Oscillations in Planar Dynamic Systems

Solution for an anti-symmetric quadratic nonlinear oscillator by a modified He’s homotopy perturbation method

“…which agrees exactly with Wang and He's solution obtained by first-order parameter expanding method [42], with Mickens's solution obtained by first-order harmonic balance method [6] and with He's solution obtained by first-order homotopy perturbation method [21]. However, this paper only considers the first-order approximation.…”

“…In particular, we will consider the solution of Eq. (1) using the harmonic balance method [6,41,44]. This method is a procedure for determining analytical approximations to the periodic solutions of differential equations using a truncated Fourier series representation.…”

“…Mickens [6,44] also used the second-order harmonic balance approximation (generalized harmonic balance method)…”

“…It is very difficult to solve nonlinear problems and, in general, it is often more difficult to get an analytic approximation than a numerical one for a given nonlinear problem. There are several methods used to find approximate solutions to nonlinear problems, such as perturbation techniques [1][2][3][4][5][6], harmonic balance based methods [6][7][8][9] or other techniques [10][11][12][13][14][15][16][17][18]. An excellent review on some asymptotic methods for strongly nonlinear equations can be found in detail in references [19] and [20].…”

Exact solution for the nonlinear pendulum

“…Acreditamos que o presente estudo possa tornar-se um exercício conveniente e frutífero para o ensino e para uma melhor compreensão do pêndulo não-linear em cursos avançados de mecânica clássica na graduação. Palavras-chave: pêndulo simples, período a grandesângulos, deslocamento angular.Perhaps one of the nonlinear systems most studied and analyzed is the simple pendulum [1][2][3][4][5][6][7][8][9][10][11][12], which is the most popular textbook example of a nonlinear system and is studied not only in advanced but also in introductory university courses of classical mechanics. The periodic motion exhibited by a simple pendulum is harmonic only for small angle oscillations [1].…”