滚珠丝杠副轴向接触刚度的研究
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摘要
滚珠丝杠副的轴向接触刚度是数控机床进给系统刚度最为薄弱的环节,对数控机床的性能有很大的影响。因此,为了能够为数控机床的控制系统提供更为精确的控制参数,需要对滚珠丝杠副的轴向接触刚度进行深入的研究。
本文以赫兹理论为基础,考虑了滚珠丝杠副中螺旋角的影响,通过对单螺母滚珠丝杠副中滚珠的受力、变形进行分析,建立了求解单螺母滚珠丝杠副轴向接触变形、轴向接触刚度的理论模型;在此基础上,通过分析双螺母预紧滚珠丝杠副的受力及位移协调关系,建立了求解双螺母滚珠丝杠副轴向接触变形、轴向接触刚度的理论模型。对所建立模型分析与研究的结果表明:增大螺旋角及预紧垫片的刚度可以提高滚珠丝杠副的轴向接触刚度。建立了分析滚珠丝杠副轴向接触变形的有限元模型,并对模型合理地划分了有限元网格,对不同载荷作用下滚珠丝杠副的轴向接触变形进行了有限元分析。有限元模型的分析结果与相应理论模型的求解结果能够较好地吻合在一起,从而验证了所建立的理论模型。
Ball screw mechanisms have been widely used in feeding mechanism of precision mechanics, especially in the numerical control machine tools, due to its performance such as high accurate positioning, high stiffness, low friction, etc. The CNC’s machining accuracy should be higher due to the development of modern manufaturing technique. In order to achieve this goal, the control of CNC’s feeding machanism should be more precise. The axial contact stiffness of ball screw should be studied, because it is the weakest link of stiffness of CNC’s feeding mechanism.
The axial contact deformation and axial contact stiffness of ball screw have been studied in this paper, based on Hertz theory and force analysis of ball screw. The main cotent is as followed.
1. The background of the task, the structure feature and working principle of ball screw are introduced. The present research situation and present production situation both in and out are summarized. The development trend of ball screw and the content of subject are presented.
2. The factors that influence the axial contact stiffness of ball screw, e.g. shape of raceway, contact angle, ratio of raceway, are introduced. The theoretical model, which is used to calculated the the number of balls of ball screw is founded, the influence of helix angle is considered in this model. Circulating mode of balls, pre-loaded mode of ball screw and the working principal of pre-loaded ball screw are introduced, the determination of preload force is described.
3. Two hypothesis are proposed to establish theoretical model, which is used to solve the axial contact deformation and axial contact stiffness of ball screw. (1) The influence of ball screw’s structure feature and manufacture error on load distribution is ignored. Consider that the load on every ball is equal. (2) The variation of contact angle is ignored, consider the contact angle is 45o unchanged. Hertz point contact theory is introduced, the method and process to solve Hertz point contact theory is discussed. The relations between axial force and normal force on ball, between axial deformation and normal deforamtion of ball are analyzed by establishing coordinate systerm. On this basis, the theoretical models which is used to solve axial contact deformation and axial contact stiffness of single-nut ball screw are established. Then, solve these models by numerical method, and draw the figures. The stiffness of single-nut ball screw becomes large as axial load and helix angle increased, which is derived from the analysis of axial contact stiffness model.
4. The axial contact deformation of double-nut ball screw is analyzed and studied, when there is only preload on double-nut ball screw(axial load is zero). On this basis, the deformation of double-nut ball screw is studied, when the axial load acts on ball screw, force equation and compatibility equation of displacement are established. The theoretical model, which is used to solve axial contact deformation and axial contact stiffness of double-nut ball screw, is derived, by the analysis of force equation and compatibility equation of displacement which have been established. Solve the theoretical model by numerical method, according to related dimensions and parameters, the result of double-nut ball screw is compared with result of sing-nut ball screw, and draw the figures. The stiffness of double-nut ball screw will be decreased as load increases. But the stiffness of double-but ball is larger than the stiffness of single-nut ball screw, and the former is steadier than the latter. The stiffness of ball screw will become larger as the helix angle and the sitffness of spacing disc increase.
5. The helix angle of ball screw if supposed to be zero, according to it’s structure feature, in this way , the finite element model of ball screw can be established reasonably. On this basis, the simplified finite element model of ball screw is established. In order to ensure the precision of calculation and save the resources of calculation, finite element meshes are generated reasonably. In this way, the meshes beside the contact point are denseness, and the meshes far away from the contact point are sparse. The contact deformation of ball screw under different axial load is analyzed by the method of finite element analysis, and the result is compared with the result which is solved by theoretical model. In this way the theoretical model of contcat deformation of ball screw is verified.
The study of this paper is helpful to the profound and system stduy on ball screw. According to this paper, the controlling parameter of CNC’s feeding mechanism will be more precise, and this will be helpful to improve CNC’s machining precision.
本文以赫兹理论为基础,考虑了滚珠丝杠副中螺旋角的影响,通过对单螺母滚珠丝杠副中滚珠的受力、变形进行分析,建立了求解单螺母滚珠丝杠副轴向接触变形、轴向接触刚度的理论模型;在此基础上,通过分析双螺母预紧滚珠丝杠副的受力及位移协调关系,建立了求解双螺母滚珠丝杠副轴向接触变形、轴向接触刚度的理论模型。对所建立模型分析与研究的结果表明:增大螺旋角及预紧垫片的刚度可以提高滚珠丝杠副的轴向接触刚度。建立了分析滚珠丝杠副轴向接触变形的有限元模型,并对模型合理地划分了有限元网格,对不同载荷作用下滚珠丝杠副的轴向接触变形进行了有限元分析。有限元模型的分析结果与相应理论模型的求解结果能够较好地吻合在一起,从而验证了所建立的理论模型。
Ball screw mechanisms have been widely used in feeding mechanism of precision mechanics, especially in the numerical control machine tools, due to its performance such as high accurate positioning, high stiffness, low friction, etc. The CNC’s machining accuracy should be higher due to the development of modern manufaturing technique. In order to achieve this goal, the control of CNC’s feeding machanism should be more precise. The axial contact stiffness of ball screw should be studied, because it is the weakest link of stiffness of CNC’s feeding mechanism.
The axial contact deformation and axial contact stiffness of ball screw have been studied in this paper, based on Hertz theory and force analysis of ball screw. The main cotent is as followed.
1. The background of the task, the structure feature and working principle of ball screw are introduced. The present research situation and present production situation both in and out are summarized. The development trend of ball screw and the content of subject are presented.
2. The factors that influence the axial contact stiffness of ball screw, e.g. shape of raceway, contact angle, ratio of raceway, are introduced. The theoretical model, which is used to calculated the the number of balls of ball screw is founded, the influence of helix angle is considered in this model. Circulating mode of balls, pre-loaded mode of ball screw and the working principal of pre-loaded ball screw are introduced, the determination of preload force is described.
3. Two hypothesis are proposed to establish theoretical model, which is used to solve the axial contact deformation and axial contact stiffness of ball screw. (1) The influence of ball screw’s structure feature and manufacture error on load distribution is ignored. Consider that the load on every ball is equal. (2) The variation of contact angle is ignored, consider the contact angle is 45o unchanged. Hertz point contact theory is introduced, the method and process to solve Hertz point contact theory is discussed. The relations between axial force and normal force on ball, between axial deformation and normal deforamtion of ball are analyzed by establishing coordinate systerm. On this basis, the theoretical models which is used to solve axial contact deformation and axial contact stiffness of single-nut ball screw are established. Then, solve these models by numerical method, and draw the figures. The stiffness of single-nut ball screw becomes large as axial load and helix angle increased, which is derived from the analysis of axial contact stiffness model.
4. The axial contact deformation of double-nut ball screw is analyzed and studied, when there is only preload on double-nut ball screw(axial load is zero). On this basis, the deformation of double-nut ball screw is studied, when the axial load acts on ball screw, force equation and compatibility equation of displacement are established. The theoretical model, which is used to solve axial contact deformation and axial contact stiffness of double-nut ball screw, is derived, by the analysis of force equation and compatibility equation of displacement which have been established. Solve the theoretical model by numerical method, according to related dimensions and parameters, the result of double-nut ball screw is compared with result of sing-nut ball screw, and draw the figures. The stiffness of double-nut ball screw will be decreased as load increases. But the stiffness of double-but ball is larger than the stiffness of single-nut ball screw, and the former is steadier than the latter. The stiffness of ball screw will become larger as the helix angle and the sitffness of spacing disc increase.
5. The helix angle of ball screw if supposed to be zero, according to it’s structure feature, in this way , the finite element model of ball screw can be established reasonably. On this basis, the simplified finite element model of ball screw is established. In order to ensure the precision of calculation and save the resources of calculation, finite element meshes are generated reasonably. In this way, the meshes beside the contact point are denseness, and the meshes far away from the contact point are sparse. The contact deformation of ball screw under different axial load is analyzed by the method of finite element analysis, and the result is compared with the result which is solved by theoretical model. In this way the theoretical model of contcat deformation of ball screw is verified.
The study of this paper is helpful to the profound and system stduy on ball screw. According to this paper, the controlling parameter of CNC’s feeding mechanism will be more precise, and this will be helpful to improve CNC’s machining precision.
引文
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[3] 姜洪奎.大导程滚珠丝杠副动力学性能及加工方法研究[D].山东大学,2007.
[4] 刘波.滚珠丝杠额定动载荷值得影响因素研究及其计算实现[D].浙江大学,2006.
[5] 刘莉.精密滚珠丝杠螺旋误差动态测试技术的研究[D].山东大学,2006.
[6] 洪宁.高速滚珠丝杠副综合性能测量系统设计[D].南京理工大学,2006.
[7] 肖正义.滚珠丝杠副的发展趋势[J].制造技术与机床,2000(4): 11-13.
[8] 郑子文.超精密机床伺服控制技术研究[D].国防科学技术大学,2001.
[9] Chin Chung Wei, Jen Fin Lin.Kinematic Analysis of the Ball Screw Mechanism Considering Variable Contact Deformations[J] . Journal of Mechanical Design, 2003(125): 717-733.
[10] D. Olaru, G. C. Puiu, L. C. Balan, V. Puiu.A New Model to Estimate Friction Torque In A Ball Screw System[J].Product Engineering, 2004 (6): 333-346.
[11] Xuesong Mei, Masaomi Tsutsumi, Tao Tao, Nuogang Sun.Study on the load distribution of ball screws with errors[J].Mechanism and Machine Theory, 2003 (38): 1257-1269.
[12] R. Whalley, M. Ebrahimi, A.A. Abdul-Ameer.Hybrid modelling of machine tool axis drives[J].International Journal of Machine Tools & Manufacture, 2005 (45): 1560-1576.
[13] Kripa K. Varanasi, Samir A. Nayfeh.The Dynamics of Lead-Screw Drives: Low-Order Modeling and Experiments[J].Journal of Dynamic Systems, Measurement, and Control, 2004(126): 388–396.
[14] Katuhiro Nakashima, Kazuki Takafuji.Stiffness of a Pre-Loaded Ball Screw[J].The Japan Society of Mechanical Engineers, 1987(11): 1898-1904.
[15] Katuhiro Nakashima, Kazuki Takafuji.Stiffness of a Ball Screw with Consideration of Deformation of the Screw, Nut and Screw Thread[J].The Japan Society of Mechanical Engineers, 1990(33): 620-626.
[16] Lin, Ming Ching, Ph.D. Design and mechanics of the ball screw mechanism[D].TheUniversity of Wisconisn-Madison, 1989.
[17] Leonard-Cristian Pop, Mircea Cretu, Liviu Morar. Methods of Evaluation of theMechanical Characteristics Influences On The Nc Ballscrew Drives DynamicBehaviour[J]. Journal of Mechanical Design, 2005(12): 336-345.
[18] Jen-Yu Liu, Meng-HUi Hsu, Fu-Chen Chen. On the design of rotating speed functions toimprove the acceleration peak value of ball-screw transmission mechanism[J]. Mechanismand Machine Theory, 2001 (36): 1035-1049.
[19] Kripa K. Varanasi, Samir A. Nayfeh. Modeling Identification and Control of BallscrewDrives[J]. Journal of Mechanical Design, 2005(15): 1685-1693.
[20] Paul I. Ro, Wonbo Shim, Sanghwa Jeong. Robust friction compensation forsubmicrometer positioning and tracking for a ball-screw-driven slide system[J]. PrecisionEngineering, 2000 (24): 160-173.
[21] Min-Seok Kim, Sung-Chong Chung. Integrated design methodology of ball-screw drivenservomechanisms with discrete controllers[J]. Mechatronics, 2006(16): 491-502.
[22] Hertz. H. On the contact of elastic solids[J]. J. Reine and Angew, 1881.
[23] Stribeck. R. Ball bearing for various loads[J]. Trans ASME, 1907.
[24] M.C. Lin, B. Ravani, S.A. Velinsky. Kinematics of the ball screw mechanismfJ]. Journalof Mechanical Design.Transactions of the ASME, 1994(116): 549-855.
[25] H. Shimoda. Stiffness analysis of ball screw[J]. Journal of the Japan Society for PrecisionEngineering, 1998 (64): 1265-1273.
[26] J.F. Cuttino, T.A. Dow, B.F. Knight. Analytical and experimental identification ofnonlinearities in a single-nut, preloaded ball screw[J]. Journal of Mechanical Design,Transactions of the ASME, 1997(119): 15-19.
[27]杜平安．滚珠直旋副滚到弹性接触分析[J]．电子科技大学学报，1994(23)：280-285．
[28]俞志平，冯志民．滚珠丝杠螺母副的有关参数及计算[J]．上海机床，1996(3)：50-53．
[29]刘晓慧，宋现春．滚珠丝杠副摩擦力矩影响因素及测试方法研究[J]．山东大学学报，2006(40)：23-28．
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