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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.
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Keywords:
- flexible multibody systems /
- impact /
- Hilber-Hughes-Taylor- method /
- Baumgarte's stabilization method
[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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[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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