This research project deals with the dynamics and control of complex mechanical systems, and addresses the order reduction problem associated with a large set of ordinary differential equations having periodic coefficients. A new approach in the design of controllers for large-scale general dynamic systems with periodically varying coefficients is suggested. The idea is based on the fact that all quasi-linear periodic systems can be replaced by similar systems whose linear parts are time invariant via the well-known Liapunov-Floquet (LF) transformations. A general technique for computation of the LF matrices for large-scale systems is presented. Hence, the control problem associated with linear systems with periodically varying coefficients is transformed to a time-invariant form. The time invariant equations are then used to develop control strategies for the time variant systems. Next, the problem of controller design for nonlinear systems with periodically varying parameters is addressed. It is proposed that the `time-dependent normal form` theory be used to simplify the quasi-linear systems resulting from the L-F transformations. For the non-critical cases (where the real parts of the eigenvalues of the linear part of the system are nonzero), the designs can be achieved via normal form transformations. In the event when linear part contains a critical uncontrollable mode, the control system design is pursued through an application of the `center manifold' theory and suitable Liapunov control functions. The practical significance of this study is demonstrated through applications to some typical problems including the controller designs for an asymmetric magnetic rotor-bearing system, columns and plates subjected to periodic loads. Large models using finite elements are constructed for these problems and the control algorithm are implemented on the reduced order systems.
