Abstract
In this article, we present a new accurate iterative and asymptotic method to construct analytical periodic solutions for a strongly nonlinear system, even if it is not Z2-symmetric. This method is applicable not only to a conservative system but also to a non-conservative system with a limit cycle response. Distinct from the general harmonic balance method, it depends on balancing a few trigonometric terms (at most five terms) in the energy equation of the nonlinear system. According to this iterative approach, the dynamic frequency is a trigonometric function that varies with time t, which represents the influence of derivatives of the higher harmonic terms in a compact form and leads to a significant reduction of calculation workload. Two examples were solved and numerical solutions are presented to illustrate the effectiveness and convenience of the method. Based on the present method, we also outline a modified energy balance method to further simplify the procedure of higher order computation. Finally, a nonlinear strength index is introduced to automatically identify the strength of nonlinearity and classify the suitable strategies.
Highlights
Nonlinear differential equations describing dynamic behaviors play a major role in mechanics and mathematics
The analysis of the literature devoted to the theory of nonlinear differential equations show that various so-called analytical, functional-analytic, numerical and numerical-analytic methods based upon successive approximations are extensively studied [1]
We present a new dynamic frequency method to solve the periodic solutions this paper, we present a new dynamic method to energy solve the periodic the solutions for for In strongly nonlinear oscillators
Summary
Nonlinear differential equations describing dynamic behaviors play a major role in mechanics and mathematics. Based on a truncated Fourier series, the harmonic balance (HB) method [13,14,15,16] is often used to determine approximate periodic solutions to a nonlinear oscillatory system. This method is applicable to nonlinear systems with non-small parameters or even with no perturbation parameter needed to exist. Based on the modified Lindstedt–Poincare (MLP) method, Chen and Tang [21] transformed the strongly nonlinear system into a small parameter system and determined the periodic response. The obtained nonlinear strength index efficiently classify the asymptotic computation
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