少齿差星轮型减速器的弹性动力学建模与模态分析
|
摘要
针对目前少齿差星轮型减速器在机械应用中行星轴承易烧毁的现象,对其进行力学分析以寻求解决的途径。基于子结构综合思想,将少齿差星轮型减速器划分为输出轴子系统、输入轴子系统、星轮轴子系统和平动星轮子系统,运用牛顿力学方法建立各子系统的运动微分方程。通过计入各轴承径向支承变形、齿轮副啮合变形以及输入轴和星轮轴上偏心套的分度误差和偏心误差等因素,构造系统各环节的变形协调条件,并将其与各子系统的运动微分方程结合,构建出少齿差星轮型减速器的弹性动力学方程。通过求解系统动力学方程的特征值问题,可获得其固有特性。以HJW-18B型星轮减速器为例,基于所建动力学模型对其进行了模态分析。结果表明,少齿差星轮型减速器的低阶固有频率远高于系统额定输入转频,一般不会引起结构谐振;系统低阶模态对应的振型表现为四类子系统的复合振动模式。在此基础上进一步进行了实验模态分析。对比发现,实验模态结果与理论仿真结果吻合较好,表明所建弹性动力学模型具有较高的计算精度,可准确揭示少齿差星轮型减速器的动态特性,从而为后续的受力分析、强度计算和结构优化提供准确的力学依据。
In order to solve the problems of teeth breakage and premature fatigue of planetary bearings in spider reducers,a methodology of dynamic modeling for spider reducers was proposed and its dynamic characteristics were analyzed. With the substructure synthesis technique,a lumped parameter elasto-dynamic model of the spider reducer was established. The transmission system was divided into four subsystems as the input shaft subsystem, output shaft subsystem,spider shaft subsystem and translational spider gear subsystem. The differential motion equations of the four subsystems were derived by using the Newtonian method. The compatibility conditions of the reducer were derived with considering the deflections of bearings and gear pairs and the index and eccentric errors of eccentric sleeves. By combining the compatibility conditions with subsystem motion equations, the governing motion equations of the reducer were formulated. Through the eigenvalue decomposition,the modal properties of the transmission system were analyzed and the lower orders of natural frequencies and corresponding vibration modes were classified. The results show that the lower orders of natural frequencies are far above the rated input rotation frequency of the reducer,so it is not possible to cause structural resonances. Meanwhile,the corresponding vibration modes behave as complicated compound vibration modes of the four subsystems. To verify the correctness of theoretical analysis,an impact modal test was carried out. The comparison of lower orders of natural frequencies reveals a good match between theoretical and experimental results. From this point of view,it can be concluded that the proposed elasto-dynamic model si of satisfactory accuracy,and hence can be used to predict the steady-state dynamic responses of the system. The present research provides a solid fundament for further dynamic design and structural optimization of spider reducers.
In order to solve the problems of teeth breakage and premature fatigue of planetary bearings in spider reducers,a methodology of dynamic modeling for spider reducers was proposed and its dynamic characteristics were analyzed. With the substructure synthesis technique,a lumped parameter elasto-dynamic model of the spider reducer was established. The transmission system was divided into four subsystems as the input shaft subsystem, output shaft subsystem,spider shaft subsystem and translational spider gear subsystem. The differential motion equations of the four subsystems were derived by using the Newtonian method. The compatibility conditions of the reducer were derived with considering the deflections of bearings and gear pairs and the index and eccentric errors of eccentric sleeves. By combining the compatibility conditions with subsystem motion equations, the governing motion equations of the reducer were formulated. Through the eigenvalue decomposition,the modal properties of the transmission system were analyzed and the lower orders of natural frequencies and corresponding vibration modes were classified. The results show that the lower orders of natural frequencies are far above the rated input rotation frequency of the reducer,so it is not possible to cause structural resonances. Meanwhile,the corresponding vibration modes behave as complicated compound vibration modes of the four subsystems. To verify the correctness of theoretical analysis,an impact modal test was carried out. The comparison of lower orders of natural frequencies reveals a good match between theoretical and experimental results. From this point of view,it can be concluded that the proposed elasto-dynamic model si of satisfactory accuracy,and hence can be used to predict the steady-state dynamic responses of the system. The present research provides a solid fundament for further dynamic design and structural optimization of spider reducers.
引文
[1]刘文,林腾蛟,李润方,等.新型少齿差减速器动态特性分析及实验研究[J].振动与冲击,2009,28(7):22-27.LIU Wen,LIN Teng-jiao,LI Run-fang,et al.Dynamic characteristic analysis and experimental research of a new type reducer with small tooth number difference[J].Journal of Vibration and Shock,2009,28(7):22-27.
[2]石万凯,刘敬,龚建春.同轴对转行星齿轮传动系统动态特性分析[J].振动与冲击,2014,33(7):99-104.SHI Wan-kai,LIU Jing,GONG Jian-chun.Dynamic characteristics analysis for a co-axial counter-rotating planetary gear transmission system[J].Journal of Vibration and Shock,2014,33(7):99-104.
[3]应海燕,杨锡和.K-H型三环减速器的研究[J].机械传动,1992,16(4):37-43.YING Hai-yan,YANG Xi-he.Research of three-ring reducer of K-H type[J].Journal of Mechanical Transmission,1992,16(4):37-43.
[4]王光建,蒋汉军,褚志刚.新型双曲柄式可调侧隙精密行星传动装置参数设计与仿真[J].机械工程学报,2011,47(5):11-18.WANG Guang-jian,JIANG Han-jun,CHU Zhi-gang.Parameters design and simulation of new double crank adjustable backlash precision planetary gears[J].Journal of Mechanical Engineering,2011,47(5):11-18.
[5]李传兵.混合少齿差星轮减速器星轮轴承寿命计算[J].安徽工业大学学报,2005,22(2):156-158.LI Chuan-bing.Life calculation of planetary wheel bearing of speed reducer with complex minor tooth difference[J].Journal of Anhui University of Technology,2005,22(2):156-158.
[6]宋轶民,张俊,郑奇斌,等.双环减速器的弹性动态静力分析[J].农业机械学报,2007,38(10):122-126.SONG Yi-min,ZHANG jun,ZHENG Qi-bing,et al.Elastoquasi-static analysis of double-ring gear reducer[J].Transactions of the Chinese Society for Agricultural Machinery,2007,38(10):122-126.
[7]张俊,宋轶民,郑奇斌.环板式减速器的弹性准静态受力分析[J].中国机械工程,2009,20(13):1594-1599.ZHANG Jun,SONG Yi-min,ZHENG Qi-bin.Elasto-quasistatic analysis of multi-ring gear reducers[J].China Mechanical Engineering,2009,20(13):1594-1599.
[8]赵自强,张春林,程爱明,等.有多重虚约束的内平动齿轮传动机构力学分析[J].北京理工大学学报,2008,28(9):768-772.ZHAO Zi-qiang,ZHANG Chun-lin,CHENG Ai-ming,et al.Force analysis on internal parallel move geartransmission with multiple redundant constrains[J].Journal of Beijing Institute of Technology,2008,28(9):768-772.
[9]张俊,谢胜龙,徐培民,等.少齿差星轮型减速器的弹性静力学分析[J].农业工程学报,2013,29(24):49-55.ZHANG Jun,XIE Sheng-long,XU Pei-min,et al.Elastostatics analysis of spider reducer with small tooth number difference[J].Transactions of the Chinese Society of Agricultural Engineering,2013,29(24):49-55.
[10]杨建明.三环传动弹性动力学的理论与实验研究[D].天津:天津大学,2001.
[11]Yang Jian-ming,Zhang Ce.Elasto-dynamics of internal gear planetary transmissions[J].Mechanism and Machine Theory,2005,40:1107-1125.
[12]宋轶民,张俊,张永新,等.计入内齿板变形的环式减速器弹性动力分析[J].机械工程学报,2006,42(9):29-32.SONG Yi-min,ZHANG Jun,ZHANG Yong-xin,et al.Elasto-dynamic Analysis of multi-ring gear reducers in consideration of deformations of gear-couplers[J].Chinese Journal of Mechanical Engineering,2006,42(9):29-32.
[13]张俊,宋轶民,张策.少齿差环板式减速器的弹性动力学分析[J].机械工程学报,2008,44(2):118-128.ZHANG Jun,SONG Yi-min,ZHANG Ce.Elasto-dynamic analysis of ring-plate gear reducer with small tooth number difference[J].Chinese Journal of Mechanical Engineering,2008,44(2):118-128.
[2]石万凯,刘敬,龚建春.同轴对转行星齿轮传动系统动态特性分析[J].振动与冲击,2014,33(7):99-104.SHI Wan-kai,LIU Jing,GONG Jian-chun.Dynamic characteristics analysis for a co-axial counter-rotating planetary gear transmission system[J].Journal of Vibration and Shock,2014,33(7):99-104.
[3]应海燕,杨锡和.K-H型三环减速器的研究[J].机械传动,1992,16(4):37-43.YING Hai-yan,YANG Xi-he.Research of three-ring reducer of K-H type[J].Journal of Mechanical Transmission,1992,16(4):37-43.
[4]王光建,蒋汉军,褚志刚.新型双曲柄式可调侧隙精密行星传动装置参数设计与仿真[J].机械工程学报,2011,47(5):11-18.WANG Guang-jian,JIANG Han-jun,CHU Zhi-gang.Parameters design and simulation of new double crank adjustable backlash precision planetary gears[J].Journal of Mechanical Engineering,2011,47(5):11-18.
[5]李传兵.混合少齿差星轮减速器星轮轴承寿命计算[J].安徽工业大学学报,2005,22(2):156-158.LI Chuan-bing.Life calculation of planetary wheel bearing of speed reducer with complex minor tooth difference[J].Journal of Anhui University of Technology,2005,22(2):156-158.
[6]宋轶民,张俊,郑奇斌,等.双环减速器的弹性动态静力分析[J].农业机械学报,2007,38(10):122-126.SONG Yi-min,ZHANG jun,ZHENG Qi-bing,et al.Elastoquasi-static analysis of double-ring gear reducer[J].Transactions of the Chinese Society for Agricultural Machinery,2007,38(10):122-126.
[7]张俊,宋轶民,郑奇斌.环板式减速器的弹性准静态受力分析[J].中国机械工程,2009,20(13):1594-1599.ZHANG Jun,SONG Yi-min,ZHENG Qi-bin.Elasto-quasistatic analysis of multi-ring gear reducers[J].China Mechanical Engineering,2009,20(13):1594-1599.
[8]赵自强,张春林,程爱明,等.有多重虚约束的内平动齿轮传动机构力学分析[J].北京理工大学学报,2008,28(9):768-772.ZHAO Zi-qiang,ZHANG Chun-lin,CHENG Ai-ming,et al.Force analysis on internal parallel move geartransmission with multiple redundant constrains[J].Journal of Beijing Institute of Technology,2008,28(9):768-772.
[9]张俊,谢胜龙,徐培民,等.少齿差星轮型减速器的弹性静力学分析[J].农业工程学报,2013,29(24):49-55.ZHANG Jun,XIE Sheng-long,XU Pei-min,et al.Elastostatics analysis of spider reducer with small tooth number difference[J].Transactions of the Chinese Society of Agricultural Engineering,2013,29(24):49-55.
[10]杨建明.三环传动弹性动力学的理论与实验研究[D].天津:天津大学,2001.
[11]Yang Jian-ming,Zhang Ce.Elasto-dynamics of internal gear planetary transmissions[J].Mechanism and Machine Theory,2005,40:1107-1125.
[12]宋轶民,张俊,张永新,等.计入内齿板变形的环式减速器弹性动力分析[J].机械工程学报,2006,42(9):29-32.SONG Yi-min,ZHANG Jun,ZHANG Yong-xin,et al.Elasto-dynamic Analysis of multi-ring gear reducers in consideration of deformations of gear-couplers[J].Chinese Journal of Mechanical Engineering,2006,42(9):29-32.
[13]张俊,宋轶民,张策.少齿差环板式减速器的弹性动力学分析[J].机械工程学报,2008,44(2):118-128.ZHANG Jun,SONG Yi-min,ZHANG Ce.Elasto-dynamic analysis of ring-plate gear reducer with small tooth number difference[J].Chinese Journal of Mechanical Engineering,2008,44(2):118-128.