Contact characteristics of cylindrical gears with curvilinear shaped teeth

Transcription

1 Mechaism ad Machie Theory 9 (004) Mechaism ad Machie Theory Cotact characteristics o cylidrical gears with curviliear shaped teeth Rui-Tag Tseg, Chug-Biau Tsay * Departmet o Mechaical Egieerig, Natioal Chiao Tug Uiversity, Hsichu 0010, Taiwa, ROC Received 18 March 00; received i revised orm 1 February 004; accepted April 004 Abstract Trasmissio errors ad bearig cotacts o a curviliear-tooth gear uder dieret assembly coditios are ivestigated by applyig the tooth cotact aalysis ad the surace topology methods. Four umerical examples are preseted to illustrate the iluece o gear assembly errors o trasmissio errors ad the directios ad orietatios o the cotact ellipses. The umerical results reveal that the trasmissio error o the curviliear-tooth gear pair is small ad the bearig cotacts are located i the middle regio o the curviliear-tooth surace eve uder axial misaligmets. Ó 004 Published by Elsevier Ltd. Keywords: Curviliear-tooth gear; Tooth cotact aalysis; Trasmissio error; Cotact ellipse 1. Itroductio The spur gear set ad the covetioal helical gear pair with parallel axes are i lie cotact, ad thus their trasmissio errors (TE) are sesitive to axial misaligmets o the gear. Whe a gear set without tooth crowig is axially misaliged, tooth edge cotact will occur ad this results i stress cocetratio o the gear, vibratio ad oise. Spur ad helical gears with tooth crowig ca prevet edge cotact iduced by axial misaligmets o the gear, ad ca also localize the bearig cotacts. Litvi et al. [1,] proposed a geeratio mechaism with a ivedegree-o-reedom system ad tooth cotact aalysis (TCA) or spur gear crowig to prevet gear edge cotact. I this study, a curviliear-tooth gear pair with poit cotacts is proposed. Gear edge cotact ca thus be avoided because the cotact paths o gear tooth suraces are * Correspodig author. Tel./ax: address: cbtsay@mail.ctu.edu.tw (C.-B. Tsay) X/$ - see rot matter Ó 004 Published by Elsevier Ltd. doi:10.101/j.mechmachtheory

2 90 R.-T. Tseg, C.-B. Tsay / Mechaism ad Machie Theory 9 (004) Nomeclature a i ; b i tool settig o rack cutter P i ði ¼ F; PÞ l i variable parameter which determies the locatio o rack cutter ði ¼ F; PÞ r i radius o pitch circle o gear i ði ¼ 1; Þ C 0 operatioal ceter distace (Fig. 1) DC variatio o ceter distace (Fig. 1) R i omial radius o the ace mill-cutter ði ¼ F; PÞ T i umber o teeth o gear i ði ¼ 1; Þ X ðiþ T ; Y ðiþ T ; ZðiÞ T coordiates o gear tooth surace i ði ¼ 1; Þ represeted i coordiate system S T (Fig. ) c ðiþ uit ormal vector o surace P i ði ¼ F; PÞ represeted i coordiate system S c R ðiþ c positio vector o surace P i ði ¼ F; PÞ represeted i coordiate system S c ðr; hþ auxiliary polar coordiate system represeted o cotact tooth taget plae (Fig. ) S i ðx i ; Y i ; Z i Þ coordiate system i ði ¼ 1; ; v; h; ; TÞ with three orthogoal axes X i, Y i, ad Z i Dc h horizotal axial misaligmet (i degrees, Fig. 1) Dc v vertical axial misaligmet (i degrees, Fig. 1) h i variable parameter which determies the locatio o rack cutter ði ¼ F; PÞ / i rotatio agle o gear i ði ¼ 1; Þ whe gear i is geerated by rack cutter / 0 i rotatio agle o gear i ði ¼ 1; Þ whe two gears mesh with each other (Fig. 1) D/ 0 ð/0 1Þ trasmissio errors (i arc-secod) ormal pressure agle o rack cutter P i ði ¼ F; PÞ w ðiþ Subscripts F, P rack cutter surace to geerate tooth suraces o piio ad gear 1, tooth suraces o piio ad gear located ear the middle regio o the tooth lak eve whe the gear set is meshed with axial misaligmets. The trasmissio error (TE) o a matig gear pair is a importat actor or gear oise ad vibratio criteria. Litvi ad Tsay [] simulated the meshig ad bearig cotact o circular arc helical gears. Litvi [4,5] ad Tsay [] ivestigated the TCA o some other types o meshig gear pairs uder dieret meshig coditios. Li et al. [7] perormed TCA or hypoid gears. Chag et al. [8] aalyzed the kiematic optimizatio o a modiied helical gear trai. Litvi et al. [9,10] proposed a surace sythesis method by utilizig a predesiged parabolic trasmissio errors uctio to absorb the trasmissio errors o a approximately liear uctio o the desiged gear pair iduced by axial misaligmets o the gear. Zhag ad Fag [11] cosidered the elastic deormatio o tooth suraces to estimate the trasmissio errors o helical gears uder a load. Umeyama et al. [1] ivestigated the loaded trasmissio errors o helical gears ad the relatioship betwee the actual cotact ratio ad eective cotact ratio. Sice the cotact type o the curviliear-tooth gear proposed herei is a poit cotact, the trasmissio error o the gear pair is ot sesitive to the axial misaligmets o the gear ad the trasmissio error is small.

3 R.-T. Tseg, C.-B. Tsay / Mechaism ad Machie Theory 9 (004) Whe two gear tooth suraces are meshed, the istataeous cotact poit o the two suraces is spread over a elliptical area due to elasticity o the tooth surace uder load. Litvi [4,5] applied pricipal directios ad pricipal curvatures to estimate the orietatios ad dimesios o cotact ellipses o the commo taget plae o the two matig tooth suraces. Jaick [1] proposed a cotact surace topology method to predict the iitial cotact patter o worm ad worm gear drives. The TCA techology has bee utilized herei to ivestigate the trasmissio errors o the curviliear-tooth gear pair uder ideal ad error assembly coditios. The cotact ellipses o the curviliear-tooth gears ca be obtaied by applyig the cotact surace topology method. The results reveal that the omial radius o the ace mill-cutter is the importat desig parameter which ilueces the dimesios o gear cotact ellipses.. Mathematical model o the curviliear-tooth gears The mathematical model o cylidrical gears with curviliear shaped teeth adopted herei was developed i our previous study [15]. I this paper, oly the mathematical models o the active tooth suraces o the piio ad the gear are expressed ad discussed. The proile o the rack cutter s ormal middle sectio is the same as that or the straight-edge rack cutter that geerates the tooth proile o ivolute gears. The right-side o the rack cutter surace P FR is used to produce the let-side o the curviliear piio tooth surace P 1L, while the let-side o the rack cutter surace P FL is used to geerate the right-side o the curviliear piio tooth surace P 1R. The gear ad piio with curviliear shaped teeth proposed i this ivestigatio are produced by the same ace mill-cutter [14]. The rack cutter with a circular-arc tooth proile is cosidered as the geeratig tool or the gear geeratio. The equatio o the circular-arc imagiary rack cutter surace P FR ad its uit ormal vector ca be expressed as [15] R ðfþ c ¼ 4 ðl F si w ðfþ ðl F si w ðfþ l F cos w ðfþ þ b F a F ta w ðfþ þ b F a F ta w ðfþ a F Þ cos h 7 F þ R F ð1 cos h F Þ 5 ð1þ Þ si h F þ R F si h F ad ðfþ c ¼ 4 si w ðfþ cos w ðfþ cos w ðfþ 7 cos h F 5; ðþ si h F where parameters a F, b F, w ðfþ ad R F are the desig parameters o the rack cutter, ad l F ad h F are surace coordiates o the rack cutter. Accordig to the geeratio mechaism described i a previous study [15], the geerated piio tooth surace ca be obtaied as ollows:

4 908 R.-T. Tseg, C.-B. Tsay / Mechaism ad Machie Theory 9 (004) ðl F ; h F ; / 1 Þ¼½l F a F ðcos w ðfþ þ ta w ðfþ si w ðfþ Þþb F si w ðfþ Š cos h F þðr F ð1 cos h F Þ r 1 / 1 Þ si w ðfþ ¼ 0; ðþ cos / 1 si / 1 0 r 1 ðcos / 1 þ / 1 si / 1 Þ R ðfþ si / 1 ¼ 1 cos / 1 0 r 1 ðsi / 1 / 1 cos / 1 Þ RðFÞ c : ð4þ The uit ormal vector o the geerated piio tooth surace ca also be obtaied by cos / 1 si w ðfþ ðfþ þ si / 1 cos w ðfþ cos h F 1 ¼ si / 1 si w ðfþ cos / 1 cos w ðfþ 7 4 cos h F 5: ð5þ cos w ðfþ si h F Eq. () is termed the equatio o meshig ad Eq. (4) represets the locus o the rack cutter suraces. The mathematical model o the let-side curviliear piio tooth surace P 1L ca be obtaied by cosiderig Eqs. () ad (4), simultaeously. Similarly, the gear tooth suraces ca be geerated by usig the same rack cuter or aother rack cutter, e.g. P P. The circular-arc imagiary rack cutter surace P PL used to produce the right-side curviliear-tooth gears ad its uit ormal vector are expressed as l P cos w ðpþ R ðpþ þ a P c ¼ ðl P si w ðpþ þ b P a P ta w ðpþ Þ cos h 7 4 P þ R P ð1 cos h P Þ 5 ðþ ðl P si w ðpþ þ b P a P ta w ðpþ Þ si h P þ R P si h P ad ðpþ c ¼ 4 si w ðpþ cos w ðpþ cos w ðpþ cos h P si h P 7 5: ð7þ Thereore, the mathematical model o the right-side o the curviliear gear tooth surace P R ca also be developed by the same process ad represeted as ollows: ðl P ; h P ; / Þ¼ ½l P a P ðcos w ðpþ R ðpþ ¼ 4 þ ta w ðpþ si w ðpþ Þþb P si w ðpþ Š cos h P þðr P ð1 cos h P Þ r / Þ si w ðpþ ¼ 0; ð8þ cos / si / 0 r ðcos / þ / si / Þ si / cos / 0 r ðsi / / cos / Þ RðPÞ c : ð9þ The uit ormal vector o the geerated gear tooth surace ca be obtaied by cos / si w ðpþ ðpþ þ si / cos w ðpþ cos h P ¼ si / si w ðpþ þ cos / cos w ðpþ 7 4 cos h P 5: ð10þ cos w ðpþ si h P

5 R.-T. Tseg, C.-B. Tsay / Mechaism ad Machie Theory 9 (004) Substitutig Eq. (8) ito Eq. (9) yields the mathematical model o the right-side curviliear gear tooth surace P R, ad Eq. (10) expresses its uit ormal vector represeted i coordiate system S ðx ; Y ; Z Þ.. Simulatio o gear meshig ad tooth cotact aalysis The model or gear meshig with assembly errors ca be simulated by chagig the settigs ad orietatios o the coordiate systems S h ðx h ; Y h ; Z h Þ ad S v ðx v ; Y v ; Z v Þ with respect to the ixed coordiate system S ðx ; Y ; Z Þ as show i Fig. 1, where coordiate systems S 1 ðx 1 ; Y 1 ; Z 1 Þ ad S ðx ; Y ; Z Þ are attached to the piio ad gear, respectively. The axes Z 1 ad Z are rotatioal axes o the piio ad gear, respectively. Coordiate systems S v ðx v ; Y v ; Z v Þ ad S h ðx h ; Y h ; Z h Þ are the reerece coordiate systems or the misaliged gear assembly simulatios. The simulatio o horizotal axial misaligmet o the gear may be achieved by rotatig the coordiate system S h ðx h ; Y h ; Z h Þ about the X h axis through a agle Dc h with respect to the coordiate system S ðx ; Y ; Z Þ. Similarly, simulatio o vertical axial misaligmet o the gear ca be perormed by rotatig the coordiate system S v ðx v ; Y v ; Z v Þ about the Y h axis through a agle Dc v with respect to coordiate system S h ðx h ; Y h ; Z h Þ. Symbol C 0 represets the operatioal ceter distace o the gear pair rotatioal axes, ad DC deotes the ceter distace variatio. The origi O o the coordiate system S ðx ; Y ; Z Þ may be displaced by a amout o DC ad the operatioal ceter distace o the meshig gear pair ca be represeted by C 0 ¼ C þ DC with respect to origi O o the ixed coordiate system S ðx ; Y ; Z Þ. / 0 1 ad /0 represet the rotatioal agles o the piio ad gear, respectively, whe they are meshed with each other. Whe applyig the tooth cotact aalysis method to calculate the trasmissio errors o the curviliear gear pair, the uit ormal ad Fig. 1. Simulatio o gear meshig with assembly errors.

6 910 R.-T. Tseg, C.-B. Tsay / Mechaism ad Machie Theory 9 (004) positio vectors o both piio ad gear tooth suraces should be represeted i the same coordiate system, say S ðx ; Y ; Z Þ. Owig to the tagecy o two cotactig gear tooth suraces, the positio vectors o the piio ad gear tooth suraces should be the same at the cotact poit ad their uit ormal vectors should be colliear to each other. Thereore, the ollowig equatios must hold [4,5]: ad R ð1þ ð1þ R ðþ ¼ 0 ð11þ ðþ ¼ 0; ð1þ where R ð1þ ad R ðþ idicate the positio vectors o the tooth suraces o piio ad gear, respectively, represeted i coordiate system S ðx ; Y ; Z Þ. ð1þ ad ðþ express the surace uit ormal vectors o piio ad gear, respectively, represeted i coordiate system S ðx ; Y ; Z Þ. Eq. (11) idicates that the piio ad gear tooth suraces have a commo cotact poit, ad Eq. (1) idicates that the uit ormal vectors o the piio ad gear suraces are colliear at their cotact poit. Sice j ð1þ j¼j ðþ j¼1, Eqs. (11) ad (1) yield a system o ive idepedet equatios with six ukows: l F, h F, l P, h P, / 0 1 ad /0, where l F ad h F are tooth surace parameters o the piio while l P ad h P are tooth surace parameters o gear. I the piio is a drivig gear, the rotatio agle / 0 1 is cosidered a kow parameter (i.e. a give value). Thereore, ive ukow parameters are solved with ive oliear equatios. The trasmissio errors o the curviliear gear pair ca be calculated by usig the ollowig equatio: D/ 0 ð/0 1 Þ¼/0 ð/0 1 Þ T 1 / 0 1 T ; ð1þ where T 1 ad T deote the tooth umbers o piio ad gear, respectively. / 0 ð/0 1Þ, which represets the actual rotatioal agle o the gear pair meshig uder dieret assembly coditios, is solved by umerical method. D/ 0 ð/0 1Þ expresses the trasmissio error o the curviliear gear pair uder the give assembly errors. 4. Cotact ellipses Whe tooth suraces are meshed with each other, their istataeous cotact poit is spread over a elliptical area owig to elastic deormatio. Fig. (a) shows that two matig suraces P 1 ad P are tagetial to each other at the istataeous cotact poit O T, where represets the commo uit ormal vector at cotact poit O T, which ca be determied by the TCA computer simulatio program. Plae T deotes the commo taget plae o the two matig suraces. The origi o coordiate system S T ðx T ; Y T ; Z T Þ ad the istataeous cotact poit O T are coicidet. The directio o axis Z T is deied to coicide with the commo uit ormal vector. Thus, the plae X T Y T is the commo taget plae. I this study, the cotact ellipses o the gear pair are obtaied by usig the surace separatio topology method. To calculate the separatio distace o two matig tooth suraces, surace coordiates o the matig curviliear-tooth gear ad piio must be trasormed to the same

7 R.-T. Tseg, C.-B. Tsay / Mechaism ad Machie Theory 9 (004) Fig.. (a) Commo taget plae ad polar coordiates. (b) Measuremets o surace separatio. coordiate P system S T ðx T ; Y T ; Z T Þ. Fig. (b) shows the separatio distaces betwee the suraces 1 ad P at poit P o the taget plae T. The separatio distace o two matig suraces ca be deied by d 1 þ d, where d 1 þ d is equal to jz ð1þ T ZðÞ T j, measured rom ay poit P o the taget plae T alog its perpedicular directio. Z ð1þ T ad ZðÞ T represet the coordiates o the Z T compoet o poits P 1 ad P, respectively. The equal distace-separatio lie or two matig suraces, which is oud by deiig a auxiliary polar coordiate system ðr; hþ, ca be represeted by the ollowig system o oliear equatios: ad X ð1þ T Y ð1þ T ¼ X ðþ T ; ð14þ ¼ Y ðþ T taðhþ ¼ Y ð1þ T X ð1þ T jz ð1þ T ZðÞ T ð15þ ð1þ j¼0:00 mm; ð17þ where X ð1þ T, Y ð1þ T, X ðþ T ad Y ðþ T are coordiates o poits P 1 ad P o the taget plae X T Y T, respectively. Eqs. (14) (17) cosist o a system o our idepedet oliear equatios with ive ukows: l F, h F, l P, h P ad h. Here l F ad h F are tooth surace parameters o the piio while l P ad h P deote tooth surace parameters o the gear. Parameter h represets the agle measured rom axis X T to axis r o the commo taget plae T. The thickess o the coatig pait used or cotact patter tests is 0.00 mm. Parameter h ca be cosidered a kow parameter with a

8 91 R.-T. Tseg, C.-B. Tsay / Mechaism ad Machie Theory 9 (004) rage o p to p. Thereore, the our ukow parameters are solved with our equatios by a umerical method. Thus, the dimesio ad shape o cotact ellipses ca be oud. 5. Numerical examples or gear meshig simulatios Example 1. The gear parameters are give as ollows: teeth umbers T 1 ¼ 18, T ¼, ormal pressure agles w ðfþ ¼ w ðpþ ¼ 0, ormal module M ¼ mm, ace width ¼ 0 mm, omial radius o ace mill-cutter R i ¼ 0 mm ði ¼ F; PÞ. I two matig gears are meshig uder ideal coditios, it meas that the misaliged agles Dc h ¼ Dc v ¼ 0:0, ad DC ¼ 0:0 mm. The bearig cotacts ad trasmissio errors or ideal meshig coditio are show i Table 1. Table 1 demostrates that the curviliear-tooth gear pair does ot iduce trasmissio errors uder ideal coditio. The parameters h F ¼ h P ¼ 0:0 show i Table 1 idicates that the cotact poits are distributed over the middle regio o the tooth lak whe the gear pair is meshed uder ideal assembly coditios. Fig. depicts the cotact path ad the cotact ellipses o the piio tooth surace uder ideal assembly coditios. The cotact ellipses are plotted whe the piio is rotated every 4 orm ) to 18. Three dieret values o the omial radius o the ace mill-cutter R i, amely 0, 50 ad 100 mm, were chose to demostrate the eects o R i o the correspodig cotact ellipses. Fig. also reveals that the legth o the major axis o the cotact ellipses is proportioal to the radius o the ace mill-cutter R i. Fig. 4 shows the relatioship betwee the radius o ace mill-cutter R i ad the ratio o the major ad mior axes o the cotact ellipse a=b is i liear. Whe the radius o a ace mill-cutter is 0, 50 ad 100 mm, respectively, the ratio a=b is.14, 9.9 ad 19.9, as show i Fig. 4. Example. Table lists the aalysis results o bearig cotacts ad trasmissio errors usig the same gear desig parameters as those give i Example 1 with Dc h ¼ Dc v ¼ 0:0, ad DC ¼ 0:5 mm (0.%). The cotact poits o the curviliear-tooth gear pair are dislocated but the matig gear pair does ot iduce trasmissio errors at all uder ceter distace variatio. Table illustrates that the cotact poits are still distributed over the middle regio o the tooth lak. Table 1 Trasmissio errors ad bearig cotacts or ideal meshig coditio / 0 1 (deg) /0 (deg) h F (deg) h P (deg) l F (mm) l P (mm) TE (arc-sec.) )10.00 ) )8.00 ) ).00 ) )4.00 ) ).00 )

9 R.-T. Tseg, C.-B. Tsay / Mechaism ad Machie Theory 9 (004) Fig.. Cotact path ad cotact ellipses o the piio tooth surace uder ideal assembly coditio. Fig. 4. Relatioship betwee the ratio a=b ad omial radius o ace mill-cuter R i. The cotact characteristic o the curviliear-tooth gears is similar to that o spur gears whe two spur gears are meshig with ceter distace variatio.

10 914 R.-T. Tseg, C.-B. Tsay / Mechaism ad Machie Theory 9 (004) Table Trasmissio errors ad bearig cotacts due to ceter distace variatio DC ¼ 0:5mm / 0 1 (deg) /0 (deg) h F (deg) h P (deg) l F (mm) l P (mm) TE (arc-sec.) )10.00 ) )8.00 ) ).00 ) )4.00 ) ).00 ) Example. The horizotal axial misaligmet o the gear Dc h ¼ 0:1 is cosidered whe two matig curviliear-tooth gears are meshed. The gear desig parameters are the same as those give i Example 1. The bearig cotacts ad trasmissio errors o the gear pair uder the prescribed horizotal axial misaligmet are give i Table. I this case the cotact poits are distributed ear the middle sectio o the tooth lak so that the edge cotact will ot occur. Sice the cotact type o the curviliear-tooth gear pair proposed here is a poit cotact, the trasmissio errors o this type o gear pairs are ot sesitive to axial misaligmets o the gear. Whe the curviliear-tooth gear pair is meshed with horizotal axial misaligmets Dc h ¼ 0:1 ad Dc h ¼ 0:1, the bearig cotacts ad cotact ellipses are depicted i Fig. 5(a) ad (b), respectively. It is oud that the bearig cotacts are localized ear the middle regio o the tooth lak. Example 4. The gear desig parameters are chose the same as those i Example 1. The gear pair with vertical axial misaligmet Dc v ¼ 0:1 is cosidered whe two matig curviliear-tooth gears are meshed. The bearig cotacts ad trasmissio errors o the gear pair uder the prescribed Table Trasmissio errors ad bearig cotacts due to horizotal axial misaligmet Dc h ¼ 0:1 / 0 1 (deg) /0 (deg) h F (deg) h P (deg) l F (mm) l P (mm) TE (arc-sec.) )10.00 ) )0.8 ) )8.00 ) )0.81 ) ).00 ).9999 )0.79 ) )4.00 ) )0.77 ) ).00 ) )0.75 ) )0.7 ) )0.71 ) ) )0.9 ) ) )0.7 ) ) )0.5 ) ) )0. ) )0.49

11 R.-T. Tseg, C.-B. Tsay / Mechaism ad Machie Theory 9 (004) Fig. 5. Cotact path ad cotact ellipses o the piio tooth surace uder horizotal axial misaligmets. vertical axial misaligmet are listed i Table 4. I this case the cotact poits are also distributed ear the middle sectio o the tooth lak. Sice the curviliear-tooth gear pair proposed here is i poit cotact, the trasmissio error o the gear pair is ot sesitive to axial misaligmets o the gear. The trasmissio errors o the curviliear-tooth gears iduced by horizotal axial misaligmets are larger tha those iduced by vertical axial misaligmets. Fig. illustrates the bearig cotacts ad cotact ellipses o the curviliear-tooth gear pair o piio tooth suraces uder vertical axial misaligmets Dc v ¼þ0:1 ad Dc v ¼ 0:1, respectively. It is oud that the bearig cotacts are also localized ear the middle regio o the tooth lak. The desig parameters o the helical gear, such as tooth umber, ormal pressure agle, ormal module ad ace width, are the same as those give i Example 1 ad helix agle is 15. Table 5 illustrates the bearig cotacts ad trasmissio errors o the helical gear pair meshed uder Table 4 Trasmissio errors ad bearig cotacts due to vertical axial misaligmet Dc v ¼ 0:1 / 0 1 (deg) /0 (deg) h F (deg) h P (deg) l F (mm) l P (mm) TE (arc-sec.) )10.00 ) )0.8 ) )0.00 )8.00 ) )0.75 ) )0.049 ).00 ) )0.8 ) )0.07 )4.00 ) )0.90 ) )0.05 ).00 ) )0.97 ) ) )0.04 ) )0.1 ) )0.19 ) )0. ) )0. ) )0.41 )

12 91 R.-T. Tseg, C.-B. Tsay / Mechaism ad Machie Theory 9 (004) Fig.. Cotact path ad cotact ellipses o the piio tooth surace uder vertical axial misaligmets. Table 5 Trasmissio errors ad bearig cotact due to vertical axial misaligmet Dc v ¼ 0:1 or the helical gear pair / 0 1 (deg) /0 (deg) Z 1 (mm) Z (mm) l F (mm) l P (mm) TE (arc-sec.) ) )4.999 )14.95 ) )8.000 ).9998 ) ) ).000 ) ) ) )4.000 ) ) ) ).000 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).77 vertical axial misaligmet Dc v ¼ 0:1. The Z 1 ad Z coordiates show i Table 5 represet the locatio o cotact poits o tooth suraces. It is oud that o the edge o gear tooth surace, say Z ¼ 15 mm, ad the tooth surace o piio are i cotiuous tagecy uder vertical axial misaligmets. Furthermore, the trasmissio errors o the helical gear are larger tha that o the curviliear-tooth gear with the same meshig coditio o axial misaligmets. The gear desig parameters are also chose the same as those o i Example 1. Whe two matig gears are meshed uder Dc h ¼ 0:1, Dc h ¼ 0:15, Dc v ¼ 0:15, Dc v ¼ 0:, ad DC ¼0:5 mm, respectively, the relatioship betwee the omial radius o the ace mill-cutter ad trasmissio errors o the curviliear-tooth gear is show i Fig. 7(a). Lie A deotes that the matig gear pair does ot iduce trasmissio errors uder ceter distace variatios. It is oud that a larger R i results i a larger TE o the gear pair uder the same axial misaligmet. Whe the

13 R.-T. Tseg, C.-B. Tsay / Mechaism ad Machie Theory 9 (004) Fig. 7. (a) ad (b) Relatioship o trasmissio errors ad omial radius o the ace mill-cutter R i uder dieret assembly coditios. curviliear-tooth gears geerated by the radius o ace mill-cutter R i ¼ 71:8 mm are meshed with vertical axial misaligmets, the trasmissio error o the gear pair approaches to zero. Fig. 7(b) idicates that dieret assembly coditios result i dieret relatioship o R i ad TE. Lie B idicates that the gear pair with the radius o ace mill-cutter R i ¼ 5:7 mm receives a zero TE uder assembly coditio B. The ratio o major ad mior axes o the cotact ellipse, a=b, as show i Fig. 4 is 7. whe the radius o ace mill-cutter is 5.7 mm. Accordig to Fig. 4, i choosig the ratio a=b less tha 10, the the largest radius o ace millcutter becomes 50.8 mm. Whe the assembly coditio o the gear pair is the same as that o Fig. 7(b) ad choosig R i less tha 0 mm, the TE o the gear pair is less tha Owig to the limit o a ace mill-cutter to geerate curviliear-tooth gears, it is clear that the diameter o the ace millcutter must be larger tha the tooth ace width. Thereore, R i must be greater tha hal o the tooth ace width.. Coclusios Based o the mathematical model o curviliear-tooth gears, the trasmissio errors ad bearig cotacts o the gear set were ivestigated by usig the TCA computer simulatio programs. The cotact ellipses o the curviliear-tooth gear pair uder dieret assembly coditios ca be estimated by utilizig the surace topology method. Accordig to the umerical results, the ollowig coclusios ca be draw:

14 918 R.-T. Tseg, C.-B. Tsay / Mechaism ad Machie Theory 9 (004) (1) The bearig cotacts are localized i the middle regio o the tooth lak ad o trasmissio errors are iduced whe a gear pair is meshed uder ideal assembly coditio. () The curviliear-tooth gear pair will ot iduce trasmissio errors while the gear pair is meshed with ceter distace variatios. () Sice the curviliear-tooth gear proposed i this paper is i poit cotact, the gear set is ot sesitive to axial misaligmets o the gear, ad the trasmissio errors are small whe the gear pair is meshed uder error assembly coditios. Tooth edge cotact ca be avoided because the cotact paths are located ear the middle regio o the tooth lak. (4) Whe the curviliear-tooth gear pair is meshed uder a o-ideal operatig coditio, or example uder axial misaligmets or variatios o ceter distaces, the dimesio o the cotact ellipses is almost the same as that o the gear pair meshig uder ideal assembly coditios. (5) The most importat desig parameter which ilueces the dimesios o gear cotact ellipses is the omial radius o ace mill-cutter. The legth o the major axis o the cotact ellipses is directly proportioal to the radius o ace mill-cutter. Ackowledgemets The authors are grateul to the Natioal Sciece Coucil o the ROC or the grat. Part o this work was perormed uder cotract No. NSC 91-1-E Reereces [1] F.L. Litvi, J. Zhag, R.F. Hadschuh, Crowed spur gears: methods or geeratio ad tooth cotact aalysis Part 1: Basic cocepts, geeratio o the piio tooth surace by a surace by a plae, ASME Joural o Mechaisms, Trasmissios ad Automatio i Desig 110 (1988) 7 4. [] F.L. Litvi, J. Zhag, R.F. Hadschuh, Crowed spur gears: methods or geeratio ad tooth cotact aalysis Part : Geeratio o the piio tooth surace by a surace o revolutio, ASME Joural o Mechaisms, Trasmissios ad Automatio i Desig 110 (1988) [] F.L. Litvi, C.B. Tsay, Helical gears with circular arc teeth: simulatio o coditios o meshig ad bearig cotact, ASME Joural o Mechaisms, Trasmissio, ad Automatio i Desig 107 (1985) [4] F.L. Litvi, Theory o Gearig, NASA Reerece Publicatio 11, Washigto, DC, [5] F.L. Litvi, Gear Geometry ad Applied Theory, PTR Pretice Hall, Eglewood Clis, NJ, [] C.B. Tsay, Helical gears with ivolute shaped teeth: geometry, computer simulatio, tooth cotact aalysis, ad stress aalysis, ASME Joural o Mechaisms, Trasmissios ad Automatio i Desig 110 (1988) [7] C.Y. Li, C.B. Tsay, Z.H. Fog, Tooth cotact aalysis o hypoid gears, Joural o the Chiese Society o Mechaical Egieers 17 () (199) [8] S.L. Chag, C.B. Tsay, C.H. Tseg, Kiematic optimizatio o a modiied helical gear trai, ASME Joural o Mechaical Desig 119 () (1997) [9] F.L. Litvi, J. Zhag, R.F. Hadschuh, J.J. Coy, Topology o modiied helical gears, Surace Topography (March) (1989) [10] F.L. Litvi, N.X. Che, C.L. Hsiao, R.F. Hadschuh, Geeratio o helical gears with ew suraces topology by applicatio o CNC machies, Gear Techology (Jauary/February) (1994) 0. [11] Y. Zhag, Z. Fag, Aalysis o trasmissio errors uder load o helical gears with modiied tooth gears, ASME Joural o Mechaical Desig 119 (1997) 10 1.

15 R.-T. Tseg, C.-B. Tsay / Mechaism ad Machie Theory 9 (004) [1] M. Umeyama, M. Kato, K. Ioue, Eects o gear dimesios ad tooth surace modiicatios o the loaded trasmissio error o a helical gear pair, ASME Joural o Mechaical Desig 10 (1998) [1] W.K. Jaick, Cotact surace topology o worm gear teeth, Gear Techology (March/April) (1988) [14] S.T. Liu, Curviliear cylidrical gears, Gear Techology (May/Jue) (1988) 8 1. [15] R.T. Tseg, C.B. Tsay, Mathematical model ad udercuttig o cylidrical gears with curviliear shaped teeth, Mechaism ad Machie Theory (001)