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An efficient method for the solution of lower rank extracted systems and analysis of the dynamics of a repeated impact oscillator.

机译：解决低等级提取系统和重复冲击振荡器动力学分析的有效方法。

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摘要

This dissertation consists of two independent parts: The first part proposes efficient numerical solutions to a class of symmetric positive definite linear systems Ax = b, called Lower Rank Extracted Systems (LRES). Such systems appear in numerical modelling of convolution type integral equations defined on arbitrary domains. We compute their solution using a recursive method called the Preconditioned Conjugate Gradient Method (PCGM).;First, we consider integral equations defined on one dimensional domains. The corresponding coefficient matrix, A is shown to be a principal submatrix of an N x N Toeplitz matrix, A. The preconditioner we pro pose is provided in terms of the inverse of a 2N x 2N circulant matrix constructed from the elements of A. The preconditioner is shown to yield clustering in the spectrum of preconditioned matrix. Our analysis further demonstrates that the computational expense to solve LRES is reduced from O(N2) to O(N log N) operations.;We generalize this approach to solve convolution type integral equations defined on a class of p-dimensional domains and achieve similar substantial reduction in the computational expense to solve p-dimensional LRES.;The second part of the dissertation models and studies the dynamics of a mass attached to a spring undergoing repeated impacts with a massive, sinusoidally oscillating table. The dissipation of energy is modelled by coefficient of restitution. First, we study the case in which the impacts are assumed to be inelastic. We prove the existence of an invariant Cantor set on which the dynamics are equivalent to the chaotic "shift map". A similar study is done for the case of plastic impacts where we show the existence of a Smale horseshoe. We also perform bifurcation studies with respect to the frequency and amplitude of the oscillations of the table; and study the dynamics of the system with "soft" springs and the case when the oscillation frequency of the table the natural frequency of the mass are equal.
机译：本文由两个独立的部分组成：第一部分为一类对称的正定线性系统Ax = b提出了有效的数值解，称为低秩提取系统（LRES）。这样的系统出现在任意域上定义的卷积型积分方程的数值模型中。我们使用称为预处理共轭梯度法（PCGM）的递归方法来计算它们的解。首先，我们考虑一维域上定义的积分方程。相应的系数矩阵A被显示为N x N Toeplitz矩阵A的主要子矩阵。我们提出的预处理器是根据由A的元素构成的2N x 2N循环矩阵的逆提供的。预处理器在预处理矩阵的光谱中显示出聚类。我们的分析进一步证明，求解LRES的计算费用从O（N2）减少为O（N log N）运算。;我们推广了这种方法来求解在p维域上定义的卷积型积分方程并获得相似的结果大大减少了解决p维LRES的计算费用。论文的第二部分研究并研究了附着在弹簧上的重物的动力学，该弹簧经受了巨大的正弦振动台的反复冲击。能量耗散由恢复系数建模。首先，我们研究假设影响是无弹性的情况。我们证明了存在不变的Cantor集，其动力学等效于混沌的“移位图”。对于塑性冲击的情况，我们进行了类似的研究，结果表明存在马累马蹄铁。我们还对桌子的振动频率和幅度进行分叉研究。并研究了带有“软”弹簧的系统的动力学特性以及表的振动频率与质量固有频率相等时的情况。

著录项

  • 作者

    Salapaka, Srinivasa Murthi.;

  • 作者单位

    University of California, Santa Barbara.;

  • 授予单位 University of California, Santa Barbara.;
  • 学科 Mathematics.;Engineering Mechanical.
  • 学位 Ph.D.
  • 年度 2002
  • 页码 217 p.
  • 总页数 217
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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