Time Domain Finite Element Analysis of Dynamic Systems

Abstract

In this study, dynamic analyses of systems are conducted using the time domain finite element analysis when initial and final conditions are specified. Dynamics of systems can be analyzed using the time-marching procedure based on Hamilton's weak principle. A new method is developed to describe the motion of a system and is extended to accommodate the initial displacement as a state variable. System matrices are constructed to evaluate time histories of the system from initial time to a specified time without the information on the velocity and momentum at each time step. Galerkin's weak principle is used to construct element matrices in the time domain and assemble the whole matrices. The Neumann boundary condition is modified to reflect the effect of initial displacement and velocity that are inherent to dynamic problems through the momentum conservation of the system. The matrices obtained thereafter conserve the symmetricity and enable us to reduce the model order by the systematic approach. Spatial propagation equation is built up, and two-point boundary conditions are used to estimate the unknown initial conditions at one end of the beam. Modal domain analysis is introduced to reduce the size of matrices. It can be seen that the matrices constructed using the time domain finite element analysis are applicable to the model reduction that is analogous to the space-domain model reduction methodology. Several numerical examples show that a consequence of the suggested method can be applied to describe the motion of various dynamic systems successfully and to validate the effect of the model reduction in the time domain.