Nonlinear Vibration of Oscillation Systems using Frequency-Amplitude Formulation

Abstract

In this paper we study the periodic solutions of free vibration of mechanical systems with third and fifth-order nonlinearity for two examples using He's Frequency-Amplitude Formulation (HFAF).The effectiveness and convenience of the method is illustrated in these examples. It will be shown that the solutions obtained with current method have a fabulous conformity with those achieved from time marching solution. HFAF is easy with powerful concepts and the high accuracy, so it can be found widely applicable in vibrations, especially strong nonlinearity oscillatory problems.

Non-linear vibration of Euler-Bernoulli beams

“…In recent years, some promising approximate analytical solutions have been proposed, such as Frequency Amplitude Formulation [13], Variational Iteration [5,6,14,17], Homotopy-Perturbation [3,4,7,24], Parametrized-Perturbation [18], Max-Min [15,19,29], Differential Transform Method [16], Adomian Decomposition Method [22], Energy Balance [23,30], etc.…”

Characteristic Behavior of Bucket Foundations

“…Modified Lindstedt-Poincaré (Ozis & Yildirim 2007), Artificial Parameter Lindstedt-Poincaré (Ramos 2009), Adomian Decomposition (Wang et al 2008;Mirgolbabaei et al 2010), Parameter-Expansion (Kimiaeifar et al 2011), Frequency Amplitude Formulation (Fereidoon et al 2011), Parametrized Perturbation (Barari et al 2011), Multiple Scale (Okuizumi & Kimura 2004), Energy Balance (Sfahani et al 2011;Bayat et al 2011),…”

A Closed Form Solution for Nonlinear Oscillators Frequencies Using Amplitude-Frequency Formulation

“…We choose u(t) = A cos(ωt) and put it into the Eqs (10) and (27), then we determine the following integral:…”