Analysis of the dynamic behavior of a mechanical system subject to unstable bifurcation

Abstract

Slender structural systems, susceptible to unstable bifurcation, generally lose stability at lower load levels than the linear buckling load of the perfect structure. In the present work, the different dynamic balance configurations in nonlinear oscillations are studied through a simple structural system given by a rigid-spring bar model with a degree of freedom. The purpose of this work is to study the different bifurcations and nonlinear oscillations through a parametric analysis of a simple structural system subject to buckling when subjected to compressive loads. To solve the calculations, computer programs such as Maple, Matlab, Visual Studio (C ++) were used, as well as Grapher to obtain the graphics. To obtain the stochastic responses of the studied model, the Legendre-Chaos polynomial will be used. Two particularities will be studied, which are the systems that present symmetrical bifurcation of the Butterfly type and the systems that present asymmetric bifurcation of the Swallowtail type, in both cases the bifurcations present an unstable initial post-critical path. Like the Butterfly-type bifurcation, Swallowtail is also significantly affected by the presence of uncertainties in the system's stiffness. Depending on the value that the uncertainty is inserted, an increase in the number of stable solutions can happen, but depending on the value it can also generate chaotic solutions. Regardless of the bifurcation analyzed, the results obtained behaved as the average of the limit results only for small values of Γ1 or for the solutions present in the pre-buckling potential valley.