南京理工大学机械原理chapter6ppt课件教学教材.ppt

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 MechanismsChapter 7 Gear TrainsChapter 8 Other Mechanisms in Common UseChapter 9 Balancing of MachineryChapter 10 Motion of Mechanical Systems and Its RegulationChapter 11 Efficiency of MachineChapter 6GEAR MECHANISMScircular gears :constant transmission ratio non-circular gears :the ratio varies In this chapter, only circular gears are considered.6.1 Types of Gear Mechanisms Depending on the relative shafts positions parallel shafts→planar 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)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 gear6.1.2 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 Fundamentals 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 1·O1P= 2 ·O2P P12 is the instant centre of the gears 1 and 2. P12P ---------- 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 normal 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 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-n——Generating line(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 the 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 渐开线方程渐开线方程Pressure angle  K FVK= NOKFVK The polar parametric equation of the involute 渐开线方程渐开线方程 with  K as a parameter is6.3.4 Gearing(啮合啮合) of Involute ProfilesInvolute gears have constant 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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α’ 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α’ =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）（齿数）Parameters of gear: the facewidth(齿宽齿宽) ---- Baddendum circle (or tip circle) （齿顶圆）（齿顶圆） ----da, radedendum circle (or root circle) （齿根圆）（齿根圆）----df, rfbase circle ----db, rban arbitrary circle (任意圆任意圆) ----di, riThe 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 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 + sipressure angle  K Different point on the involute 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).FVKαK 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 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, 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 circle.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* 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=pnReview: The normal distance between 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：一个齿轮： d=mz da=d+2ha=(z+2 ha*)mdf=d-2hf=(z-2 ha*-2 c*)mdb=dcosαha= ha*mhf=( ha*+ c*)mh=ha+hf=(2 ha*+ c*)mP=πm一对标准齿轮正常安装：一对标准齿轮正常安装：①①m 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的主要参数②②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③③至于齿形，与至于齿形，与m m，，z z，，αα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*)m6.4.3 Internal Gears The tooth thickness   the tooth space To ensure a continuous transmission, 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 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. 节圆节圆 The two reference circles are tangent and roll without slipping. 6.5.2 Centre Distance 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 are 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 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 pinion6.6 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 Ratio(重合度重合度)The meshing begins at point B2, K moves along the line of actionand 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 Usually,  >[ ]=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 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 most 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 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 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 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 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、、αα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 m、、αα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 cutter 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 of 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 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=146.8 Addendum Modification on Involute 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 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 gear6.8.2 Geometric 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 x>0>0，， h hf f比标准减小比标准减小xmxm 对负变位：对负变位：x x<0<0，， 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 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 helix(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 βb 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 of Helical Gears 1、斜齿轮的切削加工：、斜齿轮的切削加工： ①①仿形法；仿形法；②②范成法：滚齿范成法：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）（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(端面端面) (with subscript “t” ) ⊥⊥轴线的面轴线的面 The transverse profile of a helical gear is an involute. Two sets of parameters :2. On the normal plane (法面法面)（（with subscript “n”)垂直于分度圆柱面螺旋线的切线的平面垂直于分度圆柱面螺旋线的切线的平面 Parameters on the normal plane are the standard values. They are determined by cutter.刀具的齿形应于斜齿轮的法面刀具的齿形应于斜齿轮的法面齿形相同，齿形相同，→→法面上的模数和法面上的模数和压力角为标准值压力角为标准值 。

 mn is standardized.法面压力角与端面压力角法面压力角与端面压力角∵∵BD=CE 螺旋角螺旋角ββ：： 螺旋线的导程螺旋线的导程 Pz:（不同圆柱面的螺旋角不等）（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、重合度——端面重合度，与其端面重合度，与其端面齿形相同端面齿形相同的直齿轮的重合度　的直齿轮的重合度　 ——纵向重合度纵向重合度 transverse contact ratiooverlap ratio 短齿制会降低重合度。

短齿制会降低重合度短齿制斜齿轮传动短齿制斜齿轮传动即保证即保证重合度，又提高安全性重合度，又提高安全性6.9.5 Virtual Number of Teeth (当量齿数当量齿数) for Helical GearsTo calculate the strength of the tooth, the virtual gear(当量齿轮当量齿轮) of the helical gear is introduced. The virtual gear(当量齿当量齿轮轮) of the helical gear is a spur gear, the tooth profile of which is equivalent to that of a helical gear on the normal plane. 当量齿轮当量齿轮：以：以ρρ为分度圆半径，用斜为分度圆半径，用斜齿轮的齿轮的m mn n和和ααn n分别为模数和压力角作分别为模数和压力角作一虚拟的直齿轮，其齿形与斜齿轮的一虚拟的直齿轮，其齿形与斜齿轮的法面齿形最接近这个齿轮称斜齿轮法面齿形最接近这个齿轮称斜齿轮的当量齿轮，齿数的当量齿轮，齿数ZV称当量齿数。

称当量齿数 Advantage of helical gears：(1) Helical gears operate more smoothly.(2) Have higher strength. (3) The working centre distance of a helical gear pair can be adjusted by the helix angle . (4)比直齿轮小，机构更紧凑(5)制造成本与直齿轮相同 Helical gears are therefore widely used for high speed or heavy load transmission.The main disadvantage of helical gears is:They produce an axial thrust(轴向推力). They produce an axial thrust(轴向推力). 为了保证有足够的重合度和较小的轴向力，为了保证有足够的重合度和较小的轴向力，一般对螺旋角有限制一般对螺旋角有限制蜗杆传动由交错轴斜齿轮机构传动演变而来，主要传递蜗杆传动由交错轴斜齿轮机构传动演变而来，主要传递交错轴交错轴之间的运动和动力。

之间的运动和动力1、、Smooth silent operation. 2、、Greater speed reduction3、、Self-locking (usually) 6.10 Worm Gearing(蜗杆传动蜗杆传动)6.10.1Worm Gearing and its CharacteristicsNon-intersecting and non-parallel shafts4、、Lower efficiencyMain Characteristics: Worms are a kind of screw, usually righthanded (右 旋 ) for convenience of cutting, or lefthanded (左旋) if necessary. Worms are usually drivers to reduce the speed. 蜗轮蜗杆旋向的判断蜗轮蜗杆旋向的判断 蜗轮蜗轮蜗杆蜗杆旋向旋向相同相同6.10.2 Types of Worms (1) Straight sided axial worms ------ZA-worm阿基米德蜗杆阿基米德蜗杆The transverse section(横切面横切面) is Archimedes spiral(螺旋线螺旋线). The axial section is involute rack.(2) Involute helicoid worms ----ZI-worm渐开线蜗杆渐开线蜗杆 (3)Arc-contact worms (4)-----ZC-worm圆弧齿蜗圆弧齿蜗杆杆 (4) Enveloping worm(包络蜗杆包络蜗杆) -------more teeth engaged.(5)Spiroid 锥蜗杆锥蜗杆6.10.3Proper Meshing Conditions for Worm Drives The mid-plane---- The transverse plane of a worm wheel passing through the axis of the worm .The engagement(啮合) between a Archimedes worm and a worm wheel on the mid-plane corresponds to that of an involute rack and pinion. Proper Meshing Conditions for Worm Drives：6.10.4 Main Parameters and Dimensions for Worm Drives 1）The axial pressure angle of worm x1 is 20. This can be increased to 25 in power transmission or decreased to 15 or 12 in indexing devices(分度装置). 2) The module m----standard values The series of modules for worms is somehow different from those for gears. Table6-2m11.251.622.53.15Mid-diameter d1182022.42028(18)22.4(28)35.5(22.4)28(35.5)45(28) 35.5(45)56m456.381012.5Mid-diameter d1(31.5)40(50)71（40）50（63）90（50）63（80）112（63）80（100）140（71）90（112）160（90）112（140）200 Table 6.2 Modules and Reference Diameters of Worms(mm) (GB10085-88) 3)The number of threads(头数头数) of the worm Z1Z1 ,then efficiency   Z1 = 1, 2, 4, 6 are preferable. The number of teeth of a worm wheel Z2. Z2=27~804) The mid-diameter d1 of worm(蜗杆中径）蜗杆中径）d1 ——The mid-diameter of worm 蜗杆中径，蜗杆分度圆直径 d1 is standardized to reduce the number of cutters and should be chosen together with the module m. 为减少刀具，将d1与m相搭配，见表6-2m11.251.622.53.15Mid-diameter d1182022.42028(18)22.4(28)35.5(22.4)28(35.5)45(28) 35.5(45)56m456.381012.5Mid-diameter d1(31.5)40(50)71（（40））50（（63））90（（50））63（（80））112（（63））80（（100））140（（71））90（（112））160（（90））112（（140））200 d2 ——The reference diameterof worm wheel. 蜗轮分度圆直径5) The lead angle(导程角) 1 of the worm. q q——直径系数直径系数6) The centre distance a’ of the worm gear pair. 7) The transmission ratio and the rotation direction of wormwheel传动比的大小传动比的大小 判断蜗轮的转向：判断蜗轮的转向： 蜗杆左（右）旋，伸左（右）手，四指握住蜗杆转动蜗杆左（右）旋，伸左（右）手，四指握住蜗杆转动方向，拇指的反向为蜗轮的啮合点圆周速度的方向。

方向，拇指的反向为蜗轮的啮合点圆周速度的方向6.11 Bevel Gears(锥齿轮) Bevel gears are used to transmit motion and power between intersecting shafts. 6.11.1 Types and Applications of Bevel GearsThe teeth of a bevel gear are distributed on the frustum(截锥体) of a cone(圆锥体). Parameters and dimensions at the large end are taken to be standard values. The shaft angle of a bevel gear pair can be any required value. In most cases, the two shafts intersect at a right angle. There are straight, helical and spiral (直齿、斜齿、曲齿) bevel gears.6.11.2 Back Cone(背锥) and Virtual Gear(当量齿轮) of a Bevel Gear Its theoretic tooth profile should be spherical(球面的) involute. The bottom of the reference cone(分度圆锥) is called the reference circle.球面渐开线用一个锥体渐开线来近似球面渐开线用一个锥体渐开线来近似一对背锥齿轮的啮合相当于一对背锥齿轮的啮合相当于一对锥齿轮大端的啮合一对锥齿轮大端的啮合分度圆上背锥圆锥面上的渐分度圆上背锥圆锥面上的渐开线代替球面渐开线开线代替球面渐开线The back cone can be unfolded(展开) imaginarily(假想地) into a sector gear(扇形齿轮) with Z teeth and further filled up to a full gear with ZV teeth. The tooth number of the virtual gear —— the virtual number of teeth Zv of the bevel gear.当量齿数 This imaginary gear —— the virtual gear(当量齿轮) of the bevel gear. Zv should not be less than 17.6.11.3 Parameters and Dimensions of Bevel Gears 1、the reference diameter The outer cone distanceThe reference cone angle2.2.正确啮合条件：正确啮合条件： 轴角轴角 Modules of bevel gears must be chosen from Table 6-33、The transmission ratio 4、The addendum ha and dedendum hf are measured along the generatrix(母线) of the back cone.Here c*=0.2, not 0.25. ha=ha*m=1mhf=(ha*+c*)m=1.2mChapter 6 Attentions1.Parameters of standard Spur involute gear2. Calculate the working pressure angle and the radius of pitch circles with actual centre distance 3. Calculate the contact ratio4. Parameters of standard helical involute gear5. Virtual gear of helical involute gear6. Parameters of worm and worm wheel7. Parameters of bevel gear8. Virtual gear of bevel gear。