An Analytical Study on Forced Vibration of Beams Carrying a Number of Two Degrees-of-Freedom Spring–Damper–Mass Subsystems

Abstract

This paper presents an analytical solution for the forced vibration of beams carrying a number of two degrees-of-freedom (DOF) spring–damper–mass (SDM) systems. The beam is divided into a series of distinct sub-beams at the spring connection points and the point of force action. The 2DOF SDM systems are replaced with a set of effective springs with complex stiffness, and the compatibility of the placement and the force at the common interface of two adjacent sub-beams is systematically organized. Then, the boundary conditions are enforced, and the governing matrix equation is formulated. Next, the closed-form expression for the frequency response function (FRF) was determined analytically. The presented method can simultaneously consider arbitrary boundary conditions and any number of 2DOF SDM systems. Furthermore, regardless of the number of subsystems, none of the associated matrices is larger than 4 × 4, which provides a significant computational advantage. To validate the accuracy and reliability of the proposed method, some results are compared with the corresponding results obtained using the conventional finite element method (FEM); good agreement is observed between the results of the two approaches. Finally, the effects of the system parameters on the vibration transmission in the beam and subsystem vibration reduction are studied.