Stability and Controllability of Euler-Bernoulli Beams With Intelligent Constrained Layer Treatments

Abstract

This paper studies the stability and controllability of Euler-Bernoulli beams whose bending vibration is controlled through intelligent constrained layer (ICL) damping treatments proposed by Baz (1993) and Shen (1993, 1994). First of all, the homogeneous equation of motion is transformed into a first order matrix equation in the Laplace transform domain. According to the transfer function approach by Yang and Tan (1992), existence of nontrivial solutions of the matrix equation leads to a closed-form characteristic equation relating the control gain and closed-loop poles of the system. Evaluating the closed-form characteristic equation along the imaginary axis in the Laplace transform domain predicts a threshold control gain above which the system becomes unstable. In addition, the characteristic equation leads to a controllability criterion for ICL beams. Moreover, the mathematical structure of the characteristic equation facilitates a numerical algorithm to determine root loci of the system. Finally, the stability and controllability of Euler-Bernoulli beams with ICL are illustrated on three cantilever beams with displacement or slope feedback at the free end.