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1、Theory of Machines and Mechanisms (机械原理),Chapter1 Introduction Chapter2 Structural Analysis of Planar Mechanisms Chapter3 Kinematic Analysis of Mechanisms Chapter4 Planar Linkage Mechanisms Chapter 5 Cam Mechanisms Chapter 6 Gear Mechanisms Chapter 7 Gear Trains Chapter 8 Other Mechanisms in Common
2、Use Chapter 9 Balancing of Machinery Chapter 10 Motion of Mechanical Systems and Its Regulation Chapter 11 Efficiency of Machine,Chapter 6 GEAR MECHANISMS,circular gears :constant transmission ratio,non-circular gears :the ratio varies,In this chapter, only circular gears are considered.,6.1 Types o
3、f Gear Mechanisms,Depending on the relative shafts positions,parallel shaftsplanar gear mechanism (平面),nonparallel shafts spatial gear mechanism (空间),6.1.1 Planar Gear Mechanisms,External gear pair,Internal gear pair,Rack and pinion (齿条和齿轮),External gear:,Spur gear (直齿轮),Helical gear(斜齿轮),Double Hel
4、ical gear (人字齿轮),Internal gear pair:,1. Spur internal gear 2. Helical internal gear,6.1.2 Spatial Gear Mechanisms,1. Bevel gear mechanism(锥齿轮),2. Crossed helical gear mechanism(螺旋齿轮机构),3. Worm and worm wheel mechanism (蜗杆蜗轮机构),6.2 Fundamentals of Engagement of Tooth Profiles (齿廓啮合基本定律),6.2.1 Fundame
5、ntal Law of Gearing (啮合基本定律),VP1=VP2,1O1P=2 O2P,P12 is the instant centre of the gears 1 and 2.,P12,P - the pitch point(节点).,If a constant transmission ratio i12 is required.,Wherever(无论何处) the teeth contact, the common normal n-n of the tooth profiles through the contact point must intersect the ce
6、nter line at a fixed point。,Pitch circle(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. The gear pair is represented by two chain dotted(点划线) circles tangent to each other. 一对齿轮传动可看成两个节圆做纯滚动。,6.2.2
7、 Conjugate Profiles(共轭齿廓),Meshing(啮合的) profiles of teeth that can yield a desired transmission ratio(传动比) are termed conjugate profiles. 满足啮合基本定律的一对齿廓称为共轭齿廓。,轭,两头牛背上的架子称为轭，轭使两头牛同步行走。共轭即为按一定规律相配的一对。,For any specific tooth profile, we can find its conjugate profile.,Involutes(渐开线) are used most widely
8、 since they can be manufactured and assembled easily.,共轭齿廓曲线的选择,通常采用渐开线、摆线、变态摆线,1.满足定传动比的要求；2.考虑设计、制造等方面。,6.3 The Involute(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) from a fixed circle.,当一直线n-n沿一个圆的圆周作纯滚动时，
9、直线上任一点K的轨迹,rb The radius of base circle（基圆）.,n-nGenerating line(发生线),K unfolding angle (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) segment unrolled from the base circle is equal to the arc length of the base circle rolled.,NK为渐开线在K点的法线，N
10、K为曲率半径，渐开线上任一点的法线与基圆相切。,2. The length of segment NK is the radius of curvature of the involute at the point K.,A =0,3. 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 th
11、e base circle approaches infinity, 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,Pressure angle K,F,VK,=NOK,The polar parametric equation of the involute 渐开线方程 with K as a parameter is,6.3.4 Gearing(啮合) of Involute Profiles,I
12、nvolute gears have constant transmission ratio.,constant,1. Locus of contact point, the line of action(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,Since O1PN1 O2PN2,2、 Involute gears have separability(可分离性) of the centre distance.,中心距略有变动时不影响传动的大小
13、。,3、渐开线齿廓啮合的啮合角Working pressure angle 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, =渐开线在节点处啮合的压力角,6.4.1. External Gears,6.4. Standard Involute Spur Gears,The tooth number of a gear - Z （齿数）,Parameters of gear:,the fac
14、ewidth(齿宽) - B,addendum circle (or tip circle) （齿顶圆） -da, ra,dedendum circle (or root circle) （齿根圆）-df, rf,base circle -db, rb,an arbitrary circle (任意圆) -di, ri,The tooth space (齿槽) -the space between the profiles of two adjacent(相邻的) teeth.,The spacewidth(齿槽宽) ei-the arc length of the tooth space a
15、long an arbitrary circle.,The tooth thickness (齿厚) si- the arc length of the tooth along an arbitrary circle.,The pitch(齿距，周节) pi -arc distance between corresponding points of adjacent teeth along an arbitrary circle. pi = ei + si,pressure angle K,Different point on the involute has different pressu
16、re angle K .,The circle the point on which has 20 pressure angle is defined as reference circle (分度圆).,K,Any parameter on the reference circle has no subscript(下标), r, e, s, p, =20 ,pi = ei + si,p= e + s,The module m (模数)- a basic parameter on the reference circle.,Different module needs different c
17、utter. 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.,同模数不同齿数,The addendum (齿顶高) ha- radial distance between the reference circle and the addendum circle.,The de
18、dendum (齿根高) hf -radial distance between the reference circle and the dedendum circle.,The tooth depth (齿全高) h -radial distance between the dedendum circle and addendum circle.,For standard gears:,c* - the coefficient of bottom clearance(顶隙系数）,ha* - the coefficient of addendum(齿顶高系数）,ha*=1 and c*= 0
19、.25 for the normal tooth(正常齿) ha*= 0.8 and c*= 0.3 for the shorter tooth(短齿),c=c* m- bottom clearance顶隙,Base pitch(基节)pb- the pitch on the base circle.,Normal pitch(法节)pn-The distance between corresponding sides of adjacent tooth profiles along the common normal.,pb=pn,Review: The normal distance be
20、tween two involutes of the same base circle remains the same.,),pb=pn=,The fundamental parameters:,Z, m, , ha*, c*,A standard gear means: m, , ha*, c* take the standard values and e = s,一个齿轮：,d=mz da=d+2ha=(z+2 ha*)m df=d-2hf=(z-2 ha*-2 c*)m db=dcos,ha= ha*m hf=( ha*+ c*)m h=ha+hf=(2 ha*+ c*)m P=m,一
21、对标准齿轮正常安装：,m、z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m、z是决定齿轮大小的主要参数,轮齿的尺寸与m， ha* ， c* 有关与z无关,至于齿形，与m，z，有关,模数制齿轮,Involutes,straight lines,6.4.2 The Rack（齿条）,i ,pi p m,On the reference line(分度线), e=s.,For standard rack, ha=ha*m, hf=(ha*+c*)m,6.4.3 Internal Gears,The tooth thickness the tooth space,To ensure a co
22、ntinuous transmission, there must exist such situation that two pairs of teeth contact simultaneously(同时地) .,6.5.1 Proper Meshing Conditions(正确啮合条件) for Involute Gears,6.5 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N1N2上 K2在N1N2上,K1K2法向齿距,在齿轮1上： K1K2 =Pb1,在齿轮2上： K1K2 =Pb2,Pb1=Pb2,The
23、proper meshing 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,The two reference circles are tangent and roll without slipping.,6.5.2 C
24、entre Distance and Working Pressure Angle of a Gear Pair,要求：一、侧隙应该为零 二、顶隙为标准值,To keep zero backlash, it holds: 无侧隙啮合条件,the bottom clearance(顶隙) c ： the radial distance from the dedendum circle of one gear to the addendum circle of another,标准安装,标准顶隙,Therefore, when two standard gears are assembled at
25、 a centre distance a=r1+r2, not only the backlash =0, but also bottom clearance c=c*m.,Working pressure angle (啮合角),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,a只有增大,由图可知：,有侧隙,非
26、标准安装,传动比,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；节圆半径始终等于分度圆半径。,6.5.3 Meshing of r rack and pinion,6.6 Contact Ratio(重合度) of an Involute Spur Gear Set,齿轮啮合全过程,Gear 1 is a driver.,6.6.1 definition of Contact Ratio(重合度),The meshing begins at point B2,K moves along the line of action,and ends at point B1,
27、The segment B1B2 is called the actual line of action(实际啮合线).,起始啮合点：,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N1N2的交点B2。,终止啮合点：,主动轮的齿顶点与从动轮的齿根处某点接触，在啮合线N1N2上为主动轮的齿顶圆与啮合线N1N2的交点B1。,实际啮合线the actual line of action,理论啮合线theoretical line of action,齿廓工作段(the actual working profile) 齿廓非工作段,If Transmission is co
28、ntinuous,Define the contact ratio as :,齿轮传动的连续性条件,Usually, =1.2.,6.6.2 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),For internal gear pair:,For rack and pinion gear:,(2) rb1=ra1 cosa1 , rb2=ra2 cosa2,(6) If a , then tan and .
29、,(5) If Z, then . Try to use smaller m.,(4) is independent of module m.,(3) If ra , then a and . 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:,
30、Since,=20.,6.7 Manufacturing Methods of Involute Profiles,Cutting is the most common method for manufacture of gears.,There are various methods for manufacturing tooth profiles, such as die casting(模铸), precision forging(锻造), punch冲压, powder(粉末) process, cutting切削法 and so on.,6.7.1 Cutting of Tooth
31、Profiles,(1) Form (仿形法) Cutting (2) Generating (展成法或范成法) Cutting,Form Cutting,Disk milling cutter(盘形铣刀）,End milling cutter(指状铣刀),Broach(拉刀),Disk milling cutter,End milling cutters(端铣刀) are used for large modules and double helical gears.,broach(拉刀),Characteristics of the form cutting:,As the shape o
32、f an involute depends on its base radius, a different cutter is 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),(2) Generating Cutting
33、 (展成法或范成法加工),In a generating cutting method, the edges of a cutter take the form of a gear (or rack) with the same module and pressure angle as the gear to be cut.,Cutting motion,3. Generating motion展成,2. Retreat motion退刀,4. Feed motion进给,External gears can be cut with a rack-shaped shaper cutter(齿条
34、插刀). The edges of the cutter are now straight lines which can be made accurately.,优点：用一把插刀可以加工出 m、相同而齿数不同的各种齿轮(包括内齿轮)。 缺点：切削不连续，生产效率较低。,The cutting process in the shaping(插齿) of a gear is not continuous.,In mass production(大规模生产), hobbing(滚齿) is used instead of shaping.,hobbing(滚齿) 优点：用一把滚刀可以加工出 m、相
35、同而齿数不同的各种齿轮，切削连续，生产效率高。 缺点：不能加工内齿轮。,6.7.2 Cutting a Standard Gear with Standard Rack-shaped Cutter,普通齿条,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 cutter should be tangent to the reference
36、 circle of the gear,The 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 of the gear is cut by the t
37、ip 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). It is not cut by gear cutter!,6.7.3 Cutter Interference(根切现象),Some part of the involute profile near the root portion is removed in the generating cutting.,This will reduce the contact ratio as well
38、 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：ha*=0.8，a=20，Zmin=14,6.8 Addendum Modification on Involute Gears 齿轮变位,6.8.1 Intr
39、oduction of Addendum Modification,Disadvantages of standard gears:,1. Z Zmin,2. a = a,3.The strength of the pinion is much lower than that of the gear.,外啮合,无法安装；,Corrected gears变位齿轮,1、用改变刀具与轮坯径向相对位置来切制齿轮的方法称径向变位法。,变位齿轮,xm移距或变位,x移距系数或变位系数modification coefficient,规定：,正变位,零变位,负变位,Only the position of t
40、he 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 remains unchanged.,Comparison between a corrected gear and the standard gear:,1.Paramet
41、ers 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 gear,6.8.2 Geometric Dimensions of Corrected Gears,To prevent cutter interference：最小变
42、位系数,正变位,负变位,1、分度圆上的齿厚,见图：刀具节线的齿槽宽比中线齿槽宽 ，被切齿轮分度圆上的齿厚增加,在IJK中：,分度圆的齿厚：,2、齿顶高和齿根高,齿根高hf：刀具加工节线到顶 刃线之间的距离,对正变位：x0， hf比标准减小xm,对负变位：x0， hf比标准增加xm,变位齿轮的齿根圆半径：,齿顶高：,变位齿轮的分度圆与相应标准齿轮的分度圆一样， 变位齿轮的齿顶高仅决定于轮坯顶圆的大小。,为保证齿全高,由于,所以,Helical Gears for Parallel Shafts (斜齿轮传动),Crossed helical gears mechanism(螺旋齿轮机构),6.9
43、 Helical Gears for Parallel Shafts (平行轴斜齿轮传动),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).,The tooth surfaces of spur gears contact on a straight line parallel to the axes of the gears.,Vibration and noi
44、se,sudden loading and sudden unloading 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
45、the involutes on the base cylinder is a helix(螺旋) AA.,The surface profile of a helical gear is called an involute helicoid(渐开线螺旋面),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,2.端面（垂直于齿轮轴线的面）与齿廓曲面的交线为渐开线。,3.与基圆柱同轴的圆柱面与渐开线螺旋面的交线为一螺旋线。 不同面螺旋角不同,
46、斜齿轮的啮合特点：,（1）两斜齿齿廓的公法面既是两基圆柱的公切面，又是传动的啮合面,（2）两齿廓的接触线与轴线夹角 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 smooth. That is why helical gears can operate at a higher speed.,6.9.2 Parameters
47、 of Helical Gears,1、斜齿轮的切削加工：,仿形法；范成法：滚齿,（用仿形法加工斜齿轮时，铣刀是沿螺旋齿槽的方向进刀的）,1. On the transverse plane(端面) (with subscript “t” ) 轴线的面,The transverse profile of a helical gear is an involute.,Two sets of parameters :,2. On the normal plane (法面)（with subscript “n”)垂直于分度圆柱面螺旋线的切线的平面,Parameters on the normal p
48、lane are the standard values. They are determined by cutter.,刀具的齿形应于斜齿轮的法面齿形相同，法面上的模数和压力角为标准值 。,mn is standardized.,法面压力角与端面压力角,BD=CE,螺旋角：,螺旋线的导程 Pz:,（不同圆柱面的螺旋角不等）,螺旋线绕一周时它沿轴线方向前进的距离,The reference diameter and centre distance should be calculated on the transverse plane as,Reference centre distance a can be adjusted by changing the helix angle(螺旋角) . Standard helical gear drive can suit any centre distance.,6.9.3 Proper Meshing Conditions for Helical Gears,or,According to the proper meshing conditions of spur gear drive,6.9.4 Contact Ratio for a Helical Gear Pair,2、重合度