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多自由度不对称分段线性系统强迫振动的Fourier级数解法
引用本文:金基铎,关立章.多自由度不对称分段线性系统强迫振动的Fourier级数解法[J].振动工程学报,2003,16(3):373-378.
作者姓名:金基铎  关立章
作者单位:1. 沈阳航空工业学院工程力学系,沈阳,110034
2. 东北大学理学院力学系,沈阳,110006
摘    要:研究了多自由度不对称分段线性系统的强迫振动问题。把恢复力中的非线性部分表成与激振力具有同一周期的Fourier级数后,导出了以Fourier级数各谐波系数为未知数的联立方程组,把求周期解的问题最终归结为求解一组代数方程组的问题。在周期解的稳定性分析中,利用求周期解时得到的2N个相位角把一个周期区间分割成为若干个线性子区间,使得在全周期上求解周期系数方程组的问题转化成在各个子区间上求解常系数方程组的问题,因而大大地简化了建立单值矩阵的工作。

关 键 词:Fourier级数  解题方法  强迫振动  非线性振动  多自由度不对称分段线性系统  周期解  稳定性
修稿时间:2001年7月5日

Fourier Series Solution of Forced Vibration of a Multi-Degree-of-Freedom System with Piecewise-Linear Elastic Elements
Jin Jiduo.Fourier Series Solution of Forced Vibration of a Multi-Degree-of-Freedom System with Piecewise-Linear Elastic Elements[J].Journal of Vibration Engineering,2003,16(3):373-378.
Authors:Jin Jiduo
Abstract:In this paper, the forced vibrations in an unsymmetrical piecewise-linear system with multi-degree-of -freedom are analyzed by means of a method utilizing appropriate Fourier series expansion. The main features of the method are as follows : (1)Expanding the nonlinear part of the restoring forces into a Fourier series with the same period as the given excitation forces,and then, the nonlinear part is regarded as an excitation force applied. (2)Obtaining the formal solution by solving the linearized equation. (3)Determining the unknown coefficients of the Fourier expansion under the condition that the formal solution obtained above should satisfy the given conditions of the piecewise-linear characteristics of the system. To determine the stability of the periodic solution obtained, a period region is divided into several sub regions, in each of which the equation of motion of the system is linear, using the 2N phase angles obtained in the procedure of determining the periodic solution. This makes the task of establishing the monodromy matrix very simple, which corresponds to the periodic motion.
Keywords:nonlinear vibration  piecewise-linear system  forced vibration  Fourier series solution  stability of periodic solution
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