Forced response of quadratic nonlinear oscillator: comparison of various approaches  

作　　者：Wenan JIANG Guoce ZHANG Liqun CHEN 

机构地区：[1]Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University [2]College of Sciences, Shanghai University [3]Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University

出　　处：《Applied Mathematics and Mechanics(English Edition)》2015年第11期1403-1416,共14页应用数学和力学（英文版）

基　　金：Project supported by the State Key Program of National Natural Science Foundation of China(No.11232009);the National Natural Science Foundation of China(No.11572182)

摘　　要：The primary resonances of a quadratic nonlinear system under weak and strong external excitations are investigated with the emphasis on the comparison of dif- ferent analytical approximate approaches. The forced vibration of snap-through mecha- nism is treated as a quadratic nonlinear oscillator. The Lindstedt-Poincar method, the multiple-scale method, the averaging method, and the harmonic balance method are used to determine the amplitude-frequency response relationships of the steady-state responses. It is demonstrated that the zeroth-order harmonic components should be accounted in the application of the harmonic balance method. The analytical approximations are com- pared with the numerical integrations in terms of the frequency response curves and the phase portraits. Supported by the numerical results, the harmonic balance method pre- dicts that the quadratic nonlinearity bends the frequency response curves to the left. If the excitation amplitude is a second-order small quantity of the bookkeeping parameter, the steady-state responses predicted by the second-order approximation of the Lindstedt- Poincar method and the multiple-scale method agree qualitatively with the numerical results. It is demonstrated that the quadratic nonlinear system implies softening type nonlinearity for any quadratic nonlinear coefficients.The primary resonances of a quadratic nonlinear system under weak and strong external excitations are investigated with the emphasis on the comparison of dif- ferent analytical approximate approaches. The forced vibration of snap-through mecha- nism is treated as a quadratic nonlinear oscillator. The Lindstedt-Poincar method, the multiple-scale method, the averaging method, and the harmonic balance method are used to determine the amplitude-frequency response relationships of the steady-state responses. It is demonstrated that the zeroth-order harmonic components should be accounted in the application of the harmonic balance method. The analytical approximations are com- pared with the numerical integrations in terms of the frequency response curves and the phase portraits. Supported by the numerical results, the harmonic balance method pre- dicts that the quadratic nonlinearity bends the frequency response curves to the left. If the excitation amplitude is a second-order small quantity of the bookkeeping parameter, the steady-state responses predicted by the second-order approximation of the Lindstedt- Poincar method and the multiple-scale method agree qualitatively with the numerical results. It is demonstrated that the quadratic nonlinear system implies softening type nonlinearity for any quadratic nonlinear coefficients.

关 键 词：forced vibration quadratic nonlinearity primary resonance multiple-scalemethod Lindstedt-Poincar~ method averaging method harmonic balance method 

分 类 号：O322[理学—一般力学与力学基础]