The dynamic characteristic of a uniform rectangular plate with four boundary conditions and carrying three kinds of multiple concentrated element (rigidly attached point masses, linear springs and elastically mounted point masses) was investigated. Firstly, the closed-form solutions for the natural frequencies and the corresponding normal mode shapes of a rectangular ‘bare’ (or ‘unconstrained’) plate (without any attachments) with the specified boundary conditions were determined analytically. Next, by using these natural frequencies and normal mode shapes incorporated with the expansion theory, the equation of motion of the ‘constrained’ plate (carrying the three kinds of multiple concentrated element) were derived. Finally, numerical methods were used to solve this equation of motion to give the natural frequencies and mode shapes of the ‘constrained’ plate. To confirm the reliability of previous free vibration analysis results, a finite element analysis was also conducted. It was found that the results obtained from the above-mentioned two approaches were in good agreement. Compared with the conventional finite element method (FEM), the approach employed in this paper has the advantages of saving computing time and achieving better accuracy, as can be seen from the existing literature.
SUMMARYThe 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 &constrained' beam (with spring}damper}mass systems attached) was derived by using the natural frequencies and normal mode shapes of the &unconstrained' beam (without carrying any attachments) incorporated with the expansion theorem. Finally, the equation of motion for the &constrained' beam in &complex form' 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 form', the eigenvalues of the &constrained' 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.
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