Free vibration analysis of a Timoshenko beam carrying multiple spring-mass systems by using the numerical assembly technique

Abstract

From the equation of motion of the ‘bare’ Timoshenko beam (without any spring–mass systems attached), an eigenfunction in terms of four unknown integration constants is obtained. The substitution of the eigenfunction into the three compatible equations, one force–equilibrium equation and one governing equation for the sprung mass (ν=1,…,n) yields a matrix equation of the form [Bν] {Cν}=0. Similarly, when the eigenfunction is substituted into the two boundary-condition equations at the ‘left’ end and those at the ‘right’ end of the beam, one obtained [BL] {CL}=0 and [BR] {CR}=0, respectively. Assembly of the coefficient matrices [BL], [Bν] and [BR] will arrive at the eigen equation [B̄] {C̄}=0, where the elements of {C̄} are composed of the integration constants Cνi (ν=1,…,n and i=1,…,4) and the modal displacements of the sprung mass, Zν. For a Timoshenko beam carrying n spring–mass systems, the order of the overall coefficient matrix [B̄] is 5n+4. The solutions of ∣B̄∣=0 (where ∣·∣ denotes a determinant) give the natural frequencies of the ‘constrained’ beam (carrying multiple spring–mass systems) and the substitution of each corresponding values of Cνi into the associated eigenfunction at the attaching points will define the corresponding mode shapes. In the existing literature the eigen equation [B̄] {C̄}=0 was denoted in explicit form and then solved analytically or numerically. Because of the lengthy explicit mathematical expressions, the existing approach becomes impractical if the total number of the spring–mass systems is larger than ‘two’. But any number of spring–mass systems will not make trouble for the numerical assembly technique presented in this paper. Copyright © 2001 John Wiley & Sons, Ltd.