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

From the equation of motion of the 'bare' Timoshenko beam (without any spri ng-mass systems attached), an eigenfunction in terms of four unknown integr ation constants is obtained. The substitution of the eigenfunction into the three compatible equations, one force-equilibrium equation and one governi ng equation for the vth sprung mass (nu = 1,..., n) yields a matrix equatio n of the form [B-nu]{C-nu} = 0. Similarly, when the eigenfunction is substi tuted into the two boundary-condition equations at the 'left' end and those at the 'right' end of the beam, one obtained [B-L]{C-L} = 0 and [BR]{CR} = 0, respectively. Assembly of the coefficient matrices [B-L], [B-nu] and [B -R] Will arrive at the eigen equation [(B) over bar]{(C) over bar}= 0, wher e the elements of {(C) over bar} are composed of the integration constants C-nui (nu = 1,...,n and i = 1,...,4) and the modal displacements of the vth sprung mass, Z(nu). For a Timoshenko beam carrying n spring-mass systems, the order of the overall coefficient matrix [(B) over bar] is 5n + 4. The s olutions of /(B) over bar/ = 0 (where / (.) / denotes a determinant) give t he natural frequencies of the 'constrained' beam (carrying multiple spring- mass systems) and the substitution of each corresponding values of C,i into the associated eigenfunction at the attaching points will define the corre sponding mode shapes. In the existing literature the eigen equation [(B) ov er bar]{(C) over bar}= 0 was denoted in explicit form and then solved analy tically or numerically. Because of the lengthy explicit mathematical expres sions, the existing approach becomes impractical if the total number of the spring-mass systems is larger than 'two'. But any number of spring-mass sy stems will not make trouble for the numerical assembly technique presented in this paper. Copyright (C) 2001 John Wiley & Sons, Ltd.