Stability and Bifurcations of a Piecewise-Smooth Elasto-Plastic Inverted Pendulum Model : Towards an Understanding of Dynamics of Buildings under Earthquake-type Forcing

Abstract

Many dynamical systems of interesting physical phenomena are piecewise-smooth (PWS). In this dissertation, a particular PWS model, namely an elasto-plastic inverted pendulum (called the "EPT" model) is used to model an engineering structure under earthquake-like forcing. As a simplest model, we study the case of a sinusoidal forcing function. Ultimately we want to investigate the dynamics by constructing and analyzing bifurcations of periodic solutions. However, as a first step we consider the stability of equilibria of the undriven EPT model. We use a shooting method to construct periodic solutions of a simplification without plasticity, called the "elastic-torque" (ET) model. Using similar methods, we generate grazing periodic orbits for the EPT model. We observe that there are two families of periodic orbits -- one associated with the pendulum near the top (orbits near θ = 0), and the other with near the bottom (orbits near θ = π). A question of interest is: which forcing amplitudes lead to switching from the top to the bottom? We construct a "safe region" corresponding to amplitudes that do not lead to this switch.

Next, we investigate stability and bifurcations of periodic solutions to both the ET and EPT models using the more powerful continuation tool, "AUTO". However, because the EPT model is nonsmooth and degenerate, AUTO's tools sometimes fail. To repair this problem, we construct a smooth version of the EPT model, called the "ST" model; it permits the study of dynamics even when AUTO fails for the EPT model. Stability and bifurcations of periodic solutions to this final model are also investigated so that we can compare them with those of the original system. Finally, codimension-two bifurcations of our models, i.e., the loci of the special bifurcations, are constructed using two continuation parameters: the forcing amplitude, β, and the forcing frequency, Ω. Tuning these simultaneously can lead to collisions of bifurcations.