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1、南京理工大学机械原理chapter6ppt课件 Still waters run deep.流静水深流静水深,人静心深人静心深 Where there is life,there is hope。有生命必有希望。有生命必有希望Chapter1 IntroductionChapter2 Structural Analysis of Planar MechanismsChapter3 Kinematic Analysis of MechanismsChapter4 Planar Linkage MechanismsChapter 5 Cam MechanismsChapter 6 Gear Mec
2、hanismsChapter 7 Gear TrainsChapter 8 Other Mechanisms in Common UseChapter 9 Balancing of MachineryChapter 10 Motion of Mechanical Systems and Its RegulationChapter 11 Efficiency of MachineChapter 6GEAR MECHANISMScircular gears :constant transmission ratio non-circular gears :the ratio varies In th
3、is chapter,only circular gears are considered.6.1 Types of Gear Mechanisms Depending on the relative shafts positions parallel shaftsplanar gear mechanism (平面平面)nonparallel shafts spatial gear mechanism (空间)6.1.1 Planar Gear MechanismsExternal gear pair Internal gear pair Rack and pinion(齿条和齿轮齿条和齿轮)
4、External gear:Spur gear(直齿轮直齿轮)Helical gear(斜齿轮斜齿轮)Double Helical gear(人字齿轮人字齿轮)Internal gear pair:1.Spur internal gear 2.Helical internal gear6.1.2 Spatial Gear Mechanisms1.Bevel gear mechanism(锥齿轮锥齿轮)2.Crossed helical gear mechanism(螺旋齿轮机构螺旋齿轮机构)3.Worm and worm wheel mechanism (蜗杆蜗轮机构蜗杆蜗轮机构)6.2 Fu
5、ndamentals of Engagement of Tooth Profiles (齿廓啮合基本定律齿廓啮合基本定律)6.2.1 Fundamental Law of Gearing (啮合基本定律啮合基本定律)VP1=VP2 1O1P=2 O2P P12 is the instant centre of the gears 1 and 2.P12P-the pitch point(节点节点).If a constant transmission ratio i12 is required.Wherever(无论何处无论何处)the teeth contact,the common nor
6、mal n-n of the tooth profiles through the contact point must intersect the center line at a fixed point。Pitch circle(节圆节圆):the locus of the pitch point P on the motion plane of the pinion or the gear.Pitch circles are tangent and roll without slipping each other纯滚动纯滚动.The gear pair is represented by
7、 two chain dotted(点划线点划线)circles tangent to each other.一对齿轮传动可看成两个节圆一对齿轮传动可看成两个节圆做纯滚动。做纯滚动。6.2.2 Conjugate Profiles(共轭齿廓共轭齿廓)Meshing(啮啮合合的的)profiles of teeth that can yield a desired transmission ratio(传传动动比比)are termed conjugate profiles.满足啮合基本定律的一对齿廓称为共轭齿廓。满足啮合基本定律的一对齿廓称为共轭齿廓。轭轭两头牛背上的架子称为两头牛背上的架子称
8、为轭轭，轭，轭使两头牛同步行走。使两头牛同步行走。共轭共轭即为即为按一定规律相配的一对。按一定规律相配的一对。For any specific tooth profile,we can find its conjugate profile.Involutes(渐渐开开线线)are used most widely since they can be manufactured and assembled easily.共轭齿廓曲线的选择共轭齿廓曲线的选择通常采用渐开线、摆线、变态摆线通常采用渐开线、摆线、变态摆线 1.1.满足定传动比的要求；满足定传动比的要求；2.2.考虑设计、制造等方面。考虑
9、设计、制造等方面。6.3 The Involute(渐开线渐开线)and Its Properties 6.3.1 Generation of Involute An involute is the curve generated by any point on a string which is unwrapped(展开展开)from a fixed circle.当一直线当一直线n n-n n沿一个圆沿一个圆的圆周作纯滚动时，直的圆周作纯滚动时，直线上任一点线上任一点K K的轨迹的轨迹 rb The radius of base circle（基圆）（基圆）.n-nGenerating l
10、ine(发生线发生线)K unfolding angle(展角展角)of the involute at point K.6.3.2 Properties of the Involute 1.The length of the generating line(发生线发生线)segment unrolled from the base circle is equal to the arc length of the base circle rolled.NKNK为渐开线在为渐开线在K K点的法线，点的法线，NKNK为曲率半径，渐开线上任一点为曲率半径，渐开线上任一点的法线与基圆相切。的法线与基圆
11、相切。2.The length of segment NK is the radius of curvature of the involute at the point K.A=03.The normal distance between two involutes of the same base circle remains the same.4.The shape of an involute depends only on the radius of its base circle.As the radius rb of the base circle approaches infi
12、nity,the involute becomes a straight line.5.No involute exists inside its base circle.6.3.3 Equation of the Involute 渐开线方程渐开线方程Pressure angle K FVK=NOKFVK The polar parametric equation of the involute 渐开线方程渐开线方程 with K as a parameter is6.3.4 Gearing(啮合啮合)of Involute ProfilesInvolute gears have const
13、ant transmission ratio.constant 1.Locus of contact point,the line of action(啮合线啮合线)=common normal to the involute profiles =the common tangent to the base circles渐开线齿廓啮合的特点渐开线齿廓啮合的特点Since O1PN1 O2PN2,2、Involute gears have separability(可分离性可分离性)of the centre distance.中心距略有变动时不影响传动的大小。中心距略有变动时不影响传动的大小
14、。3 3、渐开线齿廓啮合的、渐开线齿廓啮合的啮合角啮合角Working pressure angle 不变不变 is defined as the angle between the velocity of pitch point P(on both gears)and the line of actionN1N2.N1N2与节圆公切线之间的夹角与节圆公切线之间的夹角=渐开线在节点处啮合的压力角渐开线在节点处啮合的压力角6.4.1.External Gears 6.4.Standard Involute Spur GearsThe tooth number of a gear-Z （齿数）（齿
15、数）Parameters of gear:the facewidth(齿宽齿宽)-Baddendum circle(or tip circle)（齿顶圆）（齿顶圆）-da,radedendum circle(or root circle)（齿根圆）（齿根圆）-df,rfbase circle-db,rban arbitrary circle(任意圆任意圆)-di,riThe tooth space(齿槽齿槽)-the space between the profiles of two adjacent(相邻的相邻的)teeth.The spacewidth(齿槽宽齿槽宽)ei-the arc
16、length of the tooth space along an arbitrary circle.The tooth thickness(齿厚齿厚)si-the arc length of the tooth along an arbitrary circle.The pitch(齿距，周节齿距，周节)pi-arc distance between corresponding points of adjacent teeth along an arbitrary circle.pi=ei+sipressure angle K Different point on the involute
17、 has different pressure angle K.The circle the point on which has 20 pressure angle is defined as reference circle(分度圆分度圆).FVKK Any parameter on the reference circle has no subscript(下标下标),r,e,s,p,=20 pi=ei+si p=e+s The module m(模数模数)-a basic parameter on the reference circle.Different module needs
18、different cutter.The module m has been standardized.m,Z,are three basic parameters of a gear.Tab 6-1 modules of Involute Cylindrical Gears (GB/T1357-1987)Modules of the first series are preferable.First series0.1,0.12,0.15,0.2,0.25,0.3,0.4,0.5,0.6,0.8,1,1.25,1.5,2,2.5,3,4,5,6,8,10,12,16,20,25,32,40,
19、Second series0.35,0.7,0.9,1.75,2.25,2.75,(3.25),3.5,(3.75),4.5,5.5,(6.5),7,9,(11),14,18,22,28,(30),36,45同模数不同齿数同模数不同齿数 The addendum(齿顶高齿顶高)ha-radial distance between the reference circle and the addendum circle.The dedendum(齿根高齿根高)hf-radial distance between the reference circle and the dedendum circ
20、le.The tooth depth(齿全高齿全高)h-radial distance between the dedendum circle and addendum circle.For standard gears:c*-the coefficient of bottom clearance(顶隙系顶隙系数）数）ha*-the coefficient of addendum(齿顶高系数）齿顶高系数）ha*=1 and c*=0.25 for the normal tooth(正常齿正常齿)ha*=0.8 and c*=0.3 for the shorter tooth(短齿短齿)c=c*
21、m-bottom clearance顶隙顶隙Base pitch(基节基节)pb-the pitch on the base circle.Normal pitch(法节法节)pn-The distance between corresponding sides of adjacent tooth profiles along the common normal.pb=pnReview:The normal distance between two involutes of the same base circle remains the same.)pb=pn=The fundamental
22、 parameters:Z,m,ha*,c*A standard gear means:m,ha*,c*take the standard values and e=s 一个齿轮：一个齿轮：d=mz da=d+2ha=(z+2 ha*)mdf=d-2hf=(z-2 ha*-2 c*)mdb=dcosha=ha*mhf=(ha*+c*)mh=ha+hf=(2 ha*+c*)mP=m一对标准齿轮正常安装：一对标准齿轮正常安装：m m、z z决定了分度圆的大小，而齿轮的大小主要取决于决定了分度圆的大小，而齿轮的大小主要取决于分度圆，因此分度圆，因此m m、z z是决定齿轮大小的主要参数是决定齿轮大小
23、的主要参数轮齿的尺寸与轮齿的尺寸与m m，h ha a*，c c*有关与有关与z z无关无关至于齿形，与至于齿形，与m m，z z，有关有关模数制齿轮模数制齿轮 Involutesstraight lines6.4.2 The Rack（齿条）（齿条）i pi p m On the reference line(分度线分度线),e=s.For standard rack,ha=ha*m,hf=(ha*+c*)m6.4.3 Internal Gears The tooth thickness the tooth space To ensure a continuous transmission,
24、there must exist such situation that two pairs of teeth contact simultaneously(同时地同时地).6.5.1 Proper Meshing Conditions(正确啮合条件正确啮合条件)for Involute Gears6.5 Gearing of Involute Spur Gears两对齿分别在K1，K2点啮合根据啮合基本定律K1在在N1N2上上K2在在N1N2上上 K1K2法向齿距法向齿距 在齿轮在齿轮1上：上：K1K2=Pb1 在齿轮在齿轮2上：上：K1K2=Pb2 Pb1=Pb2 The proper m
25、eshing condition for involute gears:The modules and pressure angles of two meshing gears are the same respectively.The centre distance a is equal to(r1+r2),the reference circles coincide with their pitch circles.节圆节圆 The two reference circles are tangent and roll without slipping.6.5.2 Centre Distan
26、ce and Working Pressure Angle of a Gear Pair要求：一、侧隙应该为零要求：一、侧隙应该为零 二、顶隙为标准值二、顶隙为标准值To keep zero backlash,it holds:无侧隙啮合条件无侧隙啮合条件the bottom clearance(顶隙顶隙)c：the radial distance from the dedendum circle of one gear to the addendum circle of another标准安装标准安装 标准顶隙标准顶隙 Therefore,when two standard gears ar
27、e assembled at a centre distance a=r1+r2,not only the backlash=0,but also bottom clearance c=c*m.Working pressure angle (啮合角啮合角)标准安装标准安装 When the actual centre distance a is larger than (r1+r2),the reference circles will separate.The two pitch circles will become larger and remain tangent.非标准安装非标准安装
28、 a a只有增大只有增大 由图可知：由图可知：有侧隙有侧隙 非标准安装非标准安装传动比传动比 中心距变大，传动比仍保持不变中心距变大，传动比仍保持不变 无论齿条离齿轮近或远，啮合角始终等于压力角；无论齿条离齿轮近或远，啮合角始终等于压力角；节圆半径始终等于分度圆半径。节圆半径始终等于分度圆半径。6.5.3 Meshing of r rack and pinion6.6 Contact Ratio(重合度重合度)of an Involute Spur Gear Set 齿齿轮轮啮啮合合全全过过程程Gear 1 is a driver.6.6.1 definition of Contact Rat
29、io(重合度重合度)The meshing begins at point B2,K moves along the line of actionand ends at point B1 The segment B1B2 is called the actual line of action(实际啮合线实际啮合线).起始啮合点：起始啮合点：从动轮的齿顶点与主动轮的齿根处某点接触，从动轮的齿顶点与主动轮的齿根处某点接触，在啮合线上为从动轮的齿顶圆与啮合线在啮合线上为从动轮的齿顶圆与啮合线N N1 1N N2 2的的交点交点B B2 2。终止啮合点：终止啮合点：主动轮的齿顶点与从动轮的齿主动轮的齿
30、顶点与从动轮的齿根处某点接触，在啮合线根处某点接触，在啮合线N N1 1N N2 2上上为主动轮的齿顶圆与啮合线为主动轮的齿顶圆与啮合线N N1 1N N2 2的交点的交点B B1 1。实际啮合线实际啮合线the actual line of action 理论啮合线理论啮合线theoretical line of action 齿廓工作段齿廓工作段(the actual working profile)齿廓非工作段齿廓非工作段 If Transmission is continuousDefine the contact ratio as:齿轮传动的连续性条件齿轮传动的连续性条件 Usual
31、ly,=1.2.6.6.2 重合度的意义重合度的意义重合度不仅是齿轮传动的连续性条件，而且是衡量齿轮承载能力和传动平稳性的重要指标。重合度表示同时参加啮合的齿对数多或多对齿啮合所占的时间比例大。6.6.3 calculation of Contact Ratio(重合度重合度)For internal gear pair:For rack and pinion gear:(2)rb1=ra1 cos a1 ,rb2=ra2 cos a2 (6)If a ,then tan and .(5)If Z,then .Try to use smaller m.(4)is independent of
32、module m.(3)If ra ,then a and .短齿制缺点短齿制缺点(1)Example.A pair of standard spur involute gears have the parameters as follow:Z1=20,Z2=50,m=10mm,actual centre distance a=350mm,=20,ha*=1,c*=0.25.Calculate the contact ratio Solution:Since=20.6.7 Manufacturing Methods of Involute Profiles Cutting is the mos
33、t common method for manufacture of gears.There are various methods for manufacturing tooth profiles,such as die casting(模铸模铸),precision forging(锻造锻造),punch冲压冲压,powder(粉末粉末)process,cutting切削法切削法 and so on.6.7.1 Cutting of Tooth Profiles(1)Form(仿形法仿形法)Cutting(2)Generating(展成法或范成法展成法或范成法)Cutting(1)Form
34、 CuttingDisk milling cutter(盘形铣刀）盘形铣刀）End milling cutter(指状铣刀指状铣刀)Broach(拉刀拉刀)Disk milling cutterEnd milling cutters(端铣刀端铣刀)are used for large modules and double helical gears.broach(拉刀拉刀)Characteristics of the form cutting:As the shape of an involute depends on its base radius,a different cutter is
35、 required for each number of teeth even for the same module and same pressure angle.Form cutting is not a good method.Its main advantage is that it can be accomplished on commonly available milling machines.(铣床铣床)(2)Generating Cutting(展成法或范成法加工展成法或范成法加工)In a generating cutting method,the edges of a
36、cutter take the form of a gear(or rack)with the same module and pressure angle as the gear to be cut.1.Cutting motion3.Generating motion展成展成2.Retreat motion退刀退刀4.Feed motion进给进给 External gears can be cut with a rack-shaped shaper cutter(齿条插刀齿条插刀).The edges of the cutter are now straight lines which
37、can be made accurately.优点优点：用一把插刀可以加工出用一把插刀可以加工出 m m、相同而齿数不同相同而齿数不同的各种齿轮的各种齿轮(包括内齿轮包括内齿轮)。缺点缺点：切削不连续，生产效率较低。切削不连续，生产效率较低。The cutting process in the shaping(插齿插齿)of a gear is not continuous.In mass production(大规模生产大规模生产),hobbing(滚齿滚齿)is used instead of shaping.hobbing(滚齿滚齿)优点优点：用一把滚刀可以加工出用一把滚刀可以加工出 m
38、 m、相同而齿数不同相同而齿数不同的各种齿轮，切削连续，生产效率高。的各种齿轮，切削连续，生产效率高。缺点缺点：不能加工内齿轮。不能加工内齿轮。6.7.2 Cutting a Standard Gear with Standard Rack-shaped Cutter普通齿条普通齿条齿条插刀齿条插刀The tip line is c*m higher than the addendum line for cutting the profile at the root to provide hf To cut a standard gear,the reference line of the c
39、utter should be tangent to the reference circle of the gearThe m,of the gear on the reference circle are the same as those on the reference line.As the s2 and e2 of the cutter on the reference line are equal to each other,the gear is cut with s=e=p/2=m/2.So，the gear is standard.The dedendum circle o
40、f the gear is cut by the tip line of the cutter.将轮坯的外圆按被切齿轮的齿顶圆直径预先加工好。The addendum circle of the gear is cut on a lathe(车床车床).It is not cut by gear cutter!6.7.3 Cutter Interference(根切现象根切现象)Some part of the involute profile near the root portion is removed in the generating cutting.This will reduce
41、 the contact ratio as well as the strength of the tooth and should be avoided or minimized.The reason for cutter interference:The addendum line of the cutter passes the limit point N1 of the line of action.To prevent cutter interference:短齿制：短齿制：ha*=0.8，a a=20，Zmin=146.8 Addendum Modification on Invo
42、lute Gears 齿轮变位齿轮变位6.8.1 Introduction of Addendum Modification Disadvantages of standard gears:1.Z Zmin2.a=a3.The strength of the pinion is much lower than that of the gear.外啮合外啮合 无法安装；无法安装；Corrected gears变位齿轮变位齿轮1、用改变刀具与轮坯径向相、用改变刀具与轮坯径向相对位置来切制齿轮的方法称对位置来切制齿轮的方法称径向变位法。径向变位法。变位齿轮变位齿轮 xm移距或变位移距或变位 x移距系
43、数或变位系数移距系数或变位系数modification coefficient 规定：规定：正变位正变位 零变位零变位 负变位负变位 Only the position of the cutter may be changed-the reference line of the cutter may not be tangent to the reference circle of the gear.The gear-cutting machines,the cutters and the transmission ratio between the cutter and the blank
44、remains unchanged.Comparison between a corrected gear and the standard gear:1.Parameters Z,m,r(=mZ/2),and rb=(r cos)are the same.2.Parameters df,da,s,e may change.This means that different portions of the same involute are employed for the profiles of the standard gear and the corrected gear6.8.2 Ge
45、ometric Dimensions of Corrected Gears To prevent cutter interference：最小变位系数最小变位系数正变位正变位 负变位负变位 1、分度圆上的齿厚、分度圆上的齿厚 见图：刀具节线的齿槽宽比中线见图：刀具节线的齿槽宽比中线齿槽宽齿槽宽 ，被切齿轮分度圆被切齿轮分度圆上的齿厚增加上的齿厚增加 在在IJK中：中：分度圆的齿厚：分度圆的齿厚：2、齿顶高和齿根高、齿顶高和齿根高 齿根高齿根高h hf f：刀具加工节线到顶：刀具加工节线到顶 刃线之间的距离刃线之间的距离 对正变位：对正变位：x x00，h hf f比标准减小比标准减小xmxm
46、对负变位：对负变位：x x00，h hf f比标准增加比标准增加xmxm 变位齿轮的齿根圆半径：变位齿轮的齿根圆半径：齿顶高：齿顶高：变位齿轮的分度圆与相应标准齿轮的分度圆一样，变位齿轮的分度圆与相应标准齿轮的分度圆一样，变位齿轮的齿顶高仅决定于轮坯顶圆的大小。变位齿轮的齿顶高仅决定于轮坯顶圆的大小。为保证齿全高为保证齿全高 由于由于所以所以Helical Gears for Parallel Shafts(斜齿轮传动)Crossed helical gears mechanism(螺旋齿轮机构)6.9 Helical Gears for Parallel Shafts (平行轴斜齿轮传动平行
47、轴斜齿轮传动)6.9.1 Generation and Characteristics of Helical Teeth Spur Gear:The tooth surface of the spur gear is therefore an involute cylinder(渐开线柱面渐开线柱面).The tooth surfaces of spur gears contact on a straight line parallel to the axes of the gears.Vibration and noisesudden loading and sudden unloading
48、 on teeth In generation of the tooth profile of a helical gear,the straight line KK on the generating plane is no longer parallel to the axis of the gear.Every point on the straight line KK will produce an involute.The curve connecting the starting points of the involutes on the base cylinder is a h
49、elix(螺旋螺旋)AA.The surface profile of a helical gear is called an involute helicoid(渐渐开线螺旋面开线螺旋面)1.The tooth surfaces of two engaging helical gears contact on a straight line inclined.渐开线螺旋面齿廓的特点渐开线螺旋面齿廓的特点2.2.端面（垂直于齿轮端面（垂直于齿轮轴线的面）与齿廓曲轴线的面）与齿廓曲面的交线为渐开线。面的交线为渐开线。3.3.与基圆柱同轴的圆柱与基圆柱同轴的圆柱面与渐开线螺旋面的交面与渐开线螺旋面
50、的交线为一螺旋线。线为一螺旋线。不同面不同面螺旋角不同螺旋角不同 斜齿轮的啮合特点：斜齿轮的啮合特点：（1）两斜齿齿廓的公法面既是）两斜齿齿廓的公法面既是两基圆柱的公切面，又是传动的两基圆柱的公切面，又是传动的啮合面啮合面（2）两齿廓的接触）两齿廓的接触线与轴线夹角线与轴线夹角 b The length of the contact line changes gradually from zero to maximum and then from maximum to zero.The loading and unloading of the teeth become gradual and