Dynamics of beam pair coupled by visco-elastic interlayer

Abstract

An exact method is presented for solving the vibration of a double-beam system subjected to harmonic excitation. The system consists of a loaded main beam and an auxiliary beam joined together using massless visco-elastic layer. The Euler-Bernoulli model is used for the transverse vibrations of beams, and the spring-dashpot represents a simplified model of viscoelastic material. The damping is assumed to be neither small nor proportional, and the forcing function can be either concentrated at any point or distributed continuously. The method involves a simple change of variables and modal analysis to decouple and to solve the governing differential equations respectively. A case study is solved in detail to demonstrate the methodology, and the frequency responses are shown in dimensionless parameters for low and high values of stiffness and  damping of the interlayer. The analysis reveals two sets of eigen-modes: (i) the odd in-phase modeswhose eigen-values and resonant peaks are independent of stiffness and damping, and (ii) the even out-of-phase modes whose eigen-values increase with raising stiffness and resonant peaks decrease with increasing damping. The closed-form solution and relevant plots (especially the three-dimensional ones) illustrate not only the principles of the vibration problem but also shed light on practical applications.