二次包络少齿差行星齿轮传动啮合特性及动力学研究
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摘要
少齿差行星齿轮传动特别是摆线针轮行星传动由于具有传动比大、结构紧凑、体积小、重量轻、传动效率高和承载能力大等优点，已经在冶金机械、起重运输机械、工程机械、化工机械和纺织机械等动力传动领域得到了广泛的应用。同时由于其多齿啮合的特点使误差均化效应明显有利于提高传动精度，近年来也越来越广泛地应用于精密传动领域。为了适应各工业领域的发展对齿轮传动高定位精度和低振动水平的要求，国内外研究学者也越来越多地关注齿轮动力学的研究。二次包络少齿差行星齿轮传动是根据齿轮传动中的二次接触现象，利用二次包络方法形成的一种新型行星齿轮传动。二次包络共轭啮合副的主要特点是双线接触且具有优良的啮合特性。系统深入地对二次包络少齿差行星齿轮传动的啮合特性和动力学性能进行研究，对于开发设计具有高效率、高精度和低振动等高性能的新型行星齿轮传动形式，具有重要的理论意义和工程实用价值。
本文主要针对二次包络少齿差行星齿轮传动的共轭啮合副、啮合特性、动力学理论分析和动力学测试等方面进行了研究。主要内容如下：
（1）根据微分几何和齿轮啮合原理，对二次包络摆线行星传动共轭啮合副的啮合特性进行了详细研究；针对第二条接触线具有优良啮合特性和双线接触对制造安装误差敏感的特点，提出了不完全摆线二次包络共轭啮合副，推导了其啮合方程和齿廓方程，对其啮合特性进行了分析；将不完全摆线二次包络啮合副、二次包络摆线啮合副和普通摆线啮合副的齿廓组成特点、接触线和诱导法曲率进行了对比分析，结果表明二次包络啮合副比普通啮合副具有更加优良的接触特性，其中不完全摆线二次包络啮合副略优于二次包络摆线啮合副；提出了抛物型二次包络共轭啮合副，给出了啮合方程和齿廓方程。
（2）对二次包络少齿差行星齿轮传动进行结构创新设计，完成4台套物理样机的试制；选取一台两级封闭式二次包络摆线行星传动样机作为研究对象，分析了其结构特点、啮合副特点和各构件的运动关系，根据行星齿轮传动调制模型，计算了两级传动的啮合频率和边带频率。
（3）对二次包络少齿差行星齿轮传动进行动力学理论分析：根据赫兹接触理论建立了二次包络摆线共轭啮合副的刚度计算模型，推导了法向刚度、力臂和扭转刚度的计算公式；计算了试验样机中二次包络摆线啮合副的轮齿扭转啮合刚度，并与普通摆线针轮行星传动进行了对比，结果表明二次包络啮合副的啮合刚度具有较大的波动值，但平均值也较大；利用集中参数法建立了试验样机五自由度扭转振动模型，推导了动力学方程，计算了自由振动下的固有频率和模态，以时变啮合刚度变化激励，用龙格-库塔法对动力学方程进行了求解，分析了样机中各构件的扭转振动，将求解出来的样机固有频率与各构件的扭转振动与采用普通摆线针轮行星传动的情况进行了对比，结果表明两种情况固有频率和模态基本相同，采用二次包络摆线啮合副的样机具有更小的振动。
（4）针对二次包络少齿差行星齿轮传动研究样机，搭建了动力学测试试验台对样机进行了试验研究，主要包括模态测试、平移振动测试、输出端盘扭转振动测试；对试验中观察到的调制现象进行了分析，并与理论结果进行了对比，试验测试结果与理论分析结果取得了较好的一致性；对输出端盘扭转振动的试验数据进行了分析，结果表明第一级齿轮传动的啮合振动在整机振动中起主要作用。
Planetary gear drive with small tooth number difference has been used widely inmany industrial areas for power transmission, such as metallurgical machinery, hoistingand conveying machinery, construction machinery, chemical machinery and textilemachinery, etc., due to its superior characteristics: large transmission ratio, compactstructure, small size, light weight, high transmission efficiency and high load capacity,etc.. In recent years, it has been also used more and more widely in precisiontransmission because of its outstanding averaging effect of errors which can lead to highprecision caused by the multi-tooth meshing. In order to meet the requirement of thehigh precision and the low vibration for the gear transmission because of thedevelopment of the industrial areas, the researchers home and abroad have paid moreand more attention to the dynamic analysis of the gear transmission. Thedouble-enveloping cycloid drive with small tooth number difference is a new type ofplanetary gear drvie which is generated by the double-enveloping method based on thesecondary action phenomenon. The main characteristics of the double-envelopingconjugated tooth pair are the double contact lines simultaneously in a certain meshingarea and the good meshing characteristics. The deep and systematical research on themeshing characteristics and the dynamic behavior of the double-enveloping planetarygear drive with small tooth number difference has the theoretical significance andengineering practice value for the development and the design of the new type of hightransmission performance planetary gear drive with high effiency, high precision andlow vibration, etc..
In this paper, the research is mainly on the conjugated surfaces, the meshingcharacteristics, the theoretical dynamic analysis and the experimental research about thedouble-enveloping planetary gear drive with small tooth number difference. The maincontents are as the following:
(1) Based on the differential geometry and gear geometry theory, the meshcharacteristics of the double-enveloping cycloid drive are studied; the incompletedouble-enveloping cycloid drive is proposed according to the superior characteristics ofthe second contact line and the sensitivity to the errors of the double contact lines, theequations of the meshing and the tooth profiles for the incomplete cycloid drive arederived, the meshing characteristics are calculated; the characteristics of the tooth profiles, the contact lines and the induced normal curvature among the incompletedouble-enveloping cycloid drive, the double-enveloping ccycloid drive and theconventional cycloid drive are compared with each other, the results show that thedouble-enveloping conjugated tooth pair has superior characteristics than theconventional conjugated tooth pair, and the meshing characteristics of the incompletedouble-enveloping cycloid drive is a little better than that of the double-envelopingcycloid drive; the parabolic type double-enveloping conjugated tooth pair is proposedand the equations of its meshing and tooth profiles are presented.
(2) The structure of the double-enveloping planetary gear drive with small toothnumber difference is creatively designed,4physical prototypes are designed andmanufactured; a physical prototype with two stages of which the second stage is thedouble-enveloping cycloid drive, is selected as the research object, the characteristics ofits structure, conjugated surfaces and the relationship among the components areanalyzed, based on the model of the sideband frequency calculation the meshingfrequencies for the two stages and the sideband frequencies are calculated.
(3) The dynamic behavior of the double-enveloping planetary gear drive with smalltooth number difference is theoretically analyzed: based on the Hertz theory, the meshstiffness calculation model of the double-enveloping cycloid conjugated tooth pair isestablished, the calculation formulas of the normal stiffness, the arm length and thetorsional stiffness are derived; the meshing stiffness of the double-envelopingconjugated tooth pair for the prototype is calculated compared with that of theconventional cycloid drive, and the comparison results show that the meshing stiffnessof the double-enveloping cycloid tooth pair has both larger average value and largerfluctuation value than that of the conventional cycloid drive; based on the lumpedparameter method, the rotation vibration model with five freedoms for the physicalprototype is established and the motion equations are derived, the natural frequenciesand the modes of the free vibration are calculated, the motion equations are solved usingRunge-Kuta method under that he variation mesh stiffness is considered as the internalexcitation, the rotation vibrations of the different parts of the physical prototype areanalyzed, the natural frequencies and the vibrations are compared with that of theconventional cycloid drive, the comparision results show that the natural frequecnciesand the vibrations of the two types of cycloid drive are almost the same, and the rotationvibration of the double-enveloping cycloid drive are smaller than that of theconventional cycloid drive.
(4) The experiment test rig is put up for the dynamic analysis of thedouble-enveloping planetary gear drive: firstly, based on the test principle, the generalscheme is designed; secondly, the types and the detail parameters of the motor, loaddevice and the test instrument are determined; then the equipments for the test aredesigned, such as cradle and the connection couplings; finally, the experiment test rig isinstalled and adjusted.
(5) The experiment test rig is put up for the dynamic experimental research on thedouble-enveloping planetary gear drive with small tooth number difference, and thedynamic tests on the prototype are carried out including modal test, translationalvibration test and rotation vibration test of the output plate; the phenomenon of themodulation sidebands is analyzed and compared with the theoretical calculation results,the comparision results show that the experimental results and the theoretical resultshave good agreement；the experiment data of the output plate rotation vibration isanalyzed and the results show that the gear mesh vibration of the first stage of theprototype is the main source of the whole gear reducer vibration.
本文主要针对二次包络少齿差行星齿轮传动的共轭啮合副、啮合特性、动力学理论分析和动力学测试等方面进行了研究。主要内容如下：
（1）根据微分几何和齿轮啮合原理，对二次包络摆线行星传动共轭啮合副的啮合特性进行了详细研究；针对第二条接触线具有优良啮合特性和双线接触对制造安装误差敏感的特点，提出了不完全摆线二次包络共轭啮合副，推导了其啮合方程和齿廓方程，对其啮合特性进行了分析；将不完全摆线二次包络啮合副、二次包络摆线啮合副和普通摆线啮合副的齿廓组成特点、接触线和诱导法曲率进行了对比分析，结果表明二次包络啮合副比普通啮合副具有更加优良的接触特性，其中不完全摆线二次包络啮合副略优于二次包络摆线啮合副；提出了抛物型二次包络共轭啮合副，给出了啮合方程和齿廓方程。
（2）对二次包络少齿差行星齿轮传动进行结构创新设计，完成4台套物理样机的试制；选取一台两级封闭式二次包络摆线行星传动样机作为研究对象，分析了其结构特点、啮合副特点和各构件的运动关系，根据行星齿轮传动调制模型，计算了两级传动的啮合频率和边带频率。
（3）对二次包络少齿差行星齿轮传动进行动力学理论分析：根据赫兹接触理论建立了二次包络摆线共轭啮合副的刚度计算模型，推导了法向刚度、力臂和扭转刚度的计算公式；计算了试验样机中二次包络摆线啮合副的轮齿扭转啮合刚度，并与普通摆线针轮行星传动进行了对比，结果表明二次包络啮合副的啮合刚度具有较大的波动值，但平均值也较大；利用集中参数法建立了试验样机五自由度扭转振动模型，推导了动力学方程，计算了自由振动下的固有频率和模态，以时变啮合刚度变化激励，用龙格-库塔法对动力学方程进行了求解，分析了样机中各构件的扭转振动，将求解出来的样机固有频率与各构件的扭转振动与采用普通摆线针轮行星传动的情况进行了对比，结果表明两种情况固有频率和模态基本相同，采用二次包络摆线啮合副的样机具有更小的振动。
（4）针对二次包络少齿差行星齿轮传动研究样机，搭建了动力学测试试验台对样机进行了试验研究，主要包括模态测试、平移振动测试、输出端盘扭转振动测试；对试验中观察到的调制现象进行了分析，并与理论结果进行了对比，试验测试结果与理论分析结果取得了较好的一致性；对输出端盘扭转振动的试验数据进行了分析，结果表明第一级齿轮传动的啮合振动在整机振动中起主要作用。
Planetary gear drive with small tooth number difference has been used widely inmany industrial areas for power transmission, such as metallurgical machinery, hoistingand conveying machinery, construction machinery, chemical machinery and textilemachinery, etc., due to its superior characteristics: large transmission ratio, compactstructure, small size, light weight, high transmission efficiency and high load capacity,etc.. In recent years, it has been also used more and more widely in precisiontransmission because of its outstanding averaging effect of errors which can lead to highprecision caused by the multi-tooth meshing. In order to meet the requirement of thehigh precision and the low vibration for the gear transmission because of thedevelopment of the industrial areas, the researchers home and abroad have paid moreand more attention to the dynamic analysis of the gear transmission. Thedouble-enveloping cycloid drive with small tooth number difference is a new type ofplanetary gear drvie which is generated by the double-enveloping method based on thesecondary action phenomenon. The main characteristics of the double-envelopingconjugated tooth pair are the double contact lines simultaneously in a certain meshingarea and the good meshing characteristics. The deep and systematical research on themeshing characteristics and the dynamic behavior of the double-enveloping planetarygear drive with small tooth number difference has the theoretical significance andengineering practice value for the development and the design of the new type of hightransmission performance planetary gear drive with high effiency, high precision andlow vibration, etc..
In this paper, the research is mainly on the conjugated surfaces, the meshingcharacteristics, the theoretical dynamic analysis and the experimental research about thedouble-enveloping planetary gear drive with small tooth number difference. The maincontents are as the following:
(1) Based on the differential geometry and gear geometry theory, the meshcharacteristics of the double-enveloping cycloid drive are studied; the incompletedouble-enveloping cycloid drive is proposed according to the superior characteristics ofthe second contact line and the sensitivity to the errors of the double contact lines, theequations of the meshing and the tooth profiles for the incomplete cycloid drive arederived, the meshing characteristics are calculated; the characteristics of the tooth profiles, the contact lines and the induced normal curvature among the incompletedouble-enveloping cycloid drive, the double-enveloping ccycloid drive and theconventional cycloid drive are compared with each other, the results show that thedouble-enveloping conjugated tooth pair has superior characteristics than theconventional conjugated tooth pair, and the meshing characteristics of the incompletedouble-enveloping cycloid drive is a little better than that of the double-envelopingcycloid drive; the parabolic type double-enveloping conjugated tooth pair is proposedand the equations of its meshing and tooth profiles are presented.
(2) The structure of the double-enveloping planetary gear drive with small toothnumber difference is creatively designed,4physical prototypes are designed andmanufactured; a physical prototype with two stages of which the second stage is thedouble-enveloping cycloid drive, is selected as the research object, the characteristics ofits structure, conjugated surfaces and the relationship among the components areanalyzed, based on the model of the sideband frequency calculation the meshingfrequencies for the two stages and the sideband frequencies are calculated.
(3) The dynamic behavior of the double-enveloping planetary gear drive with smalltooth number difference is theoretically analyzed: based on the Hertz theory, the meshstiffness calculation model of the double-enveloping cycloid conjugated tooth pair isestablished, the calculation formulas of the normal stiffness, the arm length and thetorsional stiffness are derived; the meshing stiffness of the double-envelopingconjugated tooth pair for the prototype is calculated compared with that of theconventional cycloid drive, and the comparison results show that the meshing stiffnessof the double-enveloping cycloid tooth pair has both larger average value and largerfluctuation value than that of the conventional cycloid drive; based on the lumpedparameter method, the rotation vibration model with five freedoms for the physicalprototype is established and the motion equations are derived, the natural frequenciesand the modes of the free vibration are calculated, the motion equations are solved usingRunge-Kuta method under that he variation mesh stiffness is considered as the internalexcitation, the rotation vibrations of the different parts of the physical prototype areanalyzed, the natural frequencies and the vibrations are compared with that of theconventional cycloid drive, the comparision results show that the natural frequecnciesand the vibrations of the two types of cycloid drive are almost the same, and the rotationvibration of the double-enveloping cycloid drive are smaller than that of theconventional cycloid drive.
(4) The experiment test rig is put up for the dynamic analysis of thedouble-enveloping planetary gear drive: firstly, based on the test principle, the generalscheme is designed; secondly, the types and the detail parameters of the motor, loaddevice and the test instrument are determined; then the equipments for the test aredesigned, such as cradle and the connection couplings; finally, the experiment test rig isinstalled and adjusted.
(5) The experiment test rig is put up for the dynamic experimental research on thedouble-enveloping planetary gear drive with small tooth number difference, and thedynamic tests on the prototype are carried out including modal test, translationalvibration test and rotation vibration test of the output plate; the phenomenon of themodulation sidebands is analyzed and compared with the theoretical calculation results,the comparision results show that the experimental results and the theoretical resultshave good agreement；the experiment data of the output plate rotation vibration isanalyzed and the results show that the gear mesh vibration of the first stage of theprototype is the main source of the whole gear reducer vibration.
引文
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