Mathematical observations in structural dynamics

Abstract

In dynamic structural analysis, the basic relations between forces and displacements for a beam element subjected to axial, torsional or flexural vibration are obtained either by solving the appropriate equation of motion or by using an approximate method. The exact equation leads to the dynamic stiffness matrix while the approximate method results in the superposition of elastic and inertial forces represented respectively by the stiffness and mass matrices. The common procedure in finding the natural frequencies is to set the determinant of the dynamic stiffness matrix for the system equal to zero. The approximate method leads to an eigenvalue type problem while the exact method results in a transcendental equation of trigonometric and hyperbolic functions. The natural frequencies in a region of interest are found by a systematic search in the determinntal function. The purpose of this paper is to show that the search technique cannot be applied for certain values of the argument at which the determinantal function is not defined. It is proved that the natural frequencies of any isolated member in the system are critical values for the determinantal function. A practical method is given to obviate the difficulty in order to find the natural frequencies from the determinant, including the critical values at which the dynamic stiffness matrix is not defined. Also, as part of this investigation, the mathematical relation is established between the dynamic stiffness matrix derived by the approximate finite element method and the results obtained from the exact Bernoulli-Euler equation for flexural vibration or the wave equation for axial or torsional vibration.