High Precision Methods for Solving Multi-Body Systems With Close Loop Constraints
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摘要: 含闭环约束多体系统的动力学方程是一组微分代数混合方程，它在数值求解过程中的主要难点有：①难以同时较高精度满足位移、速度、加速度约束方程，会随着误差的积累引起较大的违约；②经常存在冗余约束以及奇异构型问题，严重影响求解过程的数值性态和结果的准确性. 本文提出了一种可处理奇异构型闭环多体系统的数值求解方法，该方法摒弃了传统的约束独立性假设，在形成运动方程之前，对位置和速度进行修正，使其满足约束方程，而不是在每个积分步骤结束时纠正约束违约. 并且根据约束的几何特性可知，速度和加速度在超曲面约束法空间中的分量全部由约束决定. 通过列置换的QR分解得到约束法切空间的正交基向量，建立起系统广义速度与独立广义速度之间的关系，进而将微分代数混合方程转换为满足速度与加速度约束的纯微分方程(方程个数等于系统的广义坐标数)，这种方法可以与任何标准ODE求解器协作. 然后，将正交投影违约修正的方法应用到系统中，通过几次迭代有效地抑制系统位置和速度的违约量. 同时，依据虚功率等效原理，进一步揭示了Lagrange乘子在求解切断铰约束反力和约束反力矩中的重要物理意义. 本文所提方法可以动态识别独立约束，处理冗余约束和奇异构型问题，可高精度求解含闭环约束多体系统. 最后，通过几个数值算例验证了方法的有效性.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.
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图 2 约束生成的切空间和法空间
Figure 2. Tangent space St and normal space Sn resulting from constraints
图 6 求解铰的约束反力的拆解过程
Figure 6. The process of obtaining joint reaction forces in the derived tree-like system
图 9 滑块的违约量变化曲线(LM+Baumg.stab, ALF)
Figure 9. The curves of the slider's defaults(LM+Baumg.stab, ALF)
图 11 阻止铰点沿Y方向移动的约束反力
Figure 11. Constraint reactions to prevent joint point from moving along the Y direction
图 14 初值$\alpha_{1}=\pi / 2$时得到的C点x坐标的变化及其与解析解的差值
Figure 14. The x coordinates of point C and their differences with respect to the exact solutions with initial position $\alpha_{1}=\pi / 2$
图 15 初值$\alpha_{1}=\pi / 2$时得到的C点y坐标的变化及其与解析解的差值
Figure 15. The y coordinates of point C and their differences with respect to the exact solutions with initial position $\alpha_{1}=\pi / 2$
图 20 不同吊载作用下臂节3端部形心竖向位移及其振动频率的变化
Figure 20. Vertical displacements of the tip and their vibration frequencies for the 3rd boom under the actions of different loads
表 1 切断铰的约束
Table 1. Constraints of classical cut joints
revolute joint universal joint spherical joint prismatic joint cylinder joint rotational constraint α2=α3=0 α3=0 α1=α2=α3=0 α2=α3=0 translational constraint s1=s2=s3=0 s1=s2=s3=0 s1=s2=s3=0 s2=s3=0 s2=s3=0 
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表 2 不同分解方法的耗时(单位: s)
Table 2. Time costs of different decomposition methods (unit: s)
LU RREF pseudo inverse QR decomposition matrix dimension 4.57E-6 1.73E-3 6.24E-5 4.28E-6 20×20 1.12E-5 6.63E-3 2.10E-4 1.32E-5 40×40 2.73E-5 1.43E-2 8.45E-4 3.06E-5 60×60 2.54E-4 3.94E-2 1.49E-3 1.27E-4 100×100
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表 3 Lagrange乘子表示的切断铰约束反力
Table 3. Reaction forces of classical cut-joints expressed in terms of Lagrange multipliers
revolute joint universal joint spherical joint prismatic joint cylinder joint m2 $\lambda_2 \boldsymbol{h}_2^2-\lambda_3 \boldsymbol{h}_2^3$ $\lambda_3\left(\boldsymbol{h}_2^2 \times \boldsymbol{h}_1^1\right)$ 0 $\lambda_2 \boldsymbol{h}_1^2-\lambda_1 \boldsymbol{h}_1^1-\lambda_3 \boldsymbol{h}_1^3$ $\lambda_2 \boldsymbol{h}_2^2-\lambda_3 \boldsymbol{h}_2^3$ f2 $\lambda_4 \boldsymbol{h}_1^1+\lambda_5 \boldsymbol{h}_1^2+\lambda_6 \boldsymbol{h}_1^3$ $\lambda_4 \boldsymbol{h}_1^1+\lambda_5 \boldsymbol{h}_1^2+\lambda_6 \boldsymbol{h}_1^3$ $\lambda_4 \boldsymbol{h}_1^1+\lambda_5 \boldsymbol{h}_1^2+\lambda_6 \boldsymbol{h}_1^3$ $\lambda_5 \boldsymbol{h}_1^2+\lambda_6 \boldsymbol{h}_1^3$ $\lambda_5 \boldsymbol{h}_1^2+\lambda_6 \boldsymbol{h}_1^3$
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表 4 曲柄滑块机构的仿真参数
Table 4. Simulation parameters of the crank-slider mechanism
crankshaft length/m connecting rod length/m crankshaft mass/kg connecting rod mass/kg slider mass/kg 1 1 1 1 1
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表 5 不同数值解法的仿真结果对比
Table 5. Validities of different numerical methods to simulate the 4-bar linkage
numerical solution method initial value >numerical solution method initial value >numerical solution method initial value $\alpha_{1}=\pi / 2$ α1=0 $\alpha_{1}=\pi / 2$ α1=0 $\alpha_{1}=\pi / 2$ α1=0 LM2 diverge diverge ALF+Coord.part diverge diverge MPI(GS) diverge diverge LM2+Baumg.stab diverge diverge MPI(SVD) diverge diverge MPI(GS)+Baumg.stab diverge diverge LM2+Coord.part diverge diverge MPI(SVD)+Baumg.stab diverge diverge MPI(GS)+Coord.part diverge diverge ALF diverge diverge MPI(SVD)+Coord.part converge converge this paper converge converge
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[1] 刘延柱, 潘振宽, 戈新生. 多体系统动力学[M]. 2版. 北京: 高等教育出版社, 2014.Liu Yanzhu, Pan Zhenkuan, Ge Xinsheng. Dynamics of Multibody Systems[M]. 2nd ed. Beijing: Higher Education Press, 2014. (in Chinese) [2] 富立. 非光滑多体系统动力学LCP方法[M]. 北京: 清华大学出版社, 2016.Fu Li. Dynamics of Non-Smooth Multibody Systems: LCP Method[M]. Beijing: Tsinghua University Press, 2016. (in Chinese) [3] 齐朝晖. 多体系统动力学[M]. 北京: 科学出版社, 2008.Qi Zhaohui. Dynamics of Multibody Systems[M]. Beijing: Science Press, 2008. (in Chinese) [4] Blajer W, Kołodziejczyk K. A geometric approach to solving problems of control constraints: theory and a DAE framework[J]. Multibody System Dynamics, 2004, 11 (4): 343-364. doi: 10.1023/B:MUBO.0000040800.40045.51 [5] Pappalardo C M, Guida D. On the computational methods for solving the differential-algebraic equations of motion of multibody systems[J]. Machines, 2018, 6 (2): 20. doi: 10.3390/machines6020020 [6] Bottasso C L, Dopico D, Trainelli L. On the optimal scaling of index three DAEs in multibody dynamics[J]. Multibody System Dynamics, 2008, 19 (1): 3-20. [7] Sun W. Numerical algorithms for differential-algebraic equations of multibody dynamics[C]// 2016 16 th International Conference on Control, Automation and Systems (ICCAS). Gyeongju, Korea (South), 2016: 786-791. [8] Hussein B A, Shabana A A. Sparse matrix implicit numerical integration of the stiff differential/algebraic equations: implementation[J]. Nonlinear Dynamics, 2011, 65 (4): 369-382. doi: 10.1007/s11071-010-9898-9 [9] Hu J, Wang T. An efficient high-precision recursive dynamic algorithm for closed-loop multibody systems[J]. International Journal for Numerical Methods in Engineering, 2019, 118 (4): 181-208. doi: 10.1002/nme.6007 [10] 赵维加, 潘振宽, 王艺兵. 多体系统动力学微分/代数方程约束误差小扰动自我稳定方法[J]. 应用数学和力学, 2000, 21 (1): 94-98.Zhao Weijia, Pan Zhenkuan, Wang Yibing. An automatic constraint violation stabilization method for differential/algebraic equations of motion multibody system dynamics[J]. Applied Mathematics and Mechanics, 2000, 21 (1): 94-98. (in Chinese) [11] 刘颖, 马建敏, 苏芳, 等. 多体系统动力学方程的无违约数值计算方法[J]. 计算力学学报, 2010, 27 (5): 942-947.Liu Ying, Ma Jianmin, Su Fang, et al. Precise numerical solution for multi-body system's equations of motion based on algorithm without constraint violation[J]. Chinese Journal of Computational Mechanics, 2010, 27 (5): 942-947. (in Chinese) [12] 付士慧, 王琪. 多体系统动力学方程违约修正的数值计算方法[J]. 计算力学学报, 2007, 24 (1): 44-49.Fu Shihui, Wang Qi. A numerical method for constraint stabilization of dynamic equations of multi-body systems[J]. Chinese Journal of Computational Mechanics, 2007, 24 (1): 44-49. (in Chinese) [13] Kikuuwe R, Brogliato B. A new representation of systems with frictional unilateral constraints and its Baumgarte-like relaxation[J]. Multibody System Dynamics, 2017, 39 (3): 267-290. doi: 10.1007/s11044-015-9491-6 [14] Pogorelov D. Differential-algebraic equations in multibody system modeling[J]. Numerical Algorithms, 1998, 19 (1): 183-194. [15] 潘振宽, 赵维加, 洪嘉振, 等. 多体系统动力学微分/代数方程组数值方法[J]. 青岛大学学报, 1996, 9 (1): 83-96.Pan Zhenkuan, Zhao Weijia, Hong Jiazhen, et al. Numerical algorithms for differential/algebraic equations of motion of multibody system[J]. Journal of Qingdao University, 1996, 9 (1): 83-96. (in Chinese) [16] Pennestrì E, Vita L. Strategies for the numerical integration of DAE systems in multibody dynamics[J]. Computer Applications in Engineering Education, 2004, 12 (2): 106-116. doi: 10.1002/cae.20005 [17] Bayo E, Avello A. Singularity-free augmented Lagrangian algorithms for constrained multibody dynamics[J]. Nonlinear Dynamics, 1994, 5 (2): 209-231. doi: 10.1007/BF00045677 [18] Uchida T, Mcphee J. Triangularizing kinematic constraint equations using Gr bner bases for real-time dynamic simulation[J]. Multibody System Dynamics, 2011, 25 (3): 335-356. doi: 10.1007/s11044-010-9241-8 [19] Blajer W, Schiehlen W, Schirm W. A projective criterion to the coordinate partitioning method for multibody dynamics[J]. Archive of Applied Mechanics, 1994, 64 (2): 86-98. doi: 10.1007/BF00789100 [20] Carpinelli M, Gubitosa M, Mundo D, et al. Automated independent coordinates' switching for the solution of stiff DAEs with the linearly implicit Euler method[J]. Multibody System Dynamics, 2016, 36 (1): 67-85. doi: 10.1007/s11044-015-9455-x [21] Wehage K T, Wehage R A, Ravani B. Generalized coordinate partitioning for complex mechanisms based on kinematic substructuring[J]. Mechanism and Machine Theory, 2015, 92: 464-483. doi: 10.1016/j.mechmachtheory.2015.06.006 [22] Bauchau O A, Laulusa A. Review of contemporary approaches for constraint enforcement in multibody systems[J]. Journal of Computational and Nonlinear Dynamics, 2008, 3 (1): 011005. doi: 10.1115/1.2803258 [23] Kurdila A J, Junkins J L, Hsu S. Lyapunov stable penalty methods for imposing holonomic constraints in multibody system dynamics[J]. Nonlinear Dynamics, 1993, 4 (1): 51-58. doi: 10.1007/BF00047121 [24] Yu Q, Chen I M. A direct violation correction method in numerical simulation of constrained multibody systems[J]. Computational Mechanics, 2000, 26 (1): 52-57. doi: 10.1007/s004660000149 [25] Blajer W. A geometric unification of constrained system dynamics[J]. Multibody System Dynamics, 1997, 1 (1): 3-21. doi: 10.1023/A:1009759106323 [26] Zhang J, Liu D, Liu Y. A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix[J]. Multibody System Dynamics, 2016, 36 (1): 87-110. doi: 10.1007/s11044-015-9458-7 [27] Eich E. Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints[J]. SIAM Journal on Numerical Analysis, 1993, 30 (5): 1467-1482. doi: 10.1137/0730076 [28] Yoon S, Howe R M, Greenwood D T. Geometric elimination of constraint violations in numerical simulation of Lagrangian equations[J]. Journal of Mechanical Design, 1994, 116 (4): 1058-1064. doi: 10.1115/1.2919487 [29] Blajer W. Elimination of constraint violation and accuracy aspects in numerical simulation of multibody systems[J]. Multibody System Dynamics, 2002, 7: 265-284. doi: 10.1023/A:1015285428885 [30] Aghili F, Piedbceuf J C. Simulation of motion of constrained multibody systems based on projection operator[J]. Multibody System Dynamics, 2003, 10 (1): 3-16. doi: 10.1023/A:1024584323751 [31] Marques F, Souto A P, Flores P. On the constraints violation in forward dynamics of multibody systems[J]. Multibody System Dynamics, 2017, 39 (4): 385-419. doi: 10.1007/s11044-016-9530-y [32] Bayo E, Garcia De Jalon J, Serna M A. A modified Lagrangian formulation for the dynamic analysis of constrained mechanical systems[J]. Computer Methods in Applied Mechanics and Engineering, 1988, 71 (2): 183-195. doi: 10.1016/0045-7825(88)90085-0 [33] Blajer W. Augmented Lagrangian formulation: geometrical interpretation and application to systems with singularities and redundancy[J]. Multibody System Dynamics, 2002, 8 (2): 141-159. doi: 10.1023/A:1019581227898 [34] Zhang J, Liu D, Liu Y. A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix[J]. Multibody System Dynamics, 2016, 36 (1): 87-110. doi: 10.1007/s11044-015-9458-7 [35] Neto M A, Ambrósio J. Stabilization methods for the integration of DAE in the presence of redundant constraints[J]. Multibody System Dynamics, 2003, 10: 81-105. doi: 10.1023/A:1024567523268 -
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