Dynamic recursive decoupling method for closed-loop flexible mechanical systems

In this paper, a new method for the dynamic analysis of a closed-loop flexible kinematic mechanical system is presented. The kinematic and force models are developed using absolute reference, joint relative, and elastic coordinates as well as joint reaction forces. This recursive formulation leads to a system of loosely coupled equations of motion. In a closed-loop kinematic chain, cuts are made at selected auxiliary joints in order to form spanning tree structures. Compatibility conditions and reaction force relationships at the auxiliary joints are adjoined to the equations of open-loop mechanical systems in order to form closed-loop dynamic equations. Using the sparse matrix structure of these equations and the fact that the joint reaction forces associated with elastic degrees of freedom do not represent independent variables, a method for decoupling the joint and elastic accelerations is developed. Unlike existing recursive formulations, this method does not require inverse or factorization of large non-linear matrices. It leads to small systems of equations whose dimensions are independent of the number of elastic degrees of freedom. The application of dynamic decoupling method in dynamic analysis of closed-loop deformable multibody systems is also discussed in this paper. The use of the numerical algorithm developed in this investigation is illustrated by a closed-loop flexible four-bar mechanism.