Frequency-Dependent Element Mass Matrices

This paper considers some of the theoretical aspects of the formulation of frequency-dependent structural matrices. Two types of mass matrices are examined, the consistent mass matrix found by integrating frequency-dependent shape functions, and the mixed mass matrix found by integrating a frequency-dependent shape function against a static shape function. The coefficients in the power series expansion for the consistent mass matrix are found to be determinable from those in the expansion for the mixed mass matrix by multiplication by the appropriate constant. Both of the mass matrices are related in a similar manner to the coefficients in the frequency-dependent stiffness matrix expansion. A formulation is derived which allows one to calculate, using a shape function truncated at a given order, the mass matrix expansion truncated at twice that order. That is the terms for either of the two mass matrix expansions of order 2n are shown to be expressible using shape functions terms of order n. Finally, the terms in the matrix expansions are given by formulas which depend only on the values of the shape function terms at the boundary.