| 关于Mathieu方程周期解的讨论 |
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| 摘 要： | 形如 f″(x)+g(x)·f(x)=0的微分方程,其中 g(x)是 x 的周期函数.这类方程就是马奇耶方程.马奇耶(Mathieu)方程在实际工程中有着广泛的应用.关于它的周期解的研究,是结构动力屈曲分析的理论基础;同时也是常微分方程稳定性理论的—个重要内容.在马奇耶方程的周期解中,稳定与不稳定解的分界线即临界解是十分重要的.本文给出了临界解的求解方法,证明了临界频率方程的收敛性,讨论了某些干扰因素对临界解的影响。在实际工程中,这些干扰因素体现在结构阻尼,结构初始缺陷,结构的非线性几何点系结构的纵向惯性矩及转动惯性矩、复合材料的耦合效应等.计算结果表明,对于马奇耶方程的微小干扰,都将严重影响其临界解甚至改变解的性质.因此,在分析结构动力屈曲问题时,必须考虑问题所能包含的上述各项因素.
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| 关 键 词： | Mathieu 方程 微分方程稳定性 临界频率 收敛性 |
The Perodic Solution of Mathieu Equation |
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| Authors: | Wang Liedong |
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| Institution: | Wang Liedong |
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| Abstract: | For the mode of differential equation f″(x)+g(x)·f(x)=0,g(x) is a perodic fuction respect to x,This kind of equation is Mathieu Equation. This equation has spread of application in engineering practise.The study of it's perod- ic solution is basic fundation of dynamic stability of structure.Mattematically it is main part of stability theory of differential equation. In the perodic solution of M athieu equation the boundary between stability and instabili- ty is very important.This paper gives the method of finding critial solution,proved the cen- vergency of critial frequency;discussed some interference faction to critial solution.In engi- neering practise,these faetors are the damping of structure,nonlinearity of structure,the coupling effectness of composite materials etc.The calculating results short that the small disturbation to Mothieu Equation can influent the critical equation seriously,even though change the property of the solution.Therefore,for analysing the dynamic stabilization of the structure,these factors must be considered. |
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| Keywords: | Mathieu Equation Differential Equation Stability Critial Frequency Cenvergency |
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