Machine Design II Prof. K.Gopinath & Prof. M.M.Mayuram. Module 2 - GEARS

Transcription

1 Module 2 - GEARS Lecture 15 WORM GEARS Cotets 15.1 Worm gears a itroductio 15.2 Worm gears - geometry ad omeclature 15.3 Worm gears- tooth force aalysis 15.4 Worm gears-bedig stress aalysis 15.5 Worm gears-permissible bedig stress 15.6 Worm gears- cotact stress aalysis 15.7 Worm gears- permissible cotact stress 15.8 Worm gears -Thermal aalysis 15.1 INTRODUCTION Worm gears are used for trasmittig power betwee two o-parallel, o-itersectig shafts. High gear ratios of 200:1 ca be got. (a) (b) Fig.15.1 (a) Sigle evelopig worm gear, (b) Double evelopig worm gear.

2 Fig.15.2 The cut sectio of a worm gearbox with fis ad fa for coolig 15.2 GEOMETRY AND NOMENCLATURE Fig Nomeclature of a sigle evelopig worm gear

3 a. The geometry of a worm is similar to that of a power screw. Rotatio of the worm simulates a liearly advacig ivolute rack, Fig.15.3 b. The geometry of a worm gear is similar to that of a helical gear, except that the teeth are curved to evelop the worm. c. Evelopig the gear gives a greater area of cotact but requires extremely precise moutig. 1. As with a spur or helical gear, the pitch diameter of a worm gear is related to its circular pitch ad umber of teeth Z by the formula Zp 2 d2 (15.1) π 2. Whe the agle is 90 betwee the oitersectig shafts, the worm lead agle is equal to the gear helix agle. Agles ad have the same had. 3. The pitch diameter of a worm is ot a fuctio of its umber of threads, Z This meas that the velocity ratio of a worm gear set is determied by the ratio of gear teeth to worm threads; it is ot equal to the ratio of gear ad worm diameters. ω ω 1 Z = Z (15.2) 5. Worm gears usually have at least 24 teeth, ad the umber of gear teeth plus worm threads should be more tha 40: Z 1 + Z 2 > 40 (15.3) 6. A worm of ay pitch diameter ca be made with ay umber of threads ad ay axial pitch. 7. For maximum power trasmittig capacity, the pitch diameter of the worm should ormally be related to the shaft ceter distace by the followig equatio C C d (15.4)

4 8. Itegral worms cut directly o the shaft ca, of course, have a smaller diameter tha that of shell worms, which are made separately. 9. Shell worms are bored to slip over the shaft ad are drive by splies, key, or pi. 10. Stregth cosideratios seldom permit a shell worm to have a pitch diameter less tha d 1 = 2.4p (15.5) 11. The face width of the gear should ot exceed half the worm outside diameter. b 0.5 d a1 (15.6) 12. Lead agle λ, Lead L, ad worm pitch diameter d 1 have the followig relatioship i coectio with the screw threads. L ta λ = ( 15.7) πd To avoid iterferece, pressure agles are commoly related to the worm lead agle as idicated i Table Table 15.1 Maximum worm lead agle ad worm gear Lewis form factor for various pressure agles Pressure Agle Maximum Lead Lewis form factor Modified Lewis Φ Agle λ (degrees) y form factor Y (Degrees)

5 Table 15.2 Frequetly used stadard values of module ad axial pitch of worm or circular pitch of gear p i mm: Module m mm Axial pitch p mm Module m mm Axial pitch p mm b) Values of addedum ad tooth depth ofte coform geerally to helical gear practice but they may be strogly iflueced by maufacturig cosideratios. c) The load capacity ad durability of worm gears ca be sigificatly icreased by modifyig the desig to give predomiatly recess actio i.e. the agle of approach would be made small or zero ad the agle of recess larger. d) The axial pitch for differet stadard modules are give Table FORCE ANALYSIS Fig Worm gear force aalysis

6 a) The tagetial, axial, ad radial force compoets actig o a worm ad gear are illustrated i the Fig b) For the usual 90 shaft agle, the worm tagetial force is equal to the gear axial force ad vice versa. F 1t = F 2a (15.8) F 2t = F 1a (15.9) c) The worm ad gear radial or separatig forces are also equal, F 1r = F 2r (15.10) If the power ad speed of either the iput or output are kow, the tagetial force actig o this member ca be foud from equatio 1000 W F 1t = V (15.11) 1. I the Fig. 15.4, the drivig member is a clockwise-rotatig right had worm. 2. The force directios show ca readily be visualized by thikig of the worm as a right had screw beig tured so as to pull the ut (worm gear tooth) towards the screw head. 3. Force directios for other combiatios of worm had ad directio of rotatio ca be similarly visualized Thrust Force Aalysis. The thrust force directio for various worm ad worm wheel drive coditios are show i Fig. 15.6

7 (a) (b) Fig.15.6 (a) ad (b) Worm gears thrust force aalysis

8 The thread agle λ of a screw thread correspods to the pressure agle φ of the worm. We ca apply the force, efficiecy, ad self-lockig equatios of power screw directly to a worm ad gear set. These equatios are derived below with referece to the worm ad gear geometry. Figs.15.7 to 15.9 show i detail the forces actig o the gear. Compoets of the ormal tooth force are show solid. Compoets of the frictio force are show with the dashed lies. Fig Forces o the worm gear tooth Fig Worm drivig

9 Fig illustrates the same directios of rotatio but with the torque directio reversed (i.e., gear drivig). The cotact shifts to the other side of the gear tooth, ad the ormal load reverses. Fig.15.9 Gear drivig (Same directio of rotatio) The frictio force is always directed to oppose the slidig motio. The drivig worm is rotatig clockwise: F =F =F cosφ cosλ -f F siλ 2t 1a (15.12) F = F =F cosφ 1t 2a F = F =F siφ 2r 1r siλ +f F cosλ (15.13) (15.14) Combiig eqs. (15.12) with (15.13), we have: F 2t cosφ cosλ - f si λ = F cosφ siλ + f cosλ 1t (15.15) Combiig eqs. (15.12) with (15.14) ad (15.13) with (15.14), we have:

10 siφ F =F =F =F 2r 1r 2t 1t cosφ cosλ - f si λ siφ cosφ si λ + f cos λ (15.16) 15.4 KINEMATICS The relatioship betwee worm tagetial velocity, gear tagetial velocity, ad slidig velocity is, V V 2 1 = taλ (15.17) 15.5 EFFICIENCY Efficiecy η is the ratio of work out to work i. For the usual case of the worm servig as iput member, (15.18) The overall efficiecy of a worm gear is a little lower because of frictio losses i the bearigs ad shaft seals, ad because of churig of the lubricatig oil FRICTION ANALYSIS The coefficiet of frictio, f, varies widely depedig o variables such as the gear materials, lubricat, temperature, surface fiishes, accuracy of moutig, ad slidig velocity. The typical coefficiet of frictio of well lubricated worm gears is give i Fig

11 Fig Frictio of well lubricated worm gears, A for cast iro worm ad gear ad B for case hardeed steel worm ad phosphor broze worm gear The slidig velocity Vs is related to the worm ad gear pitch lie velocities ad to the worm lead agle by V1 V2 V= s = cosλ siλ (15.19) Fig Velocity compoets i worm gearig F1t F cos si - f F cos (15.20)

12 a) Eq shows that with a sufficietly high coefficiet of frictio, the gear tagetial force becomes zero, ad the gear set self-locks or does ot overhaul. b) With this coditio, o amout of worm torque ca produce motio. c) Self-lockig occurs, if at all, with the gear drivig. d) This is desirable i may cases ad helps i holdig the load from reversig, similar to a self-lockig power screw. The worm gear set self-locks if this force goes to zero, which happes if f cos ta (15.21) A worm gear set ca be always overhaulig or ever overhaulig, depedig o the selected value coefficiet of frictio (i.e., λ ad to a lesser extet o φ ) BENDING AND SURFACE FATIGUE STRENGTHS Worm gear capacity is ofte limited ot by fatigue stregth but by coolig capacity. The total gear tooth load F d is the product of omial load F t ad factors accoutig for impact from tooth iaccuracies ad deflectios, misaligmet, etc.). F d must be less tha the stregth the bedig fatigue ad surface fatigue stregths F b ad F w The total tooth load is called the dyamic load F d, the bedig fatigue limitig load is called stregth capacity F b, ad the surface fatigue limitig load is called the wear capacity F w. For satisfactory performace, F b F d (15. 21) ad F w F d (15.22) The dyamic load is estimated by multiplyig the omial value of gear tagetial force by velocity factor K v give i the followig Fig.15.

13 6.1+V2 F d =F2t K v =F2t (15.23) 6.1 Adaptig the Lewis equatio to the gear teeth, we have F b =[ b] bpy = [ b] bmy (15.24) Where, [σ b ] is the permissible bedig stress i bedig fatigue, i MPa, Table 15.3 Table 15.3 Permissible stress i bedig fatigue, i MPa 0.5 Material of the gear [σ b ] MPa Cetrifugally cast Cu-S broze 23.5 Alumium alloys Al-Si alloy 11.3 Z alloy 7.5 Cast iro 11.8 b is the face width i mm 0.5 d a1 p is the axial pitch i mm, Table 15.2 m is module i mm, Table 15.2 y is the Lewis form factor, Table 15.1 Y is modified Lewis form factor, Table 15.1 By assumig the presece of a adequate supply of appropriate lubricat, the followig equatio suggested by Buckigham may be used for wear stregth calculatios F w=d2bk w (15.25) F w Maximum allowable value of dyamic load uder surface fatigue coditio. d g - Pitch diameter of the gear. b - Face width of the gear.

14 K w - A material ad geometry factor with values empirically determied from the Table Table 15.4 Worm Gear Wear Factors K w Material K w (MPa) Worm Gear <10 <25 >25 Steel, 250 BHN Broze Hardeed steel (Surface 500 BHN) Broze Chill-cast Broze Cast iro Broze THERMAL CAPACITY The cotiuous rated capacity of a worm gear set is ofte limited by the ability of the housig to dissipate frictio heat without developig excessive gear ad lubricat temperatures. Normally, oil temperature must ot exceed about 200ºF (93 o C) for satisfactory operatio. The fudametal relatioship betwee temperature rise ad rate of heat dissipatio used for joural bearigs does hold good for worm gearbox. H= C A T -T H 0 a (15.26) Where H Time rate of heat dissipatio (Nm/sec) C H Heat trasfer coefficiet (Nm/sec/m 2 /ºC) A Housig exteral surface area (m 2 ) T o Oil temperature (º C) T a Ambiat air temperature (º C)

15 Surface area of A for covetioal housig desigs may be roughly estimated from the Eq 15.27, 1.7 A =14.75 C (15.27) Where A is i m 2 ad C (the distace betwee the shafts) is i m. Housig surface area ca be made far greater tha the above equatio value by icorporatig coolig fis. Rough estimates of C ca be take from the followig Fig Fig Ifluece of worm speed o heat trasfer 15.9 DESIGN GUIDELINES The desig guidelies for choosig the lead agle, pressure agle, addedum dededum, helix agle ad the miimum umber of teeth o the worm gear are give i Tables 15.5 to 15.8.

16 Table 15.5 Recommeded pressure agles ad tooth depths for worm gearig Lead agle λ i degrees Pressure agle φ i degrees Addedum h a i mm Dededum h f i mm p p p p p p p p p Helix agle Ψ i O Table 15.6 Efficiecy of worm GEAR set for f = 0.05 Efficiecy η i % Helix agle Ψ i O Efficiecy η i % Helix agle Ψ i O Efficiecy η i % Table 15.7 Miimum umber of teeth i the worm gear Pressure agle φ 14.5 o 17.5 o 20 o 22.5 o 25 o 27.5 o 30 o Z 2 miimum Table 15.8 Maximum lead agle for ormal pressure agle Normal Pressure agle φ 14.5 o 20 o 25 o 30 o Maximum lead agle λ max 16 o 25 o 35 o 45 o