Comparison between variational and bilinear time finite element methods performance in nonlinear dynamical system analysis

The purpose of this thesis is to compare the performance of two finite element formulations when applied to nonlinear dynamic system analysis. One of these formulations is a new approach to the finite element method based on Hamilton's law of varying action. That approach will be called a bilinear approach or method in this thesis. Its performance will be compared against that of a widely used in nonlinear systems dynamics variational approach to the finite element method. This study evaluates the convergence characteristics and speed (CPU time) for these two approaches in application to such single and two-degree-of-freedom dynamical systems as mass-damper-spring systems and a helicopter blade modeled as a rigid body with flap and lead-lag degrees of freedom. It is shown that the bilinear approach is slower than the variational approach in terms of CPU time. On the other hand, the bilinear approach produces steady-state solutions more often than the variational approach, particularly for systems with strong nonlinearities. Also, the program based on the bilinear approach is easier to use because the only change necessary to analyze different dynamical systems is a modification of subroutines computing inertia, damping, stiffness and forcing function arrays. This is not the case for the computer implementation of the variational method. That approach requires defining separate linear and nonlinear mass, damping and stiffness matrices. That may be very difficult for complex dynamical systems.