Vibration Suppression in a Pinned-Pinned Nonlinear Rod Using a Frictionless Slider

Abstract

We propose a new method for vibration suppression in a flexible structure using a frictionless sliding constraint. The constraint force applied by the slider is assumed known from measurements and the slider motion is prescribed to do negative work on the structure. The structure is modeled as a two-dimensional nonlinear rod with pinned-pinned boundary conditions and the slider is assumed to constrain the position of one point on the rod but not its slope. The problem is formulated using variable-length finite elements in the framework of Arbitrary Lagrange-Euler (ALE) description. The governing equations of motion are derived using the principle of virtual displacements and D’Alembert’s principle. Numerical simulation results are presented to demonstrate the effectiveness of the control strategy based on the idea of negative work. To meet the bandwidth requirement of the actuator, a nonlinear filter is placed in the feedback loop and asymptotic stability of the equilibrium configuration is established using Lyapunov stability theory.