A MATHEMATICAL MODEL FOR DESIGNING TOOTH SURFACES IN SPIRAL BEVEL GEARS AND GEAR MESHING ANALYSIS

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1 DIAGNOSTYKA, Vol. 16, No. 1 (2015) 57 A MATHEMATICAL MODEL FOR DESIGNING TOOTH SURFACES IN SPIRAL BEVEL GEARS AND GEAR MESHING ANALYSIS Jadwiga PISULA Rzeszow Universiy of Technology, The Faculy of Mechanical Engineering and Aeronauics, Deparmen of Machine Design Al. Powsańców Warszawy 12, Rzeszów. Tel , jpisula@prz.edu.pl Summary The projec involved developing a mahemaical model of machining eeh of a spiral bevel gear and a mahemaical model of he gear pair. The mahemaical model of machining was based on generaive machining wih a single indexing sysem (face milling). On he basis of ool geomery, echnological seings, kinemaics of he process based on he vecor and marix calculus and differenial machining geomery, a ooh model was buil and ooh surfaces were obained. The mahemaical model of he gear pair was developed wih he use of he gear geomery and he obained ooh surfaces of he and he gear. The model allows for he possibiliy of inroducing errors due o gear seings and olerances of he manufacuring errors in housings and oher ransmission componens. The mahemaical model of he gear pair was used o obain he conac paern and he ransmission error graph. An analysis of he resuls and he applicaion of meshing qualiy indicaors allowed us o improve he gear ransmission. This process was carried ou in an ieraive cycle by changing he se-up (by modifying echnological machining) parameers of he machined surfaces of he eeh. The applicaion of boh models, i.e. he mahemaical model of machining eeh of a spiral bevel gear and a mahemaical model of he gear pair, was presened using he example of aircraf gear 18:43. Keywords: spiral bevel gear, ooh conac analysis (TCA). MATEMATYCZNY MODEL OTRZYMYWANIA POWIERZCHNI ZĘBÓW KÓŁ STOŻKOWYCH O KOŁOWO-ŁUKOWEJ LINII ZĘBA ORAZ ANALIZY WSPÓŁPRACY PRZEKŁADNI Sreszczenie W ramach zadania opracowano maemayczny model nacinania uzębienia kół sożkowych o kołowo-łukowej linii zęba oraz model maemayczny przekładni konsrukcyjnej. Model maemayczny nacinania uzębienia doyczy obróbki obwiedniowej z podziałem przerywanym (face milling). Na podsawie geomerii narzędzia, usawień echnologicznych obrabiarki oraz kinemayki obróbki w oparciu o rachunek wekorowo-macierzowy oraz geomerię różniczkową zbudowano model nacinania uzębienia i orzymano powierzchnie zębów kół. W oparciu o geomerię przekładni oraz uzyskane powierzchnie zębów kół zbudowano model przekładni konsrukcyjnej z możliwością wprowadzenia błędów wynikających z usawień kół oraz olerancji wykonawczych korpusów i pozosałych elemenów przekładni. Model przekładni konsrukcyjnej służy do orzymywania śladu współpracy oraz określania nierównomierności przekazywania ruchu. Na podsawie wymienionych wskaźników jakości zazębienia, nasępuje dopracowanie przekładni, kóre odbywa się w cyklu ieracyjnym przez zmianę paramerów usawczych nacinania powierzchni zębów kół. Zasosowanie obu modeli przedsawiono na przykładzie przekładni loniczej 18:43. Słowa kluczowe: sożkowe koła zębae o kołowej linii zęba, analiza zazębienia przekładni. 1. INTRODUCTION Conemporary bevel gear design process is based on virual simulaion of gear machining, followed by verifying heir meshing in a design pair. Since conac paern is he main indicaor of correc gear maing, is locaion on ooh flank, shape and size are of criical imporance. Therefore, he simulaion of he designed meshing or he simulaion of he es gear mus allow for manufacuring errors and assembly errors in he gear assembly. If he desired conac paern is absen, suiable modificaions of echnological parameers are inroduced in he model generaing ooh flank surfaces. Such modificaions of gear ooh flank geomery allows us o find a saisfacory conac paern [1][2][7][10]. Through he applicaion of mahemaical models of gear ooh cuing and ooh meshing one may

2 58 DIAGNOSTYKA, Vol. 16, No. 1 (2015) eliminae many aciviies radiionally performed by muliple cuings of es gears and checking heir meshing on es machines. This leads o a significan reducion of ime and coss allocaed o he inroducion of a new bevel gear pair ino producion. A he same ime, he correc meshing can be obained as early as in he phase of he firs physically cu gear pair. This, however, requires high compeence in one of commercially available compuer-aided design sysems (and purchase hereof) or developing a suiable sofware ool. One of he possibiliies in his area is o design a numerical machining model. Correcly reproduced (in a model) engineering sysems and cuing kinemaics provide he basis for verifying he meshing of a design pair and/or making modificaions o gear eeh in order o arrive a a suiable conac area and smooh performance. To his end, a mahemaical engineering model was developed, simulaing he kinemaics of machining on a convenional machine ool and a CNC machine ool Phoenix 175HC by Gleason, aking ino accoun all machining mehods. 2. A MATHEMATICAL MODEL FOR DESIGNING TOOTH FLANK SURFACE IN A BEVEL GEAR A mahemaical model of a machine ool was developed as se of dexroroaory Caresian coordinae sysems replicaing subassemblies in a real machine ool. The sysems were marked by he leer "S followed by a subscrip denoing he subassembly represened by he sysem (Fig. 1). The applied se of coordinae sysems represens a convenional machine ool for cuing bevel eeh wih a circular pich line (e.g. 116G milling machine, 463 grinding machine). In order o develop a universal machining model, we mus consider ooh flank geomery obained by means of he envelope mehod, including shape-forming, in which he ooh flank is fully angenial o he direcion of ool acion. In envelope machining, ooh flank consiues he envelope of he surface of acion, obainable from an equaion sysem (1.1) composed of an equaion family of he surface of acion (1.2) and a meshing equaion defined on he basis of he kinemaic envelope heory by F. Livin [4]. r1 ( s, θ, ψ) 1 (1.1) n1 v1 ( s, θ, ψ) = 0 r1( s, θ, ψ) = M1 ( ψ) r ( s, θ) (1.2) where: ψ - moion parameer, n 1 - uni vecor normal o he ool surface deermined in he S 1 1 sysem, v 1 ( s, θ, ψ ) - speed of ool relaive o he gear in he S 1 sysem, r ( s, θ) - vecor equaion of he surface of acion in he S 1 sysem, M 1 ( ψ ) - ransformaion marix in he form of a produc of marices reflecing individual roaions and shifs beween inpu sysems in he echnological bevel gear model. Fig. 1. The locaion of he machined gear in he echnological model ogeher wih a se of coordinae sysems for machine ool subassemblies reflecing he se-up of echnological parameers A parameric record of he surface of acion in a coordinae sysem relaed o ool S is presened by means of equaion (1.3), where as he uni vecor normal o hese surfaces was provided in formula (1.4): cos θ ( ri ± s sin αi) r ( s, θ) sin θ ( ri s sin αi) = ± (1.3) s cos αi) cosθ cosαi n ( θ) sinθ cosα = i (1.4) m sinα i where: s, θ - parameers relaed, respecively, o he lengh of he cuing edge and he formaion of he roary conical surface (Fig. 2); α i, ri - cuers' profile angle and he radius of he cuer head (i=wk, wp), he "wk subscrip refers o he concave side of he ooh flank machined by he ool's ouer cuers, while he "wp subscrip refers o he convex side of he ooh flank machined by he ool's inner cuers ("upper" symbol for ouer cuers, "lower for inner cuers). An elaborae se of coordinae sysems modelling he echnological gear, o which nine basic seing parameers of he machine ool are inpu (excluding he geomery of he ool and he machined gear), does no allow us o presen he meshing equaion in an explici way, i.e. in he form of wo-variable parameric equaion [9]. In his siuaion, he resuling ooh flank surface is a se of poins, wih suiable indices, consiuing he soluion o he sysem of equaions (1.1). The coordinaes of he

3 DIAGNOSTYKA, Vol. 16, No. 1 (2015) 59 sough poins correspond o he coordinaes of he poins disribued across he reference grid locaed in he axial secion of he machined gear wihin he limis of he ooh's acive heigh. The surface is sreched across he resuling se of poins by inerpolaion. Thus, he inerpolaed ooh flank is subjeced o furher analysis in he design gear model [9]. model of he design gear allows us o evaluae he effec of assembly errors on qualiaive gear performance indicaors [8]. The use of he geomery of and gear flanks as he inpu in he mahemaical model of he design gear makes i possible o obain addiional informaion abou ooh conac (TCA) in an unloaded gear. These include pah of conac and he ease-off graph [2][4]. Fig. 2. Geomery of conical surfaces forming concave and convex ooh flanks [6] 3. A MATHEMATICAL MODEL OF THE DESIGN BEVEL GEAR A mahemaical model of he design gear was creaed for he purpose of ooh conac analysis (TCA) of he gear in unloaded condiion. The applicaion of he vecor and marix calculus enables a nea and generalized record of defined geomerical elemens, and presens he inerdependencies beween such elemens in a ransparen manner. Gear geomery is defined by a se of orhogonal, dexroroaory coordinae sysems reflecing he posiion of he relaive o he gear and allowing for inpu of assembly errors (Fig. 3). In Fig. 3, sysems S 1 and S 2 are relaed respecively o he and he gear, while oher sysems deermine he muual posiion of he maing gears. The engagemen is assessed in reference o a fixed coordinae sysem S f relaed o he ransmission housing. In a heoreical design gear model (wihou he inclusion of assembly errors), meshing qualiy is assessed by generaing conac paerns and he ransmission error graph. If he resuls of gear conac assessmen are unsaisfacory, new ooh flank geomery resuling from echnological parameer modificaions is inroduced, and a suiable soluion ha could provide saisfacory meshing qualiy is sough. The design gear model also includes he following changes in gear locaions: hypoid offse of he (V), axial shif of he (H), axial shif of he gear (J), and he change in he axial inersecion angle ΔΣ. This exended Fig. 3. Design gear model wih deviaions as applied: in he analysis of he effec of ooh geomery on gear meshing qualiy (for zero deviaions); and o deermine he gear's sensiiviy o assembly and manufacuring errors 4. THE RESULTS OF THE APPLICATION OF THE MATHEMATICAL MODEL OF GEAR TOOTH DESIGN AND THE MATHEMATICAL MODEL OF A DESIGN BEVEL GEAR The analysis was performed for a 18:43 gear, whose basic geomery and seing parameers of machining and ool geomery are presened in Tables 1, 2 and 3. Table 1. Basic geomerical daa of he 18/43 gear Designaion Pinion Gear Quaniy Number of eeh z Hand of spiral Righ Lef Exernal ransverse module m 2.163[mm] Pressure angle α 20 Shaf angle Σ 70 Spiral angle β 35 Mean cone disance R mm] Face widh b [mm] Exernal whole deph h 4.502[mm] 4.502[mm] Clearance c 0.40[mm] 0.40[mm] Exernal heigh of addendum Exernal heigh of ooh roo h a 2.110[mm] 1.990[mm] h f 2.392[mm] 2.512[mm] Pich angle δ Dedendum angle Addendum angle θ f θ a

4 60 DIAGNOSTYKA, Vol. 16, No. 1 (2015) Tool parameers D 0R W 2 Diameer of cuer head Widh of he blade ip Table 2. Geomerical ool daa (concave) [mm] (convex) [mm] gear (boh flanks) [mm] [mm] [mm] [mm] r Fille radius [mm] [mm] [mm] Cuer pressure α wk 16 [ ] 24 [ ] 20 [ ] angle (ouer) α wp Cuer pressure angle (inner) 20 [ ] 13 [ ] 20 [ ] Table 3. Basic seing daa for he 18:43 gear pair Basic machine seings (concave) (convex) gear (boh flanks) Cradle angle Radial disance [mm] [mm] [mm] Hypoid offse [mm] 0.04 [mm] 0.00 [mm] Machine roo angle Machine cenre o 0.00 [mm] [mm] [mm] back Sliding base [mm] [mm] [mm] Til angle [ ] Swivel angle [ ] Roll raio Based on he above geomerical and se-up daa, he mahemaical ooh cuing model was applied o find he soluion o he equaion sysem (1.1) defining he envelope of he ool's surface of acion. The soluion o he equaion sysem is numerical, herefore gear and flanks are presened as ses of poins. To faciliae calculaions and furher analyses a reference grid was inroduced, evenly dividing he ooh flank wihin he acive heigh range [5]. In addiion, he reference grid verifies if ooh undercu occurs. Tooh surfaces of he gear and he generaed in he Mahcad sofware were presened in he form of a grid of poins and he inerpolaed surface in Fig. 4 (gear) and 5 (). Fig. 4. Grid of poins (lef-hand drawing) represening he numerical soluion o he envelope equaion (1.1) defining he concave surface of gear ooh flank (he righ-hand drawing shows he inerpolaed surface) Pinion and gear surfaces obained from he mahemaical cuing model were compiled in he design gear model. A 6 mm-hick virual marking compound was assumed for deermining he conac paern. For he presened daa, conac pah, momenary and summary conac paerns and ransmission error graph were generaed (Fig. 6, 7, 8, 9). Fig. 6. Conac pah illusraed on he concave surface of he gear flank Fig. 7. Momenary gear conac paern (yellow, urquoise gear, oher pars of and gear reference cones) a) b) Fig. 8. Reference summary conac paern for a 18:43 ransmission on he drive side (a) and coas side (b) of he lef-hand gear generaed on he basis of daa presened in Tables 1, 2 and 3 Fig. 5. Grid of poins (lef-hand drawing) represening he numerical soluion o he envelope equaion (1.1) defining he convex surface of ooh flank (he righ-hand drawing shows he inerpolaed surface) Fig. 9. Transmission error graph

5 DIAGNOSTYKA, Vol. 16, No. 1 (2015) 61 The analysed unloaded gear achieves he correc conac paern boh on is drive and coas side, i.e. he paern is focused, slighly shifed in he direcion of small modules and does no reveal any edging. As expeced, he ransmission error graph is also of parabolic shape, providing uniform and smooh gear performance, and he maximum deviaion a moion ransmission poin is 5 seconds of arc. Based on he above gear meshing qualiy crieria, he presen gear design is considered correc. Table 4. Offse values for ransmission configuraion parameers Assembly errors Insance Hypoid offse V [mm] Axial shif of H [mm] Axial shif of gear J [mm] Shaf angle change ΔΣ [min] a) b) c) d) e) f) g) h) Having obained saisfacory gear meshing indicaors, i.e. a suiably shaped, sized and posiioned conac paern, and wih a ransmission error of up o 10 seconds of arc, he gear should be assessed in erms of is sensiiviy o assembly errors wihin olerance limis [4]. To his end, ooh surfaces of he wheel and he for he drive side and he coas side were compiled, aking ino accoun axial shifs defined by parameers: V, H, J and ΔΣ. Offse values for he parameers for which he posiion of he conac paern is o be verified are usually defined by he designer. Their levels resul from he olerances assumed in he performance of he assembly componens as well as he gear's load condiions. In order o perform he analysis of he gear's sensiiviy o assembly errors, V, H, J and ΔΣ values lised in Table 4 were assumed. Fig. 11. Summary conac paerns for a 18:43 ransmission on he coas side of gear ooh (lefhand gears) for he assumed deviaions V, H, J and ΔΣ. Fig. 10. Summary conac paerns for a 18:43 ransmission on he drive side of he gear ooh (lef-hand gears) for he assumed deviaions (posiive and negaive) V, H, J and ΔΣ Considering he resuls of he analysis of he gear's sensiiviy o assembly errors, i mus be concluded ha he deviaion of he axial inersecion angle is very small and has only a sligh effec on he change of he posiion of he conac paern for his gear. I will affec he amoun of backlash and ip clearance. Negaive hypoid offse V and axial shif of he H for he drive side of he engagemen lead o an unaccepable conac paern, mainly due

6 62 DIAGNOSTYKA, Vol. 16, No. 1 (2015) o edging. A similar siuaion occurs for he coas side in his case for posiive values of hypoid offse and axial shif of he, he conac paern is oo narrow and leads o edging. A gear wheel assembly error in he form J axial shif is less sensiive han verical and horizonal axial shifs of he. The conac paern shifs primarily in he direcion of ooh heigh and only slighly in he direcion of ooh widh. 5. CONCLUSIONS Mahemaical models of ooh cuing and design gear engagemen play an imporan role in designing spiral bevel gears. They allow us o deermine he correc selecion of ool geomery and echnological machining parameers. Nex, by compiling he resuling and gear wheel ooh surfaces in a design gear model and heir suiable peneraion, mahemaical models make i possible o generae conac paern he main indicaor of meshing qualiy and ransmission error graph. If meshing performance is unsaisfacory, by changing he geomery of he and he gear wheel resuling from echnological seings of he machining process and ool geomery, we may effecively shape he form of he conac paern and influence he accuracy of ransmission. Wih he correc meshing performance in place, an analysis of he sensiiviy of he gear o performance and assembly errors mus be carried ou, aking ino accoun operaing condiions of he gear. In his way we may predic gear performance in loaded condiion. The awareness of hese meshing indicaors in unloaded condiion enables he reducion of he design period and prooype-making ime, which means lower manufacuring coss. Afer making improvemens o he geomery of gear flanks using TCA (i.e. having arrived a a conac paern of expeced shape and posiion), a loaded gear ooh conac analysis (LTCA) mus be performed. LCTA may be carried ou using he finie elemen mehod (FEM) [1][3]. Models simulaing gear load are also creaed using ooh flanks obained from he mahemaical model of ooh machining. If resuls are saisfacory, he producion of prooypes should follow. For aeronauics applicaions, ess in real condiions (RIG es) mus be carried ou. LITERATURA [1] Aroni A., Bracci A., Gabiccini M., Guiggiani M. Opimizaion of he Loaded Conac Paern in Hypoid gears by Auomaic Topography modificaion. Journal of Mechanical Design, vol.131, 2009, pp [2] Fan Q., Wilcox L. New Developmens in Tooh Conac Analysis (TCA) and Loaded TCA for Spiral Bevel and Hypoid Gear Drives. Gear Technology, May 2007, pp [3] Kolivand M., Kahraman A. A load disribuion model for hypoid gears using ease-off opography and shell heory. Mechanism and Machine Theory vol. 44 (2009), pp [4] Livin F. L., Fuenes A. Gear Geomery and Applied Theory, Second Ediion. Cambridge Univ. Press, [5] Marciniec A., Pisula J., Płocica M., Sobolewski B. Projekowanie przekładni sożkowych z zasosowaniem modelowania maemaycznego i symulacji w środowisku CAD. Mechanik, No. 7/2011, sr [6] Marciniec A. Analiza i syneza zazębień przekładni sożkowych o kołowo-łukowej linii zęba. Oficyna Wydawnicza Poliechniki Rzeszowskiej, Rzeszów, 2003 [7] Pisula J., Płocica M. Mehodology of designing he geomery of he bevel gear using numerical simulaion o generae he eeh flank surfaces. Aca Mechanica e Auomaica, vol.8 no.1/2014, pp [8] Pisula J., Płocica M. Wpływ błędów monażu przekładni sożkowej na jakość zazębienia. Zeszyy Naukowe Poliechniki Śląskiej nr 1903, seria: Transpor z. 82, 2014, sr [9] Pisula J. Maemayczny model geomerii uzębienia i zazębienia kół walcowych kszałowanych obwiedniowo. Docoral disseraion, Rzeszów, [10] Simon V. Machine-Tool Seings o Reduce he Sensiiviy of Spiral Bevel Gears o Tooh Errors and Misalignmens. ASME Journal of Mechanical Design, 130(8), p , This sudy was carried ou as par of he Projec "Modern Maerial Technologies Applied in he Aerospace Indusry", No. POIG /08-00 under he Operaional Programme of Innovaive Economy (OPIE). The projec is cofinanced by he European Union hrough he European Regional Developmen Fund. Jadwiga Pisula PhD. Eng. is assisan professor a Rzeszow Universiy of Technology. She received he PhD degree in mechanical engineering and operaion of machines. Her main research areas cover power drives, especially he design and echnology of gears, mahemaical modeling, and CAD / CAM sysems.