Lyapunov-Based Vibration Control of Translational Euler-Bernoulli Beams Using the Stabilizing Effect of Beam Damping Mechanisms

Abstract

A translational cantilevered Euler-Bernoulli beam with tip mass dynamics at its free end is used to study the effect of several damping mechanisms on the stabilization of the beam displacement. Specifically, a Lyapunov-based controller utilizing a partial differential equation model of the translational beam is developed to exponentially stabilize the beam displacement while the beam support is regulated to a desired set-point position. Depending on the composition of the tip mass dynamics assumption (i.e. body-mass, point-mass, or massless). it is shown that proper combination of different damping mechanisms (i.e. strain-rate, structural, or viscous damping) guarantees exponential stability of the beam displacement. This novel Lyapunov-based approach. which is based on the energy dissipation mechanism in the beam, brings new dimensions to the stabilization problem of translational beams with tip mass dynamics. The stability analysis utilizes relatively simple mathematical tools to illustrate the exponential and asymptotic stability results. The numerical results are presented to show the effectiveness of the controller.