几种计算平面倒摆系统近似解的解析方法比较研究  

Comparative Study on Several Analytical Methods for Approximate Solutions of Planar Inverted Pendulum System

作　　者：张运鑫 镇斌[1] 李婷婷 

机构地区：[1]上海理工大学环境与建筑学院,上海

出　　处：《声学与振动》2019年第1期41-58,共18页Open Journal of Acoustics and Vibration

基　　金：国家自然科学基金(11472160).

摘　　要：倒摆是常见的非线性动力系统,具有广泛的理论和实际研究价值,而采用解析方法计算倒摆的近似周期响应是常用研究手段。由于倒摆的运动过程中蕴含丰富的非线性动力学现象,如何选取合适的解析方法并进行特定周期行为的计算通常是能否进行有效讨论的关键。常见的定量解析分析方法包括参数扰动法、同伦分析法、能量法、平均法等。本文采用这四种方法分别对平面倒摆的控制方程进行求解,并采用Maple软件进行数值模拟对比,比较几种分析方法的优缺点及适用条件。本文对非线性倒摆系统进行的定量研究有助于理解倒摆的动力学行为,对类似的非线性动力系统的定量计算有较好的参考价值。The plane inverted pendulum system is a classical model with important theory value and vast application prospect.It is usual to research the periodic solutions of an inverted pendulum by using analytical methods.Since the inverted pendulum has strong nonlinearity and can exhibit very rich nonlinear phenomena,it is the key that how to choose an appropriate analytical method to calculate the periodic motions.There are four usual methods about calculating periodic solutions of a non-linear system:parameter perturbation method,homotopy analysis method,energy method and average method.In this paper,we will apply such four methods to analyze the periodic solutions of an inverted pendulum system,respectively.By comparing with numerical results obtained by using Maple software,we demonstrate the merits and demerits of the four methods as well as their application conditions.The research presented in this paper contributes to understanding nonlinear dynamics of an inverted pendulum and provides a good reference for calculating periodic solutions of other similar nonlinear systems.

关 键 词：平面倒摆 定量分析方法 周期解 同伦法 能量法 

分 类 号：F2[经济管理—国民经济]