Hilber-Hughes-Taylor-法在接触约束多体系统动力学中的应用

郭晛; 章定国; 陈思佳

doi:10.7498/aps.66.164501

摘要

以柔性梁在重力作用下绕转动铰做大范围定轴转动，并与刚性平面发生碰撞这一动力学过程为例，对Hilber-Hughes-Taylor（HHT-）法在求解含接触约束的柔性多体系统动力学方程时的数值特性进行了研究.系统运动过程的全局动力学仿真由常微分方程组和微分-代数方程组的数值求解构成.柔性梁在无碰撞阶段系统动力学方程是一组常微分方程组.采用接触约束法模拟接触约束过程，系统的动力学方程为指标3的微分-代数方程组.采用HHT-法对的该微分-代数方程组进行求解，并与Baumgarte违约修正法进行比较.分析了HHT-法自由参数和违约修正常数对计算效率、动力学响应和系统机械能的影响，并对数值积分方法对模态截断数的敏感度以及速度约束和加速度约束的违约程度进行了分析.结果表明，违约修正常数对仿真结果影响非常明显，而HHT-法的自由参数对动力学响应的影响较小，从而避免了违约修正常数对数值积分结果的影响.HHT-法的自由参数可以消除碰撞高频模态的影响.

关键词:

Abstract

Numerical characteristics of the Hilber-Hughes-Taylor- (HHT-) method for the differential-algebraic equations (DAEs) in impact dynamics of flexible multibody systems are investigated. The research is based on a dynamic process of a flexible beam rotating about a fixed axis, whichis under the action of gravity and collides with a rigid plane. Therefore, the dynamic transformation and solution of flexible multibody system are divided into two parts. The Lagrange's equations of the second kind are used to derive the dynamic equations before and after impact, whereas the contact constraint method (CCM) is adopted to simulate the contact process. Compared with other methods, the CCM can describe the contact process accurately and avoid choosing the additional parameters. A set of the differential equations are transformed into a set of the DAEs due to the added constraint equations into impact process. Normally the dynamic equations of the flexible multibody system are index-3 DAEs. Solving a system of the index-3 DAEs directly by an integration algorithm would be subject to ill-conditioning and poor global convergence properties, so it is reasonable to find the methods that avoid both drawbacks and dependence on the constraint information. In order to solve this complex process, the HHT- method is used in the impact dynamic simulation by introducing the Gear-Gupta-Leimkuhler formulation. The coefficient of the HHT- method can be used to control the numerical dissipation, and it also represents asymptotic annihilation of the high frequency response. The smaller the value of , the more the damping is induced in the numerical solution. The Baumgarte's stabilization method is the most famous one for index-3 DAEs. Unfortunately, no general way can be adopted to determine the coefficients of the Baumgarte's stabilization method. It is the main reason for the numerical stability problems. It is necessary to study the influences of coefficients of the former two methods. Simultaneously, the simulation results from the HHT- method are compared with those from the Baumgarte's stabilization method to calculate the CCM model, and the Newmark method is used to solve the ODEs by using the continuous contact force model. The influence of the modal truncation N on the numerical method is also taken into account. Furthermore, the influences of N and the coefficient of HHT- method on the velocity and acceleration constraints in the multibody system are analyzed. Results have shown that the choice of the stabilization coefficients exerts a greater influence on the simulation results, such as the dynamic responses and the constraints, than that of the coefficient . Meanwhile, the HHT- method has an influence on the choice of coefficient and numerical damping properties. This numerical damping property can reduce the effect of high order modes induced by impact. Finally, the increase of N causes the sharpening default of both velocity and acceleration constraints.

Keywords:

作者及机构信息

1.
南京理工大学理学院, 南京 210094;

2.
浙江大学宁波理工学院, 宁波 315100

通信作者: 章定国, zhangdg419@njust.edu.cn

基金项目: 国家自然科学基金（批准号：11272155，11302192）和中央高校基本科研业务费专项资金（批准号：30917011103）资助的课题.

Authors and contacts

1.
School of Science, Nanjing University of Science and Technology, Nanjing 210094, China;

2.
Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China

Corresponding author: Zhang Ding-Guo, zhangdg419@njust.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11272155, 11302192) and the Fundamental Research Funds for Central Universities (Grant No. 30917011103).

参考文献

[1]	Petzold L R 1992 Physica D 60 269
[2]	Hilber H, Hughes T, Taylor R 1977 Earthq Eng. Struct. D 5 283
[3]	Newmark N M 1959 J. Eng. Mech. Div.-ASCE 85 67
[4]	Cardona A, Géradin M 1989 Comput. Struct. 33 801
[5]	Negrut D, Rampallir R, Ottarsson G 2007 J. Comput. Nonlin. Dyn. 2 73
[6]	Laurent O, Negrut D 2007 Electron Trans. Numer. Ana. 6 190
[7]	Chung J, Hulbert G 1993 J. Appl. Mech. 60 371
[8]	Hussein B A, Negrut D, Ahmed A 2008 Nonlinear Dynam. 54 283
[9]	Shabana A A, Hussein B A 2009 J. Sound Vib. 327 557
[10]	Hussein B A, Shabana A A 2011 Nonlinear Dynam. 65 369
[11]	Pan Z K, Zhao W J, Hong J Z, Liu Y Z 1996 Adv. Mech. 26 28 (in Chinese)[潘振宽, 赵维加, 洪嘉振, 刘延柱1996力学进展26 28]
[12]	Wang Q, Lu Q S 2011 Adv. Mech. 31 9(in Chinese)[王琪, 陆启韶2011力学进展31 9]
[13]	Ding J Y, Pan Z K 2013 Engineer. Mech. 30 380 (in Chinese)[丁洁玉, 潘振宽2013工程力学30 380]
[14]	Ma X T, Chen L P, Zhang Y Q 2009 J. Syst. Simulat. 21 6373(in Chinese)[马秀腾, 陈立平, 张云清2009系统仿真学报21 6373]
[15]	Ma X T, Zhai Y B, Luo S Q 2011 J. Southwest Jiaotong Univ. (Natural Science Edition) 33 151(in Chinese)[马秀腾, 翟彦博, 罗书强2011西南交通大学学报(自然科学版)33 151]
[16]	Zhang L, Zhang D G 2016 J. Mech. Engineer. 52 79(in Chinese)[张乐, 章定国2016机械工程学报52 79]
[17]	Zhang L, Zhang D G 2016 Nonlinear Dynam. 85 263
[18]	Kan Z Y, Peng H J, Chen B S, Zhong W X 2015 Chin. J. Computat. Mech. 32 707(in Chinese)[阚子云, 彭海军, 陈飙松, 钟万勰2015计算力学学报32 707]
[19]	Baumgarte J 1972 Comput. Method. Appl. M 1 1
[20]	Lin S T, Huang J N 2000 J. Guid. Control Dynam. 23 566
[21]	Lin S T, Huang J N 2002 J. Mech. Design 124 633
[22]	Duan Y C 2012 Ph. D. Dissertation (Nanjing:Nanjing University of Science & Technology) (in Chinese)[段玥晨2012博士学位论文(南京:南京理工大学)]
[23]	Wu S B, Zhang D G 2011 J. Vib. Engineer. 24 1(in Chinese)[吴胜宝, 章定国2011振动工程学报24 1]
[24]	Liu J Y, Hong J Z 2002 Chin. J. Solid Mech. 23 159(in Chinese)[刘锦阳, 洪嘉振2002固体力学学报23 159]
[25]	Gear C W, Gupta G K, Leumkuhler B 1985 J. Comput. Appl. Math. 12 77

施引文献

[1]	Petzold L R 1992 Physica D 60 269
[2]	Hilber H, Hughes T, Taylor R 1977 Earthq Eng. Struct. D 5 283
[3]	Newmark N M 1959 J. Eng. Mech. Div.-ASCE 85 67
[4]	Cardona A, Géradin M 1989 Comput. Struct. 33 801
[5]	Negrut D, Rampallir R, Ottarsson G 2007 J. Comput. Nonlin. Dyn. 2 73
[6]	Laurent O, Negrut D 2007 Electron Trans. Numer. Ana. 6 190
[7]	Chung J, Hulbert G 1993 J. Appl. Mech. 60 371
[8]	Hussein B A, Negrut D, Ahmed A 2008 Nonlinear Dynam. 54 283
[9]	Shabana A A, Hussein B A 2009 J. Sound Vib. 327 557
[10]	Hussein B A, Shabana A A 2011 Nonlinear Dynam. 65 369
[11]	Pan Z K, Zhao W J, Hong J Z, Liu Y Z 1996 Adv. Mech. 26 28 (in Chinese)[潘振宽, 赵维加, 洪嘉振, 刘延柱1996力学进展26 28]
[12]	Wang Q, Lu Q S 2011 Adv. Mech. 31 9(in Chinese)[王琪, 陆启韶2011力学进展31 9]
[13]	Ding J Y, Pan Z K 2013 Engineer. Mech. 30 380 (in Chinese)[丁洁玉, 潘振宽2013工程力学30 380]
[14]	Ma X T, Chen L P, Zhang Y Q 2009 J. Syst. Simulat. 21 6373(in Chinese)[马秀腾, 陈立平, 张云清2009系统仿真学报21 6373]
[15]	Ma X T, Zhai Y B, Luo S Q 2011 J. Southwest Jiaotong Univ. (Natural Science Edition) 33 151(in Chinese)[马秀腾, 翟彦博, 罗书强2011西南交通大学学报(自然科学版)33 151]
[16]	Zhang L, Zhang D G 2016 J. Mech. Engineer. 52 79(in Chinese)[张乐, 章定国2016机械工程学报52 79]
[17]	Zhang L, Zhang D G 2016 Nonlinear Dynam. 85 263
[18]	Kan Z Y, Peng H J, Chen B S, Zhong W X 2015 Chin. J. Computat. Mech. 32 707(in Chinese)[阚子云, 彭海军, 陈飙松, 钟万勰2015计算力学学报32 707]
[19]	Baumgarte J 1972 Comput. Method. Appl. M 1 1
[20]	Lin S T, Huang J N 2000 J. Guid. Control Dynam. 23 566
[21]	Lin S T, Huang J N 2002 J. Mech. Design 124 633
[22]	Duan Y C 2012 Ph. D. Dissertation (Nanjing:Nanjing University of Science & Technology) (in Chinese)[段玥晨2012博士学位论文(南京:南京理工大学)]
[23]	Wu S B, Zhang D G 2011 J. Vib. Engineer. 24 1(in Chinese)[吴胜宝, 章定国2011振动工程学报24 1]
[24]	Liu J Y, Hong J Z 2002 Chin. J. Solid Mech. 23 159(in Chinese)[刘锦阳, 洪嘉振2002固体力学学报23 159]
[25]	Gear C W, Gupta G K, Leumkuhler B 1985 J. Comput. Appl. Math. 12 77

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