Presentation on theme: "Chapter 10 Gear Mechanisms"— Presentation transcript:
1Chapter 10 Gear Mechanisms §10－1 Applications and Types of Gear Mechanisms§10－2 Fundamentals of Engagement of Tooth Profiles§10－3 The Involute and Its Properties§10－4 Terminology and Definition of Gears§10－5 Gearing of Involute Spur Gears§10－6 Introduction to Corrected Gear§10－7 Helical Gears for Parallel Shafts§10－8 Worm Gearing§10－9 Bevel Gears
2§10－1 Applications and Types of Gear Mechanisms 一、Introduction1) Gear mechanisms are widely used in all kinds of machines to transmit motion and power between rotating shafts.2) Circular gears have constant transmission ratio whereas, for non-circular gears, the ratio varies as the gears rotate.3) Depending upon the relative shafts positions, circular gear mechanisms can be divided into planar gear mechanisms and spatial gear mechanisms.4) In this chapter, only circular gears are considered.二、Types of Gear Mechanisms
4§10－2 Fundamentals of Engagement of Tooth Profiles Conjugate Profiles—— Meshing profiles of teeth that can yield a desired transmission ratio are termed conjugate profiles. (i12=ω1/ω2)O2O121C1C2一、Fundamental Law of Gearingvk1vk2The driving pinion rotates clockwise with angular velocity ω1 while the driven gear rotates counterclockwise with angular velocity ω2 . The common normal n-n intersects the center line O1O2 at point P. the point P is the instant center of velocity of the gears .ω2ω1PnKvk1k2nv12 ＝O1P ω1= O2 P ω2i12 ＝ω1/ω2＝O2 P /O1PThe transmission ratio:fundamental law of gearing:The transmission ratio of two meshing gears is inversely proportional to the ratio of two line segments cut from the center line by the common normal of the tooth profiles through the contact point.
5point P ——the pitch point. As the center distance O1O2 is constant, the position of the point P must be fixed if a constant transmission ratio i12 is required.O2O121C1C2This implies that, wherever the teeth contact, the common normal n-n of the tooth profiles through the contactpoint must intersect the centerline at a fixed point P, if aconstant transmission ratio i12 is required.r1r2ω2ω1PnKpitch circlepitch circle——The loci of P on the motion planes of both gears are called the circles.
6二、Conjugate ProfilesMeshing profiles of teeth that can yield a desired transmission ratio are termed conjugate profiles. For circular gears, the conjugate profiles are those that provide the desired constant transmission ratio. Generally speaking, for any specific tooth profile, we can find its conjugate profile. Theoretically. there is an infinity of pairs of conjugate profiles to produce any specific transmission ratio. Nevertheless, only a few curves have been used as conjugate profiles in practice. Among them, involutes are used most widely since gears using involutes as teeth profiles, or involute gears as they are called, can be manufactured and assembled easily.
7§10－3 The Involute and Its Properties 一、 Generation of Involute BK一、 Generation of InvoluteThe involute——is the curve generated by any point on a string which is unwrapped from a fixed cylinder.Generating lineBase circleOrkAθkrb二、Properties of the Involute1） AB = BK;2）The normal of an involute at any point is tangent to its base circle.3）The tangent point B of the generating line with the base circle is the curvature center of the involute at the point K. The length of the segment BK is the radius of curvature of the involute at the point K.4）The shape of an involute depends only on the radius of its base circle.
8rk= rb/cosαk θk= invαk= tanαk-αk 5）No involute exists inside its base circle.tBKOArkθkrbαkvkKO2B2A1B1O1A2αkrbO3B3∞三、Equation of the Involutecosαk= rb/rktanαk= BK/rb=AB/rb=rb(θk+αk) /rb=θk+αkrk= rb/cosαkθk= invαk= tanαk-αk
9三、Gearing of Involute Profiles 1. The transmission ratio will remain constant.O2O1ω2ω1The common normal N1N2 to the meshing involute profiles through their contact point K must be the common tangent to their base circles. The position of this common tangent remains unchanged as both gears rotate, as does the common normal to the involute profiles. This results in a fixed pitch point P. Therefore, according to the fundamental law of gearing mentioned, the transmission ratio will remain constant.rb2N2N1K/C1C2KPi12=ω1/ω2= O2P/ O1P = constant
10As shown in Fig., the transmission ratio: 2. The direction and magnitude of the reaction force does not changeO2O1ω2ω1N1N2—— trajectory of contact（line of action）α’ ——pressure anglerb2N2N1The reaction force is exerted along the line of action if there is no friction. As the position of the line of action stays unchanged during motion for an involute gear pair, the direction and magnitude of the reaction force does not change.α’K/C1C2KP3. the separability of the center distance in involute gearing△ O1N1P∽△O2N2PAs shown in Fig., the transmission ratio:i12=ω1/ω2= O2P/ O1P = rb2 /rb1A change in centre distance does not therefore affect the constant transmission ratio of an involute gear pair. This property is called the separability of the center distance in involute gearing .
11§10－4 Terminology and Definition of Gears pipAddendum circle： da、raraeiesipnhaDedendum circle： df、rfsrpbrbhTooth thickness： sirfhfSpacewidth： eiCircular pitch： pi= si + eiReference circle: Between the addendum circle and the dedendum circle, there is an important circle which is called the reference circle. Parameters on the reference circle are standardized and denoted without subscripts, such as d, s, e and p.Addendum：haBase circle： db、rbDedendum：hfNormal pitch： pn = pbTooth depth：h= ha+hf
12m=p/π (as πd = zp，then d = zp /) 二、Basic ParametersNumber of teeth： zModule：m The module m of a gear is introduced on the reference circle as a basic parameter, which is defined as:m=p/π (as πd = zp，then d = zp /)d=mz(3.25) (3.75)Second (6.5) (11)Series (30)Modules of involute cylindrical gears（GB1357－87）First Series
13Sizes of the teeth and gear are proportional to the module m.
14Coefficient of addendum: ha*，be standardized: ha*＝1 Pressure angle：αThe pressure angle α is taken as a basic parameter to determine the base circle. The pressure angle α is also standardized. It is most commonly 20°.Coefficient of addendum: ha*，be standardized: ha*＝1Coefficient of bottom clearance : c*， be standardized:c*＝0.25z、 m、α、ha*、c* are the fundamental parameters which determine the size and shape of a standard involute gear.三、Parameters of GearStandard gear：1）m ,α, ha* , c* are standardized2）e = s
16四、The Rack and Internal Gears A rack can be regarded as a special form of gear with an infinite number of teeth and its center at infinity. The radii of all circles become infinite and all circles become straight lines, such as the reference line, tip line and root line.Characteristics1) The involute tooth profile becomes a straight line too and the pressure angle remains the same at all points on the tooth profile.pbBαhahfpes2) The pitch remains unchanged on the reference line, tip line or any other line, i. e. pi= p =πm
172. Internal Gears Characteristics: df > d > da da＝ d - 2ha The teeth are distributed on the internal surface of a hollow cylinder. The tooth of an internal gear takes the shape of the tooth space of the corresponding external gear, while the tooth space of an internal geartakes the shape of tooth of thecorresponding external gear.OBsepdf > d > dada＝ d - 2hadf ＝d + 2hfrfrpbhhahfrbraNα3) To ensure that the profile of the tooth on the top is an involute curve, da>db .
18§10－5 Gearing of Involute Spur Gears 一、Proper Meshing Conditions for Involute GearsO1rb1r1O1ω1rb1ω1r1pb1PN1N2B2B1pb2pb1rb2r2O2ω2rb2r2O2ω2pb2pb1= pb2pb1> pb2
19m1cosα1=m2cosα2 m1 = m2 = m α1=α2 =α To maintain the proper meshing of two pairs of profiles at the same time, the normal distances of the teeth on both gears must be the same.rb2r2O2ω2rb1r1O1ω1pb2pb1PN1N2B2B1pb1= pb2m1cosα1=m2cosα2m1 = m2 = mα1=α2 =αThe proper meshing condition for involute gears: the modules and pressure angles of two meshing gears should be the same.
20There are two requirements in designing a gear pair. 二、Center Distance and Working Pressure Angle of a Gear PairO1There are two requirements in designing a gear pair.aω1ra1r1rb11) The backlash should be zero to prevent shock between the gears.rf1N1To obtain zero backlash of a gear pair:Zero backlashPC=C*mCs’1= e’2 s’2= e’1N22) The bottom clearance should take the standard valuerf2r2c=c*m2. Standard(reference) center distanceω2Standard mountingworking center distance a’=r’1+ r’2O2reference center distance a = r1+ r2If two gears are mounted with the reference center distance, then：
21a’cosα’= a cosα 3. Center distance a and working pressure angle α’ rb2ω2ra2O1ω1rb1ra1r1r2PN1N2aα’rb2O2ω2O1ω1rb1a’α’PN1N2r2r’2r1r’1r1’ =r1α’ =αr2’ =r23. Center distance a and working pressure angle α’α’ >αα’>αr’2 >r2r1’ >r11) Standard mounting(a’ = a)The reference circles coincide with their pitch circles.r’1=r1 r’2=r α’=α c=c*m2)Nonstandard mounting(a’ >a)The reference circles do not coincide with their pitch circles.r’1> r1 r’2> r α’>α c’>c*mrb1＋rb2 = (r1’+r2’)cosα’ rb1＋rb2 =（r1＋r2）cosαa’cosα’= a cosα
22Meshing of a rack and pinion ω1ra11） Standard mountingr1rf11The pitch line of the rack coincides with its reference line :r1’ = r1 ，α’＝αN1α’=αv22N2P2） Nonstandard mountingThe pitch line of the rack does not coincides with its reference line :r1’ = r1 ，α’＝αAs mentioned above, α’＝α, and r '＝r are characteristics of rack and pinion gearing and differ from those of two spur gears.
231. Mating Process of a Pair of Gears 三、Mating Process of a Pair of Gears and Continuous Transmission ConditionO11. Mating Process of a Pair of Gearsω2ω1rb1B2 ——meshing begins at point B2ra1N1B1 ——meshing ends at point B1PB2B1N2B1B2 ——the actual line of actionra2rb2N1N2 ——the theoretical line of actionN1、N 2 ——meshing limit pointsO2
242. Continuous Transmission Condition In order to get a continuous motion transmission, the secondpair of teeth must have meshed before the first pair moves out of contact.The condition of continuous motiontransmission is： B1B2≥pbO1N2N1KO2ω2ω1B1B2Contact ratio: = B1B2/pbpbB1B2Theoretically, if ＝1, a pair of gears can transmit continuously. Considering the manufacture tolerance, the contact ratio should be larger than 1. Actually, the contact ratio should be equal to or larger than the permissible contact ratio. 
25Equations of Contact Ratio rb1rb2O2Pεα＝ B1B2/pb ＝(PB1+P B2) /πmcosααa2αa1PB1＝B1 N1-PN1＝rb1tanαa1- rb1tanα’＝z1mcosα(tanαa1-tanα’ )/2ra1B2ra2B1PB2＝z2mcosα(tanαa2-tanα’ )/2εα＝[z1(tanαa1-tanα’) +z2(tanαa2-tanα’)]/2πThe value of the contact ratio indicates the average number of tooth pairs in contact during a cycle to share the load. The higher the contact ratio, the greater the average number of tooth pairs to share the load and the higher the capacity of the gear set to transmit the power.
27§10－6 Introduction to Corrected Gear 一、Standard gears have some disadvantagesStandard gears enjoy interchangeability and are widely used in many kinds of machines. However, they also have some disadvantages.1）It is not fit that a’≠a. When a’<a , the pair of gears can not be installed at all. When a’>a, the backlash will increase and the contact ratio will decrease.2）The curvature radius of the tooth profile and the tooth thickness of the pinion on the dedendum circle are less than those of the gear. The strength of the pinion is much lower than that of the gear, and contact time of the pinion is more than that of the gear.BasecircleReference3）When z< zmin，undercutting will occur.Cutter interference——In a generating process, it is sometimes found that the top of the cutter enters the profile of the gear and some part of the involute profile near the root portion is removed.
28二、Manufacturing Methods of Involute Profiles To improve the performance of gears, addendum modification is employed.二、Manufacturing Methods of Involute Profiles1. Cutting of Tooth Profilespinion-shaped shaper cutterrack-shaped shaper cutterThe cutting motion is the reciprocation of the cutter while the feed is the movement of the cutter toward the blank. The blank should retreat a little as the cutter goes back to prevent scraping on the finished flank by the cutter.
302. Cutting a Standard Gear with Standard Rack-shaped Cutter O1Gear blankO1’The reference line of the cutter should be tangent tothe reference circle of the gearrb’N1’O1’’N1’’rb’’raInvolutec*mrvrbReference circleN1αha*mReference lineB1B2e = s = p / 2ha= ha* m; hf =(ha*+ c*)m;P1）The addendum line of the cutter does not exceed the limit point N1’’ of the line of action, cutter interference will not occur.2）Cutter interference will occur if the addendum line of the cutter passes the limit point N1’’ of the line of action.
313. Minimum Teeth Number of Standard Gear Without Undercutting To prevent cutter interference, the point B2 should not pass point Nl ,：PN1≥PB2PN1=rsinα=mzsinα/2PB2=ha*m/sinα=mzsinα/24. Methods to Avoid UndercuttingThere are several methods to avoid undercutting：1）Decrease the coefficient of addendum depth ha*ha* zmin ha* the transmission characteristics will be influenced and the cutter will not be standard.
322）Increase the pressure angle of cutter a a zmin a rb This procedure will reduce the active length and the contact ratio will reduce too, which will also lead to rougher, noisier gear operation and the cutter will not be standard.3）Corrected gearO1The method commonly used to eliminate undercutting is to cut the gears with profile-shifted, i.e., with unequal addendum and dedendum teeth.αxminmxmN1ha*mTherefore, parameters m , a , ha* , c* , of the corrected gear remain the same as those of standard gears, but s≠e，the gear is called corrected gear (profile-shifted gear).QαPxmThe cutter will be standard.
335. Corrected gearO1Modification distance（xm）—— In cutting the corrected gear, the rackshaped cutter is located a distance xm from the position used for cutting the standard gear.x ——modification coefficientαxminmxmN1ha*mQαPxmPositive modification( x>0) ——The cutter is placed further away from the position for cutting a standard gear.positive modification gearNegative modification( x<0) ——The cutter is placed towards the axis of the blank negative modification gear
34Reference line of cutter 三、Geometric Dimensions of Corrected Gears1. Geometric dimensions are identical with that of the standard geard = mzdb = mzcosp = mReference circlePN1O1αrbKJI2. Geometric dimensions are not identical with that of the standard gearαxm1）Tooth thickness and spacewidthBase circleB2Reference line of cutterPitch line of cutterxmKIJπm/2
35Negative modification gear x<0 2）Addendum and dedendumPositive modification gear x>0Standard gear x＝0Negative modification gear x<0Reference circle
36四、 Gearing of a Corrected Gear Pair Proper meshing conditions and condition of continuous transmissionProper meshing conditions： m1= m α1=α2Condition of continuous transmission： 2. Centers distance of a pair corrected gear1) Gearing equation without backlashTo keep zero backlash for a corrected gear pair, the following relations should hold, as in the case of standard gears, i.e., sl'= e2' , s2'= el' , therefore,p'＝ s'1+ e'1 ＝ s'2+ e'2＝ s'1+ s'2
37Analysis(x1+x2) a’ aThe two pitch circles will not overlay on the two reference circlesacos＝acos a’ a2） Shifting coefficient of centers distance yDifference of the centers distance a’ with standard centers distance a ：ym = a’- ay——Shifting coefficient of centers distance
383） Shifting coefficient of addendum depth y With no backlash： Clearance be standard：If two gears mating with no backlash and remaining standard clearance, thereforea'=a'' y=x1+x2Problem： (x1+x2) > y if x1+ x2≠0 a' > a''Solution：No backlash can be assured, the depth of addendum circle is decreased.
393. Types of Corrected Gear Pairs Types of corrected gear pairs can be divided into three typesby the sum of the shifting coefficients（x1+ x2）.（1）Standard transmission（ x1+ x2＝0，and x1＝x2＝0）z1 > zmin , z2 > zmin（2）Zero transmission (height shifting gears transmission）x1+ x2＝0，and x1＝-x2≠0As x1+x2 = 0 and the above three equations a’ = a , ’=， y = 0， y = 0The pinion should be positive corrected gear( x1 >0)；the gear should be negative corrected gear（x2<0）.Two gears should not be undercutting：z1 + z2 ≥ 2zmin
40（3）Angle shifting gear transmission ( x1+x2≠0 ) 1）Positive transmission（ x1+x2 > 0 ）As x1+x2 > 0 and the above three equations a’ > a , ’ > ， y > 0， y > 0As x1+x2 > 0 z1+z2 < 2 zminSince gears are positive corrected gear, the strengths of two gears increase. But the contact ratio decreases since the working pressure angles decrease.
412）Negative transmission（ x1+x2 < 0 ） As x1+x2 < 0 and the above three equations a’ < a , ’ < ， y < 0， y > 0AS x1+x2 < 0， therefore z1+z2 > 2 zminThis transmission is contrary to positive transmission. Since gears are negative corrected the strengths of the two gears decrease.But the contact ratio increases since the working pressure angle decrease.
42§10－7 Helical Gears for Parallel Shafts Properties:Tooth profiles go into and out of contact along the whole facewidth at the same time；Sudden loading and sudden unloading on teeth；Vibration and noise are produced.Spur gearProperties:The tooth surfaces of two engaging helical gears contact on a straight line inclined to the axes of the gears；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.Helical gear
43一、 Basic Parameters of Helical Gears There are two sets of parameters for a helical gear.One set is on the transverse plane and the other set on the normal plane.The parameters on the normal plane are the standard values.To make use of the formulae for spur gear, the parameter in the equations for spur gears should be replaced by those on the transverse plane of helical gears. Therefore, it is necessary to set up relationships between both sets of parameters.（一） Basic Parameters of Helical1. Helix angleβhelix angle（β）——is the helix angle on the reference cylinder.ββrighthandedlefthanded
442. Normal module mn and transverse module mt βptpnβBπd3. Normal pressure angle n and transverse pressure angle t4. Coefficient of addendum（ h*an 、h*at）and coefficient of bottom clearance(c*n 、c*t)ha=h*anmn = h*atmthf=(h*an+cn*)mn = (h*at+ct*)mt
45(二） Sizes of helical gear Reference diameter：Center distance:Modification coefficient：二、Gearing of a pair of helical gears1. Proper Meshing Conditions for Helical Gearsor
462. Contact Ratio for a Helical Gear Pair B1B2Spur gear：BB1B2Helical gear：B1B2βbBB1B2ea --- transverse contact ratio△LLThe contact ratio of a helical gear pair is much higher than that of a spur gear pair.eb --- is the face contact ratio or overlap ratio.
47三、 Virtual Number of Teeth for Helical Gears Virtual gear——the tooth profile of the spur gear is equivalent to that of a helical gear on the normal plane. The spur gear is called the virtual gear of the helical gear. The number zv of teeth of the virtual gear is called the virtual number of teeth （zv）.The minimum number of teeth of the standard helical gear without cutter interference：zmin=zvmincos3β
48四、The main advantages and disadvantages of helical gears FnFtβ1. Main advantages：β１）Better meshing properties.２）A much higher total contact ratio.３）Being more compact means of mechanicalpower transmission.2. Main disadvantages：The helix angle results in a thrust load in addition to the usual tangential and separating loads. Fa=Ft tgb，b ，FaHerringbone gear：b = 25～35 β＝8°～20°
49§10－8 Worm GearingWorm gear drives are used to transmit motion and power between nonintersecting and non-parallel shafts, usually crossing at a right angle. ＝90一、Worm Gearing and its Characteristics1) Smooth silent operation as screw drives.2) Greater speed reduction in a single step. This means compact designs.3) If the lead angle of a worm is less than the friction angle, the back-driving is self-locking.4) Lower efficiency due to the greater relative sliding speed . The friction loss may result in overheating and serious wear. Therefore, brass is usually used as the material for the worm wheel to reduce friction and wear.
50二、 Types of Worms Archimedes worm Involute helicoid worms Cylindrical wormsArc-contact wormsTypes of WormsEnveloping wormsspiroids
51三、Proper Meshing Conditions for Worm Drives mid-plane：The transverse plane of a worm wheel passing through the axis ofthe wormThe engagement between a worm and a worm wheel on the mid-plane corresponds to that of a rack and pinionProper Meshing Conditions：The modules and pressure angles of the worm and worm wheel on the mid-plane should be equal to each other.The directions of both helices should be the same.
52四、 Main Parameters and Dimensions for Worm Drives The number of teeth The number of threads on the worm z1 : usually, z1＝1 ~ 10，the recommended value of z1:z1＝1、2、4、6。The number of teeth on the worm gear z2 is determined according to the speed ratio and the selected value of z1. For power transmission, z2＝29 ~ 70.The moduleThe series of modules for worms is somehow different from those for gears.The profile angle of worm (pressure angle)Archimedes worm ：a = 20ºIn power transmission： a = 25ºIn indexing devices： a =1 5º or 12º
534. The lead angleγ1 of the worm 5. reference diameterThe mid-diameter d1 of worm：the mid-diameter d1 ofthe worm is standardized.The reference diameter d2 of worm wheel： d2 = mz26. The center distance a of the worm gear pair
54§10－9 Bevel Gears 一、Introduction to Bevel Gears 1. Characteristics of Bevel GearsBevel gears are used to transmit motion and power between intersecting shafts. The teeth of a bevel gear are distributed on the frustum of a cone. The corresponding cylinders in cylindrical gears become cones, such as the reference cone, addendum cone and dedendum cone. The dimensions of teeth on different transverse planes are different. For convenience, 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.
552. Types and Applications or Bevel Gears are most widely used as they are easy to design and manufacture.Straight bevel gears：BevelGearsHelical bevel gears：operate smoothly and easy to design .Spiral bevel gears：operate smoothly and have higher load capacity.
56二、Back Cone and Virtual Gear of a Bevel Gear Crown gear ----d 2 = 90 ，the surface of the reference cone becomes a plane.Back cone——the cone , the element of which crosses the large end of a bevel gear and is perpendicular to the element of the reference cone.Virtual gear of the bevel gear： mv = m ； αv = α ；rv= rThe tooth profile of the virtual gear is almost the same as that of the bevel gear at the large end.Virtual number of teeth zv ：The tooth number of the virtual gearr2Crown gear=90°δ22∑PPP1δ11O2rv1O1r1
57Virtual number of teeth zv The engagement of bevel gearsThe engagement of spur gearsr2Crown gear=90°δ22∑PPP1δ11O2rv1O1r1
58三、Parameters and Dimensions of Bevel Gears Proper Meshing Conditions： m1=m2 , α1=α2The contact ratio of the bevel gear set. The virtual number of teeth zv should not be less than the minimum number of teeth of the virtual gear. zmin=zvmincosδ三、Parameters and Dimensions of Bevel GearsThe most dimensions of bevel gears are measured at the large end being standardized.1. The reference diameter is2. The transmission ratio of a gear pair is（∑＝90°）