Increasing the Strength of Standard Involute Gear Teeth with Novel Circular Root Fillet Design

Transcription

1 Ameican Jounal of Applied Science 2 (6): , 2005 ISSN Science Publication, 2005 Inceaing the Stength of Standad Involute Gea Teeth with Novel Cicula Root Fillet Deign V. Spita, Th. Cotopoulo and C. Spita Laboatoy of Machine Element, Mechanical Engineeing Depatment National Technical Univeity of Athen, Ioon Politechniou 9, 15780, Athen, Geece Abtact: In thi tudy the idea of pu gea teeth with cicula intead of the tandad tochoidal oot fillet i intoduced and invetigated numeically uing BEM. The tength of thee new teeth i tudied in compaion with the tandad deign by dicetizing the tooth bounday uing iopaametic Bounday Element. In ode to facilitate the analyi the teeth ae teated a non-dimenional auming unitay loading nomal to the pofile at thei Highet Point of Single Tooth Contact (HPSTC), o that nondimenional te v. Contact atio diagam ae plotted. The analyi demontate that the novel teeth ehibit highe bending tength (up to 70%) in cetain cae without affecting the pitting eitance ince the geomety of the load caying involute i not changed. The cicula fillet deign i paticulaly uitable in gea with a mall numbe of teeth (pinion) and thee novel gea can eplace thei eiting countepat in any mechanim without any alteation. Finally the geomety of the geneating tool (i.e. ack) i detemined in ode to be able to cut thee teeth uing a geneating method (i.e. hobbing). Key wod: Spu gea, cicula fillet, oot te, geneating ack, BEM INTRODUCTION In the field of gea tanmiion thee i a gowing need fo highe load caying capacitie and inceaed fatigue life. In ode to achieve thi, eeache focu eithe on the development of advanced mateial (Hoffman et al. [1] ) and new method of heat teatment (Townend and Bambege [2], Legge [3], Heing [4] ) o on the deign of tonge tooth pofile (Litvin et al. [5], Tai and Tai [6] ) and method of gea manufactuing (Daniewicz and Mooe [7] ). Howeve, in moden gea pactice and manufactuing the majoity of gea application ae coveed by the tandad 20 o involute teeth geneated by ack, hob o CNC cutting poce. Thi ha a numbe of impotant advantage uch a that of intechangeability, inenitivity to change in the nominal cente ditance, commecial availability and eay manufactuing by conventional method (i.e. hobbing) (Townend [8] ). On the othe hand it ha ome diadvantage, one of which i that fo a mall numbe of teeth (pactically le than 17 o 14 depending on the tip adiu of the hob) the tandad involute peent the poblem of undecutting. In undecutting the tooth fillet i geneated a the tip of the cutte emove mateial fom the involute pofile (geneated peviouly by the taight pat of the cutte), thu eulting in teeth that have a malle thickne nea thei oot, whee the citical ection i uually located weakening of the gea teeth many olution have been popoed (Fedette and Bown [10], Ciavaella and Demelio [11] ) but the mot commonly employed method i that of poitive pofile hifting (ISO 6336 [12], AGMA 2101 [13], Mabie et al. [14], Roge et al. [15] ) eulting in teeth with no undecutting, enhanced fatigue chaacteitic and change in the nominal cente ditance of the gea pai. On the othe hand thee modified teeth ehibit lowe pitting and coing eitance and lowe contat atio eulting in moe noie and vibation duing opeation (Niemann [9] ). In the peent tudy a novel deign of fillet fo pu gea teeth i peented. The popoed new teeth ae compoed of a tandad involute woking pofile fom the oute to the fom cicle of the gea and of a cicula fillet pofile fom the fom of the oot cicle of the gea eplacing the conventional tochoidal fillet pofile. Thee teeth ae fit modeled geometically and then thei behavio in bending i tudied by auming loading at thei Highet Point of Single Tooth Contact (HPSTC). The maimum fillet tee ae calculated fom vaiou numbe of teeth uing Bounday Element Analyi and the eult ae compaed with the maimum fillet te value of the coeponding tandad teeth. In ode to facilitate the modelling and the analyi of thee gea, thei geometical chaacteitic ae nomalized with epect to the module and the width of the gea and thei loading chaacteitic, which depend (Townend [8], Niemann [9] ). In ode to cope with thi Coeponding Autho: Th. Cotopoulo, Laboatoy of Machine Element, Mechanical Engineeing Depatment National Technical Univeity of Athen, Ioon Politechniou 9, 15780, Athen, Geece cot@cental.ntua.g 1058

2 on the magnitude of the eeted load and it poition at the Highet Point of Single Tooth Contact (HPSTC) on the flank, ae nomalized with epect to the magnitude of the load itelf and the contat atio of the pai epectively. Theefoe the tooth loading poblem can be olved fo thee nomalized non-dimenional teeth and then uing imple linea elationhip the tee fo the actual woking tooth can be deived. Anothe poblem elated to the capability of poducing thee novel teeth on conventional machine uing tandad geneating-type method fo gea tooth cutting (i.e. Hobbing o ack-cutting) i invetigated at the end of thi tudy. The geomety of the geneating ack with known geomety of the gea i calculated uing the Theoy of Geaing [16]. gea 1 and it poition, defining the adiu B, i (Spita [17] ): 2 B = O1B = k1 + ε 1 tg ε 1 tg 2 k1 g1 (4) Non-dimenional gea tooth modelling: Conide the pai of pu gea chematically illutated in Fig. 1 and denoted a 1 (diving) and 2 (diven). The law of geaing [8] equie that thee gea hould have the ame nominal peue angle a o and the ame module m in ode to be able to meh popely. It i alo poible that thee gea have addendum modification 1, 2 epectively and theefoe thei pitch thickne i given by the following elationhip: oi = ci π m + 2i tan αo m = oiu m (1) whee c i i the thickne coefficient of gea i,(i = 1,2), which in the geneal cae i c c 2, while oiu i the pitch thickne of the coeponding nondimenional gea fo which the module (m) and the face width (b) ae both equal to unity. Povided that no eo eit, the cente ditance O 1 O 2 i calculated: Fig. 1: Path of contact z1 + z2 a12 = m + ( ) m = a12u m (2) 2 The actual woking pitch cicle bi of gea i,(i = 1,2) hould veify the law of geaing and theefoe be equal to: z = a m = m i bi 12u biu z1 + z2 (3) Let u now conide that gea 1 and 2 evolving about thei cente O 1 and O 2 epectively ae mehing along the path of contact AB illutated in Fig. 1. Duing mehing thee ae two pai of gea teeth in contact along the egment AA and BB, thu haing the total nomal load, while thee i only a ingle uch pai when the tooth contact take place along the cental egion A B, caying the total nomal load. Point B i the Highet Point of Single Tooth Contact (HPSTC) fo Fig. 2: The geomety of the cicula fillet 1059

3 and by dividing with the module of the pai, the above equation yield in tem of non-dimenional value: = = + ( ε 1) t ( ε 1) t 2 m B 2 B u k1u gu gu k1u g1u (5) Fom the above equation it i evident that the poition of the HPSTC of a gea depend only on it geomety and on the contact atio of the pai, in which all the chaacteitic of the mating gea ae incopoated in a condened fom. The advantage offeed by thi appoach i pofound, ince the mechanical behavio of evey gea can be modeled only by uing it own geometical chaacteitic z,, c and the contat atio ε of the pai (4 vaiable) intead of uing all of the geometical chaacteitic of the mating gea (6 vaiable). Alo the ue of non-dimenional teeth futhe implifie the poblem a evey geometical featue f on the tanvee ection of a full cale gea tooth i connected to the coeponding featue f u of the tanvee ection of the non-dimenional gea tooth though the equation: f = f u.m (6) Stee can alo be calculated in non-dimenional teeth σ u (z,, C, ε) with unit loading P Nu = 1 and elated to the actual te σ uing the following equation: z o = and Sov = π c + 2 tan α o 2 S S = + 2 φ φ ov o o whee: φ = tan α α i the involute function on cicle 1 g α = co i the peue angle on cicle (8) Angle ω /2 that coepond to the ac S /2 (Fig. 2) i given by the equation: S 2 ω 2 = = Ω (9) Angle o that: ζ (Fig. 2) take value between 0 and ζ ma π ζ ma = Ω (10) z The coodinate of point A and B ae: = in ( ζ + Ω ), y co A f = ζ + Ω (11) A f PN σ = σu b m a uggeted by Roge [15] and Townend [8]. (7) B f 1 1 Geometical modeling: Conide the involute pu ( ε 1 ) : y =, ( ε 2 ) : y = (13) gea tooth of cicula fillet illutated in Fig. 2 whee tan Ω tan ( ζ + Ω ) point O i the cente of the gea, ai Oy i the ai of ymmety of the tooth and point B i the point whee the involute pofile tat (fom the fom cicle ). A i the point of tangency of the cicula fillet with the oot cicle f. Point D lying on (ε 2 ) OA epeent the cente of the cicula fillet. Line (ε 3 ) i tangent to the oot cicle at A and inteect with the line (ε 1 ) at C. The fillet i tangent to the line (ε 1 ) at point E. Since it i alway > f (Townend [8] ), the popoed cicula fillet can be implemented without eception on all pu gea ielevant of numbe of teeth o othe manufactuing paamete. A compaion of the geometical hape of a tooth of cicula fillet with that of tandad fillet i peented in Fig. 3. Fo the geometic modeling only dimenionle teeth ae eamined i.e. teeth with unit module (m) and face width (b). In dimenionle teeth the pitch Fig. 3: Supepoition of cicula fillet on a tandad adiu and the pitch thickne ae calculated: tooth 1060 = in Ω, yb = f co Ω (12) The defining equation of line (ε 1 ) and (ε 2 ) ae epectively:

4 Since (ε 3 ) (ε 2 ) and (ε 3 ) pae though point A( A, y A ) it defining equation i. Point C( C, y C ) i the inteection of (ε 1 ) and (ε 3 ) and theefoe it coodinate hould veify eq. (13) and hence: y C C tan Ω = f in tan co C = tan Ω ( ζ + Ω ) Ω + ( ζ + Ω ) (14) (15) y D 1 = co Ω + in Ω tan ζ + Ω (20) Accoding to Fig. 2 it i AC=BC and afte ubtitution and calculation we aive at the equation: f + f = Ω + Ω 2 co ζ ( in co ) fom which the value of ζ i deived a: (21) Angle ˆ BCA i calculated a: ζ = co 2 1 f + f (22) ˆ BCA π π = Ω + ζ + Ω = + ζ 2 2 Line (ε 4 ) biect the peviou angle inclination i: BCA ˆ tan tan π ζ ζ + Ω = Ω (16) ˆ BCA, o it (17) Point C( C, y C ) belong to (ε 4 ) and thu the defining equation of the line (ε 4 ) i deived a: π ζ y = tan Ω 4 2 π ζ 1 tan Ω tan Ω f in tan co ( ζ + Ω ) Ω + ( ζ + Ω ) (18) By defining the dimenion paamete eq. (22) become: 1 2S ζ = co 2 1+ S S = > 1 f (23) Equation (23) i ued fo the detemination of the angle (ζ). Point E lie below point B: In thi cae it i ζ ζ ma and the cente of the cicula fillet of the tooth i calculated following the methodology decibed below: A C A C CA = + y y = CE (24) At thi point two ditinct cae ae conideed. Point E coincide with point B(E B): In thi cae (ε 5 ) (ε 1 ) at point E B, o (ε 5 ) mut have an inclination equal to tanω and ince point B belong to (ε 5 ) it defining equation i: y = tan Ω + co Ω (19) Point D( D,y D ) hould veify both eq. (13) and (19) and theefoe ha the following coodinate: D ( ζ + Ω ) tan = co Ω + in Ω tan ζ + Ω π ζ AD = CA tan (25) The coodinate of point D( D,y D ) and E( E,y E ) ae epectively: = + AD in ζ + Ω, D f y = + AD co ζ + Ω (26) D f E = OC + CE in Ω, E y = OC + CE co Ω (27) The emaining potion of the tooth pofile between point B and E i a taight line. 1061

5 Fig. 4: BEM meh on a tooth model Tooth Contact (HPSTC) and again it i conideed to be unitay (i.e. P N = 1). The eultant nomalized te i elated to the actual te of the full-cale tooth with eq. (7). The ditance of the HPSTC fom the cente of the gea i calculated in tem of the contat atio ε and the geometical chaacteitic of the gea accoding to eq. (4) A typical meh i peented in Fig. 4. The tooth bae, ABCD, i dicetized in 43 node o 21 element and i conideed to be fied, i.e. The diplacement of all node ae zeo. Potion DE epeent the oot cicle and i dicetized in 11 node o 5 element. Potion EF i the tooth fillet coniting of 72 element o 144 node, potion FG i the involute pat of the tooth coniting of 40 element o 81 node and finally a potion GH i the tip, dicetized in 5 element o 11 node. All node belonging to the tooth pofile ae conideed to be unloaded (hoizontal and vetical taction equal to zeo) with the eception of the node maked HPSTC, which belong to the involute pat FG and on which the nomal load P N i acting. The eaon that the meh i dene at the tooth fillet i that the maimum te i epected to occu at thi aea and theefoe geate denity will enue inceaed accuacy of the eult (both the poition of the citical ection and the magnitude of the maimum developed te). Fig. 5: Non-dimenional tee fo a gea with 9 teeth Fig. 6: Non-dimenional tee fo a gea with 32 teeth Stuctual modeling: The tuctual analyi of the pu gea tooth model i caied out uing the Bounday Element method with quadatic iopaametic bounday element. The calculation of the non-dimenional tooth pofile and the geneation of the meh ae done automatically uing pecially developed oftwae. The nomal load i eeted at it Highet Point of Single the contact peue RESULTS AND DISCUSSION A compaative tudy ha been caied out between the tuctual popetie of the tandad tochoidal filleted teeth geneated by hobbing and the popoed cicula filleted involute teeth. Five ditinct cae have been eamined fo unhifted teeth with 9, 17, 24, 32 and 40 teeth epectively. Thee teeth wee conideed to be dimenionle and loaded at thei Highet Point of Single Tooth Contact (HPSTC). Since the HPSTC fo a given gea pai depend only on the geometical chaacteitic of the gea and on the contact atio of the pai, te veu contact atio diagam ae plotted. All gea wee thooughly eamined fo intefeence duing mehing and it ha been poved that thee i no uch dange when eplacing the tandad tochoidal fillet with the novel cicula fillet. In Fig. 5 the effect of the new deign to a gea with a mall numbe of teeth i invetigated. A gea with only 9 teeth peent evee undecutting and i of little pactical ue unle it i hifted by a ubtantial amount. Howeve by doing o the contat atio of the pai become malle and the hetzian contact tee ie thu educing it pitting eitance. By intoducing the concept of the cicula fillet it can be een that it fatigue eitance i inceaed by 25.26% fo tip loading (ε=1. 0) and up to 68.90% fo ε=1. 8 without affecting

6 Fig. 7: Compaion between the new and the tandad deign fo ε = 1.6 Fig. 9: Pinion-wheel pai with cicula filleted teeth in meh Fig. 8: Geneating ack/pinion with cicula fillet In Fig. 6 a gea with 32 teeth i eamined. Hee it i evident that the diffeence between the new and the eiting deign ae le impotant a the maimum deceae of the bending te i only 20% fo ε = 1.8. In all cae the new cicula fillet deign i found bette than the tandad one and thi i moe evident a the numbe of teeth deceae o a the HPSTC move lowe towad the pitch cicle. A the numbe of teeth inceae the diffeence between the two deign become malle and tend to be aymptotically zeo Fig. 10: Full path of contact of the gea tooth flank when the numbe of teeth get tend to infinity a in the cae of a ack, whee even the tandad deign give a The geneating ack i only lightly diffeent than the cicula fillet. Thi effect i illutated in Fig. 7, whee tandad one only nea it tip (which in tun geneate the maimum te veu the numbe of teeth i the oot fillet of the tooth). A it i uually pinion that plotted fo a uual gea pai with HPSTC coeponding undego the highe bending tee developed in the to a contact atio of 1.6. gea pai duing mehing, the ame geneating ack can The popoed fillet geomety ha anothe be ued to cut the mating wheel teeth, although thei advantage compaed to othe non-geneated gea tooth oot fillet geomety will not be tictly cicula a fo fom ince it can be cut on conventional hobbing the pinion. Anyway the wheel teeth will alo ehibit machine o ack-cutting machine without the need of inceaed tength compaed to the conventional one pecial tooth numbe-pecific milling tool. geneated with the tandad cicula tipped ack/hob. 1063

7 Fig. 11: Cutting poce of the cicula filleted teeth by the geneating ack and ceation of the enveloping uface In Fig. 10 the full path of contact coeponding to a pinion tooth flank with cicula filleted teeth i illutated in detail. Note that the path of contact i compoed of a taight linea pat inclined at 20 to the hoizontal at which the mehing and the powe tanmiion between gea take place and of a complicated eveing end pat at which the cutting action (i.e. Mehing with the geneating ack tooth) take place. Finally in Fig. 11 the cutting (geneating) poce of uch a fillet i illutated. Note that the taight pat of the ack emain unchanged o in fact the novel teeth ae till tandad involute teeth etaining all the well known advantage of the involute. CONCLUSIONS The effect of the novel popoed cicula fillet deign on the bending tee developed in pu gea teeth wa invetigated in compaion with the tandad deign. The new geomety incopoating the cicula fillet wa modeled fo dimenionle teeth and the maimum bending te wa calculated uing a BEM conideing unit loading at the HPSTC of the tooth. It wa demontated that the novel cicula deign upae the eiting tochoidal deign of the pu gea tooth fillet in tem of fatigue enduance without affecting the pitting eitance. Thi i moe evident on gea with a mall numbe of teeth (i.e. Pinion with le than 30 teeth). The popoed geomety doe not poduce undecut teeth even fo a mall numbe of teeth and can be eaily manufactued uing tandad involute hobbing tool with pecial tip deign. Futhemoe the novel fillet can be gound moe eaily due to thei cicula deign even with conventional, eay to manufactue tool and thu inceae the fatigue popetie of the gea pai futhe. Ameican J. App. Sci., 2 (6): , REFERENCES 1. Hoffman, G. et al., Teting P/M Mateial fo High Loading Gea Application. Intl. J. Powde Metallugy, 35: Townend, D.P. and E.N. Bambege, Suface Fatigue Life of Cabuized and Hadened M50NiL and AISI 9310 Spu Gea and Rolling Contact Tet Ba. J. Populion and Powe, Vol: Legge, G., Plama Cabuizing-Facility Deign and Opeation Data, Indutial Heating, pp: Heing, D.H., Why Vacuum Cabuizing I Effective fo Today and Tomoow: II, Indutial Heating, pp: Litvin, F.L., L. Qiming and A.L. Kapelvich, Aymmetical Modified Spu Gea Dive: Reduction of Noie, Localization of Contact, Simulation of Mehing and Ste Analyi. Compute Method in App. Mechanic and Eng., 188: Tai, M. and Y. Tai, Deign of High Contact Ratio Spu Gea Uing Quadatic Paametic Tooth Pofile. Mechanim and Machine Theoy, 33: Daniewicz, S.R. and D.H. Mooe, Inceaing the bending Fatigue Reitance of Spu Gea Teeth Uing a Peetting Poce. Intl. J. Fatigue, 20: Townend, D.P., Dudley Gea Handbookthe deign, manufactue and application of gea. McGaw-Hill, New Yok. 9. Niemann, G., Machinenelemente. Band 2, Spinge, Velag. 10. Fedette, L. and M. Bown, Gea Ste Reduction Uing Intenal Ste Relief Featue. J. Mechanical Deign, 119: Ciavaella, M. and G. Demelio, Numeical Method fo the Optimization of Specific Sliding, Ste Concentation and Fatigue Life of Gea. Intl. J. Fatigue, 21: ISO, , Calculation of the load capacity of pu and helical gea-pat 3. Calculation of tooth bending tength. 13. AGMA, 2101-C95, Fundamental ating facto and calculation method fo involute pu and helical gea. (Metic veion). Ameican Gea Manufactue Aociation. 14. Mabie, H.H., C.A. Roge and C.F. Reinholtz, Deign of Nontandad Spu Gea Cut by a Hob, Mechanim and Machine Theoy, 25: Roge, C.A., H.H. Mabie and C.F. Reinholtz, Deign of Spu Gea Geneated with Pinion Cutte. Mechanim and Machine Theoy, 25: Litvin, F.L., Gea Geomety and Applied Theoy. Pentice Hall, Englewood Cliff, NJ. 17. Spita, V.A. and T. Cotopoulo, New Concept in Numeical Modeling and Calculation of the Maimum Root Ste in Spu Gea veu Standad Method: A Compaative Study. Poc. 1t National Conf. Recent Advance in Mechanical Eng., ASME-Geek Section, Pata.