摘要：Duffing系统、Van der Pol系统以及其组合形式Duffing-Van der Pol系统是典型的非线性振子系统，这些系统模型在分析机械振动、电路振荡等问题中有非常广泛的应用。但在应用领域的工程系统中，系统受到的激励普遍是多频激励，非单频激励的情况相当少，因此研究和求解典型非线性系统在多频激励下的周期解及其振动特性具有工程应用价值。

本课题基于现有的较为成熟的非线性振动解析求解和数值求解理论和方法，对Duffing系统、Duffing-Van der Pol系统的周期响应特性及其参数影响规律展开理论和数值研究。研究过程中，通过Mathematica软件完成多频激励下Duffing方程的解析求解；而在数值求解方面，主要通过Matlab软件运用四五阶Runge-Kutta法编写程序完成多频激励下Duffing方程和Duffing-Van der Pol方程时间历程曲线、相平面图、频谱图、庞加莱截面、幅频曲线以及分岔图的绘制，进而展开多频激励下典型非线性系统振动特性的相关研究。

关键词  Duffing系统  Van der Pol系统  Duffing-Van der Pol系统  Mathematica  Matlab   Runge-Kutta   

毕业设计说明书外文摘要

Title    Study on Vibration Characteristics of Typical Nonlinear System under Multi–frequency Excitation           

Abstract：Duffing system, Van der Pol system and its combination form Duffing-Van der Pol system is a typical nonlinear oscillator system, these system models in the analysis of mechanical vibration, circuit oscillation and other issues have a very wide range of applications. However, in the actual engineering system, the excitation is mostly multi-frequency excitation rather than single frequency excitation. Therefore, it is more valuable and meaningful to study and solve the periodic solution and the vibration characteristic of the typical nonlinear system under multi-frequency excitation.

Based on the existing mathematic theory and method of nonlinear vibration analysis and numerical solution, the influence of periodic response characteristics and parameters of Duffing system and Duffing-Van der Pol system is studied. In the process of research, through the Mathematica software to complete the multi-frequency excitation Duffing equation analytic solution.In the numerical solution, the time course curve, phase plane diagram, spectrum diagram, Poincare map, amplitude-frequency curve and bifurcation diagram of Duffing equation and Duffing-Van der Pol equation are mainly based on the four- Order Runge-Kutta method to complete the preparation process. Furthermore, the research on the vibration characteristics of typical nonlinear systems under multi - frequency excitation is carried out.

Keywords   Duffing system    Van der Pol system    Duffing-Van der Pol system  Mathematica    Matlab    Runge-Kutta

目   次

1  引言 1

1.1  研究现状和意义 1

1.2  线性振动系统和非线性振动系统的区别 2

1.3  Duffing、Van der Pol以及Duffing-Van der Pol系统概述 3

2  动力学方程的求解方法 6

2.1  解析求解方法 6

2.1.1  平均法 6

2.1.2  正规摄动法 7

2.2  数值求解方法 9

2.3  相平面法 9

3  解析及数值计算结果