Simplified Hamiltonian-Based Frequency-Amplitude Formulation for Nonlinear Vibration Systems

Abstract

The Hamiltonian-based frequency formulation has been hailed as an unprecedented success for it gives a straightforward insight into a complex nonlinear vibration system with simple calculation. This paper gives a systematical analysis of the formulation, and two simplified formulations are suggested. The cubic-quintic Duffing oscillator is used as an example to show extremely simple calculation and remarkable accuracy. It can be used as a paradigm for many other applications, and the one-step solving process has cleaned up the road of the nonlinear vibration theory.

Insightful and comprehensive formularization of frequency–amplitude formula for strong or singular nonlinear oscillators

“…It is noted that the above expansion coincides with the results obtained in (31) or (32). To this end, Helmholtz-Duffing oscillator equation ( 4) can be sought in the following integrative configuration…”

“…These applications of nonlinear oscillations do not have nonlinear even functions. The frequency formulation which is Hamiltonian-based [29][30][31][32] is a modification of He's frequency formulation. 23,33 The formulation in an energy view establishes a Hamilton principle, which can clarify the energy conservation of the vibrating procedure.…”

A mathematical control for the pseudo-pull-in stability arising in a micro-electromechanical system

“…When wðuÞ ¼ 1, it turns out to be He's frequency formulation. 41 There are many other frequency formulations in literature, [43][44][45][46][47] but He-Liu's formulation 42 given in equation ( 13) is much simple and relatively accurate. Generally, h(u)/u should be larger than zero.…”

The periodic solution of a pendulum under magnetic action

“…It has periodic solution when 20,21 f ðθÞ θ > 0 (12) The square of the frequency can be obtained by He's frequency formulation 22…”