The Vibrational Motion of a Dynamical System Using Homotopy Perturbation Technique

This paper outlines the vibrational motion of a nonlinear system with a spring of linear stiffness. Homotopy perturbation technique (HPT) is used to obtain the asymptotic solution of the governing equation of motion. The numerical solution of this equation is obtained using the fourth order Runge-Kutta method (RKM). The comparison between both solutions reveals high consistency between them which confirms that, the accuracy of the obtained solution using aforementioned perturbation technique. The time history of the attained solution is represented through some plots to reveal the good effect of the different parameters of the considered system on the motion at any instant. The conditions of the stability of the attained solution are presented and discussed.


Introduction
Many problems related to mathematicians, physicists, biologists, chemists and engineers are formulated in differential equations whether linear or nonlinear.
The solutions of a linear one can be obtained easily using some of well-established methods on the contrary with nonlinear differential Equations (NDE) that we often refuge to approximate solutions. Nonlinear oscillations had shed the interest of many scientists due to that most of the problems dealing with vibrations are nonlinear, see [1] [2].
Since it is difficult to find the exact solutions of such equations, many researchers have turned their attention to obtain the approximate solutions of In [21], the authors presented a modification of HPT to get the solution of a dynamical model consists of the motion of a rod in a circular surface without slipping. The obtained results are in good agreement with the numerical ones. This problem was treated in [22] using the modified harmonic balance technique [23]. A combination of MPT and Laplace transformation to achieve the asymptotic solution of the governing equation of motion of the same problem is studied in [24], in which the stability of the obtained solutions is examined. The approximate solutions of some tested vibrating systems are obtained in [25] using a modification of HPT and Amplitude frequency formulation (AFF).
HPT is used in [26] to obtain the periodic solution of the fractional sine-Gordon equation beside the Riemann-Liouville fractional derivative. The authors obtained a relationship between the frequency and amplitude, and the impact of the order of fractional derivative on the vibration property is investigated. In [27], the authors investigated the periodic solution of the nonlinear Duffing oscillator with fractional order utilizing a modification of HPT which is the insertion of an auxiliary parameter and using two homotopy parameters. A nonlinear packaging system has been solved analytically using HPT of Li-He's in [28], in which the energy method is utilized to progress the frequency and the maximal displacement of the system.
In this paper, the solution of a nonlinear oscillating dynamical system is investigated. This system consists of a mass m 1 connected with a spring of linear stiffness and with other mass m 2 through a massless string of length l. HPT is utilized to obtain the solution of the equation of motion. This solution is graphically represented for different values of the system parameters and compared with the numerical solution of the governing equation of motion using the Runge-Kutta method [29] from fourth order. This comparison reveals high consistency between them which emphasizes the accuracy of the results obtained by HPT. The stability of the investigated model is presented and analyzed. This paper is designated as follows. In Section 2, a description of the investigated problem and the derivation of the equation of motion are presented. Section 3 sheds light on the basic idea of HPT. Section 4 is devoted to reduce the equation of motion into appropriate equation and to obtain the solution of this equation analytically using HPT. In Section 5, we are going to represent the attained solution graphically and to obtain the numerical solution using the Runge-Kutta method. The stability of the obtained solution is discussed in Section 6. Finally, the manuscript is finished with some concluding remarks.

Description of the Problem
In this section, we are going to obtain the governing equation of motion of a nonlinear oscillation system using HPT of a dynamical model. This model consists of two masses; a first one m 1 moves horizontally in which it is attached to a spring of linear stiffness k and connected with the second mass m 2 with a massless string of length l see (Figure 1). Therefore let us consider that S 0 , x and y are the natural length of the given spring, the horizontal coordinate of the centroid of m 1 and the vertical coordinate of the centroid of m 2 , respectively. Therefore, the potential and kinetic energies V and T of the system can be written in the forms ( ) where g is the gravitational acceleration, x is the extension of string after time t, dots denote to the differentiation with respect to time and ( ) An inspection of the Lagrange's function (2) shows that the investigated system has only one degree of freedom. Therefore, Lagrange's equation for conservative system may be written as Here, x and x  are the generalized coordinate and velocity of the system respectively. Making use of (2) and (3) yields to the following form of the govern-

Homotopy Perturbation Technique
This section is devoted to illustrate HPT [7] through solving the following general nonlinear differential equation Here K and B represent the general differential operator and the boundary operator respectively, ( ) f r denotes a known analytical function, Γ is the boundary of a domain Ω and u n ∂ ∂ refers to differential along the normal drawn outwards from Ω . An inspection of Equation (5), broadly speaking, the operator K can be separated into two parts; which are a linear part L and a nonlinear one N. Therefore Equation (5) can be rewritten in the form It is worthwhile to notice that according to HPT, we can construct the homo- or in an equivalent form as is a homotopy parameter and U (initial guess) is an initial approximation of Equation (5), in which it satisfies the boundary conditions.
In order to investigate the solution of (8) or (9), we express about this solution as a power series of ρ as At 1 ρ → , Equations (8) or (9) corresponds to Equation (5) and the results in the approximation to the solution of Equation (5) can be expressed as It is important to note that, series (11) is convergent for more cases. Some criteria are suggested for convergence of this series, see [7].

Method of Solution
Dividing both sides of (4) by m 1 and consider that to reduce the equation of motion (4) to a more appropriate as On the use of (12), the previous equation can be rewritten in the form ( ) ( ) Expanding the previous equation to obtain Therefore, we obtain the following equation A closer look of this equation reveals that it is a second order differential equation with high nonlinearity.
The aim of this section is to obtain the approximate solution of the governing equation of motion utilizing HPT in the presence of the following initial condi- By virtue of Equations (13) and (7), the linear part ( ) L u and nonlinear one N(u) have the forms Equation (8) can be rewritten in the form Substituting (15)- (17) into (18) to obtain Making use of (10) and (20), then equating the coefficients of similar powers of ρ in both sides to obtain Coefficient of ρ : Coefficient of 2 ρ : The previous Equations (21)

Results and Discussion
In this section, we are going to shed light on the great accuracy of the results obtained by HPT when they are compared with the numerical results of the governing equation of motion (4) using the fourth order Runge-Kutta method [26].   worthwhile to notice that these drawings have periodic forms and therefore the attained solution has a stable manner. seems to be small as in Figure 5(a) in which this difference increases in Figure   5 On the other side, this difference becomes very slightly which can be neglected as in Figure 6 and      Table 5. Error percentage of HPT for 1

Stability Analysis
In this section, we investigate the stability of the governing equation of motion (13). It is obvious from the preceding section that, this investigation will be unsuccessful in view of Equation (28). Therefore, we are going to obtain a periodic solution of (13).
It should be noticed that Equation (13) is transformed into linear and nonlinear parts as indicated in Equations (15) and (16) respectively in which 0 ω denotes a natural frequency of Equation (15). It is clear that the linear part represents a simple harmonic equation. Therefore, the stability of this part depends upon the frequency 0 ω which is always positive and consequently, the represented figures have periodic forms as expected. Therefore the system is always stable.
Now, let us focus attention on the stability of a nonlinear part in which we consider a nonlinear frequency analysis. Therefore, a nonlinear frequency 2 Ω is assumed to be in the following form , ω ρϖ ρ ϖ Ω = + + + (29) where 1 2 , , ϖ ϖ  are arbitrary parameters can be estimated.
According to the reported work [30] and HPT, we can write the approximate nonlinear frequency in the form Substitution of (29) into (20) yields Making use of (10)  Coefficient of ρ : Taking into account conditions (24), one can solve Equations (32)-(34) subsequently to get It is worthy to mention that in order to get a uniform to expand solution, the terms that produce secular terms in Equations (33) and (34)

Conclusion
The motion of a nonlinear oscillating dynamical system is studied. HPT is used