust vector.

the parameters of a possible orbit.

**1.2 Forces and their moments**

**207**

The resulting light flux pressure force *Fi*

Over the years there has been appeared numerous studies of mathematical models of the motion and possible new versions of the form of solar sails [5, 6]. The technology of large-scale designs of sunlight reflectors is still in its initial state. The first practical development of flights with a solar sail began in the 1970s of the twentieth century. Of particular concern was the planned for 1986 flight of a spaceship on a solar sail to meet with Halley's comet. The first attempt to implement the project and deploy a solar sail in space was completed in 1993; a 20-m-diameter mirror sail was successfully deployed on the "Progress M-15" cargo ship as a result of the space experiment "Znamya-2."

If a real sail is at an angle to the flow, the force vector will be directed almost normal to the plane of the sail with a good reflection coefficient. The force of light pressure on the mirror at the same time would be almost twice as much on the black sail of equal area, which completely absorbs the radiation. If the solar sail is made of black material, its thrust is twice less than perfectly mirrored. In this case, the force is not directed along the normal to the surface and the direction of flow of sunlight. Light pressure forms a central photogravitational field, which operates when spacecraft sails moving in an interplanetary space. This will allow to select optimal control during maneuvering.

There are projects using the solar sail to put the spacecraft into geosynchronous orbit in the equatorial plane or to maintain motion in the orbit plane, which is parallel to the equatorial plane and has a nonzero latitude. These latitudinal orbits can create new systems for the deployment of satellite communication systems. One of the possible tasks is the creation of a cosmic solar screen located near the Lagrange point L1 of the Sun-Earth-spacecraft system, which can be useful for monitoring the global temperature of the Earth. Many promising projects for using the solar sail are published.

The solar radiation flux creates a force locally uniform pressure field on the surface of the spacecraft sails. If the surface has a symmetry, and the point of application of the resultant coincides with the center of mass of the spacecraft design, then any initial position relative to the light flux is a state of neutral equilibrium and peace. The action of other forces, even small in size, can produce disturbing moment, which can cause rotation about the center of mass. To damp or compensate the disturbance and to maintain the sail correct position with respect to the light flux it is necessary to use an additional control force. The special sails design allows to solve control and the spacecraft stability problems. Thrust vector and point spacecraft relative to the main body can be changed or if the value of the surface properties of the solar sail and arrangement of the elements may vary with respect to the device using the additional devices.

Note the basic problems [6, 7] of engineering and realization of flights of the spacecraft with the solar sail:


Only by resolving all problems can we talk about space travel and maneuvering in reality. It should provide a sufficiently sophisticated control sail itself, as desired by changing its size, shape, and position relative to the main body. We can use the sail elements that can change the reflection coefficient of the surface of a given program. Successful construction has been recognized as the slit-like sails helicopter rotor, each blade which is rolled out from the container and can be rotated radially relative to the axis of fixing at a predetermined angle. In some projects, the spacecraft with the sail offers spinning relative to the main axis for the stability of the sail shape under the action of light pressure and disturbing forces.

The most successful may be the design of the sail system, which provides the installation of the desired orientation and control over its preservation. After creating and placing such mirror elements with certain proportions in orbit, we obtain orientation stability relative to the Sun for coplanar trajectories of the transition to a new orbit or preservation of a given final orbit. More complex options and models make it possible to program sequential control of the orbital or rotational motions of a spacecraft with a solar sail.

The efficiency of using solar sails is primarily associated with the angle of their orientation relative to the beam of rays. In a complex system of mirror surfaces, the beam path can be adjusted in such a way that the direction of the incident and reflected light beam is independent, creating new opportunities for a given direction of the thrust vector.

Only having solved all the problems, we can talk about space travel. This can be ensured by a difficult choice of controlling the elements of the sailing system, which will allow you to change the size or shape and position relative to the main body and the flow of sunlight. We can use sail elements that can change the reflection coefficient of a part of the surface according to a given program.

Of interest is the unique possibility of functioning in special zones in the vicinity of the Sun, even near the solar corona, where the sail can simultaneously play the role of a reliable heat-resistant screen that protects the main instrument compartment from overheating. This design will be indispensable for studying solar space and observing sunspots from close range. The disclosure of the sail creates a force in the direction that compensates for the force of gravity and, therefore, will change the parameters of a possible orbit.

Many options relate to near-Earth space maneuvers. The use of the solar sail is possible to put the spacecraft into a given orbit and to support further stable operation of satellite systems without additional fuel consumption.

#### **1.2 Forces and their moments**

The resulting light flux pressure force *Fi* ! is proportional to the surface *Si* area of the sail elements with a parameter (1), which is determined by a reflection coefficient *qi* and inversely proportional to the square of the distance *r* from the Sun (**Figure 1a**).

It also depends on the direction force (**Figure 1b**) of the vector normal *n* ! to the surface of the respective element *b*ð Þ *ϑ<sup>i</sup>* relative to the radial direction with angle *ϑ*.

Stability may be ensured by the torques of pressure forces about the center of mass, which can modify the value at change the settings. The main vector of the

the solar sail, which was developed in two manuscripts but was not published until 1947. Flight spacecraft using light pressure energy is no longer a fantasy but the reality of the near future [3, 4]. The first attempt of the project implementation and

Over the years there has been appeared numerous studies of mathematical models of the motion and possible new versions of the form of solar sails [5, 6]. The technology of large-scale designs of sunlight reflectors is still in its initial state. The first practical development of flights with a solar sail began in the 1970s of the twentieth century. Of particular concern was the planned for 1986 flight of a spaceship on a solar sail to meet with Halley's comet. The first attempt to implement the project and deploy a solar sail in space was completed in 1993; a 20-m-diameter mirror sail was successfully deployed on the "Progress M-15" cargo ship as a result

If a real sail is at an angle to the flow, the force vector will be directed almost normal to the plane of the sail with a good reflection coefficient. The force of light pressure on the mirror at the same time would be almost twice as much on the black sail of equal area, which completely absorbs the radiation. If the solar sail is made of black material, its thrust is twice less than perfectly mirrored. In this case, the force is not directed along the normal to the surface and the direction of flow of sunlight. Light pressure forms a central photogravitational field, which operates when spacecraft sails moving in an interplanetary space. This will allow to select optimal

There are projects using the solar sail to put the spacecraft into geosynchronous

orbit in the equatorial plane or to maintain motion in the orbit plane, which is parallel to the equatorial plane and has a nonzero latitude. These latitudinal orbits can create new systems for the deployment of satellite communication systems. One

of the possible tasks is the creation of a cosmic solar screen located near the Lagrange point L1 of the Sun-Earth-spacecraft system, which can be useful for monitoring the global temperature of the Earth. Many promising projects for using

The solar radiation flux creates a force locally uniform pressure field on the surface of the spacecraft sails. If the surface has a symmetry, and the point of application of the resultant coincides with the center of mass of the spacecraft design, then any initial position relative to the light flux is a state of neutral equilibrium and peace. The action of other forces, even small in size, can produce disturbing moment, which can cause rotation about the center of mass. To damp or compensate the disturbance and to maintain the sail correct position with respect to the light flux it is necessary to use an additional control force. The special sails design allows to solve control and the spacecraft stability problems. Thrust vector and point spacecraft relative to the main body can be changed or if the value of the surface properties of the solar sail and arrangement of the elements may vary with

Note the basic problems [6, 7] of engineering and realization of flights of the

• Take into account the restrictions on the total weight of the spacecraft with a

• Special tools for deploying sails of a large area in the working position.

deployment of a solar sail in space was made in 1993.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

of the space experiment "Znamya-2."

control during maneuvering.

the solar sail are published.

spacecraft with the solar sail:

sail at launch.

**206**

respect to the device using the additional devices.

• Creation of an effective reflective polymer film for sails.

• Packing of sails in special containers for delivery to space.

propagation of light radiation and direction from the spacecraft to the gravitational

Control using solar sails leads to complex problems and solving equations of mathematical models. There are basic versions of the equations of motion of the spacecraft in the central gravitational field, taking into account perturbations depending on the choice of the reference frame, absolute Cartesian, spherical and

Changes in Cartesian coordinates *xi* of the spacecraft center of mass in the absolute coordinate system based on the main operating force of gravity and the center of the field of light pressure at movement in three-dimensional case can be

where we have used the notations *x*, the Cartesian coordinates; *r*, the module of the radius vector; *μ*, the gravitational parameter; *U*, the force function of potential forces of the considered disturbances; and *Fi*, the nonpotential acceleration and control, including of the light pressure forces in projections on the axis coordinate

You can use the polar coordinates ð Þ *r*, *φ* in the study of movement in the orbital

where *Pi* ð Þ *i* ¼ 1, 2 is the radial and transversal components of the perturbing acceleration, which depend on the installation angle of the sail elements to implement the control law. The position relative to the flow of sunlight is taken into

The control algorithms *u t*ð Þ are numerous and are determined through the parameters of the initial and final orbits or by the tasks of maneuvering the spacecraft in the process of movement. A fixed constant angle will determine the change in the parameters of the orbit. Without taking into account all perturbations and

The movement in the central gravitational field has a solution, which in the absence of disturbing forces is determined by the initial values of the radius vector, velocity vector, and the gravitational parameter of the central body. They determine the constant Kepler elements *k* ¼ ð Þ *a*,*e*, *i*, *Ω*,*ω*, *M*<sup>0</sup> which allow us to calculate the Cartesian coordinates *xi*ð Þ*t* and components *vi*ð Þ*t* of the velocity vector for the

> *x*<sup>1</sup> ¼ *r cos u cos* ð Þ *Ω* � *sin u sin Ω cos i* , *x*<sup>2</sup> ¼ *r cos u sin* ð Þ *Ω* þ *sin u cos Ω cos i* , *x*<sup>3</sup> ¼ *r sin u sin i*, *v*<sup>1</sup> ¼ *α*ð Þ *cos u cos Ω* � *sin u sin Ω cos i*

> > �*β*ð Þ *sin u cos Ω* þ *cos u sin Ω cos i* ,

�*β*ð Þ *sin u sin Ω* þ *cos u cos Ω cos i* ,

*v*<sup>3</sup> ¼ *α sin u sin i* þ *β cos u sin i:* (4)

*v*<sup>2</sup> ¼ *α*ð Þ *cos u sin Ω* þ *sin u cos Ω cos i*

*dt <sup>r</sup>* 2

*<sup>r</sup>*<sup>2</sup> � *<sup>r</sup>*ð Þ *<sup>φ</sup>*\_ <sup>2</sup> <sup>¼</sup> *<sup>P</sup>*1, *<sup>d</sup>*

account at the corresponding pressure value far from the Sun.

controls, we can use the classical solution of the two-body problem.

unperturbed motion at any time using the following formulas:

þ *Fi*, *i* ¼ 1, 2, 3, (2)

*<sup>φ</sup>*\_ <sup>¼</sup> *<sup>P</sup>*2, (3)

cylindrical coordinates, or Kepler's elements [5, 6, 10] for the orbit.

*Problems of Control Motion of Solar Sail Spacecraft in the Photogravitational Fields*

*<sup>r</sup>*<sup>3</sup> *xi* <sup>¼</sup> *<sup>∂</sup><sup>U</sup> ∂xi*

center: in this case it may not be the same.

*DOI: http://dx.doi.org/10.5772/intechopen.89452*

described by a second-order equation:

plane:

**209**

*d*2 *xi dt*<sup>2</sup> <sup>þ</sup> *<sup>μ</sup>*

system or jet forces on the active phases of orbit.

*d*2 *r dt*<sup>2</sup> <sup>þ</sup> *<sup>μ</sup>*

**Figure 1.**

*The formation of the resultant force of the light flux acting on the surface of the sail.*

forces and the sum of the moments of all the forces *Fi* ! acting on the sail with a relative position *ρ* ! *i*

$$\overrightarrow{F} = \sum\_{i} \overrightarrow{F}\_{i} = \sum\_{i} q\_{i} \mathbb{S}\_{i} \frac{b(\mathfrak{H}\_{i})}{r^{2}} \overrightarrow{n}\ (\mathfrak{H}\_{i}), \ \overrightarrow{M} = \sum\_{i} \overrightarrow{\rho}\_{i} \times \overrightarrow{F}\_{i}(\mathfrak{H}\_{i}).\tag{1}$$

These values determine the motion of the spacecraft center of mass and rotation relative to the orbital system that accompanies motion in a central field.

The main vector of the moment of acting forces relative to the center of mass may differ from zero for light pressure forces if the elements have different areas or angles of mutual arrangement. This determines the ability to return in the right direction in the event of a change in orientation by random interference or the use of new elements of the basic layout of the spacecraft for orientation in the right direction [8, 9].

#### **2. Equations of motion**

#### **2.1 Different types of equations**

The equations of motion with allowance for disturbances can be represented in different forms, based on models of the problem of two or three bodies using convenient coordinate systems and basic parameters. Heliocentric flight to planets, asteroids, or to the Sun can be considered, to a first approximation, the motion in a photogravitational field as a two-body problem under the action of the additional light pressure of the rays on the sail surface for a fixed angle of the normal position, taking into account the influence of additional disturbances.

Motion in a photogravitational field can be considered as a two-body problem or a central force field without taking into account the influence of other forces, when the action of additional light pressure from the rays reduces the influence of gravitational interaction. This change is especially noticeable for the case of a large sail surface.

When creating orbits near the Earth or to place a spacecraft at the libration points of the Sun-Earth-spacecraft system, it is necessary to use a more general model of the photogravitational restricted three-body problem, which takes into account the movement of two main bodies, as well as the direction of the

*Problems of Control Motion of Solar Sail Spacecraft in the Photogravitational Fields DOI: http://dx.doi.org/10.5772/intechopen.89452*

propagation of light radiation and direction from the spacecraft to the gravitational center: in this case it may not be the same.

Control using solar sails leads to complex problems and solving equations of mathematical models. There are basic versions of the equations of motion of the spacecraft in the central gravitational field, taking into account perturbations depending on the choice of the reference frame, absolute Cartesian, spherical and cylindrical coordinates, or Kepler's elements [5, 6, 10] for the orbit.

Changes in Cartesian coordinates *xi* of the spacecraft center of mass in the absolute coordinate system based on the main operating force of gravity and the center of the field of light pressure at movement in three-dimensional case can be described by a second-order equation:

$$\frac{d^2\mathfrak{x}\_i}{dt^2} + \frac{\mu}{r^3}\mathfrak{x}\_i = \frac{\partial U}{\partial \mathfrak{x}\_i} + F\_i, \quad i = 1, 2, 3,\tag{2}$$

where we have used the notations *x*, the Cartesian coordinates; *r*, the module of the radius vector; *μ*, the gravitational parameter; *U*, the force function of potential forces of the considered disturbances; and *Fi*, the nonpotential acceleration and control, including of the light pressure forces in projections on the axis coordinate system or jet forces on the active phases of orbit.

You can use the polar coordinates ð Þ *r*, *φ* in the study of movement in the orbital plane:

$$\frac{d^2r}{dt^2} + \frac{\mu}{r^2} - r(\dot{\rho})^2 = P\_1, \quad \frac{d}{dt} \left(r^2 \dot{\rho}\right) = P\_2,\tag{3}$$

where *Pi* ð Þ *i* ¼ 1, 2 is the radial and transversal components of the perturbing acceleration, which depend on the installation angle of the sail elements to implement the control law. The position relative to the flow of sunlight is taken into account at the corresponding pressure value far from the Sun.

The control algorithms *u t*ð Þ are numerous and are determined through the parameters of the initial and final orbits or by the tasks of maneuvering the spacecraft in the process of movement. A fixed constant angle will determine the change in the parameters of the orbit. Without taking into account all perturbations and controls, we can use the classical solution of the two-body problem.

The movement in the central gravitational field has a solution, which in the absence of disturbing forces is determined by the initial values of the radius vector, velocity vector, and the gravitational parameter of the central body. They determine the constant Kepler elements *k* ¼ ð Þ *a*,*e*, *i*, *Ω*,*ω*, *M*<sup>0</sup> which allow us to calculate the Cartesian coordinates *xi*ð Þ*t* and components *vi*ð Þ*t* of the velocity vector for the unperturbed motion at any time using the following formulas:

$$\begin{aligned} \varkappa\_1 &= r(\cos u \cos \Omega - \sin u \sin \Omega \cos i), \\ \varkappa\_2 &= r(\cos u \sin \Omega + \sin u \cos \Omega \cos i), \end{aligned}$$

$$\begin{aligned} \varkappa\_3 &= r \sin u \sin i, \\ \nu\_1 &= a(\cos u \cos \Omega - \sin u \sin \Omega \cos i) \end{aligned}$$

$$\begin{aligned} -\beta(\sin u \cos \Omega + \cos u \sin \Omega \cos i), \\ \nu\_2 &= a(\cos u \sin \Omega + \sin u \cos \Omega \cos i) \end{aligned}$$

$$\begin{aligned} -\beta(\sin u \sin \Omega + \cos u \cos \Omega \cos i), \\ \nu\_3 &= a \sin u \sin i + \beta \cos u \sin i. \end{aligned} \tag{4}$$

forces and the sum of the moments of all the forces *Fi*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*The formation of the resultant force of the light flux acting on the surface of the sail.*

! ð Þ *ϑ<sup>i</sup>* , *M* !

These values determine the motion of the spacecraft center of mass and rotation

The main vector of the moment of acting forces relative to the center of mass may differ from zero for light pressure forces if the elements have different areas or angles of mutual arrangement. This determines the ability to return in the right direction in the event of a change in orientation by random interference or the use of new elements of the basic layout of the spacecraft for orientation in the right

The equations of motion with allowance for disturbances can be represented in

Motion in a photogravitational field can be considered as a two-body problem or a central force field without taking into account the influence of other forces, when the action of additional light pressure from the rays reduces the influence of gravitational interaction. This change is especially noticeable for the case of a large sail

When creating orbits near the Earth or to place a spacecraft at the libration points of the Sun-Earth-spacecraft system, it is necessary to use a more general model of the photogravitational restricted three-body problem, which takes into account the movement of two main bodies, as well as the direction of the

different forms, based on models of the problem of two or three bodies using convenient coordinate systems and basic parameters. Heliocentric flight to planets, asteroids, or to the Sun can be considered, to a first approximation, the motion in a photogravitational field as a two-body problem under the action of the additional light pressure of the rays on the sail surface for a fixed angle of the normal position,

taking into account the influence of additional disturbances.

relative to the orbital system that accompanies motion in a central field.

<sup>¼</sup> <sup>X</sup> *i ρ* ! *<sup>i</sup>* � *F* !

! *i*

*F* ! <sup>¼</sup> <sup>X</sup> *i F* ! *<sup>i</sup>* <sup>¼</sup> <sup>X</sup> *i qi Si b*ð Þ *ϑ<sup>i</sup> <sup>r</sup>*<sup>2</sup> *<sup>n</sup>*

a relative position *ρ*

**Figure 1.**

direction [8, 9].

surface.

**208**

**2. Equations of motion**

**2.1 Different types of equations**

! acting on the sail with

*<sup>i</sup>*ð Þ *ϑ<sup>i</sup> :* (1)

Notation used here

$$r = a(1 - e \cos E), \quad p = a(1 - e^2),$$

$$a = \sqrt{\frac{\mu}{p}} \, er^{-1} \sin \theta, \quad \beta = \sqrt{\mu p} r^{-1},$$

and Kepler's equation

$$E - e\sin E = M\_0 + n(t - t\_0) = M.\tag{5}$$

*x t* \_ðÞ¼�3*ω* 2*y*<sup>0</sup> þ

*DOI: http://dx.doi.org/10.5772/intechopen.89452*

*x*\_0 *ω* � �

*Problems of Control Motion of Solar Sail Spacecraft in the Photogravitational Fields*

*y t* \_ðÞ¼ *ω* 3*y*<sup>0</sup> þ

*z t* \_ðÞ¼ *z*\_

mathematical model of photogravitational force field [3, 7, 10].

*da*

*dt* <sup>¼</sup> *rcosϑP*3, *<sup>d</sup><sup>Ω</sup>*

*de*

*di*

*dt* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

asymptotic stability, can be obtained [6, 12, 16].

*dω dt* <sup>¼</sup> *<sup>e</sup>* �1

*dM*<sup>0</sup>

pressure far from the Sun.

**2.2 Control of motion**

**211**

þ 2*ω* 3*y*<sup>0</sup> þ

2*x*\_<sup>0</sup> *ω*

2*x*\_<sup>0</sup> *ω*

� � sin*ω<sup>t</sup>* <sup>þ</sup> *<sup>y</sup>*\_<sup>0</sup> cos*ωt*,

<sup>0</sup> cos*ωt* � *z*0*ω*sin*ωt*,

The unfolding of the sails on a circular heliocentric orbit will lead to the fact that the light pressure partially compensates the Sun's gravity. This is a reason to use a

The orbital elements for perturbed motion of the spacecraft are functions of time. We can use the differential equations of Euler, where the right-hand sides are determined by the current values of the elements *k t*ðÞ¼ ð Þ *a*,*e*, *i*, *Ω*, *ω*, *M*<sup>0</sup> and projections of the perturbing acceleration *Pi* on the axes of the orbital coordinate system:

� �,

*dt* <sup>¼</sup> *rsinu sin* �<sup>1</sup>

½ð Þ *<sup>r</sup>* <sup>þ</sup> *<sup>p</sup> sin <sup>ϑ</sup>P*<sup>2</sup> � *pcosϑP*1� � *cos i <sup>d</sup><sup>Ω</sup>*

*<sup>e</sup>*�<sup>2</sup> � <sup>1</sup> <sup>p</sup> ½ � ð Þ *pcos<sup>ϑ</sup>* � <sup>2</sup>*er <sup>P</sup>*<sup>1</sup> � ð Þ *<sup>r</sup>* <sup>þ</sup> *<sup>p</sup> sin <sup>ϑ</sup>P*<sup>2</sup> *:*

*dt* <sup>¼</sup> <sup>2</sup>*a*<sup>2</sup> *esinϑP*<sup>1</sup> <sup>þ</sup> *pr*�<sup>1</sup>

*dt* <sup>¼</sup> *p sin* ð Þ *<sup>ϑ</sup>P*<sup>1</sup> <sup>þ</sup> *cos <sup>ϑ</sup>P*<sup>2</sup> <sup>þ</sup> *cos EP*<sup>2</sup> ,

Here *Pi* ð Þ *i* ¼ 1, 2, 3 are the components of the disturbing acceleration in the projection on the axis of the orbital coordinate system. They depend on the installation angle of the sail elements for the implementation of the control law and determine a further change in the parameters of the orbit. The position relative to the stream of sunlight is taken into account with the corresponding value of light

The third and fourth equation of system shows that the plane orbit state is maintained if there is no projection of the disturbing forces in the normal to the plane. The behavior and properties of the solutions are analyzed in a dynamic system, which are simulating the controlled processes. The research methods of nonlinear continuous or discrete systems' quality of the movements, absolute or

The influence of solar pressure on the sail is determined by the angle of deviation of the normal vector to the surface from the direction of flow. If the plane of an ideal specular sail is at an angle to the rays, then the momentum transmitted to the solar sail will be directed almost perpendicular to the reflecting surface. By turning the elements of the sailing system, you can control the direction of the thrust vector. The arrangement of elements relative to the housing can be changed with the help

If the spacecraft with the folded solar sail is already delivered into orbit around the Earth or move around the Sun, the container of the sails will provide disclosure for the spacecraft new thrusters, providing a virtually unlimited supply of energy.

of electric motors, supporting their work on the basis of solar batteries.

� � cos*ω<sup>t</sup>* � <sup>2</sup>*y*\_<sup>0</sup> sin*ωt*,

*P*2

*i P*3, (8)

*dt* ,

Moving time between two points of the orbit can be determined from the above equation which is called the equation of Kepler (5).

The equations of motion while taking into account the perturbations can be represented as osculating elements *k t*ðÞ¼ ð Þ *a*,*e*, *i*, *Ω*, *ω*, *M*<sup>0</sup> . The orbit elements for the perturbed motion of the spacecraft are functions of time *k t*ð Þ. You can use Euler differential equations in which the functions on the right-hand sides of the equations are determined by the current values of the elements and the projections of the disturbing acceleration on the axis of the orbital coordinate system.

The contribution of radiation pressure is determined by the angle *ϕ* of deviation, the normal vector *n* ! from direction *r* ! <sup>0</sup> of flow. If we turn the flat mirror sail at an angle to the rays, the momentum transferred to the solar sail will be directed almost perpendicular to the reflective surface. Part of the momentum directed parallel to the sail, the photons will remain at home, so that the sail will get less than in the full disclosure of the rays. Turning the sail, we are able to control the direction of the thrust vector. However, for it to pay its value. If the vector of normal for flat sail is perpendicular to the flow of rays, the sail does not give any traction. The projections of the vector on the radial and transverse directions will be influenced by a change in the parameters of the orbit motion. The projection on the normal to the plane of the orbit will allow to change its inclination with respect to the initial position. Acceleration, which tells the stream of rays, also depends on the ratio of the area of the sail to the weight of the entire structure and the surface properties.

Equations Hill-Clohessy-Wiltshire [11–15] which managed orbital motion of the moving coordinate system in the spatial case are

$$\ddot{\mathbf{x}} + 2a\dot{\mathbf{y}} = u\_x(t),$$

$$\ddot{\mathbf{y}} - 2a\dot{\mathbf{x}} - 3a^2\mathbf{y} = u\_\mathbf{y}(t),\tag{6}$$

$$\ddot{\mathbf{z}} + a^2\mathbf{z} = u\_\mathbf{z}(t),$$

The solution nonlinear equations (6) can be presented in the form of changes or deviations from the given movement of the reference point and then add a particular solution with the selected control function. The solution of a homogeneous system can be represented as the following (7) system of six equations:

$$\mathbf{x}(t) = \left(\mathbf{x}\_0 - 2\frac{\dot{y}\_0}{\alpha}\right) - 3\alpha t \left(2\mathbf{y}\_0 + \frac{\dot{x}\_0}{\alpha}\right) + 2\left(3\mathbf{y}\_0 + \frac{2\dot{x}\_0}{\alpha}\right)\sin\alpha t + 2\frac{\dot{y}\_0}{\alpha}\cos\alpha t,$$

$$\mathbf{y}(t) = 2\left(2\mathbf{y}\_0 + \frac{\dot{x}\_0}{\alpha}\right) - \left(3\mathbf{y}\_0 + \frac{2\dot{x}\_0}{\alpha}\right)\cos\alpha t + \frac{\dot{y}\_0}{\alpha}\sin\alpha t,$$

$$\mathbf{z}(t) = \mathbf{z}\_0\cos\alpha t + \frac{\dot{z}\_0}{\alpha}\sin\alpha t,\tag{7}$$

*Problems of Control Motion of Solar Sail Spacecraft in the Photogravitational Fields DOI: http://dx.doi.org/10.5772/intechopen.89452*

$$\dot{\mathbf{x}}(t) = -3o(2\mathbf{y}\_0 + \frac{\dot{\mathbf{x}}\_0}{\alpha}) + 2o\left(3\mathbf{y}\_0 + \frac{2\dot{\mathbf{x}}\_0}{\alpha}\right)\cos\alpha t - 2\dot{\mathbf{y}}\_0\sin\alpha t,$$

$$\dot{\mathbf{y}}(t) = o\left(3\mathbf{y}\_0 + \frac{2\dot{\mathbf{x}}\_0}{\alpha}\right)\sin\alpha t + \dot{\mathbf{y}}\_0\cos\alpha t,$$

$$\dot{\mathbf{z}}(t) = \dot{\mathbf{z}}\_0\cos\alpha t - \mathbf{z}\_0\boldsymbol{\alpha}\sin\alpha t,$$

The unfolding of the sails on a circular heliocentric orbit will lead to the fact that the light pressure partially compensates the Sun's gravity. This is a reason to use a mathematical model of photogravitational force field [3, 7, 10].

The orbital elements for perturbed motion of the spacecraft are functions of time. We can use the differential equations of Euler, where the right-hand sides are determined by the current values of the elements *k t*ðÞ¼ ð Þ *a*,*e*, *i*, *Ω*, *ω*, *M*<sup>0</sup> and projections of the perturbing acceleration *Pi* on the axes of the orbital coordinate system:

$$\frac{da}{dt} = 2a^2(\sin\theta P\_1 + pr^{-1}P\_2),$$

$$\frac{de}{dt} = p(\sin\theta P\_1 + \cos\theta P\_2 + \cos E P\_2),$$

$$\frac{di}{dt} = r\cos\theta P\_3, \quad \frac{d\varOmega}{dt} = r\sin u \sin^{-1} i \ P\_3,\tag{8}$$

$$\frac{d\varOmega}{dt} = e^{-1}[(r+p)\sin\theta P\_2 - p\cos\theta P\_1] - \cos i \frac{d\varOmega}{dt},$$

$$\frac{dM\_0}{dt} = \sqrt{e^{-2}-1}[(p\cos\theta - 2er)P\_1 - (r+p)\sin\theta P\_2].$$

Here *Pi* ð Þ *i* ¼ 1, 2, 3 are the components of the disturbing acceleration in the projection on the axis of the orbital coordinate system. They depend on the installation angle of the sail elements for the implementation of the control law and determine a further change in the parameters of the orbit. The position relative to the stream of sunlight is taken into account with the corresponding value of light pressure far from the Sun.

The third and fourth equation of system shows that the plane orbit state is maintained if there is no projection of the disturbing forces in the normal to the plane. The behavior and properties of the solutions are analyzed in a dynamic system, which are simulating the controlled processes. The research methods of nonlinear continuous or discrete systems' quality of the movements, absolute or asymptotic stability, can be obtained [6, 12, 16].

#### **2.2 Control of motion**

The influence of solar pressure on the sail is determined by the angle of deviation of the normal vector to the surface from the direction of flow. If the plane of an ideal specular sail is at an angle to the rays, then the momentum transmitted to the solar sail will be directed almost perpendicular to the reflecting surface. By turning the elements of the sailing system, you can control the direction of the thrust vector. The arrangement of elements relative to the housing can be changed with the help of electric motors, supporting their work on the basis of solar batteries.

If the spacecraft with the folded solar sail is already delivered into orbit around the Earth or move around the Sun, the container of the sails will provide disclosure for the spacecraft new thrusters, providing a virtually unlimited supply of energy.

Notation used here

and Kepler's equation

the normal vector *n*

*x t*ðÞ¼ *x*<sup>0</sup> � 2

**210**

*y*\_ 0 *ω*

*y t*ðÞ¼ 2 2*y*<sup>0</sup> þ

� �

*r* ¼ *a*ð Þ 1 � *e cos E* , *p* ¼ *a* 1 � *e*

*er*�<sup>1</sup> *sin <sup>ϑ</sup>*, *<sup>β</sup>* <sup>¼</sup> ffiffiffiffiffi

Moving time between two points of the orbit can be determined from the above

The contribution of radiation pressure is determined by the angle *ϕ* of deviation,

Equations Hill-Clohessy-Wiltshire [11–15] which managed orbital motion of the

*z* ¼ *uz*ð Þ*t* ,

þ 2 3*y*<sup>0</sup> þ

2*x*\_<sup>0</sup> *ω* � �

> *z*\_ 0

2*x*\_<sup>0</sup> *ω* � �

cos*ω<sup>t</sup>* <sup>þ</sup> *<sup>y</sup>*\_

The solution nonlinear equations (6) can be presented in the form of changes or deviations from the given movement of the reference point and then add a particular solution with the selected control function. The solution of a homogeneous

*x*€ þ 2*ωy*\_ ¼ *ux*ð Þ*t* ,

angle to the rays, the momentum transferred to the solar sail will be directed almost perpendicular to the reflective surface. Part of the momentum directed parallel to the sail, the photons will remain at home, so that the sail will get less than in the full disclosure of the rays. Turning the sail, we are able to control the direction of the thrust vector. However, for it to pay its value. If the vector of normal for flat sail is perpendicular to the flow of rays, the sail does not give any traction. The projections of the vector on the radial and transverse directions will be influenced by a change in the parameters of the orbit motion. The projection on the normal to the plane of the orbit will allow to change its inclination with respect to the initial position. Acceleration, which tells the stream of rays, also depends on the ratio of the area of

The equations of motion while taking into account the perturbations can be represented as osculating elements *k t*ðÞ¼ ð Þ *a*,*e*, *i*, *Ω*, *ω*, *M*<sup>0</sup> . The orbit elements for the perturbed motion of the spacecraft are functions of time *k t*ð Þ. You can use Euler differential equations in which the functions on the right-hand sides of the equations are determined by the current values of the elements and the projections of

the disturbing acceleration on the axis of the orbital coordinate system.

the sail to the weight of the entire structure and the surface properties.

€*<sup>y</sup>* � <sup>2</sup>*ωx*\_ � <sup>3</sup>*ω*<sup>2</sup>

€*<sup>z</sup>* <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

system can be represented as the following (7) system of six equations:

*x*\_ 0 *ω* � �

� 3*y*<sup>0</sup> þ

*z t*ðÞ¼ *z*<sup>0</sup> cos*ωt* þ

� 3*ωt* 2*y*<sup>0</sup> þ

*x*\_0 *ω* � � !

*α* ¼

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

equation which is called the equation of Kepler (5).

! from direction *r*

moving coordinate system in the spatial case are

ffiffiffi *μ p* r

<sup>2</sup> � �,

*μp* p *r* �1 ,

*E* � *e sin E* ¼ *M*<sup>0</sup> þ *n t*ð Þ¼ � *t*<sup>0</sup> *M:* (5)

<sup>0</sup> of flow. If we turn the flat mirror sail at an

*y* ¼ *uy*ð Þ*t* , (6)

sin*ωt* þ 2

*<sup>ω</sup>* sin*ωt*, (7)

0 *<sup>ω</sup>* sin*ωt*, *y*\_ 0 *<sup>ω</sup>* cos*ωt*, However, the sail has one major drawback: unlike jet engines, we cannot use its thrust in any direction with the same efficiency. It is necessary to orient the sail in a special way, to achieve the desired changes in the orbital parameters of outer space.

The presence of perturbations of a periodic or random nature can change the nature of the solutions of such systems. The properties of solutions of dynamic systems are determined by the selected feedback control function. The orbital stability of trajectories or the stability with respect to a part of variables [11–13] is

*Problems of Control Motion of Solar Sail Spacecraft in the Photogravitational Fields*

The system is called stable if it returns to the equilibrium or rest state after the termination of external influences that moved it out of this position. If after the termination of the external impact of the system does not return to a state of equilibrium, it is unsustainable. Stable equilibrium position becomes asymptotically

Determining the motion of any mechanical system is often required to assess the stability and control of motion states. The strict definition of a stable equilibrium position and other solutions of dynamical systems were given in 1892 by the Russian

The movement or behavior of the solution of the dynamical system is called Lyapunov stable if small variations in the initial data from the reference phase variables selected for the study, solving a system of differential equations, lead to small deviations in the future. If the deviation over time tends to be zero, the

The system is called instable in case when even very small perturbation influence leads to large deviations or change the motion character, including the equilibrium position displacement, which is not stable if the velocity initial value is

We also consider the stability of the orbital trajectory or stability of some of the variables [6–12]. In this case, it appears that the phase trajectory and its projection onto the corresponding subspace are close enough to the base path, although the representative points can be arbitrarily disperse, away from each other over time. Periodic solution of the system is not asymptotically stable. But if in such system all multipliers' modules but one are less than one, then according to Andronov-Witt theorem [14], the trajectory of the system periodic solution is asymptotically orbital

Stability with respect to part of variables for partial differential equations involved Rumyantsev [19, 20], who published an article on the analogue of the theory of the second Lyapunov method for the stability problems for some of the variables [21]. He and his students developed methods for research on some of the

integrals, then the Arnold's theorem [12–14], all phase trajectories lie on the n-dimensional torus, and the motion are conditionally periodic system. This set is

called the equilibrium or stationary state of motion of the system.

axis of the ellipsoid of inertia in the direction of the center of gravity.

If the dynamic equations are written in the canonical form, and there are n first

In the field of action of the geopotential excluding other perturbing forces exists stable equilibrium for body position while maintaining the orientation of the major

In the Cartesian coordinate system for the main body, the known parameters of the axial moments of inertia are taken into account. In the field of the geopotential force without other perturbing forces, there are stable equilibrium positions for the

stable with the addition of dissipative forces with complete dissipation.

also considered.

**2.3 The stability of the orbital motion**

*DOI: http://dx.doi.org/10.5772/intechopen.89452*

scientist Lyapunov [6–9, 14, 18].

different from zero.

variables of stability problems.

**2.4 Control of orientation motion**

stable.

**213**

reference solution is called asymptotically stable.

When you turn the sail so that the photons are reflected back relative to the direction of orbital motion, we get an additional force that gradually accelerates the spacecraft, which will move in a spinning spiral. When you turn the sail in a different direction, you get a decrease in speed or braking on the way to the sun.

To change the inclination of the orbital plane of the spacecraft using sails, it is necessary to direct the reflected flow perpendicular to the initial plane. In addition, the elongated elliptical orbit can rotate sequentially, changing the longitude of the pericenter relative to the central body, so that over time, it approaches the orbit of an asteroid or comet for a possible encounter [17].

Even more interesting maneuvers can be performed using a solar sail in the near-Earth space, as the propagation direction of light emission in this case coincides with the direction of the center of gravity. In addition, throughout the year the Sun makes a complete revolution in the celestial sphere relative to the Earth, so be patient, you can wait for the right time of the year for optimum flexibility and translate using the spacecraft sails in the desired orbit.

We get the opportunity to control the movement by changing the direction of the thrust vector of the current light pressure when the sail plane is rotated. Vector projections of the force on the radial and transverse directions will affect the change in the basic parameters of the orbit (size, shape) in the process of movement.

The projection of the force to the normal to the orbit plane changes the inclination relative to the initial position of the orbit plane. Acceleration also depends on the ratio of the sail area to the mass of the entire structure and surface properties.

Control algorithms *u t*ð Þ are numerous. They are determined by the parameters of the orbit or the conditions and purpose of the maneuver. Motion in a gravitational field can be precisely determined if disturbing forces are absent. Then the orbit is determined by the initial values of the radius vector, velocity vector, and gravitational parameter of the central body. For a fixed position of the sail, we can consider a photogravitational field with a small perturbation, which determines the corrections to the parameters of the orbit, reducing the size of the orbit (**Figure 2a**) or increase the size of the orbit (**Figure 2b**).

#### **Figure 2.**

*Motion control is carried out by the position of the solar sail, when the vector of light pressure forces determines braking (a) or acceleration (b). In case (a) there will be a decrease in the size of the orbit; case (b) increases the size of the orbit.*

The presence of perturbations of a periodic or random nature can change the nature of the solutions of such systems. The properties of solutions of dynamic systems are determined by the selected feedback control function. The orbital stability of trajectories or the stability with respect to a part of variables [11–13] is also considered.

#### **2.3 The stability of the orbital motion**

However, the sail has one major drawback: unlike jet engines, we cannot use its thrust in any direction with the same efficiency. It is necessary to orient the sail in a special way, to achieve the desired changes in the orbital parameters of

an asteroid or comet for a possible encounter [17].

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

translate using the spacecraft sails in the desired orbit.

or increase the size of the orbit (**Figure 2b**).

**Figure 2.**

**212**

*size of the orbit.*

When you turn the sail so that the photons are reflected back relative to the direction of orbital motion, we get an additional force that gradually accelerates the spacecraft, which will move in a spinning spiral. When you turn the sail in a different direction, you get a decrease in speed or braking on the way to the sun. To change the inclination of the orbital plane of the spacecraft using sails, it is necessary to direct the reflected flow perpendicular to the initial plane. In addition, the elongated elliptical orbit can rotate sequentially, changing the longitude of the pericenter relative to the central body, so that over time, it approaches the orbit of

Even more interesting maneuvers can be performed using a solar sail in the near-Earth space, as the propagation direction of light emission in this case coincides with the direction of the center of gravity. In addition, throughout the year the Sun makes a complete revolution in the celestial sphere relative to the Earth, so be patient, you can wait for the right time of the year for optimum flexibility and

We get the opportunity to control the movement by changing the direction of the thrust vector of the current light pressure when the sail plane is rotated. Vector projections of the force on the radial and transverse directions will affect the change in the basic parameters of the orbit (size, shape) in the process of movement.

The projection of the force to the normal to the orbit plane changes the inclination relative to the initial position of the orbit plane. Acceleration also depends on the ratio of the sail area to the mass of the entire structure and surface properties. Control algorithms *u t*ð Þ are numerous. They are determined by the parameters of the orbit or the conditions and purpose of the maneuver. Motion in a gravitational field can be precisely determined if disturbing forces are absent. Then the orbit is determined by the initial values of the radius vector, velocity vector, and gravitational parameter of the central body. For a fixed position of the sail, we can consider a photogravitational field with a small perturbation, which determines the corrections to the parameters of the orbit, reducing the size of the orbit (**Figure 2a**)

*Motion control is carried out by the position of the solar sail, when the vector of light pressure forces determines braking (a) or acceleration (b). In case (a) there will be a decrease in the size of the orbit; case (b) increases the*

outer space.

The system is called stable if it returns to the equilibrium or rest state after the termination of external influences that moved it out of this position. If after the termination of the external impact of the system does not return to a state of equilibrium, it is unsustainable. Stable equilibrium position becomes asymptotically stable with the addition of dissipative forces with complete dissipation.

Determining the motion of any mechanical system is often required to assess the stability and control of motion states. The strict definition of a stable equilibrium position and other solutions of dynamical systems were given in 1892 by the Russian scientist Lyapunov [6–9, 14, 18].

The movement or behavior of the solution of the dynamical system is called Lyapunov stable if small variations in the initial data from the reference phase variables selected for the study, solving a system of differential equations, lead to small deviations in the future. If the deviation over time tends to be zero, the reference solution is called asymptotically stable.

The system is called instable in case when even very small perturbation influence leads to large deviations or change the motion character, including the equilibrium position displacement, which is not stable if the velocity initial value is different from zero.

We also consider the stability of the orbital trajectory or stability of some of the variables [6–12]. In this case, it appears that the phase trajectory and its projection onto the corresponding subspace are close enough to the base path, although the representative points can be arbitrarily disperse, away from each other over time. Periodic solution of the system is not asymptotically stable. But if in such system all multipliers' modules but one are less than one, then according to Andronov-Witt theorem [14], the trajectory of the system periodic solution is asymptotically orbital stable.

Stability with respect to part of variables for partial differential equations involved Rumyantsev [19, 20], who published an article on the analogue of the theory of the second Lyapunov method for the stability problems for some of the variables [21]. He and his students developed methods for research on some of the variables of stability problems.

If the dynamic equations are written in the canonical form, and there are n first integrals, then the Arnold's theorem [12–14], all phase trajectories lie on the n-dimensional torus, and the motion are conditionally periodic system. This set is called the equilibrium or stationary state of motion of the system.

In the field of action of the geopotential excluding other perturbing forces exists stable equilibrium for body position while maintaining the orientation of the major axis of the ellipsoid of inertia in the direction of the center of gravity.

#### **2.4 Control of orientation motion**

In the Cartesian coordinate system for the main body, the known parameters of the axial moments of inertia are taken into account. In the field of the geopotential force without other perturbing forces, there are stable equilibrium positions for the body while maintaining the orientation of the main axes (x, y, z) of the inertia ellipsoid with the main moments towards the center of attraction, taking into account the angular velocity of the orbit.

If we consider only the first linear approximation, then the equation of oscillations of the spacecraft with a small perturbation or deviation from the equilibrium position has the form, which at the next step turns into the equation of harmonic oscillations with additional control functions.

Therefore, small oscillations of the mathematical pendulum will be isochronous. We get the orbital stability of motion or stability with respect to part of the variables. For small vibrations in the vicinity of the equilibrium position, the damping effect of additional gyroscopic devices or motors can be used. The action of disturbances can be compensated by a change in the size or reflective properties of the elements of the sailing ship, as well as their relative position. This creates additional moments that can be used for control and stability.

In the case of possible oscillations of the satellite in the orbit plane while maintaining the orientation of the other major axis orthogonal to this plane, the law of change in kinetic momentum takes into account the effect of the Earth's gravitational field [9, 12, 16, 22]. This leads to the equation.

$$I\_x \ddot{\theta} = M\_x = \mathfrak{Z}a\_0^2 (I\_\mathcal{Y} - I\_x) \sin \theta \cos \theta. \tag{9}$$

**3. Conclusions**

*DOI: http://dx.doi.org/10.5772/intechopen.89452*

control.

body problem.

**215**

the quality functional [16, 17].

angular or rotation the sail geometry.

will be determined by stability conditions.

When a rigid body moves in the Earth's gravitational field, there are location options for stable orientation relative to the orbital system. Consideration of the effect of light pressure on a sailing spacecraft leads to the appearance of other possible provisions or stability conditions that can be used in the process of motion

*Problems of Control Motion of Solar Sail Spacecraft in the Photogravitational Fields*

The perturbations action can be compensated by the varying of size or reflective

In the case where the main forces can be considered the gravitational interaction with the Sun and its light pressure, you can use the model photogravitational central

In the case of movement in orbits near the Earth, the directions of the main acting forces (gravity and light pressure) do not coincide. However, as a first approximation, it can be assumed that the luminous flux determines the force of a constant value, which is directed collinearly to a straight line passing through the two main bodies of the Sun-Earth-spacecraft system in a restricted circular three-

Then amendments to control the orientation by the changing in the angular

Euler–Lagrange libration points where a small disturbing force determine the motion character and stability. Optimal control theory leads to complicated formulations of the problem for the solving of additional equations of mathematical models that can use the Pontryagin maximum principle or Bellman equation [15, 18] for different cases and tasks. There are analytical and numerical methods of research and analysis of the basic properties of the equations that allow to obtain exact or approximate solution of set of the necessary conditions of the extremum of

The particular interest is the case of placing the spacecraft in the vicinity of the

Solar sailing in space is a matter of the future. This will require sophisticated design solutions and space technology [7–12, 23–27]. A special spacecraft's control

Then we take into account corrections for attitude control due to changes of the

Thus, the nonlinear equations of motion will include a permanent disturbance, which can easily be taken into account. Of particular interest is the case of location the spacecraft in the vicinity of the libration points, where small perturbation forces

properties of spacecraft sail's elements, as well as their mutual arrangement. It creates additional torques, which can be used for a control and stability.

field to interplanetary space flights to asteroids or other planets.

position or shape of the sail can be taken into account.

uses solar sail as a motioned forcement of the thruster units.

Let is denote *<sup>ω</sup>*<sup>2</sup> <sup>¼</sup> <sup>3</sup>*ω*<sup>2</sup> <sup>0</sup> *Iy* � *Ix* ð Þ <sup>2</sup>*Iz* �<sup>1</sup> , where *Ix*,*Iy*,*Iz* are moments of inertia and *ω*<sup>0</sup> is the angular velocity motion on orbit and a new variable *φ* ¼ 2*ϑ*. Then we come to the ordinary equations of oscillations with perturbation. If we consider the first linear approximation only, the equation of oscillations of a spacecraft under a small perturbation or deviation from the equilibrium position has a form, which on the next step turns into the equation of harmonic oscillations with additional control function:

$$
\ddot{\varphi} = -\omega^2 \sin \varphi + \mathfrak{u}(\mathfrak{t}, \mathfrak{q}),
\tag{10}
$$

The period in linear approximation depends on the initial data. Therefore, small oscillations of mathematical pendulum

$$
\ddot{\boldsymbol{\rho}} + \boldsymbol{\alpha}^2 \boldsymbol{\rho} = \boldsymbol{\omega}(\mathbf{t}, \boldsymbol{\rho}) \tag{11}
$$

will be isochronous. We get the orbital stability of the movement or the stability with respect to the part of the variables. For the small oscillations in the neighborhood of the equilibrium position, it is possible to use the damping action of additional gyroscopic devices or engines [15, 17]. To damp the oscillations, a control is proposed in the form of piecewise constant functions

$$u(t,\rho) = -u\_{\text{max}} \text{sign}(\rho) \tag{12}$$

of a relay type with a switching period, which is determined by the frequency *ω*. Euler's equation (8) of rotation of a rigid solid about a center of mass show that there are three options for steady motions in the form of stationary rotations about the three principal axes of the ellipsoid of inertia when the two components of the angular velocity are equal to zero and the third is a constant [7, 8].

The Euler equation in the general case [8, 22] determines the rotation of a body under the action of moments of force relative to the center of mass. In the case of the body rotation around the instantaneous axis we need the forces moment about the axis to turn the body or to stop rotation.

*Problems of Control Motion of Solar Sail Spacecraft in the Photogravitational Fields DOI: http://dx.doi.org/10.5772/intechopen.89452*

#### **3. Conclusions**

body while maintaining the orientation of the main axes (x, y, z) of the inertia ellipsoid with the main moments towards the center of attraction, taking into

If we consider only the first linear approximation, then the equation of oscillations of the spacecraft with a small perturbation or deviation from the equilibrium position has the form, which at the next step turns into the equation of harmonic

Therefore, small oscillations of the mathematical pendulum will be isochronous. We get the orbital stability of motion or stability with respect to part of the variables. For small vibrations in the vicinity of the equilibrium position, the damping effect of additional gyroscopic devices or motors can be used. The action of disturbances can be compensated by a change in the size or reflective properties of the elements of the sailing ship, as well as their relative position. This creates additional

<sup>0</sup> *Iy* � *Ix*

*ω*<sup>0</sup> is the angular velocity motion on orbit and a new variable *φ* ¼ 2*ϑ*. Then we come to the ordinary equations of oscillations with perturbation. If we consider the first linear approximation only, the equation of oscillations of a spacecraft under a small perturbation or deviation from the equilibrium position has a form, which on the next step turns into the equation of harmonic oscillations with additional control

The period in linear approximation depends on the initial data. Therefore, small

will be isochronous. We get the orbital stability of the movement or the stability with respect to the part of the variables. For the small oscillations in the neighborhood of the equilibrium position, it is possible to use the damping action of additional gyroscopic devices or engines [15, 17]. To damp the oscillations, a control is

of a relay type with a switching period, which is determined by the frequency *ω*. Euler's equation (8) of rotation of a rigid solid about a center of mass show that there are three options for steady motions in the form of stationary rotations about the three principal axes of the ellipsoid of inertia when the two components of the

The Euler equation in the general case [8, 22] determines the rotation of a body under the action of moments of force relative to the center of mass. In the case of the body rotation around the instantaneous axis we need the forces moment about

*<sup>φ</sup>*€ <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

angular velocity are equal to zero and the third is a constant [7, 8].

*sin ϑ cos* ϑ*:* (9)

*<sup>φ</sup>*€ ¼ �*ω*<sup>2</sup> *sin <sup>φ</sup>* <sup>þ</sup> *u t*ð Þ , *<sup>φ</sup>* , (10)

*u t*ð Þ¼� , *φ umaxsign*ð Þ *φ* (12)

*φ* ¼ *u t*ð Þ , *φ* (11)

, where *Ix*,*Iy*,*Iz* are moments of inertia and

In the case of possible oscillations of the satellite in the orbit plane while maintaining the orientation of the other major axis orthogonal to this plane, the law of change in kinetic momentum takes into account the effect of the Earth's gravita-

account the angular velocity of the orbit.

oscillations with additional control functions.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

moments that can be used for control and stability.

tional field [9, 12, 16, 22]. This leads to the equation.

Let is denote *<sup>ω</sup>*<sup>2</sup> <sup>¼</sup> <sup>3</sup>*ω*<sup>2</sup>

oscillations of mathematical pendulum

proposed in the form of piecewise constant functions

the axis to turn the body or to stop rotation.

function:

**214**

*Izϑ*€ <sup>¼</sup> *Mz* <sup>¼</sup> <sup>3</sup>*ω*<sup>2</sup>

<sup>0</sup> *Iy* � *Ix* ð Þ <sup>2</sup>*Iz* �<sup>1</sup>

When a rigid body moves in the Earth's gravitational field, there are location options for stable orientation relative to the orbital system. Consideration of the effect of light pressure on a sailing spacecraft leads to the appearance of other possible provisions or stability conditions that can be used in the process of motion control.

The perturbations action can be compensated by the varying of size or reflective properties of spacecraft sail's elements, as well as their mutual arrangement. It creates additional torques, which can be used for a control and stability.

In the case where the main forces can be considered the gravitational interaction with the Sun and its light pressure, you can use the model photogravitational central field to interplanetary space flights to asteroids or other planets.

In the case of movement in orbits near the Earth, the directions of the main acting forces (gravity and light pressure) do not coincide. However, as a first approximation, it can be assumed that the luminous flux determines the force of a constant value, which is directed collinearly to a straight line passing through the two main bodies of the Sun-Earth-spacecraft system in a restricted circular threebody problem.

Then amendments to control the orientation by the changing in the angular position or shape of the sail can be taken into account.

The particular interest is the case of placing the spacecraft in the vicinity of the Euler–Lagrange libration points where a small disturbing force determine the motion character and stability. Optimal control theory leads to complicated formulations of the problem for the solving of additional equations of mathematical models that can use the Pontryagin maximum principle or Bellman equation [15, 18] for different cases and tasks. There are analytical and numerical methods of research and analysis of the basic properties of the equations that allow to obtain exact or approximate solution of set of the necessary conditions of the extremum of the quality functional [16, 17].

Solar sailing in space is a matter of the future. This will require sophisticated design solutions and space technology [7–12, 23–27]. A special spacecraft's control uses solar sail as a motioned forcement of the thruster units.

Then we take into account corrections for attitude control due to changes of the angular or rotation the sail geometry.

Thus, the nonlinear equations of motion will include a permanent disturbance, which can easily be taken into account. Of particular interest is the case of location the spacecraft in the vicinity of the libration points, where small perturbation forces will be determined by stability conditions.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

**References**

Russian)

[1] Lebedev PN. Collected Works. Moscow: Nauka; 1963 (in Russian)

with Jet Propulsion Engines. Interplanetary Travels. Moscow: Oborongiz; 1961 (in Russian)

[2] Zander FA. Problems of Spaceflights

*DOI: http://dx.doi.org/10.5772/intechopen.89452*

Dynamical Astronomy. 2003;**86**(1):

[10] Koblik VV, Polyakhova EN, Sokolov LL. Solar sail near the Sun: Point-like and extended models of radiation sources. Advances in Space Research. 2011;**48**(11):1717-1739

[11] Clohessy WH, Wiltshire RS. Terminal guidance system for satellite rendezvous. Journal of the Aerospace

Sciences. 1960;**27**(9):653-674

[12] Korolev VS, Pototskaya IY. Integration of dynamical systems and stability of solution on a part of the variables. Applied Mathematical Sciences. 2015;**9**(15):721-728. DOI: 10.12988/ams.2015.4121004

[13] Korolev VS, Pototskaya IYu. Problems of stability with respect to a part of variables. In: International Conference on Mechanics - Seventh Polyakhov's Reading; 2015. DOI: 10.1109/POLYAKHOV.2015.7106739

[14] Korolev VS. Determining movement

[15] Korolev VS. Problems of spacecraft multi-impulse trajectories modeling, International Conference on Stability and Control Processes in Memory of V. I. Zubov. In: SCP-2015—Proceedings -

[16] Martyusheva A, Oskina K, Petrov N, Polyakhova E. Solar radiation pressure influence in motion of asteroids, including near-earth objects. In:

International Conference on Mechanics - Seventh Polyakhov's Reading. 2015; DOI: 10.1109/POLYAKHOV.2015.7106756

[17] Forward RL. Light-levitated geostationary cylindrical orbits. Journal

of navigation satellites in view of disturbances. Bulletin of the Saint Petersburg State Institute of Technology.

2004;**10**(3):39-46 (in Russian)

7342072; 2015. pp. 91-94

59-80

*Problems of Control Motion of Solar Sail Spacecraft in the Photogravitational Fields*

[3] Polyakhova EN. Space Flight with a Solar Sail: Problems and Perspectives. Moscow: Nauka; 1986 (in Russian)

[4] Polyakhova EN, Koblik VV. Solar Sail – Science Fiction or Space Sailing Reality. Moscow: URSS; 2016 (in

[5] Polyakhova EN, Korolev VS. Control of the solar sail space vehicle. In: International Conference "Stability and

oscillations of nonlinear control systems". Moscow: IPU RAN; 2016.

[6] Polyakhova EN, Korolev VS. Problems of spacecraft control by solar sail. In: International Conference "Stability and Oscillations of Nonlinear

Control Systems" (Pyatnitskiy's

Conference 2016); 2016. DOI: 10.1109/

[7] Polyakhova EN, Vjuga AA, Titov VB. Orbital Space Flight in Problems with Detailed Solutions and in Numbers: A Tutorial. Moscow: URSS; 2016 (in

[8] Kirpichnikov SN, Kirpichnikova ES, Polyakhova EN, Shmyrov AS. Planar heliocentric roto-translatory motion of a spacecraft with a solar sail of complex shape. Celestial Mechanics and Dynamical Astronomy. 1995;**63**(3–4):

[9] Koblik VV, Polyakhova EN, Sokolov LL. Controlled solar sail transfers into near-Sun regions

flybys. Celestial Mechanics and

combined with planetary gravity-assist

pp. 294-297 (in Russian)

STAB.2016.7541214

Russian)

255-269

**217**

### **Author details**

Vladimir Stepanovich Korolev\*, Elena Nikolaevna Polyakhova and Irina Yurievna Pototskaya Saint-Petersburg State University, Russia

\*Address all correspondence to: vokorol@bk.ru

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Problems of Control Motion of Solar Sail Spacecraft in the Photogravitational Fields DOI: http://dx.doi.org/10.5772/intechopen.89452*

#### **References**

[1] Lebedev PN. Collected Works. Moscow: Nauka; 1963 (in Russian)

[2] Zander FA. Problems of Spaceflights with Jet Propulsion Engines. Interplanetary Travels. Moscow: Oborongiz; 1961 (in Russian)

[3] Polyakhova EN. Space Flight with a Solar Sail: Problems and Perspectives. Moscow: Nauka; 1986 (in Russian)

[4] Polyakhova EN, Koblik VV. Solar Sail – Science Fiction or Space Sailing Reality. Moscow: URSS; 2016 (in Russian)

[5] Polyakhova EN, Korolev VS. Control of the solar sail space vehicle. In: International Conference "Stability and oscillations of nonlinear control systems". Moscow: IPU RAN; 2016. pp. 294-297 (in Russian)

[6] Polyakhova EN, Korolev VS. Problems of spacecraft control by solar sail. In: International Conference "Stability and Oscillations of Nonlinear Control Systems" (Pyatnitskiy's Conference 2016); 2016. DOI: 10.1109/ STAB.2016.7541214

[7] Polyakhova EN, Vjuga AA, Titov VB. Orbital Space Flight in Problems with Detailed Solutions and in Numbers: A Tutorial. Moscow: URSS; 2016 (in Russian)

[8] Kirpichnikov SN, Kirpichnikova ES, Polyakhova EN, Shmyrov AS. Planar heliocentric roto-translatory motion of a spacecraft with a solar sail of complex shape. Celestial Mechanics and Dynamical Astronomy. 1995;**63**(3–4): 255-269

[9] Koblik VV, Polyakhova EN, Sokolov LL. Controlled solar sail transfers into near-Sun regions combined with planetary gravity-assist flybys. Celestial Mechanics and

Dynamical Astronomy. 2003;**86**(1): 59-80

[10] Koblik VV, Polyakhova EN, Sokolov LL. Solar sail near the Sun: Point-like and extended models of radiation sources. Advances in Space Research. 2011;**48**(11):1717-1739

[11] Clohessy WH, Wiltshire RS. Terminal guidance system for satellite rendezvous. Journal of the Aerospace Sciences. 1960;**27**(9):653-674

[12] Korolev VS, Pototskaya IY. Integration of dynamical systems and stability of solution on a part of the variables. Applied Mathematical Sciences. 2015;**9**(15):721-728. DOI: 10.12988/ams.2015.4121004

[13] Korolev VS, Pototskaya IYu. Problems of stability with respect to a part of variables. In: International Conference on Mechanics - Seventh Polyakhov's Reading; 2015. DOI: 10.1109/POLYAKHOV.2015.7106739

[14] Korolev VS. Determining movement of navigation satellites in view of disturbances. Bulletin of the Saint Petersburg State Institute of Technology. 2004;**10**(3):39-46 (in Russian)

[15] Korolev VS. Problems of spacecraft multi-impulse trajectories modeling, International Conference on Stability and Control Processes in Memory of V. I. Zubov. In: SCP-2015—Proceedings - 7342072; 2015. pp. 91-94

[16] Martyusheva A, Oskina K, Petrov N, Polyakhova E. Solar radiation pressure influence in motion of asteroids, including near-earth objects. In: International Conference on Mechanics - Seventh Polyakhov's Reading. 2015; DOI: 10.1109/POLYAKHOV.2015.7106756

[17] Forward RL. Light-levitated geostationary cylindrical orbits. Journal

**Author details**

**216**

Irina Yurievna Pototskaya

Saint-Petersburg State University, Russia

provided the original work is properly cited.

\*Address all correspondence to: vokorol@bk.ru

Vladimir Stepanovich Korolev\*, Elena Nikolaevna Polyakhova and

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

of the Astronautical Sciences. 1981; **29**(1):73-80

International Scientific Conference on Mechanic); 2018. С. 040014

**Chapter 12**

**Abstract**

**1. Introduction**

**219**

*Ali I. H. Al-Zughaibi*

Nonlinear Friction Model for

Identification and Effectiveness

To achieve a high level of performance, frictional effects have to be addressed by

considering an accurate friction model, such that the resulting model faithfully simulates all observed types of friction behaviour. A nonlinear friction model is developed based on observed measurement results and dynamic system analysis. The model includes a stiction effect, a linear term (viscous friction), a nonlinear term (Coulomb friction) and an extra component at low velocities (Stribeck effect). During acceleration, the magnitude of the frictional force at just beyond zero velocity decreases due to the Stribeck effect, which means the influence of friction reduces from direct contact with bearings and body into the mixed lubrication mode at low velocity. This could be due to lubricant film behaviour. In respect of acceleration and deceleration when the direction changes for the mass body, friction almost depends on this direction, while the static frictional force exhibits springlike characteristics. However, friction is not determined by current velocity alone, it also depends on the history of the relative wheel and body velocities and

movements, which are responsible for friction hysteresis behaviour.

**Keywords:** nonlinear friction model, stiction region, Stribeck effect,

Friction occurs almost everywhere. Many things, including human acts, depend on it. It is usually present in machines. Usually, friction is not required, so a great deal is done to reduce it by design or by control. Friction is often quantified by a coefficient of friction (μ), expressing the ratio of the friction force to the applied load [1]. The spearheading work of Amontons, Coulomb and Euler, who attempted to clarify the friction phenomenon regarding the mechanics of relative movement of rough surfaces in contact with one another, is mentioned by [2]. From that point forward, only sporadic consideration has been paid to the vital question of friction as a dynamic process that changes on contact. Instead, the most significant proportion of the investigation has concentrated on describing and evaluating complex mechanisms, such as adhesion and deformation that contribute to development of frictional resistance, while frequently ignoring the dynamic aspects of the issue. Consequently, despite those mechanisms being relatively well researched,

characterised and understood, no efficient and comprehensive model has emerged

viscous friction, passive suspension system model

Passive Suspension System

[18] Novoselov VS, Korolev VS. Control of a Hamiltonian system subject to disturbances. Innovations in Science. 2015;**51**:23-29 (in Russian)

[19] Rumyantsev VV. On the Stability of Stationary Motion of Satellites. Moscow: Computing Center of the USSR Academy of Sciences; 1967. 141p

[20] Rumyantsev VV. On the optimal stabilization of controlled systems. Journal of Applied Mathematics and Mechanics. 1970;**34**(3):440-456

[21] Egorov VA, Pomazanov MV. Solar Sail: The Principles of Design. Driving-Set and Flights to Asteroids. Preprints IPM Keldysh. Moscow: IPM; 1997 (in Russian)

[22] Beletsky VV. Motion of an Artificial Satellite about its Center of Mass. Moscow: Nauka; 1965 (in Russian)

[23] Forward RL. Future Magic: How Today's Science Fiction Will Become Tomorrow's Reality. New York: Avon Books; 1988

[24] Friedman L. Starsailing: Solar Sailing and Interstellar Travel. New York: Wiley Science Editions; 1988

[25] Kulakov F, Alferov G, Efimova P. Methods of remote control over space robots. International Conference on Mechanics - Seventh Polyakhov's Reading; 2015. С. 7106742. DOI: 10.1109/POLYAKHOV.2015.7106742

[26] Mcinnes CR. Solar Sailing: Technology, Dynamics and Mission Applications. Berlin, Germany: Springer-Praxis; 1999

[27] Polyakhova EN, Korolev VS. The solar sail: Current state of the problem. In: AIP Conference Proceedings (8th Polyakhov's Reading: Proceedings of the

#### **Chapter 12**

of the Astronautical Sciences. 1981;

2015;**51**:23-29 (in Russian)

Computing Center of the USSR Academy of Sciences; 1967. 141p

[18] Novoselov VS, Korolev VS. Control of a Hamiltonian system subject to disturbances. Innovations in Science.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

International Scientific Conference on

Mechanic); 2018. С. 040014

[19] Rumyantsev VV. On the Stability of Stationary Motion of Satellites. Moscow:

[20] Rumyantsev VV. On the optimal stabilization of controlled systems. Journal of Applied Mathematics and Mechanics. 1970;**34**(3):440-456

[21] Egorov VA, Pomazanov MV. Solar Sail: The Principles of Design. Driving-Set and Flights to Asteroids. Preprints IPM Keldysh. Moscow: IPM; 1997

[22] Beletsky VV. Motion of an Artificial Satellite about its Center of Mass. Moscow: Nauka; 1965 (in Russian)

[23] Forward RL. Future Magic: How Today's Science Fiction Will Become Tomorrow's Reality. New York: Avon

[24] Friedman L. Starsailing: Solar Sailing and Interstellar Travel. New York: Wiley Science Editions; 1988

[25] Kulakov F, Alferov G, Efimova P. Methods of remote control over space robots. International Conference on Mechanics - Seventh Polyakhov's Reading; 2015. С. 7106742. DOI: 10.1109/POLYAKHOV.2015.7106742

[26] Mcinnes CR. Solar Sailing: Technology, Dynamics and Mission Applications. Berlin, Germany:

[27] Polyakhova EN, Korolev VS. The solar sail: Current state of the problem. In: AIP Conference Proceedings (8th Polyakhov's Reading: Proceedings of the

Springer-Praxis; 1999

**218**

**29**(1):73-80

(in Russian)

Books; 1988

## Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness

*Ali I. H. Al-Zughaibi*

#### **Abstract**

To achieve a high level of performance, frictional effects have to be addressed by considering an accurate friction model, such that the resulting model faithfully simulates all observed types of friction behaviour. A nonlinear friction model is developed based on observed measurement results and dynamic system analysis. The model includes a stiction effect, a linear term (viscous friction), a nonlinear term (Coulomb friction) and an extra component at low velocities (Stribeck effect). During acceleration, the magnitude of the frictional force at just beyond zero velocity decreases due to the Stribeck effect, which means the influence of friction reduces from direct contact with bearings and body into the mixed lubrication mode at low velocity. This could be due to lubricant film behaviour. In respect of acceleration and deceleration when the direction changes for the mass body, friction almost depends on this direction, while the static frictional force exhibits springlike characteristics. However, friction is not determined by current velocity alone, it also depends on the history of the relative wheel and body velocities and movements, which are responsible for friction hysteresis behaviour.

**Keywords:** nonlinear friction model, stiction region, Stribeck effect, viscous friction, passive suspension system model

#### **1. Introduction**

Friction occurs almost everywhere. Many things, including human acts, depend on it. It is usually present in machines. Usually, friction is not required, so a great deal is done to reduce it by design or by control. Friction is often quantified by a coefficient of friction (μ), expressing the ratio of the friction force to the applied load [1].

The spearheading work of Amontons, Coulomb and Euler, who attempted to clarify the friction phenomenon regarding the mechanics of relative movement of rough surfaces in contact with one another, is mentioned by [2]. From that point forward, only sporadic consideration has been paid to the vital question of friction as a dynamic process that changes on contact. Instead, the most significant proportion of the investigation has concentrated on describing and evaluating complex mechanisms, such as adhesion and deformation that contribute to development of frictional resistance, while frequently ignoring the dynamic aspects of the issue. Consequently, despite those mechanisms being relatively well researched, characterised and understood, no efficient and comprehensive model has emerged

for the evolution of the friction force as a function of the states of the system, namely, time, displacement and velocity. The requirement for such a model is now becoming more urgent, since the consideration of the friction force dynamics proves essential to understanding and control of systems, including rubbing elements, from machines to earthquakes. Therefore, if it were possible to qualify and quantify this friction force dynamic, it would be a relatively simple step to treat the dynamics of a whole system comprising friction; thus, our results are consistent with their findings.

lines. Therefore, a 240 kg mass plate, used to represent a ¼-car body, is organised to move vertically via two linear bearings. Two rails, THK type HSR 35CA, 1000 mm long and parallel to each other, are used with each linear bearing. A double wishbone suspension linkage was chosen because it preserves the geometry of a wheel in an upright position independent of the suspension type used. The wheel hub is connected to the chassis, which is attached to the car body. The test rig passive suspension photograph is shown in **Figure 1**, while the schematic diagram is shown

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness*

Surawattanawan [6] conducted a simulation and experimental study for the same test rig without consideration of the real position for the spring and viscous damper (S and VD); as a result, the friction effects were ignored. However, in the author's opinion, the real inclined position of S and VD should be considered. Accordingly, the test rig design and the input type help to generate a normal force at the body bearings and a vertical force relative to body movement, as will be demonstrated by the free body diagram of test rig later. This force is responsible for generating Coulomb friction at body lubricant bearings. In addition, the mass body has been slipped on lubricant bearings; this will undoubtedly generate viscous friction. Therefore, it is essential to consider these frictions in the current study, qualified by the critical

effects of friction in any system, as well as their effects on results.

here in **Figure 2**.

*DOI: http://dx.doi.org/10.5772/intechopen.86055*

**Figure 1.**

**221**

*Photograph of the passive test rig.*

Friction is a very complicated phenomenon arising from the contact of surfaces. Experiments indicate a functional dependence upon a large variety of parameters, including sliding speed, acceleration, critical sliding distance, normal load, surface preparation and, of course, material combination. In many engineering applications, the success of models in predicting experimental results remains strongly sensitive to the friction model.

A fundamental, unresolved question in system simulation remains: what is the most appropriate way to include friction in an analytical or numerical model and what are the implications of the chosen friction model?

From a control point of view, control strategies that attempt to compensate for the effects of friction, without resorting to high gain control loops, inherently require a suitable friction model to predict and compensate for the friction. Even though no exact formula for the friction force is available, friction is commonly described in an empirical model. Nevertheless, for precision/accuracy requirement, a good friction model is also necessary to analyse stability, predict limit cycles, find controller gains, perform simulations, etc. Most existing model-based friction compensation schemes use classical friction models, such as Coulomb and viscous friction. In applications with high-precision positioning, the results are not always satisfactory. Friction is a natural phenomenon that is quite difficult to model and is not yet completely understood [3].

From a friction-type point of view, in fluid- or grease-lubricated mechanisms, friction decreases as the velocity increases away from zero. In general terms, this effect is understood. It is due to the transition from boundary lubrication to fluid lubrication. In boundary lubrication, extremely thin, perhaps monomolecular, layers of boundary lubricants that adhere to the metal surfaces separate metal parts. These lubricant additives are chosen to have low shear strength, so as to reduce friction, proper bonding and a variety of other properties such as stability, corrosion resistance or solubility in the bulk lubricant. Boundary lubricants are standard in greases and oils specified for precision machine applications. With the exception of when lubricants and the friction properties of boundary lubricants are a secondary consideration [4], therefore, this study considers transition friction.

This study found that friction helps to remove a vibration, or oscillation, from mass body displacement as the damping contributes in the test rig. That was unexpected because it always caused the system to deteriorate and friction to be incorporated with the primary target of suspension system performance. Therefore, it is vital to consider friction in this study, and this novel contribution takes into account the friction with the test rig and implements a ¼-car suspension model [5]. In addition, the author hopes to contribute towards a reconsideration of friction with conventional car suspension models.

#### **2. Why considering friction within this study?**

In the test rig, a ¼-car, to achieve the primary target of this test rig and the design requirements, the designer had to force the mass body to move in vertical

#### *Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness DOI: http://dx.doi.org/10.5772/intechopen.86055*

lines. Therefore, a 240 kg mass plate, used to represent a ¼-car body, is organised to move vertically via two linear bearings. Two rails, THK type HSR 35CA, 1000 mm long and parallel to each other, are used with each linear bearing. A double wishbone suspension linkage was chosen because it preserves the geometry of a wheel in an upright position independent of the suspension type used. The wheel hub is connected to the chassis, which is attached to the car body. The test rig passive suspension photograph is shown in **Figure 1**, while the schematic diagram is shown here in **Figure 2**.

Surawattanawan [6] conducted a simulation and experimental study for the same test rig without consideration of the real position for the spring and viscous damper (S and VD); as a result, the friction effects were ignored. However, in the author's opinion, the real inclined position of S and VD should be considered. Accordingly, the test rig design and the input type help to generate a normal force at the body bearings and a vertical force relative to body movement, as will be demonstrated by the free body diagram of test rig later. This force is responsible for generating Coulomb friction at body lubricant bearings. In addition, the mass body has been slipped on lubricant bearings; this will undoubtedly generate viscous friction. Therefore, it is essential to consider these frictions in the current study, qualified by the critical effects of friction in any system, as well as their effects on results.

**Figure 1.** *Photograph of the passive test rig.*

for the evolution of the friction force as a function of the states of the system, namely, time, displacement and velocity. The requirement for such a model is now becoming more urgent, since the consideration of the friction force dynamics proves essential to understanding and control of systems, including rubbing elements, from machines to earthquakes. Therefore, if it were possible to qualify and quantify this friction force dynamic, it would be a relatively simple step to treat the dynamics of a whole system comprising friction; thus, our results are consistent

Friction is a very complicated phenomenon arising from the contact of surfaces. Experiments indicate a functional dependence upon a large variety of parameters, including sliding speed, acceleration, critical sliding distance, normal load, surface preparation and, of course, material combination. In many engineering applications, the success of models in predicting experimental results remains strongly

A fundamental, unresolved question in system simulation remains: what is the most appropriate way to include friction in an analytical or numerical model and

From a control point of view, control strategies that attempt to compensate for

From a friction-type point of view, in fluid- or grease-lubricated mechanisms, friction decreases as the velocity increases away from zero. In general terms, this effect is understood. It is due to the transition from boundary lubrication to fluid lubrication. In boundary lubrication, extremely thin, perhaps monomolecular, layers of boundary lubricants that adhere to the metal surfaces separate metal parts. These lubricant additives are chosen to have low shear strength, so as to reduce friction, proper bonding and a variety of other properties such as stability, corrosion resistance or solubility in the bulk lubricant. Boundary lubricants are standard in greases and oils specified for precision machine applications. With the exception of when lubricants and the friction properties of boundary lubricants are a secondary

This study found that friction helps to remove a vibration, or oscillation, from mass body displacement as the damping contributes in the test rig. That was unexpected because it always caused the system to deteriorate and friction to be incorporated with the primary target of suspension system performance. Therefore, it is vital to consider friction in this study, and this novel contribution takes into account the friction with the test rig and implements a ¼-car suspension model [5]. In addition, the author hopes to contribute towards a reconsideration of friction with

In the test rig, a ¼-car, to achieve the primary target of this test rig and the design requirements, the designer had to force the mass body to move in vertical

consideration [4], therefore, this study considers transition friction.

the effects of friction, without resorting to high gain control loops, inherently require a suitable friction model to predict and compensate for the friction. Even though no exact formula for the friction force is available, friction is commonly described in an empirical model. Nevertheless, for precision/accuracy requirement, a good friction model is also necessary to analyse stability, predict limit cycles, find controller gains, perform simulations, etc. Most existing model-based friction compensation schemes use classical friction models, such as Coulomb and viscous friction. In applications with high-precision positioning, the results are not always satisfactory. Friction is a natural phenomenon that is quite difficult to model and is

with their findings.

sensitive to the friction model.

not yet completely understood [3].

conventional car suspension models.

**220**

**2. Why considering friction within this study?**

what are the implications of the chosen friction model?

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

Car body mass, Mb ¼ 240 kg. Wheel unit mass, Mw ¼ 40 kg. Tyre stiffness, kt ¼ 920000 N*=*m*:* Tyre damping rate, bt <sup>¼</sup> 3886 N*=*ms�<sup>1</sup> Suspension stiffness, ks ¼ 28900 N*=*m Suspension damping, bd <sup>¼</sup> 260 N*=*ms�<sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.86055*

following sections.

example, in **Figure 5**.

**Figure 3.**

**223**

*Typical 1 DOF test result [7].*

**4. The dynamic indicator**

mental and simulation results, as shown in **Figure 3**.

the different models and parameters used.

Following completion of the passive suspension experimental work, an attempt to model these experimental tests is by developing a passive suspension model. The simulation was achieved through developing code in C++. An issue arose in that a considerable difference was found between the body displacement observed in experiments and in the simulation results. From this aspect, the idea of considering friction force emerged. There were two clear indicators from observation measurements, which helped to quantify the friction effects; these are discussed in the

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness*

From the simulation results, it was found there are definite fluctuations in body displacement, as generally expected from a quarter-car suspension model, regarding our experience and references. Watton [7] mentioned in his book *Modelling, Monitoring and Diagnostic Techniques for Fluid Power Systems* on pages 182–186, regarding the same test rig, there was an oscillation at the car body in both experi-

It is clearly seen that the body displacement oscillates in the current simulation results, without implementing friction forces, as shown in **Figure 4**. There were differences in the periods of oscillation between **Figures 3** and **4**. This is relative to

There were no such fluctuations in the experimental results, as shown, for

**Figures 4** and **5** display the desired input, without filter, for the road and the responses of the wheel and body for the present simulation and experimental results, respectively. From these figures, it is clearly seen that the wheel displacement follows the road displacement in both experiment and simulation results, while the body travel follows the wheel with a pure delay, which will be shown in

**Figure 2.** *Schematic diagram of the test rig.*

#### **3. Mathematical model of passive suspension system**

Vehicle suspensions are designed to minimise car body acceleration X€b, within the limitation of the suspension displacement Xw � Xb and tyre deflection Xr � Xw. Hence, the vehicle response variables that need to be examined are:


Using Newton's second law, the equation of motion for the mass body passive system of ¼-car model is:

$$\mathbf{M}\_{\mathbf{b}}.\ddot{\mathbf{X}}\_{\mathbf{b}} = \left[\mathbf{k}\_{\mathbf{s}}(\mathbf{X}\_{\mathbf{w}} - \mathbf{X}\_{\mathbf{b}}) + \mathbf{b}\_{\mathbf{d}}(\dot{\mathbf{X}}\_{\mathbf{w}} - \dot{\mathbf{X}}\_{\mathbf{b}})\right] \tag{1}$$

while the dynamic equation of motion for the mass wheel is:

$$\mathbf{M}\_{\mathbf{w}} \ddot{\mathbf{X}}\_{\mathbf{w}} = -\left[\mathbf{k}\_{\mathbf{s}}(\mathbf{X}\_{\mathbf{w}} - \mathbf{X}\_{\mathbf{b}}) + \mathbf{b}\_{\mathbf{d}}(\dot{\mathbf{X}}\_{\mathbf{w}} - \dot{\mathbf{X}}\_{\mathbf{b}})\right] + \mathbf{k}\_{\mathbf{t}}(\mathbf{X}\_{\mathbf{r}} - \mathbf{X}\_{\mathbf{w}}) + \mathbf{b}\_{\mathbf{t}}(\dot{\mathbf{X}}\_{\mathbf{r}} - \dot{\mathbf{X}}\_{\mathbf{w}}) \tag{2}$$

The constant parameters taken from the test rig are as follows:

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness DOI: http://dx.doi.org/10.5772/intechopen.86055*

Car body mass, Mb ¼ 240 kg. Wheel unit mass, Mw ¼ 40 kg. Tyre stiffness, kt ¼ 920000 N*=*m*:* Tyre damping rate, bt <sup>¼</sup> 3886 N*=*ms�<sup>1</sup> Suspension stiffness, ks ¼ 28900 N*=*m Suspension damping, bd <sup>¼</sup> 260 N*=*ms�<sup>1</sup>

Following completion of the passive suspension experimental work, an attempt to model these experimental tests is by developing a passive suspension model. The simulation was achieved through developing code in C++. An issue arose in that a considerable difference was found between the body displacement observed in experiments and in the simulation results. From this aspect, the idea of considering friction force emerged. There were two clear indicators from observation measurements, which helped to quantify the friction effects; these are discussed in the following sections.

#### **4. The dynamic indicator**

From the simulation results, it was found there are definite fluctuations in body displacement, as generally expected from a quarter-car suspension model, regarding our experience and references. Watton [7] mentioned in his book *Modelling, Monitoring and Diagnostic Techniques for Fluid Power Systems* on pages 182–186, regarding the same test rig, there was an oscillation at the car body in both experimental and simulation results, as shown in **Figure 3**.

It is clearly seen that the body displacement oscillates in the current simulation results, without implementing friction forces, as shown in **Figure 4**. There were differences in the periods of oscillation between **Figures 3** and **4**. This is relative to the different models and parameters used.

There were no such fluctuations in the experimental results, as shown, for example, in **Figure 5**.

**Figures 4** and **5** display the desired input, without filter, for the road and the responses of the wheel and body for the present simulation and experimental results, respectively. From these figures, it is clearly seen that the wheel displacement follows the road displacement in both experiment and simulation results, while the body travel follows the wheel with a pure delay, which will be shown in

**Figure 3.** *Typical 1 DOF test result [7].*

**3. Mathematical model of passive suspension system**

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

1. Car body acceleration, X€<sup>b</sup>

**Figure 2.**

*Schematic diagram of the test rig.*

3. Tyre deflection, Xr � Xw

system of ¼-car model is:

**222**

2. Suspension displacement, Xw � Xb

Mw*:*X€<sup>w</sup> ¼ � ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_ <sup>b</sup>

Hence, the vehicle response variables that need to be examined are:

while the dynamic equation of motion for the mass wheel is:

The constant parameters taken from the test rig are as follows:

Vehicle suspensions are designed to minimise car body acceleration X€b, within the limitation of the suspension displacement Xw � Xb and tyre deflection Xr � Xw.

Using Newton's second law, the equation of motion for the mass body passive

(1)

(2)

Mb*:*X€<sup>b</sup> <sup>¼</sup> ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_ <sup>b</sup>

<sup>þ</sup> ktð Þþ Xr � Xw bt <sup>X</sup>\_ <sup>r</sup> � <sup>X</sup>\_ <sup>w</sup>

**Figure 4.** *Simulation results for* Xr*,*Xw and Xb *(m).*

**Figure 5.** *Experimental results for* Xr*,*Xw and Xb *(m).*

more detail in Section 5, and fluctuates in the simulation results. The author named this disparity 'a dynamic friction indicator'. This name was not unique, having been used by several authors before. From this point of view, it could be said that for the experimental work, the friction forces at body lubricant bearings are responsible for eliminating the oscillation from the body travels.

#### **5. The static indicator**

In demonstrating the measured body and wheel movements, a delay is illustrated between them when the wheel rises up or falls; the body similarly travels after pure delay. The early and later stages of the wheel rise and fall, respectively; the results to system input and the body delay are shown in **Figure 6**.

**Figure 6.**

**Figure 7.**

**225**

*Measurements of pure delay of* Xb *from* Xw *at three positions.*

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness*

*DOI: http://dx.doi.org/10.5772/intechopen.86055*

*Experimental results of difference displacements between* Xw and Xb*.*

For more convenience, the experimental data of the relative travel between the wheel and body (Xw � XbÞ was used. These are available from test rig from linear variable differential transformer (LVDT) sensors, and the result is shown in **Figure 7**. The evident noise is attributed to sensor and experimental characteristics. From this figure, it is clearly seen that there is zero difference between Xw and Xb at the start of the test or for a short period, approximately 0.3 s. This is believed to be due to data acquisition delays. The differences gradually increase; while the wheel starts to move up, the differences between Xw and Xb steadily increase until reaching the maximum. During this period the body sticks without movement (Xb ¼ 0*:*0Þ*;* when the resulting force overcomes the static friction, the body will start to move. The relative travel difference between them slowly reduces, approximately 0.5–1.5 s, until reaching zero or near zero at steady state (SS), after 1.5 s.

This observation, which the author named 'static friction indicator', leads to an investigation of the body stiction. It was found that this could be regarded as the effect of static friction force.

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness DOI: http://dx.doi.org/10.5772/intechopen.86055*

**Figure 6.** *Measurements of pure delay of* Xb *from* Xw *at three positions.*

**Figure 7.** *Experimental results of difference displacements between* Xw and Xb*.*

more detail in Section 5, and fluctuates in the simulation results. The author named this disparity 'a dynamic friction indicator'. This name was not unique, having been used by several authors before. From this point of view, it could be said that for the experimental work, the friction forces at body lubricant bearings are responsible for

In demonstrating the measured body and wheel movements, a delay is illustrated between them when the wheel rises up or falls; the body similarly travels after pure delay. The early and later stages of the wheel rise and fall, respectively;

For more convenience, the experimental data of the relative travel between the wheel and body (Xw � XbÞ was used. These are available from test rig from linear variable differential transformer (LVDT) sensors, and the result is shown in **Figure 7**. The evident noise is attributed to sensor and experimental characteristics. From this figure, it is clearly seen that there is zero difference between Xw and Xb at the start of the test or for a short period, approximately 0.3 s. This is believed to be due to data acquisition delays. The differences gradually increase; while the wheel starts to move up, the differences between Xw and Xb steadily increase until reaching the maximum. During this period the body sticks without movement (Xb ¼ 0*:*0Þ*;* when the resulting force overcomes the static friction, the body will start to move. The relative travel difference between them slowly reduces, approximately 0.5–1.5 s, until reaching zero or near zero at steady state (SS), after 1.5 s. This observation, which the author named 'static friction indicator', leads to an investigation of the body stiction. It was found that this could be regarded as the

the results to system input and the body delay are shown in **Figure 6**.

eliminating the oscillation from the body travels.

**5. The static indicator**

**Figure 4.**

**Figure 5.**

*Simulation results for* Xr*,*Xw and Xb *(m).*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*Experimental results for* Xr*,*Xw and Xb *(m).*

effect of static friction force.

**224**

To include knowledge about friction in the simulation model, consideration of conventional friction was pursued, drawing upon published information. The following section reviews the approach.

#### **6. Conventional friction model**

The traditional friction model considered the construction of a more comprehensive friction prediction model that accounts for the various aspects of this particular phenomenon so that mechanical systems with friction can be more accurately identified and, consequently, better controlled. Most of the existing modelbased friction compensation schemes use classical friction models, such as Coulomb and viscous friction. In applications with high-precision positioning and with little velocity tracking, the results are not always satisfactory. A better description of the friction phenomena for small speeds, especially when crossing zero velocity, is necessary. Friction is a natural phenomenon that is quite difficult to model and is usually modelled as a discontinuous static map between velocity and friction torque that depends on the velocity's sign. Typical examples are different combinations of Coulomb friction, viscous friction and Stribeck effect, as mentioned in [3, 8, 9]. However, there are several exciting properties observed in systems with friction that cannot be explained by static models. This is necessarily due to the fact that friction does not have an instantaneous response to a change in velocity, i.e. it has internal dynamics. Examples of these dynamic properties [3, 10] are:


The general description of friction is a kind of relation between velocity and friction force, depending on the velocity situations, described in several types of research. For example, Tustin's model consists of Coulomb and viscous friction [11]. The inclusion of the Stribeck effect, with one or more breakpoints, gives a better approximation at low velocities, as shown in **Figure 8**.

Now, in order to start establishing the real bearing friction model, it should involve the dynamic analysis of the test rig as follows:

#### **6.1 How to account for the vertical force**

The following explains in detail the main features of the friction model and will begin with how to account for the vertical force that is responsible for generating Coulomb friction by drawing a free body diagram of the test rig.

#### *6.1.1 Free body diagram of the test rig*

**Figure 9** shows the free body diagram of the test rig; the friction force acts as an internal force in the tangential direction of the contacting surfaces. This force obeys a constitutive equation, such as Coulomb's law, and acts in a direction opposite to the relative velocity. Therefore, the inclination position of S and VD and the system type inputs help to generate the kinematic bearings body friction relative to this normal force component. From **Figure 9**, the following analysis should be

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness*

*<sup>=</sup>* sin ð Þ <sup>θ</sup><sup>∓</sup> *<sup>Δ</sup><sup>θ</sup>* (3)

<sup>F</sup> <sup>¼</sup> ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_ <sup>b</sup>

conducted to account for this friction force:

**Figure 8.**

**Figure 9.**

**227**

*Free body diagram of the test rig.*

*Conventional friction model [11].*

*DOI: http://dx.doi.org/10.5772/intechopen.86055*

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness DOI: http://dx.doi.org/10.5772/intechopen.86055*

#### **Figure 8.**

To include knowledge about friction in the simulation model, consideration of conventional friction was pursued, drawing upon published information. The fol-

The traditional friction model considered the construction of a more comprehensive friction prediction model that accounts for the various aspects of this particular phenomenon so that mechanical systems with friction can be more accurately identified and, consequently, better controlled. Most of the existing modelbased friction compensation schemes use classical friction models, such as Coulomb and viscous friction. In applications with high-precision positioning and with little velocity tracking, the results are not always satisfactory. A better description of the friction phenomena for small speeds, especially when crossing zero velocity, is necessary. Friction is a natural phenomenon that is quite difficult to model and is usually modelled as a discontinuous static map between velocity and friction torque that depends on the velocity's sign. Typical examples are different combinations of Coulomb friction, viscous friction and Stribeck effect, as mentioned in [3, 8, 9]. However, there are several exciting properties observed in systems with friction that cannot be explained by static models. This is necessarily due to the fact that friction does not have an instantaneous response to a change in velocity, i.e. it has

internal dynamics. Examples of these dynamic properties [3, 10] are:

the applied force is less than the static friction breakaway force

• Stick-slip motion, which consists of limit cycle oscillation at low velocities, caused by the fact that friction is more significant at rest than during motion

• Presiding displacement which shows that friction behaves like a spring when

• Frictional lag which means that there is some hysteresis in the relationship

The general description of friction is a kind of relation between velocity and friction force, depending on the velocity situations, described in several types of research. For example, Tustin's model consists of Coulomb and viscous friction [11]. The inclusion of the Stribeck effect, with one or more breakpoints, gives a better

Now, in order to start establishing the real bearing friction model, it should

The following explains in detail the main features of the friction model and will begin with how to account for the vertical force that is responsible for generating

**Figure 9** shows the free body diagram of the test rig; the friction force acts as an internal force in the tangential direction of the contacting surfaces. This force obeys a constitutive equation, such as Coulomb's law, and acts in a direction opposite to the relative velocity. Therefore, the inclination position of S and VD and the system

lowing section reviews the approach.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

**6. Conventional friction model**

between friction and velocity

approximation at low velocities, as shown in **Figure 8**.

involve the dynamic analysis of the test rig as follows:

Coulomb friction by drawing a free body diagram of the test rig.

**6.1 How to account for the vertical force**

*6.1.1 Free body diagram of the test rig*

**226**

*Conventional friction model [11].*

#### **Figure 9.** *Free body diagram of the test rig.*

type inputs help to generate the kinematic bearings body friction relative to this normal force component. From **Figure 9**, the following analysis should be conducted to account for this friction force:

$$\mathbf{F} = \mathbf{k}\_{\mathbf{s}} (\mathbf{X}\_{\mathbf{w}} - \mathbf{X}\_{\mathbf{b}}) + \mathbf{b}\_{\mathbf{d}} (\dot{\mathbf{X}}\_{\mathbf{w}} - \dot{\mathbf{X}}\_{\mathbf{b}}) / \sin \left( \theta \mp \Delta \theta \right) \tag{3}$$

**Figure 10.** *Engineering geometry of passive units.*

$$\mathbf{F\_{nb}} = \mathbf{F} \cos(\theta \mp \Delta \theta) \tag{4}$$

$$\mathbf{F\_{nb}} = \mathbf{k\_s}(\mathbf{X\_w} - \mathbf{X\_b}) + \mathbf{b\_d}(\dot{\mathbf{X\_w}} - \dot{\mathbf{X\_b}}) / \tan\left(\theta \mp \Delta\theta\right) \tag{5}$$

$$\mathbf{F}\_{\text{fricC}} = \mu \mathbf{F}\_{\text{nb}} \tag{6}$$

A nonlinear friction model is developed based on observed measurement results and dynamic system analysis. The model includes a stiction effect, a linear term (viscous friction), a nonlinear term (Coulomb friction) and an extra component at low velocities (Stribeck effect). During acceleration, the magnitude of the frictional force at just beyond zero velocity decreases due to the Stribeck effect, which means the influence of friction reduces from direct contact with bearings and body into the mixed lubrication mode at low velocity. This could be due to lubricant film

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness*

*DOI: http://dx.doi.org/10.5772/intechopen.86055*

In respect of acceleration and deceleration when the direction changes for the mass body, friction almost depends on this direction, while the static frictional force exhibits springlike characteristics. However, friction is not determined by current velocity alone, it also depends on the history of the relative wheel and body velocities and movements, which are responsible for friction hysteresis behaviour. This model, which has now become well established, has provided a more satisfactory explanation of observed dynamic fluctuations of body mass. It will be attempted to heuristically 'fit' a dynamic model to experimentally observed results. The resulting model is not only reasonably valid for the ¼-car test rig behaviour but

The model simulates the symmetric hysteresis loops observed in the bearings'

body undergoing small amplitude ramp and step forcing inputs. As might be expected, they are capable of reproducing the more sophisticated pre-sliding behaviour in particular hysteresis. The influence of hysteresis phenomena on the dynamic response of machine elements with moving parts is not yet thoroughly examined in the literature. In other fields of engineering, where hysteretic phenomena manifest themselves, more research has been conducted. In Ref. [12], for example, adaptive modelling techniques were proposed for dynamic systems with hysteretic elements. The methods are general, but no insight into the influence of the hysteresis on the dynamics is given. Furthermore, no experimental verification is provided. Altpeter [13] made a simplified analysis of the dynamic behaviour of

the moving parts of a machine tool where hysteretic friction was present.

of relative travels of wheel and body contributes to friction hysteresis.

In this study, the friction model, despite its simplicity, can simulate all experimentally observed properties and facets of low-velocity friction force dynamics. Because of the test rig schematic and the system input signal, with historic travel, there are three circumstances depending on whether the body velocity is accelerating or decelerating. Firstly, the velocity values start from zero and just after velocity reversals, reach the highest and are restored to zero, or close to zero at SS. Secondly, the velocity starts from SS with a sharper increase than in the first stage and will extend to peak before returning to zero or near to zero at second SS. Thirdly, it starts from the second SS and after velocity reversals will touch a maximum value, twice rather than at case two, and go back down at a third SS. In the all these velocity cases, the velocity behaviours will make friction hysteretic loops, possibly because of increases in body velocity differing from decreases. The historical action

In general, this friction model considers the static, stiction region and dynamic friction, which consists of the Stribeck effect, viscous friction and Coulomb friction. The mathematical model and summary for each part will be demonstrated in the

The mathematical expression for establishing the friction model gave the constituent terms described in order to accurately represent the observed phenomena,

is also reasonably suitable for most general friction lubricant cases.

behaviour.

next step.

**229**

**7.1 Mathematical friction model**

as shown in Eq. (12).

where FfricC is Coulomb friction, μ is the friction coefficient, Fnb is the body normal force component and F is the spring and damping forces.

#### *6.1.2 Dynamic linkage angle expression*

The construction linkage angle is dynamically changed by ∓*Δθ*.

From engineering geometry of passive units, as shown in **Figure 10**, it can be found that

$$\frac{\mathbf{L\_d} - \Delta \mathbf{L\_d}}{\sin \left( 90 - \theta \right)} = \frac{\mathbf{X\_w} - \mathbf{X\_b}}{\sin \left( \Delta \theta \right)}, \theta = 45^\circ \tag{7}$$

$$\sin\left(\theta\right) = \frac{\Delta\mathcal{L}\_{\rm d}}{\mathcal{X}\_{\rm w} - \mathcal{X}\_{\rm b}}\tag{8}$$

*Δ*Ld ¼ ð Þ Xw � Xb sin ð Þθ , where *Δ*Ld is the dynamic change in S and VD length. Then,

$$\frac{\mathbf{L\_d} - (\mathbf{X\_w} - \mathbf{X\_b})\sin\left(\theta\right)}{\sin\left(\theta\right)} = \frac{\mathbf{X\_w} - \mathbf{X\_b}}{\sin\left(\Delta\theta\right)}\tag{9}$$

$$\sin \Delta \theta = \frac{(\mathbf{X\_w} - \mathbf{X\_b}) \sin \left(\theta\right)}{\mathbf{L\_d} - (\mathbf{X\_w} - \mathbf{X\_b}) \sin \left(\theta\right)} \tag{10}$$

$$
\Delta\theta = \sin^{-1}\left[\frac{(\mathbf{X}\_{\text{w}} - \mathbf{X}\_{\text{b}})\sin\left(\theta\right)}{\mathbf{L}\_{\text{d}} - (\mathbf{X}\_{\text{w}} - \mathbf{X}\_{\text{b}})\sin\left(\theta\right)}\right] \tag{11}
$$

#### **7. Nonlinear friction model**

To achieve a high level of performance, frictional effects have to be addressed by considering an accurate friction model, such that the resulting model faithfully simulates all observed types of friction behaviour.

#### *Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness DOI: http://dx.doi.org/10.5772/intechopen.86055*

A nonlinear friction model is developed based on observed measurement results and dynamic system analysis. The model includes a stiction effect, a linear term (viscous friction), a nonlinear term (Coulomb friction) and an extra component at low velocities (Stribeck effect). During acceleration, the magnitude of the frictional force at just beyond zero velocity decreases due to the Stribeck effect, which means the influence of friction reduces from direct contact with bearings and body into the mixed lubrication mode at low velocity. This could be due to lubricant film behaviour.

In respect of acceleration and deceleration when the direction changes for the mass body, friction almost depends on this direction, while the static frictional force exhibits springlike characteristics. However, friction is not determined by current velocity alone, it also depends on the history of the relative wheel and body velocities and movements, which are responsible for friction hysteresis behaviour.

This model, which has now become well established, has provided a more satisfactory explanation of observed dynamic fluctuations of body mass. It will be attempted to heuristically 'fit' a dynamic model to experimentally observed results. The resulting model is not only reasonably valid for the ¼-car test rig behaviour but is also reasonably suitable for most general friction lubricant cases.

The model simulates the symmetric hysteresis loops observed in the bearings' body undergoing small amplitude ramp and step forcing inputs. As might be expected, they are capable of reproducing the more sophisticated pre-sliding behaviour in particular hysteresis. The influence of hysteresis phenomena on the dynamic response of machine elements with moving parts is not yet thoroughly examined in the literature. In other fields of engineering, where hysteretic phenomena manifest themselves, more research has been conducted. In Ref. [12], for example, adaptive modelling techniques were proposed for dynamic systems with hysteretic elements. The methods are general, but no insight into the influence of the hysteresis on the dynamics is given. Furthermore, no experimental verification is provided. Altpeter [13] made a simplified analysis of the dynamic behaviour of the moving parts of a machine tool where hysteretic friction was present.

In this study, the friction model, despite its simplicity, can simulate all experimentally observed properties and facets of low-velocity friction force dynamics. Because of the test rig schematic and the system input signal, with historic travel, there are three circumstances depending on whether the body velocity is accelerating or decelerating. Firstly, the velocity values start from zero and just after velocity reversals, reach the highest and are restored to zero, or close to zero at SS. Secondly, the velocity starts from SS with a sharper increase than in the first stage and will extend to peak before returning to zero or near to zero at second SS. Thirdly, it starts from the second SS and after velocity reversals will touch a maximum value, twice rather than at case two, and go back down at a third SS. In the all these velocity cases, the velocity behaviours will make friction hysteretic loops, possibly because of increases in body velocity differing from decreases. The historical action of relative travels of wheel and body contributes to friction hysteresis.

In general, this friction model considers the static, stiction region and dynamic friction, which consists of the Stribeck effect, viscous friction and Coulomb friction. The mathematical model and summary for each part will be demonstrated in the next step.

#### **7.1 Mathematical friction model**

The mathematical expression for establishing the friction model gave the constituent terms described in order to accurately represent the observed phenomena, as shown in Eq. (12).

Fnb ¼ Fcosð Þ θ∓*Δθ* (4)

FfricC ¼ μFnb (6)

*<sup>=</sup>* tan ð Þ <sup>θ</sup>∓*Δ<sup>θ</sup>* (5)

sin ð Þ *<sup>Δ</sup><sup>θ</sup> ,* <sup>θ</sup> <sup>¼</sup> <sup>45</sup>° (7)

Ld � ð Þ Xw � Xb sin ð Þ<sup>θ</sup> (10)

sin ð Þ *<sup>Δ</sup><sup>θ</sup>* (9)

(8)

(11)

Fnb <sup>¼</sup> ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_ <sup>b</sup>

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

normal force component and F is the spring and damping forces.

Ld � *Δ*Ld

The construction linkage angle is dynamically changed by ∓*Δθ*.

sin 90 ð Þ � <sup>θ</sup> <sup>¼</sup> Xw � Xb

sin ð Þ¼ θ

Ld � ð Þ Xw � Xb sin ð Þθ

*6.1.2 Dynamic linkage angle expression*

*Engineering geometry of passive units.*

**7. Nonlinear friction model**

simulates all observed types of friction behaviour.

found that

**Figure 10.**

Then,

**228**

where FfricC is Coulomb friction, μ is the friction coefficient, Fnb is the body

From engineering geometry of passive units, as shown in **Figure 10**, it can be

*Δ*Ld Xw � Xb

sin ð Þ<sup>θ</sup> <sup>¼</sup> Xw � Xb

Ld � ð Þ Xw � Xb sin ð Þθ 

*Δ*Ld ¼ ð Þ Xw � Xb sin ð Þθ , where *Δ*Ld is the dynamic change in S and VD length.

sin*Δ<sup>θ</sup>* <sup>¼</sup> ð Þ Xw � Xb sin ð Þ<sup>θ</sup>

*<sup>Δ</sup><sup>θ</sup>* <sup>¼</sup> sin �<sup>1</sup> ð Þ Xw � Xb sin ð Þ<sup>θ</sup>

considering an accurate friction model, such that the resulting model faithfully

To achieve a high level of performance, frictional effects have to be addressed by

$$\mathbf{F}\_{\text{fric}} = \begin{cases} \mathbf{k}\_{\text{s}} (\mathbf{X}\_{\text{w}} - \mathbf{X}\_{\text{b}}) + \mathbf{b}\_{\text{d}} (\dot{\mathbf{X}}\_{\text{w}} - \dot{\mathbf{X}}\_{\text{b}}) \, \dot{\mathbf{X}}\_{\text{b}} = \mathbf{0}.0 \\\\ \mathbf{C}\_{\text{e}} \mathbf{e} \big( |\mathbf{x}\_{\text{b}}| \, \_{\text{e}} \mathbf{1} \big) + \left[ \frac{\mu \big( \mathbf{k}\_{\text{s}} (\mathbf{X}\_{\text{w}} - \mathbf{X}\_{\text{b}}) + \mathbf{b}\_{\text{d}} (\dot{\mathbf{X}}\_{\text{w}} - \dot{\mathbf{X}}\_{\text{b}}) \big)}{\tan \left( \theta \mp \Delta \theta \right)} \right] + \sigma\_{\text{v}} \dot{\mathbf{X}}\_{\text{b}} \, \dot{\mathbf{X}}\_{\text{b}} > \mathbf{0}.0 \\\\ \quad \quad \quad - \mathbf{C}\_{\text{e}} \mathbf{e} \big( |\dot{\mathbf{X}}\_{\text{b}}| \, \_{\text{e}} \mathbf{1} \big) + \left[ \frac{\mu \big( \mathbf{k}\_{\text{s}} (\mathbf{X}\_{\text{w}} - \mathbf{X}\_{\text{b}}) + \mathbf{b}\_{\text{d}} (\dot{\mathbf{X}}\_{\text{w}} - \dot{\mathbf{X}}\_{\text{b}}) \big)}{\tan \left( \theta \mp \Delta \theta \right)} \right] + \sigma\_{\text{v}} \dot{\mathbf{X}}\_{\text{b}} \, \dot{\mathbf{X}}\_{\text{b}} < 0.0 \end{cases} \tag{12}$$

**7.3 Dynamic friction model**

*DOI: http://dx.doi.org/10.5772/intechopen.86055*

Earlier studies (see, e.g. [8, 10, 15]) have shown that a friction model involving dynamics is necessary to describe the friction phenomena accurately. A dynamic model describing the springlike behaviour during stiction was proposed by [16]. The Dahl model is essentially Coulomb friction with a lag in the change of friction force when the direction of motion is changed. The model has many commendable features and is theoretically well understood. Questions, such as the existence and uniqueness of solutions and hysteresis effects, were studied in an interesting paper by [17]. The Dahl model does not include the Stribeck effect. An attempt to incorporate this into the Dahl model was made by [18] where the authors introduced a second-order Dahl model using linear space-invariant descriptions. The Stribeck effect in this model is only transient; however, following a velocity reversal, it is not present in the steady-state friction characteristics. The Dahl model has been used for adaptive friction compensation [19, 20], with improved performance as a result. There are also other models for dynamic friction; Armstrong-Helouvry [8] proposed a seven-parameter model. This model does not combine the different friction phenomena but is, in fact, one model for stiction and another for sliding friction. Another dynamic model suggested by [21] had been used in connection with

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness*

In this study, it was proposed that a nonlinear dynamic friction model combines the transition behaviour from stiction to the slide regime including the Stribeck effect, the Coulomb friction with consideration of the normal dynamic force at body bearings with suitable friction coefficient and the viscous friction dependent on the body velocity and appropriate viscous coefficient. This model involves arbitrary steadystate friction characteristics. The most crucial results of this model are to highlight precisely the hysteresis behaviours of friction relative to body velocity behaviour. Referring to Eq. (12), there are two forms of dynamic friction, depending on the

> � � � � tan ð Þ θ ∓*Δθ* " #

( )

From Eq. (16), it is clearly seen that dynamic friction consists of three parts. A

where FfricT is transition friction, Ce is attracting parameter, e1 is the curvature degree and the absolute body velocity value meaning the direction of velocity is not affected. The transition friction has exponential behaviour with degrees identified experimentally and completely agrees with the literature review of most research studies regarding lubricant friction, which begins from the maximum value at the sticky region and quickly dips when the body just begins to move, or the body velocity is increased. Secondly, FfricC represents Coulomb friction, which is equal to the normal

FfricC <sup>¼</sup> <sup>μ</sup> ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_

where FfricC is Coulomb friction with the opposite sign to velocity direction.

� � � � tan ð Þ θ∓ *Δθ* ( )

b

FfricT <sup>¼</sup> Cee <sup>X</sup>\_ ð Þ j j<sup>b</sup> *<sup>=</sup>*e1 (17)

b

<sup>þ</sup> *<sup>σ</sup>v*X\_ <sup>b</sup>

(16)

(18)

control by [15]. This model is not defined at zero velocity.

body velocity direction; it will be shown in detail as follows:

FfricD <sup>¼</sup> Cee <sup>X</sup>\_ ð Þ j j<sup>b</sup> *<sup>=</sup>*e1 <sup>þ</sup> <sup>μ</sup> ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_

bearing force times the friction coefficient (μ), as follows:

For X\_ <sup>b</sup> >0*:*0 the dynamic friction form is

summary is given for each: part one form is

**231**

Eq. (12) shows the friction model, which includes the two main parts of friction: static when, X\_ <sup>b</sup> <sup>¼</sup> <sup>0</sup>*:*0*,* and dynamic, when <sup>X</sup>\_ <sup>b</sup> >0*:*0. The latter is presented by two expressions, depending on the velocity direction, and is discussed in detail later. In static friction, the stiction area is solely dependent on the velocity because the body velocity should be close to zero velocity or frequently just beyond zero velocity. The static model is accounted by the force balance of the test rig when the body was motionless, while the wheel was moved and describes the static friction sufficiently accurately. However, a dynamic model is necessary which introduces an extra state which can be regarded as transition and Coulomb and viscous friction. In addition to these friction models, steady physics state is also briefly discussed in this study.

#### **7.2 Static friction model**

After a test starts, the wheel begins to move respective to the road inputs, and initially the body remains motionless. This results from the static bearing friction and is undoubtedly a stick region body, Xb ¼ 0*:*0*:* This friction component can be considered via the test rig vertical force balance ∑Fv ¼ 0*:*0*:*

For the test rig, the following conventional model represents a ¼-car without considering body friction as aforementioned by Eq. (1), the first reported implementation of friction forces within Newton's second law for a ¼-car model [14], which leads to a new dynamic equation of motion for the mass body:

$$\mathbf{M}\_{\rm b} \ddot{\mathbf{X}}\_{\rm b} = \left[ \mathbf{k}\_{\rm s} (\mathbf{X}\_{\rm w} - \mathbf{X}\_{\rm b}) + \mathbf{b}\_{\rm d} (\dot{\mathbf{X}}\_{\rm w} - \dot{\mathbf{X}}\_{\rm b}) \right] - \mathbf{F}\_{\rm fric} \tag{13}$$

As described in the short period where the body remains motionless Xb <sup>¼</sup> <sup>0</sup>*:*0 and <sup>X</sup>€<sup>b</sup> <sup>¼</sup> <sup>0</sup>*:*0, Eq. (13) becomes

$$\mathbf{0.0} = \left[\mathbf{k}\_{\mathbf{s}}(\mathbf{X}\_{\mathbf{w}} - \mathbf{X}\_{\mathbf{b}}) + \mathbf{b}\_{\mathbf{d}}(\dot{\mathbf{X}}\_{\mathbf{w}} - \dot{\mathbf{X}}\_{\mathbf{b}})\right] - \mathbf{F}\_{\text{fricS}} \tag{14}$$

then

$$\mathbf{F\_{fricS}} = \left[\mathbf{k\_s(X\_w - X\_b)} + \mathbf{b\_d(\dot{X}\_w - \dot{X}\_b)}\right] \tag{15}$$

where FfricS is the static friction, which is a function of the relative displacements and relative velocities between the wheel and body multiplied by spring stiffness and viscous damper coefficients, with direction totally dependent on the next stage X\_ <sup>b</sup> direction. This is considered as pre-sliding displacement, which exhibits how friction characteristics behave like a spring when the applied force is less than the static friction breakaway force. From the experimental work, amplitude input = 50 mm, it was found that the maximum stick friction force occasionally occurs at Xð Þ <sup>w</sup> � Xb ≤0*:*0069 and Xb ffi 0*:*0.

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness DOI: http://dx.doi.org/10.5772/intechopen.86055*

#### **7.3 Dynamic friction model**

Ffric ¼

static when, X\_

this study.

then

next stage X\_

**230**

**7.2 Static friction model**

8

>>>>>>>>>><

>>>>>>>>>>:

ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

b � � X\_

Cee <sup>X</sup>\_ ð Þ j j<sup>b</sup> *<sup>=</sup>*e1 <sup>þ</sup> <sup>μ</sup> ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_

�Cee <sup>X</sup>\_ ð Þ j j<sup>b</sup> *<sup>=</sup>*e1 <sup>þ</sup> <sup>μ</sup> ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_

<sup>b</sup> <sup>¼</sup> <sup>0</sup>*:*0*,* and dynamic, when <sup>X</sup>\_

considered via the test rig vertical force balance ∑Fv ¼ 0*:*0*:*

Xb <sup>¼</sup> <sup>0</sup>*:*0 and <sup>X</sup>€<sup>b</sup> <sup>¼</sup> <sup>0</sup>*:*0, Eq. (13) becomes

ally occurs at Xð Þ <sup>w</sup> � Xb ≤0*:*0069 and Xb ffi 0*:*0.

which leads to a new dynamic equation of motion for the mass body:

Mb*:*X€<sup>b</sup> <sup>¼</sup> ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_ <sup>b</sup>

As described in the short period where the body remains motionless

<sup>0</sup>*:*<sup>0</sup> <sup>¼</sup> ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_ <sup>b</sup>

FfricS <sup>¼</sup> ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_ <sup>b</sup>

where FfricS is the static friction, which is a function of the relative displacements and relative velocities between the wheel and body multiplied by spring stiffness and viscous damper coefficients, with direction totally dependent on the

exhibits how friction characteristics behave like a spring when the applied force is less than the static friction breakaway force. From the experimental work, amplitude input = 50 mm, it was found that the maximum stick friction force occasion-

<sup>b</sup> direction. This is considered as pre-sliding displacement, which

<sup>b</sup> ¼ 0*:*0

� � � � tan ð Þ θ∓ *Δθ* " #

� � � � tan ð Þ θ ∓*Δθ* " #

Eq. (12) shows the friction model, which includes the two main parts of friction:

After a test starts, the wheel begins to move respective to the road inputs, and initially the body remains motionless. This results from the static bearing friction and is undoubtedly a stick region body, Xb ¼ 0*:*0*:* This friction component can be

For the test rig, the following conventional model represents a ¼-car without considering body friction as aforementioned by Eq. (1), the first reported implementation of friction forces within Newton's second law for a ¼-car model [14],

� � � � � Ffric (13)

� � � � � FfricS (14)

� � � � (15)

two expressions, depending on the velocity direction, and is discussed in detail later. In static friction, the stiction area is solely dependent on the velocity because the body velocity should be close to zero velocity or frequently just beyond zero velocity. The static model is accounted by the force balance of the test rig when the body was motionless, while the wheel was moved and describes the static friction sufficiently accurately. However, a dynamic model is necessary which introduces an extra state which can be regarded as transition and Coulomb and viscous friction. In addition to these friction models, steady physics state is also briefly discussed in

b

b

<sup>þ</sup> *<sup>σ</sup>v*X\_ <sup>b</sup> <sup>X</sup>\_

<sup>þ</sup> *<sup>σ</sup>v*X\_ <sup>b</sup> <sup>X</sup>\_

<sup>b</sup> >0*:*0. The latter is presented by

<sup>b</sup> > 0*:*0

<sup>b</sup> <0*:*0

(12)

Earlier studies (see, e.g. [8, 10, 15]) have shown that a friction model involving dynamics is necessary to describe the friction phenomena accurately. A dynamic model describing the springlike behaviour during stiction was proposed by [16]. The Dahl model is essentially Coulomb friction with a lag in the change of friction force when the direction of motion is changed. The model has many commendable features and is theoretically well understood. Questions, such as the existence and uniqueness of solutions and hysteresis effects, were studied in an interesting paper by [17]. The Dahl model does not include the Stribeck effect. An attempt to incorporate this into the Dahl model was made by [18] where the authors introduced a second-order Dahl model using linear space-invariant descriptions. The Stribeck effect in this model is only transient; however, following a velocity reversal, it is not present in the steady-state friction characteristics. The Dahl model has been used for adaptive friction compensation [19, 20], with improved performance as a result. There are also other models for dynamic friction; Armstrong-Helouvry [8] proposed a seven-parameter model. This model does not combine the different friction phenomena but is, in fact, one model for stiction and another for sliding friction. Another dynamic model suggested by [21] had been used in connection with control by [15]. This model is not defined at zero velocity.

In this study, it was proposed that a nonlinear dynamic friction model combines the transition behaviour from stiction to the slide regime including the Stribeck effect, the Coulomb friction with consideration of the normal dynamic force at body bearings with suitable friction coefficient and the viscous friction dependent on the body velocity and appropriate viscous coefficient. This model involves arbitrary steadystate friction characteristics. The most crucial results of this model are to highlight precisely the hysteresis behaviours of friction relative to body velocity behaviour.

Referring to Eq. (12), there are two forms of dynamic friction, depending on the body velocity direction; it will be shown in detail as follows:

For X\_ <sup>b</sup> >0*:*0 the dynamic friction form is

$$\mathbf{F}\_{\text{fricD}} = \left\{ \mathbf{C}\_{\mathbf{e}} \mathbf{e}^{\left( \left| \dot{\mathbf{X}}\_{\text{b}} \right| / \epsilon \mathbf{1} \right)} + \left[ \frac{\mu \left( \mathbf{k}\_{\text{s}} (\mathbf{X}\_{\text{w}} - \mathbf{X}\_{\text{b}}) + \mathbf{b}\_{\text{d}} (\dot{\mathbf{X}}\_{\text{w}} - \dot{\mathbf{X}}\_{\text{b}}) \right)}{\tan \left( \theta \mp \Delta \theta \right)} \right] + \sigma\_{\text{v}} \dot{\mathbf{X}}\_{\text{b}} \right\} \tag{16}$$

From Eq. (16), it is clearly seen that dynamic friction consists of three parts. A summary is given for each: part one form is

$$\mathbf{F\_{fric}} = \mathbf{C\_e} \mathbf{e}^{\left( \left| \mathbf{\tilde{x}\_b} \right| / \epsilon \mathbf{1} \right)} \tag{17}$$

where FfricT is transition friction, Ce is attracting parameter, e1 is the curvature degree and the absolute body velocity value meaning the direction of velocity is not affected. The transition friction has exponential behaviour with degrees identified experimentally and completely agrees with the literature review of most research studies regarding lubricant friction, which begins from the maximum value at the sticky region and quickly dips when the body just begins to move, or the body velocity is increased.

Secondly, FfricC represents Coulomb friction, which is equal to the normal bearing force times the friction coefficient (μ), as follows:

$$\mathbf{F}\_{\text{fricC}} = \left\{ \frac{\mu \left( \mathbf{k}\_s (\mathbf{X}\_\mathbf{w} - \mathbf{X}\_\mathbf{b}) + \mathbf{b}\_d (\dot{\mathbf{X}}\_\mathbf{w} - \dot{\mathbf{X}}\_\mathbf{b}) \right)}{\tan \left( \theta \mp \Delta \theta \right)} \right\} \tag{18}$$

where FfricC is Coulomb friction with the opposite sign to velocity direction.

Finally, FfricV represents viscous friction, which, because there is a lubricant contact between bearing and body, is counted by multiplying the body velocity with an appropriate viscous coefficient (*σv*).

$$\mathbf{F}\_{\text{fricV}} = \sigma\_{\upsilon} \dot{\mathbf{X}}\_{\text{b}} \tag{19}$$

view, whereby it can be returned to two dominant parameters, the body velocity and the normal body force, that could be termed damping friction relative to the

For simplicity, even though the friction model, Eq. (12), reflected most of the observations measured using the system dynamics analysis and was used with the

overlooking Coulomb friction. Therefore, the simple expression of friction without

ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_ <sup>b</sup>

Cee <sup>X</sup>\_ ð Þ j j<sup>b</sup> *<sup>=</sup>*e1 <sup>þ</sup> *<sup>σ</sup>v*X\_ bX\_ <sup>b</sup> <sup>&</sup>gt;0*:*<sup>0</sup>

�Cee <sup>X</sup>\_ ð Þ j j<sup>b</sup> *<sup>=</sup>*e1 <sup>þ</sup> *<sup>σ</sup>v*X\_ bX\_ <sup>b</sup> <sup>&</sup>lt;0*:*<sup>0</sup>

In Eq. (21), this model has the same three various forms dependent on X\_ b, value and direction. Part one is the static friction, which has precisely the same shape for general friction, while the dynamic formula, damping friction, depending only on the body velocity in a different form by ignoring the Coulomb term. The interesting point is that, by implementing these simple friction forms, the simulation results also acquire a good agreement in comparison with the experimental results regarding system response parameters, which encouraged its use with the active suspension system. The question arises as to which one is more suitable for our case. Although the general friction model system, Eq. (12), gives more detail, depending on the system dynamics, and has the ability to highlight the hysteresis phenomenon that should occur with this system type, the simple friction model has lost this

However, the simple form also provides a real accord between the experimental

The RMS is defined as 'a measure of the difference between data and a model of that data'. Therefore, two measured signals, Xb and Xw � Xb*,* will be used to show

RMS accounts for the measurement and simulation with and without Coulomb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>N</sup> <sup>∑</sup> ð Þ Xw � Xb <sup>m</sup> � ð Þ Xw � Xb Sc � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>N</sup> <sup>∑</sup> ð Þ Xw � Xb <sup>m</sup> � ð Þ Xw � Xb <sup>S</sup> � �<sup>2</sup>

where RMS ð Þc and RMS ð Þ are the RMS between the measured and simulation values with and without considering Coulomb friction, respectively, Xð Þ <sup>w</sup> � Xb <sup>m</sup> is the measured relative displacement, Xð Þ <sup>w</sup> � Xb Sc and Xð Þ <sup>w</sup> � Xb <sup>S</sup> are the simulation data with and without implementing Coulomb friction and N is the total number of

From **Table 1**, the RMS results show that using the friction model considering

and simulation results for system response, with little variation relative to that gained from considering general friction. From this point of view, a mathematical

friction for relative movements between the wheel and body, as illustrated:

analysis is used, by using the residual mean square (RMS).

ð Þ RMS c ¼

ð Þ¼ RMS

sample. The RMS results are shown in **Table 1**.

Coulomb friction is more accurate.

the accuracy of considering the general or simple friction forms.

1

1

r

r

� �X\_ <sup>b</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>0</sup>

(21)

(22)

(23)

body velocity and Coulomb friction qualified to normal body force.

Coulomb friction is

hysteresis.

and

**233**

Ffric ¼

*DOI: http://dx.doi.org/10.5772/intechopen.86055*

8 >>>><

>>>>:

passive suspension model, it can still be employed in simple form through

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness*

when X\_ <sup>b</sup> < 0*:*0*,* the overall dynamic friction expression becomes

$$\mathbf{F}\_{\text{fricD}} = \left\{-\mathbf{C}\_{\text{e}}\mathbf{e}^{\left(\left|\dot{\mathbf{X}}\_{\text{b}}\right| \cdot \mathbf{e}\mathbf{1}\right)} + \left[\frac{\mu\left(\mathbf{k}\_{\text{s}}(\mathbf{X}\_{\text{w}} - \mathbf{X}\_{\text{b}}) + \mathbf{b}\_{\text{d}}(\dot{\mathbf{X}}\_{\text{w}} - \dot{\mathbf{X}}\_{\text{b}})\right)}{\tan\left(\theta \mp \Delta\theta\right)}\right] + \sigma\_{\text{v}}\dot{\mathbf{X}}\_{\text{b}}\right\} \tag{20}$$

Eq. (20) is similar to Eq. (16) as they have the same three terms but with a negative sign added in just for the transition friction term. This is because these values will describe the development friction in the opposite direction in the negative friction region.

The underlying motivation is that when the dynamic behaviour of the ¼-car model is thoroughly understood, the knowledge can be used to design appropriate feedback controllers for active suspension systems with compensation for the friction forces.

#### **7.4 Steady-state friction**

It is vital to consider the friction behaviour within SS period. From **Figure 11** of body displacement as function of time, it is clear that the historical movement demeanour, which starts to move from the stiction region, Xb ¼ 0*:*0 and <sup>X</sup>\_ <sup>b</sup> ffi <sup>0</sup>*:*0*,* is the first SS, stage (A), and then reaches the second SS, stage (B), at the mid-point of the road hydraulic actuator Xb <sup>¼</sup> <sup>0</sup>*:*085 m and <sup>X</sup>\_ <sup>b</sup> ffi <sup>0</sup>*:*0. Secondly, the body starts moving from the second SS and will reach the highest with a total amplitude Xb <sup>¼</sup> <sup>0</sup>*:*135 m and <sup>X</sup>\_ <sup>b</sup> ffi <sup>0</sup>*:*0 at the third SS, stage (C). Finally, it will start to move from the third SS stage and reach the lowest value of amplitude Xb ¼ 0*:*035 m*,* travelling twice the distance compared with the second stage. Thus, it will finally achieve the four SS (D) at Xb <sup>¼</sup> <sup>0</sup>*:*035 m and <sup>X</sup>\_ <sup>b</sup> ffi <sup>0</sup>*:*0.

At body stiction and SS station, X€<sup>b</sup> is equal to zero. Therefore, the friction at steady state should be similar to static friction as mentioned in Section 7.2.

#### **7.5 Simple friction model**

Eq. (12) gives a general form for nonlinear friction occurring at the body supported lubricant bearings. This model could be studied from a different point of

**Figure 11.** *Body displacement (*XbÞ *with time.*

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness DOI: http://dx.doi.org/10.5772/intechopen.86055*

view, whereby it can be returned to two dominant parameters, the body velocity and the normal body force, that could be termed damping friction relative to the body velocity and Coulomb friction qualified to normal body force.

For simplicity, even though the friction model, Eq. (12), reflected most of the observations measured using the system dynamics analysis and was used with the passive suspension model, it can still be employed in simple form through overlooking Coulomb friction. Therefore, the simple expression of friction without Coulomb friction is

$$\mathbf{F}\_{\text{fric}} = \begin{cases} \mathbf{k}\_{\text{s}} (\mathbf{X}\_{\text{w}} - \mathbf{X}\_{\text{b}}) + \mathbf{b}\_{\text{d}} (\dot{\mathbf{X}}\_{\text{w}} - \dot{\mathbf{X}}\_{\text{b}}) \dot{\mathbf{X}}\_{\text{b}} = \mathbf{0}.\mathbf{0} \\\\ \mathbf{C}\_{\text{e}} \mathbf{e}^{\left( \left| \dot{\mathbf{X}}\_{\text{b}} \right| / \epsilon \mathbf{1} \right)} + \sigma\_{\text{v}} \dot{\mathbf{X}}\_{\text{b}} \dot{\mathbf{X}}\_{\text{b}} > \mathbf{0}.\mathbf{0} \\\\ -\mathbf{C}\_{\text{e}} \mathbf{e}^{\left( \left| \dot{\mathbf{X}}\_{\text{b}} \right| / \epsilon \mathbf{1} \right)} + \sigma\_{\text{v}} \dot{\mathbf{X}}\_{\text{b}} \dot{\mathbf{X}}\_{\text{b}} < \mathbf{0}.\mathbf{0} \end{cases} \tag{21}$$

In Eq. (21), this model has the same three various forms dependent on X\_ b, value and direction. Part one is the static friction, which has precisely the same shape for general friction, while the dynamic formula, damping friction, depending only on the body velocity in a different form by ignoring the Coulomb term. The interesting point is that, by implementing these simple friction forms, the simulation results also acquire a good agreement in comparison with the experimental results regarding system response parameters, which encouraged its use with the active suspension system. The question arises as to which one is more suitable for our case. Although the general friction model system, Eq. (12), gives more detail, depending on the system dynamics, and has the ability to highlight the hysteresis phenomenon that should occur with this system type, the simple friction model has lost this hysteresis.

However, the simple form also provides a real accord between the experimental and simulation results for system response, with little variation relative to that gained from considering general friction. From this point of view, a mathematical analysis is used, by using the residual mean square (RMS).

The RMS is defined as 'a measure of the difference between data and a model of that data'. Therefore, two measured signals, Xb and Xw � Xb*,* will be used to show the accuracy of considering the general or simple friction forms.

RMS accounts for the measurement and simulation with and without Coulomb friction for relative movements between the wheel and body, as illustrated:

$$(\text{RMS})\mathbf{c} = \sqrt{\frac{1}{\mathbf{N}}\sum \left( (\mathbf{X}\_{\text{w}} - \mathbf{X}\_{\text{b}})\_{\text{m}} - (\mathbf{X}\_{\text{w}} - \mathbf{X}\_{\text{b}})\_{\text{Sc}} \right)^{2}}\tag{22}$$

and

Finally, FfricV represents viscous friction, which, because there is a lubricant contact between bearing and body, is counted by multiplying the body velocity

> � � � � tan ð Þ θ∓*Δθ* " #

( )

Eq. (20) is similar to Eq. (16) as they have the same three terms but with a negative sign added in just for the transition friction term. This is because these values will describe the development friction in the opposite direction in the nega-

The underlying motivation is that when the dynamic behaviour of the ¼-car model is thoroughly understood, the knowledge can be used to design appropriate feedback controllers for active suspension systems with compensation for the fric-

It is vital to consider the friction behaviour within SS period. From **Figure 11** of

<sup>X</sup>\_ <sup>b</sup> ffi <sup>0</sup>*:*0*,* is the first SS, stage (A), and then reaches the second SS, stage (B), at the mid-point of the road hydraulic actuator Xb <sup>¼</sup> <sup>0</sup>*:*085 m and <sup>X</sup>\_ <sup>b</sup> ffi <sup>0</sup>*:*0. Secondly, the body starts moving from the second SS and will reach the highest with a total amplitude Xb <sup>¼</sup> <sup>0</sup>*:*135 m and <sup>X</sup>\_ <sup>b</sup> ffi <sup>0</sup>*:*0 at the third SS, stage (C). Finally, it will start

amplitude Xb ¼ 0*:*035 m*,* travelling twice the distance compared with the second stage. Thus, it will finally achieve the four SS (D) at Xb <sup>¼</sup> <sup>0</sup>*:*035 m and <sup>X</sup>\_ <sup>b</sup> ffi <sup>0</sup>*:*0. At body stiction and SS station, X€<sup>b</sup> is equal to zero. Therefore, the friction at

steady state should be similar to static friction as mentioned in Section 7.2.

Eq. (12) gives a general form for nonlinear friction occurring at the body supported lubricant bearings. This model could be studied from a different point of

body displacement as function of time, it is clear that the historical movement demeanour, which starts to move from the stiction region, Xb ¼ 0*:*0 and

to move from the third SS stage and reach the lowest value of

when X\_ <sup>b</sup> < 0*:*0*,* the overall dynamic friction expression becomes

FfricD ¼ �Cee <sup>X</sup>\_ ð Þ j j<sup>b</sup> *<sup>=</sup>*e1 <sup>þ</sup> <sup>μ</sup> ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_

FfricV <sup>¼</sup> *<sup>σ</sup>v*X\_ <sup>b</sup> (19)

b

<sup>þ</sup> *<sup>σ</sup>v*X\_ <sup>b</sup>

(20)

with an appropriate viscous coefficient (*σv*).

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

tive friction region.

**7.4 Steady-state friction**

**7.5 Simple friction model**

*Body displacement (*XbÞ *with time.*

**Figure 11.**

**232**

tion forces.

$$(\text{RMS}) = \sqrt{\frac{1}{N} \sum \left( (\mathbf{X\_w} - \mathbf{X\_b})\_\mathbf{m} - (\mathbf{X\_w} - \mathbf{X\_b})\_\mathbf{S} \right)^2} \tag{23}$$

where RMS ð Þc and RMS ð Þ are the RMS between the measured and simulation values with and without considering Coulomb friction, respectively, Xð Þ <sup>w</sup> � Xb <sup>m</sup> is the measured relative displacement, Xð Þ <sup>w</sup> � Xb Sc and Xð Þ <sup>w</sup> � Xb <sup>S</sup> are the simulation data with and without implementing Coulomb friction and N is the total number of sample. The RMS results are shown in **Table 1**.

From **Table 1**, the RMS results show that using the friction model considering Coulomb friction is more accurate.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*


**Table 1.** *RMS results.*

### **8. Results**

#### **8.1 Friction results for general form (considering Coulomb friction)**

**Figure 12** shows friction force as a function of body velocity for the input force when amplitude = 50 mm, while the other cases when amplitude is = 30 or 70 mm. Accordingly, with the same friction behaviour, the same friction model can be used. It is apparent that the friction behaves as a hysteresis loop. Therefore, both sets of curves form a circle, enclosing a nonzero area, which is typical of dynamic friction besides the starting static friction. The loops enclosed three areas relating to velocity increases, decreases and directions. This is similar to expectations from the results of a dynamic friction model discussed in Sections 8.1 and 8.3. The upper portion of the curve shows the behaviour for increasing velocity when X\_ <sup>b</sup> > 0*:*0 in two circumstances, while the lower portion shows the behaviour for decreasing velocity when X\_ <sup>b</sup> < 0*:*0. This phenomenon may be a consequence of the dynamics of the process rather than of the nonlinearity; this phenomenon is often referred to as hysteresis. The hysteretic friction is, moreover, not a unique function of the velocity, but depends on the previous hysteresis of the movements.

when Xb >0*:*0, the friction hardly dips relative to the transition area from direct contact between body and bearings to mixed hydraulic contact. This clearly shows the Stribeck effects relative to hydraulic layer behaviour: a squeeze-film effect. Following the system inputs and velocity value when X\_ <sup>b</sup> > 0*:*0*,* the friction firstly draws a small, enclosed, positive cycle. After that, the body velocity returns to the second SS and increases to reach a maximum value before returning to the third SS with friction drawing a larger enclosed cycle in a positive direction. When X\_ <sup>b</sup> <0*:*0*,* the static values are equal to those for X\_ <sup>b</sup> >0*:*0 in the opposite direction, while the friction draws the most massive enclosed nonzero cycle with a value twice that of the larger enclosed cycle in the positive direction. This is because of the friction

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness*

value and guidance following the road input and velocity values.

**8.2 Friction results for simple form (without Coulomb friction)**

amplitude = 50 mm. It is approved that there is no hysteresis performance.

our system type.

**Figure 14.**

**235**

*Damping and Coulomb friction as a function of the body velocity.*

**Figure 13.**

*Damping friction as a function of the body velocity.*

*DOI: http://dx.doi.org/10.5772/intechopen.86055*

In considering friction, while disregarding the Coulomb effects relative to the vertical force from the force inputs and the construction of the test rig, the inclination of the spring and damper from one side and the distance between the wheel unit and body mass from another side allows a promotion friction formula, damping friction, to be obtained. Although some features of friction characteristics, the hysteresis behaviour, will have been lost in considering this friction model with the passive suspension system design, success also has been achieved close to the experimental data. **Figure 13** shows the damping friction as a function of the body velocity when

Meanwhile, **Figure 14** illustrates the association between damping and Coulomb friction. Although the damping friction is dominant, it remains vital to reflect the Coulomb friction in the general friction model, because it is responsible for bringing hysteresis performance to the model, and, as mentioned, this is quite essential to

In fact, there are two urgent situations that should be highlighted: the first is when the velocity equals zero, the body is motionless, and the friction values are similar to static friction values, as discussed in Section 7.2, while the second important situation is when the values of friction are within the SS situation, which has already been specified in the previous analysis in Section 7.4.

However, **Figure 12** shows the behaviour of friction relative to the body velocity. It is evident that the reaction in the stick region, or static friction at Xb=0.0, friction values start from zero and reach a maximum at the breakaway threshold force. From the experimental test, the breakaway force at the maximum relative displacement between Xw and Xb and the corresponding values for wheel and body velocity, accounted by Eq. (15), could be estimated. As a result, it was found to be equal to 193.8 N. Therefore, after the first positive position of static friction, because the direction of displacement moves up, whenever the body starts to move,

**Figure 12.** *Friction as function of the body velocity.*

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness DOI: http://dx.doi.org/10.5772/intechopen.86055*

**Figure 13.** *Damping friction as a function of the body velocity.*

**8. Results**

**Figure 12.**

**234**

*Friction as function of the body velocity.*

**Table 1.** *RMS results.*

**8.1 Friction results for general form (considering Coulomb friction)**

ity, but depends on the previous hysteresis of the movements.

already been specified in the previous analysis in Section 7.4.

**Figure 12** shows friction force as a function of body velocity for the input force when amplitude = 50 mm, while the other cases when amplitude is = 30 or 70 mm. Accordingly, with the same friction behaviour, the same friction model can be used. It is apparent that the friction behaves as a hysteresis loop. Therefore, both sets of curves form a circle, enclosing a nonzero area, which is typical of dynamic friction besides the starting static friction. The loops enclosed three areas relating to velocity increases, decreases and directions. This is similar to expectations from the results of a dynamic friction model discussed in Sections 8.1 and 8.3. The upper portion of the curve shows the behaviour for increasing velocity when X\_ <sup>b</sup> > 0*:*0 in two circumstances, while the lower portion shows the behaviour for decreasing velocity when X\_ <sup>b</sup> < 0*:*0. This phenomenon may be a consequence of the dynamics of the process rather than of the nonlinearity; this phenomenon is often referred to as hysteresis. The hysteretic friction is, moreover, not a unique function of the veloc-

**Signal** ð*RMS*Þc ð*RMS*Þ ð Þ Xw � Xb 0.006362 0.006366 Xb 0.096267 0.096386

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

In fact, there are two urgent situations that should be highlighted: the first is when the velocity equals zero, the body is motionless, and the friction values are similar to static friction values, as discussed in Section 7.2, while the second important situation is when the values of friction are within the SS situation, which has

However, **Figure 12** shows the behaviour of friction relative to the body velocity. It is evident that the reaction in the stick region, or static friction at Xb=0.0, friction values start from zero and reach a maximum at the breakaway threshold force. From the experimental test, the breakaway force at the maximum relative displacement between Xw and Xb and the corresponding values for wheel and body velocity, accounted by Eq. (15), could be estimated. As a result, it was found to be equal to 193.8 N. Therefore, after the first positive position of static friction, because the direction of displacement moves up, whenever the body starts to move, when Xb >0*:*0, the friction hardly dips relative to the transition area from direct contact between body and bearings to mixed hydraulic contact. This clearly shows the Stribeck effects relative to hydraulic layer behaviour: a squeeze-film effect. Following the system inputs and velocity value when X\_ <sup>b</sup> > 0*:*0*,* the friction firstly draws a small, enclosed, positive cycle. After that, the body velocity returns to the second SS and increases to reach a maximum value before returning to the third SS with friction drawing a larger enclosed cycle in a positive direction. When X\_ <sup>b</sup> <0*:*0*,* the static values are equal to those for X\_ <sup>b</sup> >0*:*0 in the opposite direction, while the friction draws the most massive enclosed nonzero cycle with a value twice that of the larger enclosed cycle in the positive direction. This is because of the friction value and guidance following the road input and velocity values.

#### **8.2 Friction results for simple form (without Coulomb friction)**

In considering friction, while disregarding the Coulomb effects relative to the vertical force from the force inputs and the construction of the test rig, the inclination of the spring and damper from one side and the distance between the wheel unit and body mass from another side allows a promotion friction formula, damping friction, to be obtained. Although some features of friction characteristics, the hysteresis behaviour, will have been lost in considering this friction model with the passive suspension system design, success also has been achieved close to the experimental data. **Figure 13** shows the damping friction as a function of the body velocity when amplitude = 50 mm. It is approved that there is no hysteresis performance.

Meanwhile, **Figure 14** illustrates the association between damping and Coulomb friction. Although the damping friction is dominant, it remains vital to reflect the Coulomb friction in the general friction model, because it is responsible for bringing hysteresis performance to the model, and, as mentioned, this is quite essential to our system type.

**Figure 14.** *Damping and Coulomb friction as a function of the body velocity.*

#### **9. Discussion**

This chapter was set up to question the aspects of friction that merit inclusion with the ¼ car model. After a brief stating of the general frictional considerations, this discussion will review and summarise the findings.

with a sharper increment than in the first stage and will be stretched to the ultimate before it returns to zero, or near to zero, at SS. Thirdly, it will begin from SS and, after velocity reversals, will reach the highest estimate, twice the time as for case two, and spine to SS. In every one of these velocity cases, the velocity behaviour will make friction hysteretic loops that could account for the increments of body speed

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness*

In general, this friction model deliberates the static, stiction region and dynamic friction, which consists of the Stribeck effect, viscous friction and Coulomb friction, which rely on the dynamic tangential force, which evolves in the test rig contact bearings. Therefore, there are general and simple friction forms as follows:

> b � � X\_

Cee <sup>X</sup>\_ ð Þ j j<sup>b</sup> *<sup>=</sup>*e1 <sup>þ</sup> <sup>μ</sup> ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_

�Cee <sup>X</sup>\_ ð Þ j j<sup>b</sup> *<sup>=</sup>*e1 <sup>þ</sup> <sup>μ</sup> ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_

<sup>b</sup> ¼ 0*:*0

� � � � tan ð Þ θ∓*Δθ* " #

� � � � tan ð Þ θ∓*Δθ* " #

This mathematical eq. (24) incorporates two primary parts of friction: static and dynamic friction. The latter as two expressions and is influenced by the velocity track. In static friction, the stiction area is exclusively not subject to the velocity because the body velocity should be close to zero velocity or just beyond zero velocity. Frequently, the static models are numbered by the strength adjustment of the test rig when the body sticks, while the wheel is moved and depicts the static friction sufficiently precisely. A dynamic model is vital to present an additional state, which can be viewed as the transition, Coulomb and viscous friction. In addition to these friction models, steady physics state is also briefly discussed in this

When be ignored the Coulomb friction, the previous nonlinear friction model

approach should be found to discover which approach obtains more accurate results by comparing with measured results. By using RMS mathematical analysis, the results shown in **Table 1** prove, as an outcome, that the friction model is more

An accurate nonlinear dynamic model for friction has been presented. The model is simple yet captures most friction phenomena that are of interest for simulated test results. The low-velocity friction characteristics are particularly

shown in Eq. (12) becomes a simple model, as illustrated in Eq. (21), despite losing some features of friction characteristics with this model, in comparison with experimental data that also obtained close results. From this aspect, another

accurate with a consideration of Coulomb friction.

b

b

<sup>þ</sup> *<sup>σ</sup><sup>v</sup>* <sup>X</sup>\_

<sup>þ</sup> *<sup>σ</sup><sup>v</sup>* <sup>X</sup>\_

<sup>b</sup> X\_

<sup>b</sup> X\_

<sup>b</sup> >0*:*0

<sup>b</sup> <0*:*0

(24)

in a variety of paths from reductions.

*DOI: http://dx.doi.org/10.5772/intechopen.86055*

**10.1 Mathematical friction model**

ksð Þþ Xw � Xb bd <sup>X</sup>\_ <sup>w</sup> � <sup>X</sup>\_

Ffric ¼

study.

8

>>>>>>>>>>>><

>>>>>>>>>>>>:

**10.2 Simple friction model**

**11. Conclusion**

**237**

Friction is a highly complex phenomenon, evolving at the contact of surfaces. Experiments demonstrate a functional addiction upon a significant change in parameters, including sliding speed, acceleration, critical sliding distance, normal load, surface preparation and material combination. In many engineering applications, the success of models in predicting experimental results remains strongly sensitive to the friction model. Friction is a natural phenomenon that is quite difficult to model and is not yet completely understood.

The investigation of the principal questions to inform the simulation framework were tested as follow: what is the most suitable technique for including friction in an analytical or numerical model, and what are the inferences of friction model superiority? The constituent elements are discussed in turn as follows:

#### **9.1 The main reasons for considering friction**

In this study, as shown in Section 3, considering and implementing the friction model within the equation of motion for the mass body is qualified for the following reasons:


#### **10. Conventional friction model**

The majority of current model-based friction compensation schemes utilise classical friction models, such as Coulomb and viscous friction. In applications with high-precision positioning and with low-velocity following, the outcomes are not generally acceptable. Typical types are different combinations of Coulomb friction, viscous friction and the Stribeck effect, as has been mentioned in several researchers' works as shown in Section 5.

In this review, the established friction model, irrespective of its extreme effortlessness, can recreate all, that we are aware of, conditionally watched properties and features of low-velocity friction force dynamics. Considering the test rig schematic and the force information, there are three conditions, depending upon whether the body speed is speeding up or decelerating. Firstly, the velocity qualities start from zero, and soon after, velocity reversals reach the most elevated level and are maintained at zero, or close to zero, at SS. Secondly, the velocity begins from SS

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness DOI: http://dx.doi.org/10.5772/intechopen.86055*

with a sharper increment than in the first stage and will be stretched to the ultimate before it returns to zero, or near to zero, at SS. Thirdly, it will begin from SS and, after velocity reversals, will reach the highest estimate, twice the time as for case two, and spine to SS. In every one of these velocity cases, the velocity behaviour will make friction hysteretic loops that could account for the increments of body speed in a variety of paths from reductions.

In general, this friction model deliberates the static, stiction region and dynamic friction, which consists of the Stribeck effect, viscous friction and Coulomb friction, which rely on the dynamic tangential force, which evolves in the test rig contact bearings. Therefore, there are general and simple friction forms as follows:

#### **10.1 Mathematical friction model**

**9. Discussion**

reasons:

term.

**236**

This chapter was set up to question the aspects of friction that merit inclusion with the ¼ car model. After a brief stating of the general frictional considerations,

Friction is a highly complex phenomenon, evolving at the contact of surfaces. Experiments demonstrate a functional addiction upon a significant change in parameters, including sliding speed, acceleration, critical sliding distance, normal load, surface preparation and material combination. In many engineering applications, the success of models in predicting experimental results remains strongly sensitive to the friction model. Friction is a natural phenomenon that is quite

The investigation of the principal questions to inform the simulation framework were tested as follow: what is the most suitable technique for including friction in an analytical or numerical model, and what are the inferences of friction model

In this study, as shown in Section 3, considering and implementing the friction model within the equation of motion for the mass body is qualified for the following

2. From the experiment test, it is clearly seen that there is no oscillation of mass body travels, while that was found with simulation model results. Therefore, a new term should be considered to overcome the issue, that is to say, a friction

3. In addition, from experimental measurements in Section 4.4, it is apparent that at the start of the test, while the wheel began to move in relation to road inputs,

The majority of current model-based friction compensation schemes utilise classical friction models, such as Coulomb and viscous friction. In applications with high-precision positioning and with low-velocity following, the outcomes are not generally acceptable. Typical types are different combinations of Coulomb friction,

In this review, the established friction model, irrespective of its extreme effortlessness, can recreate all, that we are aware of, conditionally watched properties and features of low-velocity friction force dynamics. Considering the test rig schematic and the force information, there are three conditions, depending upon whether the body speed is speeding up or decelerating. Firstly, the velocity qualities start from zero, and soon after, velocity reversals reach the most elevated level and are maintained at zero, or close to zero, at SS. Secondly, the velocity begins from SS

viscous friction and the Stribeck effect, as has been mentioned in several

1. Friction itself is crucial to find in any mechanical system. Friction exists everywhere, since degradation, precision, monitoring and control system are

this discussion will review and summarise the findings.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

difficult to model and is not yet completely understood.

**9.1 The main reasons for considering friction**

the body remained motionless for a period.

strongly affected by friction.

**10. Conventional friction model**

researchers' works as shown in Section 5.

superiority? The constituent elements are discussed in turn as follows:

$$\mathbf{F}\_{\text{fric}} = \begin{cases} \mathbf{k}\_{\text{d}} (\mathbf{X}\_{\text{w}} - \mathbf{X}\_{\text{b}}) + \mathbf{b}\_{\text{d}} (\dot{\mathbf{X}}\_{\text{w}} - \dot{\mathbf{X}}\_{\text{b}}) \, \dot{\mathbf{X}}\_{\text{b}} = \mathbf{0}. \mathbf{0} \\\\ \mathbf{C}\_{\text{e}} \mathbf{e}^{\left( |\dot{\mathbf{X}}\_{\text{b}} | / \epsilon \mathbf{1} \right)} + \left[ \frac{\mu \left( \mathbf{k}\_{\text{b}} (\mathbf{X}\_{\text{w}} - \mathbf{X}\_{\text{b}}) + \mathbf{b}\_{\text{d}} (\dot{\mathbf{X}}\_{\text{w}} - \dot{\mathbf{X}}\_{\text{b}}) \right)}{\tan \left( 0 \mp \Delta \theta \right)} \right] + \sigma\_{\text{v}} \, \dot{\mathbf{X}}\_{\text{b}} \, \dot{\mathbf{X}}\_{\text{b}} > \mathbf{0}. \mathbf{0} \\\\ -\mathbf{C}\_{\text{e}} \mathbf{e}^{\left( |\dot{\mathbf{X}}\_{\text{b}} | / \epsilon \mathbf{1} \right)} + \left[ \frac{\mu \left( \mathbf{k}\_{\text{s}} (\mathbf{X}\_{\text{w}} - \mathbf{X}\_{\text{b}}) + \mathbf{b}\_{\text{d}} (\dot{\mathbf{X}}\_{\text{w}} - \dot{\mathbf{X}}\_{\text{b}}) \right)}{\tan \left( 0 \mp \Delta \theta \right)} \right] + \sigma\_{\text{v}} \, \dot{\mathbf{X}}\_{\text{b}} \, \dot{\mathbf{X}}\_{\text{b}} < \mathbf{0}. \mathbf{0} \end{cases} \tag{24}$$

This mathematical eq. (24) incorporates two primary parts of friction: static and dynamic friction. The latter as two expressions and is influenced by the velocity track. In static friction, the stiction area is exclusively not subject to the velocity because the body velocity should be close to zero velocity or just beyond zero velocity. Frequently, the static models are numbered by the strength adjustment of the test rig when the body sticks, while the wheel is moved and depicts the static friction sufficiently precisely. A dynamic model is vital to present an additional state, which can be viewed as the transition, Coulomb and viscous friction. In addition to these friction models, steady physics state is also briefly discussed in this study.

#### **10.2 Simple friction model**

When be ignored the Coulomb friction, the previous nonlinear friction model shown in Eq. (12) becomes a simple model, as illustrated in Eq. (21), despite losing some features of friction characteristics with this model, in comparison with experimental data that also obtained close results. From this aspect, another approach should be found to discover which approach obtains more accurate results by comparing with measured results. By using RMS mathematical analysis, the results shown in **Table 1** prove, as an outcome, that the friction model is more accurate with a consideration of Coulomb friction.

#### **11. Conclusion**

An accurate nonlinear dynamic model for friction has been presented. The model is simple yet captures most friction phenomena that are of interest for simulated test results. The low-velocity friction characteristics are particularly important for high-performance pointing and tracking. The model can describe arbitrary steady-state friction characteristics. It supports hysteretic behaviour due to frictional lag and springlike behaviour in stiction and gives a different breakaway force depending on the rate of change of the applied force. All these phenomena are unified into static, steady-state and dynamic friction equations. The model can be readily used in simulations of systems with friction.

**References**

9.376053

1.3625110

4420-x (Open Access)

University; 2000

9781846283734

Media; 2012

**239**

[thesis]. Cardiff, UK: Cardiff

2005. pp. 182-186. ISBN-13:

[9] Lischinsky P et al. Friction compensation for an industrial

[7] Watton J. Chapter 3: Modelling, Monitoring and Diagnostic Techniques for Fluid Power Systems. London: Springer Science & Business Media;

[8] Armstrong-Helouvry B. Control of Machines with Friction. Vol. 128. New York: Springer Science & Business

hydraulic robot. IEEE Control Systems

[1] Nichols S. MANE 6960 Friction &

*DOI: http://dx.doi.org/10.5772/intechopen.86055*

Magazine. 1999;**19**(1):25-32. DOI:

[10] Armstrong-Hélouvry B et al. A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica. 1994;**30**(7):1083-1138. DOI: 10.1016/

[11] Tsurata K et al., editors. Genetic algorithm (GA) based modelling of nonlinear behaviour of friction of a rolling ball guide way. In: Proceedings 6th International Workshop on

Advanced Motion Control; 30 March-1 April 2000. Nagoya, Japan: IEEE; 2002

[12] Smyth AW et al. Development of adaptive modelling techniques for nonlinear hysteretic systems. International Journal of Non-Linear Mechanics. 2002;

**37**(8):1435-1451. DOI: 10.1016/ S0020-7462(02)00031-8

[13] Altpeter F. Friction modelling, identification and compensation. École Polytechnique FÉdÉrale de Lausanne; 1999. DOI: 10.5075/epfl-thesis-1988

[14] Al-Zughaibi A et al. A new insight into modelling passive suspension real test rig system, quarter race car, with considering nonlinear friction forces; ImechE, part D. Journal of Automobile Engineering. 2018;**233**(8):2257-2266. DOI: 10.1177/0954407018764942

[15] Dupont PE. Avoiding stick-slip through PD control. IEEE Transactions on Automatic Control. 1994;**39**(5): 1094-1097. DOI: 10.1109/9.284901

[16] Dahl PR. A solid friction Model. No. TOR-0158 (3107-18)-1. Segundo, CA:

[17] Bliman P-A. Mathematical study of the Dahl's friction model. European Journal of Mechanics. A, Solids. 1992; **11**(6):835-848 ISSNs: 0997-7538

Aerospace Corp El; 1968

10.1109/37.745763

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness*

0005-1098(94)90209-7

[2] Al-Bender F et al. A novel generic model at asperity level for dry friction force dynamics. Tribology Letters. 2004;**16**(1):81-93. DOI: 10.1023/B: TRIL.0000009718.60501.74

[3] De Wit CC et al. A new model for control of systems with friction. IEEE Transactions on Automatic Control. 1995;**40**(3):419-425. DOI: 10.1109/

[4] Rabinowicz E. Friction and wear of materials. Journal of Applied Mechanics. 1965;**33**(1966):479. DOI: 10.1115/

[5] Al-Zughaibi AIH. Experimental and analytical investigations of friction at lubricant bearings in passive suspension systems. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems. 2018;**94**(2): 1227-1242. DOI: 10.1007/s11071-018-

[6] Surawattanawan P. The influence of hydraulic system dynamics on the behaviour of a vehicle active suspension

Wear of Materials; 2007

It is essential to consider friction in this study, in the hope that the study creates an opening and contributes towards a reconsideration of the role of friction using the current quarter in half- and full-car suspension models.

Simulation leads to the same conclusion as proven by the experimental results obtained from the test rig test. Comparison between experimental and simulation results show that the proposed general friction model is more accurate than the conventional models (simple model).

#### **Acknowledgements**

I highly appreciate my original university, Kerbala University, and Cardiff University for giving me a chance to do this work. At the same time, I thank the technician people in the laboratory for helping.

#### **Author details**

Ali I. H. Al-Zughaibi Engineering College, Kerbala University, Karbala, Iraq

\*Address all correspondence to: ali.i@uokerbala.edu.iq

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Nonlinear Friction Model for Passive Suspension System Identification and Effectiveness DOI: http://dx.doi.org/10.5772/intechopen.86055*

#### **References**

important for high-performance pointing and tracking. The model can describe arbitrary steady-state friction characteristics. It supports hysteretic behaviour due to frictional lag and springlike behaviour in stiction and gives a different breakaway force depending on the rate of change of the applied force. All these phenomena are unified into static, steady-state and dynamic friction equations. The model can be

It is essential to consider friction in this study, in the hope that the study creates an opening and contributes towards a reconsideration of the role of friction using

Simulation leads to the same conclusion as proven by the experimental results obtained from the test rig test. Comparison between experimental and simulation results show that the proposed general friction model is more accurate than the

I highly appreciate my original university, Kerbala University, and Cardiff University for giving me a chance to do this work. At the same time, I thank the

readily used in simulations of systems with friction.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

conventional models (simple model).

technician people in the laboratory for helping.

Engineering College, Kerbala University, Karbala, Iraq

\*Address all correspondence to: ali.i@uokerbala.edu.iq

provided the original work is properly cited.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**Acknowledgements**

**Author details**

**238**

Ali I. H. Al-Zughaibi

the current quarter in half- and full-car suspension models.

[1] Nichols S. MANE 6960 Friction & Wear of Materials; 2007

[2] Al-Bender F et al. A novel generic model at asperity level for dry friction force dynamics. Tribology Letters. 2004;**16**(1):81-93. DOI: 10.1023/B: TRIL.0000009718.60501.74

[3] De Wit CC et al. A new model for control of systems with friction. IEEE Transactions on Automatic Control. 1995;**40**(3):419-425. DOI: 10.1109/ 9.376053

[4] Rabinowicz E. Friction and wear of materials. Journal of Applied Mechanics. 1965;**33**(1966):479. DOI: 10.1115/ 1.3625110

[5] Al-Zughaibi AIH. Experimental and analytical investigations of friction at lubricant bearings in passive suspension systems. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems. 2018;**94**(2): 1227-1242. DOI: 10.1007/s11071-018- 4420-x (Open Access)

[6] Surawattanawan P. The influence of hydraulic system dynamics on the behaviour of a vehicle active suspension [thesis]. Cardiff, UK: Cardiff University; 2000

[7] Watton J. Chapter 3: Modelling, Monitoring and Diagnostic Techniques for Fluid Power Systems. London: Springer Science & Business Media; 2005. pp. 182-186. ISBN-13: 9781846283734

[8] Armstrong-Helouvry B. Control of Machines with Friction. Vol. 128. New York: Springer Science & Business Media; 2012

[9] Lischinsky P et al. Friction compensation for an industrial hydraulic robot. IEEE Control Systems Magazine. 1999;**19**(1):25-32. DOI: 10.1109/37.745763

[10] Armstrong-Hélouvry B et al. A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica. 1994;**30**(7):1083-1138. DOI: 10.1016/ 0005-1098(94)90209-7

[11] Tsurata K et al., editors. Genetic algorithm (GA) based modelling of nonlinear behaviour of friction of a rolling ball guide way. In: Proceedings 6th International Workshop on Advanced Motion Control; 30 March-1 April 2000. Nagoya, Japan: IEEE; 2002

[12] Smyth AW et al. Development of adaptive modelling techniques for nonlinear hysteretic systems. International Journal of Non-Linear Mechanics. 2002; **37**(8):1435-1451. DOI: 10.1016/ S0020-7462(02)00031-8

[13] Altpeter F. Friction modelling, identification and compensation. École Polytechnique FÉdÉrale de Lausanne; 1999. DOI: 10.5075/epfl-thesis-1988

[14] Al-Zughaibi A et al. A new insight into modelling passive suspension real test rig system, quarter race car, with considering nonlinear friction forces; ImechE, part D. Journal of Automobile Engineering. 2018;**233**(8):2257-2266. DOI: 10.1177/0954407018764942

[15] Dupont PE. Avoiding stick-slip through PD control. IEEE Transactions on Automatic Control. 1994;**39**(5): 1094-1097. DOI: 10.1109/9.284901

[16] Dahl PR. A solid friction Model. No. TOR-0158 (3107-18)-1. Segundo, CA: Aerospace Corp El; 1968

[17] Bliman P-A. Mathematical study of the Dahl's friction model. European Journal of Mechanics. A, Solids. 1992; **11**(6):835-848 ISSNs: 0997-7538

**Chapter 13**

*Barun Pratiher*

**Abstract**

their operation.

**241**

Electrostatically Driven MEMS

This chapter deals with the investigation on stability and bifurcation analysis of

a highly non-linear electrically driven micro-electro-mechanical resonator has been established. A non-linear model of this system will briefly be described considering both transverse and longitudinal displacement of the resonator. A short description to explore the need of incorporating higher-order correction of electrostatic pressure has been highlighted. The pull-in results and consequences of higherorder correction on the pull-in stability will be reported. In addition, consequences of air-gap, electrostatic forcing parameter, and effective damping on non-linear phenomena have been studied to highlight the possible undesirable catastrophic failure at the unstable critical points. Basins of attractions that postulate a unique response in multi-region state for a specific initial condition will also be studied. This chapter can enable a significant adaptation to identify the locus of instability in micro-cantilever-based resonator when subjected to AC voltage polarization with the understanding of theoretical ideas for controlling the systems and optimizing

**Keywords:** micro-beam, electrostatic actuation, pull-in analysis, higher-order-electrostatic distribution, non-linear phenomena, stability

The development of electrostatically actuated micro-system has been extensively carried out by the research community in order to develop low cost and high durability, and further improve the performance of sensors and actuators for wide applications. The use of electrostatic actuation offers a simplicity in design with low-cost fabrication, fast response, the ability to achieve rotary motion, and low power consumption. However, this actuation often leads into a complex non-linear phenomenon. As a result, structural movability becomes suspicious due to pull-in occurrence for any finite air-gap thickness. Furthermore, stable deflection range due to active electrostatic actuation is always being restricted since the movable substrate or electrode gets collapsed onto the stationary plate. Thus, computing pull-in voltage is inevitable and plays a decisive factor indicating a critical voltage

under which stable operations and structural reliability may be asserted.

Generally, micro-electro mechanical system mathematically model by considering either a thin beam or a thin plate having cross-section in the order of microns

**1. Introduction and state-of-art research**

Resonator: Pull-in Behavior

and Non-linear Phenomena

[18] Bliman P, Sorine M. Friction modelling by hysteresis operators. Application to Dahl, stiction and Stribeck effects. In: Proc. Conf. on Models of Hysteresis; Trento. 1991. p. 10

[19] Walrath CD. Adaptive bearing friction compensation based on recent knowledge of dynamic friction. Automatica. 1984;**20**(6):717-727. DOI: 10.1016/0005-1098(84)90081-5

[20] Leonard NE, Krishnaprasad PS, editors. Adaptive friction compensation for bi-directional low-velocity position tracking. In: Proceedings of the 31st IEEE Conference on Decision and Control; 16-18 December 1992. Tucson, AZ, USA: IEEE; 2002

[21] Ruina A, Rice J. Stability of steady frictional slipping. Journal of Applied Mechanics. 1983;**50**(2):343-349. DOI: 10.1115/1.3167042

#### **Chapter 13**

[18] Bliman P, Sorine M. Friction modelling by hysteresis operators. Application to Dahl, stiction and Stribeck effects. In: Proc. Conf. on Models of Hysteresis; Trento. 1991. p. 10

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

[19] Walrath CD. Adaptive bearing friction compensation based on recent knowledge of dynamic friction. Automatica. 1984;**20**(6):717-727. DOI: 10.1016/0005-1098(84)90081-5

[20] Leonard NE, Krishnaprasad PS, editors. Adaptive friction compensation for bi-directional low-velocity position tracking. In: Proceedings of the 31st IEEE Conference on Decision and Control; 16-18 December 1992. Tucson,

[21] Ruina A, Rice J. Stability of steady frictional slipping. Journal of Applied Mechanics. 1983;**50**(2):343-349. DOI:

AZ, USA: IEEE; 2002

10.1115/1.3167042

**240**

## Electrostatically Driven MEMS Resonator: Pull-in Behavior and Non-linear Phenomena

*Barun Pratiher*

#### **Abstract**

This chapter deals with the investigation on stability and bifurcation analysis of a highly non-linear electrically driven micro-electro-mechanical resonator has been established. A non-linear model of this system will briefly be described considering both transverse and longitudinal displacement of the resonator. A short description to explore the need of incorporating higher-order correction of electrostatic pressure has been highlighted. The pull-in results and consequences of higherorder correction on the pull-in stability will be reported. In addition, consequences of air-gap, electrostatic forcing parameter, and effective damping on non-linear phenomena have been studied to highlight the possible undesirable catastrophic failure at the unstable critical points. Basins of attractions that postulate a unique response in multi-region state for a specific initial condition will also be studied. This chapter can enable a significant adaptation to identify the locus of instability in micro-cantilever-based resonator when subjected to AC voltage polarization with the understanding of theoretical ideas for controlling the systems and optimizing their operation.

**Keywords:** micro-beam, electrostatic actuation, pull-in analysis, higher-order-electrostatic distribution, non-linear phenomena, stability

#### **1. Introduction and state-of-art research**

The development of electrostatically actuated micro-system has been extensively carried out by the research community in order to develop low cost and high durability, and further improve the performance of sensors and actuators for wide applications. The use of electrostatic actuation offers a simplicity in design with low-cost fabrication, fast response, the ability to achieve rotary motion, and low power consumption. However, this actuation often leads into a complex non-linear phenomenon. As a result, structural movability becomes suspicious due to pull-in occurrence for any finite air-gap thickness. Furthermore, stable deflection range due to active electrostatic actuation is always being restricted since the movable substrate or electrode gets collapsed onto the stationary plate. Thus, computing pull-in voltage is inevitable and plays a decisive factor indicating a critical voltage under which stable operations and structural reliability may be asserted.

Generally, micro-electro mechanical system mathematically model by considering either a thin beam or a thin plate having cross-section in the order of microns

and length in the order of hundreds of microns with an efficient electrostatic actuation. When the range of air-gap between stationary electrode and movable electrode is relatively large, typically of the order of 10<sup>2</sup> –10<sup>1</sup> or even higher for designing small-size of electro-statically actuated devices, parallel approximation theory of capacitor becomes ill-suited and in-valid. For a high air gap, it is unavoidable to develop a large deflection model for an electrostatically actuated microbeam considering both higher order distribution of electrostatic pressure and mid-plane stretching exist. In the following section, a brief current research on modeling and dynamics of micro-electro-mechanical systems (MEMS) structures has been cited.

review papers [15–18] provided an overview of the fundamental research on modeling and dynamics of electrostatically actuated MEMS devices under working different conditions. Nayfeh et al. [19] studied that the characteristics of the pull-in phenomenon in the presence of AC loads differ from those under purely DC loads. Zhang et al. [20] furnished a survey and analysis of the electrostatic force of importance in MEMS, its physical model, scaling effect, stability, non-linearity, and reliability in details. Chao et al. [21] predicted the DC dynamics pull-in voltages of a clamped-clamped micro-beam based on a continuous model. They derived the equation of motion of the dynamics model by considering beam flexibility, inertia, residual stress, squeeze film, distributed electrostatic forces, and its electrical field fringing effects. Shao et al. [22] demonstrated the non-linear vibration behavior of a micro-mechanical clamped-lamped beam resonator under different driving conditions. They developed a non-linear model for the resonator by considering both mechanical and electrostatic non-linear effects, and the numerical simulation was verified by experimental findings. Moghimi et al. [23] investigated the non-linear oscillations of micro-beams actuated by suddenly applied electrostatic force, including the effects of electrostatic actuation, residual stress, mid-plane stretching, and fringing fields in modelling. Chatterjee and Pohit [24] introduced a non-linear model of an electrostatically actuated micro-cantilever beam considering the non-linearities of the system arising out of electric forces, geometry of the deflected beam and the inertial terms. Furthermore, one may use the review articles [15–18] as a source of information to the overall images about the electromechanical model of MEMS devices actuated by electrostatically and related dynamics. A detailed review of perturbation techniques to obtain the non-linear solution of such systems/structures can be found in [25]. A detailed description of the forced and parametrically excited

*Electrostatically Driven MEMS Resonator: Pull-in Behavior and Non-linear Phenomena*

Several researchers have studied the pull-in behavior of micro-mechanical system under various driving conditions about its static beam positions. In addition, it has been learnt that researchers are still considering simple geometry ignoring the non-linear effect or components in their mathematical model to investigate the theoretical and experimental aspects of dynamic performance of MEMS devices. Moreover, in order to highlight a proper insight and a better understanding into the MEMS devices, the accurate simulations of mechanical behaviors with a faithful mathematical model is fairly inevitable that can exhibit a more realistic shape of the bending deflection of the micro-beam and the development of resulting electrostatic pressure distribution. Here, author has been attempted to investigate the dynamic stability and bifurcation analysis of electrostatically actuated MEMS cantilever along with pull-in behavior, both statically and dynamically accounting for the effect of mid-plane stretching and non-linear distribution of electrostatic pressure. The main focus here is to investigate the assessment of the system stability and subsequent bifurcations, which usually demonstrate the locus of instability. Pull-in voltage and its response under the non-linear effects have been computed. The method of multiple scales has been used to analyze the stability and bifurcation of the steady state solutions via frequency-response characteristics, time responses,

Though the major focus of this study is to explore the non-linear behavior of an electrostatically actuated MEMS device, it is of vital importance to study the pull-in behavior of electrostatically driven MEMS device as well. Both static and dynamic pull-in have been albeit briefly discussed in the coming sub-section followed by the

systems has been highlighted in [26–28].

*DOI: http://dx.doi.org/10.5772/intechopen.88453*

and basin of attractions.

**2. Pull-in**

**243**

A number of researchers have attempted to develop numerous models over the times to improve the design characteristics and investigating related dynamics. Luo and Wang [1] investigated analytically and numerically the chaotic motion in the certain frequency band of a MEMS with capacitor non-linearity. Pamidighantam et al. [2] derived a closed-form expression for the pull-in voltage of fixed-fixed micro-beams and fixed-free micro-beams by considering axial stress, non-linear stiffening, charge re-distribution, and fringing fields. They carried out an extensive analysis of the non-linearities in a micro-mechanical clamped-lamped beam resonator. Abdel-Rahman et al. [3] presented a non-linear model of electrically actuated micro-beams with consideration of electrostatic forcing of the air-gap capacitor, restoring force of the micro-beam, and axial load applied to the micro-beam. The response of a resonant micro-beam subjected to an electrical actuation has been investigated by Younis and Nayfeh [4]. Xie et al. [5] performed the dynamic analysis of a micro-switch using invariant manifold method. They considered micro-switch as a clamped-clamped micro-beam subjected to a transverse electrostatic force. An analytical approach and resultant reduced-order model to investigate the dynamic behavior of electrically actuated micro-beam-based MEMS devices have been demonstrated by Younis et al. [6]. The natural frequency and responses of electrostatically actuated MEMS with time-varying capacitors have been investigated by Luo and Wang [7]. Authors have demonstrated that the numerically and analytically obtained predictions were in good agreement with the findings obtained experimentally. A simplified discrete spring-mass mechanical model has been considered for the dynamic analysis of MEMS device. In Teva et al. [8], a mathematical model for an electrically excited electromechanical system based on lateral resonating cantilever has been developed. The authors obtained static deflection and the frequency response of the oscillation amplitude for different voltage-polarization conditions. Kuang and Chen [9] and Najar et al. [10] studied the dynamic characteristics of nonlinear electrostatic pull-in behavior for shaped actuators in micro-electro-mechanical systems (MEMS) using the differential quadrature method (DQM). Zhang and Meng [11] analyzed the resonant responses and non-linear dynamics of idealized electrostatically actuated micro-cantilever-based devices in micro-electromechanical systems (MEMS) by using the harmonic balance (HB) method. Rhoads et al. [12] proposed a micro-beam device, which couples the inherent benefits of a resonator with purely parametric excitation with the simple geometry of a micro-beam. Krylov and Seretensky [13] developed higher-order correction to the parallel capacitor approximation of the electrostatic pressure acting on micro-structures taking into account the influence of the curvature and slope of the beam on the electrostatic pressure. The higher-order approximation has validated through a comparison with analytical solutions for simple geometries as well as numerical results. Decuzzi et al. [14] investigated the dynamic response of a micro-cantilever beam used as a transducer in a biomechanical sensor. Here, Euler-Bernoulli beam theory was introduced to model the cantilever motion of the transducer. They also considered Reynolds equation of lubrication for the analysis of hydrodynamic interactions. A number of

#### *Electrostatically Driven MEMS Resonator: Pull-in Behavior and Non-linear Phenomena DOI: http://dx.doi.org/10.5772/intechopen.88453*

review papers [15–18] provided an overview of the fundamental research on modeling and dynamics of electrostatically actuated MEMS devices under working different conditions. Nayfeh et al. [19] studied that the characteristics of the pull-in phenomenon in the presence of AC loads differ from those under purely DC loads. Zhang et al. [20] furnished a survey and analysis of the electrostatic force of importance in MEMS, its physical model, scaling effect, stability, non-linearity, and reliability in details. Chao et al. [21] predicted the DC dynamics pull-in voltages of a clamped-clamped micro-beam based on a continuous model. They derived the equation of motion of the dynamics model by considering beam flexibility, inertia, residual stress, squeeze film, distributed electrostatic forces, and its electrical field fringing effects. Shao et al. [22] demonstrated the non-linear vibration behavior of a micro-mechanical clamped-lamped beam resonator under different driving conditions. They developed a non-linear model for the resonator by considering both mechanical and electrostatic non-linear effects, and the numerical simulation was verified by experimental findings. Moghimi et al. [23] investigated the non-linear oscillations of micro-beams actuated by suddenly applied electrostatic force, including the effects of electrostatic actuation, residual stress, mid-plane stretching, and fringing fields in modelling. Chatterjee and Pohit [24] introduced a non-linear model of an electrostatically actuated micro-cantilever beam considering the non-linearities of the system arising out of electric forces, geometry of the deflected beam and the inertial terms. Furthermore, one may use the review articles [15–18] as a source of information to the overall images about the electromechanical model of MEMS devices actuated by electrostatically and related dynamics. A detailed review of perturbation techniques to obtain the non-linear solution of such systems/structures can be found in [25]. A detailed description of the forced and parametrically excited systems has been highlighted in [26–28].

Several researchers have studied the pull-in behavior of micro-mechanical system under various driving conditions about its static beam positions. In addition, it has been learnt that researchers are still considering simple geometry ignoring the non-linear effect or components in their mathematical model to investigate the theoretical and experimental aspects of dynamic performance of MEMS devices. Moreover, in order to highlight a proper insight and a better understanding into the MEMS devices, the accurate simulations of mechanical behaviors with a faithful mathematical model is fairly inevitable that can exhibit a more realistic shape of the bending deflection of the micro-beam and the development of resulting electrostatic pressure distribution. Here, author has been attempted to investigate the dynamic stability and bifurcation analysis of electrostatically actuated MEMS cantilever along with pull-in behavior, both statically and dynamically accounting for the effect of mid-plane stretching and non-linear distribution of electrostatic pressure. The main focus here is to investigate the assessment of the system stability and subsequent bifurcations, which usually demonstrate the locus of instability. Pull-in voltage and its response under the non-linear effects have been computed. The method of multiple scales has been used to analyze the stability and bifurcation of the steady state solutions via frequency-response characteristics, time responses, and basin of attractions.

#### **2. Pull-in**

and length in the order of hundreds of microns with an efficient electrostatic actuation. When the range of air-gap between stationary electrode and movable

designing small-size of electro-statically actuated devices, parallel approximation theory of capacitor becomes ill-suited and in-valid. For a high air gap, it is unavoidable to develop a large deflection model for an electrostatically actuated microbeam considering both higher order distribution of electrostatic pressure and mid-plane stretching exist. In the following section, a brief current research on modeling and dynamics of micro-electro-mechanical systems (MEMS) structures

A number of researchers have attempted to develop numerous models over the times to improve the design characteristics and investigating related dynamics. Luo and Wang [1] investigated analytically and numerically the chaotic motion in the certain frequency band of a MEMS with capacitor non-linearity. Pamidighantam et al. [2] derived a closed-form expression for the pull-in voltage of fixed-fixed micro-beams and fixed-free micro-beams by considering axial stress, non-linear stiffening, charge re-distribution, and fringing fields. They carried out an extensive analysis of the non-linearities in a micro-mechanical clamped-lamped beam resonator. Abdel-Rahman et al. [3] presented a non-linear model of electrically actuated micro-beams with consideration of electrostatic forcing of the air-gap capacitor, restoring force of the micro-beam, and axial load applied to the micro-beam. The response of a resonant micro-beam subjected to an electrical actuation has been investigated by Younis and Nayfeh [4]. Xie et al. [5] performed the dynamic analysis of a micro-switch using invariant manifold method. They considered micro-switch as a clamped-clamped micro-beam subjected to a transverse electrostatic force. An analytical approach and resultant reduced-order model to investigate the dynamic behavior of electrically actuated micro-beam-based MEMS devices have been demonstrated by Younis et al. [6]. The natural frequency and responses of electrostatically actuated MEMS with time-varying capacitors have been investigated by Luo and Wang [7]. Authors have demonstrated that the numerically and analytically obtained predictions were in good agreement with the findings obtained experimentally. A simplified discrete spring-mass mechanical model has been considered for the dynamic analysis of MEMS device. In Teva et al. [8], a mathematical model for an electrically excited electromechanical system based on lateral resonating cantilever has been developed. The authors obtained static deflection and the frequency response of the oscillation amplitude for different voltage-polarization conditions. Kuang and Chen [9] and Najar et al. [10] studied the dynamic characteristics of nonlinear electrostatic pull-in behavior for shaped actuators in micro-electro-mechanical systems (MEMS) using the differential quadrature method (DQM). Zhang and Meng [11] analyzed the resonant responses and non-linear dynamics of idealized electrostatically actuated micro-cantilever-based devices in micro-electromechanical systems (MEMS) by using the harmonic balance (HB) method. Rhoads et al. [12] proposed a micro-beam device, which couples the inherent benefits of a resonator with purely parametric excitation with the simple geometry of a micro-beam. Krylov and Seretensky [13] developed higher-order correction to the parallel capacitor approximation of the electrostatic pressure acting on micro-structures taking into account the influence of the curvature and slope of the beam on the electrostatic pressure. The higher-order approximation has validated through a comparison with analytical solutions for simple geometries as well as numerical results. Decuzzi et al. [14] investigated the dynamic response of a micro-cantilever beam used as a transducer in a biomechanical sensor. Here, Euler-Bernoulli beam theory was introduced to model the cantilever motion of the transducer. They also considered Reynolds equation of lubrication for the analysis of hydrodynamic interactions. A number of

–10<sup>1</sup> or even higher for

electrode is relatively large, typically of the order of 10<sup>2</sup>

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

has been cited.

**242**

Though the major focus of this study is to explore the non-linear behavior of an electrostatically actuated MEMS device, it is of vital importance to study the pull-in behavior of electrostatically driven MEMS device as well. Both static and dynamic pull-in have been albeit briefly discussed in the coming sub-section followed by the system's non-linear behaviors. Before proceeding with an understanding of the MEMS dynamics, especially non-linear dynamics, it is prudent to briefly explore the range of operating applied voltage under which the system model and attendant analysis are considered to be sufficiently accurate for the predictive design.

#### **2.1 Problem description**

Differential equation of motion of a continuous micro-cantilever beam subjected to AC potential difference by stationary electrode has been shown in **Figures 1** and **2**, while the associated boundary conditions are being expressed in [25, 29]. However, the electrostatic force is considered to be uniform across the width, while transverse *v x*ð Þ , *t* and axial *u x*ð Þ , *t* displacement component holds a constraint equation known as in-extensibility condition

$$\begin{aligned} &w''^2 + (1+u')^2 = 1. \\ &\ddot{w} + 2\xi\dot{w} + w'''' + (d/l)^2 \Big[ \left(w''\right)^3 + 4w'w''w'' + \left(w'\right)^2 w'''' \Big] + (d/l)^2 \\ &\left[w'\right] \left[ \left(\ddot{w}'w' + \left(\dot{w'}\right)^2\right) d\tilde{\xi} \right] - (d/l)^2 \left[w''\right] \left[ \begin{matrix} \frac{1}{7} \\ w'' \end{matrix} \Big] \left\{ \left(\ddot{w}'w' + \left(\dot{w'}\right)^2\right) d\tilde{\xi}d\tilde{\eta} \right\} \\ &= \frac{6e\_0l^4V^2}{Eh^3d^3} \left(1 + 2w + 3w^2 + HOD - \left\{\frac{\left(d/l\right)}{3}\left(2w'' + 2ww'' + w'^2 + HOD\right)\right\}\right) \Big\}. \end{aligned} \tag{1}$$

system's equation of motion. Pull-in instability as to calculate the pull-in voltage is inevitable to understand its limit to perform the desire task under a static voltage beyond which the movable electrode collapses onto the stationary electrode. As a result, the system is statically unstable as the electrostatic force overshadows the

*of higher-order correction of electrostatic pressure on the non-dimensional tip deflection* wj<sup>x</sup>¼<sup>1</sup> *with*

*Electrostatically Driven MEMS Resonator: Pull-in Behavior and Non-linear Phenomena*

<sup>x</sup>¼<sup>1</sup> *with variable <sup>α</sup> for various values of <sup>δ</sup>. (b): Effect*

In most of the communication and power-circuits systems, electrostatically actuated micro-switch works only to alternate ON and OFF conditions by tuning a bias voltage across the pull-in back and forth. Therefore, it is advisable to have a micro-system, which can operate at low actuation voltage for performing the task mostly suited in power communications. The pull-in voltage can be obtained by demonstrating the static deflection of the tip of the micro-cantilever beam directly solving the boundary value problem by setting all time derivatives in Eq. (1) equal to zero. **Figure 1** shows the tip-deflection with the voltages ranging from zero to forcing level, where the pull-in instability takes place as explained details in [25, 29]. Recalling the fact that system leads to a pull-in condition when the system's net stiffness becomes negative. Here, the pull-in condition starts at *α* equal to 1.69 that indicates 66.83 compared to the pull-in voltage 66.78 obtained in Ref. [24] and 68.5

However, the obtained static pull-in voltage may increase with increase in gap-

The loss of stability in dynamic responses occurs when the deformable electrode comes into contact with the fixed electrodes under an instantaneous electrostatic actuation that is lower than the static pull-in voltages known as dynamic pull-in phenomenon. Analysis of dynamic pull-in of an electrostatically actuated is complex due to its non-linear nature of electrostatic forces along with time integration of the momentum equations. Along with time-dependent terms, the transient part of the applied voltage is being neglected while calculating static pull-in voltage. However, calculating the actual pull-in voltage in dynamic condition when a bias alternative voltage source exists is obligatory and different as that of static pull-in voltage. Hence, there is a need to calculate the pull-in voltage called as dynamic pull-in voltage, considering the dynamics of the micro-beam instead of static state only.

when the effect of non-linear curvature is considered while calculating the electrostatic pressures. Further, the higher-order correction factor may lead to lower value of pull-in voltage, which provides a most suitable for the design of a micro-system

*<sup>l</sup>*). It can be noted that the pull-in voltage may occur at a lower

obtained in Ref. [23] considering the same design parameters.

internal resistance as restoring force.

*(a) Variations of the non-dimensional tip deflection* wj

*DOI: http://dx.doi.org/10.5772/intechopen.88453*

*=*

having significant gap-length.

length ratio (*δ* ¼ *<sup>d</sup>*<sup>0</sup>

**Figure 2.**

*variable α [25].*

**2.3 Dynamic analysis**

**245**

#### **2.2 Static analysis**

It has been practically observed that the most common failure mode considered in the design of electrostatically driven MEMS devices is static pull-in condition beyond which it leads to desterilize the system for any further applied voltage. This failure is majorly occurred in the excess of electrostatic load in comparison to the static load-bearing capacity. As a result, system undergoes a negative stiffness in the

**Figure 1.**

*(a) A pictorial diagram of micro-cantilever beam separated from a stationary electrode at a distance of d [25]. (b) Graphical representation of cantilever-based micro-structure coupled with rigid plate [29].*

*Electrostatically Driven MEMS Resonator: Pull-in Behavior and Non-linear Phenomena DOI: http://dx.doi.org/10.5772/intechopen.88453*

**Figure 2.**

system's non-linear behaviors. Before proceeding with an understanding of the MEMS dynamics, especially non-linear dynamics, it is prudent to briefly explore the range of operating applied voltage under which the system model and attendant analysis are considered to be sufficiently accurate for the predictive design.

Differential equation of motion of a continuous micro-cantilever beam subjected to AC potential difference by stationary electrode has been shown in **Figures 1** and **2**, while the associated boundary conditions are being expressed in [25, 29]. However, the electrostatic force is considered to be uniform across the width, while transverse *v x*ð Þ , *t* and axial *u x*ð Þ , *t* displacement component holds

a constraint equation known as in-extensibility condition

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*dξ*

3 7 <sup>5</sup> � ð Þ *<sup>d</sup>=<sup>l</sup>* <sup>2</sup>

*<sup>d</sup>*<sup>3</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>*<sup>w</sup>* <sup>þ</sup> <sup>3</sup>*w*<sup>2</sup> <sup>þ</sup> *HOD* � ð Þ *<sup>d</sup>=<sup>l</sup>*

*<sup>w</sup>*<sup>00</sup> ð Þ<sup>3</sup> <sup>þ</sup> <sup>4</sup>*w*<sup>0</sup>

*w*00 ð 1

2 6 4

*ξ*

ð *η*

0

3

It has been practically observed that the most common failure mode considered in the design of electrostatically driven MEMS devices is static pull-in condition beyond which it leads to desterilize the system for any further applied voltage. This failure is majorly occurred in the excess of electrostatic load in comparison to the static load-bearing capacity. As a result, system undergoes a negative stiffness in the

*(a) A pictorial diagram of micro-cantilever beam separated from a stationary electrode at a distance of d [25].*

*(b) Graphical representation of cantilever-based micro-structure coupled with rigid plate [29].*

*<sup>w</sup>*00*w*‴ <sup>þ</sup> *<sup>w</sup>*<sup>0</sup> ð Þ<sup>2</sup> *<sup>w</sup>*<sup>0000</sup> h i

*w*€0

� � � � � �

*<sup>w</sup>*<sup>0</sup> <sup>þ</sup> *<sup>w</sup>*\_<sup>0</sup> � �<sup>2</sup> n o

<sup>þ</sup> ð Þ *<sup>d</sup>=<sup>l</sup>* <sup>2</sup>

*dξdη*

<sup>2</sup>*w*<sup>00</sup> <sup>þ</sup> <sup>2</sup>*ww*<sup>00</sup> <sup>þ</sup> *<sup>w</sup>*0<sup>2</sup> <sup>þ</sup> *HOD*

3 7 5

*:*

(1)

**2.1 Problem description**

*<sup>w</sup>*0<sup>2</sup> <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>u</sup>*<sup>0</sup> ð Þ<sup>2</sup> <sup>¼</sup> <sup>1</sup>*:*

*w*0 ð *ξ*

<sup>¼</sup> <sup>6</sup>*ε*0*<sup>l</sup>* 4 *V*2

**Figure 1.**

**244**

2 6 4

0

*Eh*<sup>3</sup>

**2.2 Static analysis**

*w*€0

*<sup>w</sup>*€ <sup>þ</sup> <sup>2</sup>*ςw*\_ <sup>þ</sup> *<sup>w</sup>*<sup>0000</sup> <sup>þ</sup> ð Þ *<sup>d</sup>=<sup>l</sup>* <sup>2</sup>

*<sup>w</sup>*<sup>0</sup> <sup>þ</sup> *<sup>w</sup>*\_<sup>0</sup> � �<sup>2</sup> n o

*(a) Variations of the non-dimensional tip deflection* wj <sup>x</sup>¼<sup>1</sup> *with variable <sup>α</sup> for various values of <sup>δ</sup>. (b): Effect of higher-order correction of electrostatic pressure on the non-dimensional tip deflection* wj<sup>x</sup>¼<sup>1</sup> *with variable α [25].*

system's equation of motion. Pull-in instability as to calculate the pull-in voltage is inevitable to understand its limit to perform the desire task under a static voltage beyond which the movable electrode collapses onto the stationary electrode. As a result, the system is statically unstable as the electrostatic force overshadows the internal resistance as restoring force.

In most of the communication and power-circuits systems, electrostatically actuated micro-switch works only to alternate ON and OFF conditions by tuning a bias voltage across the pull-in back and forth. Therefore, it is advisable to have a micro-system, which can operate at low actuation voltage for performing the task mostly suited in power communications. The pull-in voltage can be obtained by demonstrating the static deflection of the tip of the micro-cantilever beam directly solving the boundary value problem by setting all time derivatives in Eq. (1) equal to zero. **Figure 1** shows the tip-deflection with the voltages ranging from zero to forcing level, where the pull-in instability takes place as explained details in [25, 29]. Recalling the fact that system leads to a pull-in condition when the system's net stiffness becomes negative. Here, the pull-in condition starts at *α* equal to 1.69 that indicates 66.83 compared to the pull-in voltage 66.78 obtained in Ref. [24] and 68.5 obtained in Ref. [23] considering the same design parameters.

However, the obtained static pull-in voltage may increase with increase in gaplength ratio (*δ* ¼ *<sup>d</sup>*<sup>0</sup>*=<sup>l</sup>*). It can be noted that the pull-in voltage may occur at a lower when the effect of non-linear curvature is considered while calculating the electrostatic pressures. Further, the higher-order correction factor may lead to lower value of pull-in voltage, which provides a most suitable for the design of a micro-system having significant gap-length.

#### **2.3 Dynamic analysis**

The loss of stability in dynamic responses occurs when the deformable electrode comes into contact with the fixed electrodes under an instantaneous electrostatic actuation that is lower than the static pull-in voltages known as dynamic pull-in phenomenon. Analysis of dynamic pull-in of an electrostatically actuated is complex due to its non-linear nature of electrostatic forces along with time integration of the momentum equations. Along with time-dependent terms, the transient part of the applied voltage is being neglected while calculating static pull-in voltage. However, calculating the actual pull-in voltage in dynamic condition when a bias alternative voltage source exists is obligatory and different as that of static pull-in voltage. Hence, there is a need to calculate the pull-in voltage called as dynamic pull-in voltage, considering the dynamics of the micro-beam instead of static state only.

In calculating the dynamic pull-in voltage, inertia and dissipative elements along with the components storing the strain energy for elastic deformation play an influential role in a dynamic condition. Here, the dynamic pull-in behavior has been depicted and investigated by directly simulating the Eq. (1) using the well-known R-K method. A qualitative phase-plane analysis has been illustrated to capture the global behavior of the response trajectories. Hence, the dynamic pull-in voltage has been illustrated in the phase portrait, i.e., in the plane of velocity ∝ displacement for the every applied voltage. The voltage turns out to be the critical voltage, i.e., pullin voltage until the trajectories lead to an intersection of the orbits with the origin. The voltage at which monoclinic orbits are passing through the saddle node or degenerate singularity point is known as dynamic pull-in voltage. It is however advised to go through the article [29] for the detail explanation of obtaining the dynamic pull-in voltage.

certain voltage leads to a closed trajectory as time approaches infinity. Hence, for every applied voltage V < VDPI (voltage in dynamic pull-in), the system shows an isolated stable closed trajectory. For any voltages greater than the dynamic pull-in voltage, the closed loop curves merge into a single curve, thus leading to an unstable domain. An effective DC contribution considering both DC and AC components has been depicted in the phase plane as shown in **Figure 4**, while the presence of AC component leads to a distortion in the solution trajectories. The presence of electrostatic actuation combining both DC and AC voltages system undergoes pseudodynamic pull-in, which is expected to all voltage-input combinations exceeding the predicted margin. However, this phenomenon generally holds true for small AC

*Electrostatically Driven MEMS Resonator: Pull-in Behavior and Non-linear Phenomena*

A comprehensive knowledge of non-linear dynamics in MEMS resonator is of great importance for the optimum design and operational stability. Thus, an under-

standing of the conditions to explore the non-linear phenomena arise; e.g., multiple-solutions; bifurcation can be implemented for further predicting chaotic responses in micro/nano-resonators. In this section, system's non-linear response at three distinct resonant conditions in the parametric space has been discussed. The non-linear phenomena have been reported here with the understanding of its instability via bifurcation. The characteristics form of the system non-linearity and electrostatic pressures and their effects on the system stability along with the effect of input voltages offer great flexibility toward designing the resonant sensors and filters. In order to obtain a detail understanding about the non-linear phenomena in

MEMS systems, one may go through the articles [1, 5, 7, 11, 12, 19, 25, 30].

*<sup>X</sup>*€ <sup>þ</sup> *<sup>X</sup>* <sup>þ</sup> *aX*\_ <sup>þ</sup> *bX*<sup>3</sup> <sup>þ</sup> *cX*\_ <sup>2</sup>

*<sup>X</sup>*€ <sup>þ</sup> *<sup>X</sup>* <sup>þ</sup> *aX*\_ <sup>þ</sup> *bX*<sup>3</sup> <sup>þ</sup> *cX*\_ <sup>2</sup>

Adopting the Galerkin's techniques and replacing *w* ¼ Φð Þ *x X*ð Þ*τ* , where Φð Þ *x* is admissible function obtained by satisfying the boundary conditions only and with similar procedures used in [25], the partial governing equation is then discretized into non-autonomous, time-dependent equation of motion with considering viscous damping effect. Expanding the non-linear electrostatic force developed due to applied voltage by Taylor series, one may obtain the following non-autonomous

Here, *X* is the non-dimensional displacement function or time modulation, while *τ* and Ω are the non-dimensional time and frequency, respectively. The expression for the co-efficient of non-autonomous (*a* � *d*, *F* � *G*,*K*) is expressed in [25]. The equation of motion is further reduced to another form of micro-system neglecting effect of higher-order electrostatic distribution pressure, and mid-plane stretching

A huge number of researchers still consider either simple lumped-spring-mass model or Euler-Bernoulli beam theory with small air-gap assumption to carry out

*<sup>X</sup>* <sup>þ</sup> *dXX*€ <sup>2</sup> <sup>¼</sup> *<sup>F</sup>* cos <sup>Ω</sup>*<sup>τ</sup>* <sup>þ</sup> *GX* cos <sup>Ω</sup>*<sup>τ</sup>* <sup>þ</sup> *KX*<sup>2</sup> cos <sup>Ω</sup>*<sup>τ</sup>* (2)

*<sup>X</sup>* <sup>þ</sup> *dXX*€ <sup>2</sup> <sup>¼</sup> *<sup>F</sup>* cos <sup>Ω</sup>*τ:* (3)

component, while inconsistency observes for a larger AC voltages.

**3. Non-linear analysis**

*DOI: http://dx.doi.org/10.5772/intechopen.88453*

**3.1 Problem description**

equation of motion.

effect.

**247**

Beyond the critical voltage, the system is found to be dynamically unstable. It has been found that the dynamic pull-in voltage is well below the static pull-in voltage nearly 80–95% of static pull-in voltage depending upon geometric configurations and physical properties **Figure 3.**

For an applied voltage less than critical one, the trajectories exhibit closed periodic orbits with steady response amplitude lower than the dynamic pull-in deflection. Hence, a periodic solution initiated always from an initial guess for a

#### **Figure 3.**

*(a) Dynamic pull-in voltage of undamped system for electrode length 45 μm [29]. (b) Time responses at pull-in voltage in [29].*

#### **Figure 4.**

*(a) Dynamic pull-in voltage under DC and combined actuation. (b) Dynamic response at pull-in voltage under combined actuation at various voltages of AC actuation.*

*Electrostatically Driven MEMS Resonator: Pull-in Behavior and Non-linear Phenomena DOI: http://dx.doi.org/10.5772/intechopen.88453*

certain voltage leads to a closed trajectory as time approaches infinity. Hence, for every applied voltage V < VDPI (voltage in dynamic pull-in), the system shows an isolated stable closed trajectory. For any voltages greater than the dynamic pull-in voltage, the closed loop curves merge into a single curve, thus leading to an unstable domain. An effective DC contribution considering both DC and AC components has been depicted in the phase plane as shown in **Figure 4**, while the presence of AC component leads to a distortion in the solution trajectories. The presence of electrostatic actuation combining both DC and AC voltages system undergoes pseudodynamic pull-in, which is expected to all voltage-input combinations exceeding the predicted margin. However, this phenomenon generally holds true for small AC component, while inconsistency observes for a larger AC voltages.

#### **3. Non-linear analysis**

In calculating the dynamic pull-in voltage, inertia and dissipative elements along

Beyond the critical voltage, the system is found to be dynamically unstable. It has been found that the dynamic pull-in voltage is well below the static pull-in voltage nearly 80–95% of static pull-in voltage depending upon geometric configu-

For an applied voltage less than critical one, the trajectories exhibit closed periodic orbits with steady response amplitude lower than the dynamic pull-in deflection. Hence, a periodic solution initiated always from an initial guess for a

*(a) Dynamic pull-in voltage of undamped system for electrode length 45 μm [29]. (b) Time responses at pull-in*

*(a) Dynamic pull-in voltage under DC and combined actuation. (b) Dynamic response at pull-in voltage*

*under combined actuation at various voltages of AC actuation.*

with the components storing the strain energy for elastic deformation play an influential role in a dynamic condition. Here, the dynamic pull-in behavior has been depicted and investigated by directly simulating the Eq. (1) using the well-known R-K method. A qualitative phase-plane analysis has been illustrated to capture the global behavior of the response trajectories. Hence, the dynamic pull-in voltage has been illustrated in the phase portrait, i.e., in the plane of velocity ∝ displacement for the every applied voltage. The voltage turns out to be the critical voltage, i.e., pullin voltage until the trajectories lead to an intersection of the orbits with the origin. The voltage at which monoclinic orbits are passing through the saddle node or degenerate singularity point is known as dynamic pull-in voltage. It is however advised to go through the article [29] for the detail explanation of obtaining the

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

dynamic pull-in voltage.

**Figure 3.**

**Figure 4.**

**246**

*voltage in [29].*

rations and physical properties **Figure 3.**

A comprehensive knowledge of non-linear dynamics in MEMS resonator is of great importance for the optimum design and operational stability. Thus, an understanding of the conditions to explore the non-linear phenomena arise; e.g., multiple-solutions; bifurcation can be implemented for further predicting chaotic responses in micro/nano-resonators. In this section, system's non-linear response at three distinct resonant conditions in the parametric space has been discussed. The non-linear phenomena have been reported here with the understanding of its instability via bifurcation. The characteristics form of the system non-linearity and electrostatic pressures and their effects on the system stability along with the effect of input voltages offer great flexibility toward designing the resonant sensors and filters. In order to obtain a detail understanding about the non-linear phenomena in MEMS systems, one may go through the articles [1, 5, 7, 11, 12, 19, 25, 30].

#### **3.1 Problem description**

Adopting the Galerkin's techniques and replacing *w* ¼ Φð Þ *x X*ð Þ*τ* , where Φð Þ *x* is admissible function obtained by satisfying the boundary conditions only and with similar procedures used in [25], the partial governing equation is then discretized into non-autonomous, time-dependent equation of motion with considering viscous damping effect. Expanding the non-linear electrostatic force developed due to applied voltage by Taylor series, one may obtain the following non-autonomous equation of motion.

$$\begin{aligned} \ddot{X} + X + a\dot{X} + bX^3 + c\dot{X}^2X + d\ddot{X}X^2 \\ = F\cos\Omega\tau + GX\cos\Omega\tau + KX^2\cos\Omega\tau \end{aligned} \tag{2}$$

Here, *X* is the non-dimensional displacement function or time modulation, while *τ* and Ω are the non-dimensional time and frequency, respectively. The expression for the co-efficient of non-autonomous (*a* � *d*, *F* � *G*,*K*) is expressed in [25]. The equation of motion is further reduced to another form of micro-system neglecting effect of higher-order electrostatic distribution pressure, and mid-plane stretching effect.

$$
\ddot{X} + X + a\dot{X} + b\dot{X}^3 + c\dot{X}^2\dot{X} + d\ddot{\mathcal{X}}\dot{X}^2 = F\cos\Omega\pi. \tag{3}
$$

A huge number of researchers still consider either simple lumped-spring-mass model or Euler-Bernoulli beam theory with small air-gap assumption to carry out

the theoretical and experimental investigation of dynamic performance of MEMS devices considering the mid-plane stretching effect.

$$
\ddot{X} + X + a\dot{X} + bX^3 = F\cos\Omega\pi.\tag{4}
$$

It may be observed that these terms exist when Ω ≈ 1, Ω ≈ 3, or Ω ≈2. In the following sub-sections, three resonance conditions, i.e., primary resonance, parametric resonance condition, and third-order sub-harmonic conditions have been briefly discussed. A details derivation and explanation is being carried

*Electrostatically Driven MEMS Resonator: Pull-in Behavior and Non-linear Phenomena*

Here, the resonance condition occurs when the frequency of applied voltage becomes equal to that of one of the natural frequencies, i.e., fundamental natural frequency. Following reduced equations are obtained as given below replacing *X* ¼ *a T*ð Þ<sup>0</sup> *ei<sup>ϕ</sup>*ð Þ *<sup>T</sup>*<sup>0</sup> . A detail explanation about to obtain these reduced equations is being

<sup>8</sup>*aϕ*<sup>0</sup> <sup>¼</sup> <sup>8</sup>*a<sup>σ</sup>* � ð Þ <sup>3</sup>*<sup>b</sup>* � <sup>3</sup>*<sup>c</sup>* <sup>þ</sup> *<sup>d</sup>* ð Þ <sup>≃</sup>*<sup>μ</sup> <sup>a</sup>*<sup>3</sup> <sup>þ</sup> <sup>4</sup>*<sup>F</sup>* cos *<sup>ϕ</sup>* <sup>þ</sup> <sup>3</sup>*Ka*<sup>2</sup> cos *<sup>ϕ</sup>:* (10)

Here, system only exhibits only non-trivial responses, i.e., *a* 6¼ 0 obtained from (Eqs. 9–10). Dynamic responses are being determined by solving the set of algebraic equations obtained by converting differential equations into set of algebraic equations under steady state conditions, i.e., *a*<sup>0</sup> ¼ 0, and *ϕ*<sup>0</sup> ¼ 0. Here, stability of the steady state responses has been analyzed by investigating eigenvalues of the Jacobian matrix, which has been obtained by perturbing the algebraic equations with

Here, the condition at which the resonator has been excited with a frequency of the applied alternative voltage nearly equal to the fundamental frequency of the resonator is being discussed. In this vibrating state, the amplitude of vibration is always found to be a non-zero solution, while both stable and unstable non-trivial solutions are being observed. The study of bifurcation is being carried out on to see the losses the stability of system when a parameter passes through a critical value called a bifurcation point. The sudden change in amplitude undergoes catastrophic failure of the system. The graphical illustration of the vibration amplitude with

**Figure 5a** represents a typical frequency response curves for a specific air-gap ð Þ ≃*μ* between resonator and stationary. It is noteworthy that amplitude of vibration becomes increasing with increase in forcing frequency. The non-linear mode of operation possesses most likely hardening when the effective structural nonlinearity becomes *μ* ¼ þ2*:*0. In this condition, restoring forces due to geometric non-linearity overshadows the inertia effect of the device that leads to hardening spring effect. When the effective structural non-linearity becomes *μ* ¼ �2*:*0, vice versa effect is observed. For sweeping up the frequency moving toward E from point D, system undergoes a sudden jump down when the frequency reaches to its critical value regarded as saddle-node fixed bifurcation point. Similarly, a sudden upward jump in the amplitude from lower to higher amplitude undergoes for a sweeping down frequency. With further increase in frequency, the amplitude of vibratory motion decreases and follows the path DA. Hence, with decrease in frequency leads to the lower amplitude of responses from a higher value continuously.

*a* ¼ *ao* þ *a*<sup>1</sup> and *γ* ¼ *γ*<sup>0</sup> þ *γ*1, where *a*0, *γ*<sup>0</sup> are the singular points.

varying the system control parameter has been constructed.

<sup>8</sup>*a*<sup>0</sup> ¼ �8*ζ<sup>a</sup>* <sup>þ</sup> <sup>4</sup>*<sup>F</sup>* sin *<sup>ϕ</sup>* <sup>þ</sup> *Ka*<sup>2</sup> sin *<sup>ϕ</sup>*, (9)

out in [25].

*3.2.1 Primary resonance condition*

*DOI: http://dx.doi.org/10.5772/intechopen.88453*

*3.2.1.1 Reduced order model*

*3.2.1.2 Results and discussions*

**249**

carried out in [25].

#### **3.2 Bifurcation and stability**

Equation of motions [Eqs. (2)–(4)] for various MEMS devices comprise linear and non-linear terms, direct forced, parametric term, and non-linear parametric terms due to non-linear electrostatic actuation. Since, the temporal equation of motion holds non-linear terms; it is difficult to find closed form solution. Hence, one may go for approximate solution by using the perturbation method. Here, method of multiple scales as explained in [25, 29–33] is used to obtain the set of algebraic equations turning into non-autonomous equations of motion for three resonance conditions, viz. primary resonance, parametric resonance condition, and third-order sub-harmonic conditions are being expressed under steady state conditions. The procedures used to derive the reduced order equation are similar to those explained in [25, 29–33]. Based on numerical values of the coefficients of the damping, forcing, and non-linear terms, they are one order less than the coefficients of the linear terms, which have a value of unity in this case and as result, in the following technique, co-efficient are expressed as *a* ¼ 2*ες*, *b* ¼ *εb*, *c* ¼ *εc*, *d* ¼ *εd*, *F* ¼ *εF*, *G* ¼ *εG*, and *K* ¼ *εK* for sake of simplicity. By using method of multiple scales with the procedure as explained in [25, 30, 32, 33], substituting *Tn* ¼ *<sup>ε</sup><sup>n</sup>τ*, *<sup>n</sup>* <sup>¼</sup> 0, 1, 2, 3<sup>⋯</sup> and displacement *<sup>X</sup>*ð Þ¼ *<sup>τ</sup>*; *<sup>ε</sup> <sup>X</sup>*0ð Þþ *<sup>T</sup>*0, *<sup>T</sup>*<sup>1</sup> *<sup>ε</sup>X*1ð Þþ *<sup>T</sup>*0, *<sup>T</sup>*<sup>1</sup> *<sup>O</sup> <sup>ε</sup>*<sup>2</sup> ð Þ in Eq. (2) and equating the coefficients of like powers of *ε*, one may obtain the following expressions:

Order

$$\varepsilon^0: D\_0^{\;2}X\_0 + X\_0 = 0,\tag{5}$$

Order

$$\begin{aligned} & \varepsilon^1: D\_0 \mathbf{^2 X\_1} + \mathbf{X\_1} \\ &= -2D\_0 D\_1 \mathbf{X\_0} - 2i\zeta \mathbf{X\_0} - b\mathbf{X\_0^3} - c \left( D\_0^2 X\_0 \right) \mathbf{X\_0} - d(D\_0 \mathbf{X\_0})^2 \mathbf{X\_0^2} + F \cos \Omega T\_0 \\ &+ G \mathbf{X\_0} \cos \Omega T\_0 + K \mathbf{X\_0^2} \cos \Omega T\_0. \end{aligned} \tag{6}$$

General solutions of Eq. (5) can be written as

$$X\_0 = A(T\_1) \exp\left(i T\_0\right) + \overline{A}(T\_1) \exp\left(-i T\_0\right). \tag{7}$$

After substituting Eq. (7) into Eq. (8), we have

$$\begin{split} D\_0 \,^2 X\_1 + X\_1 &= -2i D\_1 A \exp\left(i T\_0\right) - 2i \zeta A \exp\left(i T\_0\right) - 3(b - c + d) A^2 \overline{A} \exp\left(i T\_0\right) - \\ \left(b - c + d\right) A^3 \exp\left(3i T\_0\right) &+ \frac{F}{2} \exp\left(i \Omega T\_0\right) + \frac{G}{2} A \exp\left(i \Omega + 1\right) T\_0 + \frac{G}{2} \overline{A} \exp\left(i \Omega - 1\right) T\_0 + \\ \frac{K}{2} A^2 \exp\left(i \Omega + 2\right) T\_0 + \frac{K}{2} A^2 \exp\left(i \Omega - 2\right) T\_0 + \frac{K}{2} A \overline{A} \exp\left(i \Omega T\_0\right) + c c. \end{split} \tag{8}$$

Any solution from the above equation may lead to an unbounded solution due to the existence of small divisor and secular terms in the equation. The terms associated *eiT*<sup>0</sup> or ≈ *ei*Ω*T*<sup>0</sup> , ≈e*<sup>i</sup>*ð Þ <sup>Ω</sup>�<sup>1</sup> *<sup>T</sup>*<sup>0</sup> , ≈e*<sup>i</sup>*ð Þ <sup>Ω</sup>�<sup>2</sup> *<sup>T</sup>*<sup>0</sup> are known as small divisor and secular terms. These terms are required to be eliminated to obtain any bounded solution.

It may be observed that these terms exist when Ω ≈ 1, Ω ≈ 3, or Ω ≈2. In the following sub-sections, three resonance conditions, i.e., primary resonance, parametric resonance condition, and third-order sub-harmonic conditions have been briefly discussed. A details derivation and explanation is being carried out in [25].

#### *3.2.1 Primary resonance condition*

#### *3.2.1.1 Reduced order model*

the theoretical and experimental investigation of dynamic performance of MEMS

Equation of motions [Eqs. (2)–(4)] for various MEMS devices comprise linear and non-linear terms, direct forced, parametric term, and non-linear parametric terms due to non-linear electrostatic actuation. Since, the temporal equation of motion holds non-linear terms; it is difficult to find closed form solution. Hence, one may go for approximate solution by using the perturbation method. Here, method of multiple scales as explained in [25, 29–33] is used to obtain the set of algebraic equations turning into non-autonomous equations of motion for three resonance conditions, viz. primary resonance, parametric resonance condition, and third-order sub-harmonic conditions are being expressed under steady state conditions. The procedures used to derive the reduced order equation are similar to those explained in [25, 29–33]. Based on numerical values of the coefficients of the damping, forcing, and non-linear terms, they are one order less than the coefficients of the linear terms, which have a value of unity in this case and as result, in the following technique, co-efficient are expressed as *a* ¼ 2*ες*, *b* ¼ *εb*, *c* ¼ *εc*, *d* ¼ *εd*, *F* ¼ *εF*, *G* ¼ *εG*, and *K* ¼ *εK* for sake of simplicity. By using method of multiple scales with the procedure as explained in [25, 30, 32, 33], substituting *Tn* ¼ *<sup>ε</sup><sup>n</sup>τ*, *<sup>n</sup>* <sup>¼</sup> 0, 1, 2, 3<sup>⋯</sup> and displacement *<sup>X</sup>*ð Þ¼ *<sup>τ</sup>*; *<sup>ε</sup> <sup>X</sup>*0ð Þþ *<sup>T</sup>*0, *<sup>T</sup>*<sup>1</sup> *<sup>ε</sup>X*1ð Þþ *<sup>T</sup>*0, *<sup>T</sup>*<sup>1</sup> *<sup>O</sup> <sup>ε</sup>*<sup>2</sup> ð Þ in Eq. (2) and equating the coefficients of like powers of *ε*, one may obtain the

> *ε*<sup>0</sup> : *D*<sup>0</sup> 2

> > <sup>0</sup> � *c D*<sup>2</sup>

*<sup>X</sup>*<sup>1</sup> <sup>þ</sup> *<sup>X</sup>*<sup>1</sup> ¼ �2*iD*1*<sup>A</sup>* exp ð Þ� *iT*<sup>0</sup> <sup>2</sup>*iζ<sup>A</sup>* exp ð Þ� *iT*<sup>0</sup> <sup>3</sup>ð Þ *<sup>b</sup>* � *<sup>c</sup>* <sup>þ</sup> *<sup>d</sup> <sup>A</sup>*<sup>2</sup>

<sup>2</sup> exp ð Þþ *<sup>i</sup>*Ω*T*<sup>0</sup>

*<sup>A</sup>*<sup>2</sup> exp *<sup>i</sup>*ð Þ <sup>Ω</sup> � <sup>2</sup> *<sup>T</sup>*<sup>0</sup> <sup>þ</sup>

<sup>0</sup> *X*<sup>0</sup>

*G* 2

> *K* 2

Any solution from the above equation may lead to an unbounded solution due to the existence of small divisor and secular terms in the equation. The terms associated *eiT*<sup>0</sup> or ≈ *ei*Ω*T*<sup>0</sup> , ≈e*<sup>i</sup>*ð Þ <sup>Ω</sup>�<sup>1</sup> *<sup>T</sup>*<sup>0</sup> , ≈e*<sup>i</sup>*ð Þ <sup>Ω</sup>�<sup>2</sup> *<sup>T</sup>*<sup>0</sup> are known as small divisor and secular terms. These terms are required to be eliminated to obtain any bounded solution.

*<sup>X</sup>*<sup>0</sup> � *d D*ð Þ <sup>0</sup>*X*<sup>0</sup>

*X*<sup>0</sup> ¼ *A T*ð Þ<sup>1</sup> exp ð Þþ *iT*<sup>0</sup> *A T*ð Þ<sup>1</sup> exp ð Þ �*iT*<sup>0</sup> *:* (7)

*A* exp *i*ð Þ Ω þ 1 *T*<sup>0</sup> þ

*AA* exp ð Þþ *i*Ω*T*<sup>0</sup> *cc:*

<sup>0</sup> cos Ω*T*0*:* (6)

*<sup>X</sup>*€ <sup>þ</sup> *<sup>X</sup>* <sup>þ</sup> *aX*\_ <sup>þ</sup> *bX*<sup>3</sup> <sup>¼</sup> *<sup>F</sup>* cos <sup>Ω</sup>*τ:* (4)

*X*<sup>0</sup> þ *X*<sup>0</sup> ¼ 0, (5)

2 *X*2

<sup>0</sup> þ *F* cos Ω*T*<sup>0</sup>

*A* exp ð Þ� *iT*<sup>0</sup>

*A* exp *i*ð Þ Ω � 1 *T*0þ

(8)

*G* 2

devices considering the mid-plane stretching effect.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

**3.2 Bifurcation and stability**

following expressions:

¼ �2*D*<sup>0</sup> *<sup>D</sup>*1*X*<sup>0</sup> � <sup>2</sup>*iζX*<sup>0</sup> � *bX*<sup>3</sup>

General solutions of Eq. (5) can be written as

After substituting Eq. (7) into Eq. (8), we have

*F*

*K* 2

<sup>þ</sup> *GX*<sup>0</sup> cos <sup>Ω</sup>*T*<sup>0</sup> <sup>þ</sup> *KX*<sup>2</sup>

ð Þ *<sup>b</sup>* � *<sup>c</sup>* <sup>þ</sup> *<sup>d</sup> <sup>A</sup>*<sup>3</sup> exp 3ð Þþ *iT*<sup>0</sup>

*<sup>A</sup>*<sup>2</sup> exp *<sup>i</sup>*ð Þ <sup>Ω</sup> <sup>þ</sup> <sup>2</sup> *<sup>T</sup>*<sup>0</sup> <sup>þ</sup>

Order

Order

*ε*<sup>1</sup> : *D*<sup>0</sup> 2 *X*<sup>1</sup> þ *X*<sup>1</sup>

*D*<sup>0</sup> 2

*K* 2

**248**

Here, the resonance condition occurs when the frequency of applied voltage becomes equal to that of one of the natural frequencies, i.e., fundamental natural frequency. Following reduced equations are obtained as given below replacing *X* ¼ *a T*ð Þ<sup>0</sup> *ei<sup>ϕ</sup>*ð Þ *<sup>T</sup>*<sup>0</sup> . A detail explanation about to obtain these reduced equations is being carried out in [25].

$$
\delta \mathbf{8} a' = -8\zeta a + 4F\sin\phi + Ka^2\sin\phi,\tag{9}
$$

$$8a\phi'=8a\sigma-(\Im b-\Im c+d)(\simeq\mu)a^3+4F\cos\phi+\Im a^2\cos\phi.\tag{10}$$

Here, system only exhibits only non-trivial responses, i.e., *a* 6¼ 0 obtained from (Eqs. 9–10). Dynamic responses are being determined by solving the set of algebraic equations obtained by converting differential equations into set of algebraic equations under steady state conditions, i.e., *a*<sup>0</sup> ¼ 0, and *ϕ*<sup>0</sup> ¼ 0. Here, stability of the steady state responses has been analyzed by investigating eigenvalues of the Jacobian matrix, which has been obtained by perturbing the algebraic equations with *a* ¼ *ao* þ *a*<sup>1</sup> and *γ* ¼ *γ*<sup>0</sup> þ *γ*1, where *a*0, *γ*<sup>0</sup> are the singular points.

#### *3.2.1.2 Results and discussions*

Here, the condition at which the resonator has been excited with a frequency of the applied alternative voltage nearly equal to the fundamental frequency of the resonator is being discussed. In this vibrating state, the amplitude of vibration is always found to be a non-zero solution, while both stable and unstable non-trivial solutions are being observed. The study of bifurcation is being carried out on to see the losses the stability of system when a parameter passes through a critical value called a bifurcation point. The sudden change in amplitude undergoes catastrophic failure of the system. The graphical illustration of the vibration amplitude with varying the system control parameter has been constructed.

**Figure 5a** represents a typical frequency response curves for a specific air-gap ð Þ ≃*μ* between resonator and stationary. It is noteworthy that amplitude of vibration becomes increasing with increase in forcing frequency. The non-linear mode of operation possesses most likely hardening when the effective structural nonlinearity becomes *μ* ¼ þ2*:*0. In this condition, restoring forces due to geometric non-linearity overshadows the inertia effect of the device that leads to hardening spring effect. When the effective structural non-linearity becomes *μ* ¼ �2*:*0, vice versa effect is observed. For sweeping up the frequency moving toward E from point D, system undergoes a sudden jump down when the frequency reaches to its critical value regarded as saddle-node fixed bifurcation point. Similarly, a sudden upward jump in the amplitude from lower to higher amplitude undergoes for a sweeping down frequency. With further increase in frequency, the amplitude of vibratory motion decreases and follows the path DA. Hence, with decrease in frequency leads to the lower amplitude of responses from a higher value continuously.

With the experimental investigation, it has been noted that these jump phenomena may lead to mechanical crack across the width of the beam. The growth of the crack may further propagate with the experiences of subsequent jump up and down in response amplitude (**Figures 6** and **7**).

It is observed that multiple solutions exist at some frequencies in the entire frequency range from 0.5 to 1.5. Being existence of multiple solutions, it is desirable to check whether all solutions are found to be stable or unstable or mixed solution. For a specific initial condition, trajectories have been drawn in the plane of amplitude and phase as time goes infinity. It is being observed that the system possesses the condition of bi-stability at some regions. Thus, in this region, wrong selection of initial conditions mostly results the wrong output response. It is thus keyed to opt out an appropriate condition for a specific solution that can prevail physically by the

*Electrostatically Driven MEMS Resonator: Pull-in Behavior and Non-linear Phenomena*

Here, the resonance condition occurs when the frequency of applied voltage becomes twice the natural frequencies, i.e., fundamental natural frequency. Fol-

Here, system possesses both trivial *a* ¼ 0 and non-trivial *a* 6¼ 0 responses determined by solving the reduced non-linear algebraic equations at steady state condition by using Newton's method, simultaneously. The stability of the steady state responses has also here obtained by replacing *a* with *ao* þ *a*1, *γ* with *γ*<sup>0</sup> þ *γ*1, respectively, and then investigating the eigenvalues of the resulting Jacobian

The electrostatically actuated micro- beam is vibrating with a frequency of the applied voltage nearly equal to the twice the fundamental frequency of the resonator. Unlike, primary resonance case, here, the system possesses both trivial and non-trivial solutions. Here, the vibration amplitude may vary from zero to non-zero value and vice-versa depending upon the state of vibration being considered. Depending upon the selected values of control parameters, the trivial and nontrivial solutions are noticed as stable and unstable for a specific frequency of the AC voltages. Sub-critical pitchfork bifurcation leads to sudden change in amplitude. This discontinuity in amplitude results catastrophic failure of the system.

Approximate solutions obtained by using the method of multiple scales have been compared with those found by numerically solving the temporal Eq. (2). Time

response finally moves to stable non-trivial fixed point response. Responses mostly obtained by solving the temporal equation of motion are being in good agreement

Similarly, one may have following reduced equations when the frequency of

<sup>8</sup>*a*<sup>0</sup> ¼ �8*ζ<sup>a</sup>* <sup>þ</sup> *Ka*<sup>2</sup> sin *<sup>ϕ</sup>*, (13)

applied voltage is nearly equal to that of three times of natural frequency

response clearly shows that the trajectory initiated from the unstable trivial

with those findings by perturbation technique.

**3.4 Sub-harmonic resonance case (Θ ≈ 3)**

4*a*<sup>0</sup> ¼ �4*ζ a* þ *Ha* sin *ϕ*, (11)

<sup>8</sup>*aϕ*<sup>0</sup> <sup>¼</sup> <sup>8</sup>*a<sup>σ</sup>* � ð Þ <sup>3</sup>*<sup>b</sup>* � <sup>3</sup>*<sup>c</sup>* <sup>þ</sup> *<sup>d</sup>* ð Þ <sup>≃</sup> *<sup>μ</sup> <sup>a</sup>*<sup>3</sup> <sup>þ</sup> <sup>4</sup>*<sup>H</sup>* cos *<sup>ϕ</sup>:* (12)

system.

matrix (J).

**251**

*3.3.1.1 Results and discussions*

*3.3.1 Reduced order model*

**3.3 Principal parametric resonance (Θ ≈ 2)**

*DOI: http://dx.doi.org/10.5772/intechopen.88453*

lowing reduced equations are obtained as given below:

**Figure 5.** *(a) Frequency response curves for ς* ¼ *0:1*, *F* ¼ *1:2*,*K* ¼ *0:12.(b) Basins of attraction at bi-stable point [25].*

#### **Figure 6.**

*(a) Frequency characteristics curves for ς* ¼ *0:1*, *H* ¼ *1:5: (b) Time histories at unstable point [25].*

#### **Figure 7.**

*(a) Frequency characteristics curve for ς* ¼ *0:1*, *and K* ¼ *1:5. (b) Frequency characteristics curve for d=l* ¼ *0:2*, *ς* ¼ *0:2 and K* ¼ *1:0.*

*Electrostatically Driven MEMS Resonator: Pull-in Behavior and Non-linear Phenomena DOI: http://dx.doi.org/10.5772/intechopen.88453*

It is observed that multiple solutions exist at some frequencies in the entire frequency range from 0.5 to 1.5. Being existence of multiple solutions, it is desirable to check whether all solutions are found to be stable or unstable or mixed solution. For a specific initial condition, trajectories have been drawn in the plane of amplitude and phase as time goes infinity. It is being observed that the system possesses the condition of bi-stability at some regions. Thus, in this region, wrong selection of initial conditions mostly results the wrong output response. It is thus keyed to opt out an appropriate condition for a specific solution that can prevail physically by the system.

#### **3.3 Principal parametric resonance (Θ ≈ 2)**

#### *3.3.1 Reduced order model*

With the experimental investigation, it has been noted that these jump phenomena may lead to mechanical crack across the width of the beam. The growth of the crack may further propagate with the experiences of subsequent jump up and

*(a) Frequency response curves for ς* ¼ *0:1*, *F* ¼ *1:2*,*K* ¼ *0:12.(b) Basins of attraction at bi-stable point [25].*

*(a) Frequency characteristics curves for ς* ¼ *0:1*, *H* ¼ *1:5: (b) Time histories at unstable point [25].*

*(a) Frequency characteristics curve for ς* ¼ *0:1*, *and K* ¼ *1:5. (b) Frequency characteristics curve for d=l* ¼

down in response amplitude (**Figures 6** and **7**).

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

**Figure 6.**

**Figure 7.**

**250**

*0:2*, *ς* ¼ *0:2 and K* ¼ *1:0.*

**Figure 5.**

Here, the resonance condition occurs when the frequency of applied voltage becomes twice the natural frequencies, i.e., fundamental natural frequency. Following reduced equations are obtained as given below:

$$4a' = -4\zeta a + H a \sin \phi,\tag{11}$$

$$8a\,\phi'=8a\sigma-(\Im b-\Im c+d)(\simeq\mu)a^3+4H\cos\phi.\tag{12}$$

Here, system possesses both trivial *a* ¼ 0 and non-trivial *a* 6¼ 0 responses determined by solving the reduced non-linear algebraic equations at steady state condition by using Newton's method, simultaneously. The stability of the steady state responses has also here obtained by replacing *a* with *ao* þ *a*1, *γ* with *γ*<sup>0</sup> þ *γ*1, respectively, and then investigating the eigenvalues of the resulting Jacobian matrix (J).

#### *3.3.1.1 Results and discussions*

The electrostatically actuated micro- beam is vibrating with a frequency of the applied voltage nearly equal to the twice the fundamental frequency of the resonator. Unlike, primary resonance case, here, the system possesses both trivial and non-trivial solutions. Here, the vibration amplitude may vary from zero to non-zero value and vice-versa depending upon the state of vibration being considered. Depending upon the selected values of control parameters, the trivial and nontrivial solutions are noticed as stable and unstable for a specific frequency of the AC voltages. Sub-critical pitchfork bifurcation leads to sudden change in amplitude. This discontinuity in amplitude results catastrophic failure of the system.

Approximate solutions obtained by using the method of multiple scales have been compared with those found by numerically solving the temporal Eq. (2). Time response clearly shows that the trajectory initiated from the unstable trivial response finally moves to stable non-trivial fixed point response. Responses mostly obtained by solving the temporal equation of motion are being in good agreement with those findings by perturbation technique.

#### **3.4 Sub-harmonic resonance case (Θ ≈ 3)**

Similarly, one may have following reduced equations when the frequency of applied voltage is nearly equal to that of three times of natural frequency

$$
\mathbf{8a'} = -\mathbf{8\zeta}a + \mathbf{K}a^2 \sin \phi,\tag{13}
$$

$$8a\phi'=8a\sigma-(\Im b-\Im c+d)(\simeq\mu)a^3+\Im K a^2 \cos\phi.\tag{14}$$

**References**

2002;**7**:31-49

2002;**12**:458-464

2002;**12**:759-766

[1] Luo ACJ, Wang FY. Chaotic motion in a Micor-electro-mechanical system with non-linearity from capacitors. Communications in Nonlinear Science and Numerical Simulation.

*DOI: http://dx.doi.org/10.5772/intechopen.88453*

[9] Kuang JH, Chen CJ. Dynamic characteristics of shaped microactuators solved using the differential quadrature method. Journal of

2004;**14**:647-655

*Electrostatically Driven MEMS Resonator: Pull-in Behavior and Non-linear Phenomena*

2005;**15**:419-429

Micromechanics and Microengineering.

[10] Najar F, Houra S, El-Borgi S, Abdel-Rahman E, Nayfeh A. Modeling and

electrostatic microactuators. Journal of Micromechanics and Microengineering.

[11] Zhang W, Meng G. Nonlinear dynamical system of micro-cantilever under combined parametric and forcing excitations in MEMS. Sensors and Actuators A. 2005;**119**:291-299

[12] Rhoads JF, Shaw SW, Turner KL. The nonlinear response of resonant microbeam systems with purelyparametric electrostatic actuation. Journal of Micromechanics and Microengineering. 2006;**16**:890-899

[13] Krylov S, Seretensky S. Higher order correction of electrostatic pressure and its influence on the pull-in behaviour of

Micromechanics and Microengineering.

[14] Decuzzi P, Granaldi A, Pascazio G. Dynamic response of microcantileverbased sensors in a fluidic chamber. Journal of Applied Physics. 2007;**101**:

[15] Batra R, Porfiri M, Spinello D. Review of modelling electrostatically actuated micro-electromechanical systems. Smart Materials and Structures. 2007;**16**:23-31

[16] Fargas MA, Costa CR, Shakel AM. Modeling the Electrostatic Actuation of MEMS: State of the Art 2005. Barcelona, Spain: Institute of Industrial and Control

microstructures. Journal of

2006;**16**:1382-1396

Engineering; 2005

024303

design of variable-geometry

[2] Pamidighantam S, Puers R, Baert K, Tilmans H. Pull-in voltage analysis of electrostatically actuated beam structures with fixed–fixed and fixed–free end conditions. Journal of Micromechanics and Microengineering.

[3] Abdel-Rahman EM, Younis MI, Nayfeh AH. Characterization of the mechanical behavior of an electrically actuated microbeam. Journal of

Micromechanics and Microengineering.

[4] Younis M, Nayfeh A. Study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dynamics. 2003;**31**:91-117

[5] Xie W, Lee H, Lim S. Nonlinear dynamic analysis of MEMS switches by nonlinear modal analysis. Nonlinear

[6] Younis MI, Abdel-Rahman EM, Nayfeh AH. A reduced-order model for electrically actuated microbeam-based MEMS. Journal of Microelectromechanical

[7] Luo ACJ, Wang F-E. Nonlinear dynamics of a micro-electro-mechanical system with time-varying capacitors. Journal of Vibration and Acoustic. 2000;

[8] Teva J, Abadal G, Davis ZJ, Verd J, Borrise X, Boisen A, et al. On the electromechanical modeling of a resonating nano-cantilever-based transducer. Ultramicroscopy. 2004;**100**:

Dynamics. 2003;**3**:243-256

Systems. 2003;**12**:672-680

**126**:77-83

225-232

**253**

Similar to the previous resonance case, here also system possesses both trivial *a* ¼ 0 and non-trivial *a* 6¼ 0 responses obtained by solving the equations obtained after setting *a*<sup>0</sup> and *γ*<sup>0</sup> equal to zero using similar Newton's method for different system parameters. Similar procedure one may follow to find out the stability of the steady state response of this case by investigating the nature of the equilibrium points.

This resonance takes place when the frequency of applied voltage is nearly equal to thrice the fundamental frequency of the resonator. Here, the amplitude of vibration may shift from non-zero to zero value depending on the initial operating point. In this resonance case, trivial solutions are found to be stable for any frequency and control parameters. The loss of stability of the system depends on the position of critical point and selection of control parameters, while the system can bring down to stable condition by simply choosing the frequency and other system parameters, appropriately. The bifurcation present here is known as addle-node bifurcation point. The jump length is found to be increased with increase in control parameters. Similarly, it has been observed that jump length will increase with increase in both forcing parameter and damping.

#### **4. Conclusions**

The investigation on stability and bifurcation analysis of a highly non-linear electrically driven MEMS resonator along with pull-in behavior has been established. A non-linear mathematical model has been briefly described accounting off midplane stretching and non-linear electrostatic pressure under both DC and AC actuation. A short description of perturbation method to study the steady state responses has been highlighted. The pull-in results and consequences of non-linear effects on dynamics responses have been reported. Non-linear phenomenon has been studied to highlight the possible undesirable catastrophic failure at the unstable critical points, i.e., bifurcation. Basins of attractions that postulate a unique response in multi-region state for a specific initial condition has been demonstrated. This chapter enables a significant understanding about the locus of instability in micro-cantilever-based resonator when subjected to DC and AC potentials. A theoretical understanding for controlling the systems and optimizing their operation is being reported here.

#### **Author details**

Barun Pratiher

Department of Mechanical Engineering, Indian Institute of Technology Jodhpur, India

\*Address all correspondence to: barun@iitj.ac.in

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Electrostatically Driven MEMS Resonator: Pull-in Behavior and Non-linear Phenomena DOI: http://dx.doi.org/10.5772/intechopen.88453*

#### **References**

<sup>8</sup>*aϕ*<sup>0</sup> <sup>¼</sup> <sup>8</sup>*a<sup>σ</sup>* � ð Þ <sup>3</sup>*<sup>b</sup>* � <sup>3</sup>*<sup>c</sup>* <sup>þ</sup> *<sup>d</sup>* ð Þ <sup>≃</sup>*<sup>μ</sup> <sup>a</sup>*<sup>3</sup> <sup>þ</sup> <sup>3</sup>*Ka*<sup>2</sup> cos *<sup>ϕ</sup>:* (14)

Similar to the previous resonance case, here also system possesses both trivial *a* ¼ 0 and non-trivial *a* 6¼ 0 responses obtained by solving the equations obtained after setting *a*<sup>0</sup> and *γ*<sup>0</sup> equal to zero using similar Newton's method for different system parameters. Similar procedure one may follow to find out the stability of the steady state response of this case by investigating the nature of the equilibrium

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

This resonance takes place when the frequency of applied voltage is nearly equal to thrice the fundamental frequency of the resonator. Here, the amplitude of vibration may shift from non-zero to zero value depending on the initial operating point. In this resonance case, trivial solutions are found to be stable for any frequency and control parameters. The loss of stability of the system depends on the position of critical point and selection of control parameters, while the system can bring down to stable condition by simply choosing the frequency and other system parameters, appropriately. The bifurcation present here is known as addle-node bifurcation point. The jump length is found to be increased with increase in control parameters. Similarly, it has been observed that jump length will increase with increase in both

The investigation on stability and bifurcation analysis of a highly non-linear electrically driven MEMS resonator along with pull-in behavior has been established. A non-linear mathematical model has been briefly described accounting off midplane stretching and non-linear electrostatic pressure under both DC and AC actuation. A short description of perturbation method to study the steady state responses has been highlighted. The pull-in results and consequences of non-linear effects on dynamics responses have been reported. Non-linear phenomenon has been studied to highlight the possible undesirable catastrophic failure at the unstable critical points, i.e., bifurcation. Basins of attractions that postulate a unique response in multi-region state for a specific initial condition has been demonstrated. This chapter enables a significant understanding about the locus of instability in micro-cantilever-based resonator when subjected to DC and AC potentials. A theoretical understanding for controlling the systems and optimizing their operation is being reported here.

Department of Mechanical Engineering, Indian Institute of Technology Jodhpur,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: barun@iitj.ac.in

provided the original work is properly cited.

points.

forcing parameter and damping.

**4. Conclusions**

**Author details**

Barun Pratiher

India

**252**

[1] Luo ACJ, Wang FY. Chaotic motion in a Micor-electro-mechanical system with non-linearity from capacitors. Communications in Nonlinear Science and Numerical Simulation. 2002;**7**:31-49

[2] Pamidighantam S, Puers R, Baert K, Tilmans H. Pull-in voltage analysis of electrostatically actuated beam structures with fixed–fixed and fixed–free end conditions. Journal of Micromechanics and Microengineering. 2002;**12**:458-464

[3] Abdel-Rahman EM, Younis MI, Nayfeh AH. Characterization of the mechanical behavior of an electrically actuated microbeam. Journal of Micromechanics and Microengineering. 2002;**12**:759-766

[4] Younis M, Nayfeh A. Study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dynamics. 2003;**31**:91-117

[5] Xie W, Lee H, Lim S. Nonlinear dynamic analysis of MEMS switches by nonlinear modal analysis. Nonlinear Dynamics. 2003;**3**:243-256

[6] Younis MI, Abdel-Rahman EM, Nayfeh AH. A reduced-order model for electrically actuated microbeam-based MEMS. Journal of Microelectromechanical Systems. 2003;**12**:672-680

[7] Luo ACJ, Wang F-E. Nonlinear dynamics of a micro-electro-mechanical system with time-varying capacitors. Journal of Vibration and Acoustic. 2000; **126**:77-83

[8] Teva J, Abadal G, Davis ZJ, Verd J, Borrise X, Boisen A, et al. On the electromechanical modeling of a resonating nano-cantilever-based transducer. Ultramicroscopy. 2004;**100**: 225-232

[9] Kuang JH, Chen CJ. Dynamic characteristics of shaped microactuators solved using the differential quadrature method. Journal of Micromechanics and Microengineering. 2004;**14**:647-655

[10] Najar F, Houra S, El-Borgi S, Abdel-Rahman E, Nayfeh A. Modeling and design of variable-geometry electrostatic microactuators. Journal of Micromechanics and Microengineering. 2005;**15**:419-429

[11] Zhang W, Meng G. Nonlinear dynamical system of micro-cantilever under combined parametric and forcing excitations in MEMS. Sensors and Actuators A. 2005;**119**:291-299

[12] Rhoads JF, Shaw SW, Turner KL. The nonlinear response of resonant microbeam systems with purelyparametric electrostatic actuation. Journal of Micromechanics and Microengineering. 2006;**16**:890-899

[13] Krylov S, Seretensky S. Higher order correction of electrostatic pressure and its influence on the pull-in behaviour of microstructures. Journal of Micromechanics and Microengineering. 2006;**16**:1382-1396

[14] Decuzzi P, Granaldi A, Pascazio G. Dynamic response of microcantileverbased sensors in a fluidic chamber. Journal of Applied Physics. 2007;**101**: 024303

[15] Batra R, Porfiri M, Spinello D. Review of modelling electrostatically actuated micro-electromechanical systems. Smart Materials and Structures. 2007;**16**:23-31

[16] Fargas MA, Costa CR, Shakel AM. Modeling the Electrostatic Actuation of MEMS: State of the Art 2005. Barcelona, Spain: Institute of Industrial and Control Engineering; 2005

[17] Lin RM, Wang WJ. Structural dynamics of microsystems—Current state of research and future directions. Mechanical Systems and Signal Processing. 2006;**20**:1015-1043

[18] Rhoads J, Shaw SW, Turner KL. Nonlinear dynamics and its applications in micro- and Nano-resonators. In: Proceedings of DSCC 2008, ASME Dynamic Systems and Control Conference. Ann Arbor, Michigan, USA; October 20-22 2008

[19] Nayfeh A, Younis MI, Abdel-Rahman EM. Dynamic pull-in phenomenon in MEMS resonators. Nonlinear Dynamics. 2007;**48**:153-163

[20] Zhang WM, Meng G, Chen D. Stability, nonlinearity and reliability of electrostatically actuated MEMS devices. Sensors. 2007;**7**:760-796

[21] Chao PCP, Chiu C, Liu TH. DC dynamics pull-in predictions for a generalized clamped–clamped microbeam based on a continuous model and bifurcation analysis. Journal of Micromechanics and Microengineering. 2008;**18**:115008

[22] Shao L, Palaniapan M, Tan W. The nonlinearity cancellation phenomenon in micromechanical resonators. Journal of Micromechanics and Microengineering. 2008;**18**:065014

[23] Moghimi ZM, Ahmadian M, Rashidian B. Semi-analytic solutions to nonlinear vibrations of microbeams under suddenly applied voltages. Journal of Sound and Vibration. 2009; **325**:382-396

[24] Chatterjee S, Pohit G. A large deflection model for the pull-in analysis of electrostatically actuated microcantilever beams. Journal of Sound and Vibration. 2009;**322**:969-986

[25] Pratiher B. Stability and bifurcation analysis of an electrostatically controlled highly deformable microcantileverbased resonator. Nonlinear Dynamics. 2014;**78**(3):1781-1800

**Chapter 14**

**Abstract**

**1. Introduction**

**255**

Nonlinear Oxygen Transport with

In a recent paper by the authors, a well-known governing nonlinear PDE used to model oxygen transport was formulated in a generalized coordinate system where the Laplacian was expressed in metric tensor form. A reduction of the PDE to a simpler problem, subject to specific integrability conditions, was shown, and in the present work, a novel approximate analytical solution is obtained in terms of the degenerate Weierstrass P function using a compatibility relation through the factorization of the reduced almost linear ode and subject to similar boundary conditions for a microfluidic channel used in recent work by the authors. A specific form of the initial equation which was reduced has been used by Nair and coworkers describing the intraluminal problem of oxygen transport in large capillaries or arterioles and more recent work by the corresponding author describing the release of adenosine triphosphate (ATP) in micro-channels. In the present problem, a channel with a central core, rich in red blood cells, and with a thin plasma region

**Keywords:** almost linear ODE, Poiseuille flow, oxygen transport, Painlevé analysis

Various biophysical phenomena are modeled using nonlinear differential equa-

tions. Such is the case of a model used by Nair et al. [1–3] to describe oxygen transport in large capillaries [1–3]. This model incorporates two regions of blood flow. One is a core region of the blood with RBCs present, and in this core, oxygen dissociates into blood oxyhemoglobin. The velocity of blood in the core region is a function of the plasma velocity and rate of oxygen dissociating. The second region is a thin strip of flowing plasma with no RBCs at the wall of the micro-fluidic channel. The appropriate Robin condition and no-flux conditions are incorporated at and around a permeable membrane with oxygen transport through the membrane. Since there are two distinct regions of flow of liquid, RBCs with plasma and plasma alone, it is necessary to match the rate of change of partial pressure of oxygen at the common boundary of the liquid in each of the two regions. This kind of model has been used previously by Moschandreou et al. [4] studying the influence of tissue metabolism and capillary oxygen supply on arteriolar oxygen transport. In that study a numerical approach was used to solve the governing equations. In the present work we seek an analytical solution for a different highly nonlinear problem

Poiseuille Hemodynamic Flow

in a Micro-Channel

*Terry E. Moschandreou and Keith C. Afas*

near the boundary wall, free of RBCs is considered.

[26] Nayfeh AH, Mook DT. Nonlinear Oscillations. New York: Wiley-VCH; 1995

[27] Cartmell MP. Introduction to Linear, Parametric and Nonlinear Vibrations. London: Chapman and Hall; 1990

[28] Nayfeh AH, Balachandran B. Applied Nonlinear Dynamics: Analytical. Computational and Experimental Methods: Wiley; 1995

[29] Harsha CS, Prasanth CSR, Pratiher B. Prediction of pull-in phenomena and structural stability analysis of an electrostatically actuated microswitch. Acta Mechanica. 2016; **227**(9):2577-2594

[30] Pratiher B. Tuning the nonlinear behaviour of resonant MEMS sensors actuated electrically. Procedia Engineering. 2012;**47**:9-12

[31] Harsha CS, Prasanth CSR, Pratiher B. Modeling and non-linear responses of MEMS capacitive accelerometer. MATEC Web of Conferences. 2014;**16**:04003

[32] Harsha CS, Prasanth CSR, Pratiher B. Electrostatic pull-in analysis of a nonuniform micro-resonator undergoing large elastic deflection. Journal of Mechanical Engineering Sciences. 2018;**232**:3337-3350

[33] Harsha CS, Prasanth CSR, Pratiher B. Effect of squeeze film damping and AC actuation voltage on pull-in phenomenon of electrostatically actuated microswitch. Procedia Engineering. 2016;**144**:891-899

#### **Chapter 14**

[17] Lin RM, Wang WJ. Structural dynamics of microsystems—Current state of research and future directions.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

highly deformable microcantileverbased resonator. Nonlinear Dynamics.

[26] Nayfeh AH, Mook DT. Nonlinear Oscillations. New York: Wiley-VCH;

[27] Cartmell MP. Introduction to Linear, Parametric and Nonlinear Vibrations. London: Chapman and Hall;

[28] Nayfeh AH, Balachandran B. Applied Nonlinear Dynamics: Analytical. Computational and Experimental Methods: Wiley; 1995

[29] Harsha CS, Prasanth CSR, Pratiher B. Prediction of pull-in phenomena and structural stability analysis of an electrostatically actuated microswitch. Acta Mechanica. 2016;

[30] Pratiher B. Tuning the nonlinear behaviour of resonant MEMS sensors actuated electrically. Procedia Engineering. 2012;**47**:9-12

[31] Harsha CS, Prasanth CSR, Pratiher B. Modeling and non-linear responses of MEMS capacitive accelerometer. MATEC Web of Conferences. 2014;**16**:04003

[32] Harsha CS, Prasanth CSR,

[33] Harsha CS, Prasanth CSR, Pratiher B. Effect of squeeze film damping and AC actuation voltage on pull-in phenomenon of electrostatically

actuated microswitch. Procedia Engineering. 2016;**144**:891-899

Pratiher B. Electrostatic pull-in analysis of a nonuniform micro-resonator undergoing large elastic deflection. Journal of Mechanical Engineering Sciences. 2018;**232**:3337-3350

**227**(9):2577-2594

2014;**78**(3):1781-1800

1995

1990

Mechanical Systems and Signal Processing. 2006;**20**:1015-1043

USA; October 20-22 2008

[19] Nayfeh A, Younis MI,

[18] Rhoads J, Shaw SW, Turner KL. Nonlinear dynamics and its applications in micro- and Nano-resonators. In: Proceedings of DSCC 2008, ASME Dynamic Systems and Control Conference. Ann Arbor, Michigan,

Abdel-Rahman EM. Dynamic pull-in phenomenon in MEMS resonators. Nonlinear Dynamics. 2007;**48**:153-163

[20] Zhang WM, Meng G, Chen D. Stability, nonlinearity and reliability of electrostatically actuated MEMS devices. Sensors. 2007;**7**:760-796

[21] Chao PCP, Chiu C, Liu TH. DC dynamics pull-in predictions for a generalized clamped–clamped microbeam based on a continuous model and

Micromechanics and Microengineering.

[22] Shao L, Palaniapan M, Tan W. The nonlinearity cancellation phenomenon in micromechanical resonators. Journal

Microengineering. 2008;**18**:065014

[23] Moghimi ZM, Ahmadian M, Rashidian B. Semi-analytic solutions to nonlinear vibrations of microbeams under suddenly applied voltages. Journal of Sound and Vibration. 2009;

[24] Chatterjee S, Pohit G. A large deflection model for the pull-in analysis

[25] Pratiher B. Stability and bifurcation analysis of an electrostatically controlled

of electrostatically actuated microcantilever beams. Journal of Sound and Vibration. 2009;**322**:969-986

bifurcation analysis. Journal of

2008;**18**:115008

**325**:382-396

**254**

of Micromechanics and

## Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel

*Terry E. Moschandreou and Keith C. Afas*

#### **Abstract**

In a recent paper by the authors, a well-known governing nonlinear PDE used to model oxygen transport was formulated in a generalized coordinate system where the Laplacian was expressed in metric tensor form. A reduction of the PDE to a simpler problem, subject to specific integrability conditions, was shown, and in the present work, a novel approximate analytical solution is obtained in terms of the degenerate Weierstrass P function using a compatibility relation through the factorization of the reduced almost linear ode and subject to similar boundary conditions for a microfluidic channel used in recent work by the authors. A specific form of the initial equation which was reduced has been used by Nair and coworkers describing the intraluminal problem of oxygen transport in large capillaries or arterioles and more recent work by the corresponding author describing the release of adenosine triphosphate (ATP) in micro-channels. In the present problem, a channel with a central core, rich in red blood cells, and with a thin plasma region near the boundary wall, free of RBCs is considered.

**Keywords:** almost linear ODE, Poiseuille flow, oxygen transport, Painlevé analysis

#### **1. Introduction**

Various biophysical phenomena are modeled using nonlinear differential equations. Such is the case of a model used by Nair et al. [1–3] to describe oxygen transport in large capillaries [1–3]. This model incorporates two regions of blood flow. One is a core region of the blood with RBCs present, and in this core, oxygen dissociates into blood oxyhemoglobin. The velocity of blood in the core region is a function of the plasma velocity and rate of oxygen dissociating. The second region is a thin strip of flowing plasma with no RBCs at the wall of the micro-fluidic channel. The appropriate Robin condition and no-flux conditions are incorporated at and around a permeable membrane with oxygen transport through the membrane. Since there are two distinct regions of flow of liquid, RBCs with plasma and plasma alone, it is necessary to match the rate of change of partial pressure of oxygen at the common boundary of the liquid in each of the two regions. This kind of model has been used previously by Moschandreou et al. [4] studying the influence of tissue metabolism and capillary oxygen supply on arteriolar oxygen transport. In that study a numerical approach was used to solve the governing equations. In the present work we seek an analytical solution for a different highly nonlinear problem in a micro-fluidic channel. Flows that are fully developed at inlet were studied by Ng [5] who studied oscillatory dispersion in a tube with chemical species undergoing linear reversible and irreversible reactions at the tube wall. Relative importance to the present work is that fully developed flow occurs at inlet of channel with boundary conditions specified on the wall. No inlet mass transport is specified at inlet of channel similar to [5] but unlike [6], for example. Using Painlevé analysis [7], certain nonlinear second-order ordinary differential equations (ODE) can be factorized and solved. We consider the model of Nair et al. [1–3], where the nonlinear PDE reduced in [8], to an "almost linear" second-order ode is considered. It is well-known that the Weierstrass P function in its series form is problematic to use in computational work due to very slow convergence of numerical methods. It is the aim of the present work to show that a degenerate form of the special function can be used as in [9] in the reduction of the nonlinear PDE in [1–3]. A thorough and recent review of oxygen control with microfluidics has been carried out in [10] and all of its references within. In this work we see how the microscale can be leveraged for oxygen control of RBCs.

#### **2. General tensorial mass transport**

Regardless of the kinematics of a surface (dynamic or stationary), all surfaces, *S* ¼ ∂Ω, enclosing a solid volume, Ω, obey the following intuitive conservation relation for an enclosed observable mass, *mo* of some arbitrary substance:

$$\frac{d}{dt}m\_o + \int\_S \mathbf{j}\_o \cdot \mathbf{dS} = \Sigma,\tag{1}$$

*d dt* ð Ω

*DOI: http://dx.doi.org/10.5772/intechopen.90575*

convention) by the tensorial equation:

where **<sup>N</sup>** <sup>¼</sup> **<sup>Z</sup>***<sup>i</sup>*

ð Ω

ð Ω

*∂ ∂t*

surface integral terms into one:

ð Ω

where **<sup>v</sup>***<sup>o</sup>* <sup>¼</sup> *vi*

**257**

*∂ ∂t*

the terms under one volume integral:

*∂ ∂t*

*ρ<sup>o</sup> d*Ω þ

term into a tensorial formation, by recognizing *j*

*ρ<sup>o</sup> d*Ω þ

ð Ω

*∂ ∂t*

> *∂ ∂t*

tive formulas and Fick's first law, we obtain the flux to be

these into the differential conservation relation, we obtain

*<sup>ρ</sup><sup>o</sup>* <sup>þ</sup> <sup>∇</sup>*<sup>i</sup> <sup>V</sup><sup>i</sup>*

*∂ ∂t*

form of the conservation relation:

*j i <sup>o</sup>* <sup>¼</sup> *<sup>v</sup><sup>i</sup>*

<sup>⊥</sup>*ρ<sup>o</sup>* <sup>þ</sup> *<sup>v</sup><sup>i</sup>*

ð *S Vi* <sup>⊥</sup>*ρ<sup>o</sup>* þ *j i o* � �*Ni dS* �

*<sup>ρ</sup><sup>o</sup>* <sup>þ</sup> <sup>∇</sup>*<sup>i</sup> <sup>V</sup><sup>i</sup>*

integral, and we obtain the differential form of the conservation relation:

*<sup>ρ</sup><sup>o</sup>* <sup>þ</sup> <sup>∇</sup>*<sup>i</sup> <sup>V</sup><sup>i</sup>*

ð *S*

*ρ<sup>o</sup> d*Ω þ

ð *S*

*ψ d*Ω ¼

ð Ω

*Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel*

*∂ ∂t*

also named, the *surface velocity*. This is encapsulated (using Einstein summation

*<sup>C</sup>*<sup>~</sup> <sup>¼</sup> **<sup>V</sup>**<sup>⊥</sup> � **<sup>N</sup>** <sup>¼</sup> *<sup>V</sup><sup>i</sup>*

basis for the coordinate space. Thus, we can simplify our conservation relation:

ð *S* **j** *<sup>o</sup>* � **dS** �

We can also simplify the vector surface element, **dS** ¼ **N***dS*, and convert the flux

ð *S j o* � �*<sup>i</sup>*

Finally, we will use the definition of the surface velocity and combine the two

We can use Gauss' divergence theorem on the surface integral term and unite all

<sup>⊥</sup>*ρ<sup>o</sup>* þ *j i o* � � � *<sup>σ</sup> <sup>d</sup>*<sup>Ω</sup> <sup>¼</sup> <sup>0</sup>*:*

Using the localization theorem, the integrand must be zero inside the volume

<sup>⊥</sup>*ρ<sup>o</sup>* þ *j i o*

We can simplify the equation, further by considering a particular form of the flux. In this, we consider advective flux (flux due to bulk movement of an observable's mass) and diffusive flux (flux due to a concentration gradient). Using advec-

*<sup>o</sup>ρ<sup>o</sup>* � *Do*∇*<sup>i</sup>*

*<sup>o</sup>ρ<sup>o</sup>* � *Do*∇*<sup>i</sup>*

We simplify the equation, expanding the covariant derivative to obtain the final

� � � *<sup>σ</sup>* <sup>¼</sup> <sup>0</sup>*:*

*<sup>o</sup>***Z***<sup>i</sup>* is the velocity of the observable within the volume. Substituting

*ρo*

*<sup>C</sup>*~*ρ<sup>o</sup> dS* <sup>þ</sup>

*<sup>C</sup>*~*ρ<sup>o</sup> dS* <sup>þ</sup>

*ψ d*Ω þ

where *C*~ is defined as the normal projection of the surface's perpendicular speed,

ð *S*

<sup>⊥</sup>*Ni*,

ð Ω

*<sup>o</sup>* ¼ *j o* � �*<sup>i</sup>* **Z***i*:

*Ni dS* �

ð Ω ð Ω

*σ d*Ω ¼ 0,

*σ d*Ω ¼ 0*:*

*σ d*Ω ¼ 0*:*

� � � *<sup>σ</sup>* <sup>¼</sup> <sup>0</sup>*:* (4)

*ρo*, (5)

*Ni* is the unit normal to the surface and **Z***<sup>i</sup>* is the contravariant

*C*~*ψ dS*, (3)

where **j** *<sup>o</sup>* is the flux of the observable out or into the surface and Σ represents the net increase or decrease in the observable's mass.

The relation states intuitively that *any change of the observable's mass within the solid, plus all observable mass entering or leaving the boundary, should represent the net change in the mass of the object.*

Any mass transport can be derived from the above relation converted into the differential form. We first recognize that the observable's mass can be represented through the observable's density:

$$m\_{\boldsymbol{\sigma}} = \int\_{\Omega} \rho\_{\boldsymbol{\sigma}} \quad d\Omega,$$

In addition, we can make the same statement about the net equilibrium constant. Suppose there is a local equilibrium density, *σ*, such that

$$
\Sigma = \int\_{\Omega} \sigma \quad d\Omega.
$$

We then can obtain a full integral form of the conservation relation:

$$\frac{d}{dt}\int\_{\Omega} \rho\_o \quad d\Omega + \int\_{\mathcal{S}} \mathbf{j}\_o \cdot \mathbf{dS} - \int\_{\Omega} \sigma \quad d\Omega = 0. \tag{2}$$

In general, it can be shown that for a general surface that is moving, the time derivative of a volume integral defined over the dynamic volume enclosed by the surface can be summarized as [11]

*Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel DOI: http://dx.doi.org/10.5772/intechopen.90575*

$$
\frac{d}{dt} \int\_{\Omega} \boldsymbol{\psi} \quad d\Omega = \int\_{\Omega} \frac{\partial}{\partial t} \boldsymbol{\psi} \quad d\Omega + \int\_{S} \bar{\mathbf{C}} \boldsymbol{\psi} \quad d\mathbf{S}, \tag{3}
$$

where *C*~ is defined as the normal projection of the surface's perpendicular speed, also named, the *surface velocity*. This is encapsulated (using Einstein summation convention) by the tensorial equation:

$$
\bar{\mathbf{C}} = \mathbf{V}\_{\perp} \cdot \mathbf{N} = V\_{\perp}^{i} N\_{i\*}
$$

where **<sup>N</sup>** <sup>¼</sup> **<sup>Z</sup>***<sup>i</sup> Ni* is the unit normal to the surface and **Z***<sup>i</sup>* is the contravariant basis for the coordinate space. Thus, we can simplify our conservation relation:

$$
\int\_{\Omega} \frac{\partial}{\partial t} \rho\_o \quad d\Omega + \int\_{\mathcal{S}} \bar{\mathbf{C}} \rho\_o \quad d\mathbf{S} + \int\_{\mathcal{S}} \mathbf{j}\_o \cdot \mathbf{dS} - \int\_{\Omega} \sigma \quad d\Omega = \mathbf{0},
$$

We can also simplify the vector surface element, **dS** ¼ **N***dS*, and convert the flux term into a tensorial formation, by recognizing *j <sup>o</sup>* ¼ *j o* � �*<sup>i</sup>* **Z***i*:

$$\int\_{\Omega} \frac{\partial}{\partial t} \rho\_o \quad d\Omega + \int\_{\mathcal{S}} \tilde{\mathcal{C}} \rho\_o \quad d\mathbb{S} + \int\_{\mathcal{S}} \left(j\_o\right)^i N\_i \quad d\mathbb{S} - \int\_{\Omega} \sigma \quad d\Omega = \mathbf{0} \dots$$

Finally, we will use the definition of the surface velocity and combine the two surface integral terms into one:

$$\int\_{\Omega} \frac{\partial}{\partial t} \rho\_o \quad d\Omega + \int\_{\mathcal{S}} \left( V\_{\perp}^i \rho\_o + j\_o^i \right) N\_i \quad d\mathcal{S} - \int\_{\Omega} \sigma \quad d\Omega = \mathbf{0}.$$

We can use Gauss' divergence theorem on the surface integral term and unite all the terms under one volume integral:

$$\int\_{\Omega} \frac{\partial}{\partial t} \rho\_o + \nabla\_i \left( V\_\perp^i \rho\_o + j\_o^i \right) - \sigma \quad d\Omega = \mathbf{0}.$$

Using the localization theorem, the integrand must be zero inside the volume integral, and we obtain the differential form of the conservation relation:

$$\frac{\partial}{\partial t}\rho\_o + \nabla\_i \left(V^i\_\perp \rho\_o + j^i\_o\right) - \sigma = 0. \tag{4}$$

We can simplify the equation, further by considering a particular form of the flux. In this, we consider advective flux (flux due to bulk movement of an observable's mass) and diffusive flux (flux due to a concentration gradient). Using advective formulas and Fick's first law, we obtain the flux to be

$$j\_o^i = v\_o^i \rho\_o - D\_o \nabla^i \rho\_o,\tag{5}$$

where **<sup>v</sup>***<sup>o</sup>* <sup>¼</sup> *vi <sup>o</sup>***Z***<sup>i</sup>* is the velocity of the observable within the volume. Substituting these into the differential conservation relation, we obtain

$$\frac{\partial}{\partial t}\rho\_o + \nabla\_i\left(V^i{}\_\perp\rho\_o + \nu^i\_o\rho\_o - D\_o\nabla^i\rho\_o\right) - \sigma = \mathbf{0}.$$

We simplify the equation, expanding the covariant derivative to obtain the final form of the conservation relation:

in a micro-fluidic channel. Flows that are fully developed at inlet were studied by Ng [5] who studied oscillatory dispersion in a tube with chemical species undergoing linear reversible and irreversible reactions at the tube wall. Relative importance to the present work is that fully developed flow occurs at inlet of channel with boundary conditions specified on the wall. No inlet mass transport is specified at inlet of channel similar to [5] but unlike [6], for example. Using Painlevé analysis [7], certain nonlinear second-order ordinary differential equations (ODE) can be factorized and solved. We consider the model of Nair et al. [1–3], where the nonlinear PDE reduced in [8], to an "almost linear" second-order ode is considered. It is well-known that the Weierstrass P function in its series form is problematic to use in computational work due to very slow convergence of numerical methods. It is the aim of the present work to show that a degenerate form of the special function can be used as in [9] in the reduction of the nonlinear PDE in [1–3]. A thorough and recent review of oxygen control with microfluidics has been carried out in [10] and all of its references within. In this work we see how the microscale can be leveraged

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

Regardless of the kinematics of a surface (dynamic or stationary), all surfaces, *S* ¼ ∂Ω, enclosing a solid volume, Ω, obey the following intuitive conservation relation for an enclosed observable mass, *mo* of some arbitrary substance:

The relation states intuitively that *any change of the observable's mass within the solid, plus all observable mass entering or leaving the boundary, should represent the net*

Any mass transport can be derived from the above relation converted into the differential form. We first recognize that the observable's mass can be represented

In addition, we can make the same statement about the net equilibrium con-

In general, it can be shown that for a general surface that is moving, the time derivative of a volume integral defined over the dynamic volume enclosed by the

*ρ<sup>o</sup> d*Ω,

*σ d*Ω*:*

ð Ω

*mo* ¼ ð Ω

> Σ ¼ ð Ω

We then can obtain a full integral form of the conservation relation:

ð *S* **j** *<sup>o</sup>* � **dS** �

stant. Suppose there is a local equilibrium density, *σ*, such that

*ρ<sup>o</sup> d*Ω þ

*<sup>o</sup>* is the flux of the observable out or into the surface and Σ represents the

*<sup>o</sup>* � **dS** ¼ Σ, (1)

*σ d*Ω ¼ 0*:* (2)

ð *S* **j**

*d dt mo* <sup>þ</sup>

for oxygen control of RBCs.

*change in the mass of the object.*

through the observable's density:

where **j**

**2. General tensorial mass transport**

net increase or decrease in the observable's mass.

*d dt* ð Ω

surface can be summarized as [11]

**256**

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

$$\frac{\partial}{\partial t}\rho\_o + \nabla\_i\left(\left(V^i\_\perp + v^i\_o\right)\rho\_o\right) = \nabla\_i\left(D\_o\nabla^i\rho\_o\right) + \sigma. \tag{6}$$

1 ffiffiffiffiffiffiffiffiffi <sup>∣</sup>*Zjk*<sup>∣</sup> <sup>p</sup> *vO*<sup>2</sup> ð Þ*<sup>z</sup>*

equation of

pressure.

**Figure 1.**

**259**

greatly to obtain the final form:

the geometry of the problem:

*vp*ð Þþ 1 � *HT vRBCHT*

*∂ ∂z*

*DOI: http://dx.doi.org/10.5772/intechopen.90575*

From here, if we assume the coordinates *Z*<sup>1</sup>

**3. Governing equation for oxygen transport**

*KRBC Kp*

� � � � *∂PO*<sup>2</sup>

*Geometry of micro-channel with permeable membrane centered at the top of the channel.*

*vO*<sup>2</sup> ð Þ*<sup>z</sup>*

ffiffiffiffiffiffiffiffiffi ∣*Zjk*∣ q

*ρO*2 � � <sup>¼</sup> <sup>1</sup>

*Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel*

*<sup>∂</sup>ρO*<sup>2</sup>

*vO*<sup>2</sup> ð Þ*<sup>z</sup>*

*<sup>∂</sup><sup>z</sup>* <sup>¼</sup> *DO*<sup>2</sup>

*<sup>∂</sup>ρO*<sup>2</sup>

ffiffiffiffiffiffiffiffiffi <sup>∣</sup>*Zjk*<sup>∣</sup> <sup>p</sup> *DO*<sup>2</sup>

*∂*

In addition, we assume that *ρO*<sup>2</sup> does not depend on *x*. This produces the final

*<sup>∂</sup><sup>z</sup>* <sup>¼</sup> *DO*<sup>2</sup>

As can be extrapolated from above, the general form of the nonlinear PDE for consideration defining oxygen transport in core region with Poiseuille hemodynamic flow is given by Eq. (13). The boundary conditions are shown in Section 10.1 of the Appendix, the velocity profile is shown in Section 10.2, and **Figure 1** shows

> <sup>1</sup> <sup>þ</sup> ½ � *HbT KRBC*

There is a core region of blood flow with RBCs and plasma and a cell-free region with only plasma flowing. In the plasma region near the wall, the governing equation is as in Eq. (13), without the second term in the square brackets. In general the geometry of the problem can be either a tube or a channel, and the Laplacian is generalized in [8]. In the present work, we confine the problem to a channel flow. The blood plasma velocity is *vp*, and *vRBC* is the velocity of RBCs together with plasma in the cell-rich region. The velocity of the RBCs is lower due to the slip

*dSO*<sup>2</sup> *dPO*<sup>2</sup>

In addition, for all the above equations, since by Boyle's law, at a constant temperature, all instances of oxygen density can be equivalently replaced by oxygen

*<sup>∂</sup>Z<sup>i</sup> <sup>Z</sup>i*<sup>ℓ</sup> *<sup>∂</sup>ρO*<sup>2</sup>

*∂*2 *ρO*2

*∂Z*<sup>ℓ</sup>

*∂ ∂Z<sup>i</sup>*

ffiffiffiffiffiffiffiffiffi ∣*Zjk*∣ q

*<sup>Z</sup>i*<sup>ℓ</sup> *<sup>∂</sup> <sup>∂</sup>Z*<sup>ℓ</sup> *<sup>ρ</sup>O*<sup>2</sup>

� �*:*

, *<sup>Z</sup>*<sup>2</sup> � � <sup>¼</sup> ð Þ *<sup>x</sup>*, *<sup>y</sup>* , then we can simplify

� �*:* (11)

*<sup>∂</sup>y*<sup>2</sup> *:* (12)

*<sup>∂</sup><sup>z</sup>* <sup>¼</sup> *Dp*∇<sup>2</sup>

*PO*<sup>2</sup> (13)

This form can also be put into an invariant tensorial form by utilizing the invariant time derivative operator ∇\_ � � from the calculus of moving surfaces [11]:

$$
\dot{\nabla}\rho\_o + \mathbf{V}\_\perp^i \nabla\_i \rho\_o + \nabla\_i \left( \left( \mathbf{V}\_\perp^i + v\_o^i \right) \rho\_o \right) = \nabla\_i \left( \mathbf{D}\_o \nabla^i \rho\_o \right) + \sigma. \tag{7}
$$

This equation can also be put into a vector form:

$$
\vec{\nabla}\rho\_o + V\_\perp \cdot \overrightarrow{\nabla}\rho\_o + \overrightarrow{\nabla} \cdot ((V\_\perp + v\_o)\rho\_o) = \overrightarrow{\nabla} \cdot \left(D\_o \overrightarrow{\nabla}\rho\_o\right) + \sigma. \tag{8}
$$

#### **2.1 Application to oxygen transport**

We consider a biological application of the observable's mass transport equation to microfluidic arterial oxygen transport. In this case, *o* ¼ *O*2. For this, we are required to make a few assumptions:


Using the above relations, we reduce the conservation relation to

$$
\nabla\_i \left( \boldsymbol{\nu}\_{O\_2}^i \rho\_{O\_2} \right) = D\_{O\_2} \nabla\_i \nabla^i \rho\_{O\_2}.\tag{9}
$$

Both of the operators can be expanded using the Voss-Weyl formula [11] and restated in terms of partial derivatives with respect to the spatial coordinates, *Z<sup>i</sup>* , and the spatial metric tensor, *Zij*:

$$\frac{1}{\sqrt{|\mathbb{Z}\_{jk}|}} \frac{\partial}{\partial \mathbf{Z}^i} \left( \sqrt{|\mathbb{Z}\_{jk}|} v^i\_{O\_2} \rho\_o \right) = \frac{1}{\sqrt{|\mathbb{Z}\_{jk}|}} D\_{O\_2} \frac{\partial}{\partial \mathbf{Z}^i} \left( \sqrt{|\mathbb{Z}\_{jk}|} \mathbf{Z}^{i\ell} \frac{\partial}{\partial \mathbf{Z}^\ell} \rho\_{O\_1} \right). \tag{10}$$

We assume for a moment that the coordinate system chosen is some general axial coordinate system consisting of two arbitrary coordinates, *Z*<sup>1</sup> , *Z*<sup>2</sup> � �, and a third coordinate corresponding to the standard *z* coordinate found in cylindrical and Euclidean coordinate systems.

This forms a three-dimensional coordinate system of *Z*<sup>1</sup> , *Z*<sup>2</sup> , *z* � �*:* We first assume that the velocity of the observable is only along the *z* coordinate. This means that the term on the left is greatly simplified:

$$\frac{1}{\sqrt{|Z\_{jk}|}}\frac{\partial}{\partial \mathbf{z}}\left(\sqrt{|Z\_{jk}|}(v\_{\mathrm{O\_2}})\_{\mathrm{z}}\rho\_{\mathrm{O\_2}}\right) = \frac{1}{\sqrt{|Z\_{jk}|}}D\_{\mathrm{O\_2}}\frac{\partial}{\partial \mathbf{Z}^i}\left(\sqrt{|Z\_{jk}|}\mathbf{Z}^{i\ell}\frac{\partial}{\partial \mathbf{Z}^\ell}\rho\_{\mathrm{O\_2}}\right).$$

We then assume that the velocity of the observable does not depend on the zcoordinate. This means that the particle moving along a streamline parallel to the length of the tube will not accelerate. This will produce the equation:

*Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel DOI: http://dx.doi.org/10.5772/intechopen.90575*

$$\frac{1}{\sqrt{|Z\_{jk}|}} (\boldsymbol{\nu}\_{O\_2})\_x \frac{\partial}{\partial \mathbf{z}} \left( \sqrt{|Z\_{jk}|} \boldsymbol{\rho}\_{O\_2} \right) = \frac{1}{\sqrt{|Z\_{jk}|}} D\_{O\_2} \frac{\partial}{\partial \mathbf{Z}^i} \left( \sqrt{|Z\_{jk}|} \mathbf{Z}^{i\ell} \frac{\partial}{\partial \mathbf{Z}^{\ell}} \boldsymbol{\rho}\_{O\_2} \right).$$

From here, if we assume the coordinates *Z*<sup>1</sup> , *<sup>Z</sup>*<sup>2</sup> � � <sup>¼</sup> ð Þ *<sup>x</sup>*, *<sup>y</sup>* , then we can simplify greatly to obtain the final form:

$$(\nu\_{O\_2})\_x \frac{\partial \rho\_{O\_2}}{\partial \mathbf{z}} = D\_{O\_2} \frac{\partial}{\partial \mathbf{Z}^i} \left( Z^{i\ell} \frac{\partial \rho\_{O\_2}}{\partial \mathbf{Z}^\ell} \right). \tag{11}$$

In addition, we assume that *ρO*<sup>2</sup> does not depend on *x*. This produces the final equation of

$$(v\_{O\_2})\_x \frac{\partial \rho\_{O\_2}}{\partial \mathbf{z}} = D\_{O\_2} \frac{\partial^2 \rho\_{O\_2}}{\partial \mathbf{y}^2}. \tag{12}$$

In addition, for all the above equations, since by Boyle's law, at a constant temperature, all instances of oxygen density can be equivalently replaced by oxygen pressure.

#### **3. Governing equation for oxygen transport**

As can be extrapolated from above, the general form of the nonlinear PDE for consideration defining oxygen transport in core region with Poiseuille hemodynamic flow is given by Eq. (13). The boundary conditions are shown in Section 10.1 of the Appendix, the velocity profile is shown in Section 10.2, and **Figure 1** shows the geometry of the problem:

$$
\left[\upsilon\_p(\mathbf{1} - H\_T) + \upsilon\_{RBC} H\_T \frac{K\_{RBC}}{K\_p} \left(\mathbf{1} + \frac{[Hb\_T]}{K\_{RBC}} \frac{dSO\_2}{dPO\_2}\right)\right] \frac{dPO\_2}{d\mathbf{z}} = D\_p \nabla^2 P O\_2 \tag{13}
$$

There is a core region of blood flow with RBCs and plasma and a cell-free region with only plasma flowing. In the plasma region near the wall, the governing equation is as in Eq. (13), without the second term in the square brackets. In general the geometry of the problem can be either a tube or a channel, and the Laplacian is generalized in [8]. In the present work, we confine the problem to a channel flow. The blood plasma velocity is *vp*, and *vRBC* is the velocity of RBCs together with plasma in the cell-rich region. The velocity of the RBCs is lower due to the slip

**Figure 1.** *Geometry of micro-channel with permeable membrane centered at the top of the channel.*

*∂ ∂t*

<sup>∇</sup>\_ *<sup>ρ</sup><sup>o</sup>* <sup>þ</sup> *<sup>V</sup><sup>i</sup>*

<sup>∇</sup>\_ *<sup>ρ</sup><sup>o</sup>* <sup>þ</sup> *<sup>V</sup>*<sup>⊥</sup> � <sup>∇</sup>

**2.1 Application to oxygen transport**

required to make a few assumptions:

and the spatial metric tensor, *Zij*:

ffiffiffiffiffiffiffiffiffi ∣*Zjk*∣ q

� �

*vi O*<sup>2</sup> *ρo*

*∂ ∂Z<sup>i</sup>*

Euclidean coordinate systems.

*∂ ∂z*

1 ffiffiffiffiffiffiffiffiffi ∣*Zjk*∣ p

**258**

the term on the left is greatly simplified:

ffiffiffiffiffiffiffiffiffi ∣*Zjk*∣ q

1 ffiffiffiffiffiffiffiffiffi ∣*Zjk*∣ p

*σ* ¼ 0*:*

This would necessarily imply *V<sup>i</sup>*

not dependent on time. This means that *<sup>∂</sup>*

*<sup>ρ</sup><sup>o</sup>* <sup>þ</sup> <sup>∇</sup>*<sup>i</sup> <sup>V</sup><sup>i</sup>*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

This equation can also be put into a vector form:

! *ρ<sup>o</sup>* þ ∇ !

⊥∇*iρ<sup>o</sup>* <sup>þ</sup> <sup>∇</sup>*<sup>i</sup> <sup>V</sup><sup>i</sup>*

<sup>⊥</sup> <sup>þ</sup> *<sup>v</sup><sup>i</sup> o* � �*ρ<sup>o</sup>*

� � <sup>¼</sup> <sup>∇</sup>*<sup>i</sup> Do*∇*<sup>i</sup>*

This form can also be put into an invariant tensorial form by utilizing the invariant time derivative operator ∇\_ � � from the calculus of moving surfaces [11]:

> <sup>⊥</sup> <sup>þ</sup> *vi o* � �*ρ<sup>o</sup>*

� ð Þ *V*<sup>⊥</sup> þ *vo ρ<sup>o</sup>* ð Þ¼ ∇

We consider a biological application of the observable's mass transport equation

(A1) We first tentatively assume that the arteriole's surface is stationary and **not**.

(A3) In addition, we restrict our studies to microfluidic environments which are in equilibrium. This would imply that the net local density change is zero, or

<sup>¼</sup> *DO*2∇*i*∇*<sup>i</sup>*

*∂ ∂Z<sup>i</sup>*

ffiffiffiffiffiffiffiffiffi ∣*Zjk*∣ q

*∂ ∂Z<sup>i</sup>* *<sup>Z</sup><sup>i</sup>*<sup>ℓ</sup> *<sup>∂</sup> <sup>∂</sup>Z*<sup>ℓ</sup> *<sup>ρ</sup><sup>O</sup>*<sup>2</sup>

, *Z*<sup>2</sup>

ffiffiffiffiffiffiffiffiffi ∣*Zjk*∣ q

*<sup>Z</sup><sup>i</sup>*<sup>ℓ</sup> *<sup>∂</sup> <sup>∂</sup>Z*<sup>ℓ</sup> *<sup>ρ</sup><sup>O</sup>*<sup>2</sup>

� �

� �

Both of the operators can be expanded using the Voss-Weyl formula [11] and restated in terms of partial derivatives with respect to the spatial coordinates, *Z<sup>i</sup>*

We assume for a moment that the coordinate system chosen is some general

coordinate corresponding to the standard *z* coordinate found in cylindrical and

that the velocity of the observable is only along the *z* coordinate. This means that

<sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffi <sup>∣</sup>*Zjk*<sup>∣</sup> <sup>p</sup> *DO*<sup>2</sup>

We then assume that the velocity of the observable does not depend on the zcoordinate. This means that the particle moving along a streamline parallel to the

to microfluidic arterial oxygen transport. In this case, *o* ¼ *O*2. For this, we are

<sup>⊥</sup>**Z***<sup>i</sup>* ¼ **0***:* (A2) We also consider steady-state solutions, by assuming that the density is

(A4) We also assume that the diffusion constant is a constant.

Using the above relations, we reduce the conservation relation to

<sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffi <sup>∣</sup>*Zjk*<sup>∣</sup> <sup>p</sup> *DO*<sup>2</sup>

axial coordinate system consisting of two arbitrary coordinates, *Z*<sup>1</sup>

This forms a three-dimensional coordinate system of *Z*<sup>1</sup>

*vO*<sup>2</sup> ð Þ*zρ<sup>O</sup>*<sup>2</sup> � �

length of the tube will not accelerate. This will produce the equation:

∇*<sup>i</sup> v<sup>i</sup> O*2 *ρO*2 � �

� � <sup>¼</sup> <sup>∇</sup>*<sup>i</sup> Do*∇*<sup>i</sup>*

!

*<sup>∂</sup><sup>t</sup> ρ<sup>O</sup>*<sup>2</sup> ¼ 0*:*

� *Do*∇ ! *ρo* � �

*ρo*

� � <sup>þ</sup> *<sup>σ</sup>:* (6)

� � <sup>þ</sup> *<sup>σ</sup>:* (7)

*ρ<sup>O</sup>*<sup>2</sup> *:* (9)

,

*:* (10)

, *Z*<sup>2</sup> � �, and a third

*:*

, *z* � �*:* We first assume

þ *σ:* (8)

*ρo*

between plasma and RBCs. The distribution of RBCs is such that the hematocrit is higher at the center of the channel and lower near the wall. The term *dSO*<sup>2</sup> *dPO*<sup>2</sup> is the slope of the oxyhemoglobin dissociation curve and is a highly nonlinear function of oxygen tension *PO*<sup>2</sup> [1–3]. The dissociation curve is approximated by the Hill equation [4, 6], where an empirical constant N is used and a constant *P*<sup>50</sup> appears which is the oxygen tension that yields 50% oxygen saturation. ½ � *HbT* is the total heme concentration, *Dp* is the given oxygen diffusion coefficient in plasma and has units of *μm*2*=s* and *KRBC*, *Kp* are the solubilities of *O*<sup>2</sup> in the RBCs and plasma, respectively, and have units of *M=mmHg*. The values of the respective parameters are taken from [8].

In connection with [8], we let

$$
\Sigma(PO\_2) = \text{const} + \text{coeff}\frac{dSO\_2}{dPO\_2}.
$$

This allows for a transformation of Eq. (13) into Eq. (14) where "const" and "coeff" are constants derived in [8].

We thus have a PDE of the form:

$$
\Delta(PO\_2)\frac{\partial PO\_2}{\partial \mathbf{z}} = \frac{1}{v\_p}\nabla^2 PO\_2. \tag{14}
$$

Isolating Σ terms and allowing a zero separation constant, we obtain

*Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel*

*dz* þ

*dL*<sup>2</sup> *dz* � �*vp* � *<sup>L</sup>*1∇<sup>2</sup>

> Σ�<sup>1</sup> *<sup>d</sup>*Σ<sup>2</sup> <sup>¼</sup> <sup>0</sup>

*dz* <sup>þ</sup> <sup>2</sup> <sup>¼</sup> <sup>0</sup> � �, (21)

*PO*<sup>2</sup> ¼ *A*1*PL*<sup>1</sup> þ *A*2, (22)

,

, *L*<sup>2</sup> ¼ *C*1*zL*<sup>1</sup>

� � <sup>¼</sup> *<sup>c</sup>* � *<sup>d</sup>*k k**<sup>r</sup>**

In the present work, we consider the general case where the forms are chosen for

1 3 *mi* � �<sup>2</sup>

( )

where *C*, *C*1, and *mi* are constants. *C*<sup>1</sup> is a free constant, *C* is to be determined using boundary conditions, and *mi* is a fixed known constant. The reason for this choice of functions *L*1ð Þ*z* and *L*2ð Þ*z* will be made apparent in Sections 4 and 5. The

The equation to solve is a PDE related to Eq. (18) and condition Eq. (20), for *L*<sup>1</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*zC*<sup>1</sup> ð Þ *Cz* <sup>þ</sup> <sup>1</sup>*=*3*mi <sup>C</sup>*g*<sup>P</sup>*

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*zC*<sup>1</sup> ð Þ *Cz* <sup>þ</sup> <sup>1</sup>*=*3*mi <sup>C</sup>*

*<sup>ξ</sup>* <sup>¼</sup> *<sup>ξ</sup>*ð Þ¼ *<sup>r</sup> <sup>c</sup>* � *dr*<sup>2</sup> � �, (24)

*Prr* ¼ �<sup>1</sup> *<sup>c</sup>* � *dr*<sup>2</sup> � �f2*C Cz* ð Þ <sup>þ</sup> <sup>1</sup>*=*3*mi <sup>P</sup>*<sup>2</sup> þ f2*zC*<sup>1</sup> ð Þ *Cz* <sup>þ</sup> <sup>1</sup>*=*3*mi*

h i*:*

( )

It has been shown in [8] that for *L*2ð Þ¼ *z* 0, Eq. (13) can be reduced to

*<sup>P</sup>* <sup>þ</sup> 2*vpP*<sup>2</sup> <sup>¼</sup> 0, *dL*<sup>1</sup>

*P* ¼ 0*:* (18)

*:* (20)

<sup>2</sup> is Poiseuille flow, and

(23)

*<sup>Z</sup>*~*ij*ð Þ <sup>∇</sup>*iP* ð Þ <sup>∇</sup>*iP* 6¼ <sup>0</sup>*:* (19)

ð Þ *PL*<sup>1</sup> <sup>þ</sup> *<sup>L</sup>*<sup>2</sup> *<sup>P</sup> dL*<sup>1</sup>

where *L*<sup>1</sup> 6¼ 0 and *P* must obey the metric condition

∇2

*L*<sup>1</sup> ¼ �<sup>1</sup> *Cz* þ

constant *mi* is chosen as in **Figure 1**, and <sup>1</sup> is a free constant.

**4. Transformation of associated equation**

þ *C*<sup>1</sup> ð Þ *Cz* þ 1*=*3*mi*

þ *zC*<sup>1</sup> *C*<sup>1</sup> ð Þ *Cz* þ 1*=*3*mi*

where <sup>2</sup> is a separation constant, *vp Z*~

where *A*1, *A*<sup>2</sup> are arbitrary constants.

*L*<sup>1</sup> and *L*2:

and *L*<sup>2</sup> defined above:

Let

**261**

*d*Σ�<sup>1</sup> *<sup>d</sup>*<sup>Σ</sup> 6¼ 0, *<sup>d</sup>*<sup>2</sup>

Also, Σ�<sup>1</sup> must obey the conditions

*DOI: http://dx.doi.org/10.5772/intechopen.90575*

Choosing a separation form of *PO*<sup>2</sup>

$$PO\_2 = PO\_2(\tilde{Z}, z),\tag{15}$$

where *<sup>Z</sup>*<sup>~</sup> <sup>¼</sup> *<sup>Z</sup>*~<sup>1</sup> , *<sup>Z</sup>*~<sup>2</sup> � � indicates a semi-general coordinate system; we assume the following form of the solution:

$$
\Sigma(\mathcal{PO}\_2(\tilde{Z}, z)) = P(\tilde{Z})L\_1(z) + L\_2(z). \tag{16}
$$

Let *<sup>Z</sup>*~*ij* be the metric tensor of the coordinate system composed of *<sup>Z</sup>*~<sup>1</sup> , *<sup>Z</sup>*~<sup>2</sup> � �. As derived in [8], we express the Laplacian of an arbitrary function, *ψ Z*~ � �, in terms of the metric tensor in curvilinear coordinates, i.e.,

$$\nabla\_i \nabla^i \mu \left(\tilde{Z}\right) = \nabla^2 \mu \left(\tilde{Z}\right) = \frac{1}{\sqrt{|\tilde{Z}\_{jk}|}} \frac{\partial}{\partial \tilde{Z}^j} \left(\sqrt{|\tilde{Z}\_{jk}|} \tilde{Z}^{i\ell} \frac{\partial \Psi}{\partial \tilde{Z}^\ell}\right). \tag{17}$$

Applying this definition to ∇<sup>2</sup>Σ�<sup>1</sup> ð Þ *PL*<sup>1</sup> þ *L*<sup>2</sup> similar to [8], we show that

$$\mathcal{L}(PL\_1+L\_2)\frac{\partial}{\partial \mathbf{z}}\Sigma^{-1}(PL\_1+L\_2) = \frac{1}{\nu\_p} \left[ L\_1 \frac{d\Sigma^{-1}}{d\Sigma} \nabla^2 P + L\_1 \tilde{Z}^{\tilde{\mathcal{V}}}(\nabla\_i P)(\nabla\_i P) \frac{d^2 \Sigma^{-1}}{d\Sigma^2} \right].$$

Rearranging in terms of *<sup>d</sup>*Σ�<sup>1</sup> *<sup>d</sup>*<sup>Σ</sup> , we obtain

$$\int \left[ (PL\_1 + L\_2) \left( P \frac{dL\_1}{dz} + \frac{dL\_2}{dz} \right) v\_p - L\_1 \nabla^2 P \right] \frac{d \Sigma^{-1}}{d \Sigma} = \left[ L\_1 \tilde{Z}^{\dagger j} (\nabla\_i P) (\nabla\_i P) \right] \frac{d^2 \Sigma^{-1}}{d \Sigma^2}.$$

*Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel DOI: http://dx.doi.org/10.5772/intechopen.90575*

Isolating Σ terms and allowing a zero separation constant, we obtain

$$(PL\_1 + L\_2) \left( P \frac{dL\_1}{dz} + \frac{dL\_2}{dz} \right) v\_p - L\_1 \nabla^2 P = \mathbf{0}.\tag{18}$$

where *L*<sup>1</sup> 6¼ 0 and *P* must obey the metric condition

$$
\tilde{Z}^{\tilde{\mathcal{Y}}} (\nabla\_i P)(\nabla\_i P) \neq \mathbf{0}.\tag{19}
$$

Also, Σ�<sup>1</sup> must obey the conditions

between plasma and RBCs. The distribution of RBCs is such that the hematocrit is

slope of the oxyhemoglobin dissociation curve and is a highly nonlinear function of oxygen tension *PO*<sup>2</sup> [1–3]. The dissociation curve is approximated by the Hill equation [4, 6], where an empirical constant N is used and a constant *P*<sup>50</sup> appears which is the oxygen tension that yields 50% oxygen saturation. ½ � *HbT* is the total heme concentration, *Dp* is the given oxygen diffusion coefficient in plasma and has units of *μm*2*=s* and *KRBC*, *Kp* are the solubilities of *O*<sup>2</sup> in the RBCs and plasma, respectively, and have units of *M=mmHg*. The values of the respective parameters

<sup>Σ</sup>ð Þ¼ *PO*<sup>2</sup> const <sup>þ</sup> coeff *dSO*<sup>2</sup>

This allows for a transformation of Eq. (13) into Eq. (14) where "const" and

*∂PO*<sup>2</sup> *<sup>∂</sup><sup>z</sup>* <sup>¼</sup> <sup>1</sup> *vp* ∇2

Σð Þ *PO*<sup>2</sup>

<sup>Σ</sup> *PO*<sup>2</sup> *<sup>Z</sup>*~, *<sup>z</sup>* � � � � <sup>¼</sup> *<sup>P</sup> <sup>Z</sup>*<sup>~</sup>

derived in [8], we express the Laplacian of an arbitrary function, *ψ Z*~

*ψ Z*~

ð Þ¼ *PL*<sup>1</sup> þ *L*<sup>2</sup>

*dL*<sup>2</sup> *dz*

� � *d*Σ�<sup>1</sup>

*dz* þ

� �

Let *<sup>Z</sup>*~*ij* be the metric tensor of the coordinate system composed of *<sup>Z</sup>*~<sup>1</sup>

� � <sup>¼</sup> <sup>1</sup>

1 *vp L*1 *d*Σ�<sup>1</sup> *d*Σ ∇2

*<sup>d</sup>*<sup>Σ</sup> , we obtain

*vp* � *<sup>L</sup>*1∇<sup>2</sup>

*P*

ffiffiffiffiffiffiffiffiffi ∣*Z*~*jk*∣ q

*∂ ∂Z*~*i* *dPO*<sup>2</sup> *:*

*PO*<sup>2</sup> <sup>¼</sup> *PO*<sup>2</sup> *<sup>Z</sup>*~, *<sup>z</sup>* � �, (15)

ffiffiffiffiffiffiffiffiffi ∣*Z*~*jk*∣ q

ð Þ *PL*<sup>1</sup> þ *L*<sup>2</sup> similar to [8], we show that

" #

*<sup>P</sup>* <sup>þ</sup> *<sup>L</sup>*1*Z*~*ij*

*<sup>d</sup>*<sup>Σ</sup> <sup>¼</sup> *<sup>L</sup>*1*Z*~*ij*

� �

� �*L*1ð Þþ *<sup>z</sup> <sup>L</sup>*2ð Þ*<sup>z</sup> :* (16)

*<sup>Z</sup>*~*<sup>i</sup>*<sup>ℓ</sup> *<sup>∂</sup><sup>ψ</sup> ∂Z*~<sup>ℓ</sup>

ð Þ <sup>∇</sup>*iP* ð Þ <sup>∇</sup>*iP <sup>d</sup>*<sup>2</sup>

ð Þ ∇*iP* ð Þ ∇*iP* h i *d*<sup>2</sup>

, *<sup>Z</sup>*~<sup>2</sup> � �

*:* (17)

Σ�<sup>1</sup> *d*Σ<sup>2</sup>

> Σ�<sup>1</sup> *<sup>d</sup>*Σ<sup>2</sup> *:*

*:*

� �, in terms of

. As

indicates a semi-general coordinate system; we assume the

*PO*2*:* (14)

*dPO*<sup>2</sup> is the

higher at the center of the channel and lower near the wall. The term *dSO*<sup>2</sup>

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

are taken from [8].

where *<sup>Z</sup>*<sup>~</sup> <sup>¼</sup> *<sup>Z</sup>*~<sup>1</sup>

ð Þ *PL*<sup>1</sup> þ *L*<sup>2</sup>

**260**

In connection with [8], we let

"coeff" are constants derived in [8]. We thus have a PDE of the form:

Choosing a separation form of *PO*<sup>2</sup>

, *<sup>Z</sup>*~<sup>2</sup> � �

∇*i*∇*<sup>i</sup> ψ Z*~ � � <sup>¼</sup> <sup>∇</sup><sup>2</sup>

Applying this definition to ∇<sup>2</sup>Σ�<sup>1</sup>

*∂ ∂z* Σ�<sup>1</sup>

Rearranging in terms of *<sup>d</sup>*Σ�<sup>1</sup>

ð Þ *PL*<sup>1</sup> <sup>þ</sup> *<sup>L</sup>*<sup>2</sup> *<sup>P</sup> dL*<sup>1</sup>

the metric tensor in curvilinear coordinates, i.e.,

following form of the solution:

$$\left\{\frac{d\Sigma^{-1}}{d\Sigma} \neq 0, \frac{d^2\Sigma^{-1}}{d\Sigma^2} = 0\right\}.\tag{20}$$

It has been shown in [8] that for *L*2ð Þ¼ *z* 0, Eq. (13) can be reduced to

$$\left\{\nabla^2 P + \mathbb{S}\_2 v\_p P^2 = 0, \frac{dL\_1}{dz} + \mathbb{S}\_2 = \mathbf{0}\right\},\tag{21}$$

where <sup>2</sup> is a separation constant, *vp Z*~ � � <sup>¼</sup> *<sup>c</sup>* � *<sup>d</sup>*k k**<sup>r</sup>** <sup>2</sup> is Poiseuille flow, and

$$PO\_2 = A\_1 PL\_1 + A\_2,\tag{22}$$

where *A*1, *A*<sup>2</sup> are arbitrary constants.

In the present work, we consider the general case where the forms are chosen for *L*<sup>1</sup> and *L*2:

$$\left\{ L\_1 = -\mathbb{S}\_1 \left( \mathbf{C} \mathbf{z} + \frac{\mathbf{1}}{3} m\_i \right)^2, L\_2 = \mathbf{C}\_1 \mathbf{z} L\_1 \right\},$$

where *C*, *C*1, and *mi* are constants. *C*<sup>1</sup> is a free constant, *C* is to be determined using boundary conditions, and *mi* is a fixed known constant. The reason for this choice of functions *L*1ð Þ*z* and *L*2ð Þ*z* will be made apparent in Sections 4 and 5. The constant *mi* is chosen as in **Figure 1**, and <sup>1</sup> is a free constant.

#### **4. Transformation of associated equation**

The equation to solve is a PDE related to Eq. (18) and condition Eq. (20), for *L*<sup>1</sup> and *L*<sup>2</sup> defined above:

$$\begin{split} P\_{rr} &= -\mathbb{S}\_{1} \left( \mathbf{c} - dr^{2} \right) \{ 2\, \mathrm{C} (\mathrm{Cz} + 1/3m\_{i})P^{2} + \{ 2\mathrm{zC}\_{1} (\mathrm{Cz} + 1/3m\_{i}) \\ &+ \mathrm{C}\_{1} \left( \mathrm{Cz} + 1/3m\_{i} \right)^{2} + 2\mathrm{zC}\_{1} \left( \mathrm{Cz} + 1/3m\_{i} \right) \mathrm{C} \} P \\ &+ \mathrm{zC}\_{1} \left[ \mathrm{C}\_{1} \left( \mathrm{Cz} + 1/3m\_{i} \right)^{2} + 2\mathrm{zC}\_{1} \left( \mathrm{Cz} + 1/3m\_{i} \right) \mathrm{C} \right]. \end{split} \tag{23}$$

Let

$$
\xi = \xi(r) = \left(c - dr^2\right),
\tag{24}
$$

The transformed equation becomes

$$\begin{aligned} -2d\frac{\partial P}{\partial \xi} + 4d(c - \xi)\frac{\partial^2 P}{\partial \xi^2} &= -\mathbb{S}\_1 \xi \{ 2CMP^2 + \left( 2x\mathcal{C}\_1 M + \mathcal{C}\_1 M^2 + 2x\mathcal{C}\_1 MC \right) P \\ &+ x\mathcal{C}\_1 \left( \mathcal{C}\_1 M^2 + 2x\mathcal{C}\_1 MC \right), \end{aligned} \tag{25}$$

where *<sup>M</sup>* <sup>¼</sup> *Cz* <sup>þ</sup> <sup>1</sup> <sup>3</sup> *mi*, *<sup>z</sup>* <sup>¼</sup> *<sup>M</sup>*�<sup>1</sup> <sup>3</sup>*mi <sup>C</sup>* , and *C*<sup>1</sup> ¼��ð Þ <sup>1</sup> <sup>1</sup>*=<sup>n</sup>* ¼ �ϵ, small, for *<sup>n</sup>*= 5, 4, 3, 2.

#### **5. General form of nonlinear equation**

The following form of the nonlinear nonhomogeneous PDE, Eq. (25), is considered:

$$\frac{\partial^2 Y(\xi, x)}{\partial \xi^2} + F\_1(\xi) \frac{\partial Y(\xi, x)}{\partial \xi} + F\_2(\xi, x) Y^2(\xi, x) - (\text{e}/2) F\_3(\xi, b\_0(x)) Y(\xi, x) + \\ \tag{26}$$

$$(\text{1/2}) G(\xi, x; \epsilon^2) = \mathbf{0},$$

where

$$F\_1(\xi) = -\left(2\mathcal{L} - 2\xi\right)^{-1},\tag{27}$$

It can be verified that as <sup>1</sup> ! 0 and ϵ ! 0, the solution obtained from Eq. (26)

The functions *λ ξ*ð Þ , *z* ,*ϕ ξ*ð Þ , *z* and function *b*0ð Þ*z* chosen to be large in magnitude in the supremum sense for all *z* values are selected so that the transformed equation, Eq.(26), through Eqs. (29)–(31) is written as one of the Painlevé classifications of

There are two independent canonical forms for this equation [7], one of which is

<sup>2</sup> ð Þ *ξ*, *z e*

The functions *F*1, *F*<sup>2</sup> and *F*<sup>3</sup> should satisfy the compatibility relation [7]:

The previous equation, Eq.(35), after substitution of Eqs. (27)–(29) reduces

<sup>2</sup> ¼ � *λ ξ*ð Þ , *<sup>z</sup> <sup>F</sup>*2ð Þ *<sup>ξ</sup>*, *<sup>z</sup>*

*∂*2 *<sup>∂</sup>ξ*<sup>2</sup> *λ ξ*ð Þ , *z*

*λ ξ*ð Þ , *<sup>z</sup>* � *<sup>F</sup>*1ð Þ*<sup>ξ</sup> <sup>∂</sup>*

ð Þþ *<sup>Z</sup>* <sup>6</sup>*b*<sup>2</sup>

�2 5 Ð *F*1ð Þ*ξ dξ*

<sup>0</sup> ¼ 0*:* (32)

, (33)

<sup>6</sup> *:* (34)

*<sup>∂</sup><sup>ξ</sup> λ ξ*ð Þ , *z*

*λ ξ*ð Þ , *<sup>z</sup> :* (35)

is a linear function in *ξ* and independent of *z*, and one possible solution is a decreasing function on the height of the channel, which is intuitive as the *PO*<sup>2</sup> should drop at the wall of the channel (see **Figures 1** and **2**). In an approximating sense, though, the other terms will contribute in Eq. (26) as shown below.

*through* �*7950 microns (bottom) in intervals of 95 microns for lower human hematocrit.*

*Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel*

*DOI: http://dx.doi.org/10.5772/intechopen.90575*

*Oxygen tension PO*<sup>2</sup> *[mmHg] versus channel height in microns at axial distance z = mi =* �*7000 (at top)*

According to Estevez et al. [7], the functions must be of the form

*λ ξ*ð Þ¼ , *<sup>z</sup> <sup>F</sup>*�1*=*<sup>5</sup>

*ϕ ξ*ð Þ , *z*

*F*3ð Þ¼ *ξ*, *b*0ð Þ*z* 2*b*0ð Þ*z F*2ð Þ *ξ*, *z λ ξ*ð Þ� , *z*

considerably to the following for *F*3:

the second-order differential equations [7].

and

**263**

**Figure 2.**

*d*2 *W Z*ð Þ *dZ*<sup>2</sup> � <sup>6</sup>*W*<sup>2</sup>

$$F\_2(\xi, z) = -1/4 \frac{\mathbb{S}\_1(2\text{CM})\xi}{d(c - \xi)},\tag{28}$$

and *<sup>G</sup> <sup>ξ</sup>*, *<sup>z</sup>*; <sup>ϵ</sup><sup>2</sup> ð Þ involve a small parameter by choice of *<sup>L</sup>*<sup>1</sup> and *<sup>L</sup>*<sup>2</sup> and can be made arbitrarily small, whereas *F*3ð Þ *ξ*, *b*0ð Þ*z* can be made large due to choice of *b*0ð Þ*z* .

In light of work in [7], upon substitution of *F*<sup>1</sup> and *F*<sup>2</sup> for *λ* as defined in Eq. (33):

$$\lambda(\xi, z) = \frac{1}{\sqrt[\xi]{-\mathbf{1}/4} \frac{\mathbb{S}\_1(2\mathbb{C}\mathbf{M})\xi}{d(c-\xi)}} \frac{1}{\sqrt[\xi]{2c-2\xi}},\tag{29}$$

and

$$\begin{split} Y(\xi, \mathbf{z}) &= \lambda(\xi, \mathbf{z}) (W(Z(\xi, \mathbf{z})) - b\_0(\mathbf{z})) + \epsilon \mathsf{M}(\mathbf{c} - dt^2) \times \\ &\quad [\mathbf{z}/\mathfrak{Z} \ (\mathsf{9C} \mathbf{z} + \mathsf{m}\_i)] t^{-2} \times F\_3^{-1}(\xi, b\_0(\mathbf{z})), \end{split} \tag{30}$$

where *Z* is defined such that

$$d\mathcal{Z} = \phi(\xi, \boldsymbol{z}) \partial \xi,\tag{31}$$

where *F*�<sup>1</sup> <sup>3</sup> ¼ 1*=F*<sup>3</sup> is defined by Eq. (35), *ϕ ξ*ð Þ , *z* is defined by Eq. (34), and *W Z*ð Þ is defined by Eq. (32). It follows that substitution of Eq. (30) for *Y* into Eq. (26) will cancel part of the coefficient of the *Y* term leaving the first term on the right side of Eq. (30). (The second term on the right side of Eq. (30) is the simplification of the last term on the right-hand side of Eq. (25) except the *F*�<sup>1</sup> 3 term). Next, *ε*<sup>2</sup> terms will cancel in Eq. (26) leaving a form of the equation which is homogeneous similar to that in [7] (where we have assumed that sup*zb*0ð Þ*z* is large for *z* far down the channel), i.e.:

$$\frac{\partial^2 Y(\xi, z)}{\partial \xi^2} + F\_1(\xi) \frac{\partial Y(\xi, z)}{\partial \xi} + F\_2(\xi, z) Y^2(\xi, z) + F\_3(\xi, z) Y(\xi, z) = 0.$$

*Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel DOI: http://dx.doi.org/10.5772/intechopen.90575*

#### **Figure 2.**

The transformed equation becomes

*P*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

<sup>3</sup> *mi*, *<sup>z</sup>* <sup>¼</sup> *<sup>M</sup>*�<sup>1</sup>

**5. General form of nonlinear equation**

*<sup>∂</sup>ξ*<sup>2</sup> <sup>þ</sup> *<sup>F</sup>*1ð Þ*<sup>ξ</sup> <sup>∂</sup>Y*ð Þ *<sup>ξ</sup>*, *<sup>z</sup>*

where *Z* is defined such that

for *z* far down the channel), i.e.:

*<sup>∂</sup>ξ*<sup>2</sup> <sup>þ</sup> *<sup>F</sup>*1ð Þ*<sup>ξ</sup> <sup>∂</sup>Y*ð Þ *<sup>ξ</sup>*, *<sup>z</sup>*

*∂*2 *Y*ð Þ *ξ*, *z*

**262**

<sup>3</sup>*mi*

*<sup>∂</sup><sup>ξ</sup>* <sup>þ</sup> *<sup>F</sup>*2ð Þ *<sup>ξ</sup>*, *<sup>z</sup> <sup>Y</sup>*<sup>2</sup>

*λ ξ*ð Þ¼ , *z*

5 q

*z=*3 9ð Þ *Cz* þ *mi* ½ �*t*

ð Þ <sup>1</sup>*=*<sup>2</sup> *<sup>G</sup> <sup>ξ</sup>*, *<sup>z</sup>*; <sup>ϵ</sup><sup>2</sup> � � <sup>¼</sup> 0,

*<sup>F</sup>*1ð Þ¼� *<sup>ξ</sup>* ð Þ <sup>2</sup>*<sup>c</sup>* � <sup>2</sup>*<sup>ξ</sup>* �<sup>1</sup>

*<sup>F</sup>*2ð Þ¼� *<sup>ξ</sup>*, *<sup>z</sup>* <sup>1</sup>*=*<sup>4</sup> <sup>1</sup> ð Þ <sup>2</sup>*CM <sup>ξ</sup>*

arbitrarily small, whereas *F*3ð Þ *ξ*, *b*0ð Þ*z* can be made large due to choice of *b*0ð Þ*z* . In light of work in [7], upon substitution of *F*<sup>1</sup> and *F*<sup>2</sup> for *λ* as defined in Eq. (33):

> 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi �1*=*<sup>4</sup> <sup>1</sup> ð Þ <sup>2</sup>*CM <sup>ξ</sup> d c*ð Þ �*ξ*

*<sup>Y</sup>*ð Þ¼ *<sup>ξ</sup>*, *<sup>z</sup> λ ξ*ð Þ , *<sup>z</sup>* <sup>ð</sup>*W Z*ð Þ� ð Þ *<sup>ξ</sup>*, *<sup>z</sup> <sup>b</sup>*0ð Þ*<sup>z</sup>* Þ þ <sup>ϵ</sup>*M c* � *d t*<sup>2</sup> � ��

�<sup>2</sup> � *<sup>F</sup>*�<sup>1</sup>

<sup>3</sup> ¼ 1*=F*<sup>3</sup> is defined by Eq. (35), *ϕ ξ*ð Þ , *z* is defined by Eq. (34), and

*W Z*ð Þ is defined by Eq. (32). It follows that substitution of Eq. (30) for *Y* into Eq. (26) will cancel part of the coefficient of the *Y* term leaving the first term on the

term). Next, *ε*<sup>2</sup> terms will cancel in Eq. (26) leaving a form of the equation which is homogeneous similar to that in [7] (where we have assumed that sup*zb*0ð Þ*z* is large

right side of Eq. (30). (The second term on the right side of Eq. (30) is the simplification of the last term on the right-hand side of Eq. (25) except the *F*�<sup>1</sup>

*<sup>∂</sup><sup>ξ</sup>* <sup>þ</sup> *<sup>F</sup>*2ð Þ *<sup>ξ</sup>*, *<sup>z</sup> <sup>Y</sup>*<sup>2</sup>

and *<sup>G</sup> <sup>ξ</sup>*, *<sup>z</sup>*; <sup>ϵ</sup><sup>2</sup> ð Þ involve a small parameter by choice of *<sup>L</sup>*<sup>1</sup> and *<sup>L</sup>*<sup>2</sup> and can be made

*<sup>∂</sup>ξ*<sup>2</sup> ¼ �1*ξ*f2*CMP*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*zC*1*<sup>M</sup>* <sup>þ</sup> *<sup>C</sup>*1*M*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*zC*1*MC* � �*<sup>P</sup>*

(25)

(26)

<sup>1</sup>*=<sup>n</sup>* ¼ �ϵ, small, for *<sup>n</sup>*= 5, 4, 3, 2.

, (27)

*d c*ð Þ � *<sup>ξ</sup>* , (28)

<sup>2</sup>*<sup>c</sup>* � <sup>2</sup>*<sup>ξ</sup>* <sup>p</sup><sup>5</sup> , (29)

<sup>3</sup> ð Þ *<sup>ξ</sup>*, *<sup>b</sup>*0ð Þ*<sup>z</sup>* , (30)

3

*<sup>∂</sup><sup>Z</sup>* <sup>¼</sup> *ϕ ξ*ð Þ , *<sup>z</sup> <sup>∂</sup>ξ*, (31)

ð Þþ *ξ*, *z F*3ð Þ *ξ*, *z Y*ð Þ¼ *ξ*, *z* 0*:*

<sup>þ</sup> *zC*<sup>1</sup> *<sup>C</sup>*1*M*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*zC*1*MC* � �, <sup>g</sup>

*<sup>C</sup>* , and *C*<sup>1</sup> ¼��ð Þ <sup>1</sup>

The following form of the nonlinear nonhomogeneous PDE, Eq. (25), is considered:

ð Þ� *ξ*, *z* ð Þ ϵ*=*2 *F*3ð Þ *ξ*, *b*0ð Þ*z Y*ð Þþ *ξ*, *z*

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>∂</sup><sup>ξ</sup>* <sup>þ</sup> <sup>4</sup>*d c*ð Þ � *<sup>ξ</sup> <sup>∂</sup>*<sup>2</sup>

where *<sup>M</sup>* <sup>¼</sup> *Cz* <sup>þ</sup> <sup>1</sup>

�2*<sup>d</sup> <sup>∂</sup><sup>P</sup>*

*∂*2 *Y*ð Þ *ξ*, *z*

where

and

where *F*�<sup>1</sup>

*Oxygen tension PO*<sup>2</sup> *[mmHg] versus channel height in microns at axial distance z = mi =* �*7000 (at top) through* �*7950 microns (bottom) in intervals of 95 microns for lower human hematocrit.*

It can be verified that as <sup>1</sup> ! 0 and ϵ ! 0, the solution obtained from Eq. (26) is a linear function in *ξ* and independent of *z*, and one possible solution is a decreasing function on the height of the channel, which is intuitive as the *PO*<sup>2</sup> should drop at the wall of the channel (see **Figures 1** and **2**). In an approximating sense, though, the other terms will contribute in Eq. (26) as shown below.

The functions *λ ξ*ð Þ , *z* ,*ϕ ξ*ð Þ , *z* and function *b*0ð Þ*z* chosen to be large in magnitude in the supremum sense for all *z* values are selected so that the transformed equation, Eq.(26), through Eqs. (29)–(31) is written as one of the Painlevé classifications of the second-order differential equations [7].

There are two independent canonical forms for this equation [7], one of which is

$$\frac{d^2W(Z)}{dZ^2} - \epsilon W^2(Z) + \epsilon b\_0^2 = 0. \tag{32}$$

According to Estevez et al. [7], the functions must be of the form

$$\lambda(\xi, z) = F\_2^{-1/5}(\xi, z)e^{-\frac{2}{5}\int F\_1(\xi)d\xi},\tag{33}$$

and

$$\left(\phi(\xi, z)\right)^{2} = -\frac{\lambda(\xi, z)F\_{2}(\xi, z)}{6}.\tag{34}$$

The functions *F*1, *F*<sup>2</sup> and *F*<sup>3</sup> should satisfy the compatibility relation [7]:

$$F\_3(\xi, b\_0(z)) = 2b\_0(z)F\_2(\xi, z)\lambda(\xi, z) - \frac{\frac{\partial^2}{\partial \xi^2}\lambda(\xi, z)}{\lambda(\xi, z)} - \frac{F\_1(\xi)\frac{\partial}{\partial \xi}\lambda(\xi, z)}{\lambda(\xi, z)}\tag{35}$$

The previous equation, Eq.(35), after substitution of Eqs. (27)–(29) reduces considerably to the following for *F*3:

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

$$F\mathfrak{3}(\xi, b\_0(\mathfrak{z})) = \left(-\mathfrak{z}\mathfrak{0}\left(\mathfrak{c} - \mathfrak{f}\right)\mathfrak{z}^2\right)^{-1} \times$$

$$\left\{12c - 7\xi + 2\mathfrak{z}b\_0(\mathfrak{z})\,\xi^2\,\sqrt[3]{2}\left(-\frac{\mathfrak{S}1\xi\mathsf{C}\mathsf{M}}{d(\mathfrak{c} - \mathfrak{f})}\right)^{4/5} (2c - 2\xi)^{4/5}\right\}.\tag{36}$$

for *z* with a large number multiplied with itself maintains the equality (i.e., *z* ! *αz*, for *α* large and positive). The solution is valid for *z* downstream, and we exclude the interval 0, *mi* ½ � in **Figure 1** with no inlet *PO*2 value specified as a boundary condition at

*Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel*

The final form of the general solution to Eq. (25), downstream (defined on the

*K r*ð Þþ , *z Q* � ��<sup>2</sup>

*<sup>M</sup>*1ð Þ¼ � *<sup>r</sup>*, *<sup>z</sup>* ð Þ <sup>2</sup>*CM* <sup>1</sup> *<sup>c</sup>* � *dr*<sup>2</sup> � �<sup>Þ</sup> � �<sup>1</sup>*=*<sup>5</sup>

*N r*ð Þ¼ *<sup>c</sup>* � *dr*<sup>2</sup>

� �

<sup>3</sup> ð Þþ *ξ*, *b*0ð Þ*z*

(38)

*<sup>M</sup>*1ð Þ *<sup>r</sup>*, *<sup>z</sup>* <sup>g</sup>,

vuut , (39)

*<sup>r</sup>*<sup>2</sup> , (41)

, (40)

� 24*<sup>=</sup>*<sup>5</sup>

þ 1 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi �ð Þ *CM* <sup>1</sup> *<sup>c</sup>* � *dr*<sup>2</sup> � � �ð Þ <sup>2</sup>*CM* <sup>1</sup> *<sup>c</sup>* � *dr*<sup>2</sup> � � � � <sup>1</sup>*=*<sup>5</sup>

*<sup>M</sup>*1ð Þ *<sup>r</sup>*, *<sup>z</sup>* �

*z* ¼ 0 as this would give erroneous results in the upstream region.

*P r*ð Þ¼ , *<sup>z</sup>* <sup>2</sup>*CM*f*N r*ð Þ<sup>ϵ</sup> � <sup>½</sup>*z*ð Þ <sup>9</sup>*Cz* <sup>þ</sup> *mi <sup>=</sup>*6*C*� � *<sup>F</sup>*�<sup>1</sup>

2

<sup>10</sup>4*z*ð Þ �*S*<sup>1</sup> <sup>1</sup>*=*<sup>5</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>3</sup>*<sup>C</sup> <sup>=</sup>*16*C*4*=*<sup>5</sup> h i*M*1*=*<sup>5</sup> ð Þ <sup>2</sup>*CM* �<sup>1</sup> <sup>1</sup>

� � in **Figure 1**) using Eq. (30) is

<sup>4</sup> � sin *<sup>π</sup>*

*K r*ð Þ¼ , *<sup>z</sup>* <sup>0</sup>*:*<sup>29</sup> *<sup>c</sup>* � *dr*<sup>2</sup> � �

ð Þ <sup>2</sup>*CM* �<sup>1</sup> *<sup>π</sup>*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.90575*

interval *mi*, *mf*

where

**Figure 4.**

**265**

*2CM versus 0:<sup>01040</sup>* <sup>∗</sup> ð Þ *2CM <sup>9</sup>=<sup>5</sup>*

*.*

The two forms of the solution for *F*3 (one being the coefficient of *P* simplified in Eq. (25) and the other Eq. (36)) are equated to each other and compared:

$$- (\mathbf{1}/50) \left( \mathfrak{U} 2c - 7\xi + 25b\_{00} \mathfrak{x} \mathcal{M} 2^{2/5} \xi^2 \left( -\frac{\mathbb{S}\_1 \xi \mathcal{C} \mathcal{M}}{d(c-\xi)} \right)^{4/5} (2c-2\xi)^{4/5} (-\mathbb{S}\_1)^{1/5} (\mathsf{C}\mathcal{M})^{-4/5} \right) \times 1$$

$$(c-\xi)^{-1} \xi^{-2} = \mathcal{M} \frac{\mathbb{S}\_1 \xi}{c-\xi} \frac{z(2+3\mathcal{C})+m\_i}{2} \sim \mathcal{M} \frac{\mathbb{S}\_1 \xi}{c-\xi} \frac{z(2+3\mathcal{C})}{2}.$$

For *z* large *mi=*2 is dropped from the total expression. The *ξ*<sup>4</sup>*=*<sup>5</sup> term is scaled by multiplying by a factor of 4 to obtain *ξ* approximately (see **Figure 3**). Here *<sup>b</sup>*<sup>00</sup> <sup>¼</sup> ð Þ <sup>1</sup>*=*<sup>8</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>3</sup>*<sup>C</sup> <sup>=</sup>* <sup>2</sup> ð Þ �*S*<sup>1</sup> <sup>1</sup>*=*<sup>5</sup> � � h i

$$b\_0(\mathbf{z}) = b\_{00} \mathbf{z} \frac{M(-\mathbf{S1})^{1/5}}{\left(\mathbf{C}\mathbf{M}\right)^{4/5}}.\tag{37}$$

Since *z* can be large downstream, even for some small 1, then the term 12*c* � 7*ξ* drops out. This results in equality of the two forms of F3 presented. A final substitution

**Figure 3.** *ξ versus 4ξ<sup>4</sup>=5.*

*Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel DOI: http://dx.doi.org/10.5772/intechopen.90575*

for *z* with a large number multiplied with itself maintains the equality (i.e., *z* ! *αz*, for *α* large and positive). The solution is valid for *z* downstream, and we exclude the interval 0, *mi* ½ � in **Figure 1** with no inlet *PO*2 value specified as a boundary condition at *z* ¼ 0 as this would give erroneous results in the upstream region.

The final form of the general solution to Eq. (25), downstream (defined on the interval *mi*, *mf* � � in **Figure 1**) using Eq. (30) is

$$\begin{split}P(r,z)&=2\text{CM}\{\mathbf{N}(r)\mathbf{e}\times\left[\mathbf{z}(\mathbf{9}\,\mathbf{C}z+m\_{i})/\mathbf{6C}\right]\times\mathbf{F}\_{3}^{-1}(\xi,b\_{0}(\mathbf{z}))+\\ &\quad \underbrace{\left(2\text{CM}\right)^{-1}\frac{\pi^{2}}{4}\left(-\sin\left(\frac{\pi}{2}K(r,z)+\mathbf{Q}\right)^{-2}+\frac{1}{3}\right)\cdot\mathbf{2}^{4/5}}\_{M\_{1}(r,z)}-\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(38)\\ \mathbf{10}^{4}z(-\text{S1})^{1/5}\left[\left(2+\mathbf{3C}\right)/\mathbf{16C}^{4/5}\right]\mathbf{M}^{1/5}\left(2\mathbf{C}\mathbf{M}\right)^{-1}\frac{1}{M\_{1}(r,z)}\},\end{split}\tag{38}$$

where

*<sup>F</sup>*3ð Þ¼ � *<sup>ξ</sup>*, *<sup>b</sup>*0ð Þ*<sup>z</sup>* <sup>50</sup>ð Þ *<sup>c</sup>* � *<sup>ξ</sup> <sup>ξ</sup>*<sup>2</sup> � ��<sup>1</sup>

( )

� *<sup>S</sup>*1*ξCM d c*ð Þ � *ξ* � �4*=*<sup>5</sup>

The two forms of the solution for *F*3 (one being the coefficient of *P* simplified in

!

2 p5

Eq. (25) and the other Eq. (36)) are equated to each other and compared:

by multiplying by a factor of 4 to obtain *ξ* approximately (see **Figure 3**).

*b*0ð Þ¼ *z b*00*z*

*<sup>ξ</sup>*<sup>2</sup> � <sup>1</sup> *<sup>ξ</sup>CM d c*ð Þ � *ξ* � �<sup>4</sup>*=*<sup>5</sup>

*z*ð Þþ 2 þ 3*C mi*

For *z* large *mi=*2 is dropped from the total expression. The *ξ*<sup>4</sup>*=*<sup>5</sup> term is scaled

Since *z* can be large downstream, even for some small 1, then the term 12*c* � 7*ξ* drops out. This results in equality of the two forms of F3 presented. A final substitution

<sup>12</sup>*<sup>c</sup>* � <sup>7</sup> *<sup>ξ</sup>* <sup>þ</sup> <sup>25</sup>*b*0ð Þ*<sup>z</sup> <sup>ξ</sup>*<sup>2</sup> ffiffi

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*<sup>ξ</sup>*�<sup>2</sup> <sup>¼</sup> *<sup>M</sup>* <sup>1</sup> *<sup>ξ</sup>*

*c* � *ξ*

�ð Þ <sup>1</sup>*=*<sup>50</sup> <sup>12</sup>*<sup>c</sup>* � <sup>7</sup> *<sup>ξ</sup>* <sup>þ</sup> <sup>25</sup>*b*00*zM*2<sup>2</sup>*=*<sup>5</sup>

ð Þ *<sup>c</sup>* � *<sup>ξ</sup>* �<sup>1</sup>

**Figure 3.** *ξ versus 4ξ<sup>4</sup>=5.*

**264**

Here *<sup>b</sup>*<sup>00</sup> <sup>¼</sup> ð Þ <sup>1</sup>*=*<sup>8</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>3</sup>*<sup>C</sup> <sup>=</sup>* <sup>2</sup> ð Þ �*S*<sup>1</sup> <sup>1</sup>*=*<sup>5</sup> � � h i

�

ð Þ <sup>2</sup>*<sup>c</sup>* � <sup>2</sup>*<sup>ξ</sup>* <sup>4</sup>*=*<sup>5</sup>

ð Þ <sup>2</sup>*<sup>c</sup>* � <sup>2</sup>*<sup>ξ</sup>* <sup>4</sup>*=*<sup>5</sup>

*c* � *ξ*

<sup>2</sup> � *<sup>M</sup>* <sup>1</sup> *<sup>ξ</sup>*

*<sup>M</sup>*ð Þ �*S*<sup>1</sup> <sup>1</sup>*=*<sup>5</sup>

ð Þ �<sup>1</sup> 1*=*5

*z*ð Þ 2 þ 3*C* <sup>2</sup> *:*

ð Þ *CM* <sup>4</sup>*=*<sup>5</sup> *:* (37)

*:* (36)

ð Þ *CM* �4*=*<sup>5</sup>

�

$$K(r,z) = 0.29 \left(c - dr^2\right) \sqrt{\frac{-(\text{CM})\mathbb{S}\_1 \left(c - dr^2\right)}{\left(-(\text{2CM})\mathbb{S}\_1 \left(c - dr^2\right)\right)^{1/\xi}}},\tag{39}$$

$$\mathbf{M}\_1(r, z) = \left(-(2\mathbf{C}\mathbf{M})\mathbb{S}\_1(\varepsilon - dr^2)\right)^{1/5},\tag{40}$$

$$N(r) = \frac{c - dr^2}{r^2},\tag{41}$$

**Figure 4.** *2CM versus 0:<sup>01040</sup>* <sup>∗</sup> ð Þ *2CM <sup>9</sup>=<sup>5</sup>*

*.*

*C* and *Q* are constants and ϵ ¼ �ð Þ <sup>1</sup> 1*=n* , small, for *n* = 5, 4, 3, 2.

As an approximation, if we divide Eq. (26) by *z*, using Eq. (30), for large *z* downstream in the channel, it can be seen that the following equation emerges:

$$\frac{\partial^2}{\partial \xi^2} (\lambda \mathcal{W}) + F\_1(\xi) \frac{\partial}{\partial \xi} (\lambda \mathcal{W}) = F\_2(\xi) \lambda^2 \mathcal{W}^2 - F\_3(\xi) (\lambda \mathcal{W}) \dots$$

Now the *<sup>z</sup>* part appearing in *<sup>λ</sup>* is 1*=*ð Þ <sup>2</sup>*CM* <sup>1</sup>*=*<sup>5</sup> . Multiplying the equation above by ð Þ <sup>2</sup>*CM* <sup>6</sup>*=*<sup>5</sup> will result in the left side of the equation to consist of two 2*CM* terms, and the right-hand side of the equation to have two 2ð Þ *CM* <sup>9</sup>*=*<sup>5</sup> terms where *<sup>F</sup>*3ð Þ*<sup>ξ</sup>* has a ð Þ <sup>2</sup>*CM* <sup>4</sup>*=*<sup>5</sup> term from Eq. (36). In Eq. (38) the sin ð Þ *<sup>π</sup>=*2*K r*ð Þþ*<sup>Q</sup>* �<sup>2</sup> *<sup>M</sup>*1ð Þ *<sup>r</sup>*, *<sup>z</sup>* term will oscillate to zero as *z* approaches minus infinity.

determined and is shown in **Table 1** for low and high hematocrit. The value of *Dp* is obtained from Table 1 in [6] and *c* ¼ 1250 and *d* ¼ 0*:*5. For high hematocrit as shown in the Appendix, the solution incorporates different values of *c* and *d*. Also we let <sup>1</sup> ¼ �0*:*<sup>911</sup> � <sup>10</sup>�<sup>8</sup> in the core region and <sup>1</sup> ¼ �0*:*<sup>211</sup> � <sup>10</sup>�<sup>3</sup> the in plasma layer. Here there are two different separation constants for two different regions. The fourth equation used was that the change in flux at the wall at the edge of

**Constants** *C Q* CP**<sup>1</sup>** CP**<sup>2</sup>** High hematocrit = 0.45 �0.1009687192 6.261616672 �107 1.111333619 1.542724257 Low hematocrit t = 0.15 �0.09824747318 2063.831966 1.111333619 1.542724258

*Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel*

tion shown in **Figure 1** (i.e., impermeable membrane) at *r* ¼ 0 was satisfied exactly for all *z* for the core region solution. This is the fifth boundary condition. There is no

inlet *PO*<sup>2</sup> specified at *z* ¼ 0. The oxygen tension in core is *PO*2ð Þ¼ core

of approximately 150 mmHg at *r* ¼ 0, *z* ¼ *mi* ¼ �7000 microns.

*System of constants for associated system of four unknowns solved in maple.*

*W Z*ð Þ¼ <sup>1</sup>

The development of *η* in the neighborhood of *u* ¼ 0 is

*η* ¼ 1 þ

*<sup>η</sup>* � <sup>1</sup> <sup>¼</sup> *<sup>u</sup>π<sup>i</sup>*

ð Þ *<sup>η</sup>* � <sup>1</sup> <sup>2</sup> ¼ � *<sup>u</sup>*<sup>2</sup>*π*<sup>2</sup>

*uπi ω* þ 1 2!

*<sup>ω</sup>* <sup>1</sup> <sup>þ</sup>

*<sup>ω</sup>*<sup>2</sup> <sup>1</sup> <sup>þ</sup>

1 2! *uπi ω* þ 1 3!

*uπi ω* þ

is zero of the second order at all the points *u* ¼ 2*μω μ*ð ¼ 0, � 1, � 2, � 3, *::*).

As in [9], we consider the following expression:

*<sup>Z</sup>*<sup>2</sup> <sup>þ</sup><sup>X</sup> *w*

1 ð Þ *Z* � *w*

and derive a function of *η* which behaves like the Weierstrass P function at *η* ¼ 0.

*uπi ω* � �<sup>2</sup>

� �<sup>2</sup> "

7 12

" #

<sup>2</sup> � <sup>1</sup>

*η* ¼ exp ð Þ *uπi=ω* , (45)

*uπi ω*

*uπi ω* � �<sup>2</sup>

þ …

*<sup>w</sup>*<sup>2</sup> *:* (44)

þ … , (46)

� (47)

, (48)

� � is zero. In addition the no-flux condi-

*P* which is in the form of Eq. (22) and gives a *PO*<sup>2</sup>

the plasma layer far downstream *z*< < *mf*

*DOI: http://dx.doi.org/10.5772/intechopen.90575*

<sup>33</sup>*:*<sup>07440049</sup> <sup>þ</sup> <sup>3</sup>*:*63804058210�<sup>7</sup>

**7. Weierstrass elliptic function**

Observe that the function

or

**267**

**Table 1.**

The Weierstrass P function is defined as

A best line of fit can be made by scaling the term 2ð Þ *CM* <sup>9</sup>*=*<sup>5</sup> . Multiplying this by approximately 0.01040 results in a best approximation as shown in **Figure 4**. This allows us to cancel almost all *z* dependence in the equation except oscillating term and get a homogeneous "almost" *z*-independent equation as in [7]. Hence, the PDE of Eq. (26) can be reduced to an ode in *ξ* of similar form but homogeneous as *z* approaches minus infinity.

#### **6. Boundary and matching conditions at wall and core-plasma interface**

We employ the Robin boundary condition Eq. (58) in Appendix and derivative matching conditions at the interface of plasma and RBC core regions, respectively, at *z* ¼ *mi* and *z* ¼ *mf* , shown in **Figure 1**, to determine constants *Q* and *C* as well as two additional constants for linear solution in plasma layer. The solution to the linear part of Eq. (13) defined in the plasma layer, i.e.,

$$v\_p \frac{\partial PO\_2(plasma)}{\partial \mathbf{z}} = D\_p \nabla^2 PO\_2(plasma),\tag{42}$$

is

$$PO\_2(plasma) = CP\_1 
odot U
\left(1/2c\sqrt{-\frac{\mathbb{S}\_1}{D\_p}} \frac{1}{\sqrt{d}}, r\sqrt[4]{-4\frac{\mathbb{S}\_1 d}{D\_p}}\right) + \dotsb$$

$$CP\_2 
odot V
\left(1/2c\sqrt{-\frac{\mathbb{S}\_1}{D\_p}} \frac{1}{\sqrt{d}}, r\sqrt[4]{-4\frac{\mathbb{S}\_1 d}{D\_p}}\right),$$

where CylinderU and CylinderV are parabolic cylinder functions and *CP*<sup>1</sup> and *CP*<sup>2</sup> are constants to be determined. <sup>1</sup> is a separation constant for Eq. (42). The following boundary matching condition is utilized at the interface of the plasma layer (1 micron in height) and RBC core region (49 microns in height):

$$\frac{\partial PO\_2(plasma)}{\partial r} = \frac{\partial PO\_2(core)}{\partial r},\tag{43}$$

at *z* ¼ *mi* and *z* ¼ *mf* , respectively (see **Figure 1**). Four equations in four unknowns were solved in Maple 18, where two constants are from each of the two solutions in plasma and core regions, respectively. The total of eight constants was

*Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel DOI: http://dx.doi.org/10.5772/intechopen.90575*


**Table 1.**

*C* and *Q* are constants and ϵ ¼ �ð Þ <sup>1</sup>

*<sup>∂</sup>ξ*<sup>2</sup> ð Þþ *<sup>λ</sup><sup>W</sup> <sup>F</sup>*1ð Þ*<sup>ξ</sup> <sup>∂</sup>*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

Now the *<sup>z</sup>* part appearing in *<sup>λ</sup>* is 1*=*ð Þ <sup>2</sup>*CM* <sup>1</sup>*=*<sup>5</sup>

ð Þ <sup>2</sup>*CM* <sup>4</sup>*=*<sup>5</sup> term from Eq. (36). In Eq. (38) the sin ð Þ *<sup>π</sup>=*2*K r*ð Þþ*<sup>Q</sup>* �<sup>2</sup>

linear part of Eq. (13) defined in the plasma layer, i.e.,

*PO*2ð Þ¼ *plasma CP*1*CylinderU* 1*=*2*c*

*CP*2*CylinderV* 1*=*2*c*

*<sup>∂</sup>PO*2ð Þ *plasma*

*<sup>∂</sup><sup>z</sup>* <sup>¼</sup> *Dp*∇<sup>2</sup>

*vp*

A best line of fit can be made by scaling the term 2ð Þ *CM* <sup>9</sup>*=*<sup>5</sup>

*∂ξ*

*∂*2

zero as *z* approaches minus infinity.

approaches minus infinity.

is

**266**

1*=n*

As an approximation, if we divide Eq. (26) by *z*, using Eq. (30), for large *z* downstream in the channel, it can be seen that the following equation emerges:

ð Þ¼ *<sup>λ</sup><sup>W</sup> <sup>F</sup>*2ð Þ*<sup>ξ</sup> <sup>λ</sup>*<sup>2</sup>

ð Þ <sup>2</sup>*CM* <sup>6</sup>*=*<sup>5</sup> will result in the left side of the equation to consist of two 2*CM* terms, and the right-hand side of the equation to have two 2ð Þ *CM* <sup>9</sup>*=*<sup>5</sup> terms where *<sup>F</sup>*3ð Þ*<sup>ξ</sup>* has a

approximately 0.01040 results in a best approximation as shown in **Figure 4**. This allows us to cancel almost all *z* dependence in the equation except oscillating term and get a homogeneous "almost" *z*-independent equation as in [7]. Hence, the PDE of Eq. (26) can be reduced to an ode in *ξ* of similar form but homogeneous as *z*

**6. Boundary and matching conditions at wall and core-plasma interface**

We employ the Robin boundary condition Eq. (58) in Appendix and derivative matching conditions at the interface of plasma and RBC core regions, respectively, at *z* ¼ *mi* and *z* ¼ *mf* , shown in **Figure 1**, to determine constants *Q* and *C* as well as two additional constants for linear solution in plasma layer. The solution to the

> ffiffiffiffiffiffiffiffiffiffi � 1 *Dp*

> > 4

1 ffiffiffi *d* p ,*r*

! s

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi �<sup>4</sup> <sup>1</sup> *<sup>d</sup> Dp*

s

1 ffiffiffi *d* p ,*r*

! s

ffiffiffiffiffiffiffiffiffiffiffiffi � <sup>1</sup> *Dp*

where CylinderU and CylinderV are parabolic cylinder functions and *CP*<sup>1</sup> and *CP*<sup>2</sup> are constants to be determined. <sup>1</sup> is a separation constant for Eq. (42). The following boundary matching condition is utilized at the interface of the plasma

*<sup>∂</sup><sup>r</sup>* <sup>¼</sup> *<sup>∂</sup>PO*2ð Þ *core*

at *z* ¼ *mi* and *z* ¼ *mf* , respectively (see **Figure 1**). Four equations in four unknowns were solved in Maple 18, where two constants are from each of the two solutions in plasma and core regions, respectively. The total of eight constants was

layer (1 micron in height) and RBC core region (49 microns in height):

*<sup>∂</sup>PO*2ð Þ *plasma*

s

, small, for *n* = 5, 4, 3, 2.

*<sup>W</sup>*<sup>2</sup> � *<sup>F</sup>*3ð Þ*<sup>ξ</sup>* ð Þ *<sup>λ</sup><sup>W</sup> :*

. Multiplying the equation above by

*<sup>M</sup>*1ð Þ *<sup>r</sup>*, *<sup>z</sup>* term will oscillate to

*PO*2ð Þ *plasma* , (42)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi �<sup>4</sup> <sup>1</sup> *<sup>d</sup> Dp*

,

*<sup>∂</sup><sup>r</sup>* , (43)

þ

. Multiplying this by

*System of constants for associated system of four unknowns solved in maple.*

determined and is shown in **Table 1** for low and high hematocrit. The value of *Dp* is obtained from Table 1 in [6] and *c* ¼ 1250 and *d* ¼ 0*:*5. For high hematocrit as shown in the Appendix, the solution incorporates different values of *c* and *d*. Also we let <sup>1</sup> ¼ �0*:*<sup>911</sup> � <sup>10</sup>�<sup>8</sup> in the core region and <sup>1</sup> ¼ �0*:*<sup>211</sup> � <sup>10</sup>�<sup>3</sup> the in plasma layer. Here there are two different separation constants for two different regions.

The fourth equation used was that the change in flux at the wall at the edge of the plasma layer far downstream *z*< < *mf* � � is zero. In addition the no-flux condition shown in **Figure 1** (i.e., impermeable membrane) at *r* ¼ 0 was satisfied exactly for all *z* for the core region solution. This is the fifth boundary condition. There is no inlet *PO*<sup>2</sup> specified at *z* ¼ 0. The oxygen tension in core is *PO*2ð Þ¼ core <sup>33</sup>*:*<sup>07440049</sup> <sup>þ</sup> <sup>3</sup>*:*63804058210�<sup>7</sup> *P* which is in the form of Eq. (22) and gives a *PO*<sup>2</sup> of approximately 150 mmHg at *r* ¼ 0, *z* ¼ *mi* ¼ �7000 microns.

#### **7. Weierstrass elliptic function**

The Weierstrass P function is defined as

$$W(Z) = \frac{1}{Z^2} + \sum\_{w} \frac{1}{\left(Z - w\right)^2} - \frac{1}{w^2}.\tag{44}$$

As in [9], we consider the following expression:

$$\eta = \exp\left(u\pi i/o\right),\tag{45}$$

and derive a function of *η* which behaves like the Weierstrass P function at *η* ¼ 0. The development of *η* in the neighborhood of *u* ¼ 0 is

$$\eta = \mathbf{1} + \frac{u\pi i}{\alpha} + \frac{\mathbf{1}}{2!} \left(\frac{u\pi i}{\alpha}\right)^2 + \dots,\tag{46}$$

or

$$\eta - 1 = \frac{u\pi i}{\omega} \left[ 1 + \frac{1}{2!} \frac{u\pi i}{\omega} + \frac{1}{3!} \left( \frac{u\pi i}{\omega} \right)^2 \right] \tag{47}$$

Observe that the function

$$\left(\left(\eta - 1\right)^{2} = -\frac{u^{2}\pi^{2}}{a^{2}}\left[1 + \frac{u\pi i}{a} + \frac{7}{12}\left(\frac{u\pi i}{a}\right)^{2} + \dots\right],\tag{48}$$

is zero of the second order at all the points *u* ¼ 2*μω μ*ð ¼ 0, � 1, � 2, � 3, *::*).

Consider a function *J*ð Þ*η* such that *J*ð Þ*η* 6¼ 0 for *η* ¼ 1, the function

$$\frac{J(\eta)}{\left(\eta-1\right)^{2}},\tag{49}$$

of the permeable membrane downstream. It is in this region that there is a significant concentration of ATP released. It is important to mention that a rapid decrease in oxygen saturation is one means to produce ATP; it is also accomplished by means

*Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel*

A well-known governing nonlinear PDE used to model oxygen transport was

formulated in a recent paper in a generalized coordinate system where the Laplacian is expressed in metric tensor form. A reduction of the PDE to simpler problem subject to specific integrability conditions was shown there. A reduced almost linear ode was derived, and in the present paper, a solution has been obtained using a well-known factorization method for the second-order ode where a compatibility equation has been used in equating it to a specific form of the original differential equation. Approximate oxygen tension profiles have been determined downstream in a micro-channel in the vicinity of a permeable membrane with an oxygen supply on the other side of the membrane. Although it is expected that ATP will be released as blood flows past the permeable membrane downstream, it has been shown mathematically that this is the case and increases in hematocrit produce more ATP. Future work remains to apply tensor equations for a

moving arterial surface as generalized at the start of the present work.

For cell-free region as shown in **Figure 1**, we have that

*vp ∂PO*<sup>2</sup>

public, commercial, or not-for-profit sectors.

This research did not receive any specific grant from funding agencies in the

*<sup>∂</sup><sup>z</sup>* <sup>¼</sup> *Dp*∇<sup>2</sup>

In this case there is a central core region where oxygen dissociates to form *HbO*2.

*dSO*<sup>2</sup> *dPO*<sup>2</sup>

where *vp* is the velocity of plasma, given at the end of the Appendix.

Therefore, the velocity in core region, i.e., *<sup>v</sup>*<sup>ℓ</sup>*O*<sup>2</sup> <sup>¼</sup> *f vp <sup>q</sup><sup>i</sup>* , *dSO*<sup>2</sup>

The model consists of the following partial differential equation:

<sup>1</sup> <sup>þ</sup> ½ � *HbT KRBC*

*KRBC Kp*

*∂PO*<sup>2</sup>

*PO*2, (52)

*dPO*<sup>2</sup> .

*PO*2*:* (53)

*<sup>∂</sup><sup>z</sup>* <sup>¼</sup> *Dp*∇<sup>2</sup>

of applying shear stress on the RBC as shown in [12].

*DOI: http://dx.doi.org/10.5772/intechopen.90575*

**9. Conclusion**

**Funding**

**Appendix**

**A.1 Cell free**

**A.2 Core region**

**269**

*vp*ð Þþ 1 � *HT vRBCHT*

is infinite of the second order for all values *u* ¼ 0 and *u* ¼ 2*μω*. This behavior at these points is the same as the Weierstrass P function. We write *<sup>J</sup>*ð Þ¼ *<sup>η</sup> <sup>a</sup>* <sup>þ</sup> *<sup>b</sup><sup>η</sup>* <sup>þ</sup> *<sup>c</sup>η*<sup>2</sup> where *<sup>a</sup>*, *<sup>b</sup>*,*<sup>c</sup>* are constants. Since *<sup>η</sup>*<sup>2</sup> <sup>¼</sup> *<sup>e</sup>* 2*uπi <sup>ω</sup>* , we have

$$\frac{f(\eta)}{\left(\eta - 1\right)^{2}} = $$

$$\frac{a + b\left[1 + \frac{u\pi i}{o\nu} + \frac{1}{2!} \left(\frac{u\pi i}{o\nu}\right)^{2} + \dots\right] + c\left[1 + \frac{2u\pi i}{o\nu} + \frac{1}{2!} \left(\frac{2u\pi i}{o\nu}\right)^{2}\right]}{-\frac{u^{2}\pi^{2}}{o\nu^{2}}\left[1 + \frac{u\pi i}{o\nu} + \frac{1}{3}\left(\frac{u\pi i}{o\nu}\right)^{2} + \dots\right]}$$

$$-\frac{\alpha^{2}}{\pi^{2}}\left[\frac{a + b\left(1 + \frac{u\pi i}{o\nu} + \frac{1}{2}\left(\frac{u\pi i}{o\nu}\right)^{2} + \dots\right) + c\left[1 + \frac{2u\pi i}{o\nu} + \dots\right]}{u^{2}}\right] \left[1 - \frac{u\pi i}{o\nu} + u^{2}\right].\tag{50}$$

As in [9] the Weierstrass P function is shown to have the following degenerate form which is used in Section 5:

$$\frac{J(\eta)}{\left(\eta - 1\right)^{2}} = \left(\pi/2\alpha\right)^{2} \left[\sin^{-2}\left(u\pi/2\alpha\right) - \mathbf{1}/3\right].\tag{51}$$

#### **8. Results and discussion**

The present work shows that near the inlet of the permeable membrane (see **Figure 1**), there is a significant drop at the wall of the channel in *PO*<sup>2</sup> as compared to downstream values as shown in **Figures 2** and **5**. It is worthy to note that the structure of the solution obtained in terms of degenerate Weierstrass P function forms a near constant solution across the height of a micro-fluidic channel downstream at end of the membrane region. This is seen in both **Figures 2** and **5**, and in **Figure 5**, the contours flatten out as the flow of blood proceeds far downstream. The release of ATP has been shown to be caused by a change in oxygen saturation [6]. It is concluded that there is a significant decrease in oxygen tension to the right

**Figure 5.**

*Contour plot for channel from z0* ¼ �*7000 to z* ¼ �*7900 microns. Vertical axis is r variation across the channel. Horizontal axis is z variation.*

of the permeable membrane downstream. It is in this region that there is a significant concentration of ATP released. It is important to mention that a rapid decrease in oxygen saturation is one means to produce ATP; it is also accomplished by means of applying shear stress on the RBC as shown in [12].

### **9. Conclusion**

Consider a function *J*ð Þ*η* such that *J*ð Þ*η* 6¼ 0 for *η* ¼ 1, the function

is infinite of the second order for all values *u* ¼ 0 and *u* ¼ 2*μω*.

We write *<sup>J</sup>*ð Þ¼ *<sup>η</sup> <sup>a</sup>* <sup>þ</sup> *<sup>b</sup><sup>η</sup>* <sup>þ</sup> *<sup>c</sup>η*<sup>2</sup> where *<sup>a</sup>*, *<sup>b</sup>*,*<sup>c</sup>* are constants.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*uπi ω* � �<sup>2</sup>

*uπi ω* � �<sup>2</sup>

ð Þ *<sup>η</sup>* � <sup>1</sup> <sup>2</sup> <sup>¼</sup> ð Þ *<sup>π</sup>=*2*<sup>ω</sup>* <sup>2</sup> sin �<sup>2</sup>

!

" #

� *<sup>u</sup>*<sup>2</sup>*π*<sup>2</sup> *<sup>ω</sup>*<sup>2</sup> <sup>1</sup> <sup>þ</sup>

Since *<sup>η</sup>*<sup>2</sup> <sup>¼</sup> *<sup>e</sup>*

*J*ð Þ*η* ð Þ *<sup>η</sup>* � <sup>1</sup> <sup>2</sup> <sup>¼</sup>

> � *ω*2 *π*2

**Figure 5.**

**268**

2*uπi*

*a* þ *b* 1 þ

*a* þ *b* 1 þ

form which is used in Section 5:

**8. Results and discussion**

*channel. Horizontal axis is z variation.*

*<sup>ω</sup>* , we have

*uπi ω* þ 1 2!

*uπi ω* þ 1 2

*J*ð Þ*η*

This behavior at these points is the same as the Weierstrass P function.

þ …

þ …

As in [9] the Weierstrass P function is shown to have the following degenerate

The present work shows that near the inlet of the permeable membrane (see **Figure 1**), there is a significant drop at the wall of the channel in *PO*<sup>2</sup> as compared to downstream values as shown in **Figures 2** and **5**. It is worthy to note that the structure of the solution obtained in terms of degenerate Weierstrass P function forms a near constant solution across the height of a micro-fluidic channel downstream at end of the membrane region. This is seen in both **Figures 2** and **5**, and in **Figure 5**, the contours flatten out as the flow of blood proceeds far downstream. The release of ATP has been shown to be caused by a change in oxygen saturation [6]. It is concluded that there is a significant decrease in oxygen tension to the right

*Contour plot for channel from z0* ¼ �*7000 to z* ¼ �*7900 microns. Vertical axis is r variation across the*

*uπi ω* þ 1 3

� � " #

þ *c* 1 þ

*uπi ω* � �<sup>2</sup>

þ *c* 1 þ

" #

2*uπi ω* þ

þ …

2*uπi <sup>ω</sup>* <sup>þ</sup> …

*<sup>u</sup>*<sup>2</sup> <sup>1</sup> � *<sup>u</sup>π<sup>i</sup>*

1 2!

ð Þ� *<sup>u</sup>π=*2*<sup>ω</sup>* <sup>1</sup>*=*<sup>3</sup> � �*:* (51)

� �<sup>2</sup> " #

2*uπi ω*

> *<sup>ω</sup>* <sup>þ</sup> *<sup>u</sup>*<sup>2</sup> � �

*:*

(50)

*J*ð Þ*η*

ð Þ *<sup>η</sup>* � <sup>1</sup> <sup>2</sup> , (49)

A well-known governing nonlinear PDE used to model oxygen transport was formulated in a recent paper in a generalized coordinate system where the Laplacian is expressed in metric tensor form. A reduction of the PDE to simpler problem subject to specific integrability conditions was shown there. A reduced almost linear ode was derived, and in the present paper, a solution has been obtained using a well-known factorization method for the second-order ode where a compatibility equation has been used in equating it to a specific form of the original differential equation. Approximate oxygen tension profiles have been determined downstream in a micro-channel in the vicinity of a permeable membrane with an oxygen supply on the other side of the membrane. Although it is expected that ATP will be released as blood flows past the permeable membrane downstream, it has been shown mathematically that this is the case and increases in hematocrit produce more ATP. Future work remains to apply tensor equations for a moving arterial surface as generalized at the start of the present work.

#### **Funding**

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

#### **Appendix**

#### **A.1 Cell free**

For cell-free region as shown in **Figure 1**, we have that

$$
v\_p \frac{\partial PO\_2}{\partial \mathbf{z}} = D\_p \nabla^2 PO\_2,\tag{52}$$

where *vp* is the velocity of plasma, given at the end of the Appendix.

#### **A.2 Core region**

In this case there is a central core region where oxygen dissociates to form *HbO*2. Therefore, the velocity in core region, i.e., *<sup>v</sup>*<sup>ℓ</sup>*O*<sup>2</sup> <sup>¼</sup> *f vp <sup>q</sup><sup>i</sup>* , *dSO*<sup>2</sup> *dPO*<sup>2</sup> . The model consists of the following partial differential equation:

$$
\left[v\_p(\mathbf{1} - H\_T) + v\_{RBC}H\_T \frac{K\_{RBC}}{K\_p} \left(\mathbf{1} + \frac{[Hb\_T]}{K\_{RBC}} \frac{dSO\_2}{dPO\_2}\right)\right] \frac{\partial PO\_2}{\partial \mathbf{z}} = D\_p \nabla^2 PO\_2. \tag{53}
$$

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

#### **A.3 Boundary conditions**

The boundary conditions are defined as follows:

$$PO\_2(r, m\_i) = (PO\_2)\_i, \ r \in (0, h), \tag{54}$$

The boundary condition is

*DOI: http://dx.doi.org/10.5772/intechopen.90575*

*∂PO*<sup>2</sup> *∂r* � � � � *r*¼*h*

**A.4 Velocity profile function**

*vp*ð Þ¼ **<sup>x</sup>**^ 3 130 ð Þ� <sup>10</sup><sup>6</sup>

**Author details**

Ontario, Canada

**271**

Terry E. Moschandreou<sup>1</sup>

London, Ontario, Canada

<sup>4</sup>*w h*<sup>3</sup> <sup>þ</sup> *<sup>μ</sup>p=μ<sup>c</sup>*

found in Table 1 in [6].

<sup>¼</sup> *DmKm DpKp*

The following velocity profile is used in channel:

� �*y*<sup>3</sup> *i*

stream with a resulting higher production of ATP.

\*Address all correspondence to: tmoschan@uwo.ca

provided the original work is properly cited.

� �

where ð Þ *PO*<sup>2</sup> *<sup>o</sup>* is the *PO*<sup>2</sup> level on the other side of the membrane.

*Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel*

ð Þ *PO*<sup>2</sup> *<sup>o</sup>* � *PO*2ð Þ *h*, *z τ*

All of these conditions act on the entire system, and constants appearing are

*<sup>h</sup>*<sup>2</sup> � *<sup>y</sup>*<sup>2</sup> *i* � � <sup>þ</sup> *<sup>μ</sup>p=μ<sup>c</sup> <sup>y</sup>*<sup>2</sup>

ous results (**Figures 2** and **5**) were based on this data. **Figure 6** is based on a discharge hematocrit of approximately 40% higher with relative apparent viscosity equal to 1.7. It is apparent from the graph that in comparison to **Figure 2** with lower discharge hematocrit that there is an increase in the drop of *PO*<sup>2</sup> profiles down-

\* and Keith C. Afas2

1 Department of Applied Mathematics, Faculty of Science, Western University,

2 Medical Biophysics, Faculty of Medical Science, Western University, London,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Relative apparent viscosity, *μp=μ<sup>c</sup>* � 1, for low discharge hematocrit. The previ-

(

� �, *<sup>z</sup>*<sup>∈</sup> *mi*, *mf*

*<sup>h</sup>*<sup>2</sup> � **<sup>x</sup>**^<sup>2</sup> � �, *yi* ≤ ∣**x**^∣ ≤*<sup>h</sup>*

*<sup>i</sup>* � **<sup>x</sup>**^<sup>2</sup> � � <sup>0</sup>≤ ∣**x**^∣ ≤ *yi*

� �, (58)

(59)

1.Pre-membrane

$$\left. \frac{\partial PO\_2}{\partial r} \right|\_{r=h} = 0, z \in (0, m\_i). \tag{55}$$

2.Bottom of membrane region

$$\left. \frac{\partial PO\_2}{\partial r} \right|\_{r=0} = 0, z \in (0, L). \tag{56}$$

3.Post-membrane

$$\left. \frac{\partial PO\_2}{\partial r} \right|\_{r=h} = 0, z \in \left( m\_f, L \right). \tag{57}$$

The only region not covered by the above three flux equations is the region of *PO*<sup>2</sup> occupying *z*∈ *mi*, *mf* at *<sup>r</sup>* <sup>¼</sup> *<sup>h</sup>* the membrane. This region is governed by a Robin condition specified in [8].

#### **Figure 6.**

*Oxygen tension PO2 versus channel height in microns at axial distance z* ¼ *mi =* �*7000 (at top) through z* ¼ *mf =* �*7700 microns (at the bottom) for higher human hematocrit in intervals of approximately 95 microns.*

*Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel DOI: http://dx.doi.org/10.5772/intechopen.90575*

The boundary condition is

**A.3 Boundary conditions**

1.Pre-membrane

3.Post-membrane

*PO*<sup>2</sup> occupying *z*∈ *mi*, *mf*

**Figure 6.**

**270**

Robin condition specified in [8].

2.Bottom of membrane region

The boundary conditions are defined as follows:

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*PO*2ð Þ¼ *r*, *mi* ð Þ *PO*<sup>2</sup> *<sup>i</sup>*

*∂PO*<sup>2</sup> *∂r r*¼*h*

*∂PO*<sup>2</sup> *∂r r*¼0

*∂PO*<sup>2</sup> *∂r r*¼*h*

The only region not covered by the above three flux equations is the region of

*Oxygen tension PO2 versus channel height in microns at axial distance z* ¼ *mi =* �*7000 (at top) through z* ¼ *mf =* �*7700 microns (at the bottom) for higher human hematocrit in intervals of approximately 95 microns.*

at *<sup>r</sup>* <sup>¼</sup> *<sup>h</sup>* the membrane. This region is governed by a

, *r*∈ð Þ 0, *h* , (54)

¼ 0, *z*∈ð Þ 0, *mi :* (55)

¼ 0, *z*∈ð Þ 0, *L :* (56)

<sup>¼</sup> 0, *<sup>z</sup>*<sup>∈</sup> *mf* , *<sup>L</sup> :* (57)

$$\left. \frac{\partial \text{PO}\_2}{\partial r} \right|\_{r=h} = \frac{D\_m K\_m}{D\_p K\_p} \left( \frac{(\text{PO}\_2)\_o - \text{PO}\_2(h, z)}{\tau} \right), z \in \left( m\_i, m\_f \right), \tag{58}$$

where ð Þ *PO*<sup>2</sup> *<sup>o</sup>* is the *PO*<sup>2</sup> level on the other side of the membrane.

All of these conditions act on the entire system, and constants appearing are found in Table 1 in [6].

#### **A.4 Velocity profile function**

The following velocity profile is used in channel:

$$w\_p(\hat{\mathbf{x}}) = \frac{\mathbf{3}(\mathbf{1}\mathbf{3}0) \cdot \mathbf{10}^6}{4w\left(h^3 + \left(\mu\_p/\mu\_c\right)p\_i^3\right)} \begin{cases} \left(h^2 - \hat{\mathbf{x}}^2\right), & y\_i \le |\hat{\mathbf{x}}| \le h\\ \left(h^2 - y\_i^2\right) + \mu\_p/\mu\_c(y\_i^2 - \hat{\mathbf{x}}^2) & \mathbf{0} \le |\hat{\mathbf{x}}| \le y\_i \end{cases} \tag{59}$$

Relative apparent viscosity, *μp=μ<sup>c</sup>* � 1, for low discharge hematocrit. The previous results (**Figures 2** and **5**) were based on this data. **Figure 6** is based on a discharge hematocrit of approximately 40% higher with relative apparent viscosity equal to 1.7. It is apparent from the graph that in comparison to **Figure 2** with lower discharge hematocrit that there is an increase in the drop of *PO*<sup>2</sup> profiles downstream with a resulting higher production of ATP.

#### **Author details**

Terry E. Moschandreou<sup>1</sup> \* and Keith C. Afas2

1 Department of Applied Mathematics, Faculty of Science, Western University, London, Ontario, Canada

2 Medical Biophysics, Faculty of Medical Science, Western University, London, Ontario, Canada

\*Address all correspondence to: tmoschan@uwo.ca

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Nair PK, Huang NS, Olson JS. A simple model for prediction of oxygen transport rates by flowing blood in large capillaries. Microvascular Research. 1990;**39**:203-211

[2] Nair PK, Hellums JD, Olson JS. Prediction of oxygen transport rates in blood flowing in large capillaries. Microvascular Research. 1989;**36**: 269-285

[3] Nair PK. Simulation of Oxygen Transport in Capillaries. Rice University; 1988

[4] Moschandreou TE, Ellis CG, Goldman D. Influence of tissue metabolism and capillary oxygen supply on arteriolar oxygen transport: A computational model. Mathematical Biosciences. 2011;**232**(1):1-10

[5] Ng C-O. Dispersion in steady and oscillatory flows through a tube with reversible and irreversible wall reactions. Proceedings of the Royal Society A. 2006;**462**:481-515

[6] Sove RJ, Ghonaim N, Goldman D, Ellis CG. A computational model of a microfluidic device to measure the dynamics of oxygen-dependent ATP release from erythrocytes. PLoS One. 2013;**8**(11):1-9

[7] Estevez PG, Kuru S, Negro J, Nieto LM. Factorization of a class of almost linear second-order differential equations. Journal of Physics A: Mathematical and Theoretical. 2007;**40**: 9819-9824

[8] Afas KC, Moschandreou TE. Analytic multiplicative separation and existence investigation of non-linear oxygen transport with Poiseuille hemodynamic flow in a semi-generalized co-ordinate system. International Journal of Differential Equations and Applications. 2015;**14**:281-311

[9] Hancock H. Lectures on the Theory of Elliptic Functions. Dover; 2004

[10] Brennan MD, Rexius-Hall ML, Elgass LJ, Eddington DT. Oxygen control with microfluidics. Lab on a Chip. 2014;**22**:1-14

[11] Grinfeld P. Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer; 2010

[12] Wan J, Ristenpart WD, Stone HA. Dynamics of shear-induced ATP release from red blood cells. PNAS;**105**(43): 16432-16437

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University; 1988

2013;**8**(11):1-9

9819-9824

2015;**14**:281-311

**272**

269-285

[1] Nair PK, Huang NS, Olson JS. A simple model for prediction of oxygen transport rates by flowing blood in large capillaries. Microvascular Research.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

[9] Hancock H. Lectures on the Theory of Elliptic Functions. Dover; 2004

[10] Brennan MD, Rexius-Hall ML, Elgass LJ, Eddington DT. Oxygen control with microfluidics. Lab on a

[11] Grinfeld P. Introduction to Tensor Analysis and the Calculus of Moving

[12] Wan J, Ristenpart WD, Stone HA. Dynamics of shear-induced ATP release from red blood cells. PNAS;**105**(43):

Chip. 2014;**22**:1-14

16432-16437

Surfaces. Springer; 2010

[2] Nair PK, Hellums JD, Olson JS. Prediction of oxygen transport rates in blood flowing in large capillaries. Microvascular Research. 1989;**36**:

[3] Nair PK. Simulation of Oxygen Transport in Capillaries. Rice

[4] Moschandreou TE, Ellis CG, Goldman D. Influence of tissue

metabolism and capillary oxygen supply on arteriolar oxygen transport: A computational model. Mathematical Biosciences. 2011;**232**(1):1-10

[5] Ng C-O. Dispersion in steady and oscillatory flows through a tube with reversible and irreversible wall reactions. Proceedings of the Royal Society A. 2006;**462**:481-515

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### *Edited by Walter Legnani and Terry E. Moschandreou*

The editors of this book have incorporated contributions from a diverse group of leading researchers in the field of nonlinear systems. To enrich the scope of the content, this book contains a valuable selection of works on fractional differential equations.The book aims to provide an overview of the current knowledge on nonlinear systems and some aspects of fractional calculus. The main subject areas are divided into two theoretical and applied sections.

Nonlinear systems are useful for researchers in mathematics, applied mathematics, and physics, as well as graduate students who are studying these systems with reference to their theory and application. This book is also an ideal complement to the specific literature on engineering, biology, health science, and other applied science areas.

The opportunity given by IntechOpen to offer this book under the open access system contributes to disseminating the field of nonlinear systems to a wide range of researchers.

Published in London, UK © 2020 IntechOpen © FotoMak / iStock

Nonlinear Systems -Theoretical Aspects and Recent Applications

IntechOpen Book Series

Nonlinear Systems, Volume 2

Nonlinear Systems

Theoretical Aspects and Recent Applications

*Edited by Walter Legnani* 

*and Terry E. Moschandreou*