Systematic computer-aided analysis of dynamic systems

In this paper an automated numerical-symbolical analysis concept for dynamic systems in engineering mechanics is outlined. Besides the computerized generation of symbolic equations of motion, the subsequent analysis is also performed by means of computer algebra in combination with well-established numerical methods. Methodology The first step in the analysis is to replace the considered real-world system by a simplified mechanical system and to determine relevante parameter ranges. The derivation of a mathematical model is the next step. For multibody systems, this task can be symbolically performed by the special-purpose symbol manipulation package N EWEUL, [1]. Via MAPLE the obtained second order system can easily be transformed into the standard state space representation of dynamic systems. It follows the numerical determination of periodic solutions. In order to analyze the stability of motion a local discrete version of the continuous mathematical model has to be derived in the neighborhood of the periodic solution to be examined. This local approximation of a Poincarti Map is generated by a combined numerical-symbolical procedure, which results in numerical coefficients of a truncated Taylor Series expansion. The st ability analysis simplifies to the evaluation of the stabil-it y of fixed points of the map. By means of an iterative procedure, an approximation for the bifurcation parameter value can be calculated on the basis of the linear map. Linear stability analysis, however, fails for non-linear systems with critical eigenvalues, i.e. for which a bifurcation takes place. Permission to copy without fee all or part of this material is granted provided that the copiaa are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is givan that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to rspublish, requires a fee and/or specific permission. It can be shown that for this case the system can be reduced to a low-dimensional critical subsystem without loosing information about the stability properties. This reduction process is performed by the computer algebra package MAPLE. Approximation of the Poincar6 Map For studying periodic motions it is advantageous to replace the ordinary differential equation by a discretized map: t~+l = g(~~). The stability is then determined by the stability of the fixed point of the map. In order to apply analytical procedures, a closed symbolic formulation is necessary. Therefore, a local approximation for g(~) is …