Dynamic stability and chaos of system with piecewise linear stiffness

Abstract

The dynamic stability behaviour of a single-degree-of-freedom system with piecewise linear stiffness is considered. The analytical expression representing the divergence of perturbed trajectories is derived. The mechanism triggering dynamic instability of trajectories and the cause of chaotic behavior are then studied. Liapunov exponents are used as a quantitative measure of system stability. Numerical results including bifurcation diagrams and largest Liapunov exponents of a system with symmetric bilinear stiffness are presented. Several different types of bifurcation and nonlinear phenomena are identified, including pitchfork, fold, flip, boundary crisis, and intermittency of type 3. To illustrate the initial-condition dependent nature of the problem, basins of attraction of multiple steady-state responses are determined on the phase plane using the simple cell mapping method.