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Stability, modal couplings and local bifurcations in bending -torsion forced vibrations of a three-dimensional elastic rod.

机译：三维弹性杆在弯曲扭转强迫振动中的稳定性，模态耦合和局部分叉。

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摘要

Numerical and experimental studies of the coupled bending-torsion forced vibration in a 3-D thin elastica are conducted. The elastic rod is under the clamped-free boundary condition, with its long axis aligned in the vertical direction, and is subjected to a sinusoidal base excitation. Particular interest is focused on the phenomena associated at the transition of a planar motion to a nonplanar one.;The numerical study focuses on the stability of the planar motion and the related local bifurcations. The model developed by Joseph P. Cusumano from Cornell University is adopted. The structure of the instability band is observed in the stability diagram. A similar phenomenon shown by the Mathieu equation with a resonating coefficient is demonstrated to explain this unusual feature. Under resonance excitation, the modal convergence test at and right after the planar instability suggests a dominant modal coupling among the first bending mode, the resonance bending mode and the first torsional mode.;The local bifurcation at the planar instability is studied by the continuation method and the underlying symmetry structure of the model. A subcritical pitchfork bifurcation is identified at the loss of planar stability. Other bifurcation scenarios for a family of period one solutions are also given; these are the saddle-node, the pitchfork and the secondary Hopf bifurcations. Low-dimensional chaos is discovered by numerical simulations. The chaotic orbit is evolved from a set of 2-tori undergoing what is known as the torus doubling bifurcations. From the Poincare map, Fast Fourier Transform, Lyapunov exponents and the correlation dimension calculations, detailed descriptions of the phenomena are presented.;The experiment incorporating the constrained layer damping shows two similar features found from the numerical work. These features are, therefore, believed intrinsic to the vibration of beams with thin cross section. The first feature is suggested by the stability transition curves for beams under different damping treatments. A dominant modal coupling involving only lower bending modes and the torsional motion is observed at the loss of planar stability. The second feature is the low-dimensional chaotic motion right after the planar instability. The active degree of freedom of the chaotic motion is estimated from the delay-embedding map.
机译：进行了三维薄弹性体中弯扭耦合强迫振动的数值和实验研究。弹性杆处于不受夹紧的边界条件，其长轴在垂直方向上对齐，并受到正弦基础激励。特别关注的是平面运动向非平面运动过渡时的相关现象。数值研究集中于平面运动的稳定性和相关的局部分叉。采用了康奈尔大学的Joseph P. Cusumano开发的模型。在稳定性图中观察到不稳定带的结构。 Mathieu方程显示的具有共振系数的类似现象被证明可以解释这一异常特征。在共振激励下，平面不稳定性及其后的模态收敛测试表明，第一弯曲模式，共振弯曲模式和第一扭转模式之间存在主要的模态耦合。以及模型的基本对称结构。在失去平面稳定性的情况下，发现了亚临界干草叉分叉。还给出了针对一期周期解决方案的其他分叉方案；它们是鞍形节点，干草叉和次要Hopf分支。通过数值模拟发现了低维混沌。混沌轨道是从一组2 tori演化而来的，该2 tori经历了所谓的环形加倍分叉。从庞加莱图，快速傅立叶变换，李雅普诺夫指数和相关维数计算，给出了对现象的详细描述。结合约束层阻尼的实验显示出从数值工作中发现的两个相似特征。因此，这些特征被认为是具有薄截面的梁的振动所固有的。第一个特征是梁在不同阻尼处理下的稳定性过渡曲线所暗示的。在失去平面稳定性的情况下观察到主要只涉及较低弯曲模式和扭转运动的主要模态耦合。第二个特征是紧随平面不稳定之后的低维混沌运​​动。从延迟嵌入图估计混沌运动的主动自由度。

著录项

  • 作者

    Lin, DerChyan Bill.;

  • 作者单位

    The Pennsylvania State University.;

  • 授予单位 The Pennsylvania State University.;
  • 学科 Engineering Mechanical.;Applied Mechanics.
  • 学位 Ph.D.
  • 年度 1992
  • 页码 184 p.
  • 总页数 184
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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