Multibody elastodynamic modeling of parallel manipulators based on the Lagrangian equations without Lagrangian multipliers

Due to the complexity of the multi constraint and multi closed-loop of parallel mechanisms (PMs), their dynamic equations require additional Lagrangian multipliers and constraint equations, which is unfavorable for solving the dynamic equations. In this paper, a multibody elastodynamic modeling of PMs based on the Lagrangian equations is proposed to fix this issue. First, we decompose the closed-loop PMs into open-loop ones by cutting open at the joints, and we propose the subassembly element that considers the flexible links and joints to avoid the singularity of the traditional constraint equations. Second, we extract the nonsingular independent coordinates for these nodes that are constrained by joints and rigid element through multi-point constraint theory and singularity analysis, and then summarized to establish the global independent generalized displacement coordinates (IGDCs). Third, we formulate the Lagrangian equations through the global IGDCs by closing the open-loop of the mechanism. The Guyuan, improved reduction system, subspace iteration method, and precision integration method are respectively presented to analyze the natural frequencies and dynamic response of the mechanism. The proposed method is computationally efficient because the computation concerning the Lagrangian multipliers are not required. Finally, the 3PRRR PM is presented to implement the proposed method.