Tooth Surface Design for Variable Transmission Ratio Bevel Gearing

Transcription

1 Applied Mathematics, 05, 6, Published Online September 05 in SciRes. Tth Surface Design fr Variable Transmissin Rati Bevel Gearing Nan Yang,*, Dawei Zhang, Yanling Tian Schl f Mechanical Engineering, Tianjin Universit, Tianjin, China Tianjin Ke Labratr f the Design and Intelligent Cntrl f the Advanced Mechatrnical Sstem, Schl f Mechanical Engineering, Tianjin Universit f Technlg, Tianjin, China * 79nzw@63.cm Received 3 Octber 04; accepted 5 September 05; published 8 September 05 Cpright 05 b authrs and Scientific Research Publishing Inc. This wrk is licensed under the Creative Cmmns Attributin Internatinal License (CC BY). Abstract A nvel CAD methd fr tth prfile based n the gearing feature has been prpsed in this paper, and this methd culd be applied t the variable transmissin rati and three-dimensin situatin. The tw pitch curves f the tw gears can be generated due t the given transmissin rati. The tth prfile curves are frmed in each crdinate sstem f its pitch curve b the gearing prcess. Finall, the tth prfile culd be extracted and translated int a real dimensin b CAD methd. It is prvided a simpl thught fr tth prfile design t avid s much cmplicated mathematical reasning. Kewrds Cmputer-Aided Design, Cmputatinal Gemetr, Gear Design, Gear Gemetr. Intrductin Bevel gearings are used in the situatin f driving between the unparallel axes and have a wide applicatin field. Because the gearing mtin is s cmplex, the mathematical ther is ver deep and the calculatins are multifarius []. In this paper, an apparent and effective methd is prpsed, althugh it desn t need unfathmable mathematical ther. At first, based n the gearing feature, a pair f pitch curves (frm gear and ) are generated. Tth prfile can be arbitraril drawn arund the pitch curve ; tth prfile is uniquel frmed b the mtin f the tth prfile alng the pitch curve. The mtin f the tth prfile divides the space int tw regins (scanned b gear and nt scanned b gear ) and their bundar line. The regin nt scanned b gear can be the re- * Crrespnding authr. Hw t cite this paper: Yang, N., Zhang, D.W. and Tian, Y.L. (05) Tth Surface Design fr Variable Transmissin Rati Bevel Gearing. Applied Mathematics, 6,

2 gin where gear exists; therwise the tth prfile shuld be the bundar line. The bundar line culd be extracted b digital image prcessing [] and translated int a real dimensin finall. The gear is btained just like the paper-cut, s the cmplex prcess f envelping line slutin f mathematical meaning culd be avided.. The Presentatin f the Prblem The aim gears t be design are a pair f bevel gearings whse axes are vertical t each ther as shwn in Figure. The transmissin rati: = dt z ( sin ( )) dt = z () i c t where: z = 9, z = 8, c = 0.3 ; i is the transmissin rati; z, z are teeth number; c is adjusting cnstant; t, t are the rtatin angle f gear and. The mathematical relatinship between t and t is shwn in Figure. Gear γ Gear Figure. The psitin relatinship f the pair f bevel gearing, where γ is the π angle between the tw gear s axes, and γ = t Figure. The relatinship between t and t accrding t Equatin (), where t = 0.5 t cst is the integratin f Equatin (). ( ( )) t 686

3 3. The Slutin f the Tw Pitch Curves The pitch curves are tw fictitius clsed curves n the tw gears which have pure rlling behavir. Suppse that gear (the axis is OY) lives in crdinate sstem {O} and dn t mve, n the ther hand, gear lives in crdinate sstem {A}, and its rtating axis is OA. OA cntrartates based n OZ axis in XOY plane (see frm + Z t Z ), while gear cntrartates based n OA (see frm A t O), as shwn in Figure 3. The {O} crdinate f a pint K n the pitch curve in {A} is {x,, z}. After gearing t and t angle, K mves t K' whse {O} crdinate is {x', ', z'}. S: where: t, t x x B = z z ( + ) ( + ) ( ) ( ) ( ) ( ) ( t ) ( t ) ( ) ( ) ( ) ( ) cs t t sin t t cs t sin t 0 B = sin t + t cs t + t 0 0 cs t sin t sin t cs t sin cs 0 0 are the small rtating steps f axis OA and gear. Accrding t the meanings f pitch curve: Accrding t Equatin (): x x =. z z () (3) (4) x z ( B I) = 0 (5) A K O X Z Figure 3. The tw crdinate sstems f the tw gears. Gear in crdinate sstem {O} and gear in crdinate sstem π {A} OZ OA because f γ =. 687

4 where: Because: S, the equatins have infinite slutins: when t, t 0 : where transmissin rati t i = t. 0 0 I = ( B I) DET = 0. (6) t t cs t + tan t sin x z = t t sin t + tan t sin ( t ) cs x i z = sin ( t ) i The intersecting line G between the bevel surface f Equatin (8) and spherical surface x + + z = a (8 ) (7) (8) is the pitch curve f gear in {O}. The equatins f the intersecting line G are: ( t ) cs + i x sin ( t ) = a. z + i i + i The equatins f the pitch curve G f gear in {A} can be als btained: + i x i sin ( t ) = a. z i + i cs( t ) + i The pitch curves G and G is shwn in Figure 4. (9) (0) 688

5 Z G x G 4. The Design f the Tth Prfiles 4.. The Generatin f Gears Figure 4. The tw pitch curves G and G. When the transmissin rati is given, the tw pitch curves are determined uniquel. Hwever, the tth prfile curves f the tw gears are nt unique. When ne is given, the ther is determined. We can design a threedimensin curve based n G s pitch curve t be its tth prfile curve: where: ( ) sin zt β = H, H is adjusting cefficient,,, x xcs β + z sin β s s = sin t xsin + + z cs z cst xsin + + z cs ( β β) ( β β) s xz cme frm Equatin (0). a S, G s tth prfile curve S is shwn in Figure 5. Suppse that n the spherical surface, the S rlls purel alng the pitch curve G (G and G are tangent), then the regin where S desn t sweep shuld be the regin where gear exist. The black regin where is swept b S (see frm Z t Z + ) is shwn in Figure 6, and the new pitch curve f gear (S) appears. 4.. The Extractin f Tth Prfile Curve The black regin (in Figure 6) swept b S is prjected n XOY plane, as shwn in Figure 7. The tth prfile curve S can be btained b image prcessing technlg. As shwn in Figure 7, the clr regin is cmpsed b civil pints and the white regin is cmpsed b utside pints. The bundar line between the clr regin and white regin can be fund and extracted rderl in a given directin b image prcessing. OA and OB are calibratin lines t translate pixel size int real size; the clr regin is cmpsed b civil pints and the white regin is cmpsed b utside pints. We prvide (this prvisin reall makes sense) that these pints are bundar pints: if there is at least an utside pint n the psitin a, c, e, g f pint M as shwn in Figure 8, then the pint M is bundar pint. S, there are three tpes f pints: utside pint 0, civil pint, bundar pint. We can find a bundar pint as a beginning pint alng an scan line, and name its address P. Search all the eight psitins () 689

6 G G S Figure 5. G s tth prfile curve S. S G Figure 6. The black regin swept b S. 690

7 A inner pints O B Calibratin line S uter pints Figure 7. Inner pints and uter pints. h a b g M c f e d Figure 8. The eight psitins (a, b, c,, h) f pint M. (a, b, c, d, e, f, g, h) f P (see in Figure 8), name the first bundar pint s address P, and let the cntent f P be 3 which wn t be searched anmre. P replaces P, and find the next P in the same wa till return t the beginning pint. Therefre, at last, all the bundar pints are 3. This prcess is called rdinal extractin, as shwn in Figure 9. The flw f the rdinal extractin f bundar line is shwn in Figure The Translatin f Pixel Size int Real Size The tth prfile S which has been extracted is shwn in Figure (a). But S is pixel meaning, s it shuld be translated int real size. The relatinship between pixel size and real size is shwn in Equatin (). x = xp OA N = p OB N where xp, p are pixel crdinate; x, are real crdinate; OA, OB are the length f the lines which we knw; N A, N B are the pixel number in OA and OB. S in a real crdinate is shwn in Figure (b). The XOY prjectins f the fur psitins f S gearing with S are shwn in Figure Discussin Will the tw gears run steadil? Will their prfile curves leave each ther? These are nt in the scpe f this paper A B () 69

8 P P P P 3 0 P P 3 P 3 3 P Figure 9. The rdinal extractin f bundar line. Chse a pint as the beginning pint, and its psitin P Are the pints amng a->h psitins near P pints? N Y Let the psitin f the first pint be P Let the cntent f P be 3 ; P replaces P END Let the cntent f P be 3 Figure 0. The flw f rdinal extractin f bundar line. 69

9 S (a) Figure. (a) The extractin f S and (b) S in a real crdinate. S (b) S S S S (a) (b) S S S S (c) (d) Figure. The XOY prjectin f fur psitin f S gearing with S. (a) t = t = 0 ; (b) t = ; t = 90 ; (c) t = 360 ; t = 80 ; (d) t = 57 ; t =

10 (which culd be knwn frm [3]-[6]). But a gemetrical answer can be given. At an ecumenical psitin f gearing, via a tangent pint f tw tth prfile curve, draw a nrmal line f a tth prfile curve. The directins f the nrmal line and the press frce at this pint are same. If there are sme nrmal lines upn the rtating axis and sme nrmal line belw the rtating axis, that is t sa, the mments f the press frces f the rtating axis are cmplete negativel and psitivel, and then the gearing is stead reasnabl. Fr example, as shwn in Figure 3, draw three nrmal lines f gear via the tangent pints, the mments f press frce F, F are clckwise and mment f F3 is cunterclckwise. But this is nl a gemetrical precnditin t be stead; the detailed mechanical analzatin is nt in the scpe f this paper Three-Dimensin Situatin In the three-dimensin situatin, S is a space curve n the spherical surface (the S mentined befre is its prjectin n XOY). Given a radius a (frm Equatin (8 )), there will be a pair f gears S and S, as shwn in Figure 4. Therefre, the part f the spherical surface encircled b S r S is a cut sectin f a bevel gearing crrespnding radius a. All the cut sectins crrespnding different radius a cmpse the whle bevel gearing. S F F F3 S Figure 3. The nrmal lines via the tangent pints. S z S x Figure 4. The cut sectins f the tw bevel gearings n the spherical surface. 694

11 5. Cnclusin In this paper, a nvel design methd f space bevel gearing based n gearing feature is prpsed. The tw pitch curves are generated based n the transmissin rati, and then a tth prfile culd be designed based n its pitch curve independentl. Accrding t its mtin, the secnd tth prfile culd be fund and translated int a real size b image prcessing technlg. Cmparing t cmplicated methd [7]-[], this methd culd avid cmplex calculatin and exceptinal slutin, and culd almst design an gear r rack. Hwever, this methd is als limited in precisin, if the image is ver small, the size errr will arise due t lack f pixels []. S, the image must be zmed in enugh. Acknwledgements This wrk was supprted b China Pst-Dctral Fundatin N. 0M5057, Tianjin Municipal Educatin Cmmissin Grant N. 0040, and Tianjin Municipal Science and Technlg Cmmissin Ke Grant N. 4JCZDJC Cnflict f Interests Nne. References [] Jia, J.M., Ga, B. and Zha, D.L. (008) Analsis Methd fr Nncircular Bevel Gearing Based n Gedesic Curvature Preserving Mapping. Chinese Jurnal f Mechanical Engineering, 44, [] Zhang, D.F. (009) MATLAB Digital Image Prcessing. China Machine Press, Beijing. [3] Osman, T. and Velex, Ph. (00) Static, Dnamic Simulatins f Mild Abrasive Wear in Wide-Faced Slid Spur and Helical Gears. Mechanism and Machine Ther, 45, [4] Wiener, D. (000) Mdificatin Methds fr Bevel Gearings. Antriebstechnik, 39, [5] Beach, R. (004) Hpid and Bevel Gearing. Mtin Sstem Design, 46, [6] Sheveleva, G.I., Vlkv, A.E. and Msdvedev, V.I. (003) Calculating the Cntact Pressures in Bevel Gear Drives with Different Tth Mdels. Jurnal f Machiner Manufacture and Reliabilit, [7] Yang, F.C., Feng, J.X. and Zhang, H.C. (05) Pwer Flw and Efficienc Analsis f Multi-Flw Planetar Gear Trains. Mechanism and Machine Ther, 9, [8] Pedrer, J.I., Pleguezuels, M., Artés, M. and Antna, J.A. (00) Lad Distributin Mdel alng the Line f Cntact fr Invlute External Gears. Mechanism and Machine Ther, 45, [9] Nagamt, A.F.H., Litvin, F.L., Gnzalez-Perez, I. and Haasaka, K. (00) Cmputerized Design f Mdified Helical Gears Finished b Plunge Shaving. Cmputer Methds in Applied Mechanics and Engineering, 99, [0] Feng, Z.P., Zu, M.J. and Chu, F.L. (00) Applicatin f Regularizatin Dimensin t Gear Damage Assessment. Mechanical Sstems and Signal Prcessing, 4, [] Ottewill, J.R., Neild, S.A. and Wilsn, R.E. (00) An Investigatin int the Effect f Tth Prfile Errrs n Gear Rattle. Jurnal f Sund and Vibratin, 39, [] Sng, C.S., Zhu, C.C., Liu, H.J. and Ni, G.X. (05) Dnamic Analsis and Experimental Stud f a Marine Gearbx with Crssed Bevelid Gears. Mechanism and Machine Ther, 9,