Two-loop Implicit Integration Method Based on Backward Differential Formulation for Differential-algebraic Equations of Multibody System Dynamics

doi:10.3901/JME.2016.07.079

Abstract

Abstract: An iterative method is needed when implicit integration method is used to integrate the independent coordinates of differential-algebraic equations (DAEs) which come from multibody system dynamics. If the iterative method is Newton’s method, numerical differentiation is needed to obtain the Jacobian matrix. A fixed-point iterative method without the using of the Jacobian matrix can simplify the progress. The nonlinear algebraic constraint equations in DAEs are also solved by iteration to obtain the dependent coordinates. A two-loop structure is designed to manage those two iterations. The numerical accuracy of the integration method influences the numbers of iteration of implicit integration method which is called as the outer loop. Backward differential formulation (BDF) is introduced as the integration method into the two-loop method. New strategy is proposed to constitute the new two-loop method. The numbers of iteration of outer loop in the new two-loop method are reduced, and the computational efficiency is improved. The new two-loop method can solve the DAEs of multibody system dynamics well, and its universality is good. Numerical examples are provided.

Key words: BDF, DAEs, iteration, Jacobian matrix, multibody system dynamics

CLC Number:

O313

ZHANG Le, ZHANG Dingguo. Two-loop Implicit Integration Method Based on Backward Differential Formulation for Differential-algebraic Equations of Multibody System Dynamics[J]. Journal of Mechanical Engineering, 2016, 52(7): 79-87.

References

[1] 洪嘉振. 计算多体系统动力学[M]. 北京：高等教育出版社,1999.
HONG Jiazhen. Computational dynamics of multibody systems[M]. Beijing：Higher Education Press,1999.
[2] EBERHARD P,SCHIEHLEN W. Computational dynamics of multibody systems：History,formalisms,and applications[J]. Journal of Computational and Nonlinear Dynamics,2006,1(1)：3-12.
[3] PETZOLD L R. Numerical solution of differential-algebraic equations in mechanical systems simulation[J]. Physica D：Nonlinear Phenomena,1992,60(1)：269-279.
[4] SHABANA A A. Computational dynamics[M]. John Wiley & Sons,2009.
[5] WASFY T M,NOOR A K. Computational strategies for flexible multibody systems[J]. Applied Mechanics Reviews,2003,56(6)：553-613.
[6] 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]. 力学进展,1996,26(1)：28-40.
PAN Zhenkuan,ZHAO Weijia,HONG Jiazhen,et al. On numercial algorithms for differential/algebraic equation of multibody systems[J]. Advances in Mechanics,1996,26(1)：28-40.
[7] 王琪,陆启韶. 多体系统 Lagrange 方程数值算法的研究进展[J]. 力学进展,2001,31(1)：9-17.
WANG Qi,LU Qishao. Advances in the numerical methods for Lagrange’s equations of multibody systems[J]. Advances in Mechanics,2001,31(1)：9-17.
[8] 王琪,庄方方,郭易圆,等. 非光滑多体系统动力学数值算法的研究进展[J]. 力学进展,2013,43(1)：101-111.
WANG Qi,ZHUANG Fangfang,GUO Yiyuan,et al. Advances in the research on numerical methods for non-smooth dynamics of multibody systems[J]. Advances in Mechanics,2013,43(1)：101-111.
[9] 姚廷强,迟毅林,黄亚宇. 柔性多体系统动力学新型广义-α数值分析方法[J]. 机械工程学报,2009,45(10)：53-60.
YAO Tingqiang,CHI Yilin,HUANG Yayu. New generalized-α algorithms for multibody dynamics[J]. Journal of Mechanical Engineering,2009,45(10)：53-60.
[10] 刘颖,马建敏. 多体系统动力学方程的基于离散零空间理论的 Newmark 积分算法[J]. 机械工程学报,2012,48(5)：87-91.
LIU Ying,MA Jianmin. Discrete null space method for the Newmark integration of multibody dynamic systems[J]. Journal of Mechanical Engineering,2012,48(5)：87-91.
[11] WU J L,ZHANG Y Q. The dynamic analysis of multibody systems with uncertain parameters using interval method[J]. Applied Mechanics and Materials,2012,152：1555-1561.
[12] ASCHER U M,PETZOLD L R. Computer methods for ordinary differential equations and differential-algebraic equations[M]. Philadelphia：Society for Industrial and Applied Mathematics,1998.
[13] SHABANA A A,HUSSEIN B A. A two-loop sparse matrix numerical integration procedure for the solution of differential/algebraic equations：Application to multibody systems[J]. Journal of Sound and Vibration,2009,327(3)：557-563.
[14] 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.
[15] BAUMGARTE J. Stabilization of constraints and integrals of motion in dynamical systems[J]. Computer Methods in Applied Mechanics and Engineering,1972,1(1)：1-16.
[16] GEAR C W,LEIMKUHLER B,GUPTA G K. Automatic integration of Euler-Lagrange equations with constraints[J]. Journal of Computational and Applied Mathematics,1985,12：77-90.
[17] FISETTE P,VANEGHEM B. Numerical integration of multibody system dynamic equations using the coordinate partitioning method in an implicit Newmark scheme[J]. Computer Methods in Applied Mechanics and Engineering,1996,135(1)：85-105.
[18] HAUG E J,NEGRUT D,IANCU M. A state-space-based implicit integration algorithm for differential-algebraic equations of multibody dynamics[J]. Journal of Structural Mechanics,1997,25(3)：311-334.
[19] McGRATHh J F,RAMPALLI R. Implicit integration with coordinate partitioning[J]. Applied Mathematics and Computation,2000,111(1)：7-31.
[20] SANDU A,NEGRUT D,HAUG E J,et al. A Rosenbrock-Nystrom state space implicit approach for the dynamic analysis of mechanical systems：I—theoretical formulation[J]. Proceedings of the Institution of Mechanical Engineers,Part K：Journal of Multi-body Dynamics,2003,217(4)：263-271.
[21] NEGRUT D,HAUG E J,GERMAN H C. An implicit Runge-Kutta method for integration of differential algebraic equations of multibody dynamics[J]. Multibody System Dynamics,2003,9(2)：121-142.
[22] NEGRUT D,RAMPALLI R,OTTARSSON G,et al. On an implementation of the Hilber-Hughes-Taylor method in the context of index 3 differential-algebraic equations of multibody dynamics (DETC2005-85096)[J]. Journal of Computational and Nonlinear Dynamics,2007,2(1)：73-85.
[23] HUSSEIN B,NEGRUT D,SHABANA A A. Implicit and explicit integration in the solution of the absolute nodal coordinate differential/algebraic equations[J]. Nonlinear Dynamics,2008,54(4)：283-296.
[24] 袁兆鼎,费景高,刘德贵. 刚性常微分方程初值问题的数值解法[M]. 北京：科学出版社,1987.
YUAN Zhaoding,FEI Jinggao,LIU Degui. Numerical methods for the initial value problems of stiff ordinary differential equations[M]. Beijing：Science Press,1987.
[25] CAMPANELLI M,BERZERI M,SHABANA A A. Performance of the incremental and non-incremental finite element formulations in flexible multibody problems[J]. Journal of Mechanical Design,2000,122(4)：498-507.
[26] BERZERI M,SHABANA A A. Development of simple models for the elastic forces in the absolute nodal co-ordinate formulation[J]. Journal of Sound and Vibration,2000,235(4)：539-565.
[27] MIKKOLA A,SHABANA A A,SANCHEZ-REBOLLO C,et al. Comparison between ANCF and B-spline surfaces[J]. Multibody System Dynamics,2013,30(2)：119-138.