A Jacobian formulation for efficient simulation of multibody chain dynamics

This research proposes an efficient analysis method with an implicit integration method for multibody chain dynamics. Absolute Cartesian coordinates which are related to the body reference frame are used as generalized coordinates. Contact between a sprocket and chain links is formulated as a circle to circle contact. Chain links are connected by bushing elements. Newton-Raphson method is used to solve the combined equations of motion, constraints, and implicit integration formulas. Jacobian matrix must be calculated and LU decomposed with the implicit integration method in Newton-Raphson method. Since Cartesian coordinates with respect to the body reference frame are used as the generalized coordinates and the contact and bushing elements exist between two bodies, the Jacobian matrix for these elements depended on only the relative motion of two adjacent bodies. Therefore, the Jacobian matrix hardly changes unless the relative motion changes significantly. This research investigates the Jacobian matrix to find small-changing and large-changing submatrices. Investigation showed that 65.4 % and 34.6 % of the Jacobian matrix belong to the small changing and the large changing submatrices, respectively. The small changing submatrices are calculated and LU decomposed only once at the initial time and reused at every time step afterwards. The large changing submatrices are updated and LU decomposed at every time step. As the sprocket rotates, the generalized coordinate sequence of the Jacobian matrix for the links is shifted one by one to keep the Jacobian matrix change of the small change submatrices small until the end of simulation time. The proposed method is implemented and its results are compared to those obtained from the original method which updates and LU decomposes the whole Jacobian matrix at every time step. Numerical experiments showed that the proposed method is about 11 times faster and yields almost identical results to the original method.