第12次課-Chapter 6Gear Mechanisms

1、6.5 6.5 Gearing of InvoluteGearing of Involute Spur Gears Spur Gears To be discussedTo be discussed6.3 The Involute(漸開線漸開線) and Its Properties 6.4 Standard Involute Spur Gears (標準漸開線直齒輪）標準漸開線直齒輪）6.2 Fundamentals of Engagement of Tooth Profiles(齒廓嚙合基本定律齒廓嚙合基本定律) 6.1 Types of Gear MechanismsREVIEW of

2、what we have learnt6.4 Standard Involute6.4 Standard Involute Spur Gears Spur Gears (標準漸開線直齒輪）標準漸開線直齒輪）1. External Gears（外齒輪） OThe tooth number (齒數) of a gear - z z The facewidth(齒寬) - BBOBrfraOAddendum circle (or tip circle) （齒頂圓） -da, raDedendum circle (or root circle) （齒根圓）-df, rfBase circle -db,

3、 rbAn arbitrary circle (任意圓) -di, riBrbrfraOriBrbrfraOriThe tooth space (齒槽) -the space between the profiles of two adjacent(相鄰的) teeth.The spacewidthspacewidth （齒槽寬） ei -the arc length of the tooth space along an arbitrary circleThe tooth thicknesstooth thickness （齒厚）（齒厚） si - the arc length of the

4、 tooth along an arbitrary circleseriOiiiThe pitchpitch(齒距，周節(jié)) pi -arc distance between corresponding points of adjacent teeth along an arbitrary circle. pi = ei + siseriOiiiReference circle:Reference circle: Different point on the involute has different pressure angle K . The circle the point on whi

5、ch has 20pressure angle is defined as reference circle (分度圓).OABKxyrbkrkkFVkkAny parameter on the reference circle has no subscript(下標), i.e., d，r, e, s, pseprbrhahhfNOcosrrb pi = ei + si p= e + s KKbrrcosThe module m (模數)- a basic parameter on the reference circle.iizpdmpmzzpdzpd mzzpdseprbrNOm, z,

6、 are three basic parameters of a gear.Different module needs different cutter. The module m has been standardized.mzd pesmseprbrNOTab 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,

7、 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, 45The addendum （齒頂高）（齒頂高） ha- radial distance between the reference circle and the addendum circle.seprbrhahhfNO

8、The dedendum （齒根高）（齒根高） hf -radial distance between the reference circle and the dedendum circle.The tooth depth （齒全高）（齒全高） h -radial distance between the dedendum circle and addendum circle.For standard gears:mse5.0hh maa*hhcmfa()*c* - the coefficient of bottom clearance(頂隙系數）頂隙系數）ha* - the coeffic

9、ient of addendum(齒頂高系數）齒頂高系數） mchha)2(*ha*=1 and c*= 0.25 for the normal tooth(正常齒正常齒)ha*= 0.8 and c*= 0.3 for the shorter tooth(短齒短齒) c=c* m- bottom clearance(頂隙頂隙)OOBB1212PKN2N11a22C*mCC C*mmhzhddaaa)2(2*mchzhddaff)22(2*rbOBrserahahhfNpfrZ=18 不同模數的齒輪Base pitchBase pitch( (基節(jié)基節(jié)) ) pb- the pitch on

10、the base circle. Normal pitchNormal pitch( (法節(jié)法節(jié)) ) pn-the distance between corresponding sides of adjacent tooth profiles along the common normal.prOppbrbncoscospzdzdppbbnprOppbrbnrbOBrserahahhfNpfrThe fundamental parameters: z, m, , ha*, c*A standard gear means: m, , ha*, c* take the standard valu

11、es and e = s 2. The Rack (齒條）齒條）CirclesStraight linesInvolutesStraight linespeshafBhpbi pi p m peshafBhpbThe characteristic of rack:peshafBhpb On the reference line(分度線), e=s. For standard rack, ha=ha*m, hf=(ha*+c*)mhhOs eprbrarrfBfa3. Internal Gears3. Internal Gears The tooth the tooth space mhzhdd

12、aaa)2(2*mchzhddaff)22(2*6.5 6.5 Gearing of InvoluteGearing of Involute Spur Gears Spur Gears 6.3 The Involute(漸開線漸開線) and Its Properties 6.4 Standard Involute Spur Gears (標準漸開線直齒輪）標準漸開線直齒輪）6.2 Fundamentals of Engagement of Tooth Profiles(齒廓嚙合基本定律齒廓嚙合基本定律) 6.1 Types of Gear MechanismsCONTENTS WILL BE

13、 DISCUSSEDONO12rb1b2NKPr126.5 Gearing of Involute Spur Gears 1.The process of gearing1.The process of gearing Suppose gear 1 is a driverSuppose gear 1 is a driverONO12rb1b2NPr12B2a2rThe meshing The meshing begins at point Bbegins at point B2 2ONO12rb1b2NKPr12ONO12rb1b2NPr12B12Bar1Moves along the Mov

14、es along the line of actionline of action( (嚙合線）嚙合線）Ends at point BEnds at point B1 1The segment B1B2 is called the actual line of action(實際嚙合線實際嚙合線). N1N2 is the limit of B1B2 and called the theoretical line of action(理論嚙合線理論嚙合線).ONO12rb1b2NKPr12B12BDECE is the actualactual working profileworking p

15、rofile of the gear 1. DF is the actual actual working profileworking profile of the gear 2.ONO12rb1b2NKPr12B12BCDEFN1PMLrb1r11a22Nrr22b2r2O1O2. Proper Meshing Conditions2. Proper Meshing Conditions(正確嚙合條件正確嚙合條件) For Involute Gears, To ensure a continuous transmission,there must exist such situation

16、that two pairs of teeth contact simultaneously . To maintain the proper meshing of two pairs of profiles at the same time, M1L1 = M2L2 , M1L1=Pn1 , M2L2=Pn2, Pb1=Pb2N1PMLrb1r11a22Nrr22b2r2O1ON1PMLrb1r11a22Nrr22b2r2O1Omm1122coscosThe proper meshing condition for involute gears: The modules and pressu

17、re angles of two meshing gears are the same respectivelymm1122coscos202121標準值mmm3. Continuous Meshing Conditions3. Continuous Meshing Conditions (連續(xù)傳動連續(xù)傳動條件條件) of an Involuteof an Involute Spur Gear Set Spur Gear Set The proper meshing condition is not a sufficient condition to ensure a continuous t

18、ransmission.2121mmmTransmission is continuous ifOOpbBB1212PKN2N111aa22npBB21OOpbBB1212PKN2N111aa22Define the contact ratiocontact ratio( (接觸比）接觸比） as : bpBB21Usually, =1.2.B1B2=PB1+PB2PB1=N1B1-PN1 , PB2=N2B2-PN2cosmPbOOpbBB1212PKN2N111aa22) tan(tan) tan(tan212211aazzFor external gear pair： depends o

19、n z z1 1 , z, z2 2 , , a1 a1 , , a2 a2 and and For internal gear pair:) tan(tan) tan(tan212211aazz2sin*2tantan211aahzB22r1=11=B1*hamVP2N1Nb1ra1rrFor rack and pinion: (1) rb1=ra1 cosa1 , rb2=ra2 cosa2 (5) If a , then and (4) If z, then . Try to use smaller m (3) is independent of module m (2) If ra ,

20、 then a and （短齒制缺點）短齒制缺點）) tan(tan) tan(tan212211aazzThe value of the contact ratio indicates the average number of tooth pairs in contact during a cycle to share the load. OOpbBB1212PKN2N111aa22singledoublerpnrb2B1Pb1B2singlenpOOpbBB1212PKN2N111aa224. Centre Distance and Working Pressure 4. Centre

21、Distance and Working Pressure AngleAngle( (嚙合角嚙合角) ) of a Gear Pair of a Gear PairTwo requirements in designing a gear pair: (1) The backlash(齒側隙齒側隙) b should be zero；(2) The bottom clearance(頂隙頂隙) c O12r1r2s1e1s2e2e1s12e2s(1) The backlash(齒側隙齒側隙) b should be zeroTo keep zero backlash(b=0), it holds

22、: sese1221(2) the bottom clearance(頂隙) c ( the radial distance from the dedendum circle of one gear to the addendum circle of another ) should take the standard valuemccOOBB1212PKN2N11a22m*CCBoth the backlash b and bottom clear-ance c are related to the centre distanceb1r11Or1Nar222Ob2rN21r2r2Ob22N1

23、r1O1r1N2rab1r2r1ar12r2For an external gear For an external gear pairpair ,When the centre distance a is equal to (r1+r2), the reference circles coincide with their pitch circles.The two reference circles are tangent and roll without slipping. b1r11ON22Ob2rN21r2r2Ob22N1r1O1r1N2rab1r2r1ar12raa212r1r1=

24、2r=aara1rf1ra2m*cm*c1O1OOO222frIf both gears are standard, When a=r1+r2 = =m(z1+z2)/ 2 , we have sseesseem111122222,0bsese1221mcrrrrrracaffa212121When two standard gears are assembled at a centre distance a a = =r r1 1+r+r2 2 , not only the backlash b=0, but also bottom clearance c=c*m. The centre d

25、istance a=ra=r1 1+r+r2 2 is called the reference centre distancereference centre distance( (標準中心距標準中心距). ).1= 2= working pressure angleworking pressure angle (嚙合角嚙合角) is defined as the angle between the velocity of pitch point P (on both gears) and the line of action（嚙合線）（嚙合線） .OOpbBB1212PKN2N111aa22If a is larger than (r1+r2), the reference circlesreference circles will separate,The two pitch circlespitch circles will become larger and remain tangent.2r2Ob21r