Development of a Modal Solution Library for the Transverse Vibration of a Thick Beam

Abstract

Abstract The authors apply a method of obtaining the modal solutions for the free lateral vibrations of a thick rectangular orthotropic plate where through-the-thickness shear and rotary inertia are important. The approach (Ayorinde, 1995) utilizes optimized modes and is based on the application of the Rayleigh method to the Timoshenko beam. The plate solution was synthesized from those of beams in the two transverse directions. With as few as three optimized modes this method is of comparable accuracy with other methods employing 36 (six per direction) modes. and thus presents a significant advantage. Nevertheless, obtaining the thick beam frequencies involves the numerical solution of complicated hyperbolic equations, which takes much more computer time than for a thin beam. There are many important procedures that require the results of this “forward” problem. Examples are the material identification and damage characterization of thick plates. Since such procedures could utilize literally thousands of trial solutions of the forward problem, methods of significantly expediting the forward solution are of great importance. In this work an intensive investigation has been made of tile parameter space of elastic constants and geometries for thick plates for the behavior of the fundamental (2.2) mode, which is usually the most important structural mode, obtaining definitive guidance that the feasible approach is the setting up of a solution library for a thick beam rather than for a plate directly. The thick beam solution library is actually developed and an interpolation algorithm set up. The library is utilized for the forward plate problem. Results with and without the library are compared, and they confirm the accuracy and significant time-economy of the beam solution library approach.