From frequency-dependent mass and stiffness matrices to the dynamic response of elastic systems

More than three decades ago, Przemieniecki introduced a formulation for the free vibration analysis of bar and beam elements based on a power series of frequencies. In the present paper, the authors generalize this formulation for the analysis of the dynamic response of elastic systems submitted to arbitrary nodal loads as well as initial displacements. Based on the mode-superposition method, a set of coupled, higher-order di€erential equations of motion is transformed into a set of uncoupled second-order di€erential equations, which may be integrated by means of standard procedures. Motivation for this theoretical achievement is the hybrid boundary element method, which has been developed by the authors for time-dependent as well as frequency-dependent problems. This formulation, as a generalization of PianÕs previous achievements for ®nite elements, yields a sti€ness matrix for which only boundary integrals are required, for arbitrary domain shapes and any number of degrees of freedom. The use of higher-order frequency terms drastically improves numerical accuracy. The introduced modal assessment of the dynamic problem is applicable to any kind of ®nite element for which a generalized sti€ness matrix is available. Some academic examples illustrate the theory