三维弹性接触问题的数值模拟技术及其应用研究
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摘要
接触问题是工程实践中一类常见、复杂、典型的非线性问题。经典的数学方法和力学方法对大多数接触问题都难以解决，因此需要使用数值计算方法进行求解。有限元数值计算方法的发展，尤其以有限单元法为核心的现代CAE技术的快速发展使有限元数值模拟技术得到了广泛的应用。本文着重采用计算机数值模拟和经验解析解分析对比的方法，对汽车后桥传动用B7517E圆锥滚子轴承进行三维数值模拟研究。汽车后桥传动系统主要以圆锥滚子轴承作为支承，而且为了提高整个系统的支承刚度，在装配时就使锥轴承具有一定的预紧度。
本文在对接触问题的研究和发展现状及接触问题的基本理论进行介绍的基础上，对汽车后桥传动锥轴承弹性接触问题的有限元数值求解方法进行了理论分析。通过对锥轴承的特征参数、接触应力和变形及负荷分布特性的综合分析，得出了圆锥滚子轴承的经典解析解存在局限性的结论。而采用数值模拟技术则可以相对避免解析方法的不足。
本文针对汽车后桥传动锥轴承的接触弹性变形特点、接触应力及负荷分布规律，采用三维实体单元及三维接触单元对圆锥滚子轴承实体模型进行特殊的有限元网格划分，将接触单元按柔体一柔体处理，采用面一面接触分析模式，建立了符合实际的有限元模型。在此基础上，应用商用软件ANSYS对汽车后桥传动锥轴承进行三维有限元模拟与研究，获得了汽车后桥传动锥轴承在两种不同情况下的求解结果。
根据有限元数值模拟结果对经典解析解中接触弹性变形量与作用负荷的关系进行了修正，利用修正过的接触弹性变形公式对汽车后桥传动锥轴承的接触应力和变形及负荷分布规律在无游隙及负游隙两种情况下进行研究，并将结果与经典解析解进行了对比分析。本文的研究结果及研究方法对汽车后桥锥轴承的工程应用、可靠性分析、优化设计等具有重要参考，对研究开发新型锥轴承具有一定的应用价值。
Contact problems belong to typical non-linear problems, familiar and specially complicated problems in engineering. It is helpless for classical mathematics and mechanics method to solve contact problems; Resorting numerical compute method is necessary. Due to the development of finite element method and CAE(Computer Aided Engineering) technology based on finite element method, finite element numerical simulation technology has been applied widely. The paper puts emphasis on three-dimensional numerical simulation research for the B7517E conical roller bearing in automotive transmission by comparing computer numerical simulation with classical result. Conical roller bearings are the most important supporting parts in automotive rear bridge, interfered initially.
After introducing research & development and basic theory of contact problems, theoretical analysis of finite element numerical solution method is made for contact problems of conical roller bearings in automotive rear bridges. According to synthetical analysis on the feature of parameters, contact pressure & formation and load distribution of conical roller bearings, the limitation is discovered, based on classical solution. Contrarily, numerical simulation & analysis is able to overcome these problems.
Special grids is adopted for solid model of conical roller bearings by applying for three-dimensional solid and contact elements, aiming at its characteristic, stress and load distribution of elastic contact problems. Finite element model built in the paper is fit for objective practice, with flexible-flexible contact bodies and surfaces -surfaces contact mode. In term of that, the solution result in two different condition is achieved by ANSYS software.
Deep research and comparing analysis between the classical solution and the finite element result is made on contact stress & formation and load distribution of conical roller bearings by revised classical formula of elastic contact problems, according to finite element solution in two different interfering conditions. The
finial studied outcome and method adopted will bring important effect on engineering application, reliable analysis, optimization design etc. of conical roller bearings, at the same time, certain application value for developing new type of bearings.
本文在对接触问题的研究和发展现状及接触问题的基本理论进行介绍的基础上，对汽车后桥传动锥轴承弹性接触问题的有限元数值求解方法进行了理论分析。通过对锥轴承的特征参数、接触应力和变形及负荷分布特性的综合分析，得出了圆锥滚子轴承的经典解析解存在局限性的结论。而采用数值模拟技术则可以相对避免解析方法的不足。
本文针对汽车后桥传动锥轴承的接触弹性变形特点、接触应力及负荷分布规律，采用三维实体单元及三维接触单元对圆锥滚子轴承实体模型进行特殊的有限元网格划分，将接触单元按柔体一柔体处理，采用面一面接触分析模式，建立了符合实际的有限元模型。在此基础上，应用商用软件ANSYS对汽车后桥传动锥轴承进行三维有限元模拟与研究，获得了汽车后桥传动锥轴承在两种不同情况下的求解结果。
根据有限元数值模拟结果对经典解析解中接触弹性变形量与作用负荷的关系进行了修正，利用修正过的接触弹性变形公式对汽车后桥传动锥轴承的接触应力和变形及负荷分布规律在无游隙及负游隙两种情况下进行研究，并将结果与经典解析解进行了对比分析。本文的研究结果及研究方法对汽车后桥锥轴承的工程应用、可靠性分析、优化设计等具有重要参考，对研究开发新型锥轴承具有一定的应用价值。
Contact problems belong to typical non-linear problems, familiar and specially complicated problems in engineering. It is helpless for classical mathematics and mechanics method to solve contact problems; Resorting numerical compute method is necessary. Due to the development of finite element method and CAE(Computer Aided Engineering) technology based on finite element method, finite element numerical simulation technology has been applied widely. The paper puts emphasis on three-dimensional numerical simulation research for the B7517E conical roller bearing in automotive transmission by comparing computer numerical simulation with classical result. Conical roller bearings are the most important supporting parts in automotive rear bridge, interfered initially.
After introducing research & development and basic theory of contact problems, theoretical analysis of finite element numerical solution method is made for contact problems of conical roller bearings in automotive rear bridges. According to synthetical analysis on the feature of parameters, contact pressure & formation and load distribution of conical roller bearings, the limitation is discovered, based on classical solution. Contrarily, numerical simulation & analysis is able to overcome these problems.
Special grids is adopted for solid model of conical roller bearings by applying for three-dimensional solid and contact elements, aiming at its characteristic, stress and load distribution of elastic contact problems. Finite element model built in the paper is fit for objective practice, with flexible-flexible contact bodies and surfaces -surfaces contact mode. In term of that, the solution result in two different condition is achieved by ANSYS software.
Deep research and comparing analysis between the classical solution and the finite element result is made on contact stress & formation and load distribution of conical roller bearings by revised classical formula of elastic contact problems, according to finite element solution in two different interfering conditions. The
finial studied outcome and method adopted will bring important effect on engineering application, reliable analysis, optimization design etc. of conical roller bearings, at the same time, certain application value for developing new type of bearings.
引文
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[3] Chan S K, Tuba I S.A finite element method for contact problems of solid bodies-Part Ⅰ. Theory and Validation. Int J Meth Sci, 1971: 13: 615—625
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[5] Tsuta T, Yamaji S. Finite element analysis of contact problem. Theory and Practice in Finite Element Structure Analysis, Japan: University of Tokyo Press, 1973
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[7] Okamoto N, Nakazawa M. Finite element incremental contact analysis with various frictional conditions. Int J Num Meth Engng, 1979, 14:337—357
[8] Campos L T, Oden J T, Kikuchi N. A numerical analysis of a class of contact problems with friction in elastostatics. Computer Meth Appl Mech and Engng, 1982, 34:261—268
[9] Ohtakake K, Oden J T, Kikuchi N. Analysis of certain unilateral problem in von karman plate theory by a penalty method—Part Ⅰ: a variational priciple with penalty. Comput&Meth Appl Mech and Engng, 1980, 24:187—213
[10] Kikuchi N, Song Y J. Penalty/finite element approximation of a class of unilateral problems in linear elasticity. Quart Appl Math, 1981, 39(1):1—22
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Meth. Eng, 1993, 9
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[15] 崔俊芝．计算机辅助工程的现在和未来[J]．计算机辅助设计与制造，2000，6：3—7
[16] 王有智．CAE技术的过去、现在和未来[J]．机械设计与制造工程，2000，29(3)：1—2
[17] 尹柏生．有限元分析系统的发展现状与展望[J]．计算机世界，2000，3：1—2
[18] 杨先海，褚金奎等．有限元数值模拟技术及工程应用[J]．机械设计与制造．2003．3：107—108
[19] 王自勤．计算机辅助工程(CAE)技术及其应用．贵州工业大学学报(自然科学版)，2001，30(4)：16—19
[20] Robert G. Dubensky. Effectiveness of CAE in the Simultaneous Engineering Process SAE960519
[21] 冯登台．接触力学的发展概况．力学进展，1987，17(4)：431—446
[22] V. M. Alexandrov, D. A. Pozharskii. Three-dimensional contact problems. Dordrecht: Kluwer Academic, c2001
[23] K. L. Johnson. Contact mechanics. Cambridge[Cambridgeshirel]: Cambridge University Press, 1985
[24] 程昌钧等．弹性力学．北京：高等教育出版社，1996，6
[25] 陆明万，罗学富．弹性理论基础．北京：清华大学出版社，1990，2
[26] 李润方，龚剑霞．接触问题数值方法及其在机械设计中的应用．重庆：重庆大学出版社，1991，2
[27] 李景湧．有限元法．北京：北京邮电大学出版社，1999，2
[28] 朱加铭．有限元与边界元法．哈尔滨：哈尔滨工程大学出版社，2002，2
[29] 吴迪光．变分法．北京：高等教育出版社，1987，4
[30] 赵经文，王宏钰．结构有限元分析．哈尔滨：哈工大出版社，1988，4
[31] (美)艾金，J.E.有限元法的应用与实现．北京：科学出版社，1992，8
[32] 徐士良．计算机常用算法．北京：清华大学出版社，1995，11
[33] 王勖成，邵敏．有限元基本原理和数值方法．清华大学出版社，1995
[34] 张世禄．计算方法．成都：电子科技大学出版社，1999，9
[35] 吴大为．计算机数值方法试验．北京：化学工业出版社，2000，4
[36] 署恒木，仝兴华．工程有限单元法．东营：石油大学出版社，2003，1
[37] 彼得·艾伯哈特，胡斌．现代接触动力学．南京：东南大学出版社，2003，6
[38] 王勖成．有限单元法．北京：清华大学出版社，2003，7
[39] 王国强．实用工程数值模拟技术及其在ANSYS上的实践．西安：西北工业大学出版社，2000
[40] ANSYS, Inc.ANSYS Workbook Release 5.6. SAS, IP Inc., 1999
[41] ANSYS, Inc. ANSYS Commands Reference. SAS, IP Inc., 1999
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[44] ANSYS, Inc. ANSYS Verification Manual. SAS, IP Inc., 1999
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[46] 美国ANSYS公司北京办事处．ANSYS入门手册(上、下)．1998
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[48] 美国ANSYS公司北京办事处．ANSYS建模与分网手册(上、下)．1998
[49] 美国ANSYS公司北京办事处．ANSYS非线性分析指南(上、下)．1998
[50] 美国ANSYS公司北京办事处．ANSYS高级分析指南(上、下)．1998
[51] ANSYS公司北京办事处．ANSYS APDL使用指南[M]．2000
[52] 易日．使用ANSYS6.0进行静力学分析．北京：北京大学出版社，2002，9
[53] 李皓月，周田朋．ANSYS工程计算应用教程．北京：中国铁道出版社，
2003，1
[54] 刘国庆，杨庆东．ANSYS工程应用教程．北京：中国铁道出版社，2003，1
[55] 艾斯曼，哈斯巴跟等．滚动轴承设计与应用手册．武汉：华中工学院出版社，1973
[56] 万长森．滚动轴承的分析方法．北京：机械工业出版社，1987
[57] 余俊．滚动轴承计算．北京：高等教育出版社．1993，7
[58] 丁长安，周晓文．圆锥滚子轴承的受载变形特性．轴承，1996，7：6—11
[59] 张国方等．圆锥滚子负荷分布计算．专用汽车，1996，4：54—57
[60] 大岛宏之．汽车传动装置用圆锥滚子轴承新技术．国外轴承技术，1996，3：11—15
[61] 濮良贵，纪名刚．机械设计．北京：高等教育出版社．2001，6
[62] 李靖华，王进戈等．机械设计．重庆：重庆大学出版社，2002，6
[2] Parson B, Wilson E A. A method for determining the contact stresses resulting from interference fits. ASME J Engng for Industry, 1970: 208-218
[3] Chan S K, Tuba I S.A finite element method for contact problems of solid bodies-Part Ⅰ. Theory and Validation. Int J Meth Sci, 1971: 13: 615—625
[4] Ohte S. Finite element analysis of elastic contact problems. Bull JSME 1973, 16: 797—804
[5] Tsuta T, Yamaji S. Finite element analysis of contact problem. Theory and Practice in Finite Element Structure Analysis, Japan: University of Tokyo Press, 1973
[6] Fredriksson B. Finite element solution of surface nonlinearities in structural mechanics with special emphasis to contact and fracture mechanics problems. Computers&Structures. 1976, 6:281—290
[7] Okamoto N, Nakazawa M. Finite element incremental contact analysis with various frictional conditions. Int J Num Meth Engng, 1979, 14:337—357
[8] Campos L T, Oden J T, Kikuchi N. A numerical analysis of a class of contact problems with friction in elastostatics. Computer Meth Appl Mech and Engng, 1982, 34:261—268
[9] Ohtakake K, Oden J T, Kikuchi N. Analysis of certain unilateral problem in von karman plate theory by a penalty method—Part Ⅰ: a variational priciple with penalty. Comput&Meth Appl Mech and Engng, 1980, 24:187—213
[10] Kikuchi N, Song Y J. Penalty/finite element approximation of a class of unilateral problems in linear elasticity. Quart Appl Math, 1981, 39(1):1—22
[11] P Wriggers, G Zavarise. Application of Augmented Lagrangian Techniques for Non-linear Constitutive Laws in Contact Interfaces. Commu. Nummer.
Meth. Eng, 1993, 9
[12] G Zsvarise, P Wriggers, B A Scheffler. On Augmented Lagrangian Algorithms for Thermo-mechanical Contact problems with Friction. Int.J. Numer. Meth. Engng, 1995, 38
[13] J H Heegaard, A Curnier. An Augmented Lagrangian Method for Discreet Large-slip Contact Problems. Int. J. Numer. Meth. Eng, 1993, 36
[14] J C Simo, T A Laursen. An Augmented Lagrangian Treatment of Contact Problems Involving Friction, Comp&Stru, 1992, 42(1)
[15] 崔俊芝．计算机辅助工程的现在和未来[J]．计算机辅助设计与制造，2000，6：3—7
[16] 王有智．CAE技术的过去、现在和未来[J]．机械设计与制造工程，2000，29(3)：1—2
[17] 尹柏生．有限元分析系统的发展现状与展望[J]．计算机世界，2000，3：1—2
[18] 杨先海，褚金奎等．有限元数值模拟技术及工程应用[J]．机械设计与制造．2003．3：107—108
[19] 王自勤．计算机辅助工程(CAE)技术及其应用．贵州工业大学学报(自然科学版)，2001，30(4)：16—19
[20] Robert G. Dubensky. Effectiveness of CAE in the Simultaneous Engineering Process SAE960519
[21] 冯登台．接触力学的发展概况．力学进展，1987，17(4)：431—446
[22] V. M. Alexandrov, D. A. Pozharskii. Three-dimensional contact problems. Dordrecht: Kluwer Academic, c2001
[23] K. L. Johnson. Contact mechanics. Cambridge[Cambridgeshirel]: Cambridge University Press, 1985
[24] 程昌钧等．弹性力学．北京：高等教育出版社，1996，6
[25] 陆明万，罗学富．弹性理论基础．北京：清华大学出版社，1990，2
[26] 李润方，龚剑霞．接触问题数值方法及其在机械设计中的应用．重庆：重庆大学出版社，1991，2
[27] 李景湧．有限元法．北京：北京邮电大学出版社，1999，2
[28] 朱加铭．有限元与边界元法．哈尔滨：哈尔滨工程大学出版社，2002，2
[29] 吴迪光．变分法．北京：高等教育出版社，1987，4
[30] 赵经文，王宏钰．结构有限元分析．哈尔滨：哈工大出版社，1988，4
[31] (美)艾金，J.E.有限元法的应用与实现．北京：科学出版社，1992，8
[32] 徐士良．计算机常用算法．北京：清华大学出版社，1995，11
[33] 王勖成，邵敏．有限元基本原理和数值方法．清华大学出版社，1995
[34] 张世禄．计算方法．成都：电子科技大学出版社，1999，9
[35] 吴大为．计算机数值方法试验．北京：化学工业出版社，2000，4
[36] 署恒木，仝兴华．工程有限单元法．东营：石油大学出版社，2003，1
[37] 彼得·艾伯哈特，胡斌．现代接触动力学．南京：东南大学出版社，2003，6
[38] 王勖成．有限单元法．北京：清华大学出版社，2003，7
[39] 王国强．实用工程数值模拟技术及其在ANSYS上的实践．西安：西北工业大学出版社，2000
[40] ANSYS, Inc.ANSYS Workbook Release 5.6. SAS, IP Inc., 1999
[41] ANSYS, Inc. ANSYS Commands Reference. SAS, IP Inc., 1999
[42] ANSYS, Inc. ANSYS Element Reference. SAS, IP Inc., 1999
[43] ANSYS, Inc. ANSYS Basic Analysis Procedures Guide. SAS, IP Inc.1999
[44] ANSYS, Inc. ANSYS Verification Manual. SAS, IP Inc., 1999
[45] ANSYS, Inc. ANSYS Tutorials for Release 5.6. SAS, IP Inc., 1999
[46] 美国ANSYS公司北京办事处．ANSYS入门手册(上、下)．1998
[47] 美国ANSYS公司北京办事处．ANSYS基本过程手册(上、下)．1998
[48] 美国ANSYS公司北京办事处．ANSYS建模与分网手册(上、下)．1998
[49] 美国ANSYS公司北京办事处．ANSYS非线性分析指南(上、下)．1998
[50] 美国ANSYS公司北京办事处．ANSYS高级分析指南(上、下)．1998
[51] ANSYS公司北京办事处．ANSYS APDL使用指南[M]．2000
[52] 易日．使用ANSYS6.0进行静力学分析．北京：北京大学出版社，2002，9
[53] 李皓月，周田朋．ANSYS工程计算应用教程．北京：中国铁道出版社，
2003，1
[54] 刘国庆，杨庆东．ANSYS工程应用教程．北京：中国铁道出版社，2003，1
[55] 艾斯曼，哈斯巴跟等．滚动轴承设计与应用手册．武汉：华中工学院出版社，1973
[56] 万长森．滚动轴承的分析方法．北京：机械工业出版社，1987
[57] 余俊．滚动轴承计算．北京：高等教育出版社．1993，7
[58] 丁长安，周晓文．圆锥滚子轴承的受载变形特性．轴承，1996，7：6—11
[59] 张国方等．圆锥滚子负荷分布计算．专用汽车，1996，4：54—57
[60] 大岛宏之．汽车传动装置用圆锥滚子轴承新技术．国外轴承技术，1996，3：11—15
[61] 濮良贵，纪名刚．机械设计．北京：高等教育出版社．2001，6
[62] 李靖华，王进戈等．机械设计．重庆：重庆大学出版社，2002，6