Divide-and-conquer-based approach for the reaction uniqueness analysis in overconstrained multibody systems

For rigid multibody systems with redundant constraints, mathematical modeling and physical interpretation of the obtained results are impeded due to the nonuniqueness of the calculated reactions, which—in the case of load-dependent joint friction—may additionally lead to unrealistic simulated motion. It makes the uniqueness analysis crucial for assessing the fidelity of the results. The developed methods so far for the uniqueness examination—based on the modified mobility equation, the constraint matrix, or the free-body diagram—are not well suited for multibody systems described by relative coordinates. The novel method discussed in this paper breaks this limitation. The proposed approach is based on the divide-and-conquer algorithm (DCA)—a low-order recursive method for dynamic simulations of complex multibody systems. The devised method may be used for checking the joint-reaction uniqueness of holonomic systems with ideal constraints that fulfill some additional assumptions. The reaction-uniqueness analysis is performed when the main pass of the DCA is completed. An eight-step algorithm is proposed. In the case of the single-joint connections, it is sufficient to study the appropriate equations of motion. However, if the multijoint connection is present, then one of the numerical methods—known from the constraint-matrix-based or the free-body-diagram-based approach—has to be used, namely the rank-comparison, QR-decomposition, SVD, or nullspace methods; all of these approaches are discussed. To illustrate the devised method, a spatial parallelogram mechanism with a triple pendulum is analyzed.