A piecewise linear suspension bridge model: nonlinear dynamics and orbit continuation

摘要

The effect of harmonic excitation on suspension bridges is examined as a first step towards the understanding of the effect of wind, and possibly certain kinds of earthquake, excitation on such structures. The Lazer—McKenna suspension bridge model is studied completely for the first time by using a methodology that has been successfully applied to models of rocking blocks and other free-standing rigid structures. An unexpectedly rich dynamical structure is revealed in this way. Conditions for the existence of asymptotic periodic responses are established, via a complicated nonlinear transcendental equation. A two-part Poincare map is derived to study the orbital stability of such solutions. Numerical results are presented which illustrate the application of the analytical procedure to find and classify stable and unstable solutions, as well as determine bifurcation points accurately. The richness of the possible dynamics is then illustrated by a menagerie of solutions which exhibit fold and flip bifurcations, period doubling, period adding, and sub- and superharmonic coexistence of solutions. The solutions are shown both in the phase plane and as Poincare map fixed points under parameter continuation using the package AUTO. Such results illustrate the possibility of the coexistence of 'dangerous', large-amplitude responses at the same point of parameter space as 'safe' solutions. The feasibility of experimental verification of the results is discussed.