Homotopy Analysis Method for Large-Amplitude Free Vibrations of Strongly Nonlinear Generalized Duffing Oscillators

In this study, the homotopy analysis method (HAM) is used to solve the generalized Duffing equation. Both the frequencies and periodic solutions of the nonlinear Duffing equation can be explicitly and analytically formulated. Accuracy and validity of the proposed techniques are then verified by comparing the numerical results obtained based on the HAM and numerical integration method. Numerical simulations are extended for even very strong nonlinearities and very good correlations which achieved between the results. Besides, the optimal HAM approach is introduced to accelerate the convergence of solutions.


Introduction
Nonlinear Duffing equation is a simple mathematical model which describes the resonance and chaotic phenomenon.In science and engineering many nonlinear vibration problems can be transformed into the Duffing equation to research [1].From a certain point, the real significance of the research on nonlinear Duffing system attracts a lot of scholars and several ingenious analytical methods have been developed for dealing with the nonlinear Duffing oscillator, such as the modified perturbation methods [2,3], improved harmonic balance methods [4], energy balance method [5,6], the frequency-amplitude formulation [7,8].Meanwhile, the homotopy analysis method (HAM) [9] proposed by Liao has been proved to be one of the efficient analytical techniques in solving a variety of nonlinear Duffing problems.By the HAM, Hoseini et al. [10] study free vibrations of tapered beams and give an accurate analytical solution for the thirdorder Duffing equations; Qian et al. [11] obtain accurate analytical solutions for the fifth-order Duffing equations by considering vibrations of a restrained cantilever beam.For the seventh-order Duffing equations Qian et al. [12] get accurate analytical solutions by researching vibrations of an electrostatically actuated microbeam.
Thus, the prime objective of this paper is to explore the utility of the HAM for the generalized Duffing equa-tion.All odd-type analytical results can be then involved in the generalized solution.In what follows, Section 2 presents natural frequency of the system obtained as a function of the initial amplitude and the general solution for any arbitrary odd power of .In addition, the optimal HAM approach used to accelerate the convergence of solutions is also provided and discussed.In Section 3 two numerical examples are presented to examine the accuracy and validity of the proposed technique.In Section 4 the numerical results of the HAM are presented and compared with the numerical integration solutions.Finally, a conclusion summarizes the research findings in Section 5. n

Solution Methodology
In this section, we apply the HAM to solve the following nonlinear Duffing oscillator: where is displacement and 3 5 7 (n is the odd number) are arbitrary constants.Subject to the following initial conditions: where A is an arbitrary constant.
where a dot denotes differentiation with respect to     , and  is the nonlinear frequency.
It is known that free oscillation of a conservative system without damping is a periodic motion and a harmonic function is the simplest type of periodic motion.So it can be expressed by the following base functions: (5) Taking into consideration the initial conditions in Equation (4), we choose the initial guess of   u  for the zeroth-order deformation equation as follows: Thus, the auxiliary linear operator of a conservative system can be selected as The auxiliary linear operator L is chosen in such a way that all solutions of the corresponding high-order formation equations exist and can be expressed by the general form of the base function.According to Equation (4), the nonlinear operator is written as: ; ; Then considering the homotopy function, we obtain the zeroth-order deformation equation as: where are, respectively, embedding and convergence-control parameters.As q changes from 0 to 1, varies from the initial guess Then we make use of the Taylor series expansion to get It is known that if is properly chosen, the power se  at ries solutions in Equ ions (10) and ( 11) can be converged at 1 q  .So Equations (10) and ( 11) then become For the sake of simplicity, we define the following vectors: , , , .
By differentiating the zeroth-order deformation equatio n (9) m times with respect to q, then the resulting equation is divided m! and setting 0 q  , it can be found the mth-order deformation equation with the initial conditions: in which Because odd nonlinearity of considered conserv sy ative stem, m R can also be written as: where   m  wing is an integer that depends on the lin-m .an Follo the rule of solution expression d ear operator L , the terms of cos should not exist in m R of Equation ( 17), otherwise the so-called secular s such as cos term   will appear in the final solutions.
Therefore, their c ficients are set to zero as follows: The solutions of  in Equ and ( 22) can be det vely.Fo va , , n  and ations (17) r the given The problem is solved and the general solution can be obtained based on the HAM in Section 2. lues of  

3,5 i i  
A , we have the periodic solutions by the abovementioned analytical approach, where and can be determined by using the initial co giv n Equation ( 18).e, -o in which where 3 1,3 3 5 We know there are many opti which can be able to achieve faster convergent homoto mal HAM approaches, For 2 m  , we substitute the solutions of 22) to yield py-series solutions [13,14].In theory, we can define the exact residual error of the mth-order of approximation as From Equation (21), are derived in the following: where is the exact nonlinear fre (1) deri d by using the numerical integration technique.he nonlinear Duffing oscillaform.So in this section, we quency of Equation It can b ound ∆ m embraces the unknown convergencecontrol parameter  .As ∆ m decreases more rapidly to zero, the speed of the convergence for the corresponding homotopy-series so tion is faster [14].The corresponding value of the convergence-control parameter  at the given order of approximation m can be optimized and selected by minimizing the residual error ∆ m .

Numerical Results
where the coefficients can be readily derived using Equation (21).In addition,  2, 3,5, 7,9 where all coefficients can be computed from Equation ( 2According to Equations ( 24) and ( 25), the corresponding third-order analytical approximation for Equation ( 27) is The higher-order appro ximations for  and   u  ca can get some approximations for n be derived in a similar manner.Secondly we consider the nonlinear Duffing oscillator for 7 n  : We  and   u  according to Section 2 in a similar manner.For 1 m  , one obtains For , one arrives at   where all coefficients and can Equation (21).rresp r analytical approximation for Equation (40) is 3,5, ,13 where The higher-order approximations for  and   u  can be derived in a similar manner.

Numerical Discussion
In order to demonstrate the effectiveness of the presentethod, the asymptotic analytical solutions obtained ared to th ed analytical and numer pr ing m by the HAM are directly comp e publish ical integration solutions.ect of the convergence-to optimize and draw the curve to obtain the optimal rgence-control er .By Figure 1, we get the optimal convergence-control parameters in Table 3 for Modes 1-6.In Table 3

Conclusion
In this study, the HAM has intensively studied the generalized Duffing equation.The general frequencies and periodic solutions are presented for any arbitrary oddtype of nonlinearity.The purposes of this paper are not only to formulate the asymptotic approximate solutions for the nonlinear Duffing oscillators, but also to furnish a guidance to establish the higher-order asymptotic analytical approximations if necessary.Moreover, it is found that the accuracy of the HAM is affected by the selection ies corresponding to various parameters in Equation (27) for with the numerical integration solutions even if the amplitude is larger.


to the physical frequency  .

Figure 2 .Figure 3 .
Figure 2. Comparison of the approx Phase portra diagram; (b) Time history response.imate and numerical integration solutions for A = 10.(a) it