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电磁场中载流薄板的分岔和混沌电子书
摘 要
分岔和混沌是非线性系统最重要而又最基本的特性，几乎在所有涉及非线性科学的领域中，都存在着分岔现象和混沌运动。随着现代高新科技的发展，板、壳等结构元件处于电磁场环境中的情况已是屡见不鲜，这种电磁场与力学场相互耦合的一个基本特征就是非线性特性。论文在分析和总结非线性微分动力系统分岔和混沌研究现状的基础上，对电磁场中横向磁场和机械载荷共同作用下载流矩形薄板的非线性行为进行了研究。主要内容如下：首先，简要介绍了混沌运动的国内外研究现状，论文研究的主要内容以及判定通向混沌运动的Melnikov 方法，并给出了几种常用的Melnikov函数。其次，针对横向电磁场中三边简支一边自由和一对边简支一对边固定载流矩形薄板，建立了单模态下薄板的非线性运动方程。利用Melnikov函数法对其单模态位移模式下的混沌运动进行了分析。得到了非线性动力系统在无扰动情况下的平衡点及在有扰动的情况下，发生Smale 马蹄意义下的混沌的临界条件。并利用Matlab 程序模拟了满足此临界条件的非线性系统的混沌运动。最后，对横向电磁场中三边简支一边自由和一对边简支一对边固定载流矩形薄板，建立了双模态下薄板的非线性运动方程。利用平均法求得板在双模态下的分岔点，并讨论了分岔点的稳定性情况。进一步从理论上讨论了利用单、双模态位移模式模拟非线性行为的差异。
关键词 薄板；Menikov 函数；平均法；电磁场；分岔；混沌；临界载荷；电流密度
目 录
摘要···············································································································I
Abstract ·········································································································· II
第 1 章 绪论···································································································· 1
1.1 课题的提出及意义··············································································· 1
1.2 混沌运动的研究现状及前景······························································· 2
1.2.1 混沌研究简介················································································· 2
1.2.2 薄板混沌运动简介········································································· 4
1.3 论文的结构安排··················································································· 6
第2 章 混沌研究的基本方法········································································· 7
2.1 Melnikov 方法简介·············································································· 7
2.2 Melnikov 函数及混沌判断依据的推导··············································· 7
2.3 常用的Melnikov 函数······································································· 12
2.4 本章小结···························································································· 14
第3 章 载流