FFT Multi-harmonic Balance Method And Its Applications

With the widely utilization of nonlinear vibration isolator in engineeringpractices, approachs for solving nonlinear differential equations and ascertainingmechanism of nonlinear vibration isolation systems (NVIS) become one of hot topicsof nonlinear dynamics.The research in solving nonlinear differential equations with analytic methods isalways one of the main concerns of nonlinear dynamics. It is impossible to find oneuniversal analytic method due to the complicated characteristics of nonlineardynamical systems. Many analytical approximation methods have been proposed anddeveloped, such as L-P method, harmonic balance method etc. Each method has itslimitations. Therefore, it is necessary and worthwise to explore some effective andfeasible approximate analytical methods.In this dissertation, we focus on the study of analytical approximate methods forsingle degree of freedom nonlinear oscillation equations. Due to the fact that thesolutions obtained by using the harmonic balance (HB) method are incorrect in somecircumstances, a new analytical approximate method, i.e. FFT multi-harmonic balancemethod, was proposed based on multi-harmonic balances and dominant frequenciesidentified by Fast Foourier Transform (FFT). Then this method was applied to studythe typical characteristics of the nonlinear dynamical system, such as jump responses,coexistent motions, multi-periodic motions and quasi-periodic responses. Based onthe full consideration of the complexity of the nonlinear responses, the definition oftransmissibility of NVIS was modified and improved. Numerical simulations of theDuffing-type NVIS revealed that the modified transmissibility can be used to evaluatethe performance of NVIS.The FFT multi-harmonic balance method, first-order harmonic balance methodand numerical method were compared in analyzing typical characteristics of NVIS.Results show that:1) the response of nonlinear systems may contain several dominantfrequencies, which may not be multipled to exciting frequency. The FFTmulti-harmonic balance method and multiple-harmonic balance method are limited tothe systems dominated by the exciting frequency.2) The analytical solutions ofnonlinear dynamical systems with multi-periodic motions, quasi-periodic motions etc.can be obtained by FFT multi-harmonic balance method with high accuracy.3) By means of FFT multi-harmonic balance method and modified transmissibility of NVIS,the approximate solutions of performance of the system were acquired, and themethod is an effective tool for theoretical analysis of NVIS.In addition, the free vibration response of damped nonlinear system was studied.We discover that the free vibration frequency can drift away in the decay oscillationof nonlinear systems, which was consistent with the experimental observation of thetail wings of airoplanes observed by an astronautics institute. This paper only revealsthe phenomena and discusses its positive significance of frequency drift, with hope toattract more research in this field.