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

From the equation of motion of the `bareʹ 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 th sprung mass ( =1; : : : ; n) yields a matrix equation of the form [B ]fC g=0. Similarly, when the eigenfunction is substituted into the two boundary-condition equations at the `leftʹ end and those at the `rightʹ end of the beam, one obtained [BL]fCLg=0 and [BR]fCRg=0, respectively. Assembly of the coe cient matrices [BL]; [B ] and [BR] will arrive at the eigen equation [ B]f C g=0, where the elements of f C g are composed of the integration constants C i ( =1; : : : ; n and i=1; : : : ; 4) and the modal displacements of the th sprung mass, Z . For a Timoshenko beam carrying n spring{mass systems, the order of the overall coe cient matrix [B ] is 5n + 4. The solutions of jB j=0 (where j j denotes a determinant) give the natural frequencies of the `constrainedʹ 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 de ne the corresponding mode shapes. In the existing literature the eigen equation [ B]f C g=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 `twoʹ. But any number of spring{mass systems will not make trouble for the numerical assembly technique presented in this paper.