A symbolic approach for automatic generation of the equations of motion of multibody systems

This paper describes a collection of methods and procedures for the automatic generation of the equations of motion of multibody systems using general-purpose Computer Algebra Software. A brief review of existing symbolic multibody systems is given, and advantages and disadvantages of symbolic approaches compared with numerical ones are discussed. Then, a set of methods for symbolic modeling of multibody systems is explained. The first step of the modeling procedure consists of the description of the multibody system, by defining objects ( such as points, vectors, rigid bodies, forces and torques, special objects) and the relationships between them ( kinematic chains, constraints). The second step is the derivation of the equations of motion, which can be performed in a quasiautomatic way. A further step is the linearization of the equations and the calculation of the system's frequency response functions. By way of example, a dynamic model of the motorcycle is developed, obtaining the nonlinear equations of motion in a dependent coordinates' formulation. Next, the equations of motion are linearized and reduced to an independent formulation, reobtaining the well known Sharp's model of the straight running of the motorcycle. Root loci and frequency response functions are also calculated. This example demonstrates the power of the given symbolic procedures and shows how a model suitable for stability, handling and control analysis can be developed quickly and easily. The procedure described in this paper has been implemented in a Maple package called 'MBSymba', which is available on the web page www.dim.unipd.it/lot/mbsymba.html.