Poincaré非线性振动理论在连续介质力学中的推广(Ⅰ)--基本理论与方法<sup>*</sup>

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Poincaré非线性振动理论在连续介质力学中的推广(Ⅰ)--基本理论与方法^*

基金项目: * 中国科学院科学基金资助课题

摘要: 本文将离散介质的Poincaré非线性振动理论^[1]向连续介质力学推广,做了初步尝试。首先讨论在非共振与共振情况下,连续介质线性强迫振动周期解,及其周期解存在条件。进而运用线性理论结果,将Poincaré理论中的主要结论推广到连续介质非线性振动问题中去。此外,本文提出并建议用偏微分方程直接摄动与加权积分方法,计算共振区内的周期解。

Abstract: In this paper we extend Poincaré's nonlinear oscillation theory of discrete system to continuum mechanics.First we investigate the existence conditions of periodic solution for linear continuum system in the states of resonance and non-resonance.By applying the results of linear theory,we prove that the main conclusion of Poincaré's nonlinear oscillation theory can be extended to continuum mechanics.Besides,in this paper a new method is suggested to calculate the periodic solution in the states of both resonance and nonresonance by means of the direct perturbation of partial differential equation and weighted integration.

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