A tube dynamics perspective governing stability transitions: An example based on snap-through buckling
A tube dynamics perspective governing stability transitions: An example based on snap-through buckling
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DOI：
10.1016/j.ijmecsci.2017.10.040
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发表时间：
2017-05
影响因子：
7.3
通讯作者：
Jun-Hao Zhong;L. Virgin;S. Ross
中科院分区：
文献类型：
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作者：
Jun-Hao Zhong;L. Virgin;S. Ross
工程结构的平衡构形，能够承受一定的载荷条件，通常与潜在势能的局部最小值有关。然而，在非线性背景下，可能存在其他均衡，这带来了过渡到替代（远程）最小值的可能性。也就是说，如果有足够的扰动，结构可能会突然弯曲成另一种形状。本文认为，这种转变（通常通过鞍点）发生的动态机制。建立了浅拱/屈曲梁的双模态哈密顿量。由此产生的形式的势能-两个稳定的威尔斯连接的秩-1鞍点-显示了类似的天体力学或化学中的分子重新配置的共振跃迁，而在这里的过渡对应于两个稳定的结构配置之间的切换。然后，从汉密尔顿方程，平衡确定和线性化的运动方程的鞍。在计算与线性化相关联的系数矩阵的特征值和特征向量之后，给出了辛变换，该辛变换将哈密顿量转化为标准形式并简化了方程，使我们能够使用被称为管动力学的概念框架。讨论了相空间中平衡区的流动以及位置空间中的不变流形管。此外，我们考虑到在管动力学框架中增加阻尼，这导致了比以前探索的更丰富的过渡动力学行为。
The equilibrium configuration of an engineering structure, able to withstand a certain loading condition, is usually associated with a local minimum of the underlying potential energy. However, in the nonlinear context, there may be other equilibria present, and this brings with it the possibility of a transition to an alternative (remote) minimum. That is, given a sufficient disturbance, the structure might buckle, perhaps suddenly, to another shape. This paper considers the dynamic mechanisms under which such transitions (typically via saddle points) occur. A two-mode Hamiltonian is developed for a shallow arch/buckled beam. The resulting form of the potential energy—two stable wells connected by rank-1 saddle points—shows an analogy with resonance transitions in celestial mechanics or molecular reconfigurations in chemistry, whereas here the transition corresponds to switching between two stable structural configurations. Then, from Hamilton’s equations, the equilibria are determined and linearization of the equations of motion about the saddle is obtained. After computing the eigenvalues and eigenvectors of the coefficient matrix associated with the linearization, a symplectic transformation is given which puts the Hamiltonian into normal form and simplifies the equations, allowing us to use the conceptual framework known as tube dynamics. The flow in the equilibrium region of phase space as well as the invariant manifold tubes in position space are discussed. Also, we account for the addition of damping in the tube dynamics framework, which leads to a richer set of behaviors in transition dynamics than previously explored.