The Calculation And Analysis Of Nonlinear Vibration Characteristics With Duffing Equation

Duffing equation is a typical class of equations in nonlinear vibration system, many practical non-linear mathematical model of the vibration problem can be transformed into the representative equation to study.In some ways,the study of non-linear Duffing system is the foundation to research into many complex dynamic systems.In this paper,the harmonic balance method of slow-varying parameters was adopted.The hypothetical solution was incorporated into the Duffing equation,then differential equations was derived and its stability was analyzed.Nonlinear vibration analysis of the average method of the slow-varying parameters way was applied to harmonic balance method,in order to analyze the stability of forced vibrations with the harmonic balance method.Theory of analytic method(harmonic balance method, average method) and numerical integration method were used to obtain equations of free vibration,forced vibration,chaotic vibration,calculate the time response and spectral analysis,draw the amplitude-frequency characteristic curve to make the analysis of stability,comparison and verification.On this basis,we analyzed the existence of chaos phenomenon and explained the phenomena and laws of vibration.This article is divided into four chapters as follows:The first chapter introduced the importance of study on nonlinear vibration system,several common equations and solving methods of theoretical research in nonlinear vibration,focused on the production of Duffing equation and the significance of research on Duffing equation,and introduced a number of important research results.Chapterâ…¡,we studied free vibration,forced vibration of the main resonance,the forced fraction harmonic resonance vibration and forced vibration resonance of the chaotic vibrations through the numerical calculation,and drawn the corresponding numerical results map and waveform chart,etc.Chapterâ…¢,using harmonic balance method and average method to obtain analytic solution of Duffing equation to determine the stability of solution respectively, at the same time,compare and verify the results with the results of second chapter. Validated to prove that the theoretical results and numerical calculation results are basically the same,which notes the above theoretical analysis is correct.Chapterâ…£,conclusions.