1. Introduction

In various designs, parts that transmit any motion are often used. Any such design often consists of an input and an output link connected by a force line. With the perception of the load, either compression (stretching) or twisting takes place here. Usually, such elements are checked for longitudinal stability, according to Euler's criterion [1], or for the ultimate twisting. However, such devices often perceive variable loads, for example, wind, shock, etc., at which various vibrations occur. In this regard, it is advisable to evaluate the dynamic stability, which can manifest itself in the form of self-oscillatory regimes, both for the whole truss structure and for its elements, or in the form of sudden destruction. The proposed work is devoted to the study of the loss of dynamic stability of the elements of a truss or frame.

The stability problem of the movement of mathematics and mechanics has been studied since the nineteenth century. To solve such problems, the criteria and theories of Routh E., Gurwitz A., Lyapunov A., Chetayev N., Mikhailov A., Nyquist H., Bolotin V., Popov E. [2–8], etc. are used.

In the last years, many developments have been made, both in the theory and applications of the subject. However, accurate analytical solutions in the calculations of vibrations of a structural element were obtained in rare cases. Typically, calculations are performed approximately. Simplifications are made when choosing a design scheme for the mechanism. In such cases, negligible features of the system are neglected, and the main parameters that determine the nature of the phenomenon are distinguished.

dΩ<sup>j</sup> dt <sup>¼</sup> <sup>X</sup><sup>n</sup> i¼1

Then the investigation reduces to solving equations

Dynamic Stability of Open Two-Link Mechanical Structures

DOI: http://dx.doi.org/10.5772/intechopen.91045

v ∂2 u <sup>∂</sup>t<sup>2</sup> � <sup>∂</sup>

u xð Þ¼ , <sup>t</sup> <sup>X</sup><sup>∞</sup>

d dt

oscillations occur (i.e., stability is lost), and at what not.

∂T ∂q\_ � �

i¼1

whose solution is sought in the form

determined from the initial conditions.

the second kind has the form

after some transformations.

Here

and space.

failure.

5

dξi

speed of the technological object.

ple, using the equation [22, 23]

∂Ω<sup>j</sup> ∂ξi fi

where ξ<sup>i</sup> are the coordinates of the system, fi = d ξi/dt, t is the time, and Ω<sup>j</sup> is the

If such a path seems natural for specialists in control systems, then for the specialists-mechanics, it may seem unusual, since they often solve the problem of determining the change of coordinates and the shape of oscillations [16–21]. Such processes are usually investigated by methods of the theory of elasticity, for exam-

> <sup>∂</sup><sup>x</sup> Ef <sup>∂</sup><sup>u</sup> ∂x � �

> > Hiθð Þ x sin pi

where ν and E are mass and elastic characteristics of the mechanical highway, f is cross-sectional area, Q is intensity of external load, and Hi, θ, pi, and α<sup>i</sup> are constants

In the case of using the Lagrange equation of the second kind, the oscillations of the kinetic (T) and potential energy (U) are considered. The Lagrange equation of

� ∂T

<sup>Q</sup> ¼ � <sup>∂</sup><sup>U</sup>

These Eqs. (5, 6) which are the basis of many papers on the dynamics of machines, for example, [21, 24, 25], etc., allow, under given boundary conditions, to estimate the change in the displacements of rod section, pipe string, etc. in time

On the one hand, such information is redundant if it is necessary to take into account the interconnection of a large number of factors. For example, to assess the performance of the system, it is enough to know under what conditions self-

On the other hand, due to the lack of explicit information about the stresses developed in the dynamic process, it is difficult to estimate the probability of part

The parameters of the movement of the mechanism are determined from Eq. (5)

t þ α<sup>i</sup>

, (1)

¼ Q xð Þ , t , (3)

� �: (4)

<sup>∂</sup><sup>q</sup> <sup>¼</sup> <sup>Q</sup>: (5)

<sup>∂</sup><sup>q</sup> : (6)

dt <sup>¼</sup> fi <sup>t</sup>, <sup>ξ</sup>1, <sup>ξ</sup>2, … , <sup>ξ</sup><sup>n</sup> ð Þ: (2)

In most cases, a method is specified in which parts of complex geometric shape (springs, crankshafts, etc.) are considered as equivalent straight bar or nonlinear elastic elements are replaced by linear elements. This approach allows replacing a mechanical system with concentrated masses with a system with distributed parameters [9]. Thus, simplifications are allowed that lead to the loss of objective data.

Some publications [10, 11] provide solutions to such problems by an approximate method with the replacement of the corresponding functional equations by suitable finite-dimensional difference schemes. As a result, the authors come to the problem of optimal control of the approximating system, which is described by equations in finite differences or the system of ordinary differential equations [9]. Then there is a need to consider the maximum principle and evaluate approximation methods. Such questions have not enough yet been investigated.

Some specialists of mechanical, for example, the authors of the Encyclopedia of Engineering Industry, Fedosov E., Krasovsky A., Popov E., propose to evaluate the stability of mechanical systems with distributed parameters by dispersion relations, i.e., according to the internal properties of the physical process. Here we use differential equations with variable coefficients that characterize the process under consideration. In this case, the solution of differential equations should be sought by numerical methods [12].

The condition for the stable operation of a system with distributed parameters was formulated in [13, 14]. The mathematical essence of the stability condition is formulated as follows:

If in the subspace W<sup>φ</sup> = 0 the process φ 0 is stable under integrally small perturbations with respect to the measure ||ρ||, and in the subspace W<sup>φ</sup> < 0 – asymptotically stable under integrally small perturbations with measure ||ρ||, then in a neighborhood ZR for any δ (ε, t0) > 0, there exists a number such that for t ≥ T it is true ρ [φ (, t)] < 2δ, if ρ [φ (, t0)] < δ and ρ [h (x)] < δ. Here, φ are the parameters of the process; h (x) is a vector function of admissible solutions.

At present, it has not been possible to find scientific publications in which stability criteria are sufficiently clearly formulated in the study of open two-link mechanical systems with distributed parameters in the presence of significant nonlinearities.

These mechanisms are widely used by technicians. Therefore, it is very important to develop such methods that would make it possible to more accurately mathematically formalize the functioning processes and determine the zones of stable and unstable operation of these mechanisms.

In this regard, the proposed work attempts to consider in more detail the stability issues of these mechanical systems.

## 2. Statement of the problem

The reliability of the functioning of the noted mechanisms under external variable loads is largely determined by the speeds of the links and the stresses in the force lines. Therefore, there is a need to study the Equations [15]

Dynamic Stability of Open Two-Link Mechanical Structures DOI: http://dx.doi.org/10.5772/intechopen.91045

$$\frac{d\Omega\_{\!\!\!}}{dt} = \sum\_{i=1}^{n} \frac{\partial \Omega\_{\!\!\!}}{\partial \xi\_{i}} f\_{\!\!\!/},\tag{1}$$

where ξ<sup>i</sup> are the coordinates of the system, fi = d ξi/dt, t is the time, and Ω<sup>j</sup> is the speed of the technological object.

Then the investigation reduces to solving equations

$$\frac{d\xi\_i}{dt} = f\_i(t, \xi\_1, \xi\_2, \dots, \xi\_n). \tag{2}$$

If such a path seems natural for specialists in control systems, then for the specialists-mechanics, it may seem unusual, since they often solve the problem of determining the change of coordinates and the shape of oscillations [16–21]. Such processes are usually investigated by methods of the theory of elasticity, for example, using the equation [22, 23]

$$
\upsilon \frac{\partial^2 u}{\partial t^2} - \frac{\partial}{\partial \mathbf{x}} \left( E \mathbf{f} \frac{\partial u}{\partial \mathbf{x}} \right) = Q(\mathbf{x}, t), \tag{3}
$$

whose solution is sought in the form

$$u(\mathbf{x},t) = \sum\_{i=1}^{\infty} H\_i \Theta(\mathbf{x}) \sin \left(p\_i t + \alpha\_i\right). \tag{4}$$

where ν and E are mass and elastic characteristics of the mechanical highway, f is cross-sectional area, Q is intensity of external load, and Hi, θ, pi, and α<sup>i</sup> are constants determined from the initial conditions.

In the case of using the Lagrange equation of the second kind, the oscillations of the kinetic (T) and potential energy (U) are considered. The Lagrange equation of the second kind has the form

$$\frac{d}{dt}\left(\frac{\partial T}{\partial \dot{q}}\right) - \frac{\partial T}{\partial q} = Q. \tag{5}$$

Here

In the last years, many developments have been made, both in the theory and applications of the subject. However, accurate analytical solutions in the calculations of vibrations of a structural element were obtained in rare cases. Typically, calculations are performed approximately. Simplifications are made when choosing

In most cases, a method is specified in which parts of complex geometric shape (springs, crankshafts, etc.) are considered as equivalent straight bar or nonlinear elastic elements are replaced by linear elements. This approach allows replacing a mechanical system with concentrated masses with a system with distributed parameters [9]. Thus, simplifications are allowed that lead to the loss of

The condition for the stable operation of a system with distributed parameters was formulated in [13, 14]. The mathematical essence of the stability condition

These mechanisms are widely used by technicians. Therefore, it is very impor-

tant to develop such methods that would make it possible to more accurately mathematically formalize the functioning processes and determine the zones of

In this regard, the proposed work attempts to consider in more detail the

The reliability of the functioning of the noted mechanisms under external variable loads is largely determined by the speeds of the links and the stresses in the force lines. Therefore, there is a need to study the Equations [15]

stable and unstable operation of these mechanisms.

stability issues of these mechanical systems.

2. Statement of the problem

If in the subspace W<sup>φ</sup> = 0 the process φ 0 is stable under integrally small perturbations with respect to the measure ||ρ||, and in the subspace W<sup>φ</sup> < 0 – asymptotically stable under integrally small perturbations with measure ||ρ||, then in a neighborhood ZR for any δ (ε, t0) > 0, there exists a number such that for t ≥ T it is true ρ [φ (, t)] < 2δ, if ρ [φ (, t0)] < δ and ρ [h (x)] < δ. Here, φ are the parameters of the process; h (x) is a vector function of admissible solutions. At present, it has not been possible to find scientific publications in which stability criteria are sufficiently clearly formulated in the study of open two-link mechanical systems with distributed parameters in the presence of significant

a design scheme for the mechanism. In such cases, negligible features of the system are neglected, and the main parameters that determine the nature of the

Some publications [10, 11] provide solutions to such problems by an approximate method with the replacement of the corresponding functional equations by suitable finite-dimensional difference schemes. As a result, the authors come to the problem of optimal control of the approximating system, which is described by equations in finite differences or the system of ordinary differential equations [9]. Then there is a need to consider the maximum principle and evaluate approximation methods. Such questions have not enough yet been investigated. Some specialists of mechanical, for example, the authors of the Encyclopedia of Engineering Industry, Fedosov E., Krasovsky A., Popov E., propose to evaluate the stability of mechanical systems with distributed parameters by dispersion relations, i.e., according to the internal properties of the physical process. Here we use differential equations with variable coefficients that characterize the process under consideration. In this case, the solution of differential equations should be sought

phenomenon are distinguished.

Truss and Frames - Recent Advances and New Perspectives

by numerical methods [12].

is formulated as follows:

nonlinearities.

4

objective data.

$$Q = -\frac{\partial U}{\partial q}.\tag{6}$$

The parameters of the movement of the mechanism are determined from Eq. (5) after some transformations.

These Eqs. (5, 6) which are the basis of many papers on the dynamics of machines, for example, [21, 24, 25], etc., allow, under given boundary conditions, to estimate the change in the displacements of rod section, pipe string, etc. in time and space.

On the one hand, such information is redundant if it is necessary to take into account the interconnection of a large number of factors. For example, to assess the performance of the system, it is enough to know under what conditions selfoscillations occur (i.e., stability is lost), and at what not.

On the other hand, due to the lack of explicit information about the stresses developed in the dynamic process, it is difficult to estimate the probability of part failure.

In the above approaches, such methods of solving problems are specified in which a linear relationship between stresses and displacements of points of a solid body is adopted. According to the accepted linear dependence, these quantities are recalculated. These approaches may not always be applicable, since it is known from rheology that the elastic modulus can depend on the vibration frequency [26, 27].

In addition, depending on the stresses, the rod can be bent and thereby change the peculiarities of the formation of force factors at the links of the mechanism. At the same time, various nonlinear effects, including the essential ones, such as backlash, have a significant impact on the functioning. In this regard, there is a need to develop a method where the oscillations are clearly taken into account speeds and voltages, as well of various nonlinearities.

## 3. Basic equations

To solve this problem, it is assumed that in dynamics the elements of a truss or frame can be represented as models in Figure 1.

In accordance with the theory of strength of materials [1], part of the links of a mechanical system can be represented as a separate element, on which, in addition to external forces, bond reactions act. Therefore, during vibrations, the ends of such an element move with certain speeds, and force factors (Fc, Mr) are the corresponding resulting factors. To this we add viscous resistance (h), which we will consider as resistance to the movement of a particular unit from the side of the entire or adjacent part of the structure to this element (Figure 1).

When considering longitudinal vibrations in a straight solid rod, we use the equation quantity of motion in differential form for the case of the absence of mass forces [28]

$$
\rho \frac{\partial v}{\partial t} = \frac{\partial \sigma}{\partial x} \tag{7}
$$

1 ρ ∂σ <sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>E</sup> ρ ∂2 u

We integrate the Eq. (9) by x, assuming that for х = 0, σ = const:

∂σ ∂x dx ¼ ð x

0

∂u ∂x � � � � x � ∂u ∂x � � � � x¼0

¼ � <sup>∂</sup><sup>σ</sup>

When considering torsional vibrations, we assume that the movement of the

Denote the current value of the stress σ<sup>x</sup> by σ. Given that the surface forces acting on each point of the cross section of the elementary volume are directed in the opposite direction from the direction of the speed of movement, we rewrite the

∂2 u

1 E ð x

1

Dynamic Stability of Open Two-Link Mechanical Structures

DOI: http://dx.doi.org/10.5772/intechopen.91045

We finally obtain

resulting equation in the form

We therefore have

stresses of rod cross section.

7

0

<sup>E</sup> ð Þ¼ <sup>σ</sup><sup>x</sup> � <sup>σ</sup><sup>0</sup>

ρ ∂υ ∂t

1 E ∂σ <sup>∂</sup><sup>t</sup> ¼ � <sup>∂</sup><sup>υ</sup>

sections is absent and the elastic vibrations are described by the equation

∂2 φ <sup>∂</sup>t<sup>2</sup> <sup>¼</sup> <sup>G</sup> ρ ∂2 φ

coordinate (Figure 1b) and G is the shear modulus of the material.

elementary section of a rod with an outer radius r at ρ = const [28, 29]

ρr ∂Ω <sup>∂</sup><sup>t</sup> ¼ � <sup>∂</sup><sup>τ</sup>

From comparison Eqs. (14) and (15), we arrive at the equation

G ∂2 φ <sup>∂</sup>x<sup>2</sup> ¼ � <sup>1</sup> r ∂τ

We integrate this equation over coordinate x. We finally obtain

<sup>∂</sup><sup>x</sup> <sup>¼</sup> rG <sup>∂</sup><sup>φ</sup> ∂x � � � � 0

rG <sup>∂</sup><sup>φ</sup>

where φ and x are the angle of rotation of the section of the rod and the

In addition, we use the equation quantity of motion in differential form for an

where Ω is the speed of rod cross section (Ω = ∂φ/∂t) and τ is maximum shear

<sup>∂</sup>x<sup>2</sup> : (9)

<sup>∂</sup>x<sup>2</sup> dx: (10)

<sup>∂</sup><sup>x</sup> : (12)

<sup>∂</sup><sup>x</sup> : (13)

<sup>∂</sup>x<sup>2</sup> : (14)

<sup>∂</sup><sup>x</sup> , (15)

<sup>∂</sup><sup>x</sup> : (16)

� ð Þ¼� τ<sup>x</sup> � τ<sup>0</sup> τ þ Bτ: (17)

: (11)

and the equation of longitudinal oscillations [27]

$$\frac{\partial^2 u}{\partial t^2} = \frac{E}{\rho} \frac{\partial^2 u}{\partial x^2}. \tag{8}$$

where υ is a speed of longitudinal displacement (υ = ∂u/∂t), u is the displacement along the x-axis, σ are longitudinal (normal) stresses, ρ is density of the material, and E is modulus of elasticity.

Let us assume at this stage that E = const and ρ = const. We determine the derivative ∂υ/∂t from Eq. (7) and substitute it in the left side of Eq. (8). We therefore have

Figure 1. Models of a rod with a mass: (a) with longitudinal vibrations; (b) with torsional vibrations.

Dynamic Stability of Open Two-Link Mechanical Structures DOI: http://dx.doi.org/10.5772/intechopen.91045

$$\frac{1}{\rho} \frac{\partial \sigma}{\partial \mathbf{x}} = \frac{E}{\rho} \frac{\partial^2 u}{\partial \mathbf{x}^2}. \tag{9}$$

We integrate the Eq. (9) by x, assuming that for х = 0, σ = const:

$$\frac{1}{E} \int\_{0}^{\chi} \frac{\partial \sigma}{\partial \mathbf{x}} d\mathbf{x} = \int\_{0}^{\chi} \frac{\partial^{2} u}{\partial \mathbf{x}^{2}} d\mathbf{x}. \tag{10}$$

We finally obtain

In the above approaches, such methods of solving problems are specified in which a linear relationship between stresses and displacements of points of a solid body is adopted. According to the accepted linear dependence, these quantities are recalculated. These approaches may not always be applicable, since it is known from rheology that the elastic modulus can depend on the vibration frequency [26, 27]. In addition, depending on the stresses, the rod can be bent and thereby change the peculiarities of the formation of force factors at the links of the mechanism. At the same time, various nonlinear effects, including the essential ones, such as backlash, have a significant impact on the functioning. In this regard, there is a need to develop a method where the oscillations are clearly taken into account speeds

To solve this problem, it is assumed that in dynamics the elements of a truss or

In accordance with the theory of strength of materials [1], part of the links of a mechanical system can be represented as a separate element, on which, in addition to external forces, bond reactions act. Therefore, during vibrations, the ends of such an element move with certain speeds, and force factors (Fc, Mr) are the corresponding resulting factors. To this we add viscous resistance (h), which we will consider as resistance to the movement of a particular unit from the side of the

When considering longitudinal vibrations in a straight solid rod, we use the equation quantity of motion in differential form for the case of the absence of mass

(7)

<sup>∂</sup>x<sup>2</sup> : (8)

ρ ∂υ ∂t ¼ ∂σ ∂x

∂2 u <sup>∂</sup>t<sup>2</sup> <sup>¼</sup> <sup>E</sup> ρ ∂2 u

Models of a rod with a mass: (a) with longitudinal vibrations; (b) with torsional vibrations.

where υ is a speed of longitudinal displacement (υ = ∂u/∂t), u is the displacement along the x-axis, σ are longitudinal (normal) stresses, ρ is density of the material,

Let us assume at this stage that E = const and ρ = const. We determine the derivative ∂υ/∂t from Eq. (7) and substitute it in the left side of Eq. (8). We

and voltages, as well of various nonlinearities.

Truss and Frames - Recent Advances and New Perspectives

frame can be represented as models in Figure 1.

entire or adjacent part of the structure to this element (Figure 1).

and the equation of longitudinal oscillations [27]

3. Basic equations

forces [28]

and E is modulus of elasticity.

therefore have

Figure 1.

6

$$\frac{1}{E}(\sigma\_{\mathbf{x}} - \sigma\_0) = \frac{\partial u}{\partial \mathbf{x}}\Big|\_{\mathbf{x}} - \frac{\partial u}{\partial \mathbf{x}}\Big|\_{\mathbf{x}=\mathbf{0}}.\tag{11}$$

Denote the current value of the stress σ<sup>x</sup> by σ. Given that the surface forces acting on each point of the cross section of the elementary volume are directed in the opposite direction from the direction of the speed of movement, we rewrite the resulting equation in the form

$$
\rho \frac{\partial \nu}{\partial t} = -\frac{\partial \sigma}{\partial \mathbf{x}}.\tag{12}
$$

We therefore have

$$\frac{1}{E}\frac{\partial \sigma}{\partial t} = -\frac{\partial \upsilon}{\partial \mathbf{x}}.\tag{13}$$

When considering torsional vibrations, we assume that the movement of the sections is absent and the elastic vibrations are described by the equation

$$\frac{\partial^2 \Phi}{\partial t^2} = \frac{G}{\rho} \frac{\partial^2 \Phi}{\partial \mathbf{x}^2}. \tag{14}$$

where φ and x are the angle of rotation of the section of the rod and the coordinate (Figure 1b) and G is the shear modulus of the material.

In addition, we use the equation quantity of motion in differential form for an elementary section of a rod with an outer radius r at ρ = const [28, 29]

$$
\rho r \frac{\partial \Omega}{\partial t} = -\frac{\partial \pi}{\partial \mathbf{x}},
\tag{15}
$$

where Ω is the speed of rod cross section (Ω = ∂φ/∂t) and τ is maximum shear stresses of rod cross section.

From comparison Eqs. (14) and (15), we arrive at the equation

$$G\frac{\partial^2 \Phi}{\partial \mathbf{x}^2} = -\frac{1}{r}\frac{\partial \mathbf{r}}{\partial \mathbf{x}}.\tag{16}$$

We integrate this equation over coordinate x. We finally obtain

$$rG\frac{\partial\mathfrak{q}}{\partial\mathfrak{x}} = rG\frac{\partial\mathfrak{q}}{\partial\mathfrak{x}}\bigg|\_{0} - (\mathfrak{r}\_{\mathfrak{x}} - \mathfrak{r}\_{0}) = -\mathfrak{r} + B\_{\mathfrak{r}}.\tag{17}$$

Here B<sup>τ</sup> is constant characterizing the stress at the initial conditions x = x<sup>0</sup> and t = t0.

We differentiate the derived Eq. (17) over t. We have

$$
\sigma G \frac{\partial \Omega}{\partial t} = -\frac{\partial \pi}{\partial t}. \tag{18}
$$

Here it is believed that there is a body that, under the action of stress, is elastically deformed and at the same time can flow. When stress is applied when t=t1, the springs are instantly deformed by magnitudes σ/Е<sup>1</sup> and σ/Е2, and the

Here, E<sup>1</sup> and E<sup>2</sup> are the isothermal and adiabatic modulus of elasticity,

<sup>D</sup> � <sup>d</sup>

1 τε 

> 1 τε

> > σðÞ¼ s θ0E<sup>2</sup>

σ D þ

σð Þs s þ

like deformation θ (t) = θ<sup>0</sup> 1(t), is written in the form [32]

<sup>D</sup><sup>σ</sup> <sup>¼</sup> <sup>E</sup>2D<sup>θ</sup> � <sup>σ</sup> � <sup>E</sup>1<sup>θ</sup>

dt ; τε <sup>¼</sup> <sup>η</sup>

¼ θE<sup>2</sup> D þ

Passing under zero initial conditions to Laplace transformations [29], we rewrite

¼ θð Þs E<sup>2</sup> s þ

Here, we replaced the operator D with a complex variable with (D = s), and

The Laplace image of the stress change from (18), taking into account the jump-

We define the original by means of residues relative to the poles. We then have

From this expression, it follows that when t<sup>1</sup> = 0, i.e., at the time of a jump-like change in the relative deformation of the rod, the stress is σ(0) = σ<sup>0</sup> Е2, but then with time the stress decreases, relaxes, at t<sup>2</sup> > τε to the value σ(t2) = θ<sup>0</sup> Е1. This conclusion is mathematically obtained in [29], and the process is illustrated by the

τε

s þ 1=keτε sð Þ s þ 1=τε

¼ θ0E<sup>1</sup> 1 þ

E2 E1 � 1 exp � <sup>t</sup>

Here τε is the relaxation time under the condition of constant deformation.

τε

E2

1 keτε

> 1 keτε

dt <sup>¼</sup> <sup>E</sup>1<sup>θ</sup> <sup>þ</sup> <sup>η</sup>

dθ

dt : (20)

: (21)

: (22)

: (23)

: (24)

: (25)

:

τε

(26)

piston starts to move evenly with speed (dσ/dt)/η. The differential equation is written in the form

Dynamic Stability of Open Two-Link Mechanical Structures

DOI: http://dx.doi.org/10.5772/intechopen.91045

We transform Eq. (20) into an operator form

We perform another transformation

respectively.

where

Here ke = Е2/Е1.

the Eq. (23) in the form

<sup>s</sup> <sup>=</sup> <sup>u</sup> <sup>+</sup> jv; <sup>j</sup> = (�1)1/2.

σðÞ¼ t θ0E<sup>2</sup>

graph in Figure 2.

9

1 ke <sup>þ</sup> <sup>1</sup> � <sup>1</sup> ke exp � <sup>t</sup>

<sup>σ</sup> <sup>þ</sup> <sup>η</sup> E2 dσ

The system of Eqs. (12), (13), (15), and (18), first published in manuscript [29], makes it possible to describe changes in stresses in the elementary volume and velocity of movement of the elementary sections of solid-state lines. These are also applicable of the elementary sections of the solid (of the frames and of the trusses).

It should be noted that the process of motion transmission in systems with hydraulic lines is characterized by the equations [30].

$$\frac{\partial \nu}{\partial t} = -\frac{1}{\rho\_0} \frac{\partial P}{\partial \mathbf{x}} - 2\pi\_0 \rho\_0 r\_0; \ \frac{\partial P}{\partial t} = -\kappa \frac{\partial \nu}{\partial \mathbf{x}},\tag{19}$$

where ρ<sup>0</sup> is the initial density of the medium, P is line pressure, κ is reduced modulus of elasticity of the line, τ<sup>0</sup> is shear stress on the pipe wall, and r<sup>0</sup> is the radius of the pipe section.

Thus, the transfer of motion in solid and liquid media can be described by similar equations. This is shown in the analysis of the hydraulic drive operation [29].

### 4. Analysis of the Zener model

The equations of motion for the elementary volume of a substance with relatively low speeds of displacement are obtained above. However, in the event of any abrupt changes caused by either external influences or rapidly occurring vibration phenomena, there is a need for a deeper study of the process of motion transmission in mechanical systems [31].

A number of Maxwell, Voigt, and Zener phenomenological models have been developed for this problem. We consider the more general Zener model [26, 27] (Figure 2).

Figure 2. Zener rheological model: θ is deformation; η is viscosity; σ is normal stress; t is time.

Dynamic Stability of Open Two-Link Mechanical Structures DOI: http://dx.doi.org/10.5772/intechopen.91045

Here it is believed that there is a body that, under the action of stress, is elastically deformed and at the same time can flow. When stress is applied when t=t1, the springs are instantly deformed by magnitudes σ/Е<sup>1</sup> and σ/Е2, and the piston starts to move evenly with speed (dσ/dt)/η.

The differential equation is written in the form

$$
\sigma + \frac{\eta}{E\_2} \frac{d\sigma}{dt} = E\_1 \Theta + \eta \frac{d\Theta}{dt} \,. \tag{20}
$$

Here, E<sup>1</sup> and E<sup>2</sup> are the isothermal and adiabatic modulus of elasticity, respectively.

We transform Eq. (20) into an operator form

$$D\sigma = E\_2 D\theta - \frac{\sigma - E\_1 \theta}{\tau\_c}.\tag{21}$$

where

Here B<sup>τ</sup> is constant characterizing the stress at the initial conditions

rG <sup>∂</sup><sup>Ω</sup>

makes it possible to describe changes in stresses in the elementary volume and velocity of movement of the elementary sections of solid-state lines. These are also applicable of the elementary sections of the solid (of the frames and of the trusses). It should be noted that the process of motion transmission in systems with

<sup>∂</sup><sup>x</sup> � <sup>2</sup>τ0ρ0r0;

where ρ<sup>0</sup> is the initial density of the medium, P is line pressure, κ is reduced modulus of elasticity of the line, τ<sup>0</sup> is shear stress on the pipe wall, and r<sup>0</sup> is the

Thus, the transfer of motion in solid and liquid media can be described by similar equations. This is shown in the analysis of the hydraulic drive operation [29].

The equations of motion for the elementary volume of a substance with relatively low speeds of displacement are obtained above. However, in the event of any abrupt changes caused by either external influences or rapidly occurring vibration phenomena, there is a need for a deeper study of the process of motion

A number of Maxwell, Voigt, and Zener phenomenological models have been developed for this problem. We consider the more general Zener model

Zener rheological model: θ is deformation; η is viscosity; σ is normal stress; t is time.

<sup>∂</sup><sup>x</sup> ¼ � <sup>∂</sup><sup>τ</sup> ∂t

The system of Eqs. (12), (13), (15), and (18), first published in manuscript [29],

∂P <sup>∂</sup><sup>t</sup> ¼ �<sup>κ</sup> ∂υ

: (18)

<sup>∂</sup><sup>x</sup> , (19)

We differentiate the derived Eq. (17) over t. We have

Truss and Frames - Recent Advances and New Perspectives

hydraulic lines is characterized by the equations [30].

¼ � <sup>1</sup> ρ0 ∂P

∂υ ∂t

radius of the pipe section.

[26, 27] (Figure 2).

Figure 2.

8

4. Analysis of the Zener model

transmission in mechanical systems [31].

x = x<sup>0</sup> and t = t0.

$$D \equiv \frac{d}{dt}; \; \tau\_{\varepsilon} = \frac{\eta}{E\_2}. \tag{22}$$

Here τε is the relaxation time under the condition of constant deformation. We perform another transformation

$$
\sigma \left( D + \frac{1}{\tau\_{\varepsilon}} \right) = \theta E\_2 \left( D + \frac{1}{k\_{\varepsilon} \tau\_{\varepsilon}} \right). \tag{23}
$$

Here ke = Е2/Е1.

Passing under zero initial conditions to Laplace transformations [29], we rewrite the Eq. (23) in the form

$$
\sigma(s)\left(s+\frac{1}{\tau\_{\varepsilon}}\right) = \theta(s)E\_2\left(s+\frac{1}{k\_{\varepsilon}\tau\_{\varepsilon}}\right).\tag{24}
$$

Here, we replaced the operator D with a complex variable with (D = s), and <sup>s</sup> <sup>=</sup> <sup>u</sup> <sup>+</sup> jv; <sup>j</sup> = (�1)1/2.

The Laplace image of the stress change from (18), taking into account the jumplike deformation θ (t) = θ<sup>0</sup> 1(t), is written in the form [32]

$$\sigma(\mathbf{s}) = \theta\_0 E\_2 \frac{\mathbf{s} + \mathbf{1}/k\_\varepsilon \tau\_\varepsilon}{\mathbf{s}(\mathbf{s} + \mathbf{1}/\tau\_\varepsilon)}. \tag{25}$$

We define the original by means of residues relative to the poles. We then have

$$
\sigma(t) = \Theta\_0 E\_2 \left[ \frac{\mathbf{1}}{k\_\varepsilon} + \left( \mathbf{1} - \frac{\mathbf{1}}{k\_\varepsilon} \right) \exp\left( -\frac{t}{\tau\_\varepsilon} \right) \right] = \Theta\_0 E\_1 \left[ \mathbf{1} + \left( \frac{E\_2}{E\_1} - \mathbf{1} \right) \exp\left( -\frac{t}{\tau\_\varepsilon} \right) \right]. \tag{26}
$$

From this expression, it follows that when t<sup>1</sup> = 0, i.e., at the time of a jump-like change in the relative deformation of the rod, the stress is σ(0) = σ<sup>0</sup> Е2, but then with time the stress decreases, relaxes, at t<sup>2</sup> > τε to the value σ(t2) = θ<sup>0</sup> Е1. This conclusion is mathematically obtained in [29], and the process is illustrated by the graph in Figure 2.

Obviously, the elastic modulus Е<sup>2</sup> corresponds to the adiabatic deformation process, and Е<sup>1</sup> corresponds to the isothermal process.

Physically, this can be represented as follows. Initially, an adiabatic, without heat transfer, convergence of atoms in metal crystals takes place, but at the same time the entire atomic system becomes unbalanced—non-equilibrium. In order for the system to reach an equilibrium state, a relaxation time τε = η/Е<sup>2</sup> is necessary, when the atoms, having received their share of thermal energy, occupy a new position.

Note. In physics [33], elastic oscillations in some cases are interpreted as the motion of a phonon gas. In this case, the relaxation of the internal energy in the crystal lattice is described by the kinematic equation for phonons. Acoustic relaxation is always accompanied by sound absorption, its dispersion, and the dependence of the speed of sound on frequency. The physical encyclopedia for solid dielectrics suggests estimating the relaxation constant from the phonon lifetime

$$
\pi\_{\mathfrak{r}} \cong \mathfrak{r}\_f = \mathfrak{Z}\lambda / \left(\mathbb{C}\mathfrak{c}\_{av}^2\right),
\tag{27}
$$

The modulus function E(ω) is equal to the isothermal modulus of elasticity.

The modulus function E(ω) is equal to the adiabatic modulus of elasticity. From Eq. (23), it follows that the ratio σ(ω)/θ(ω) describes some complex function, which can be called the function of the generalized elastic modulus Eω(s),

<sup>s</sup> <sup>þ</sup> ð Þ keτε �<sup>1</sup>

of elasticity is Est = 1.617 \* 10<sup>5</sup> MPa, and dynamic is Е<sup>ω</sup> = 2.058 \* 10<sup>5</sup> MPa.

where Eu(ω) and Eυ(ω) are, respectively, the values of the function Eω(ω) along

It follows from the above that the value of the elastic modulus strongly depends on the experimental conditions, in particular, on the oscillation frequency, and on the relaxation spectrum. So, for casting steel 1Х15Н15М2К3ВТ, the static modulus

Since alloys, such as steel, contain different phases, it is likely that each of them will have its own combination of Е2, Е1, and τε. In this case, you may have to take into account the average relaxation time, the width relaxation of spectrum, etc.

The interaction of the elastic wave is largely with the interface of the media due to the wave impedance, which is determined by the relation [34]. It follows that

� σ

where a<sup>1</sup> is the velocity of propagation of longitudinal oscillations in the

study the wave resistance in operator form or mechanical impedance

Referring to Formula (34) and taking into account that

Θ ¼

s

When considering harmonic oscillations propagating along a line, we usually

Zbð Þ¼ <sup>j</sup><sup>ω</sup> <sup>σ</sup>ð Þ <sup>j</sup><sup>ω</sup>

ffiffiffi E ρ

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � μ ð Þ 1 þ μ ð Þ 1 � 2μ

a<sup>1</sup> ¼ Θ

�1

j j Eð Þ ω ¼ E2: (31)

<sup>s</sup> <sup>þ</sup> ð Þ τε �<sup>1</sup> <sup>¼</sup> Euð Þþ <sup>ω</sup> jEυð Þ <sup>ω</sup> , (32)

, then this may have a certain impact

<sup>υ</sup> <sup>¼</sup> <sup>ρ</sup>a1, (33)

<sup>υ</sup>ð Þ <sup>j</sup><sup>ω</sup> : (34)

, (35)

, (36)

The second case: when ω > 1/τε, we have

DOI: http://dx.doi.org/10.5772/intechopen.91045

Dynamic Stability of Open Two-Link Mechanical Structures

EωðÞ¼ s E<sup>2</sup>

If the frequency value is equal to ω = (τε)

on the properties of the mechanical system.

5. Derivation of the initial equations

Let us turn further to the longitudinal oscillations.

the real and imaginary axes of the complex plane (U, jV).

i.e., we have

medium.

where [34]

11

where C is the lattice heat capacity, λ is thermal conductivity coefficient, and cav is the average value of the speed of sound.

If the deformation will change according to the harmonic law, then the stress will also change according to the harmonic law, but with a slightly different amplitude and phase advance, depending on frequency.

The ratio modulus σ(ω)/θ(ω) and phase shift φ are calculated from (23) using expressions [29].

$$\left|\frac{\sigma(\boldsymbol{\alpha})}{\Theta(\boldsymbol{\alpha})}\right| = E\_1 \frac{\sqrt{\mathbf{1} + \left(k\_\varepsilon \boldsymbol{\tau}\_\varepsilon \boldsymbol{\alpha}\right)^2}}{\sqrt{\mathbf{1} + \left(\boldsymbol{\tau}\_\varepsilon \boldsymbol{\alpha}\right)^2}};\tag{28}$$

$$\Phi = \operatorname{arcctg}(k\_{\epsilon}\tau\_{\epsilon}a) - \operatorname{arcctg}(\tau\_{\epsilon}a). \tag{29}$$

where ω is the circular oscillation frequency.

The peculiarity of the passage of a harmonic signal through a metal is illustrated in Figure 3.

Here are graphs of the functions E = E(ω) and φ = φ(ω). And here is τ<sup>0</sup> <sup>ε</sup> = ke τε. The physical meaning of the function E(ω) is illustrated by two cases. The first case: when ω < 1/τ<sup>0</sup> <sup>ε</sup>, we have

Figure 3. Passage of a harmonic signal through a metal.

Obviously, the elastic modulus Е<sup>2</sup> corresponds to the adiabatic deformation

Physically, this can be represented as follows. Initially, an adiabatic, without heat transfer, convergence of atoms in metal crystals takes place, but at the same time the entire atomic system becomes unbalanced—non-equilibrium. In order for the system to reach an equilibrium state, a relaxation time τε = η/Е<sup>2</sup> is necessary, when the atoms, having received their share of thermal energy, occupy a new

Note. In physics [33], elastic oscillations in some cases are interpreted as the motion of a phonon gas. In this case, the relaxation of the internal energy in the crystal lattice is described by the kinematic equation for phonons. Acoustic relaxation is always accompanied by sound absorption, its dispersion, and the dependence of the speed of sound on frequency. The physical encyclopedia for solid dielectrics

τε ffi <sup>τ</sup><sup>f</sup> <sup>¼</sup> <sup>3</sup>λ<sup>=</sup> Cc<sup>2</sup>

where C is the lattice heat capacity, λ is thermal conductivity coefficient, and cav

If the deformation will change according to the harmonic law, then the stress will also change according to the harmonic law, but with a slightly different ampli-

The ratio modulus σ(ω)/θ(ω) and phase shift φ are calculated from (23) using

q

The peculiarity of the passage of a harmonic signal through a metal is illustrated

Here are graphs of the functions E = E(ω) and φ = φ(ω). And here is τ<sup>0</sup>

The physical meaning of the function E(ω) is illustrated by two cases.

<sup>ε</sup>, we have

av

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> ð Þ keτεω <sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> ð Þ τεω <sup>2</sup>

ϕ ¼ arctg kð Þ� <sup>e</sup>τεω arctgð Þ τεω : (29)

� �, (27)

<sup>q</sup> ; (28)

j j Eð Þ ω ¼ E1: (30)

<sup>ε</sup> = ke τε.

suggests estimating the relaxation constant from the phonon lifetime

process, and Е<sup>1</sup> corresponds to the isothermal process.

Truss and Frames - Recent Advances and New Perspectives

is the average value of the speed of sound.

tude and phase advance, depending on frequency.

σð Þ ω θð Þ ω

� � � � <sup>¼</sup> <sup>E</sup><sup>1</sup>

� � � �

where ω is the circular oscillation frequency.

The first case: when ω < 1/τ<sup>0</sup>

Passage of a harmonic signal through a metal.

position.

expressions [29].

in Figure 3.

Figure 3.

10

The modulus function E(ω) is equal to the isothermal modulus of elasticity. The second case: when ω > 1/τε, we have

$$|E(\alpha)| = E\_2. \tag{31}$$

The modulus function E(ω) is equal to the adiabatic modulus of elasticity.

From Eq. (23), it follows that the ratio σ(ω)/θ(ω) describes some complex function, which can be called the function of the generalized elastic modulus Eω(s), i.e., we have

$$E\_{\boldsymbol{\alpha}}(\boldsymbol{\varepsilon}) = E\_2 \frac{\boldsymbol{s} + (k\_{\boldsymbol{\varepsilon}} \boldsymbol{\varepsilon}\_{\boldsymbol{\varepsilon}})^{-1}}{\boldsymbol{s} + (\boldsymbol{\varepsilon}\_{\boldsymbol{\varepsilon}})^{-1}} = E\_{\boldsymbol{u}}(\boldsymbol{\alpha}) + jE\_{\boldsymbol{\upsilon}}(\boldsymbol{\alpha}), \tag{32}$$

where Eu(ω) and Eυ(ω) are, respectively, the values of the function Eω(ω) along the real and imaginary axes of the complex plane (U, jV).

It follows from the above that the value of the elastic modulus strongly depends on the experimental conditions, in particular, on the oscillation frequency, and on the relaxation spectrum. So, for casting steel 1Х15Н15М2К3ВТ, the static modulus of elasticity is Est = 1.617 \* 10<sup>5</sup> MPa, and dynamic is Е<sup>ω</sup> = 2.058 \* 10<sup>5</sup> MPa.

If the frequency value is equal to ω = (τε) �1 , then this may have a certain impact on the properties of the mechanical system.

Since alloys, such as steel, contain different phases, it is likely that each of them will have its own combination of Е2, Е1, and τε. In this case, you may have to take into account the average relaxation time, the width relaxation of spectrum, etc.
