We applied an approach to obtain the natural frequency of the generalized Duffing oscillator and a nonlinear oscillator with a restoring force which is the function of a noninteger power exponent of deflection . This approach is based on involved parameters, initial conditions, and collocation points. For any arbitrary power of , the approximate frequency analysis is carried out between the natural frequency and amplitude. The solution procedure is simple, and the results obtained are valid for the whole solution domain. 1. Introduction Although a large amount of the efforts on dynamical systems are related to second-order differential equations, some dynamical systems can be described by nonlinear (second-order) differential equations. Attention in nonlinear oscillator equations involving the second temporal derivative of displacement has recently been focused on the existence of periodic solutions. The study of nonlinear periodic oscillator is of interest to many researchers and various methods of solution have been suggested. Several approaches have been proposed to deal with different kinds of oscillator equations, for example, [1–7]. He in  used Hamiltonian method to calculate the analytical approximate periodic solutions of nonlinear oscillator equations. The approximations to the periodic solution and the angular frequency obtained by He were not accurate enough. Yildirim et al.  and Khan et al. , respectively, applied a higher order Hamiltonian formulation combined with parameters for nonlinear oscillators. Our concern in this work is the derivation of amplitude-frequency relationship for the nonlinear oscillator equations and . The attention here has been restricted primarily to odd positive integer power for the first equation and rational powers greater than unity for the second equation. There are examples of systems, however, for which these exponents can be of noninteger order, for instance, the flexible elements of vibration isolators made of wire-mesh and felt materials, cable isolators, and radially loaded rubber cylinder. In the present work, the mentioned parameters are the undetermined values in the assumed solution. In the parameters technique, the motion has been assumed as where , , and are the angular frequency of motion and Fourier coefficients, respectively. The method in this approach to obtain the parameters is quite different from the method in He’s Hamiltonian technique. Hence, the present technique is not similar to He’s Hamiltonian technique. Finally, the paper provides some accurate results for the angular
N. A. Khan, M. Jamil, S. A. Ali, and A. Nadeem Khan, “Solutions of the force-free duffing-van der pol oscillator equation,” International Journal of Differential Equations, vol. 2011, Article ID 852919, 9 pages, 2011.
A. Barari, A. Kimiaeifar, M. G. Nejad, M. Motevalli, and M. G. Sfahani, “A closed form solution for nonlinear oscillators frequencies using amplitude-frequency formulation,” Shock and Vibration, vol. 19, no. 6, pp. 1415–1426, 2012.
D. Younesian, H. Askari, Z. Saadatnia, and M. K. Yazdi, “Frequency analysis of strongly nonlinear generalized Duffing oscillators using He's frequency-amplitude formulation and He's energy balance method,” Computers and Mathematics & Applications, vol. 59, no. 9, pp. 3222–3228, 2010.
A. Yildirim, Z. Saadatnia, H. Askari, Y. Khan, and M. K. Yazdi, “Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach,” Applied Mathematics Letters, vol. 24, no. 12, pp. 2042–2051, 2011.
N. A. Khan, M. Jamil, and A. Ara, “Multiple-parameter Hamiltonian approach for higher accurate approximations of a nonlinear oscillator with discontinuity,” International Journal of Differential Equations, vol. 2011, Article ID 649748, 7 pages, 2011.
S. S. Ganji, D. D. Ganji, H. Babazadeh, and N. Sadoughi, “Application of amplitude-frequency formulation to nonlinear oscillation system of the motion of a rigid rod rocking back,” Mathematical Methods in the Applied Sciences, vol. 33, no. 2, pp. 157–166, 2010.