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多体系统指标2运动方程HHT方法违约校正

马秀腾 翟彦博 谢守勇

马秀腾, 翟彦博, 谢守勇. 多体系统指标2运动方程HHT方法违约校正[J]. 力学学报, 2017, 49(1): 175-181. doi: 10.6052/0459-1879-16-275
引用本文: 马秀腾, 翟彦博, 谢守勇. 多体系统指标2运动方程HHT方法违约校正[J]. 力学学报, 2017, 49(1): 175-181. doi: 10.6052/0459-1879-16-275
Ma Xiuteng, Zhai Yanbo, Xie Shouyong. HHT METHOD WITH CONSTRAINTS VIOLATION CORRECTION IN THE INDEX 2 EQUATIONS OF MOTION FOR MULTIBODY SYSTEMS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(1): 175-181. doi: 10.6052/0459-1879-16-275
Citation: Ma Xiuteng, Zhai Yanbo, Xie Shouyong. HHT METHOD WITH CONSTRAINTS VIOLATION CORRECTION IN THE INDEX 2 EQUATIONS OF MOTION FOR MULTIBODY SYSTEMS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(1): 175-181. doi: 10.6052/0459-1879-16-275

多体系统指标2运动方程HHT方法违约校正

doi: 10.6052/0459-1879-16-275
基金项目: 

国家自然科学基金 51605391

和现代汽车零部件技术湖北省重点实验室开放基金 2012-04

详细信息
    通讯作者:

    马秀腾,副教授,主要研究方向:多体系统动力学建模与仿真算法.E-mail:maxt@swu.edu.cn

  • 中图分类号: O313.7

HHT METHOD WITH CONSTRAINTS VIOLATION CORRECTION IN THE INDEX 2 EQUATIONS OF MOTION FOR MULTIBODY SYSTEMS

  • 摘要: 采用Cartesian绝对坐标建模方法,完整约束多体系统运动方程是指标3的微分——代数方程(differentialalgebraic equations,DAEs),数值求解指标3的DAEs属于高指标问题,通过对位置约束方程求导,可使运动方程的指标降为2.位置约束方程求导得到的是速度约束方程.直接求解指标3的运动方程,速度约束方程得不到满足,而且高指标DAEs的数值求解存在一些问题.论文首先采用HHT(Hilber——Hughes——Taylor)直接积分方法求解降指标得到的指标2运动方程,此时速度约束方程参与离散计算,从机器精度上讲速度约束自然得到满足,而位置约束方程没有参与计算,存在“违约”.针对违约问题,采用基于Moore——Penrose广义逆理论的违约校正方法,消除位置约束方程的违约.指标2运动方程HHT方法违约校正,将HHT方法和违约校正方法很好地结合,在数值求解指标2运动方程的过程中,位置约束方程和速度约束方程都不存在违约问题,而且新方法没有引入新的未知数向量,离散得到的非线性方程组的方程数量与原指标2运动方程的方程数量相同,求解规模没有扩大.新方法的实用和有效性通过算例的数值实验得到验证,数值实验也说明新方法保持了HHT方法本身具有的数值阻尼可以控制和二阶精度的特性.最后从非线性方程组的求解规模和计算速度上与其他方法进行了比较分析,说明新方法的优势所在.

     

  • 图  1  曲柄滑块机构简图

    Figure  1.  Sketch of slider crank mechanism

    图  2  机构$\theta $角时间历程

    Figure  2.  Time evolution of $\theta $ for mechanism

    图  3  速度约束方程违约曲线

    Figure  3.  Evolution of velocity constraint equations violation

    图  4  位置约束方程违约曲线

    Figure  4.  Evolution of position constraint equations violation

    图  5  速度约束方程违约曲线

    Figure  5.  Evolution of velocity constraint equations violation

    图  6  位置约束方程违约曲线

    Figure  6.  Evolution of position constraint equations violation

    图  7  求解过程中系统能量的变化曲线

    Figure  7.  Energy of the system during integration

    图  8  精度

    Figure  8.  Order of accuracy

  • [1] Brenan KE, Campbell SL, Petzold LR. Numerical Solution of Initial- Value Problems in Differential Algebraic Equations. 2nd edn. Philadelphia:SIAM, 1996
    [2] 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. 力学进展, 1996, 26(1):28-40 http://www.cnki.com.cn/Article/CJFDTOTAL-QDDD199601013.htm

    Pan Zhenkuan, Zhao Weijia, Hong Jiazhen, et al. On numerical algorithms for differential/algebraic equations of motion of multibody systems. Advances in Mechanics, 1996, 26(1):28-40(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-QDDD199601013.htm
    [3] 王琪, 陆启韶. 多体系统Lagrange方程数值算法的研究进展. 力学进展, 2001, 31(1):9-17 http://www.cnki.com.cn/Article/CJFDTOTAL-LXJZ200101001.htm

    Wang Qi, Lu Qishao. Advances in the numerical methods for Lagrange's equations of multibody systems. Advances in Mechanics, 2001, 31(1):9-17(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-LXJZ200101001.htm
    [4] 赵维加, 潘振宽. 多体系统Euler-Lagrange方程的最小二乘法与违约修正. 力学学报, 2002, 34(4):594-603 http://www.cnki.com.cn/Article/CJFDTOTAL-LXXB200204013.htm

    Zhao Weijia, Pan Zhenkuan. Least square algorithms and constraint stabilization for Euler-Lagrange equations of multibody system dynamics. Chinese Journal of Theoretical and Applied Mechanics, 2002, 34(4):594-603(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-LXXB200204013.htm
    [5] Bauchau OA, Laulusa A. Review of contemporary approaches for constraint enforcement in multibody systems. ASME Journal of Computational and Nonlinear Dynamics, 2008, 3(1):11005 doi: 10.1115/1.2803258
    [6] Arnold M. DAE aspects of multibody system dynamics. Report No. 01. Martin-Luther-Universität Halle-Wittenberg, 2016
    [7] Simeon B. Computational Flexible Multibody Dynamics:A Differential- Algebraic Approach. Berlin Heidelberg:Springer-Verlag, 2013
    [8] Negrut D, Jay LO, Khude N. A discussion of low order numerical integration formulas for rigid and flexible multibody dynamics. ASME Journal of Computational and Nonlinear Dynamics, 2009, 4(2):21008 doi: 10.1115/1.3079784
    [9] 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. ASME Journal of Computational and Nonlinear Dynamics, 2007, 2:73-85 doi: 10.1115/1.2389231
    [10] 丁洁玉, 潘振宽. 多体系统动力学刚性方程广义α投影法. 中国科学:物理学 力学 天文学. 2013, 43(4):572-578 http://www.cnki.com.cn/Article/CJFDTOTAL-JGXK201304028.htm

    Ding Jieyu, Pan Zhenkuan. Generalized-α projection method for stiff dynamic equations of multibody systems. Scientia Sinica Physica, Mechanica & Astronomica, 2013, 43(4):572-578(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-JGXK201304028.htm
    [11] 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. 机械工程学报, 2009, 45(10):53-60 doi: 10.3901/JME.2009.10.053

    Yao Tingqiang, Chi Yilin, Huang Yayu. New generalized-α algorithms for multibody dynamics. Journal of Mechanical Engineering, 2009, 45(10):53-60(in Chinese) doi: 10.3901/JME.2009.10.053
    [12] 丁洁玉, 潘振宽. 多体系统动力学微分-代数方程广义α投影法. 工程力学, 2013, 30(4):380-384 http://www.cnki.com.cn/Article/CJFDTOTAL-GCLX201304055.htm

    Ding Jieyu, Pan Zhenkuan. Generalized-α projection method for differential-algebraic equations of multibody dynamics. Engineering Mechanics, 2013, 30(4):380-384(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-GCLX201304055.htm
    [13] 马秀腾, 翟彦博, 罗书强. 基于θ1方法的多体动力学数值算法研究. 力学学报, 2011, 43(5):931-938 http://lxxb.cstam.org.cn/CN/abstract/abstract142692.shtml

    Ma Xiuteng, Zhai Yanbo, Luo Shuqiang. Numerical method of multibody dynamics based on θ1 method. Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(5):931-938(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract142692.shtml
    [14] 马秀腾, 翟彦博, 罗书强. 多体动力学超定运动方程广义α求解新算法. 力学学报, 2012, 44(5):948-952 http://lxxb.cstam.org.cn/CN/abstract/abstract143674.shtml

    Ma Xiuteng, Zhai Yanbo, Luo Shuqiang. New generalized-α method for over-determined motion equations in multibody dynamics. Chinese Journal of Theoretical and Applied Mechanics, 2012, 44(5):948-952(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract143674.shtml
    [15] Jay LO, Negrut D. A second order extension of the generalized-α method for constrained systems in mechanics//Bottasso C L, ed. Multibody Dynamics:Computational Methods and Applications, Springer Science & Business Media B.V. 2009. 143-158
    [16] Lunk C, Simeon B. Solving constrained mechanical systems by the family of Newmark and α-methods. ZAMM, 2006, 86(10):772-784 doi: 10.1002/(ISSN)1521-4001
    [17] 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. 机械工程学报, 2012, 48(5):87-91 doi: 10.3901/JME.2012.05.087

    Liu Ying, Ma Jianmin. Discrete null space method for the Newmark integration of multibody dynamic systems. Journal of Mechanical Engineering, 2012, 48(5):87-91(in Chinese) doi: 10.3901/JME.2012.05.087
    [18] Orlandea NV. Multibody systems history of ADAMS. ASME Journal of Computational and Nonlinear Dynamics, 2016, 11(6):60301 doi: 10.1115/1.4034296
    [19] Gear CW, Gupta GK, Leimkuhler B. Automatic integration of Euler-Lagrange equations with constraints. Journal of Computational and Applied Mathematics, 1985, 12 & 13:77-90 http://cn.bing.com/academic/profile?id=c71b60b05b640d28c7818eaa4545e1aa&encoded=0&v=paper_preview&mkt=zh-cn
    [20] Arnold M, Hante S. Implementation details of a generalized-α DAE Lie group method. ASME Journal of Computational and Nonlinear Dynamics, 2016, 12(2):021002 doi: 10.1115/1.4033441
    [21] Arnold M, Cardona A, Brüls O. A Lie algebra approach to Lie group time integration of constrained systems//Betsch P, ed. Structure-preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics. Switzerland:Springer International Publishing, 2016:91-158
    [22] 陈立平, 张云清, 任卫群等. 机械系统动力学分析及ADAMS应用教程. 北京:清华大学出版社, 2005

    Chen Liping, Zhang Yunqing, Ren Weiqun, et al. Dynamic Analysis of Mechanical Systems and ADAMS Application. Beijing:Tsinghua University Press, 2005(in Chinese)
    [23] Shampine LF, Reichelt MW, Kierzenka JA. Solving index-1 DAEs in MATLAB and Simulink. SIAM Review, 1999, 42(3):538-552 http://cn.bing.com/academic/profile?id=3d212205dcbea8df1f6815abb4a22b3f&encoded=0&v=paper_preview&mkt=zh-cn
    [24] Yoon S, Howe RM, Greenwood DT. Geometric elimination of constraint violations in numerical simulation of Lagrangian equations. ASME Journal of Mechanical Design, 1994, 116(4):1058-1064 doi: 10.1115/1.2919487
    [25] Yu Q, Cheng I. A direct violation correction method in numerical simulation of constrained multibody systems. Computational Mechanics, 2000, 26:52-57 doi: 10.1007/s004660000149
    [26] 于清, 洪嘉振. 受约束多体系统一种新的违约校正方法. 力学学报, 1998, 30(3):300-306 http://lxxb.cstam.org.cn/CN/abstract/abstract139888.shtml

    Yu Qing, Hong Jiazhen. A new violation correction method for constraint multibody systems. Chinese Journal of Theoretical and Applied Mechanics, 1998, 30(3):300-306(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract139888.shtml
    [27] Nikravesh PE. Initial condition correction in multibody dynamics. Multibody System Dynamics, 2007, 18(1):107-115 doi: 10.1007/s11044-007-9069-z
    [28] Marques F, Souto AP, Flores P. On the constraints violation in forward dynamics of multibody systems. Multibody System Dynamics, online. DOI:10.1007/s11044-016-9530-y. 2016
    [29] Flores P. A new approach to eliminate the constraints violation at the position and velocity levels in constrained mechanical multibody systems//Flores P, Viadero F, eds. New Trends in Mechanism and Machine Science, 5th European Conference on Mechanism Science, Guimaraes, Portugal, September 16-20, 2014. Switzerland:Springer International Publishing, 2015. 385-393
    [30] Gear CW. Differential-algebraic equation index transformations. SIAM Journal on Scientific and Statistical Computing, 1988, 9(1):39-47 doi: 10.1137/0909004
    [31] Bottasso CL, Bauchau OA, Cardona A. Time-step-size-independent conditioning and sensitivity to perturbations in the numerical solution of index three differential algebraic equations. SIAM Journal on Scientific Computing, 2007, 29(1):397-414 doi: 10.1137/050638503
    [32] Marsden JE, West M. Discrete mechanics and variational integrators. Acta Numerica, 2001, 10:357-514 doi: 10.1017/S096249290100006X
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出版历程
  • 收稿日期:  2016-09-29
  • 修回日期:  2016-11-14
  • 网络出版日期:  2016-11-30
  • 刊出日期:  2017-01-18

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