Dynamics of Multibody Systems Using Virtual Work and Symbolic Programming

Two different Maple programs have been developed to generate automatically the symbolic kinematic and dynamic equations for rigid and flexible multibody systems, given only a description of the system as input. Kinematic equations are generated using graph-theoretic methods in one program, and by a recursive formulation in the second. Virtual work and virtual power methods are used to develop the dynamic equations in terms of joint coordinates. These dynamic equations are reduced to a minimal set, i.e., one equation per system degree of freedom, by using an orthogonal complement based on partitioning of virtual displacements or speeds. The symbolic dynamic equations are easily solved for the actuator loads in an inverse dynamic analysis, especially if the actuated joint coordinates are selected as independent variables. This orthogonal complement approach also offers certain advantages for the forward dynamic simulation of rigid and flexible systems; these advantages are outlined in the first example. In the second example, the inverse dynamics problem is solved for the Gough-Stewart platform, a complex example of a parallel manipulator, and compared to previous results in the literature.