Non-linear dynamics of the extended Lazer-McKenna bridge oscillation model

Abstract

We examine the dynamics of two simple coupled non-linear ordinary di erential equations (ODEs) ® rst introduced by Lazer and McKenna (SIAM Review 32(4), 537± 578, 1990) . Using the numerical continuation package AUTO, we obtain multiple coexistence of periodic motions, period-doubling sequences and the onset of`beats' solutions via torus bifurcations. We discuss the implications of these results for the modelling of suspension bridge dynamics.

Canard-Like Explosion of Limit Cycles in Two-Dimensional Piecewise-Linear Models of FitzHugh–Nagumo Type

“…The dependence of κ on η is not linear as (24) and Figure 12(B) show. In Figure 13(A) we present the curves κ(η) for various values of and two values of α: α = 4 (left) and α = 2 (right).…”

“…For the parameters we used, the Hopf bifurcation is supercritical (subcritical) for α > 3 (α < 3) (see Appendix A). Consequently, the Hopf of suspension bridges [24], and indeed impacting systems in general (see the recent book by di Bernardo et al [25]). Many of the techniques for analyzing mechanical systems have been extended to the study of oscillatory electronic circuits and particularly the analysis of nonsmooth bifurcations such as those of border-collision type [26].…”

Mathematical Models for Suspension Bridges

“…Further numerical results by Doole-Hogan [96] and McKenna-Tuama [195] show that a purely vertical periodic forcing in (4.2) (that is, D 0 and ¤ 0) may create a torsional response: high frequency vertical forcing can result in a periodic motion that is predominantly torsional. They also considered different nonlinear restoring forces, in particular…”

Bifurcation and stability properties of periodic solutions to two nonlinear spring-mass systems

“…The results on existence, uniqueness, multiplicity, bifurcation, and stability of periodic solutions are consistent with the nonlinear behavior of some suspension bridges; see [2], [4], [6], [8], and [13], for example.…”