VIBRATION OF A DOUBLE-BEAM SYSTEM

An exact method is presented for solving the vibration of a double-beam system subject to harmonic excitation. The system consists of a main beam with an applied force, and an auxiliary beam, with a distributed spring k and dashpot c in parallel between the two beams. The viscous damping and the applied forcing function can be completely arbitrary. The damping is assumed to be neither small nor proportional, and the forcing function can be either concentrated at any point or distributed. The Euler–Bernoulli model is used for the transverse vibrations of beams, and the spring–dashpot represents a simplified model of viscoelastic material. 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 (k/k0) and damping (c/c0). The plots show that each natural mode consists of two submodes: (1) the in-phase submode whose natural frequencies and resonant peaks are independent of stiffness and damping, and (2) the out-of-phase submode whose natural frequencies are increased with increasing stiffness and resonant peaks are decreased with increasing damping. The closed-form solution and the plots, especially the three-dimensional ones, not only illustrate the principles of the vibration problem but also shed light on practical applications.