多自由度系统及结构非线性自由振动的摄动谐波平衡法

The Method of Perturbation—Harmonic Balance for Analysing Nonlinear Free Vibration of MDOF Systems and Structures

本文在谐波平衡法的基础上,引入摄动的思想,得出了解多自由度系统及结构的非线性自由振动的新方法。其解的形式为小参数和谐波的级数形式,因此,其解不会遗漏任何项,方程为线性的代数方程;利用线性变换,将系数矩阵变换为对角阵,一旦求出线性模态,就可得其解,比线性化迭代法优越得多。算例表明,本文方法对于小振幅有较高的精度,对于较大振幅其结果也是令人满意的。

In this paper,the perturbation technique is introduced into the method of harmonic balance. A new method used for analysing nonlinear free vibration of multidegree-of-freedom systems and structures is obtained. The form of solution is expanded into a series of Small parameters and harmonics, so no term will be lost in the solution and the algebric equations are linear. With the linear transformations,the matrices of the equations become diagonal. As soon as the modes related to linear vibration are found ,the solution can be obtained. This method is superior to the method of linearized iteration. The examples show that the method has high accuracy for small amplitude problems and the results for rather large amplitude are satisfactory.