A Tube Dynamics Perspective Governing Stability Transitions: An Example Based on Snap-through Buckling

摘要

The equilibrium configuration of an engineering structure, able to withstanda certain loading condition, is usually associated with a local minimum of theunderlying potential energy. However, in the nonlinear context, there may beother equilibria present, and this brings with it the possibility of atransition to an alternative (remote) minimum. That is, given a sufficientdisturbance, 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 ashallow arch/buckled beam. The resulting form of the potential energy---twostable wells connected by rank-1 saddle points---shows an analogy withresonance transitions in celestial mechanics or molecular reconfigurations inchemistry, whereas here the transition corresponds to switching between twostable structural configurations. Then, from Hamilton's equations, theanalytical equilibria are determined and linearization of the equations ofmotion about the saddle is obtained. After computing the eigenvalues andeigenvectors of the coefficient matrix associated with the linearization, asymplectic transformation is given which puts the Hamiltonian into normal formand simplifies the equations, allowing us to use the conceptual framework knownas tube dynamics. The flow in the equilibrium region of phase space as well asthe invariant manifold tubes in position space are discussed. Also, we accountfor the addition of damping in the tube dynamics framework, which leads to aricher set of behaviors in transition dynamics than previously explored.