基于几何误差不确定性的滚动导轨运动误差研究
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摘要
产品设计阶段,滚动导轨几何误差是未知的,具有不确定性,其运动精度亦是不确定的。为了研究设计阶段滚动导轨几何误差不确定对其运动误差的影响规律,提出将不确定的几何误差等效为滚动体弹性变形的运动误差分析方法。采用区间参数和小位移旋量描述滚动导轨的几何误差,并采用蒙特卡罗模拟(Monte Carlo simulation, MCS)方法进行不确定模拟仿真,获得满足约束的几何误差分布范围。基于此,将滚动导轨几何误差等效为滚动体的弹性变形并根据赫兹接触理论建立滚动导轨的静力平衡方程,从而实现了滚动导轨运动误差的不确定性分析。以某一滚动导轨为例,基于提出的方法分析了滚动导轨几何误差不确定性下其运动误差的分布规律,并分析了不同直线度公差、预紧力以及外载荷对导轨运动误差的影响规律。
In product design stage, the geometric error of rolling guide is unknown and uncertain, and its motion precision is also uncertain. To research the influence of uncertain geometric error on the motion error of rolling guide, an analysis method of motion error with the uncertain geometric error being equivalent to the elastic deformation of rolling balls is proposed. Interval parameters and small displacement torsor are used to describe the uncertainties of geometric error, and Monte Carlo simulation(MCS) method is used to obtain distribution range of geometric error restricted by constraint inequalities. Taking the distribution range of geometric error torsor parameters as modelling samples, static force equilibrium equations of rolling guide system are established with geometrical error and motion error of rolling guide being equivalently mapped into the elastic deformation of the rolling elements via the Hertz contact theory. With a case of certain rolling guide, the changing rules of rolling guide motion error in the conditions of uncertainty of straightness error of rolling guide are obtained, and the effects of straightness tolerance, preloads and external loads under different design requirements on motion error are analysed.
In product design stage, the geometric error of rolling guide is unknown and uncertain, and its motion precision is also uncertain. To research the influence of uncertain geometric error on the motion error of rolling guide, an analysis method of motion error with the uncertain geometric error being equivalent to the elastic deformation of rolling balls is proposed. Interval parameters and small displacement torsor are used to describe the uncertainties of geometric error, and Monte Carlo simulation(MCS) method is used to obtain distribution range of geometric error restricted by constraint inequalities. Taking the distribution range of geometric error torsor parameters as modelling samples, static force equilibrium equations of rolling guide system are established with geometrical error and motion error of rolling guide being equivalently mapped into the elastic deformation of the rolling elements via the Hertz contact theory. With a case of certain rolling guide, the changing rules of rolling guide motion error in the conditions of uncertainty of straightness error of rolling guide are obtained, and the effects of straightness tolerance, preloads and external loads under different design requirements on motion error are analysed.
引文
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[16]崔鑫磊,李威.基于蒙特卡洛模拟的6-UPS并联机构的误差分析[J].机械设计与制造,2016(6):89-92.CUI Xinlei,LI Wei.Error analysis of 6-UPS parallel mechanisms based on Monte Carlo simulation[J].Machinery Design&Manufacture,2016(6):89-92.
[17]吴兆强.基于小位移旋量的公差模拟建模及公差分析[J].机械设计与制造,2010(1):205-207.WU Zhaoqiang.Stimulated tolerances modeling based on small displacement torsors and tolerances analysis[J].Machinery Design&Manufacture,2010(1):205-207.
[18]BOURDET P,MATHIEU L,LARTIGUE C,et al.The concept of the small displacement torsor in metrology[J].Adv.Math.Tools Metrol.,1996,40:110-122.
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[20]贾振元,王福吉.机械制造技术基础[M].北京:科学出版社,2011.JIA Zhenyuan,WANG Fuji.Foundation of machine manufacturing technology[M].Beijing:Science Press,2011.
[21]MAJDA P.Modeling of geometric errors of linear guideway and their influence on joint kinematic error in machine tools[J].Precision Engineering,2012,36:369-378.
[2]程强,刘广博,刘志峰,等.基于敏感度分析的机床关键性几何误差源识别方法[J].机械工程学报,2012,48(7):171-179.CHENG Qiang,LIU Guangbo,LIU Zhifeng,et al.An identification approach for key geometric error sources of machine tool based on sensitivity analysis[J].Journal of Mechanical Engineering,2012,48(7):171-179.
[3]胡鹏浩,张建坤,马晓庆,等.3-PUU并联测量机滑块运动误差分析[J].机械工程学报,2015,51(18):1-6.HU Penghao,ZHANG Jiankun,MA Xiaoqing,et al.Analysis of slider motion error on 3-PUU parallel coordinate measuring maching[J].Journal of Mechanical Engineering,2015,51(18):1-6.
[4]要小鹏,殷国富,李光明.数控机床进给轴综合误差解耦建模与补偿研究[J].机械工程学报,2016,52(1):184-192.YAO Xiaopeng,YIN Guofu,LI Guangming.Positioning error of feed axis decouple-separating modeling and compensating research for CNC machine tools[J].Journal of Mechanical Engineering,2016,52(1):184-192.
[5]OHTA H,TANAKA K.Vertical stiffness of preloaded linear guideway type ball bearing incorporating the flexibility of the carriage and rail[J].Journal of Tribology-Transactions of the ASME,2010,132(1):1-9.
[6]JEONG J,KANG E,JEONG J.Equivalent stiffness modeling of linear motion guideways for stage systems[J].International Journal of Precision Engineering and Manufacturing,2014,15(9):1987-1993.
[7]KHIM G,OH J S,PARK C H.Analysis of 5-DOF motion errors influenced by the guide rails of an aerostatic linear motion stage[J].International Journal of Precision Engineering and Manufacturing,2014,15(2):283-290.
[8]KHIM G,PARK C H,OH J S.A method of calculating motion error in a linear motion bearing stage[J].The Scientific World Journal,2015:696417.
[9]RACKWITZ R,FlESSLER B.Structural reliability under combined random load sequences[J].Computers&Structures,1978,9(5):489-494.
[10]YANG Z Z.Reliability-based robust optimization design of automobile components with non-normal distribution parameters[J].Chinese Journal of Mechanical Engineering,2013,26(4):823-830.
[11]刘成武,李连升,钱林方.随机与区间不确定下基于近似灵敏度的序列多学科可靠性设计优化[J].机械工程学报,2015,51(21):174-184.LIU Chengwu,LI Liansheng,QIAN Linfang.Sequential multidisciplinary reliability design and optimization based on approximate sensitivity method under random and interval uncertainties[J].Journal of Mechanical Engineering,2015,51(21):174-184.
[12]吕辉,于德介,陈宁,等.基于可靠性的混合不确定参数汽车盘式制动器振动稳定性分析[J].机械工程学报,2014,50(6):112-119.LU(5)(5)Hui,YU Dejie,CHEN Ning,et al.Reliability analysis for vibration stability of automotive disc brake system with hybrid uncertain parameters[J].Journal of Mechanical Engineering,2014,50(6):112-119.
[13]黎凯,杨旭静,郑娟.基于参数和代理模型不确定性的冲压稳健性设计优化[J].中国机械工程,2015,26(23):3234-3239.LI Kai,YANG Xujing,ZHENG Juan.Robust design optimization for stamping based on parametric and metamodel uncertainty[J].China Mechanical Engineering,2015,26(23):3234-3239.
[14]熊芬芬.稳健优化设计中代理模型不确定性的研究[J].机械工程学报,2014,50(19):136-143.XIONG Fenfen.Robust design optimization considering metamodel uncertainty[J].Journal of Mechanical Engineering,2014,50(19):136-143.
[15]余治民,刘子建,董思科,等.基于蒙特卡洛模拟与响应面方法的公差建模[J].中国机械工程,2015,26(4):427-434.YU Zhimin,LIU Zijian,DONG Sike,et al.Tolerance modeling based on Monte-Carlo simulation and response surface method[J].China Mechanical Engineering,2015,26(4):427-434.
[16]崔鑫磊,李威.基于蒙特卡洛模拟的6-UPS并联机构的误差分析[J].机械设计与制造,2016(6):89-92.CUI Xinlei,LI Wei.Error analysis of 6-UPS parallel mechanisms based on Monte Carlo simulation[J].Machinery Design&Manufacture,2016(6):89-92.
[17]吴兆强.基于小位移旋量的公差模拟建模及公差分析[J].机械设计与制造,2010(1):205-207.WU Zhaoqiang.Stimulated tolerances modeling based on small displacement torsors and tolerances analysis[J].Machinery Design&Manufacture,2010(1):205-207.
[18]BOURDET P,MATHIEU L,LARTIGUE C,et al.The concept of the small displacement torsor in metrology[J].Adv.Math.Tools Metrol.,1996,40:110-122.
[19]ZHONG X,YANG R Q,ZHOU B.Accuracy analysis of assembly success rate with Monte Carlo simulations[J].Journal of Donghua University,2003,20(4):128-131.
[20]贾振元,王福吉.机械制造技术基础[M].北京:科学出版社,2011.JIA Zhenyuan,WANG Fuji.Foundation of machine manufacturing technology[M].Beijing:Science Press,2011.
[21]MAJDA P.Modeling of geometric errors of linear guideway and their influence on joint kinematic error in machine tools[J].Precision Engineering,2012,36:369-378.