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A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix

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Abstract

The constraint violation problem of spatial multibody systems is analyzed in this paper. The mass matrix is singular in the equations of motion when the Euler parameters with the normalization constraints are used to describe the orientation of the spatial rigid body. The constrained and weighted least-squares-based geometrical projection method is implemented to suppress the constraint violation during numerical integration, and the explicit correction formulation can be obtained by the block matrix inversion scheme. The mass matrix weighted correction formulation gives the physically consistent energy norm, but it needs the mass matrix to be positive definite. To extend the physically consistent correction formulation for solving spatial multibody systems’ constraint violation problems with a singular mass matrix, a Modified Mass-Orthogonal Projection Method (MMOPM) and a Generalized Physical Orthogonal Projection Method (GPOPM) are proposed. MMOPM modifies the mass matrix directly by adding a penalty factor matrix which appears in the mass-orthogonal projection method and leads to a positive definite weight matrix that satisfies the block matrix inversion scheme condition. GPOPM is a generalization of the physical orthogonal projection method where the constrained least-squares method is weighted by the positive semi-definite mass matrix and the correction formulation is given by using the generalized block matrix inversion scheme. Numerical results show the feasibility and accuracy of the presented MMOPM and GPOPM. The constraints in position and velocity can reach machine precision during numerical integration. The elimination of violation of position constrains may require few iterations, while the violation of velocity constraints is removed in one step, and GPOPM is more accurate in velocity correction than MMOPM.

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Acknowledgements

This work was supported by the Training Program of the Major Research Plan of the National Natural Science Foundation of China (Project No. 91016026) and National Natural Science Foundation of China (Project No. 11302023). The authors express thanks to anonymous referees for their valuable comments and suggestions, which led to a significant improvement over an early version of the paper.

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  1. Department of Engineering Mechanics, AML, Tsinghua University, Beijing, 100084, People’s Republic of China

    Jie Zhang & Yinghua Liu

  2. Department of Applied Mechanics, University of Science and Technology Beijing, Beijing, 100083, People’s Republic of China

    Donghuan Liu

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  1. Jie Zhang
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Correspondence to Yinghua Liu.

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Zhang, J., Liu, D. & Liu, Y. A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix. Multibody Syst Dyn 36, 87–110 (2016). https://doi.org/10.1007/s11044-015-9458-7

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