基于曲面包络逼近原理的点啮合齿面整体设计方法
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摘要
基于曲面活动标架理论,提出了点啮合齿面整体设计的曲面包络逼近方法:以第一齿面包络特征线为样条曲线,以第二齿面沿接触点迹线的齿面局部结构作为曲面插值条件,通过第一齿面运动不变量描述的运动变换,构造完全由第一齿面的运动不变量描述的第一齿面共轭运动空间作为第二齿面整体优化设计空间,确保齿面设计方法的可全局优化性和可加工性。根据点啮合齿面加载接触实验数据,以空载啮合时两齿面间隙作为预测和控制重载情况下的接触区形状和大小的设计目标参数,建立点啮合齿面整体优化设计的变分不等式模型。与以往点啮合齿面设计和齿面接触分析(TCA)方法相比,该齿面设计方法摆脱了以往齿面接触分析(TCA)中以特定机床加工参数作为齿面重构参数而给齿面优化设计带来的不必要的限制,能在满足预定的加载的啮合特性要求的同时,确保齿面能用盘铣刀通过多轴联动数控加工方法以展成法加工。仿真实例计算和分析表明:本齿面设计方法能形成完整的弧齿锥齿轮齿面,该齿面在满载啮合时,重合度达到2.122,传动误差幅值降低至0.063°,具有优良的加载啮合特性。
A method(envelope-approximation theory)based on the surface moving frame was presented to design the global properties of point-contact tooth surface.The presented method proposed a universal approach to reconstruct the second tooth surface with motion invariants of the first tooth surface.The proposal reconstruction approach was formulated for approximating the conjugate surface of the first tooth surface to the first tooth surface under the movement transformation described with motion invariants of the first tooth surface.The basis function of the previously mentioned approximation was taken as the charac-teristic curve on the first tooth surface,the interpolation condition was designated as the local structure of the second tooth surface along the contact curve.Based on the experiment data of tooth contact,a variational inequality model for the global optimal design of tooth surfaces was established to determine the loaded meshing characteristics of the second tooth surface according to the position,orientation and dimension of the contact area on the first tooth surface under load-deformation condition.Different from the previous studies about tooth contact analysis,the presented method,freeing the tooth surface optimization design from unnecessary restrictions of the specific machine tool processing parameters,can satisfy the predesigned loaded meshing characteristics and ensure that the tooth surface can be generated by multi-axis flank milling with a conical cutter.A numerical example was given to validate that above-mentioned design method can construct unbroken spiral bevel gear tooth surface with excellent loaded meshing characteristics,for exanple,the loaded contact ratio was increased to 2.122 and the loaded transmission errors amplitude was lowered to 0.063°.
A method(envelope-approximation theory)based on the surface moving frame was presented to design the global properties of point-contact tooth surface.The presented method proposed a universal approach to reconstruct the second tooth surface with motion invariants of the first tooth surface.The proposal reconstruction approach was formulated for approximating the conjugate surface of the first tooth surface to the first tooth surface under the movement transformation described with motion invariants of the first tooth surface.The basis function of the previously mentioned approximation was taken as the charac-teristic curve on the first tooth surface,the interpolation condition was designated as the local structure of the second tooth surface along the contact curve.Based on the experiment data of tooth contact,a variational inequality model for the global optimal design of tooth surfaces was established to determine the loaded meshing characteristics of the second tooth surface according to the position,orientation and dimension of the contact area on the first tooth surface under load-deformation condition.Different from the previous studies about tooth contact analysis,the presented method,freeing the tooth surface optimization design from unnecessary restrictions of the specific machine tool processing parameters,can satisfy the predesigned loaded meshing characteristics and ensure that the tooth surface can be generated by multi-axis flank milling with a conical cutter.A numerical example was given to validate that above-mentioned design method can construct unbroken spiral bevel gear tooth surface with excellent loaded meshing characteristics,for exanple,the loaded contact ratio was increased to 2.122 and the loaded transmission errors amplitude was lowered to 0.063°.
引文
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[2]吴训成.基于功能需求的弧齿锥齿轮齿面主动设计与先进制造技术研究[D].西安:西安交通大学,2000.WU Xuncheng.Research on function active tooth surface design and advanced manufacturing technology for the curved tooth bevel gears[D].Xi'an:Xi'an Jiaotong University,2000.(in Chinese)
[3]邓效忠,蒋闯.在共轭基础上的准双曲面齿轮主动修形设计[J].河南科技大学学报(自然科学版),2012,33(5):32-36.DENG Xiaozhong,JIANG Chuang.Function-oriented modification design for tooth surfaces of hypoid gears based on conjugate theory[J].Journal of Henan University of Science and Technology(Natural Science),2012,33(5):32-36.(in Chinese)
[4]邓效忠,李天兴,李聚波,等.汽车驱动桥准双曲面齿轮齿面测量误差精确计算[J].中国机械工程,2012,23(12):1502-1509.DENG Xiaozhong,LI Tianxing,LI Jubo,et al.Accurate calculation of tooth surface deviation based on automobile driving axle hypoid gear[J].China Mechanical Engineering,2012,23(12):1502-1509.(in Chinese)
[5]方宗德,曹雪梅,沈云波,等.弧线齿面齿轮的齿面设计与加工[J].航空动力学报,2010,25(1):224-227.FANG Zongde,CAO Xuemei,SHEN Yunbo,et al.Tooth surface design and manufacture for arcuate tooth trace face-gear[J].Journal of Aerospace Power,2010,25(1):224-227.(in Chinese)
[6]曹雪梅,方宗德,许浩,等.弧齿锥齿轮的齿面主动设计及试验验证[J].机械工程学报,2008,44(7):209-214.CAO Xuemei,FANG Zongde,XU Hao,et al.Function-oriented active tooth surface design of spiral bevel gears and experimental tests[J].Journal of Mechanical Engineering,2008,44(7):209-214.(in Chinese)
[7]苏智剑,吴序堂,毛世民,等.基于齿面参数化表示的准双曲面齿轮的设计[J].西安交通大学学报,2005,39(1):17-20.SU Zhijian,WU Xutang,MAO Shimin,et al.Design of hypoid gear tooth surface represented by non-uniform rational B-spline polynomial[J].Journal of Xi'an Jiaotong University,2005,39(1):17-20.(in Chinese)
[8]SHIH Y P,CHEN S D.A flank correction methodology for a five-axis CNC gear profile grinding machine[J].Mechanism and Machine Theory,2012,47(1):31-45.
[9]GABICCINI M,BRACCI A.On the estimation of continuous mappings from cradle-style to 6-axis machines for facemilled hypoid gear generation[J].Mechanism and Machine Theory,2011,46(10):1492-1506.
[10]ARTONI A,GABICCINI M.Ease-off based compensation of tooth surface deviations for spiral bevel and hypoid gears:only the pinion needs corrections[J].Mechanism and Machine Theory,2013,61(3):84-101.
[11]郭晓东,张卫青,张明德,等.螺旋锥齿轮切齿调整参数的精确反调[J].重庆大学学报,2011,34(3):32-37.GUO Xiaodong,ZHANG Weiqing,ZHANG Mingde,et al.The machine setting reverse method of spiral bevel gear[J].Journal of Chongqing University,2011,34(3):32-37.(in Chinese)
[12]李天兴,邓效忠,李聚波,等.螺旋锥齿轮齿面误差分析与自动反馈修正[J].航空动力学报,2011,26(5):1194-1200.LI Tianxing,DENG Xiaozhong,LI Jubo,et al.Automatic feedback correction and deviation analysis for tooth surface of spiral bevel and hypoid gear[J].Journal of Aerospace Power,2011,26(5):1194-1200.(in Chinese)
[13]周凯红,唐进元,曾韬.基于预定啮合特性的点啮合齿轮的CNC制造技术研究[J].机械工程学报,2009,45(9):173-182.ZHOU Kaihong,TANG Jinyuan,ZENG Tao.CNC manufacturing technology of point meshing gear based on predetermined meshing characteristics[J].Journal of Mechanical Engineering,2009,45(9):173-182.(in Chinese)
[14]周凯红.基于预定啮合特性的螺旋锥齿轮点啮合齿面设计及CNC制造技术研究[D].长沙:中南大学,2009.ZHOU Kaihong.Research on meshing characteristicsbased design and CNC technology of point-contact spiral bevel gears tooth surface[D].Changsha:Central South University,2009.(in Chinese)
[15]ZHOU Kaihong,TANG Jinyuan.Envelope-approximation theory of manufacture technology for point-contact tooth surface on six-axis CNC hypoid generator[J].Mechanism and Machine Theory,2011,46(6):806-819.
[16]周凯红,唐进元,严宏志.基于预定啮合特性的点啮合齿面设计方法[J].航空动力学报,2009,24(11):2612-2617.ZHOU Kaihong,TANG Jinyuan,YAN Hongzhi.Research on meshing characteristics-based design of point-contact tooth surface[J].Journal of Aerospace Power,2009,24(11):2612-2617.(in Chinese)
[17]WAGAI P,KAHRAMAN A.Influence of tooth profile modification on helical gear durability[J].Journal of Mechanical Design,2002,124(3):501-510.
[18]邓效忠,方宗德,魏冰阳,等.高重合度弧齿锥齿轮的性能分析与实验研究[J].航空动力学报,2003,18(6):744-748.DENG Xiaozhong,FANG Zongde,WEI Bingyang,et al.Analysis of meshing behavior and experiments of spiral bevel gears with high contact ratio[J].Journal of Aerospace Power,2003,18(6):744-748.(in Chinese)
[2]吴训成.基于功能需求的弧齿锥齿轮齿面主动设计与先进制造技术研究[D].西安:西安交通大学,2000.WU Xuncheng.Research on function active tooth surface design and advanced manufacturing technology for the curved tooth bevel gears[D].Xi'an:Xi'an Jiaotong University,2000.(in Chinese)
[3]邓效忠,蒋闯.在共轭基础上的准双曲面齿轮主动修形设计[J].河南科技大学学报(自然科学版),2012,33(5):32-36.DENG Xiaozhong,JIANG Chuang.Function-oriented modification design for tooth surfaces of hypoid gears based on conjugate theory[J].Journal of Henan University of Science and Technology(Natural Science),2012,33(5):32-36.(in Chinese)
[4]邓效忠,李天兴,李聚波,等.汽车驱动桥准双曲面齿轮齿面测量误差精确计算[J].中国机械工程,2012,23(12):1502-1509.DENG Xiaozhong,LI Tianxing,LI Jubo,et al.Accurate calculation of tooth surface deviation based on automobile driving axle hypoid gear[J].China Mechanical Engineering,2012,23(12):1502-1509.(in Chinese)
[5]方宗德,曹雪梅,沈云波,等.弧线齿面齿轮的齿面设计与加工[J].航空动力学报,2010,25(1):224-227.FANG Zongde,CAO Xuemei,SHEN Yunbo,et al.Tooth surface design and manufacture for arcuate tooth trace face-gear[J].Journal of Aerospace Power,2010,25(1):224-227.(in Chinese)
[6]曹雪梅,方宗德,许浩,等.弧齿锥齿轮的齿面主动设计及试验验证[J].机械工程学报,2008,44(7):209-214.CAO Xuemei,FANG Zongde,XU Hao,et al.Function-oriented active tooth surface design of spiral bevel gears and experimental tests[J].Journal of Mechanical Engineering,2008,44(7):209-214.(in Chinese)
[7]苏智剑,吴序堂,毛世民,等.基于齿面参数化表示的准双曲面齿轮的设计[J].西安交通大学学报,2005,39(1):17-20.SU Zhijian,WU Xutang,MAO Shimin,et al.Design of hypoid gear tooth surface represented by non-uniform rational B-spline polynomial[J].Journal of Xi'an Jiaotong University,2005,39(1):17-20.(in Chinese)
[8]SHIH Y P,CHEN S D.A flank correction methodology for a five-axis CNC gear profile grinding machine[J].Mechanism and Machine Theory,2012,47(1):31-45.
[9]GABICCINI M,BRACCI A.On the estimation of continuous mappings from cradle-style to 6-axis machines for facemilled hypoid gear generation[J].Mechanism and Machine Theory,2011,46(10):1492-1506.
[10]ARTONI A,GABICCINI M.Ease-off based compensation of tooth surface deviations for spiral bevel and hypoid gears:only the pinion needs corrections[J].Mechanism and Machine Theory,2013,61(3):84-101.
[11]郭晓东,张卫青,张明德,等.螺旋锥齿轮切齿调整参数的精确反调[J].重庆大学学报,2011,34(3):32-37.GUO Xiaodong,ZHANG Weiqing,ZHANG Mingde,et al.The machine setting reverse method of spiral bevel gear[J].Journal of Chongqing University,2011,34(3):32-37.(in Chinese)
[12]李天兴,邓效忠,李聚波,等.螺旋锥齿轮齿面误差分析与自动反馈修正[J].航空动力学报,2011,26(5):1194-1200.LI Tianxing,DENG Xiaozhong,LI Jubo,et al.Automatic feedback correction and deviation analysis for tooth surface of spiral bevel and hypoid gear[J].Journal of Aerospace Power,2011,26(5):1194-1200.(in Chinese)
[13]周凯红,唐进元,曾韬.基于预定啮合特性的点啮合齿轮的CNC制造技术研究[J].机械工程学报,2009,45(9):173-182.ZHOU Kaihong,TANG Jinyuan,ZENG Tao.CNC manufacturing technology of point meshing gear based on predetermined meshing characteristics[J].Journal of Mechanical Engineering,2009,45(9):173-182.(in Chinese)
[14]周凯红.基于预定啮合特性的螺旋锥齿轮点啮合齿面设计及CNC制造技术研究[D].长沙:中南大学,2009.ZHOU Kaihong.Research on meshing characteristicsbased design and CNC technology of point-contact spiral bevel gears tooth surface[D].Changsha:Central South University,2009.(in Chinese)
[15]ZHOU Kaihong,TANG Jinyuan.Envelope-approximation theory of manufacture technology for point-contact tooth surface on six-axis CNC hypoid generator[J].Mechanism and Machine Theory,2011,46(6):806-819.
[16]周凯红,唐进元,严宏志.基于预定啮合特性的点啮合齿面设计方法[J].航空动力学报,2009,24(11):2612-2617.ZHOU Kaihong,TANG Jinyuan,YAN Hongzhi.Research on meshing characteristics-based design of point-contact tooth surface[J].Journal of Aerospace Power,2009,24(11):2612-2617.(in Chinese)
[17]WAGAI P,KAHRAMAN A.Influence of tooth profile modification on helical gear durability[J].Journal of Mechanical Design,2002,124(3):501-510.
[18]邓效忠,方宗德,魏冰阳,等.高重合度弧齿锥齿轮的性能分析与实验研究[J].航空动力学报,2003,18(6):744-748.DENG Xiaozhong,FANG Zongde,WEI Bingyang,et al.Analysis of meshing behavior and experiments of spiral bevel gears with high contact ratio[J].Journal of Aerospace Power,2003,18(6):744-748.(in Chinese)