三自由度高速混联内螺纹铣削机床研究与开发
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摘要
目前小直径内螺纹的加工普遍存在加工困难、效率低、精度差和缺乏专用设备等问题。高速螺纹铣削可高效、低耗、高精度地完成各种不同材料零件上内螺纹的加工,但目前高速螺纹铣削在我国的应用还存在许多问题。本文采用平面5R并联机构与串联机构混联方式构建了一种新型中小直径内螺纹高速铣削机床,对并联机构运动学与整机静动态特性进行了深入系统的研究,为内螺纹高速铣削工艺的应用奠定了基础。
本文分析了内螺纹高速铣削的工作原理与特点；提出了一种基于平面5R并联机构的三自由度混联螺纹铣削运动实现方案,建立了平面5R并联机构的运动学模型,推导了其逆向运动学速度雅可比矩阵,对并联机构的工作空间和性能指标进行了深入研究,提出了驱动杆型位组合的原则,完成了内螺纹铣削公转圆运动的位置反解。综合平面5R并联机构的工作空间和性能指标因素优化了5R机构杆长参数；确定了该混联机床的主要技术参数,划分了机床各主要功能模块,对各模块的主要构件进行了力学分析设计、外购件选型和校核计算,完成了混联机床的开发,最后利用PRO/E软件对混联机床的各个模块进行三维实体设计和虚拟装配。
在ADAMS环境下对混联机构进行了运动学仿真分析,验证了理论计算正确性和机床设计的合理性,为混联螺纹铣削机床的实际设计制造和应用提供了理论依据。
利用有限元分析软件ANSYS对3自由度混联机床整机结构进行了静力学分析,获得了整机变形和等效应力云图,进而对整机结构进行了静刚度分析；通过模态分析获得了机床各阶固有频率和相应振型特性,并对各阶振型进行了分析,得出结构的薄弱环节,为再次优化设计提供了理论依据。分析结果表明：设计的混联螺纹机床的刚度能够满足螺纹铣削运动的要求,整机结构大部分区域的安全系数均较大,强度能满足危险工况下加工的要求；机床受迫振动的激振频率避开了机床的固有频率,该机床可以投入实际应用。
At present, there are many problems in the process of small-diameter internal thread machining such as hard processing, low efficiency, low precision and lack of special equipment etc.. High-speed thread milling can manufacture internal threads in different materials and parts with high efficiency, low consumption and high precision. However, there are many problems in the application of high-speed thread milling in China. This article develops a new high-speed small-diameter internal thread milling machine with a hybrid approach of planar 5R parallel mechanism and series machine tool, studies systematically the parallel mechanism kinematics and static and dynamic characteristics of complete appliance, and lays a foundation for the application of internal thread milling process.
This article studies the working principle and features of high-speed internal thread milling, proposes the kinematic implementation sheme of 3-dof small diameter, high-speed hybrid thread milling based on the planar 5R parallel mechanism, establishes the kinematics model of planar 5R parallel mechanism, deduces its inverse kinematics velocity Jacobian matrix, and studies the work space and performance index of parallel mechanism, raises the combination principles of the type and position of drive rods, and figures out the inverse position solution of circular movement. This article optimizes the rod length of 5R mechanism under the workspace and performance index of 5R parallel mechanism, confirms the main technical parameters of the hybrid machine, divides the main functional modules under the direction of modularization design, takes the mechanical analysis of the main components of each module, design selection and check calculation, and then develops the hybrid machine. Finaly PRO/E is used for the three-dimensional physical design and virtual assembly of hybrid machine modules.
Kinematics simulation analysis of the hybrid institutions is performed with ADAMS, which verifies the validity of theoretical calculation and the rationality of machine design and provides the theoretical basis for practical design, manufacture and application of the machine.
The statics and modal analysis of 3-DOF hybrid machine tool is completed with ANSYS, from which deformation and distribution of equivalent stress pattern are obtained, and then the strength and stiffness of the machine tool are analyzed. Through analysis, the natural frequency and vibration characteristics of each order are obtained, vibration mode of each order is analyzed, and then the weakness of the machine is obtained. It provides a theoretical basis for further optimization. The results show that the hybrid machine tool is expected to meet the requirements of actual thread milling trajectory and the safety coefficient in most regions of the whole structural is high, the intensity meets the processing requirements under dangerous working conditions; The Excited frequency of forced vibration diverge from the Natural frequency of the machine tool, and the machine tool can be put on the practical application.
本文分析了内螺纹高速铣削的工作原理与特点；提出了一种基于平面5R并联机构的三自由度混联螺纹铣削运动实现方案,建立了平面5R并联机构的运动学模型,推导了其逆向运动学速度雅可比矩阵,对并联机构的工作空间和性能指标进行了深入研究,提出了驱动杆型位组合的原则,完成了内螺纹铣削公转圆运动的位置反解。综合平面5R并联机构的工作空间和性能指标因素优化了5R机构杆长参数；确定了该混联机床的主要技术参数,划分了机床各主要功能模块,对各模块的主要构件进行了力学分析设计、外购件选型和校核计算,完成了混联机床的开发,最后利用PRO/E软件对混联机床的各个模块进行三维实体设计和虚拟装配。
在ADAMS环境下对混联机构进行了运动学仿真分析,验证了理论计算正确性和机床设计的合理性,为混联螺纹铣削机床的实际设计制造和应用提供了理论依据。
利用有限元分析软件ANSYS对3自由度混联机床整机结构进行了静力学分析,获得了整机变形和等效应力云图,进而对整机结构进行了静刚度分析；通过模态分析获得了机床各阶固有频率和相应振型特性,并对各阶振型进行了分析,得出结构的薄弱环节,为再次优化设计提供了理论依据。分析结果表明：设计的混联螺纹机床的刚度能够满足螺纹铣削运动的要求,整机结构大部分区域的安全系数均较大,强度能满足危险工况下加工的要求；机床受迫振动的激振频率避开了机床的固有频率,该机床可以投入实际应用。
At present, there are many problems in the process of small-diameter internal thread machining such as hard processing, low efficiency, low precision and lack of special equipment etc.. High-speed thread milling can manufacture internal threads in different materials and parts with high efficiency, low consumption and high precision. However, there are many problems in the application of high-speed thread milling in China. This article develops a new high-speed small-diameter internal thread milling machine with a hybrid approach of planar 5R parallel mechanism and series machine tool, studies systematically the parallel mechanism kinematics and static and dynamic characteristics of complete appliance, and lays a foundation for the application of internal thread milling process.
This article studies the working principle and features of high-speed internal thread milling, proposes the kinematic implementation sheme of 3-dof small diameter, high-speed hybrid thread milling based on the planar 5R parallel mechanism, establishes the kinematics model of planar 5R parallel mechanism, deduces its inverse kinematics velocity Jacobian matrix, and studies the work space and performance index of parallel mechanism, raises the combination principles of the type and position of drive rods, and figures out the inverse position solution of circular movement. This article optimizes the rod length of 5R mechanism under the workspace and performance index of 5R parallel mechanism, confirms the main technical parameters of the hybrid machine, divides the main functional modules under the direction of modularization design, takes the mechanical analysis of the main components of each module, design selection and check calculation, and then develops the hybrid machine. Finaly PRO/E is used for the three-dimensional physical design and virtual assembly of hybrid machine modules.
Kinematics simulation analysis of the hybrid institutions is performed with ADAMS, which verifies the validity of theoretical calculation and the rationality of machine design and provides the theoretical basis for practical design, manufacture and application of the machine.
The statics and modal analysis of 3-DOF hybrid machine tool is completed with ANSYS, from which deformation and distribution of equivalent stress pattern are obtained, and then the strength and stiffness of the machine tool are analyzed. Through analysis, the natural frequency and vibration characteristics of each order are obtained, vibration mode of each order is analyzed, and then the weakness of the machine is obtained. It provides a theoretical basis for further optimization. The results show that the hybrid machine tool is expected to meet the requirements of actual thread milling trajectory and the safety coefficient in most regions of the whole structural is high, the intensity meets the processing requirements under dangerous working conditions; The Excited frequency of forced vibration diverge from the Natural frequency of the machine tool, and the machine tool can be put on the practical application.
引文
[1]冯启高,马利杰.内螺纹加工新技术[J].机械研究与应用,2005,(4)：2-3
[2]周凯,陶真,毛得柱.虚拟轴机床的发展趋势-混联机床[J].现代制造工程,2002,(03)：5-7.
[3]J. P. Merlet.An initiative for the kinematics study of parallel manipulators[J].Fundamental issues and future research directions for parallel mechanisms and manipulators,2002,Quebee City,CA.2002: 1012-1016.
[4]王小兵.一种新型混联机床的应用基础研究[D].成都：电子科技大学,2006：2.
[5]Yi,XianZhong.Gao,De-Li.Tool-path planning for five-axis lomputer numerical control machining of helical rotors with whirlmind milling method.Shi you,Daxue xue bao,2004(8):66-69.
[6]李春渝.旋风铣削螺纹径向误差分析与对策[J].机械工艺师,1997,(5)：3.
[7]谭立新.旋风铣削加工质量分析[J].新技术新工艺,2002(3)：15-17.
[8]杜娟,戴建中,张美清.大尺寸螺纹孔的旋风铣削加工[J].一重技术,1998(2)：36-3.
[9]Yoshida,Yoshitaroh.Design system for thread rotary cutter profile in consideration of interference. Nippon kikai Gakkai Ronbunshu,1995(6):484-491.
[10]Baer.CAD of worms and their machining tools.Computers craphics,1990(14):405-411.
[11]Bar.Curuatures of the enleloped helicoids.Mechanism and Mechine Theory,1997(32):111-120.
[12]邬昌峰,高荣慧.并联机床的特点、研究现状及展望[J].机械,2003,(30)：1-3.
[13]H.MacCallion and D. T. Pham.The Analysis of a Six Degree of Freedom Workstation for Mechanised Assembly[J].Proceedings of the 5th World Congress on Theory of Machines and Mechanisms,1979:123-130.
[14]段广洪,李铁民.并联机器的起源和发展[J].WMEN,2006,(1),41-48.
[15]黄玉美,高峰,史文浩等.混联式数控机床的发展[J].制造技术与机床,2001,(8)：8-9.
[16]杨斌久.少自由度并联机床约束链的若干问题研究[D].东北大学：东北大学,2006：21.
[17]Jun Wu,Jinsong Wang,Tiemin Li,Liping Wang. Dynamic analysis of the 2-DOF planar parallel manipulator of a heavy duty hybrid machine tool[J] The International Journal of Advanced Manufacturing Technology,2007,34, (3-4).
[18]Lin,Chuen Sen,Lee,Terry.A Classification Method for Input Joints of Planar Five-Bar Mechanisms[J]. Mechanism and Machine Theory,2003,38:1307-1322.
[19]Xinjun Liu,Jinsong Wang And Gunter Pritschow.Performance Atlases and Optimum Design of Planar 5R Symmetrical Parallel Mechanisms[J].Mechanism and Machine Theory,2006,41:119-144.
[20]Karger,Adolf.Architecture Singular Planar Parallel Manipulators[J].Mechanism and Machine Theory,2003,38(11):1149-1164.
[21]Xinjun Liu,Jin song Wang,Gunter Pritschow.Kinematics,Singularity and Workspace of Planar 5R Symmetrical Parallel Mechanisms[J].Mechanism and Machine Theory,2006,41:145-169.
[22]刘辛军.并联机器人机构尺寸与性能关系分析及其设计理论研究[D].燕山大学,1999：25-33.
[23]张立杰.两自由度并联机器人的性能分析及尺寸优化[D].燕山大学,2006：12-26.
[24]刘善增,平面2自由度并联机器人的动力学设计[J].机械科学与技术,2008,(2)：2-3.
[25]S.Kock,W.Schumacher.A Parallel X-Y Manipulator with Actuation Redundancy for High-Speed and Active-Stiffness Applications.In:Proceedings of the 1998 Ieee Int.Conf.On Roboitcs and Automation,1998,Belgium:2295-2300.
[26]屠凤莲.2-PRR并联机床的设计理论及方法研究[D].河北：河北工业大学,2006：8-15.
[27]高发伟.高速铣削加工技术概述[J].广西轻工业,2008,(11)：3-5.
[28]谭加才,谭立新,傅彩明.旋风铣削刀具的设计与有限元结构分析[J].工具技术.2007,(3)：3-6.
[29]梁海,黄华剑.螺纹铣刀在数控加工中心上的应用[J].现代制造工程,2006,(10)：2.
[30]高莹.曲轴螺孔加工工艺优化[D].上海：同济大学,2007：33-34.
[31]李海滨,朱姗姗,杨义虎,邱元庆.铣削加工螺纹刀具的选择[J].机床与液压,2008,(9)：2-4.
[32]王晓霞,金属切削原理与刀具[M].航空工业出版社,2000,5：31-34.
[33]D.McCloy.Some Comparisons of Serial-Driver and Parallel Driver Manipulator[J].Robotica,1990,8: 355-362.
[34]Xinjun Liu,Jin song Wang,Gunter Pritschow. Kinematics,Singularity and Workspace of Planar 5R Symmetrical Parallel Mechanisms[J].Mechanism and Machine Theory,2006,41:145-169.
[35]C. M. Gosselin, J. Angeles. Singularity Analysis of Closed-Loop Kinematic Chains[J]. IEEE Trans. Robotics& Auto,1990,6(3):281-290.
[36]F.Gao,X.Q.Zhang,Y.S.Zhao,et al.A Physical Model of the Solution Space and the Atlases of the Reachable Workspaces for 2-DOF Parallel Plane Wrists[J].Mechanism and Machine theory,1996,31: 173-184.
[37]J.J.Cervantes,Sanchez,J.C. Hernandez Rodriguez,On the 5R Spherical,Symmetric Manipulator: Workspace and Singularity Characterization[J].Mechanism and Machine Theory,2004,39:409-429.
[38]J.K.Salisbury,J.J.Craig.Articulated Hands:Force Control and Kinematic Issues[J].Int.J.Robot.l982, 1(1):4-12.
[39]张顺心,范顺成,高峰,许晓云,张建军.多功能模块化串并联机床结构的研究[J].机械设计与研究,2003,(1)：2-4.
[40]徐宏玲.模块化组织形成及运行机理研究[D].成都：西南财经大学,2006：35-37.
[41]陈定方,罗亚波.虚拟设计[M].机械工业出版社,2002：12-23.
[42]孙训方.材料力学[M].北京：高等教育出版社,2002：55-62.
[43]姜大志,应强,孙俊兰.滚动直线导轨的受力分析与载荷计算[J].机床与液压,2008,(4)：3-5.
[44]Talaslidis D G, Manolis G D, Paraskevopoulos E,et,al. R iskanalysis of industrial structure under extreme transient loads[J].Soil Dynamics and Earthquake Engineering,2004,24(4):435-448.
[45]诸乃雄.机床动态设计原理与应用[M].上海：同济大学出版社,1987：21-23.
[46]唐金松主编.简明机械设计手册第二版[M].上海：上海科学技术出版社,1992：5-23.
[47]韩玉龙.Pro/ENGINEERWildfire组建设计与运动仿真专业教程[M].北京：清华大学出版社,2004：12-34.
[48]王冰.五轴混联机床的关键技术研究[D].秦皇岛：燕山大学,2005：12-14.
[49]李军,邢俊文,覃文洁等著.ADAMS实例教程.北京：北京理工大学出版社,2002：7.
[50]蒋洪平.XH715立式加工中心床身模态分析与结构改进[J].制造技术与机床,2008,(09)：2-4.
[51]Yan LH,Leung TP,LiDB,etal.Theoretical and experimental study of modal strain analysis[J].Sound and Vibration,1996,191 (2):251-260.
[52]商跃进.有限元原理与ANSYS应用指南[M].北京：清华大学出版社,2005：24.
[53]王勖成,邵敏.有限单元法基本原理和数值方法(第二版)[M].北京：清华大学出版社,1997：38-73
[54]王崧,董春敏等.有限元分析-ANSYS理论与应用[M].北京：电子工业出版社,2005：6.
[55]张朝晖,李树奎.ANSY11.0有限元分析理论与工程应用[M].北京：电子工业出版社,2008：3.
[56]张洪信,赵清海.ANSYS有限元分析完全自学手册[M].北京：机械工业出版社,2008：24.
[57]谭雪松,甘露萍等.Pro/ENGINEER中文版基础教程[M].北京：人民邮电出版社,2005：139-181.
[58]杜平安.结构有限元分析的形状处理方法[J].机械与电子,2000,(1)：26-27.
[59]黄幼玲.机床主轴系统静动态性能分析与优化设计[D].南京：河海大学,2007：1-2.
[60]姚鹏.机床零件结合面动态特性参数的识别与研究[D].沈阳：东北大学,2005：15-20.
[61]黄玉美等.虚拟样机整机结构特性边界元仿真[M].北京：机械工业出版社,2004：152-173.
[62]唐金松.简明机械设计手册[M].上海：上海科学技术出版社,2000,10：89.
[63]陈庆堂.基于ANSYS的XK713数控铣床有限元分析及优化设计[D].南京：东南大学,2005：5-20.
[64]魏永庚,王知行.并联机床结构静刚度的有限元分析[J],机械与电子,2003,(1)：16-19
[65]庄茁,张帆等.非线性有限元分析与实例[M].北京：科学出版社,2005：36-37
[2]周凯,陶真,毛得柱.虚拟轴机床的发展趋势-混联机床[J].现代制造工程,2002,(03)：5-7.
[3]J. P. Merlet.An initiative for the kinematics study of parallel manipulators[J].Fundamental issues and future research directions for parallel mechanisms and manipulators,2002,Quebee City,CA.2002: 1012-1016.
[4]王小兵.一种新型混联机床的应用基础研究[D].成都：电子科技大学,2006：2.
[5]Yi,XianZhong.Gao,De-Li.Tool-path planning for five-axis lomputer numerical control machining of helical rotors with whirlmind milling method.Shi you,Daxue xue bao,2004(8):66-69.
[6]李春渝.旋风铣削螺纹径向误差分析与对策[J].机械工艺师,1997,(5)：3.
[7]谭立新.旋风铣削加工质量分析[J].新技术新工艺,2002(3)：15-17.
[8]杜娟,戴建中,张美清.大尺寸螺纹孔的旋风铣削加工[J].一重技术,1998(2)：36-3.
[9]Yoshida,Yoshitaroh.Design system for thread rotary cutter profile in consideration of interference. Nippon kikai Gakkai Ronbunshu,1995(6):484-491.
[10]Baer.CAD of worms and their machining tools.Computers craphics,1990(14):405-411.
[11]Bar.Curuatures of the enleloped helicoids.Mechanism and Mechine Theory,1997(32):111-120.
[12]邬昌峰,高荣慧.并联机床的特点、研究现状及展望[J].机械,2003,(30)：1-3.
[13]H.MacCallion and D. T. Pham.The Analysis of a Six Degree of Freedom Workstation for Mechanised Assembly[J].Proceedings of the 5th World Congress on Theory of Machines and Mechanisms,1979:123-130.
[14]段广洪,李铁民.并联机器的起源和发展[J].WMEN,2006,(1),41-48.
[15]黄玉美,高峰,史文浩等.混联式数控机床的发展[J].制造技术与机床,2001,(8)：8-9.
[16]杨斌久.少自由度并联机床约束链的若干问题研究[D].东北大学：东北大学,2006：21.
[17]Jun Wu,Jinsong Wang,Tiemin Li,Liping Wang. Dynamic analysis of the 2-DOF planar parallel manipulator of a heavy duty hybrid machine tool[J] The International Journal of Advanced Manufacturing Technology,2007,34, (3-4).
[18]Lin,Chuen Sen,Lee,Terry.A Classification Method for Input Joints of Planar Five-Bar Mechanisms[J]. Mechanism and Machine Theory,2003,38:1307-1322.
[19]Xinjun Liu,Jinsong Wang And Gunter Pritschow.Performance Atlases and Optimum Design of Planar 5R Symmetrical Parallel Mechanisms[J].Mechanism and Machine Theory,2006,41:119-144.
[20]Karger,Adolf.Architecture Singular Planar Parallel Manipulators[J].Mechanism and Machine Theory,2003,38(11):1149-1164.
[21]Xinjun Liu,Jin song Wang,Gunter Pritschow.Kinematics,Singularity and Workspace of Planar 5R Symmetrical Parallel Mechanisms[J].Mechanism and Machine Theory,2006,41:145-169.
[22]刘辛军.并联机器人机构尺寸与性能关系分析及其设计理论研究[D].燕山大学,1999：25-33.
[23]张立杰.两自由度并联机器人的性能分析及尺寸优化[D].燕山大学,2006：12-26.
[24]刘善增,平面2自由度并联机器人的动力学设计[J].机械科学与技术,2008,(2)：2-3.
[25]S.Kock,W.Schumacher.A Parallel X-Y Manipulator with Actuation Redundancy for High-Speed and Active-Stiffness Applications.In:Proceedings of the 1998 Ieee Int.Conf.On Roboitcs and Automation,1998,Belgium:2295-2300.
[26]屠凤莲.2-PRR并联机床的设计理论及方法研究[D].河北：河北工业大学,2006：8-15.
[27]高发伟.高速铣削加工技术概述[J].广西轻工业,2008,(11)：3-5.
[28]谭加才,谭立新,傅彩明.旋风铣削刀具的设计与有限元结构分析[J].工具技术.2007,(3)：3-6.
[29]梁海,黄华剑.螺纹铣刀在数控加工中心上的应用[J].现代制造工程,2006,(10)：2.
[30]高莹.曲轴螺孔加工工艺优化[D].上海：同济大学,2007：33-34.
[31]李海滨,朱姗姗,杨义虎,邱元庆.铣削加工螺纹刀具的选择[J].机床与液压,2008,(9)：2-4.
[32]王晓霞,金属切削原理与刀具[M].航空工业出版社,2000,5：31-34.
[33]D.McCloy.Some Comparisons of Serial-Driver and Parallel Driver Manipulator[J].Robotica,1990,8: 355-362.
[34]Xinjun Liu,Jin song Wang,Gunter Pritschow. Kinematics,Singularity and Workspace of Planar 5R Symmetrical Parallel Mechanisms[J].Mechanism and Machine Theory,2006,41:145-169.
[35]C. M. Gosselin, J. Angeles. Singularity Analysis of Closed-Loop Kinematic Chains[J]. IEEE Trans. Robotics& Auto,1990,6(3):281-290.
[36]F.Gao,X.Q.Zhang,Y.S.Zhao,et al.A Physical Model of the Solution Space and the Atlases of the Reachable Workspaces for 2-DOF Parallel Plane Wrists[J].Mechanism and Machine theory,1996,31: 173-184.
[37]J.J.Cervantes,Sanchez,J.C. Hernandez Rodriguez,On the 5R Spherical,Symmetric Manipulator: Workspace and Singularity Characterization[J].Mechanism and Machine Theory,2004,39:409-429.
[38]J.K.Salisbury,J.J.Craig.Articulated Hands:Force Control and Kinematic Issues[J].Int.J.Robot.l982, 1(1):4-12.
[39]张顺心,范顺成,高峰,许晓云,张建军.多功能模块化串并联机床结构的研究[J].机械设计与研究,2003,(1)：2-4.
[40]徐宏玲.模块化组织形成及运行机理研究[D].成都：西南财经大学,2006：35-37.
[41]陈定方,罗亚波.虚拟设计[M].机械工业出版社,2002：12-23.
[42]孙训方.材料力学[M].北京：高等教育出版社,2002：55-62.
[43]姜大志,应强,孙俊兰.滚动直线导轨的受力分析与载荷计算[J].机床与液压,2008,(4)：3-5.
[44]Talaslidis D G, Manolis G D, Paraskevopoulos E,et,al. R iskanalysis of industrial structure under extreme transient loads[J].Soil Dynamics and Earthquake Engineering,2004,24(4):435-448.
[45]诸乃雄.机床动态设计原理与应用[M].上海：同济大学出版社,1987：21-23.
[46]唐金松主编.简明机械设计手册第二版[M].上海：上海科学技术出版社,1992：5-23.
[47]韩玉龙.Pro/ENGINEERWildfire组建设计与运动仿真专业教程[M].北京：清华大学出版社,2004：12-34.
[48]王冰.五轴混联机床的关键技术研究[D].秦皇岛：燕山大学,2005：12-14.
[49]李军,邢俊文,覃文洁等著.ADAMS实例教程.北京：北京理工大学出版社,2002：7.
[50]蒋洪平.XH715立式加工中心床身模态分析与结构改进[J].制造技术与机床,2008,(09)：2-4.
[51]Yan LH,Leung TP,LiDB,etal.Theoretical and experimental study of modal strain analysis[J].Sound and Vibration,1996,191 (2):251-260.
[52]商跃进.有限元原理与ANSYS应用指南[M].北京：清华大学出版社,2005：24.
[53]王勖成,邵敏.有限单元法基本原理和数值方法(第二版)[M].北京：清华大学出版社,1997：38-73
[54]王崧,董春敏等.有限元分析-ANSYS理论与应用[M].北京：电子工业出版社,2005：6.
[55]张朝晖,李树奎.ANSY11.0有限元分析理论与工程应用[M].北京：电子工业出版社,2008：3.
[56]张洪信,赵清海.ANSYS有限元分析完全自学手册[M].北京：机械工业出版社,2008：24.
[57]谭雪松,甘露萍等.Pro/ENGINEER中文版基础教程[M].北京：人民邮电出版社,2005：139-181.
[58]杜平安.结构有限元分析的形状处理方法[J].机械与电子,2000,(1)：26-27.
[59]黄幼玲.机床主轴系统静动态性能分析与优化设计[D].南京：河海大学,2007：1-2.
[60]姚鹏.机床零件结合面动态特性参数的识别与研究[D].沈阳：东北大学,2005：15-20.
[61]黄玉美等.虚拟样机整机结构特性边界元仿真[M].北京：机械工业出版社,2004：152-173.
[62]唐金松.简明机械设计手册[M].上海：上海科学技术出版社,2000,10：89.
[63]陈庆堂.基于ANSYS的XK713数控铣床有限元分析及优化设计[D].南京：东南大学,2005：5-20.
[64]魏永庚,王知行.并联机床结构静刚度的有限元分析[J],机械与电子,2003,(1)：16-19
[65]庄茁,张帆等.非线性有限元分析与实例[M].北京：科学出版社,2005：36-37