On the eigenvalues of a uniform cantilever beam carrying any number of spring-damper-mass systems

The eigenvalues of a uniform cantilever beam carrying any number of spring}damper}mass systems with arbitrary magnitudes and locations were determined by means of the analytical-and-numerical-combined method (ANCM). First of all, each spring}damper}mass system was replaced by a massless e!ective spring with spring constant k%&&, which is the main point that the ANCM is available for the present problem. Next, the equation of motion for the &constrainedʹ beam (with spring}damper}mass systems attached) was derived by using the natural frequencies and normal mode shapes of the &unconstrainedʹ beam (without carrying any attachments) incorporated with the expansion theorem. Finally, the equation of motion for the &constrainedʹ beam in &complex formʹ is separated into the real and the imaginary parts. From either part, a set of simultaneous equations were obtained. Since the simultaneous equations are in &real formʹ, the eigenvalues of the &constrainedʹ beam were determined with the conventional numerical methods. To con"rm the reliability of the presented theory, all the numerical results obtained from the ANCM were compared with the corresponding ones obtained from the conventional "nite element method (FEM) and good agreement was achieved. Because the order of the property matrices for the equation of motion derived by using the ANCM is much lower than that by using the conventional FEM, the storing memory and the CPU time required by the ANCM are much less than those required by the FEM. Besides, the solution of the equation of motion derived from the ANCM can always be obtained with the general personal computers, but that from the FEM can sometimes be obtained only with the computers of workstations or main frames when the total degrees of freedom exceeding a certain limit