非线性振动系统周期运动及其稳定性的数值研究

§1引言确定型非线性振动系统的运动可分类如下: 1.非定常运动;2.定常运动:(1)周期运动,(2)各态历经运动,(3)浑沌运动。其中非定常运动是一暂态过程,会随着时间的增长逐步衰减乃至实际上消失。定常运动中的各态历经运动,指系统至少有两个互不通约(即其比值为无理数)的振动频率,因此运动虽然局

NUMERICAL TREATMENTS OF A PERIODIC MOTION AND ITS STABILITY OF NONLINEAR OSCILLATION SYSTEMS

Some recent developments of the numerical methods for evaluting a periodic motion of nonliear oscillation systems and for deforming the stability of such a motion are reviewed in this paper. We lay the emphasize on the direct numerical method, which is based on the shooting method combined with, in most cases, a direct numerical integration. With this method we can treat a great variety of nonlinear oscillation problems (including those in the strong nonlinear systems) with a desired accuracy. The another aspect of this method is to decide the stability of the obtained periodic solutions and to drawing stability charts of a linear periodic system by use of the Floquet theory with numerical calculations.Some examples are given in this paper. Another method is the fast Galerkin's method. This method combines a standard Galcrkin method with the FFT technique and therefore is more effective than the standrcl one. The point mapping method is only briefly mentioned, because this method is in principle more suitable for investigating the global property and the chaotic behaviour of a system than for treating its periodic solutions.