The Presence of Chaos in a Viscoelastic Harmonically Forced Von Mises Truss

Abstract

This work investigates how viscoelasticity affects the dynamic behavior of a lumped-parameter model of a bistable von Mises truss. The system is controlled by a linear first-order equation and a second-order nonlinear Duffing equation with a quadratic nonlinearity that governs mechanical behavior. The second-order equation controls mechanical oscillations, while the linear first-order equation controls viscoelastic force evolution. Combined, the two equations form a third-order jerk equation that controls system dynamics. Viscoelasticity adds time scales and degrees of freedom to material behavior, distinguishing it from viscosity-only systems. Due to harmonic excitation, the system exhibits varied dynamic responses from periodic to quasiperiodic to chaotic. We explore the dynamics of a harmonically forced von Mises truss with viscous damping to address this purpose. We demonstrate this system's rich dynamic behavior due to driving amplitude changes. This helps explain viscoelastic system behavior. A viscoelastic unit replaces the viscous damper, and we show that, although viscous damping merely changes how fast the trajectory decays to an attractor, viscoelasticity modifies both the energy landscape and the rate of decay. In a conventional linear solid model, three viscoelastic parameters control the system's behavior instead of one, as in pure viscous damping. This adds degrees of freedom that affect system dynamics. We present the parameter space for chaotic behavior and the shift from regular to irregular motion. Finally, Melnikov's criteria identify the regular-chaotic threshold. The system's viscous and elastic components affect the chaotic threshold amplitude.