High Precision Methods for Solving Multi-Body Systems With Close Loop Constraints

Zhao Tianjiao; Qi Zhaohui; Wang Tianyu; Xu Jinshuai

doi:10.21656/1000-0887.450230

Volume 47 Issue 8

Aug. 2026

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Article Navigation > Applied Mathematics and Mechanics > 2026 > 47(8): 1068-1085

Zhao Tianjiao, Qi Zhaohui, Wang Tianyu, Xu Jinshuai. High Precision Methods for Solving Multi-Body Systems With Close Loop Constraints[J]. Applied Mathematics and Mechanics, 2026, 47(8): 1068-1085. doi: 10.21656/1000-0887.450230

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High Precision Methods for Solving Multi-Body Systems With Close Loop Constraints

doi: 10.21656/1000-0887.450230

1.
Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian, Liaoning 116024, P.R. China
2.
Dalian University of Technology, Dalian, Liaoning 116024, P.R. China

Received Date: 2024-08-14
Rev Recd Date: 2025-02-11
Publish Date: 2026-08-01

Abstract

Abstract

The dynamics equations for multi-body systems with closed-loop constraints are a set of differential algebraic mixed equations. The main difficulties in their numerical solution are as follow: it is difficult to satisfy the displacement, velocity and acceleration constraint equations with high precision at the same time, which will cause large defaults with the accumulation of errors; there are often redundant constraints and singular configuration problems, which seriously affect the numerical behavior of the solution process and the accuracy of the results. Herein, a numerical method for solving singular configuration closed-loop multibody systems was presented, without the traditional constraint independence assumption. The correction of position and velocity were done to satisfy the constraint equations before the motion equation formulation, rather than the correction of the constraint default at the end of each integration step. This method works with any standard ODE solver. According to the geometric characteristics of constraints, the components of velocity and acceleration in the hypersurface constraint space were all determined by constraints. The orthogonal basis vectors of the constrained tangent space were obtained through QR decomposition of the column permutation. The relationship between the generalized velocity of the system and the independent generalized velocity was established, and the default values of the generalized velocity and the generalized acceleration were filtered out in the dynamic calculation process, and the differential algebraic mixed equation was transformed into a pure differential equation satisfying the velocity and acceleration constraints (with the number of equations equal to the generalized coordinate number of the system). Then, the orthogonal projection default correction method was applied to the system, and the default magnitude of the system position and speed was effectively suppressed by several iterations. At the same time, according to the principle of virtual power equivalence, the physical significance of the Laplace multiplier in solving the constraint reaction torque of the cut hinge was further revealed. Numerical examples show that, the precision of displacement, velocity and acceleration constraints of the proposed method is higher than that of the traditional augmented method and the penalty function method. The proposed method can deal with redundant constraints and singular configuration problems, and can be programmed to solve multi-body systems with closed-loop constraints with high precision. The proposed method can identify the independent constraints dynamically and modify the motion equation accordingly. Numerical examples demonstrate the effectiveness of the proposed method.
- closed-loop constraint,
- multi-body system,
- mixed differential algebraic equation,
- joint constraint force,
- correction of default

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References

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