Consistent mass Matrix for Distributed Mass Systems

A technique is developed for constructing a consistent mass matrix for vibration analysis of general structural configurations. The approach is consistent with standard procedures for stiffness matrix formulation of structural problems, but accounts for the actual mass distribution within the structure in a manner similar to the Rayleigh-Ritz formulation. The basic equation for computing the mass matrix coefficients is given a physical interpretation. It is shown that the natural frequencies obtained by using the consistent mass matrix are upper bounds to the exact solution. The procedure is applicable to general dynamic response analysis and is demonstrably superior to the usual procedure of physical mass lumping by application to frequency analysis of free-free and simply-supported prismatic beams with uniformly distributed mass. It is observed that if lateral translation coordinates alone are used to represent the distortion of a uniform beam, the number of individual beam segments must exceed by one the number of natural mode frequencies it is desired to closely approximate.