Modeling and numerical simulation of the nonlinear dynamics of the forced planar string pendulum

Abstract

The string pendulum consists of a mass attached to the end of an inextensible string which is fastened to a support. Analyzing the dynamics of such forced supports is motivated by understanding the behavior of suspension bridges or of tethered structures during earthquakes. Applying an external forcing to the pendulum's support can cause the pendulum string to go from taut to slack states and vice versa, and is capable of exhibiting interesting periodic or chaotic dynamics. The inextensibility of the string and its capacity to go slack make simulation and analysis of the system complicated. The string pendulum system is thus formulated here as a piecewise-smooth dynamical system using the method of Lagrange multipliers to obtain a system of differential algebraic equations (DAE) for the taut state. In order to find a formulation for the forced string pendulum system, we first turn to similar but simpler pendulum systems, such as the classic rigid pendulum, the elastic spring pendulum and the elastic spring pendulum with piecewise constant stiffness. We perform a perturbation analysis for both the unforced and forced cases of the spring pendulum approximation, which shows that, for large stiffness, this is a reasonable model of the system. We also show that the spring pendulum with piecewise constant stiffness can be a good approximation of the string pendulum, in the limit of a large extension constant and a low compression constant. We indicate the behavior and stability of this simplified model by using numerical computations of the system's Lyapunov exponents. We then provide a comparison of the spring pendulum with piecewise constant stiffness with the formulation of the taut-slack pendulum using the DAE for the taut states and derived switching conditions to the slack states.