A parametric generalized coordinate formulation for mechanical joint elements in dynamic systems

Abstract

The efficient and easy computational implementation of multibody dynamic formulations is an important issue. In this paper, a parametric generalized coordinate formulation is proposed as a new approach to formulate joint constraint equations. By introducing the parametric generalized coordinates in the constraint equations, the complexity of the equations is significantly reduced. The number of arithmetic operations required to compute the derivatives of the constraints is drastically decreased depending on the type of joint involved (i.e. especially when the proposed method is applied to joints having lower degrees of freedom). Furthermore, the second derivatives of the constraint equations tend to have a large portion of zero sub-matrices, which simplifies the system matrix of the governing equations and allows for application of efficient sparse matrix techniques. Although the proposed approach has the drawback of a larger set of coordinates than a conventional Cartesian coordinate approach resulting in slightly less efficient computation, a systematic and easy formulation is achieved, which may relieve the implementation complexity for program developers. The dynamic analysis of a multibody slider-crank system is carried out to demonstrate the validity of the proposed formulation.