非线性振动系统周期解的数值分析

凌复华

非线性振动系统周期解的数值分析

凌复华

上海交通大学工程力学系

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出版历程
- 收稿日期: 1982-07-28
- 刊出日期: 1983-08-15

A Numerical Treatment of the Periodic Solutions of Non-Linear Vibration Systems

Ling

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai

摘要

摘要: 用直接数值积分法求非线性振动系统的周期解,求解时对初始条件进行迭代,使它与终点条件相一致.积分时间区间(即周期)或运动方程中的某些参数,也可在迭代过程中随同变化,积分方法是变步长的. 用这种“打靶”法求周期解,所需计算工作量相对较少.其中误差主要来源于数值积分,故不难估计并控制它足够小.这种方法可处理各种类型的振动问题,如单自由度和多自由度系统的自由无阻尼振动、强迫振动、自激振动和参数振动等等;也能求得不稳定解和那些对参数变动十分敏感的解.解的稳定性根据相关的周期系数微分方程来研究.求共振曲线或其他振动特性曲线时,利用插值方法并自动调节步长来定出迭代始值. 为了阐明这种方法的通用性,计算了若干例子.非线性的描述可用解析函数或任何其他形式,例如分段线性函数.文中还就所得周期解指出了非线性振动的一些值得注意的性质.部分计算结果与已有的近似解或实验结果作了比较.

Abstract: Direct numerical integration can be used to find the periodic solutions for the equations of motion of nonlinear vibration systems. The initial conditions are iterated so that they coincide with the terminal conditions. The time interval of the integration (i.e.,the period) and certain parameters of the equations of motion can be included in the iterations.The integration method has a variable steplength.This shooting method can produce periodic solutions with a shorter computer time. The only error occurs in the numerical integration and it can therefore be estimated and made small enough. Using this method one can treat a variety of vibration problems, such as free conservative, forced, parameter-excited and self-sustained vibrations with one or several degrees-of-freedom. Unstable solutions and those which are sensitive to parameter changes can also be calculated.The stability of the solutions is investigated based on the theory of differential equations with periodic coefficients. The extrapolation method and the procedure of automatic steplength control are used to estimate the initial values of iterations by determining the resonance curve and other vibration characteristics.Some examples have been calculated to illustrate the applicability of the method. The non-linearity way be expressed by an analytical function or any other functions, such as a piecewise linear function. Several remarkable features of nonlinear vibrations are presented through the periodic solutions obtained. Finally, some results are compared with those obtained by other approximation methods and experiments.

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