Comparative analysis of methods of getting inconsistent mass matrices
Engineering Education # 12, December 2013
Several various methods could be applied for description of mass-inertia properties of the finite element; each method leads to a different mass matrix. In particular, one could use the method of lumped masses, various shape functions both matching with (consistent) or distinct from shape functions for a stiffness matrix. Other approaches are also possible. The purpose of this work was to analyze some non-trivial approaches to determination of a mass matrix, such as non-standard shape functions, oscillation modes of a bar element, a method of combining different mass matrices with weighting factors. The article compares five different mass matrices by the example of a bar finite element and identifies inherent errors of these matrices obtained during determination of eigenfrequencies of the fixed bar. A system of equations for the finite element method for all mass matrices was solved analytically in the trigonometric series defined at the nodal points. It was shown that calculation of eigenfrequency based on inconsistent mass matrices could provide higher accuracy in comparison with diagonal and consistent matrices.