第六章 齒輪傳動

1、第六章 Toothed gearing 齒輪傳動3-1 General considerations 概述3-2 Materials, types of failure and design criteria 齒輪材料、失效形式和計算準則3-3 Loads in cylindrical gearing 圓柱齒輪傳動載荷計算3-4 Strength of spur cylindrical gearing 直齒圓柱齒輪傳動強度計算3-5 Strength of helical cylindrical gearing 斜齒圓柱齒輪傳動強度計算 Toothed gearing design examp

2、le 齒輪傳動設計實例3-6 Straight bevel gearing 直齒圓錐齒輪傳動6-1 General considerations 概述v Characteristics and applications 特點和應用v Types of the toothed gearing 齒輪分類一、Characteristics and applications 特點和應用1、 Advantages 優(yōu)點 Long service life and high reliability 工作可靠，壽命長 Constant speed ratio 傳動比恒定 High efficiency 工作

3、效率高 Small overall size 結構緊湊2、 Shortcomings 缺點 High cost 成本高 Noise in operation at considerable speeds or low precision in manufacture 制造精度低或者高速運轉時，振動噪音大 They are unsuitable for duty over medium and long distances 不適合大的中心距3、Applications 應用二、 Types of toothed gearing齒輪分類1、 According to the difference

4、between the relative positions of the shafts： 按軸布置方式分 Toothed gearing between parallel shafts 平行軸齒輪傳動 （ Cylindrical gearing圓柱齒輪傳動 ）Spur Cylindrical gearing 直齒圓柱齒輪傳動Helical Cylindrical gearing 斜齒圓柱齒輪傳動Herringbone gearing 人字齒圓柱齒輪傳動 Toothed gearing between intersecting shafts 相交軸齒輪傳動 （ Bevel gearing 圓錐

5、齒輪傳動） Toothed gearing between crossed shafts ： 交錯軸齒輪傳動 直齒圓錐齒輪傳動 斜齒圓錐齒輪傳動 曲齒圓錐齒輪傳動Crossed helical gearing 螺旋齒輪傳動Skew bevel gearingStraight bevel gearingSpiral bevel gearing2、 According to the difference operating conditions： 按工作條件分 Closed gearing 閉式齒輪傳動 Open gearing 開式齒輪傳動 Partly open gearing 半開式齒輪傳動

6、 3-2 Materials, types of failures and design criteria of gears 齒輪材料、失效形式和計算準則 Materials of gears 齒輪材料 Types of failures 失效形式 Design criteria 計算準則一、 Materials of gear 齒輪材料 Requirements 基本要求： Enough impact toughness 一定的沖擊韌性(Ensure sufficient beam strength in bending under variable load )（保證變載荷下，彎曲疲勞強度

7、較高) Enough hardness 一定的硬度（ Ensure sufficient endurance of their surface layers 保證抗齒面失效的能力） High manufacturability and low cost 工藝性好、成本低1、 Forged steel 鍛鋼 （ Wide application 常用） Under ordinary service condition一般場合： Lower hardness of tooth surface 軟齒面（HB350）齒輪材料Processing procedure 工藝過程： Medium-carbo

8、n steels(45) 中碳鋼 、Medium-carbon alloy steels(40Cr) 中碳合金鋼 Structural improvement and normalization 調質或正火 Cutting 切齒 For gears with lower hardness of tooth surface, the hardness difference between small gear tooth surface and large gear one is required: 軟齒面大小齒輪之間要求具有一定的硬度差： HBHB1 HB23050 Critical gear

9、s 重要應用：vProcessing procedure 工藝過程： Cutting teeth 切齒 Hardening 淬火 Grinding 磨削v 2、 Cast steel鑄鋼 （For gears of large diameters 尺寸較大、不容易鍛造時選用 da400mm）Higher hardness of tooth surface 選硬齒面（HB350）齒輪材料3、 Cast iron鑄鐵 Grey iron灰口鑄鐵（HT200HT400)：Easy to manufacture and low cost工藝性好、價格低；However, the strength an

10、d toughness is considerably less than that of steels 但沖擊韌性差、強度低 For low-speed, large-size gearing without impact. 低速、無沖擊、大尺寸齒輪 High-strength cast iron 球墨鑄鐵（QT5005、 QT6002等)：High strength and toughnessTaking the place of cast steel gears強度高、沖擊韌性好、代替鑄鋼4、 Plastics非金屬材料（ Caprolon and polyformaldehyde et

11、c 夾布膠木和尼龍等, ）：Self-lubrication and noiseless operation具有自潤滑性、工作噪音低For gears of high speed, light load and low accuracy requirement 用于高速、小功率、精度要求不高時 二、Types of failure 失效形式 Tooth breakage輪齒折斷 Failure of tooth surface layers 齒面損傷： Fatigue pitting, seizing, abrasive wear and plastic flow of material 點蝕

12、、膠合、磨損和塑性變形 三、Design criteria 計算準則v1、 Close gearing閉式齒輪傳動： The principal types of failure are pitting, tooth breakage and seizing of the teeth. And seizing of the teeth occurs under heavy load and calculation in checking for the possibility of tooth seizure is not effective. Hence, only calculations

13、 of the contacting fatigue strength and bending fatigue strength are carried out.以點蝕、輪齒折斷和膠合失效形式為主，其中膠合發(fā)生于重載，而且計算方法不成熟，所以一般只進行齒面接觸疲勞強度和齒根彎曲疲勞強度計算。 Low hardness gearing軟齒面齒輪傳動：The principal type of failure is pitting以點蝕失效為主Designing on the basic of the calculating of contact fatigue strength of the w

14、orking surfaces of the teeth按齒面接觸疲勞強度設計Checking the bending fatigue strength of the teeth root按齒根彎曲疲勞強度校核vHigh hardness gearing： 硬齒面齒輪傳動vThe principal type of failure is tooth breakage 以輪齒折斷失效為主Designing on the basic of the calculating of the bending fatigue strength of the teeth root按齒根彎曲疲勞強度設計Chec

15、king the contact fatigue strength of the working surfaces of the teeth按齒面接觸疲勞強度校核2、 Open gearing 開式齒輪傳動： The principal types of failure are abrasive wear and tooth breakage. Up till now, calculation for an assessment of the service life of gears subject to substantial wear is not effective. Hence, t

16、he design is carried out on the basic of the calculations of the bending fatigue strength , then module m is increased by 10 to 15% to take into account the influence of wear.以磨損和輪齒折斷失效為主，其中磨損無可靠計算方法，所以按彎曲疲勞強度設計，然后考慮磨損因素，將模數m增加1015。 6-3 Loads in cylindrical gearing圓柱齒輪傳動載荷計算Forces in spur cylindrica

17、l gearing 直齒圓柱齒輪傳動受力分析 Forces in helical cylindrical gearing 斜齒圓柱齒輪傳動受力分析 計算載荷一、 Forces in spur cylindrical gearing直齒圓柱齒輪傳動受力分析1、Relationship between forces acting on driving and driven gearscosd2TcosFFFtgFFFd2Td2TFFFFFFFF2111t1n2n1t1r2r12211t2t1r2r1t2t1n2n1、力的大小、各力間關系Magnitude of forces回轉方向相同從動輪（驅動

18、力）：與轉方向相反主動輪（阻力）：與回：中心由嚙合點指向各自回轉力的方向、trF:Fforces ofDirection 3Driving gear（ resistance force): Its direction is opposite to the direction of rotation of driving gear. Driven gear (driving force): Its direction is same with the direction of rotation of driven gear. Directing from the pitch point to t

19、he rotating centre二、 Forces in helical cylindrical gearing 斜齒圓柱齒輪傳動受力分析a2a1r2r1t2t1n2n1FFFFFFFF1、各力間關系Relationship between forces acting on the driving and driven gearsResolve into tb11n11nt1n2n1t1a2a1nt1tt1r2r12211t2t1coscosd2Tcoscosd2TcoscosFFFtgFFFcostgFtgFFFd2Td2TFF2、力的大小Magnitude of forces反從動輪：

20、與拇指指向相同主動輪：與拇指指向相方向：方向回轉方向：四指握拳右旋伸右手旋向：左旋伸左手，同前和、力的方向aatrF:FFF3Direction of forcesAs beforeHelix direction: left hand for left-hand helix gear and right hand for right-hand helix gear. Rotating direction: the bending direction of four fingers.The same direction of thumbing The opposite direction of

21、thumbing 三、 Design load 計算載荷 Fnd=KFn KLoad factor 載荷系數（ Take into account the influence of multiform factors on the load 考慮各種因素對載荷實際影響）KA Service factor 使用系數Motor原動機Operating machine工作機性能不穩(wěn)附加外部動載荷Unsteady performanceLoad grow instantly載荷瞬時增大Additional external dynamicsKAKv Dynamic load factor 動載系數Er

22、rors in manufacture and assembly 制造安裝誤差Deflections of the teeth under load 輪齒變形Errors in engagement 嚙合誤差 Instantaneous angular acceleration 瞬時角加速度： Additional internal dynamic load附加內部動載荷Load grow instantly載荷瞬時增大KvDecrease in velocity, increase in accuracy of gear manufacture速度v降低,精度增高Modification o

23、f the tooth face profile 齒頂修形Decrease in dynamic load 動載荷減小KV DropsKFactor of load distribution among the teeth齒間載荷分配系數Errors in manufacture由于誤差K載荷在各嚙合齒對間分配不均No uniform load distribution among the teeth某對齒所受載荷高于平均值Load acting on some gear is more than the average loadK Factor of the nonuniform loadi

24、ng along the face width 齒向載荷分配系數Bending and torsion deformation of the shaft軸彎曲、扭轉變形Elastic displacement of bearings軸承彈性位移Errors in manufacture and assembly制造安裝誤差齒輪副相傾斜、輪齒扭曲The gears are in misalignment with respect to each otherThe load is distributed nonuniformly along the face width載荷沿接觸線分布不均Load

25、 acting on some part of the teeth along the face width increase 載荷局部增大K Making the teeth with a tapered crown 鼓形齒Proper face width齒寬合理Load concentration can be reduced by: Locating the gears favorably between the bearings 齒輪布置合理Increasing the rigidity of the shaft and bearings 增加軸和軸承座剛度Raising the a

26、ccuracy of manufacture and assembly 提高制造、安裝精度Load concentration is reduced偏載降低 K Decreased由上述五點Influence of bending deformation of the shaft on KInfluence of torsion deformation of the shaft on KInfluence of total deformation of the shaft on K6-4 Strength calculations for spur gears 直齒圓柱齒輪傳動強度計算 Con

27、tact fatigue strength calculation 齒面接觸疲勞強度計算 Beam fatigue strength calculation 齒根彎曲疲勞強度一、 Contact fatigue strength calculation齒面接觸疲勞強度計算1、 Formulas 公式推導The Hertz formula and strength condition ： 赫茲公式與強度條件 21222121H111.-1-11LFHEE Characteristic point 計算特征點： Pitch point 節(jié)點（ Due to pitting begins at th

28、e line passing through this point where load is transmitted by a single pair of teeth. Another reason is to simplify calculation. 此處為單對齒嚙合，最易發(fā)生點蝕；此處計算過程簡化.）The contact stress depend on load and the radii of curvature of the contact point of the teeth. And the contact stress varies along with the cha

29、nge of the contact point. Equivalent radius of curvature綜合曲率半徑Radius of curvature of gearEquivalent radius of curvature Influences of radius of curvatureChanges of contact stress Length of the line of contact 接觸線長度LFace width of tooth 齒寬b. ZbL2端面重合度重合度系數.34 Zaa Load.cosd2KTKFF11nContact ratio factor

30、The face contact ratio HEHZZuubdKTZtgEE節(jié)點區(qū)域系數彈性系數12cos2-1-112112222121H Substituting 、 into ：Factor taking into account the mechanical properties of the gear materialsFactor taking into account the shape of the contacting surfaces （校核公式）HHEuubdKTZZZ12211H For checking HdHEdduudKTZZZdb12db311H11代入則齒寬

31、系數設 設計公式）(12d3211HHEdZZZuuKTFace width factorWe obtain:For designIt is substituted into following equation 1、 Formula1） Basic hypotheses 基本假設 Gear tooth is treated as a cantilever and the whole force is applied to the top of the tooth 輪齒視為懸臂梁，全部載荷作用于齒頂 二、 Beam fatigue strength of tooth root 齒根彎曲疲勞強度

32、The strength is determined for the side of the teeth subject to tensile stress 以受拉邊為計算依據 The dangerous cross section is taken by 30tangent to the axis of the tooth 危險截面AB（30切線法）Only considering peripheralcomponent of the force ( compression stress caused by vertical component is less than 10% of ben

33、ding stress)僅計水平力FncosF（僅為b百分之幾）bmdT2YYcosmscosm6bmdT2coscos6bSF6bSWcoscosFcosFmWm11FaFFa2F11F2tF2FtFnbmaxF000齒形系數代入：llll2） Introduction 推導過程 On the dangerous cross section： 危險截面處Tooth form factor計入Load factor 載荷系數KStress concentration factor應力修正系數YSa(應力集中）Contact ratio factor重合度系數Y（Load is transmit

34、ted by more than one pair of teeth 載荷不只一對齒承擔）設計公式）代入：及將（校核公式）( YYYZ2KTm YYYZm2KTmd YYYmbd2KTYYK3FSaFa211FSaFa2131F111FSaFa11FSaF0bbbZdbFor designFor checkingSubstituting b and d1 into above formula: Module increases 模數m增大 F decrease： Contact ratio factor 重合度系數a75. 025. 0Y2、 NotesBending strength mai

35、nly depend on m.彎曲強度主要取決于m（For power drives:動力傳動m1.52mm）FSaFa YY Tooth form factor YFa is independent of m ,only depend on tooth form齒形系數YFa：P59 圖3-21 與模數m無關，僅取決于齒形（Z，x ） Z YFa F1 F2 Usually F1 F2* Bending strength of the pinion and gear should be calculated respectively or carried out for the tooth

36、ed gear with the higher ratio of大小齒輪彎曲強度應該分別計算，或取 中大者計算。FSaFa YY F Allowable bending stress 許用彎曲應力 Flim Bending endurance limit of gears tested with 1 of fatigue possibilities失效概率為1時，試驗齒輪彎曲疲勞強度 P67 圖328. SFSafety margin 安全系數（一般SF1） KFNService life factor 壽命系數 KFNf（N） P65 圖326FFNFlimFSK Z1（ Number of

37、 the teeth of the pinion 小齒輪齒數） Low hardness close gearing: ( Overall size of the drive mainly depends on contact strength, while bending strength of the drive is sufficient.) 閉式軟齒面齒輪： （傳動尺寸主要取決于接觸強度，彎曲強度較富裕）When d1=const and bending strength is ensured, it is suggested that a larger number of the t

38、eeth of the pinion should be taken.在d1=const時，并滿足彎曲強度前提下，盡量取多的齒數（Z1，m相應， d1=mZ1）齒根彎曲應力變化規(guī)律單側受載雙側受載6-5 Strength for helical cylinder gearing 斜齒圓柱齒輪傳動強度計算 Contacting fatigue strength of the tooth surface 齒面接觸疲勞強度計算 Bending strength 彎曲強度計算 一、Contacting fatigue strength of the tooth surface 齒面接觸疲勞強度計算1、

39、 Formula 按節(jié)點處法面當量直齒圓柱齒輪計算。.1sind2cos112cossindcos2cossindcos?1.-1-11LFt1bn1n2n1n1n2bt2bt2n2bt1bt1n1222121HuuEEH .1sind2cos112cossindcos2cossindcos?1.-1-11LFt1bn1n2n1n1n2bt2bt2n2bt1bt1n1222121HuuEEHuZZdddd121212n1n2The length of the line of contact接觸線長度L（ Varies）b2min1Z11.)(134Z.cos ZbLL則時，取軸向重合度取Axi

40、al contacting ratio.coscos1d2KTKFFbt11n F Substituting 、 、代入into， and favorable influence of the line of contact inclined to the root of the teeth on the contacting strength of the gears is considered by helix anger factor此外，計入螺旋角系數以考慮接觸線傾斜對接觸強度的有利影響cos Z HEZZ節(jié)點區(qū)域系數彈性系數H211tt2b222121HZu1ubd2KTZtgcos

41、2cos E-1E-11Factor taking into account the mechanical properties of the gear materialsFactor taking into account the shape of the contacting surfaces代入1dbd 設計公式）(ZZZZu1u2KTd32HHEd11 校核公式）(u1ubd2KTZZZZH211HEHFor checkingFor design2、Notes Helical gears are stronger than spur gears for the following re

42、ason斜齒輪相對于直齒輪承載能力:n L Advantages is not obvious 優(yōu)點不明顯 Fa Service life of the bearing 軸承壽命Usually 825（Herringbone gears： 2035）二、 Bending strength 彎曲強度計算1、 Formula ：Helical teeth can be assumed to conformed with the tooth of a equivalent spur gear in a cross section normal to the tooth. The strength c

43、alculation is done against the equivalent spur gear按法面當量直齒圓柱齒輪計算 Basic parameters of the equivalent spur gear 當量齒輪的基本參數：Module mn、Equivalent number of teeth 當量齒數cosZZcosZZ32V231V1Favorable influence of the line of contact inclined to the root of the teeth on the bending strength of the gears is take

44、n into account by helix anger factor Y 引入螺旋角系數Y考慮接觸線傾斜對彎曲強度的有利影響校核公式斜齒輪：直齒輪：FSaFan11FFSaFa11FYYYYmbd2KTYYYmbd2KTSpur gearHelical gearFor checking By analogy with spur gearing： 采用比較法 設計公式）(YYYYZcos2KTm3FSaFa21b21nFor design:cosZmddb1n11代入dinto above formula:SubstitutingYFa、 YSa Taken by equivalent nu

45、mber of teeth 據當量齒數查取75. 0Y75. 0Y111201Yoo取取螺旋角系數cosZZ3V2、 NotesHelix anger factor Helical gears are stronger than spur gears in bending strength 斜齒輪和直齒輪相比，彎曲強度增大. （Y1，ZvYFa F ）6-6 Spur bevel gearing drive 直齒圓錐齒輪傳動 （Usually =901+2） Special features of the bevel gearing as compared with cylindrical g

46、earing 和圓柱齒輪相比，錐齒輪的特點 Geometrical relationship and basic parameters of the equivalent gear for bevel gearing 幾何關系和當量齒輪基本參數 Forces in bevel gearing 受力分析 Contact strength 接觸強度 Bending strength 彎曲強度 Notes 說明 Problems 思考題一、 Special features of the bevel gearing as compared with cylindrical gearing 和圓柱齒輪

47、相比，錐齒輪的特點： Lower machining accuracy ,Higher level of noise Suitable for low peripheral velocities, Outside diameter of bevel gears should not be large. 制造精度低,振動噪音大，圓周速度不宜過高,大錐齒輪直徑不宜過大。 The tooth dimensions of bevel gears are specified at the large end on the back cone because it is more convenient to make measurements here 為了檢驗和確定輪廓尺寸方便，規(guī)定大端參數為標準值； For strength calculations, the bevelgears are replaced by spur gears of a pitch diameter and module equal to those at the middle cross section of the bevel gears 強度計算轉化為錐齒輪齒寬中點處當量直齒圓柱齒輪進行。21 b R d m 1 d 1