AN ANALYSIS OF THE EFFECT OF THE APPLICATION OF HELICAL MOTION AND ASSEMBLY ERRORS ON THE MESHING OF A SPIRAL BEVEL GEAR USING DUPLEX HELICAL METHOD

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1 ADVANCES IN MANUFACTURING SCIENCE AND TECHNOLOGY Vol. 40, No. 1, 2016 DOI: /ams AN ANALYSIS OF THE EFFECT OF THE APPLICATION OF HELICAL MOTION AND ASSEMBLY ERRORS ON THE MESHING OF A SPIRAL BEVEL GEAR USING DUPLEX HELICAL METHOD Jadwiga Pisula S u m m a r y This aricle presens he effec of he helical moion parameer, specific for he Duplex Helical mehod, on he obained pinion ooh flanks and on he correcness of conac in a spiral bevel gear. Furhermore, he gear's sensiiviy o assembly errors is demonsraed. The analyses are based on he designed mahemaical model of cuing bevel eeh wih a circular pich line and he mahemaical model of he design gear. The resuls are presened for he seleced gear wih he raio 20:37. Keywords: spiral bevel gears, Duplex Helical mehod, conac paern Analiza wpływu ruchu śrubowego i błędów monażu na współpracę przekładni sożkowej o kołowej linii zęba wykonanej meodą Duplex Helical S r e s z c z e n i e W pracy przedsawiono analizę wpływu parameru ruchu śrubowego, charakerysycznego dla meody Duplex Helical, na dokładność wykonania powierzchni bocznej zęba zębnika oraz poprawność współpracy zazębienia przekładni sożkowej o kołowej linii zęba. Usalono ponado wrażliwość ej przekładni na błędy monażu. Podsawą analizy jes opracowany model maemayczny nacinania uzębienia sożkowego o kołowej linii zęba i model maemayczny przekładni konsrukcyjnej. Uzyskano wyniki dla wybranej przekładni o przełożeniu 20:37. Słowa kluczowe: koła zębae sożkowe o kołowej linii zęba, meoda Duplex Helical, ślad syku 1. Inroducion The Duplex Helical mehod involves cuing (milling) or grinding spiral bevel gears by means of a helical machining moion. I is a dual wo-sided mehod, i.e. boh sides of he gear wheel and pinion eeh are cu simulaneously using wosided cuer heads or grinding wheels. There are wo varians of he Duplex Helical mehod: formae (for gear wheels wih a large number of eeh) and generaed approach. In he formae approach, he gear wheel is cu using he formae (nongeneraed) mehod, whereas he pinion is generaed; he generaed approach Address: Jadwiga PISULA, PhD Eng., Rzeszow Universiy of Technology, Deparmen of Mechanical Engineering, Powsańców Warszawy 8, Rzeszów, jpisula@prz.edu.pl

2 20 J. Pisula involves generaing boh pars of he ransmission. An obvious advanage of he Duplex Helical mehod is shorer machining ime. However, he applicaion of he mehod is limied o machine ools which provide helical moion [1, 2]; i may be used for fabricaing gears wih small modules and a raio of up o 2.5. The appropriae selecion of echnological parameers for cuing gear wheels by means of he Duplex Helical mehod, so ha he gear saisfies requiremens arising from design assumpions in erms of a specific conac paern and moion graph, is very difficul. In addiion, simulaneous cuing of he enire ooh space increases he gear's sensiiviy o machining errors and assembly errors. The applicaion of helical moion, accomplished as an addiional moion of he able wih he fas headsock of he machined gear wheel, allows us o achieve he correc gear meshing deermined by means of conac paern and moion graph. 2. A model of echnological gear including he applicaion of he Duplex Helical mehod A mahemaical model of spiral bevel gear cuing was used in order o deermine he effec of he helical moion parameer on simulaneously obained convex and concave surfaces of he ooh space. The mahemaical model of ooh surfaces involves deermining an envelope of he family of ool's surfaces of acion by means of a kinemaic mehod [3]. The mehod makes use of kinemaic relaionships beween maing surfaces, in his specific case beween he family of he ool's surfaces of acion and he resuling concave or convex ooh surface. The kinemaic relaionship beween he maing surfaces is ranscribed in he form of a meshing equaion. The unknown envelope is he soluion o he meshing equaion supplemened wih he equaion of he family of he ool's surfaces of acion [3 6]. The mahemaical model of spiral bevel gear cuing involves defining a se of dexroroaory Caresian coordinae sysems reflecing geomerical and kinemaic relaionships beween he machine ool's main assemblies. Fig. 1 presens coordinae sysems relaed o he sub-assemblies of a convenional machine ool, including he applicaion of an addiional moion modifying he machining moion. Individual sysems are marked wih an upper-case S i, he index of which refers o a specific sub-assembly of he machine ool reflecing he posiion or moion relaive o anoher sub-assembly. The helical machining moion is performed by he axial feed of he able wih an aached fas headsock of he machined pinion, which is suiably coupled wih he machining moion performed by he cradle [2].

3 An analysis of he effec Fig. 1. A se of coordinae sysems reflecing geomerical and kinemaic relaionships in he echnological gear (S1 sysem relaed o he pinion (or gear S2) being machined, S sysem relaed o he ool, Sm fixed coordinae sysem; he remaining ones are auxiliary sysems) Mahemaical noaion of he surface makes use of elemens of differenial geomery and vecor calculus. Vecor equaions of he ool's acion surface, i.e. a wo-sided face cuer head and a cup grinding wheel, define conical surfaces obained by roaing cuing edges of inner and ouer cuers or he axial profile of he grinding wheel around ool axes (Fig. 2). The equaions are shown in formulae (1) and (2) for edges cuing respecively, concave and convex wheel ooh spaces. cos θ ( rwk + s sin αwk ) r ( wk) ( s, θ ) sin θ ( rwk s sin αwk ) = + (1) s cosαwk

4 22 J. Pisula for s1 s s2, θ1 θ θ2 cos θ ( rwp s sin αwp ) r ( wp) ( s, θ ) = sin θ ( rwp s sin αwp ) (2) s cosα wp, where: s, θ curvilinear coordinaes of surfaces shaping concave and convex ooh space surfaces, αwk, α wp ool acion surface profile angles, r wk,r wp radial locaions of head cuers or of he grinding wheel, r nominal radius of he head, W o 2 head cuer spacing or grinding wheel face widh. Fig. 2. Tool acion surfaces machining he concave and he convex side of bevel gear ooh space (parameers described in he ex) For conical surfaces of he ool's cuing concave and convex ooh space surfaces, normal uni vecors were deermined, which were presened in equaions (3) and (4) respecively. cosθ cosα wk n ( wk )( θ ) sinθ cosα = wk (3) sinα wk cosθ cosα wp n (wp) ( θ ) = sinθ cosαwp (4) sinα wp

5 An analysis of he effec In order o deermine he concave and convex ooh space surface using a wo-sided approach, equaions of families of he ool's acion surface are required. Such equaions are obainable by esablishing he ool's acion surface equaions in he coordinae sysem relaed o he machine ool body or he gear being machined (5). This necessiaes he use of coordinae sysems' ransformaion marix described by he equaion (6). As he marix includes roaions and coordinae sysem ranslaions, for convenience i was presened as a 4x4 square marix. Consequenly, parameric equaions of he ool's acion surface and he family of ool acion surfaces were expanded o uniform noaion, i.e. 4x1 vecors (marked in he equaion as (5)). ( ) ( ) ( s,, ) [ 4 1] ( ) (, θ ψ = ψ s θ ) x [ 4 x 1] r M r (5) 1 1 ( ) ( ) ( ) M ψ = M ψ ψ M M ψ M ψ M M M (6) 1 1r 1 rh hm mk kc cd d where: ( ψ ) M coordinae sysems ransformaion marix, ψ moion 1 parameer, r 1 (,, ) [ 4x1 ], (, ) [ 4x1] equaion of he ool's acion surface family 1 ( s, θ, ψ ) equaions ( s,θ ) s θ ψ r s θ uniform noaion of he parameric r and ool's acion surface r, M ji elemenary ransformaion marices represening roaions and ranslaions of uniform coordinae sysems, he lower index specifies he direcion of he ransformaion from sysem S i o sysem S j. The surface angen o each surface of he family of surfaces consiues he envelope of he family of surfaces. The envelope of he family of he ool's acion surfaces is he unknown ooh flank. Such envelope is obainable by solving equaion sysem (7), comprising he family of he ool's acion surfaces and he meshing equaion. The meshing equaion included in equaion sysem (7) expresses he basic principle of he kinemaics of maing surfaces, according o which he relaive velociy vecor of hose surfaces is perpendicular o heir normal common a he poin of conac [3]. This is he kinemaic approach o deermining envelopes, in which a relaive velociy vecor of he gear and he ool may be deermined on he basis of he knowledge of machining kinemaics of he analysed cuing mehod. r n 1 ( s, θ, ψ ) ( 1 ) 1 v1 ( s, θ, ψ ) = 0 (7)

6 24 J. Pisula where: ( s θ ψ ) r 1,, parameric equaion of he family of ool's acion surfaces, n 1 normal uni vecor in he pinion-relaed sysem S 1, ( 1 ) v 1 ( s, θ, ψ ) relaive velociy vecor of he ool in relaion o he pinion being machined in he pinionrelaed sysem S 1. If we subsiue 2 for index 1, he dependences will refer o he gear wheel. Thus deermined laeral surface (flank) consiues a se of poins being he numerical soluion o he equaion sysem (7). The number and locaion of he poins defining he ooh flank corresponds o he poins on he reference grid locaed in he axial secion of he pinion's or he gear wheel's ooh. Having arrived a a finie se of soluions (values) corresponding o he coordinaes of he ooh's laeral surface, we may conduc an inerpolaion o quickly find soluions for he remaining "inernal" poins on he ooh's laeral surface. Thus obained surfaces will be used in he analysis of he effec of machining parameers on ooh flank geomery and o deermine meshing parameers. 3. A mahemaical model of bevel gear meshing A mahemaical model of design bevel gear wih he inclusion of errors in he performance of gear componens and assembly errors was used in he gear conac analysis. Figure 3 presens a se of orhogonal dexroroaory coordinae sysems reflecing kinemaic and geomerical inerdependences in he design bevel gear. Parameers H, V and J define assembly errors corresponding, respecively, o: a shif in he pinion's posiion owards is axis (H), a shif in he posiion of he pinion axis in he direcion perpendicular o he gear wheel axis (V) and a shif in he posiion of he gear wheel owards is axis (J). Parameer Σ is he difference beween he acual axis inersecion angle and is nominal value. As he surfaces of pinion's and gear wheel's eeh mesh, specific conac condiions on which he gear's conac analysis is based are me. Namely, a momenary poin of conac M common for boh surfaces, he coordinaes of momenary poin of conac M are equal on boh surfaces (8) and normal uni vecors of boh surfaces overlap a he poin of conac (9). M ( 1) M ( 2) r = r (8) M ( 1) M ( 2) n = n (9) In addiion, he condiion of he locaion of he relaive velociy vecor of he poin of conac of boh surfaces on a common angen plane consiues a meshing equaion included in formula (7), inroducing a normal vecor a he poin

7 An analysis of he effec of conac on one of maing planes and relaive velociy of he poin of conac of one surface in relaion o anoher. Thus defined sysem of equaions allows us o evaluae conac in he bevel gear. The aricle presen he gear conac analysis as oal conac paern. Fig. 3. A se of coordinae sysems reflecing geomerical and kinemaic relaionships in he design gear, including assembly errors: H, V, J, Σ (S1 pinion-relaed sysem, S2 gear wheel-relaed sysem, Sf fixed coordinae sysem; oher sysems are auxiliary) 4. An analysis of he effec of helical moion parameer on ooh flank The analyses shown here were performed for a 20:37 bevel gear, for which basic geomery and se machining parameers and ool geomery using he Duplex Helical mehod are presened in Tables 1, 2 and 3.

8 26 J. Pisula Table 1. Basic geomerical daa of he 20/37 gear Quaniy Designaion Pinion Gear Number of eeh z Hand of spiral Lef Righ Exernal ransverse module 2.6 mm Pressure angle 20 Shaf angle Σ 90 Spiral angle Mean cone disance R mm Face widh b 16.0 mm Exernal whole deph h mm mm Clearance c mm mm Exernal heigh of addendum h mm mm Exernal heigh of ooh roo h mm mm Pich angle Dedendum angle Addendum angle Table 2. Geomerical ool daa Tool parameers Pinion Gear Diameer of cuer head 88.9 mm 88.9 mm Widh of he blade ip mm mm Fille radius mm mm Cuer pressure angle (ouer) Cuer pressure angle (inner) Table 3. The basic seing daa for he gear and he pinion processing Basic machine seings Designaion Pinion (boh flanks) Gear (boh flanks) Cradle angle q Radial disance U mm mm Hypoid offse a mm 0 mm Machine roo angle δ m Machine cenre o back Xp mm 0 mm Sliding base Xb mm mm Til angle i Swivel angle j Roll raio iod Helical moion ph mm/rad 0 mm/rad

9 An analysis of he effec On he basis of daa gahered in Tables 1, 2 and 3 and he mahemaical model of he machined gear, concave and convex surfaces of he pinion and he gear wheel were generaed as a se of poins. The presen sudy analyses he effec of a change in helical moion parameer p H on he resulan concave and convex surface so he pinion. The resuls are shown in Figures 4 and 5. Figure 4 conains a image of wo ooh space surfaces of he pinion as a se of poins. Laeral surfaces of he ooh space were marked red and were reaed as reference surfaces, i.e. surfaces obained on he basis of daa conained in Tables 1-3. Nex, helical moion parameer was changed o p H = [mm/rad] (he absolue value of he parameer was increased by 0.06). The effec of he change in he parameer was presened on he pinion's ooh space surfaces marked blue (Fig. 4). As differences in poin locaions are sligh, he naure of he changes in he helical moion parameer should be inerpreed as follows: a rise in he helical moion parameer causes larger quaniies of maerial o be colleced on he convex surface from he small modules and he dedendum, and on he concave surface from he small modules and he addendum, while he peneraion boundary of he nominal surface and he alered surface lies close o he diagonal of he surface from relevan verices (Fig. 4 indicaes he peneraion boundary of he surfaces by a change in he colour of surface poins). Fig. 4. The effec of he change in he helical moion parameer on he locaion of laeral surfaces of ooh space relaive o he reference surface. Red poins define reference surfaces for he convex and concave sides of he pinion's ooh space (ph = mm/rad), blue poins for he convex and concave sides of he pinion's ooh space afer he change in parameer ph = 3, mm/rad Figure 5 conains a quaniaive presenaion of he effec of he change in he helical moion parameer separaely for he concave on convex surface of he pinion. These are graphs of he arc differenial according o he formula (10) from he angular and radial coordinae of he polar coordinaes of he poins of he nominal and he alered surface. Posiive values of he arc differenial indicae ha he maerial is colleced from he surface, whereas negaive values show ha he maerial remains on he surface.

10 28 J. Pisula ( _ 0 φ _ 1 ) = R φ (10) i i V i V where: arc differenial, R relevan radial coordinae of he poins from he i cylindrical coordinaes, φ i _V 0, φ i _V 1 angular coordinaes of he poins of he reference surface and he alered surface. Fig. 5. The effec of he change in he helical moion parameer on he obained side surfaces of pinion's ooh space presened in he arc differenial graph deermined from polar coordinaes of he poins of he reference surface and he alered surface 5. An analysis of he bevel gear's sensiiviy o assembly errors The analysis was conduced for he perfec gear and a gear wih inenionally inroduced assembly errors, he values of which were presened in Table 4. The gear's conac qualiy was presened by means of a diagram of he conac paern on he gear wheel's ooh flanks (Fig. 6). The conac paern was obained hrough peneraion by pinion's surface o he deph of 5 m, which corresponds o he hickness of he ink used in he gear conac es [3]. Table 4. Offse values for ransmission configuraion parameers Assembly errors Insance a) b) c) d) Hypoid offse V, mm Axial shif of pinion H, mm Axial shif of gear J, mm Shaf angle change Σ, min

11 An analysis of he effec a) Gear drive side (convex side) Gear coas side (concave side) b) c) d) e) Fig. 6. Conac paern diagrams presened form he concave and convex surfaces of he gear wheel ooh 20:37 for he following insances: a) he perfec gear, b) a gear wih a deviaion of V = 0.05 mm, c) a gear wih he deviaion of H = 0.05 mm, d) a gear wih he deviaion of J = 0.1 mm, e) a gear wih he deviaion of DS = 5/60 deg

12 30 J. Pisula Changes in he locaion of he conac paern presened on he gear wheel's laeral surfaces for he analysed gear were as follows: for a posiive hypoid offse V, he conac paern shifs in he direcion of he addendum and small modules for he drive side of he ooh of he gear wheel, and in he direcion of he addendum and large modules for he coas side of he ooh of he gear wheel, if a posiive offse of pinion axis H is used, he conac paern shifs owards he addendum for boh gear wheel ooh surfaces, if a posiive offse of gear wheel axis J is used, he conac paern shifs owards he dedendum and slighly in he direcion of large modules for boh gear wheel ooh surfaces, if a posiive (incremenal) axis inersecion error Σ is inroduced, he conac paern shifs owards he addendum for he drive side of he gear wheel ooh, and owards he dedendum for he coas side of he gear wheel ooh. The shifs in he conac paern on he drive and he coas side are closely inerrelaed. This is a consequence of simulaneously machining of boh ooh space surfaces for he gear and he pinion in he Duplex Helical mehod applied. 6. Conclusions An analysis of he geomery of he laeral surface of he ooh space obained wih he Duplex Helical mehod using helical moion and he resuls of he simulaion of he conac beween he surfaces provides he basis for he gear's accepance a he design sage and, more generally, offers guidance as o he design of a new gear according o he discussed approach. The approach is ready for implemenaion on convenional machine ools. The inroducion of a non-linear change in he helical moion will enable addiional modificaions o ooh surfaces. Obviously, i will be possible only by means of numerically-conrolled machine ools which allow operaors o inpu non-linear moion funcions. Manufacurers' ineres in he Duplex Helical mehod seems quie raional. I is moivaed mainly by economical producion and he avoidance of problems arising a he boom of he ooh space in dual wo-sided mehods. Researchers also seem o be ineresed in he discussed mehod, which is confirmed in publicaions such as [2-9]. The resuls of he sudies will help in finding he correc way o design a spiral bevel gear. References [1] J. KLINGELNBERG (Hrsg.): Kegelräder. Grundlagen, Anwendungen. Springer- Verlag, Berlin Heidelberg 2008.

13 An analysis of he effec [2] Z. WÓJCIK: Obrabiarki do uzębień kół sożkowych. Wydawnicwo Naukowo- Techniczne, Warszawa [3] F.L. LITVIN, A. FUENTES: Gear geomery and applied heory, Second Ediion, Cambridge Univ. Press, Cambridge [4] Z.-H. FONG, Y.-P. SHIH, G. LIN: Mahemaical model for a universal Face hobbing hypoid gear generaor. Journal of Mechanical Design, 129(2007)1, [5] A. MARCINIEC: Analiza i syneza zazębień przekładni sożkowych o kołowołukowej linii zęba. Oficyna Wydawnicza Poliechniki Rzeszowskiej, Rzeszów [6] J. PISULA, M. PŁOCICA: Mehodology of designing he geomery of he bevel gear using numerical simulaion o generae he eeh flank surfaces. Aca Mechanica e Auomaica, 8(2014)1, 5-8. [7] I. GONZALEZ-PEREZ, A. FUENTES, K. HAYASAKA: Compuerized design and ooh conac analysis of spiral bevel gears generaed by he duplex helical mehod. Proc. ASME Iner. Design Conference Engineering Technical Conferences and Compuers and Informaion in Engineering Conference, Washingon [8] J. PISULA, M. PŁOCICA: Numerical model of bevel gears cuing by duplex helical mehod. Key Engineering Maerials, 490(2011), [9] Y. ZHANG, H. YAN, T. ZENG: Cuing principle and ooh conac analysis of spiral bevel and hypoid gears generaed by duplex helical mehod. Journal of Mechanical Engineering, 51(2015). Received in January 2016

14 Address: Jadwiga PISULA, PhD Eng., Rzeszow Universiy of Technology, Deparmen of Mechanical Engineering, Powsańców Warszawy 8, Rzeszów,