Direct integration of the equations of multibody dynamics using central differences and linearization

Abstract

A methodology for integrating rigid body dynamics for the analysis of multibody systems is presented. The novelty lies in the fact that the equation system is solved directly by means of central differences as a second-order integration method. To obtain the best achievable convergence, the equilibrium is solved iteratively by the exact Newton method. Thus, it is possible to achieve the system solution directly without having to reduce the differential order. This decreases the number of unknowns. In return, it is necessary to linearize the equations. The rotation of each element is described by parameterization under a unit quaternion. In this paper the necessary developments for the modelization of the spherical and rotational joints are included. The constraints imposed by these joints, as well as the quaternion norm, are introduced into the model through a null space matrix. The reactions produced by these constraints are also eliminated from the system by using null space. Several examples are analyzed through the implementation of the methodology in Octave. The accuracy of the method is verified with results obtained from commercial software. The examples include benchmark problems.