Elementary Information on Gears

Transcription

1 Elementry Informtion on Gers The Role Gers re Plying Gers re some of the most importnt elements use in mchinery. There re few mechnicl evices tht o not hve the nee to trnsmit power n motion between rotting shfts. Gers not only o this most stisfctorily, but cn o so with uniform motion n relibility. In ition, they spn the entire rnge of pplictions from lrge to smll. To summrie:. Gers offer positive trnsmission of power.. Gers rnge in sie from smll miniture instrument instlltions, tht mesure in only severl millimeters in imeter, to huge powerful gers in turbine rives tht re severl meters in meter.. Gers cn provie position trnsmission with very high ngulr or liner ccurcy, such s use in servomechnisms n precision instruments.. Gers cn couple power n motion between shfts whose xes re prllel, intersecting or skew.. Ger esigns re stnrie in ccornce with sie n shpe which provies for wiespre interchngebility. This introuction is written s n i for the esigner who is beginner or only superficilly knowlegeble bout gering. It provies funmentl, theoreticl n prcticl informtion. When you select KHK proucts for your pplictions plese utilie it long with KHK009 ctlog.

2 Tble of Contents Ger Types n Terminology.... Type of Gers.... s n Terminology... Ger Trins Single - Stge Ger Trin... Double - Stge Ger Trin Involute Gering Moule Sies n Stnrs The Involute Curve... 0 Meshing of Involute Gering... The Generting of Spur Ger... Unercutting... Profile Shifting... Clcultion of Ger Dimensions Spur Gers... Internl Gers... 8 Helicl Gers... Bevel Gers... 8 Screw Gers... Cylinricl Worm Ger Pir... 6

3 Ger Types n Terminology. Type of gers (b) Spur Rck In ccornce with the orienttion of xes, there re three ctegories of gers:. Prllel xes gers. Intersecting xes gers. Nonprllel n nonintersecting xes gers Spur n helicl gers re the prllel xes gers. Bevel gers re the intersecting xes gers. Screw or crosse helicl gers n worm gers hnle the thir ctegory. Tble. Lists the ger types per xes orienttion. Tble. Types of gers n their ctegories Ctegories of gers Prllel xes gers Intersecting xes gers Nonprllel n nonintersecting xes gers Types of gers Spur ger Spur rck Internl ger Helicl ger Helicl rck Double helicl ger Stright bevel ger Spirl bevel ger Zerol bevel ger Worm ger Screw ger Efficiency(%) 98.0 ~ ~ ~ ~ 9.0 This is liner shpe ger which cn mesh with spur ger with ny number of teeth. The spur rck is portion of spur ger with n infinite rius. (c) Internl Ger This is cylinricl shpe ger but with the teeth insie the circulr ring. It cn mesh with spur ger. Internl gers re often use in plnetry ger systems n lso in ger couplings. () Helicl Ger This is cylinricl shpe ger with helicoi teeth. Helicl gers cn ber more lo thn spur gers, n work more quietly. They re wiely use in inustry. A negtive is the xil thrust force the helix form cuses. Fig.. Spur rck Fig.. Internl ger n spur ger Fig.. Helicl ger Also, inclue in tble. Is the theoreticl efficiency rnge of the vrious ger types. These figures o not inclue bering n lubricnt losses. Also, they ssume iel mounting in regr to xis orienttion n center istnce. Inclusion of these relistic consiertions will owngre the efficiency numbers. () Prllel Axes Gers (e) Helicl Rck This is liner shpe ger which meshes with helicl ger. Agin, it cn be regre s portion of helicl ger with infinite rius. Fig.. Helicl rck () Spur Ger (f) Double Helicl Ger This is cylinricl shpe ger in which the teeth re prllel to the xis. It hs the lrgest pplictions n, lso, it is the esiest to mnufcture. Fig.. Spur ger This is ger with both lefthn n right-hn helicl teeth. The ouble helicl form blnces the inherent thrust forces. Fig..6 Double helicl ger

4 () Intersecting Axes Gers (b) Screw Ger (Crosse Helicl Ger) () Stright Bevel Ger This is ger in which the teeth hve tpere conicl elements tht hve the sme irection s the pitch cone bse line (genertrix). The stright bevel ger is both the simplest to prouce n the most wiely pplie in the bevel ger fmily. Fig..7 Stright bevel ger A pir of cylinricl gers use to rive non-prllel n nonintersecting shfts where the teeth of one or both members of the pir re of screw form. Screw gers re use in the combintion of screw ger / screw ger, or screw ger / spur ger. Screw gers ssure smooth, quiet opertion. However, they re not suitble for trnsmission of high horsepower. Fig.. Screw ger (b) Spirl Bevel Ger () Other Specil Gers This is bevel ger with helicl ngle of spirl teeth. It is much more complex to mnufcture, but offers higher strength n lower noise. (c) Zerol Bevel Ger Fig..8 Spirl bevel ger () Fce Ger This is pseuobevel ger tht is limite to 90 O intersecting xes. The fce ger is circulr isc with ring of teeth cut in its sie fce; hence the nme fce ger Zerol bevel ger is specil cse of spirl bevel ger. It is spirl bevel with spirl ngle of ero. It hs the chrcteristics of both the stright n spirl bevel gers. The forces cting upon the tooth re the sme s for stright bevel ger. () Nonprllell n Nonintersecting Axes Gers Fig..9 Zerol bevel ger (b) Enveloping Worm Ger Pir This worm ger pir uses specil worm shpe in tht it prtilly envelops the worm wheel s viewe in the irection of the worm wheel xis. Its big vntge over the stnr worm is much higher lo cpcity. However, the worm wheel is very complicte to esign n prouce. Fig.. Enveloping worm ger pir (c) Hypoi Ger Fig.. Fce ger () Worm Ger Pir Worm ger pir is the nme for meshe worm n worm wheel. The outstning feture is tht it offers very lrge ger rtio in single mesh. It lso provies quiet n smooth ction. However, trnsmission efficiency is very poor. Fig..0 Worm ger pir This is evition from bevel ger tht originte s specil evelopment for the utomobile inustry. This permitte the rive to the rer xle to be nonintersecting, n thus llowe the uto boy to be lowere. It looks very much like the spirl bevel ger. However, it is complicte to esign n is the most ifficult to prouce on bevel ger genertor. Fig.. Hypoi ger

5 . s n Terminology Tble. through.6 inicte the symbols n the terminology use in this ctlog. JIS B 0:999 n JIS B00:999 cncel n replce former JIS B0 (symbols) n JIS B00 (vocbulry) respectively. This revision hs been me to conform to Interntionl Stnr Orgnition (ISO) Stnr. Tble. Liner imensions n circulr imensions Terms Centre istnce Reference pitch Trnsverse pitch Norml pitch Axil pitch Bse pitch Trnsverse bse pitch Norml bse pitch Tooth epth Aenum Deenum Chorl height Constnt chor height Working epth Tooth thickness Norml tooth thickness Trnsverse tooth thickness Crest with Bse thickness Chorl tooth thickness Constnt chor Spn mesurement over k teeth Tooth spce Tip n root clernce Circumferentil bcklsh Norml bcklsh Ril bcklsh Angulr bcklsh Fcewith Effective fcewith Le Length of pth of contct Length of pproch pth Length of recess pth Overlp length Reference imeter Pitch imeter Tip imeter Bse imeter Root imeter Center reference imeter Inner tip imeter Reference rius Pitch rius Tip rius Bse rius Root rius Rius of curvture of tooth profile Cone istnce Bck cone istnce s p p t p n p x p b p bt p bn h h h f h h c h' s s n s t s s b s s c W e c jt j n j r j θ b b' p g g f g g β ' b f m i r r' r r b r f r R Rv Tble. Angulr imensions Terms Reference pressure ngle Working pressure ngle Cutter pressure ngle Trnsverse pressure ngle Norml pressure ngle Axil pressure ngle Trnsverse working pressure ngle Tip pressure ngle Norml working pressure ngle Reference cyliner helix ngle Pitch cyliner helix ngle Men spirl ngle Tip cyliner helix ngle Bse cyliner helix ngle Reference cyliner le ngle Pitch cyliner le ngle Tip cyliner le ngle Bse cyliner le ngle Shft ngle Reference cone ngle Pitch ngle Tip ngle Root ngle Aenum ngle Deenum ngle Trnsverse ngle of trnsmission Overlp ngle Totl ngle of trnsmission Tooth thickness hlf ngle Tip tooth thickness hlf ngle Spcewith hlf ngle Angulr pitch of crown ger Involute function Tble. Sie numbers, rtios & spee terms Terms Number of teeth Equivlent number of teeth Number of thres, or number of teeth in pinion Ger rtio Trnsmission rtio Moule Trnsverse moule Norml moule Axil moule Dimetrl pitch Trnsverse contct rtio Overlp rtio Totl contct rtio Angulr spee Tngentil spee Rottionl spee Profile shift coefficient Norml profile shift coefficient Trnsverse profile shift coefficient Center istnce moifiction coefficient s ' o t n x ' t ' n β β' β m β β b γ γ' γ γ b Σ δ δ' δ δ f θ θ f ζ ζ β ζ γ ψ ψ η τ inv s v u i m m t m n m x P ε ε β ε γ ω v n x x n x t y

6 Tble. Others Terms Tngentil force Axil force Ril force Pin imeter Iel pin imeter Mesurement over rollers (pin) Pressure ngle t pin center Coefficient of friction Circulr thickness fctor Tble.6 Accurcy/Error terms Terms Single pitch evition Pitch evition Totl cumultive pitch evition Totl profile evition Runout Totl helix evition s F t F x F r p ' p M φ μ Κ s f pt f v or f pu F p F F r F b A numericl subscript is use to istinguish "pinion" from "ger" (Exmple:, ), "worm" from "worm wheel", "rive ger" from "riven ger", n so forth. Tble.7 inictes the Greek lphbet, the interntiol phonetic lphbet. Tble.7 The Greek lphbet Upper cse letters Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω Lower cse letters β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ σ τ υ φ χ ψ ω Spelling Alph Bet Gmm Delt Epsilon Zet Et Thet lot Kpp Lmb Mu Nu Xi Omicron Pi Rho Sigm Tu Upsilon Phi Chi Psi Omeg 6

7 Ger Trins The objective of gers is to provie esire motion, either rottion or liner. This is ccomplishe through either simple ger pir or more involve n complex system of severl ger meshes. Also, relte to this is the esire spee, irection of rottion n the shft rrngement.. Single-Stge Ger Trin A meshe ger is the bsic form of single-stge ger trin. It consists of n numbers of teeth on the river n riven gers, n their respective rottions, n & n. The trnsmission rtio is then: Trnsmission rtio = n = (.) n Ger trins cn be clssifie into three types: Trnsmission rtio <, incresing : n < n Trnsmission rtio =, equl spees: n = n Trnsmission rtio >, reucing : n > n Figure. illustrtes four bsic forms. For the very common cses of spur n bevel ger meshes, Figures.(A) n (B), the irection of rottion of river n riven gers re reverse. In the cse of n internl ger mesh, Figure.(C), both gers hve the sme irection of rottion. In the cse of worm mesh, Figure.(D), the rottion irection of is etermine by its helix hn. Ger Ger Ger Ger (,n ) (,n ) (,n ) (,n ) (A) A Pir of spur gers (B) Bevel gers Ger Ger Right-hn worm ger Left-hn worm ger (,n ) (,n ) (,n ) (,n ) (C) Spur ger n internl ger Right-hn worm wheel (D) Worm ger pir Left-hn worm wheel (,n ) (,n ) Fig.. Single-stge ger trins 7

8 In ition to these four bsic forms, the combintion of rck n pinion cn be consiere specific type. The isplcement of rck, l, for rottion θ of the mting pinion is: l = θ pm (.) 60 where: pm is the reference pitch is the number of teeth of the pinion. In the ouble-stge ger trin, Figure., ger rottes in the sme irection s ger. If gers n hve the sme number of teeth, then the trin simplifies s in Figure.. In this rrngement, ger is known s n iler, which hs no effect on the trnsmission rtio. The trnsmission rtio is then: Trnsmission Rtio = = (.). Double-Stge Ger Trin A ouble-stge ger trin uses two single-stges in series. Figure. represents the bsic form of n externl ger oublestge ger trin. Let the first ger in the first stge be the river. Then the trnsmission rtio of the ouble-stge ger trin is: Trnsmission Rtio = n n = (.) n n Ger Ger Ger (,n ) (,n ) (,n ) In this rrngement, n = n Ger Ger Ger Ger (,n ) (,n ) (,n ) (,n ) Fig.. Single-stge ger trin with n iler Fig.. Double-stge ger trin 8

9 Involute Gering The invoute profile is the one most commonly use toy for ger-tooth forms tht re use to trnsmit power. The beuty of involute gering is its ese of mnufcture n its smooth meshing espite the mislignment of center istnce in some egree.. Moule Sies n Stnrs Moule m represents the sie of involute ger tooth. The unit of moule is mm. Moule is converte to pitch p, by the fctor π. p = πm (.) Tble. is extrcte from JIS B which efines the tooth profile n imensions of involute gers. It ivies the stnr moule into three series. Figure. shows the comprtive sie of vrious rck teeth. M M. M M. M M Tble. Stnr vlues of moule Series Series Series Series Series NOTE: The preferre choices re in the series orer beginning with. unit: mm Series M M6 M0 Fig.. Comprtive sie of vrious rck teeth Dimetrl Pitch P(D.P.), the unit to enote the sie of the gertooth, is use in the USA, the UK, etc. The trnsformtion from Dimetrl Pitch P(D.P.) to moule m is ccomplishe by the following eqution: m =. / P 9

10 b Elementry Informtion on Gers p b p p / h f. The Involute Curve Figure. shows n element of involute curve. The efinition of involute curve is the curve trce by point on stright line which rolls without slipping on the circle. The circle is clle the bse circle of the involutes. We cn see, from Figure., the length of bse circle rc c equls the length of stright line bc. h h bc r tn = = b θ oc = θ (rin) (.) r b Moule m Reference pressure ngle = 0 Aenum h = m Deenum h f.m Tooth epth h.m Working epth h' =.00m Tip n root clernce c 0. m Reference pitch p = πm Bse pitch p b = p cos Reference imeter = m Bse imeter b = cos Fig.. The tooth profile n imension of stnr rck The θ in Figure. cn be expresse s inv +, then Formul (.) will become: inv = tn - (.) Function of, or inv, is known s involute function. Involute function is very importnt in ger esign. Involute function vlues cn be obtine from pproprite tbles. With the center of the bse circle O t the origin of coorinte system, the involute curve cn be expresse by vlues of x n y s follows: Pitch, p, is lso use to represent tooth sie when specil esire spcing is wnte, such s to get n integrl fee in mechnism. In this cse, pitch is chosen tht is n integer or specil frctionl vlue. This is often the choice in esigning position control systems. Most involute ger teeth hve the stnr whole epth n stnr pressure ngle = 0. Figure. shows the tooth profile of full epth stnr rck tooth n mting ger. It hs n enum of h = m n eenum h f.m. If tooth epth is shorter thn full epth teeth it is clle stub tooth; n if eeper thn full epth teeth it is high epth tooth. The most wiely use stub tooth hs n enum h = 0.8m n eenum h f = m. Stub teeth hve more strength thn full epth ger, but contct rtio is reuce. On the other hn, high tooth cn increse contct rtio. In the stnr involute ger, pitch (p) times the number of teeth becomes the length of reference circle: π = πm (.) Reference imeter () is then: = m (.) x = r cos ( inv ) r b = cos cos ( inv ) y = r sin ( inv ) r b = sin ( inv ) cos where, r = r b / cos (.6) The rwings of involute tooth-form cn be esily crete with this eqution. O y r r b c θ inv b x Fig.. The involute curve 0

11 . Meshing of Involute Ger Figure. shows pir of stnr gers meshing together. The contct point of the two involutes, s Figure. shows, slies long the common tngent of the two bse circles s rottion occurs. The common tngent is clle the line of contct, or line of ction. A pir of gers cn only mesh correctly if the pitches n the pressure ngle re the sme. Tht the pressure ngles must be ienticl becomes obvious from the following eqution for bse pitch: p b = πmcos (.7) Thus, if the pressure ngles re ifferent, the bse pitches cnnot be ienticl. The contct length b shown Figure. is escribe s "Length of pth of contct. O O The contct rtio cn be expresse by the following eqution: Length of pth of contct b Trnsverse Contct rtio ε = Bse pitch p b (.8) It is goo prctice to mintin trnsverse contct rtio of. or greter. Uner no circumstces shoul the rtio rop below.. Moule m n the pressure ngle re the key items in the meshing of gers.. The Generting of Spur Ger Involute gers cn be reily generte by rck type cutters. The hob is in effect rck cutter. Ger genertion is lso ccomplishe with ger type cutters using shper or plner mchine. Figure. illustrtes how n involute ger tooth profile is generte. It shows how the pitch line of rck cutter rolling on pitch circle genertes spur ger. Ger shpers with pinion cutters cn lso be use to generte involute gers. Ger shpers cn not only generte externl gers but lso generte internl gers. Rck form tool b length of pss of Contct O O b b sin I b O O O Fig.. The generting of stnr spur ger ( = 0, = 0, x = 0 ) Fig.. The meshing of involute ger

12 . Unercutting Unercutting is the phenomenon tht some of tooth eenum is cut by the ege of generting tool. In cse gers with smll number of teeth is generte s is seen in Figure., unercut occurs when the cutting is me eeper thn interfering point I. The conition for no unercutting in stnr spur ger is given by the expression: m m sin (.9) n the minimum number of teeth is: = (.0) sin For pressure ngle 0 egrees, the minimum number of teeth free of unercutting is 7. However, the gers with 6 teeth or uner cn be usble if its strength or contct rtio pose ny ill effect..6 Profile Shifting As Figure. shows, ger with 0 egrees of pressure ngle n 0 teeth will hve huge unercut volume. To prevent unercut, positive correction must be introuce. A positive correction, s in Figure.6, cn prevent unercut. Unercutting will get worse if negtive correction is pplie. See Figure.7. The extr fee of ger cutter (xm) in Figures.6 n.7 is the mount of shift or correction. An x is the profile shift coefficient. The conition to prevent unercut in spur ger is: Tip pressure ngle Tip tooth thickness hlf ngle Crest with m m - xm sin (.) The number of teeth without unercut will be: (-x) = (.) sin The profile shift coefficient without unercut is: x = - sin (.) Profile shift is not merely use to prevent unercut. It cn be use to just center istnce between two gers. If positive correction is pplie, such s to prevent unercut in pinion, the tooth tip is shrpene. Tble. presents the clcultion of top ln thickness ( Crest with ). Tble. The clcultions of top ln thickness ( Crest with ) Formul Exmple m = = 0 = 6 x = + 0. = cos - b b = = 7. ψ = π x tn + + (inv - inv ) inv = inv = (rin) ψ =.98 s ψ. ( rin) s =.076 Rck form tool Rck form tool xm ψ xm S sin () (ψ) b O b O (S) Fig..6 Generting of positive shifte spur ger Fig..7 The generting of negtive shifte spur ger Fig..8 Top ln thickness ( = 0, = 0, x = +0. ) ( = 0, = 0, x = - 0. ) ( Crest with )

13 f Elementry Informtion on Gers Clcultion of Ger Dimensions The following shoul be tken into consiertion in ue orer t the erly stge of esigning: To clculte the require strength To clculte the imensions To clclte the tooth thickness To clculte the necessry mount of bcklsh To clculte the forces to be cting on the ger To consier wht kin of lubriction is necessry n pproprite. to etermine the specifictions, the mterils to be use, n the egree of ccurcy. in orer to provie the necessry t for the ger shping. in orer to provie the necessry t for cutting n grining. to provie the necessry informtion useful for selecting the proper shfts n berings. The explntion is given, herefter, s to items necessry for the esign of gers. The clcultion of the imentions comes first. The imentions re to be clculte in ccornce with the funmentl specifictions of ech type of gers. The processes of turning etc. re to be crrie out on the bsis of tht t. b f b. Spur Gers O O () Stnr Spur Ger Figure. shows the meshing of stnr spur gers. The meshing of stnr spur gers mens reference circles of two gers contct n roll with ech other. The clcultion formuls re in Tble.. Fig.. The meshing of stnr spur gers ( = 0, =, =, x = x = 0 ) Tble. The clcultion of stnr spur gers Moule Reference pressure ngle Number of teeth Center istnce Reference imeter Bse imeter Aenum Tooth epth Tip imeter Root imeter m b h h f ( + ) m m cos.00m.m + m -.m NOTE : The subscripts n of n enote pinion n ger. Formul NOTE Pinion Exmple Ger

14 All clculte vlues in Tble. re bse upon given moule (m) n number of teeth ( n ). If inste moule (m), center istnce () n spee rtio (i) re given, then the number of teeth, n, woul be clculte with the formuls s shown in Tble.. Tble. The clcultion of number of teeth Formul Exmple Moule Center istnce Trnsmission rtio m i Sum of of teeth + m Number of teeth + i + i ( + ) i Note tht the number of teeth probbly will not be integer vlues by clcultion with the formuls in Tble.. In tht cse, it will be necessry to resort to profile shifting or to employ helicl gers to obtin s ner trnsmission rtio s possible.

15 () Profile Shifte Spur Ger Figure. shows the meshing of pir of profile shifte gers. The key items in profile shifte gers re the operting (working) pitch imeters (') n the working (operting) pressure ngle ('). These vlues re obtinble from the moifie center istnce n the following formuls: ' = + ' = + ' = cos - b + b In the meshing of profile shifte gers, it is the operting pitch circle tht re in contct n roll on ech other tht portrys ger ction. Tble. presents the clcultion where the profile shiht coefficient hs been set t x n x t the beginning. This clcultion is bse on the ie tht the mount of the tip n root clernce shoul be 0. m. (.) O ' b f ' Fig.. The meshing of profile shifte gers ( = 0, =, =, x = + 0.6, x = ) f b ' ' O Tble. The clcultion of profile shifte spur ger () 6 7 Moule Reference pressure ngle Number of teeth Profile shift coefficient Involute function ' Working pressure ngle Center istnce moifiction coefficient m x inv ' ' y Formul tn x + x + inv + Fin from Involute Function Tble + cos cos' - Exmple Pinion () Ger () Center istnce Reference imeter Bse imeter b + m cos + y m Working pitch imeter Aenum Tooth epth Tip imeter Root imeter ' h h h f b cos' ( + y-x ) m ( + y- x ) m {. + y - ( x + x )}m + h -h A stnr spur ger is, ccoring to Tble., profile shifte ger with 0 coefficient of shift; tht is, x = x = 0.

16 Tble. is the inverse formul of items from to 8 of Tble.. Tble. The clcultion of profile shifte spur ger () Center istnce Center istnce moifiction coefficient Working pressure ngle Sum of profile shift coefficient Profile shift coefficient y ' x + x x m - + Formul cos cos - y + + ( + ) (inv'- inv) tn Exmple There re severl theories concerning how to istribute the sum of profile shift coefficient (x + x ) into pinion (x ) n ger (x ) seprtely. BSS (British) n DIN (Germn) stnrs re the most often use. In the exmple bove, the tooth pinion ws given sufficient correction to prevent unercut, n the resiul profile shift ws given to the mting ger. 6

17 () Rck n Spur Ger Tble. presents the metho for clculting the mesh of rck n spur ger. Figure.() shows the the meshing of stnr ger n rck. In this meshing, the reference sircle of the ger touches the pitch lin of the rck. Figure.() shows profile shifte spur ger, with positive correction xm, meshe with rck. The spur ger hs lrger pitch rius thn stnr, by the mount xm. Also, the pitch line of the rck hs shifte outwr by the mount xm. Tble. presents the clcultion of meshe profile shifte spur ger n rck. If the profile shift coefficient x is 0, then it is the cse of stnr ger meshe with the rck. Tble. The clcultion of imensions of profile shifte spur ger n rck Moule Reference pressure ngle Number of teeth Profile shift coefficient Height of pitch line Working pressure ngle Mounting istnce Reference imeter Bse imeter Working pitch imeter Aenum Tooth epth Tip imeter Root imeter m x H ' b ' h h f m m cos b cos' + H + xm m ( + x ). m + h - h Formul Spur ger Exmple Rck One rottion of the spur ger will isplce the rck l one circumferentil length of the ger's reference circle, per the formul: l = pm (.) The rck isplcement, l, is not chnge in ny wy by the profile shifting. Eqution (.) remins pplicble for ny mount of profile shift. b H b H xm Fig..() The meshing of stnr spur ger n rck ( = 0, =, x = 0 ) Fig..() The meshing of profile shifte spur ger n rck ( = 0, =, x = ) 7

18 . Internl Gers () Internl Ger Clcultions Figure. presents the mesh of n internl ger n externl ger. Of vitl importnce is the working pitch imeters (' ) n working pressure ngle ('). They cn be erive from center istnce (' ) n Equtions (.). ' ' O b ' = ' = ' = cos b - b (.) O f Tble.6 shows the clcultion steps. It will become stnr ger clcultion if x = x = 0. Fig.. The meshing of internl ger n externl ger ( = 0, = 6, =, x = x = + 0. ) Tble.6 The clcultion of profile shifte internl ger n externl ger () Moule Reference pressure ngle Number of teeth Profile shift coefficient Involute function ' Working pressure ngle Center istnce moifiction coefficient Center istnce Reference imeter Bse imeter m x inv' ' y b tn Formul + inv Fin from involute Function Tble - - m cos x -x - cos cos' + y m - Externl ger Exmple Internl ger Working pitch imeter ' b cos ' Aenum Tooth epth h h h ( + x ) m ( - x ) m. m Tip imeter + h - h Root imeter f f - h + h

19 If the center istnce () is given, x n x woul be obtine from the inverse clcultion from item to item 8 of Tble.6. These inverse formuls re in Tble.7. Tble.7 The clcultion of profile shifte internl ger n externl ger () Center istnce Center istnce moifiction coefficient Working pressure ngle Difference of profile shift coefficient Profile shift coefficient y ' x - x x - m - cos - Formul Pinion cutters re often use in cutting internl gers n externl gers. The ctul vlue of tooth epth n root imeter, fter cutting, will be slightly ifferent from the clcultion. Tht is becuse the cutter hs profile shift coefficient. In orer to get correct tooth profile, the profile shift coefficient of cutter shoul be tken into consiertion. () Interference In Internl Gers Three ifferent types of interference cn occur with internl gers: () Involute Interference, (b) Trochoi Interference, n (c) Trimming Interference. () Involute Interference This occurs between the eenum of the externl ger n the enum of the internl ger. It is prevlent when the number of teeth of the externl ger is smll. Involute interference cn be voie by the conitions cite below: - (.) tn' where tooth. tn is the pressure ngle t tip of the internl ger b = cos - (.) ' : working pressure ngle cos y - + ' = cos - ( - ) m cos (.6) Equition (.) is true only if the tip imeter of the internl ger is bigger thn the bse circle: b (.7) For stnr internl ger, where = 0, Eqution (.7) is vli only if the number of teeth is >. ( - ) (inv '- inv ) tn (b) Trochoi Interference This refers to n interference occurring t the enum of the externl ger n the eenum of the internl ger uring recess tooth ction. It tens to hppen when the ifference between the numbers of teeth of the two gers is smll. Eqution (.8) presents the conition for voiing trochoil interference. Here θ + inv '- inv θ = cos - + inv -inv θ = cos - Exmple r -r - r + r -r r θ (.8) where is the pressure ngle of the spur ger tooth tip: = cos - (.0) = cos - In the meshing of n externl ger n stnr internl ger = 0, trochoi interference is voie if the ifference of the number of teeth, -, is lrger thn 9. b b ' (.9) 9

20 (c) Trimming Interference This occurs in the ril irection in tht it prevents pulling the gers prt. Thus, the mesh must be ssemble by sliing the gers together with n xil motion. It tens to hppen when the numbers of teeth of the two gers re very close. Eqution (.) inictes how to prevent this type of interference. θ + inv -inv' θ = sin - θ = sin - ( θ + inv -inv' ) -(cos / cos ) -( / ) (cos / cos ) - ( / ) - (.) (.) This type of interference cn occur in the process of cutting n internl ger with pinion cutter. Shoul tht hppen, there is nger of breking the tooling. Tble.8() shows the limit for the pinion cutter to prevent trimming interference when cutting stnr internl ger, with pressure ngle 0 = 0, n no profile shift, i.e., x 0 = 0. Tble.8() The limit to prevent n internl ger From trimming interference =0 x 0 = x = There will be n involute interference between the internl ger n the pinion cutter if the number of teeth of the pinion cutter rnges from to ( 0 = to ). Tble.8() shows the limit for profile shifte pinion cutter to prevent trimming interference while cutting stnr internl ger. The correction (x 0 ) is the mgnitue of shift which ws ssume to be: x 0 = Tble.8() The limit to prevent n internl ger from trimming interference 0 x 0 0 x 0 0 x = 0, x = There will be n involute interference between the internl ger n the pinion cutter if the number of teeth of the pinion cutter rnges from to 9 ( 0 = to 9). θ Interference θ Interference Interference θ θ Involute interference Trochoi interference Fig.. Involute interference n trochoi interference Fig..6 Trimming interference 0

21 . Helicl Gers A helicl ger such s shown in Figure.7 is cylinricl ger in which the teeth flnk re helicoi. The helix ngle in reference cyliner is b, n the isplcement of one rottion is the le, p. The tooth profile of helicl ger is n involute curve from n xil view, or in the plne perpeniculr to the xis. The helicl ger hs two kins of tooth profiles one is bse on norml system, the other is bse on n trnsverse system. Pitch mesure perpeniculr to teeth is clle norml pitch, p n. An p n ivie by p is then norml moule, m n. hob if moule m n n pressure ngle n re constnt, no mtter wht the vlue of helix ngle b. It is not tht simple in the trnsverse system. The ger hob esign must be ltere in ccornce with the chnging of helix ngle b, even when the moule m t n the pressure ngle t re the sme. Obviously, the mnufcturing of helicl gers is esier with the norml system thn with the trnsverse system in the plne perpeniculr to the xis. In meshing helicl gers, they must hve the sme helix ngle but with opposite hns. p n m n = (.) π The tooth profile of helicl ger with pplie norml moule, m n, n norml pressure ngle n belongs to norml system. In the xil view, the pitch on the reference is clle the trnsverse pitch, p t. An p t ivie by p is the trnsverse moule, m t. p t m t = (.) π These trnsverse moule m t n trnsverse pressure ngle t re the bsic configurtion of trnsverse system helicl ger. In the norml system, helicl gers cn be cut by the sme ger p x π Length of reference circle Reference imeter p t p n β Helix ngle p = π / tn β Le Fig..7 Funmentl reltionship of helicl ger (Right-hn)

22 () Norml System Helicl Ger In the norml system, the clcultion of profile shifte helicl ger, the working pitch imeter ' n trnsverse working pressure ngle ' t is one per Equtions (.). Tht is becuse meshing of the helicl gers in the trnsverse plne is just like spur gers n the clcultion is similr. ' = ' = ' t = cos b + b (.) Tble.9 shows the clcultion of profile shifte helicl gers in the norml system. If norml profile shift coefficients x n, x n re ero, they become stnr gers. Tble.9 The clcultion of profile shifte helicl ger in the norml system () Norml moule Norml pressure ngle Reference cyliner helix ngle Number of teeth &helicl hn Trnsverse pressure ngle Norml profile shift coefficient Involute function ' t Trnsverse woking pressure ngle Center istnce moifiction coefficient m n n β t x n inv ' t ' t y tn - Formul tn n x n + x n + inv t + Fin from involute Function Tble + cos β tn n cos β cos t cos' t - Pinion (L) Exmple Ger 60(R) 0 0 Center istnce + cos β + y m n Reference imeter Bse imeter b m n cos β cos t Working pitch imeter ' b cos' t Aenum Tooth epth Tip imeter Root imeter h h h f ( + y-x n ) m n ( + y- x n ) m n {. + y- (x n +x n )} m n + h -h

23 If center istnce,, is given, the norml profile shift coefficients x n n x n cn be clculte from Tble.0. These re the inverse equtions from items to 0 of Tble.9. Tble.0 The clcultions of profile shifte helicl ger in the norml system () Center istnce Center istnce moifiction coefficient Trnsverse working pressure ngle Sum of profile shift coefficient Norml profile shift coefficient y ' t x n + x n x n - cos - Formul The trnsformtion from norml system to trnsverse system is ccomplishe by the following equtions: m n + cos β cos t + y cos β + ( + ) (inv' t - inv t ) tn n Exmple x t = x n cos β m t = m n cos β t = tn - tn n cos β (.6)

24 () Trnsverse System Helicl Ger Tble. shows the clcultion of profile shifte helicl gers in trnsverse system. They become stnr if xt = xt = 0. Tble. The clcultion of profile shifte helicl ger in the trnsverse system () Trnsverse moule Trnsverse pressure ngle Reference cyliner helix ngle Number of teeth & helicl hn Trnsverse profile shift coefficient Involute function ' t Trnsverse working pressure ngle Center istnce moifiction coefficient m t t β x t inv' t ' t y Formul tn t x t + x t + inv t + Fin from Involute Function Tble + cos t cos' t - Exmple Pinion Ger 0 0 (L) 60 (R) Center istnce Reference imeter Bse imeter b + + y m t m t cos t Working pitch imeter ' b cos' t Aenum Tooth epth Tip imeter Root imeter h h h f ( + y-x t ) m t ( + y -x t ) m t {. + y - (x t + x t )} m t + h - h Tble. presents the inverse clcultion of items to 9 of Tble.. Tble. The clcultion of profile shifte helicl ger in the trnsverse system () Center istnce Center istnce moifiction coefficient y m t - + Formul Exmple Trnsverse working pressure ngle ' t cos - y + cos t +.97 Sum of profile shift coefficient Trnsverse profile shift coefficient x t + x t x t ( + ) (inv' t - inv t ) tn t The trnsformtion from trnsverse to norml system is escribe by the following equtions: x n = x t cosβ m n = m t cosβ n = tn - (tn t cosβ ) (.7)

25 () Sunerln Double Helicl Ger A representtive ppliction of trnsverse system is ouble helicl ger, or herringbone ger, me with the Sunerln mchine. The trnsverse pressure ngle, t, n helix ngle, β, re specifie s 0 n., respectively. The only ifferences from the trnsverse system equtions of Tble. re those for enum n tooth epth. Tble. presents equtions for Sunerln ger. Tble. The clcultion of ouble helicl ger of SUNDERLAND tooth profile Trnsverse moule Trnsverse pressure ngle Reference cyliner helix ngle Number of teeth Trnsverse profile shift coefficient Involute function ' t Trnsverse working pressure ngle Center istnce moifiction coefficient m t t β x t inv' t ' t y Formul tn t x t + x t + inv t + Fin from Involute Function Tble + cos t cos' t - Exmple Pinion Ger Center istnce Reference imeter Bse imeter b + m t cos t + y m t Working pitch imeter ' b cos' t Aenum Tooth epth Tip imeter Root imeter h h h f ( y -x t ) m t ( y - x t ) m t { y - (x t + x t )}m t + h - h

26 () Helicl Rck Viewe in the trnsverse plne, the meshing of helicl rck n ger is the sme s spur ger n rck. Tble. presents the clcultion exmples for mte helicl rck with norml moule n norml pressure ngle. Similrily, Tble. presents exmples for helicl rck in the trnsverse system (i.e., perpeniculr to ger xis). Tble. The clcultion of helicl rck in the norml system 6 7 Norml moule Norml pressure ngle Reference cyliner helix ngle Number of teeth & helicl hn Norml profile shift coefficient Pitch line height Trnsverse pressure ngle m n n β x n H t tn n tn - cosβ Formul Exmple Ger Rck ' 9" 0 (R) - (L) Mounting istnce m n cosβ + H + x n m n Reference imeter Bse imeter Aenum Tooth epth Tip imeter Root imeter b h h f m n cosβ cos t m n ( + x n ).m n + h - h The formuls of stnr helicl rck re similr to those of Tble. with only the norml profile shift coefficient x n = 0. To mesh helicl ger to helicl rck, they must hve the sme helix ngle but with opposite hns. The isplcement of the helicl rck, l, for one rottion of the mting ger is the prouct of the trnsverse pitch n number of teeth. πm n l = (.8) cosβ Accoring to the equtions of Tble., let trnsverse pitch p t = 8 mm n isplcement l = 60 mm. The trnsverse pitch n the isplcement coul be moifie into integers, if the helix ngle were chosen properly. 6

27 Tble. The clcultion of helicl rck in the trnsverse system 6 Trnsverse moule Trnsverse pressure ngle Reference cyliner helix ngle Number of teeth & helicl hn Trnsverse profile shift coefficient Pitch line height m t t β x t H Formul Exmple Ger Rck ' 9" 0 (R) - (L) Mounting istnce Reference imeter Bse imeter Aenum Tooth epth Tip imeter Root imeter b h h f m t m t cos t m t ( + x t ).m t + h -h + H + x t m t In the meshing of trnsverse system helicl rck n helicl ger, the movement, l, for one turn of the helicl ger is the trnsverse pitch multiplie by the number of teeth. l = πm t (.9) 7

28 . Bevel Gers Bevel gers, whose pitch surfces re cones, re use to rive intersecting xes. Bevel gers re clssifie ccoring to their type of the tooth forms into Stright Bevel Ger, Spirl Bevel Ger, Zerol Bevel Ger, Skew Bevel Ger etc. The meshing of bevel gers mens pitch cone of two gers contct n roll with ech other. Let n be pinion n ger tooth numbers; shft ngle S; n reference cone ngles δ n δ ; then: tn δ = tn δ = sin S + cos S sin S + cos S Generlly, shft ngle S = 90 is most use. Other ngles (Figure.8) re sometimes use. Then, it is clle bevel ger in nonright ngle rive. The 90 cse is clle bevel ger in right ngle rive. When S = 90, Eqution (.0) becomes: (.0) δ = tn - δ = tn - (.) m Miter gers re bevel gers with S = 90 n =. Their trnsmission rtio / =. δ δ S Figure.9 epicts the meshing of bevel gers. The meshing must be consiere in pirs. It is becuse the reference cone ngles δ n δ re restricte by the ger rtio /. In the fcil view, which is norml to the contct line of pitch cones, the meshing of bevel gers ppers to be similr to the meshing of spur gers. m Fig..8 The reference cone ngle of bevel ger R b Rv δ δ Rv Fig..9 The meshing of bevel gers 8

29 () Gleson Stright Bevel Gers A stright bevel ger is simple form of bevel ger hving stright teeth which, if extene inwr, woul come together t the intersection of the shft xes. Stright bevel gers cn be groupe into the Gleson type n the stnr type. In this section, we iscuss the Gleson stright bevel ger. The Gleson Compny efine the tooth profile s: tooth epth h =.88m; tip n root clernce c = 0.88m; n working epth h' =.000m. R b The chrcteristics re: Design specifie profile shifte gers: In the Gleson system, the pinion is positive shifte n the ger is negtive shifte. The reson is to istribute the proper strength between the two gers. Miter gers, thus, o not nee ny shifte tooth profile. i δ 90 - δ The tip n root clernce is esigne to be prllel: The fce cone of the blnk is turn prllel to the root cone of the mte in orer to eliminte possible fillet interference t the smll ens of the teeth. X X b h h f h Tble.6 shows the minimum number of teeth to prevent unercut in the Gleson system t the shft ngle S = 90. θ f θ δ f δ δ Tble.6 The minimum numbers of teeth to prevent unercut Pressure ngle 0 ( ) Combintion of number of teeth / (. ) 9/9 n higher 8/9 n higher 7/ n higher 6/ n higher /0 n higher /7 n higher 6/6 n higher /7 n higher /0 n higher /0 n higher / n higher Tble.7 presents equtions for esigning stright bevel gers in the Gleson system. The menings of the imensions n ngles re shown in Figure.0 bove. All the equtions in Tble.7 cn lso be pplie to bevel gers with ny shft ngle. The stright bevel ger with crowning in the Gleson system is clle Coniflex ger. It is mnufcture by specil Gleson Coniflex mchine. It cn successfully eliminte poor tooth contct ue to improper mounting n ssembly. Fig..0 Dimentions n ngles of bevel gers 9

30 Tle.7 The clcultions of stright bevel gers of the gleson system 6 Shft ngle Moule Reference pressure ngle Number of teeth Reference imeter Reference cone ngle S m δ m Formul sins tn - + coss Exmple Pinion () Ger () δ S - δ 7 8 Cone istnce Fcewith R b sinδ It shoul not excee R/ or 0m h.000m - h 9 0 Aenum Deenum Deenum ngle h h f θ f 0.0m +.88m - h tn - ( h f / R ) 0.60m cosδ cosδ Aenum ngle θ θ θ f θ f Tip ngle Root ngle Tip imeter Pitch pex to crown δ δ f X δ + θ δ - θ f + h cosδ R cosδ - h sinδ c 8.90c.77c 08.c 7 Axil fcewith X b b cosδ cosθ c c 8 Inner tip imeter i - b sinδ cosθ.80c c The first chrcteristics of Gleson Stright Bevel Ger is its profile shifte tooth. From Figure., we cn see the tooth profile of Gleson Stright Bevel Ger n the sme of Stnr Stright Bevel Ger. Fig.. The tooth profile of stright bevel gers Gleson stright bevel ger Stnr stright bevel ger 0

31 () Stnr Stright Bevel Gers A bevel ger with no profile shifte tooth is stnr stright bevel ger. The pplicble equtions re in Tble.8. Tble.8 Clcultion of stnr stright bevel gers Shft ngle Moule Reference pressure ngle Number of teeth Reference imeter S m m Formul Exmple Pinion () Ger () Reference cone ngle δ tn - sins + coss δ S - δ Cone istnce Fcewith Aenum Deenum Deenum ngle Aenum ngle Tip ngle Root ngle Tip cimeter Pitch pex to crown R b h h f θ f θ δ δ f X sinδ It shoul not excee R/ or 0m.00m.m tn - ( h f / R ) tn - ( h / R ) δ + θ δ - θ f + h cosδ R cosδ - h sinδ c 8.680c.680c c 7 Axil fcewith X b b cosδ cosθ 9.70c c 8 Inner tip imeter i - b sinδ cosθ.990c 08.80c These equtions cn lso be pplie to bevel ger sets with other thn 90 shft ngle.

32 () Gleson Spirl Bevel Gers A spirl bevel ger is one with spirl tooth flnk s in Figure.. The spirl is generlly consistent with the curve of cutter with the imeter c. The spirl ngle b is the ngle between genertrix element of the pitch cone n the tooth flnk. The spirl ngle just t the tooth flnk center is clle men spirl ngle b m. In prctice, spirl ngle mens men spirl ngle. c β m All equtions in Tble. re eicte for the mnufcturing metho of Spre Ble or of Single Sie from Gleson. If ger is not cut per the Gleson system, the equtions will be ifferent from these. The tooth profile of Gleson spirl bevel ger shown here hs the tooth epth h =.888m; tip n root clernce c = 0.88m; n working epth h' =.700m. These Gleson spirl bevel gers belong to stub ger system. This is pplicble to gers with moules m >.. R b b/ b/ Tble.9 shows the minimum number of teeth to voi unercut in the Gleson system with shft ngle S = 90 n pressure ngle n = 0. δ R v Fig.. spirl bevel ger (Left-hn) Tble.9 Pressure ngle 0 The minimum numbers of teeth to prevent unercut 7/7 n higher Combintion of numbers of teeth / β = 6/8 n higher /9 n higher /0 n higher / n higher /6 n higher If the number of teeth is less thn, Tble.0 is use to etermine the ger sies. Tble.0 Dimentions for pinions with number of teeth less thn Number of teeth in pinion Number of teeth in ger Working epth Tooth epth Ger enum Pinion enum Tooth thickness of ger s h' h h h n higher n higher n higher n higher n higher n higher Norml pressure ngle Spirl ngle Shft ngle n β S 0 ~ 0 90 NOTE: All vlues in the tble re bse on m =.

33 Tble. shows the clcultions of spirl bevel gers of the Gleson system Tble. The clcultions of spirl bevel gers of the Gleson system Shft ngle Moule Norml pressure ngle Men spirl ngle Number of teeth n spirl hn Trnsverse pressure ngle Reference imeter Reference cone ngle S m n β m t δ tn ( - m tn - Formul tn n cosβ ) sins + coss Pinion 0 (L) Exmple Ger 00 (R) δ S - δ 9 0 Cone istnce Fcewith R b sinδ It shoul be less thn 0.R or 0m h.700m - h Aenum Deenum Deenum ngle h h f θ f 0.90m 0.60m + cosδ cosδ.888m - h tn - ( h f / R ) c c Aenum ngle θ θ θ f θ f Tip ngle Root ngle Tip imeter Pitch pex to crown δ δ f X δ + θ δ - θ f + h cosδ Rcosδ - h sinδ c 8.670c.990c 08.00c 9 Axil fcewith X b b cosδ cosθ 7.60c c 0 Inner tip imeter i - b sinδ cosθ 6.00c 8. All equtions in Tble. re lso pplicble to Gleson bevel gers with ny shft ngle. A spirl bevel ger set requires mtching of hns; left-hn n right-hn s pir. Figure. is left-hn Zerol bevel ger. () Gleson Zerol Bevel Gers When the spirl ngle bm = 0, the bevel ger is clle Zerol bevel ger. The clcultion equtions of Tble.7 for Gleson stright bevel gers re pplicble. They lso shoul tke cre gin of the rule of hns; left n right of pir must be mtche. Fig.. Left-hn erol bevel ger

34 . Screw Gers Screw gering inclues vrious types of gers use to rive nonprllel n nonintersecting shfts where the teeth of one or both members of the pir re of screw form. Figure. shows the meshing of screw gers. Two screw gers cn only mesh together uner the conitions tht norml moules (m n ) n (m n ) n norml pressure ngles ( n, n ) re the sme. Let pir of screw gers hve the shft ngle S n helix ngles b n b : If they hve the sme hns, then: S = β + β If they hve the opposite hns, then: S = β - β or S = β - β (.) If the screw gers were profile shifte, the meshing woul become little more complex. Let β', β' represent the working pitch cyliner; S Ger (Right-hn) (Left-hn) β β β Ger (Right-hn) Fig.. Screw gers of nonprllel n nonintersecting xes β S If they hve the sme hns, then: S = β' + β' If they hve the opposite hns, then: S = β' - β' or S = β' - β' (.) Tble. presents equtions for profile shifte screw ger pir. When the norml profile shift coefficients xn = x n = 0, the equtions n clcultions re the sme s for stnr gers.

35 Tble. The equtions for screw ger pir on nonprllel n Nonintersecting xes in the norml system Norml moule Norml pressure ngle Reference cyliner helix ngle Number of teeth & helicl hn Number of teeth of n Equivlent spur ger m n n β v cos β Formul Exmple Pinion Ger (R) (R) Trnsverse pressure ngle Norml profile shift coefficient t x n tn - tn n cosβ Involute function ' n Norml working pressure ngle inv' n ' n tn n x n + x n + inv n v + v Fin from involute function tble Trnsverse working pressure ngle ' t tn - tn' n cosβ Center istnce moifiction coefficient y ( v + v ) cos n cos' - n 0.977c Center istnce + + y m n cosβ cosβ 67.9c Reference imeter Bse imeter b m n cosβ cos t c 8.8c.6c 76.6c Working pitch imeter ' ' c 8.69c 6 Working helix ngle β' tn - ' tnβ Shft ngle Σ β' + β' or β' - β' Aenum Tooth epth Tip imeter Root imeter h h h f ( + y - x n ) m n ( + y - x n ) m n {. + y - ( x n + x n )}m n + h - h 0.079c 6.066c.7880c c c 76.88c Stnr screw gers hve reltions s follows: ' = ' = β' = β β' = β (.)

36 .6 Cylinricl Worm Ger Pir Cylinricl worms my be consiere cylinricl type gers with screw thres. Generlly, the mesh hs 90 O shft ngle. The number of thres in the worm is equivlent to the number of teeth in ger of screw type ger mesh. Thus, onethre worm is equivlent to one-tooth ger; n two-thres equivlent to two-teeth, etc. Referring to Figure., for reference cyliner le ngle g, mesure on the pitch cyliner, ech rottion of the worm mkes the thre vnce one le p. There re four worm tooth profiles in JIS B 7, s efine on the right. Tble. Axil moule of cylinricl worm ger pir γ pt Type I Worm: The tooth profile is trpeoil on the xil plne. Type II Worm: The tooth profile is trpeoi on the plne norml to the spce. Type III Worm: The tooth profile which is obtine by inclining the xis of the milling or grining, of which cutter shpe is trpeoil on the cutter xis, by the le ngle to the worm xis. Type IV Worm: The tooth profile is of involute curve on the plne of rottion. Type III worm is the most populr. In this type, the norml pressure ngle n hs the tenency to become smller thn tht of the cutter, 0. Per JIS, Type III worm uses xil moule mx n cutter pressure ngle 0 = 0 s the moule n pressure ngle. A specil worm hob is require to cut Type III worm wheel. Stnr vlues of xil moule, mx, re presente in Tble.. Becuse the worm mesh couples nonprllel n nonintersecting xes, the xil plne of worm oes not correspon with the xil plne of worm wheel. The xil plne of worm correspons with the trsnsverse plne of worm wheel. The trnsverse plne of worm correspons with the xil plne of worm wheel. The common plne of the worm n worm wheel is the norml plne. Using the norml moule, mn, is most populr. Then, n orinry hob cn be use to cut the worm wheel. Tble. presents the reltionships mong worm n worm wheel xil plne, trnsverse plne, norml plne, moule, pressure ngle, pitch n le. t π p x x p n p n n Tble. The reltions of cross sections of worm ger pir Worm Axil plne Norml plne Trnsverse plne p = π tn γ Fig.. Cylinricl worm (Right-hn) β m n m x = m n m t = cosγ m n sinγ x = tn - tn n n t = tn cosγ - tn n sinγ p x = πm x p n = πm n p t = πm t p = πm x p = πm n cosγ p = πm t tnγ Trnsverse plne Norml plne Axil plne Worm wheel NOTE: The trnsverse plne is the plne perpeniculr to the xis. 6

37 Reference to Figure. cn help the unerstning of the reltionships in Tble.. They re similr to the reltions in Formuls (.6) n (.7) tht the helix ngle b be substitute by (90 g). We cn consier tht worm with le ngle g is lmost the sme s helicl ger with helix ngle (90 g). γ f r i () Axil Moule Worm Ger Pir Tble. presents the equtions, for imensions shown in Figure.6, for worm gers with xil moule, mx, n norml pressure ngle n = 0. f t Tble. The clcultions of xil moule system worm ger pir 6 Axil moule Norml pressure ngle of thres, no. of teeth Reference imeter Reference cyliner le ngle Profile shift coefficient m x n γ x t Formul (Qm x ) NOTE m x tn - m x Fig..6 Dimentions of cylinricl worm ger pir Exmple Worm Wheel (0 ) Double(R) * 0 (R) Center istnce + + x t m x Aenum Tooth epth h h h.00 m x (.00 + x t ) m x. m x Tip imeter Throt imeter t + h + h + m x NOTE + h Throt surfce rius r i - h Root imeter f f - h t - h * Double-three right-hn worm. NOTE : Dimeter fctor, Q, mens reference imeter of worm,, over xil moule, m x. Q = m x NOTE : There re severl clcultion methos of worm wheel tip imeter besies those in Tble.. NOTE : The fcewith of worm, b, woul be sufficient if: b = pm x ( ) NOTE : Effective fcewith of worm wheel b' = m x Q +. So the ctul fcewith of b b' +. m x woul be enough. 7

38 () Norml Moule System Worm Ger Pir The equtions for norml moule system worm gers re bse on norml moule, m n, n norml pressure ngle, n = 0. See Tble.6. Tble The clcultions of norml moule system worm ger pir Norml moule Norml pressure ngle of thres, of teeth Reference imeter of worm Reference cyliner le ngle Reference imeter of worm wheel Norml profile shift coefficient m n n γ x n sin - m n m n cosγ Formul Exmple Worm Worm Wheel ( 0 ) Double(R) * 0 (R) Center istnce + + x n m n Aenum Tooth epth h h h.00 m n (.00 + x n ) m n. m n Tip imeter Throt imeter t + h + h + m n + h Throt surfce rius r i - h Root imeter f f - h t - h * Double-three right-hn worm. NOTE: All notes re the sme s those of Tble.. () Crowning of the Tooth Crowning is criticlly importnt to worm gers. Not only cn it eliminte bnorml tooth contct ue to incorrect ssembly, but it lso provies for the forming of n oil film, which enhnces the lubriction effect of the mesh. This cn fvorbly impct enurnce n trnsmission efficiency of the worm mesh. There re four methos of crowning worm ger pir: by cutting it with hob whose reference imeter is slightly lrger thn tht of the worm. This is shown in Figure.7. This cretes teeth contct in the center region with spce for oil film formtion. Hob Worm () Cut Worm Wheel with Hob Cutter of Greter Reference Dimeter thn the Worm. A crownless worm wheel results when it is me by using hob tht hs n ienticl pitch imeter s tht of the worm. This crownless worm wheel is very ifficult to ssemble correctly. Proper tooth contct n complete oil film re usully not possible. However, it is reltively esy to obtin crowne worm wheel Fig..7 The metho of using greter imeter hob 8