Periodic Solution of Nonlinear Conservative Systems

*Akuro Big-Alabo and Chinwuba Victor Ossia*

## **Abstract**

Conservative systems represent a large number of naturally occurring and artificially designed scientific and engineering systems. A key consideration in the theory and application of nonlinear conservative systems is the solution of the governing nonlinear ordinary differential equation. This chapter surveys the recent approximate analytical schemes for the periodic solution of nonlinear conservative systems and presents a recently proposed approximate analytical algorithm called continuous piecewise linearization method (CPLM). The advantage of the CPLM over other analytical schemes is that it combines simplicity and accuracy for strong nonlinear and large-amplitude oscillations irrespective of the complexity of the nonlinear restoring force. Hence, CPLM solutions for typical nonlinear Hamiltonian systems are presented and discussed. Also, the CPLM solution for an example of a non-Hamiltonian conservative oscillator was presented. The chapter is aimed at showcasing the potential and benefits of the CPLM as a reliable and easily implementable scheme for the periodic solution of conservative systems.

**Keywords:** Hamiltonian system, conservative system, nonlinear vibration, continuous piecewise linearization method, periodic solution, nonnatural system, perturbation method

#### **1. Introduction**

#### **1.1 Hamiltonian and non-Hamiltonian conservative systems**

Conservative systems can be defined as oscillating or vibrating systems in which the total energy content of the system remains constant. In order words, the total energy in the system is conserved. Ideally, such a system will continue to be in periodic oscillatory motion ad infinitum because the energy content of the system does not diminish due to the absence of dissipative force or increase due to additional energy input. However, for real cases where dissipative mechanisms such as friction or viscous damping cannot be completely eliminated, a conservative system can be thought of as one in which the energy dissipated is negligible during the time range under consideration. For example, the first few seconds of the oscillation of a simple pendulum may be considered conservative since the effect of air friction is negligible, but in the long run, the initial energy content is gradually dissipated until the pendulum comes to a halt. Other examples of practical conservative systems include mass-spring oscillator, structural elements (i.e., beams, plates, and shells), slider-crank mechanism [1], human eardrum [2], relativistic oscillator [3],

planetary orbits around the sun [3], and current-carrying conductor in the electric field of an infinite rod [4]. Hence, a large number of oscillating physical systems can be studied as conservative systems.

approximate analytical schemes for the periodic solution of nonlinear conservative oscillators. It should be noted that an approximate analytic method for nonlinear oscillators is considered adequate if it gives accurate predictions for the frequency-

Approximate analytical techniques to solve the nonlinear ODE governing the oscillations of a conservative system have been formulated for at least 100 years and can be classified as perturbation and nonperturbation methods. The first attempts were based on perturbation theory and are referred to as *classical perturbation methods*. The perturbation methods are formulated based on the concept that an unknown nonlinear system can be studied by introducing a small disturbance to a known linear system in equilibrium. For this reason, the classical perturbation methods (see Nayfeh [9] for a comprehensive treatment of classical perturbation methods) depend on the assumption of a small parameter. The problem with the small parameter assumption is that it has a small range of validity and only produces reliable solutions for cases of small-amplitude oscillations and weak nonlinearity. Nevertheless, the classical perturbation methods are still very relevant today for

More recently, in the last four decades, a number of approximate analytical schemes have been proposed. Most of these recent schemes are nonperturbation methods, but some recent perturbation methods that attempt to improve on their classical counterparts have been formulated too. The recent perturbation methods include *δ*-method [10], Homotopy perturbation method [11] and its variants [12–17], modified Lindstedt-Poincare method [18–21], book-keeping parameter method [22], iteration perturbation method [23], parameterized perturbation method [24], perturbation incremental method [25], and linearized perturbation method [26]. A review article on some of the recent perturbation methods has been published by He [27]. The main point of the recent perturbation methods is to deal with the issue of the small parameter in order to formulate solutions that are applicable to small- and large-amplitude oscillations and also weak and strong nonlinear oscillations. Although the higher order approximations of the recent perturbation methods have been very successful in producing accurate estimates of the frequency-amplitude response, the same cannot be said of their estimation of the oscillation history. Studies [7, 28] have shown that the higher order approximations of the recent perturbation methods produce large unbounded errors in the oscillation history during large-amplitude oscillations and are, therefore, not better than the classical perturbation methods in this regard. A plausible explanation for this observation is that it occurs because perturbation methods are based on asymptotic series that are inherently divergent for amplitudes greater than unity [28]. Therefore, it may not be possible to formulate perturbation schemes that would correctly predict the oscillation history of large-

In contrast to the perturbation methods, the nonperturbation methods do not use any small or artificial parameter. Examples include Adomian decomposition method [29], Homotopy analysis method [30], Variational iteration method [31], Energy balance method [2] and its modifications [32–34], He Chengtian's interpolation method [27] also called max-min approach, amplitude-frequency formulation [35], Hamiltonian approach [36], global error minimization method [37], Harmonic balance method [4] and its modifications [38–42], cubication methods [43–46], variational methods [47–49], differential transform method [50], and continuous piecewise linearization method [8]. Nonperturbation methods also have various limitations. For instance, a study [51] showed that the Adomian decomposition method does not converge to the correct solution in some cases, and the study proposed an optimal convergence acceleration parameter to deal with this issue.

amplitude response and the oscillation history as well [7, 8].

*Periodic Solution of Nonlinear Conservative Systems DOI: http://dx.doi.org/10.5772/intechopen.90282*

introducing and investigating various nonlinear concepts.

amplitude vibrations.

**237**

At any point in time, the energy of a conservative system is composed of kinetic (*T*) and potential (*V*) energies except at critical points where the total energy may be only kinetic (*Tmax*) or potential (*Vmax*). Generally, it is expected that *T* ¼ *T q*ð Þ , *q*\_ and *V* ¼ *V q*ð Þ , *q*\_ , where *q* is the generalized displacement. Naturally, *q* and *q*\_ are not expected to form a product in the function *T q*ð Þ , *q*\_ , but in some cases, they do*:* Therefore, two types of conservative systems are distinguished namely: natural and nonnatural conservative systems. The natural conservative systems are those in which the kinetic energy can be expressed as a pure quadratic function of velocity, i.e., does not contain a product of the velocity and displacement. They are also known as *Hamiltonian systems* because they admit a Hamiltonian function (*H q*ð Þ¼ , *q*\_ *T q*ð Þþ , *q*\_ *V q*ð Þ , *q*\_ ) that is always constant at any point in time. While this definition of Hamiltonian systems is a physical one, a mathematical definition has been discussed by Jordan and Smith [3]. Examples of Hamiltonian systems include mass-spring oscillator, simple pendulum, and a mass attached to the mid-point of an elastic spring. On the other hand, there are conservative systems in which the kinetic energy cannot be expressed as a pure quadratic function of the velocity because the kinetic energy expression contains a product of velocity and displacement. This second group of conservative systems is referred to as nonnatural because their kinetic energy is not a pure quadratic function of velocity. Although the total energy in such systems is conserved, their Hamiltonian function (*H q*ð Þ , *q*\_ ) is not constant [4]. Hence, the nonnatural conservative systems may be referred to as *non-Hamiltonian conservative systems*. Examples of this category of conservative systems abound in artificial systems and include slider-crank mechanism [1], particle sliding on a vertical rotating parabola [4], pendulum attached to massless rolling wheel [4], rigid rod rocking on a circular surface without slip [4], and circular sector oscillator [5]. An important quality of the non-Hamiltonian conservative systems is that their vibration equation, which is normally derived by the Lagrangian approach, does not conform to the standard representation of conservative systems that clearly shows the restoring force. Rather, the derived vibration equation has a quadratic velocity term, which represents a coordinatedependent parameter rather than a dissipative parameter.

#### **1.2 Recent advances in solution schemes for nonlinear conservative oscillators**

Exact analytical solutions for the nonlinear vibration models of conservative systems can be derived only in very few situations, and the solutions are usually derived in terms of special functions. Alternatively, highly accurate numerical solutions can be obtained for the nonlinear vibration model of any conservative system. However, as it is well recognized among the nonlinear science community, numerical solutions often have the limitations of lack of physical insight and convergence issues. Furthermore, there is the possibility of obtaining inaccurate convergent solutions for a nonlinear ordinary differential equation (ODE) [6], thus necessitating the independent verification of the convergent numerical solution by another numerical or analytical method. These limitations have driven the search for approximate analytical schemes capable of providing periodic solutions to nonlinear conservative oscillators. It can be rightly concluded that this search has been very fruitful considering the many approximate analytical schemes that now appear in the nonlinear science literature. The purpose of this section is to provide a brief survey of some of the notable achievements in the development of

#### *Periodic Solution of Nonlinear Conservative Systems DOI: http://dx.doi.org/10.5772/intechopen.90282*

planetary orbits around the sun [3], and current-carrying conductor in the electric field of an infinite rod [4]. Hence, a large number of oscillating physical systems can

At any point in time, the energy of a conservative system is composed of kinetic (*T*) and potential (*V*) energies except at critical points where the total energy may be only kinetic (*Tmax*) or potential (*Vmax*). Generally, it is expected that *T* ¼ *T q*ð Þ , *q*\_ and *V* ¼ *V q*ð Þ , *q*\_ , where *q* is the generalized displacement. Naturally, *q* and *q*\_ are not expected to form a product in the function *T q*ð Þ , *q*\_ , but in some cases, they do*:* Therefore, two types of conservative systems are distinguished namely: natural and nonnatural conservative systems. The natural conservative systems are those in which the kinetic energy can be expressed as a pure quadratic function of velocity, i.e., does not contain a product of the velocity and displacement. They are also known as *Hamiltonian systems* because they admit a Hamiltonian function

(*H q*ð Þ¼ , *q*\_ *T q*ð Þþ , *q*\_ *V q*ð Þ , *q*\_ ) that is always constant at any point in time. While this definition of Hamiltonian systems is a physical one, a mathematical definition has been discussed by Jordan and Smith [3]. Examples of Hamiltonian systems include mass-spring oscillator, simple pendulum, and a mass attached to the mid-point of an elastic spring. On the other hand, there are conservative systems in which the kinetic energy cannot be expressed as a pure quadratic function of the velocity because the kinetic energy expression contains a product of velocity and displacement. This second group of conservative systems is referred to as nonnatural because their kinetic energy is not a pure quadratic function of velocity.

Although the total energy in such systems is conserved, their Hamiltonian function (*H q*ð Þ , *q*\_ ) is not constant [4]. Hence, the nonnatural conservative systems may be referred to as *non-Hamiltonian conservative systems*. Examples of this category of conservative systems abound in artificial systems and include slider-crank mechanism [1], particle sliding on a vertical rotating parabola [4], pendulum attached to massless rolling wheel [4], rigid rod rocking on a circular surface without slip [4], and circular sector oscillator [5]. An important quality of the non-Hamiltonian conservative systems is that their vibration equation, which is normally derived by the Lagrangian approach, does not conform to the standard representation of conservative systems that clearly shows the restoring force. Rather, the derived vibration equation has a quadratic velocity term, which represents a coordinate-

**1.2 Recent advances in solution schemes for nonlinear conservative oscillators**

Exact analytical solutions for the nonlinear vibration models of conservative systems can be derived only in very few situations, and the solutions are usually derived in terms of special functions. Alternatively, highly accurate numerical solutions can be obtained for the nonlinear vibration model of any conservative system. However, as it is well recognized among the nonlinear science community, numerical solutions often have the limitations of lack of physical insight and convergence issues. Furthermore, there is the possibility of obtaining inaccurate convergent solutions for a nonlinear ordinary differential equation (ODE) [6], thus necessitating the independent verification of the convergent numerical solution by another numerical or analytical method. These limitations have driven the search for approximate analytical schemes capable of providing periodic solutions to nonlinear conservative oscillators. It can be rightly concluded that this search has been very fruitful considering the many approximate analytical schemes that now appear in the nonlinear science literature. The purpose of this section is to provide a brief survey of some of the notable achievements in the development of

dependent parameter rather than a dissipative parameter.

**236**

be studied as conservative systems.

*Progress in Relativity*

approximate analytical schemes for the periodic solution of nonlinear conservative oscillators. It should be noted that an approximate analytic method for nonlinear oscillators is considered adequate if it gives accurate predictions for the frequencyamplitude response and the oscillation history as well [7, 8].

Approximate analytical techniques to solve the nonlinear ODE governing the oscillations of a conservative system have been formulated for at least 100 years and can be classified as perturbation and nonperturbation methods. The first attempts were based on perturbation theory and are referred to as *classical perturbation methods*. The perturbation methods are formulated based on the concept that an unknown nonlinear system can be studied by introducing a small disturbance to a known linear system in equilibrium. For this reason, the classical perturbation methods (see Nayfeh [9] for a comprehensive treatment of classical perturbation methods) depend on the assumption of a small parameter. The problem with the small parameter assumption is that it has a small range of validity and only produces reliable solutions for cases of small-amplitude oscillations and weak nonlinearity. Nevertheless, the classical perturbation methods are still very relevant today for introducing and investigating various nonlinear concepts.

More recently, in the last four decades, a number of approximate analytical schemes have been proposed. Most of these recent schemes are nonperturbation methods, but some recent perturbation methods that attempt to improve on their classical counterparts have been formulated too. The recent perturbation methods include *δ*-method [10], Homotopy perturbation method [11] and its variants [12–17], modified Lindstedt-Poincare method [18–21], book-keeping parameter method [22], iteration perturbation method [23], parameterized perturbation method [24], perturbation incremental method [25], and linearized perturbation method [26]. A review article on some of the recent perturbation methods has been published by He [27]. The main point of the recent perturbation methods is to deal with the issue of the small parameter in order to formulate solutions that are applicable to small- and large-amplitude oscillations and also weak and strong nonlinear oscillations. Although the higher order approximations of the recent perturbation methods have been very successful in producing accurate estimates of the frequency-amplitude response, the same cannot be said of their estimation of the oscillation history. Studies [7, 28] have shown that the higher order approximations of the recent perturbation methods produce large unbounded errors in the oscillation history during large-amplitude oscillations and are, therefore, not better than the classical perturbation methods in this regard. A plausible explanation for this observation is that it occurs because perturbation methods are based on asymptotic series that are inherently divergent for amplitudes greater than unity [28]. Therefore, it may not be possible to formulate perturbation schemes that would correctly predict the oscillation history of largeamplitude vibrations.

In contrast to the perturbation methods, the nonperturbation methods do not use any small or artificial parameter. Examples include Adomian decomposition method [29], Homotopy analysis method [30], Variational iteration method [31], Energy balance method [2] and its modifications [32–34], He Chengtian's interpolation method [27] also called max-min approach, amplitude-frequency formulation [35], Hamiltonian approach [36], global error minimization method [37], Harmonic balance method [4] and its modifications [38–42], cubication methods [43–46], variational methods [47–49], differential transform method [50], and continuous piecewise linearization method [8]. Nonperturbation methods also have various limitations. For instance, a study [51] showed that the Adomian decomposition method does not converge to the correct solution in some cases, and the study proposed an optimal convergence acceleration parameter to deal with this issue.

Also, methods that rely on a simple harmonic approximation of the oscillation history, such as the energy balance method, amplitude-frequency formulation, Hamiltonian approach, max-min approach, and variational methods, can only give reliable estimate of the frequency-amplitude response. Sometimes these methods perform poorly in predicting the oscillation history during large-amplitude and/or strong nonlinear vibrations. Other methods that usually require high-order approximations, such as Adomian decomposition method, harmonic balance method, and variational iteration method, present algebraic complexities in their determination of higher order solutions and may be impractical for oscillators with highly complex nonlinearities such as the slider-crank mechanism [1] and the bifilar pendulum [52]. Furthermore, it has been observed that higher order estimates do not always improve the solution of the oscillation history [27]. Finally, some nonperturbation methods are heuristic in nature (e.g., energy balance method and variational methods) and require experience to choose the initial trial function and the condition for error minimization [8].

The continuous piecewise linearization method (CPLM) is an iterative analytic algorithm that was formulated to overcome most of the above challenges by providing simple and accurate solutions for the oscillation history and frequencyamplitude response of Duffing-type oscillators. In another study [53], the CPLM was modified in order to generalize it so that it can handle more complicated nonlinear conservative oscillators. Interestingly, the CPLM does not require higher order approximations or any small, artificial, or embedded parameter. Also, the algorithm is inherently stable, straightforward, and based on closed-form analytical approximations. This chapter is aimed at presenting the generalized CPLM algorithm as a veritable approach for accurate periodic solution of Hamiltonian and non-Hamiltonian conservative oscillators with complex nonlinearity. As is shown later, the CPLM retains the same simplicity in its implementation irrespective of the complexity of the nonlinear conservative system.

**2.2 Mathematical formulation of the continuous piecewise linearization**

numbering on the horizontal axis represents the boundary points of the

negative. To account for this possibility, the linearized force for the *s*

Since *Frs*ð Þ *u* is an approximation of *f u*ð Þ for the *s*

The standard form for representation of a nonlinear conservative oscillator

where *f u*ð Þ is the nonlinear restoring force as shown in **Figure 1**. In **Figure 1**, the

point at *r* and endpoint at *s* ¼ *r* þ 1. **Figure 1** shows that for each discretization, the slope of the linear approximation of the restoring force can either be positive or

where *Krs* ¼ *f u*ð Þ�*<sup>s</sup> f u*ð Þ*<sup>r</sup>* ½ �*=*ð Þ *us* � *ur* is the linear slope of *Frs*ð Þ *u* and *Fr* ¼ *f u*ð Þ*<sup>r</sup>* .

Eq. (10) is a nonhomogeneous linear ODE and its solution depends on whether

If *Krs* > 0, the solution for the displacement and velocity can be expressed as:

ing Eq. (2) in (1) gives the approximate equation of motion for each discretization

*th* discretization represents a general discretization with start

*Frs*ð Þ¼� *u Krs* j jð Þþ *u* � *ur Fr*, (2)

*u*€ � *Krs* j j*u* ¼ � *Krs* j j*ur* � *Fr:* (3)

*u t*ðÞ¼ *Rrs* sin ð Þþ *ωrst* þ Φ*rs Crs* (4a) *u t* \_ðÞ¼ *ωrsRrs* cosð Þ *ωrst* þ Φ*rs* , (4b)

*u*€ þ *f u*ð Þ¼ 0, (1)

*th* discretization

*th* discretization, then substitut-

**method**

**Figure 1.**

discretization. The *s*

can be expressed as:

the sign is positive or negative.

*2.2.1 Solution for positive linearized stiffness*

as follows:

**239**

moving in the *u*-direction is given as:

*Discretization of the restoring force of a typical nonlinear oscillator.*

*Periodic Solution of Nonlinear Conservative Systems DOI: http://dx.doi.org/10.5772/intechopen.90282*
