变位直齿轮副齿顶修形参数设计
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摘要
该文以变位直齿轮副的齿顶修形为研究对象,考虑齿轮啮合的非线性接触、修正基体刚度以及延长啮合的影响,建立了考虑齿顶修形的变位直齿轮副时变啮合刚度解析模型,并通过有限元方法验证了该模型的正确性;以对刚度进行快速傅里叶变换(FFT)得到的前五阶幅值之和最小为设计目标,获得了齿顶修形的最优参数范围,并通过有限元模型进行应力分析,反证了该范围的正确性。研究结果表明:基于啮合刚度FFT前5阶幅值之和最小的设计方法可以更为高效地计算并锁定最优修形参数的范围,通过有限元进行应力分析可进一步验证齿顶修形的最优参数;齿轮齿顶修形后,刚度谐波量和应力均明显减小,有助于降低齿轮系统的振动和噪声。研究结果可为变位直齿轮副齿顶修形设计提供理论方法与依据。
An analytical model is developed for a spur gear pair with an addendum modification to concern the tip relief when calculating time-varying mesh stiffness, by considering the effects of nonlinear contact stiffness, of revised fillet-foundation stiffness and of extended tooth contact. The proposed model is verified by the finite element(FE) method. The ranges of optimum parameters for tip relief are acquired by making the sum of the first five-step amplitude of fast Fourier transform(FFT) for mesh stiffness minimum. Furthermore, stress analysis is conducted to prove the optimal modification parameters based on the FE method. Results showed that the design method based on the sum of the first five-step amplitude of FFT for mesh stiffness minimum can determine efficiently the optimal modification parameter range and the stress analysis based on the finite element method can further verify the optimal parameters of tip relief. The fluctuation of stiffness and stress decrease obviously after tip relief, which can contribute to the reduction of vibration and noise in the meshing process of gear pairs. The results can provide a theoretical basis for the tip relief design of a spur gear pair with an addendum modification.
An analytical model is developed for a spur gear pair with an addendum modification to concern the tip relief when calculating time-varying mesh stiffness, by considering the effects of nonlinear contact stiffness, of revised fillet-foundation stiffness and of extended tooth contact. The proposed model is verified by the finite element(FE) method. The ranges of optimum parameters for tip relief are acquired by making the sum of the first five-step amplitude of fast Fourier transform(FFT) for mesh stiffness minimum. Furthermore, stress analysis is conducted to prove the optimal modification parameters based on the FE method. Results showed that the design method based on the sum of the first five-step amplitude of FFT for mesh stiffness minimum can determine efficiently the optimal modification parameter range and the stress analysis based on the finite element method can further verify the optimal parameters of tip relief. The fluctuation of stiffness and stress decrease obviously after tip relief, which can contribute to the reduction of vibration and noise in the meshing process of gear pairs. The results can provide a theoretical basis for the tip relief design of a spur gear pair with an addendum modification.
引文
[1]李润方.齿轮传动的刚度分析和修形方法[M].重庆:重庆大学出版社,1998:262―286.Li Runfang.Stiffness analysis and modification of gear transmission[M].Chongqing:Chongqing University Press,1998:262―286.(in Chinese)
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[3]孙志礼,冷兴聚,魏延刚,等.机械设计[M].沈阳:东北大学出版社,2000:102―126.Sun Zhili,Leng Xingju,Wei Yangang,et al.Mechanical design[M].Shenyang:Northeastern University Press,2000:102―126.(in Chinese)
[4]杜飞.两级行星齿轮箱的轮齿修形方法研究[D].重庆大学,2015.Du Fei.Research on tooth modification method of two-stage planetary gearbox[D].Chongqing University,2015.(in Chinese)
[5]魏永祥,陈建军,王敏娟.随机参数齿轮-转子系统扭转振动固有频率分析[J].工程力学,2011,28(4):172―177.Wei Yongxiang,Chen Jianjun,Wang Minjuan.Natural frequency analysis for torsional vibration of a gear-rotor system with random parameters[J].Engineering Mechanics,2011,28(4):172―177.(in Chinese)
[6]魏永祥,陈建军,马洪波.随机参数齿轮系统的非线性动力响应分析[J].工程力学,2012,29(11):319―324.Wei Yongxiang,Chen Jianjun,Ma Hongbo.Analysis of nonlinear dynamic response of gear-rotor with random parameters[J].Engineering Mechanics,2012,29(11):319―324.(in Chinese)
[7]唐进元,陈思雨,钟掘.一种改进的齿轮非线性动力学模型[J].工程力学,2008,25(1):217―223.Tang Jinyuan,Chen Siyu,Zhong Jue.An improved nonlinear model for a spur gear pair system[J].Engineering Mechanics,2008,25(1):217―223.(in Chinese)
[8]张希康.渐开线齿轮齿顶修缘的计算与加工[J].南通市机械研究所,1985,9(6):22―25.Zhang Xikang.Calculation and machining of involute gear with tip modification[J].Nantong Machinery Research Institute,1985,9(6):22―25.(in Chinese)
[9]汤鱼,常山,何玉龙,等.齿廓修形对齿面接触力影响的研究[J].冷加工通用机械,2010(4):7―9.Tang Yu,Chang Shan,He Yulong,et al.Study on influence of tooth profile modification to contact force[J].Cold Work and General Machinery,2010(4):7―9.(in Chinese)
[10]尚振国,刘辉,谢忠东,等.齿轮修形原理及应用[J].大连水产学院学报,2009,24(增刊1),220―221.Shang Zhenguo,Liu Hui,Xie Zhongdong,et al.Principle and application of gear modification[J].Journal of Dalian Fisheries University,2009,24(Suppl1),220―221.(in Chinese)
[11]马辉,逄旭,宋溶泽,等.考虑齿顶修缘的齿轮-转子系统振动响应分析[J].机械工程学报,2014,50(7):39―45.Ma Hui,Pang Xu,Song Rongze,et al.Vibration response analysis of a geared rotor system considering the tip relief[J].Journal of Mechanical Engineering,2014,50(7):39―45.(in Chinese)
[12]Chen Z G,Shao Y M.Mesh stiffness calculation of a spur gear pair with tooth profile modification and tooth root crack[J].Mechanism and Machine Theory,2013,62:63―74.
[13]Ma H,Zeng J,Feng R J,et al.An improved analytical model for mesh stiffness calculation of spur gears with tip relief[J].Mechanism and Machine Theory,2016,98:64―80.
[14]Diez-Ibarbia A,Fernandez-del-Rincon A,Iglesias M,et al.Efficiency assessment in spur gears with shifting and profile modifications[J].Mechanisms and Machine Science,2017,43:193―201.
[15]Pedrero J,Pleguezuelos M,B.Sánchez M.On the determination of the meshing stiffness and the load sharing of spur gears[J].Mechanisms and Machine Science,2017,43:221―229.
[16]Chen Z G,Zhai W M,Shao Y M,et al.Mesh stiffness evaluation of an internal spur gear pair with tooth profile shift[J].Science China Technological Science,2016,59(9):1328―1339.
[17]Ma H,Yang J,Song R Z,et al.Effects of tip relief on vibration responses of a geared rotor system[J].Proceedings of the Institution of Mechanical Engineers,Part C:Journal of Mechanical Engineering Science,2014,228(7):1132―1154.
[18]Ma H,Pang X,Wang Q B,et al.Commentary on effects of tip relief on vibration responses of a geared rotor system[J].Proceedings of the Institution of Mechanical Engineers,Part C:Journal of Mechanical Engineering Science,2017,23(11):2159―2169.
[19]Lin T J,He Z Y.Analytical method for coupled transmission error of helical gear system with machining errors,assembly errors and tooth modifications[J].Mechanical Systems&Signal Processing,2017,91:167―182.
[20]Wang S S,Zhu C C,Song C S,et al.Effects of gear modifications on the dynamic characteristics of wind turbine gearbox considering elastic support of the gearbox[J].Journal of Mechanical Science&Technology,2017,31(3):1079―1088.
[21]Wei J,Zhang A Q,Qin D T,et al.A coupling dynamics analysis method for a multistage planetary gear system[J].Mechanism and Machine Theory,2017,110:27―49.
[22]岳喜铮,丁问司,丁康,等.行星轮系渐开线变位齿轮时变啮合刚度数值计算[J].重庆理工大学学报,2016,30(6):38―44.Yue Xizheng,Ding Wensi,Ding Kang,et al.Numerical calculation of planetary gear train involute modified gear time-varying meshing stiffness[J].Journal of Chongqing University of Technology(Natural Science),2016,30(6):38―44.(in Chinese)
[23]乔福瑞.基于ANSYS的齿轮接触应力与啮合刚度研究[D].大连:大连理工大学,2013.Qiao Furui.Research of contact stress and meshing stiffness of gear with profile error[D].Dalian:Dalian University of Technology,2013.(in Chinese)
[24]魏静,孙清超,孙伟,等.大型风电齿轮箱系统耦合动态特性研究[J].振动与冲击,2012,31(08):16―23.Wei Jing,Sun Qingchao,Sun Wei,et al.Dynamical coupling characteristics of a large wind turbine gearbox transmission system[J].Journal of Vibration and Shock,2012,31(08):16―23.(in Chinese)
[25]Weck M,Mauer G.Optimum tooth flank corrections for helical gears[J].Journal of Mechanical Design,1990,112(4):584.
[26]Hedlund J,Lehtovaara A.A parameterized numerical model for the evaluation of gear mesh stiffness variation of a helical gear pair[J].Drive System Technique,2012,222(7):1321―1327.
[2]李树军.机械原理[M].沈阳:东北大学出版社,2002:102―122.Li Shujun.Mechanical principle[M].Shenyang:Northeastern University Press,2002:102―122.(in Chinese)
[3]孙志礼,冷兴聚,魏延刚,等.机械设计[M].沈阳:东北大学出版社,2000:102―126.Sun Zhili,Leng Xingju,Wei Yangang,et al.Mechanical design[M].Shenyang:Northeastern University Press,2000:102―126.(in Chinese)
[4]杜飞.两级行星齿轮箱的轮齿修形方法研究[D].重庆大学,2015.Du Fei.Research on tooth modification method of two-stage planetary gearbox[D].Chongqing University,2015.(in Chinese)
[5]魏永祥,陈建军,王敏娟.随机参数齿轮-转子系统扭转振动固有频率分析[J].工程力学,2011,28(4):172―177.Wei Yongxiang,Chen Jianjun,Wang Minjuan.Natural frequency analysis for torsional vibration of a gear-rotor system with random parameters[J].Engineering Mechanics,2011,28(4):172―177.(in Chinese)
[6]魏永祥,陈建军,马洪波.随机参数齿轮系统的非线性动力响应分析[J].工程力学,2012,29(11):319―324.Wei Yongxiang,Chen Jianjun,Ma Hongbo.Analysis of nonlinear dynamic response of gear-rotor with random parameters[J].Engineering Mechanics,2012,29(11):319―324.(in Chinese)
[7]唐进元,陈思雨,钟掘.一种改进的齿轮非线性动力学模型[J].工程力学,2008,25(1):217―223.Tang Jinyuan,Chen Siyu,Zhong Jue.An improved nonlinear model for a spur gear pair system[J].Engineering Mechanics,2008,25(1):217―223.(in Chinese)
[8]张希康.渐开线齿轮齿顶修缘的计算与加工[J].南通市机械研究所,1985,9(6):22―25.Zhang Xikang.Calculation and machining of involute gear with tip modification[J].Nantong Machinery Research Institute,1985,9(6):22―25.(in Chinese)
[9]汤鱼,常山,何玉龙,等.齿廓修形对齿面接触力影响的研究[J].冷加工通用机械,2010(4):7―9.Tang Yu,Chang Shan,He Yulong,et al.Study on influence of tooth profile modification to contact force[J].Cold Work and General Machinery,2010(4):7―9.(in Chinese)
[10]尚振国,刘辉,谢忠东,等.齿轮修形原理及应用[J].大连水产学院学报,2009,24(增刊1),220―221.Shang Zhenguo,Liu Hui,Xie Zhongdong,et al.Principle and application of gear modification[J].Journal of Dalian Fisheries University,2009,24(Suppl1),220―221.(in Chinese)
[11]马辉,逄旭,宋溶泽,等.考虑齿顶修缘的齿轮-转子系统振动响应分析[J].机械工程学报,2014,50(7):39―45.Ma Hui,Pang Xu,Song Rongze,et al.Vibration response analysis of a geared rotor system considering the tip relief[J].Journal of Mechanical Engineering,2014,50(7):39―45.(in Chinese)
[12]Chen Z G,Shao Y M.Mesh stiffness calculation of a spur gear pair with tooth profile modification and tooth root crack[J].Mechanism and Machine Theory,2013,62:63―74.
[13]Ma H,Zeng J,Feng R J,et al.An improved analytical model for mesh stiffness calculation of spur gears with tip relief[J].Mechanism and Machine Theory,2016,98:64―80.
[14]Diez-Ibarbia A,Fernandez-del-Rincon A,Iglesias M,et al.Efficiency assessment in spur gears with shifting and profile modifications[J].Mechanisms and Machine Science,2017,43:193―201.
[15]Pedrero J,Pleguezuelos M,B.Sánchez M.On the determination of the meshing stiffness and the load sharing of spur gears[J].Mechanisms and Machine Science,2017,43:221―229.
[16]Chen Z G,Zhai W M,Shao Y M,et al.Mesh stiffness evaluation of an internal spur gear pair with tooth profile shift[J].Science China Technological Science,2016,59(9):1328―1339.
[17]Ma H,Yang J,Song R Z,et al.Effects of tip relief on vibration responses of a geared rotor system[J].Proceedings of the Institution of Mechanical Engineers,Part C:Journal of Mechanical Engineering Science,2014,228(7):1132―1154.
[18]Ma H,Pang X,Wang Q B,et al.Commentary on effects of tip relief on vibration responses of a geared rotor system[J].Proceedings of the Institution of Mechanical Engineers,Part C:Journal of Mechanical Engineering Science,2017,23(11):2159―2169.
[19]Lin T J,He Z Y.Analytical method for coupled transmission error of helical gear system with machining errors,assembly errors and tooth modifications[J].Mechanical Systems&Signal Processing,2017,91:167―182.
[20]Wang S S,Zhu C C,Song C S,et al.Effects of gear modifications on the dynamic characteristics of wind turbine gearbox considering elastic support of the gearbox[J].Journal of Mechanical Science&Technology,2017,31(3):1079―1088.
[21]Wei J,Zhang A Q,Qin D T,et al.A coupling dynamics analysis method for a multistage planetary gear system[J].Mechanism and Machine Theory,2017,110:27―49.
[22]岳喜铮,丁问司,丁康,等.行星轮系渐开线变位齿轮时变啮合刚度数值计算[J].重庆理工大学学报,2016,30(6):38―44.Yue Xizheng,Ding Wensi,Ding Kang,et al.Numerical calculation of planetary gear train involute modified gear time-varying meshing stiffness[J].Journal of Chongqing University of Technology(Natural Science),2016,30(6):38―44.(in Chinese)
[23]乔福瑞.基于ANSYS的齿轮接触应力与啮合刚度研究[D].大连:大连理工大学,2013.Qiao Furui.Research of contact stress and meshing stiffness of gear with profile error[D].Dalian:Dalian University of Technology,2013.(in Chinese)
[24]魏静,孙清超,孙伟,等.大型风电齿轮箱系统耦合动态特性研究[J].振动与冲击,2012,31(08):16―23.Wei Jing,Sun Qingchao,Sun Wei,et al.Dynamical coupling characteristics of a large wind turbine gearbox transmission system[J].Journal of Vibration and Shock,2012,31(08):16―23.(in Chinese)
[25]Weck M,Mauer G.Optimum tooth flank corrections for helical gears[J].Journal of Mechanical Design,1990,112(4):584.
[26]Hedlund J,Lehtovaara A.A parameterized numerical model for the evaluation of gear mesh stiffness variation of a helical gear pair[J].Drive System Technique,2012,222(7):1321―1327.