求解共轭齿形的轮转曲线等距偏移法
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摘要
针对传统共轭齿形求解方法无法解决奇异点的问题,提出轮转曲线等距偏移法。分析了共轭曲线的等距偏移特性,推导出轮转曲线等距偏移线方程,基于该方程进行了圆弧齿廓的共轭齿形计算;采用圆弧逼近方式,以轮转曲线等距偏移线族求解任意齿廓的共轭齿形,并以含齿顶尖点的渐开线齿廓曲线为例进行共轭齿形计算;讨论了该方法的原理性误差,优化了该方法的求解精度。计算验证表明:相比于传统共轭齿形求解方法,轮转曲线等距偏移法能解决奇异点问题,且无需求解啮合方程,在齿廓曲线曲率半径变化率较小且存在奇异点时,用该方法求解共轭齿形优势明显。
In order to solve the problem of singular point in the traditional algorithm for solving conjugate gear profile,an equidistant offset algorithm of rotation curve was proposed.The equidistant offset characteristics of conjugate curves were analyzed and the equations of equidistant offset line of rotation curve were derived.By way of arc approximation,equidistant offset line family of rotation curve was used to solve conjugate curves of any gear profile,and the involute profile curve with a cuspidal point of addendum was taken as an example.The principle error of this algorithm was analyzed,and a method to solve precise conjugate points was proposed.Compared with the traditional method of solving conjugate gear profile,the equidistant offset algorithm of rotation curve has no singular point problem,and there is no need to solve the meshing equation.It has obvious advantages in solving conjugate gear profile with singular points and small change rate of curvature radius.
In order to solve the problem of singular point in the traditional algorithm for solving conjugate gear profile,an equidistant offset algorithm of rotation curve was proposed.The equidistant offset characteristics of conjugate curves were analyzed and the equations of equidistant offset line of rotation curve were derived.By way of arc approximation,equidistant offset line family of rotation curve was used to solve conjugate curves of any gear profile,and the involute profile curve with a cuspidal point of addendum was taken as an example.The principle error of this algorithm was analyzed,and a method to solve precise conjugate points was proposed.Compared with the traditional method of solving conjugate gear profile,the equidistant offset algorithm of rotation curve has no singular point problem,and there is no need to solve the meshing equation.It has obvious advantages in solving conjugate gear profile with singular points and small change rate of curvature radius.
引文
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[2]Litvin F L,Fuentes A.Gear geometry and applied theory[M].Cambridge:Cambridge University Press,2004.
[3]王家序,袁攀,李俊阳,等.基于不同啮合原理的谐波传动齿廓研究[J].华中科技大学学报(自然科学版),2017,45(3):58-64.WANG Jiaxu,YUAN Pan,LI Junyang,et al.Tooth profile research of harmonic drive based on different engagement theory[J].Journal of Huazhong University of Science and Technology(Natural Science Edition),2017,45(3):58-64.(in Chinese)
[4]Chiang C J,Fong Z H.Design of form milling cutters with multiple inserts for screw rotors[J].Mechanism and Machine Theory,2010,45(11):1613-1627.
[5]Wu Y R,Fong Z H,Zhang Z X.Simulation of a cylindrical form grinding process by the radial-ray shooting(RRS)method[J].Mechanism and Machine Theory,2010,45(2):261-272.
[6]李阁强,张龙飞,韩伟锋,等.双圆弧斜齿齿轮泵脉动特性分析及齿形设计[J].中国机械工程,2018,29(2):186-192.LI Geqiang,ZHANG Longfei,HAN Weifeng,et al.Pulsation characteristic analysis and tooth profile design of double-circular-arc helical gear pumps[J].China Mechanical Engineering,2018,29(2):186-192.
[7]Wen Q,Du Q G,Zhai X C.An analytical method for calculating the tooth surface contact stress of spur gears with tip relief[J].International Journal of Mechanical Sciences,2019,151:170-180.
[8]Radzevich S P.On the accuracy of precision involute hobs:an analytical approach[J].Journal of Manufacturing Processes,2007,9(2):121-136.
[9]Zhou Y S.Form grinding technology for the mold of powder metallurgy gears[J].Chinese Journal of Mechanical Engineering,2005,41(1):162.
[10]Wu Y R,Fong Z H,Zhang Z X.Simulation of a cylindrical form grinding process by the radial-ray shooting(RRS)method[J].Mechanism and Machine Theory,2010,45(2):261-272.
[11]李国龙,孙孟辉,李先广,等.螺旋面截形尖点矢量离散法及其在双圆弧滚刀过渡曲面最小化中的应用[J].机械工程学报,2011,47(17):187-192.LI Guolong,SUN Menghui,LI Xianguang,et al.Method of discreting cusp vectors of helicoid and its application in minimization of the relief grinding of double circular-arc gear hob[J].Journal of Mechanical Engineering,2011,47(17):187-192.(in Chinese)
[12]李国龙,林超,李先广,等.成形磨齿砂轮包络计算的双参数点矢量族法[J].重庆大学学报,2013,36(4):11-18.LI Guolong,LIN Chao,LI Xianguang,et al.Envelope to a two-parameter family of point vectors method for surface swept by wheel during 5-axis gear form grinding[J].Journal of Chongqing University(Natural Science Edition),2013,36(4):11-18.(in Chinese)
[13]He K,Li G L,Li X G.The second envelope method of point-vector and its application on worm wheel grinding modified gear[J].The International Journal of Advanced Manufacturing Technology,2017,88(9/10/11/12):3175-3184.
[14]Ji W T,Yao L G,Zhang J.Mathematical modeling and characteristics analysis for the nutation gear drive based on error parameters[J].Journal of Chongqing University(English Edition),2016,15(4):149-158.
[15]宋朝省,樊荣,刘立斌.小角度交错轴变厚齿轮齿根应力及影响因素分析[J].重庆大学学报(自然科学版),2014,37(1):1-6.SONG Chaosheng,FAN Rong,LIU Libin.Analysis on tooth root stress and influencing factors of crossed beveloid gears with small shaft angle[J].Journal of Chongqing University(Natural Science Edition),2014,37(1):1-6.(in Chinese)