USE OF THE ANALYTICAL-AND-NUMERICAL-COMBINED METHOD IN THE FREE VIBRATION ANALYSIS OF A RECTANGULAR PLATE WITH ANY NUMBER OF POINT MASSES AND TRANSLATIONAL SPRINGS

Abstract

By means of the analytical-and-numerical-combined method (ANCM), the natural frequencies and the corresponding mode shapes of a uniform rectangular flat plate carrying any number of point masses and translational springs are determined. The boundary (supported) conditions of the plate and the magnitudes and locations of the concentrated elements are arbitrary. First, the closed form solution of the natural frequencies and the normal mode shapes of the “unconstrained” plate (without any concentrated elements attached) are obtained. Second, based on the closed form solution and using the mode superposition technique, the eigenvalue equation of the “constrained” plate (with any number of concentrated elements attached) is derived. Finally, the eigenvalue equation is solved numerically to give the desired natural frequencies and mode shapes of the “constrained” plate. Under the condition that the accuracies of the sought natural frequencies are approximately the same, the order of the eigenvalue equation derived from the ANCM is much lower than that derived from the traditional finite element method (FEM). Thus the CPU time required by the ANCM is much less than that required by the FEM. Furthermore, the ANCM is also superior to the pure analytical (closed form solution) method, since the former (ANCM) is available for free vibration analysis of a uniform rectangular flat plate carrying any number of concentrated elements, but the latter is usually practical only for single concentrated elements.