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Vibration and Stability of Continuous Systems: New Parametric Instability Analysis and Spatial Discretization Method

机译：连续系统的振动和稳定性：新的参数不稳定性分析和空间离散化方法

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摘要

Parametric instability in a system is caused by periodically varying coefficients in its governing differential equations. Parametric instability regions of a second-order non-dispersive distributed structural system in this work are obtained using the wave solution and the fixed point theory without spatially discretizing the governing partial differential equation. The parametric instability regions are classified as period-1 and period- i (i > 1) instability regions, where the former is analytically obtained, and the latter can be numerically calculated using bifurcation diagrams. The parametric instability phenomenon is characterized by a bounded displacement and an unbounded vibratory energy, due to formation of infinitely compressed shock-like waves. Parametric instability in a taut string with a periodically moving boundary is then investigated. The free linear vibration of the taut string is studied first, and three corresponding nonlinear models are introduced next. It is shown that the responses and vibratory energies of the nonlinear models are close to those of the linear model, which indicates that the parametric instability in the linear model can also exist in the nonlinear models.;A new global spatial discretization method for one- and two-dimensional continuous systems is investigated. General formulations for one- and two-dimensional systems that can achieve uniform convergence are established, whose displacements are divided into internal terms and boundary-induced terms. For one-dimensional systems, natural frequencies, mode shapes, harmonic steady-state responses, and transient responses of a rod and a tensioned Euler-Bernoulli beam are calculated using the new method and the assumed modes method, and results are compared with those from exact analyses. The new method gives better results than the assumed modes method in calculating eigensolutions and responses of a system, and it can use sinusoidal functions as trial functions for the internal term rather than possibly complicated eigenfunctions in exact analyses. For two-dimensional systems, natural frequencies and dynamic responses of a rectangular Kirchhoff plate that has three simply-supported boundaries and one free boundary with an attached Euler-Bernoulli beam are calculated using both the new method and the assumed modes method, and compared with results from the finite element method and the finite difference method, respectively. Advantages of the new method over local spatial discretization methods are fewer degrees of freedom and less computational effort, and those over the assumed modes method are better numerical property, a faster calculation speed, and much higher accuracy in calculation of high-order spatial derivatives of the displacement.
机译：系统中的参数不稳定性是由控制微分方程中的系数周期性变化引起的。这项工作中的二阶非分散分布式结构系统的参数不稳定区域是使用波动解和不动点理论获得的，而没有在空间上离散控制性偏微分方程。参数不稳定区域可分为周期1和周期i（i> 1）不稳定区域，其中前者是通过解析获得的，后者可以使用分叉图进行数值计算。由于形成了无限压缩的类似冲击波，因此参数不稳定现象的特征在于有界位移和无界振动能量。然后研究了具有周期性移动边界的拉紧弦中的参数不稳定性。首先研究了拉紧弦的自由线性振动，然后介绍了三个相应的非线性模型。结果表明，非线性模型的响应和振动能量与线性模型的响应和振动能量接近，这表明线性模型中的参数不稳定性也可以存在于非线性模型中。并研究了二维连续系统。建立了可以实现均匀收敛的一维和二维系统的一般公式，将其位移分为内部项和边界引起的项。对于一维系统，使用新方法和假定模态方法计算杆和张拉的Euler-Bernoulli梁的固有频率，模态，谐波稳态响应和瞬态响应，并将结果与确切的分析。在计算系统的本征解和响应时，新方法比假定的模式方法具有更好的结果，并且可以将正弦函数用作内部项的试验函数，而不是在精确分析中可能使用复杂的本征函数。对于二维系统，使用新方法和假定模式方法计算了矩形基尔霍夫板的固有频率和动态响应，该矩形基尔霍夫板具有三个简单支撑的边界和一个自由边界，并附加了欧拉-伯努利梁，并与分别来自有限元法和有限差分法。与局部空间离散化方法相比，新方法的优点是自由度较小，计算量较小，而与假定模式方法相比，新方法具有更好的数值特性，更快的计算速度以及更高的计算高阶空间导数的准确性。位移。

著录项

  • 作者

    Wu, Kai.;

  • 作者单位

    University of Maryland, Baltimore County.;

  • 授予单位 University of Maryland, Baltimore County.;
  • 学科 Mechanical engineering.;Engineering.
  • 学位 Ph.D.
  • 年度 2018
  • 页码 300 p.
  • 总页数 300
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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