Multibody Systems Containing Active Elements: Generation of Linearized System Equations, System Analysis and Order-Reduction

Abstract

The nucleus of a general purpose computer program for the dynamics of vehicle-guideway-interaction is an algorithm which automatically generates the linearized equations of motion of arbitrary configured multibody systems connected by linear elements. Every body of the system can be rigid or flexible and arbitrary translational and rotational modes of motion can be allowed. Also, kinematic constraints are used for reducing the system order to the minimal number of (independent) degrees of freedom (DOF).

Besides of the usual basic coupling elements as linear spring-damper Systems (in parallel or in series), a broad class of more general elements can be modelled: dynamic element including active control devices (sensors, actuators and (dynamic) feedback).

The resulting (state-) equations can be evaluated directly with respect to a number of important qualitative and structural system properties such as stability, controllability, observability and transfer functions. It is recommended to base all computations on the eigenvalues and -vectors of the system matrix F.

The results also give the basic information for the necessary order-reduction in order to obtain managable models for system design as well as for routine simulation purposes. After comparing several existing methods for order reduction with respect to their feasibility our recommendation for dynamic condensation can be summarized as follows:

• i) for weak coupling and/or fast decaying eigenmodes to discard the associated transient dynamics, and

• ii) for strong coupling to use a condensation procedure based on the second-order multibody equations [approximate generalized kinematic constraints).

The main advantages are the correct reproduction of the steady-state values, the possibility of adaptation of the kept eigenvalues, as well as the close preservation of the physical structure of the models in connection with relative numerical simplicity.

The use of the prescribed procedures of model-synthesis, -analysis and -order-reduction is demonstrated on a typical levitated ground vehicle model (electrodynamic repulsion system), where the order can be reduced from more than 70 down to about 10 (!) states without considerable deterioration of accuracy.