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The simulation of mechanical system dynamics by the Integrated Mechanisms Program (IMP) has depended on a unique time integration algorithm based on modal analysis of the linearized equations of motion and using variable time-steps based on monitoring force imbalance. Since this achieves time steps suited to the frequency of the response rather than to the highest natural frequency of the system modeled, this often results in orders of magnitude improvement in overall efficiency without instability or loss of accuracy. Experience has shown, however, that this approach does not behave robustly if the system simulated has two or more repeated eigenvalues. This happens for real mechanical system models, either instantaneously when passing through a critical geometry, or on a continuing basis such as with repeated zero eigenvalues for rigid-body modes. The primary goal of this research, therefore, was to investigate and cure this difficulty without loss of the advantages of the approach.
As theoretical background, methods of solution of the eigenvalue problem, including eigenvectors, is reviewed. Mechanical system examples having repeated eigenvalues are shown for which the dynamic equations can only be diagonalized if the system matrix is nondefective. For defective systems, it is shown where the dynamic equations cannot be uncoupled, but take on the Jordan canonical form.
No eigen-solution software could be found which solves for a complete set of eigenvectors in the presence of repeated eigenvalues. Therefore, after review of the theory, a more robust numerical algorithm was employed for obtaining eigenvectors, including generalized eigenvectors when necessary, and is presented in detail. Based on these, the modal response equations are extended to include the solution of systems with these characteristics.
To demonstrate the validity of the simulation technique developed in this research, numerical solutions are presented and compared to Runge-Kutta solutions for three mechanical system examples. These include a system with continuing repeated zero eigenvalues (a rigid mode), a non-linear system with repeated non-zero eigenvalues appearing at certain critical geometric positions, and a problem containing repeated zero and repeated non-zero eigenvalues. All are solved with good accuracy and high numerical efficiency.