Contact characteristics of beveloid gears

Transcription

1 Mehaism ad Mahie Theory 7 (2002) 50 Cotat harateristis o beveloid gears Chia-Chag Liu, Chug-Biau Tsay * Departmet o Mehaial Egieerig, Natioal Chiao Tug Uiversity, Hsihu, 0010, Taiwa Reeived 19 Otober 2000;aepted 4 November 2001 Abstrat This study ivestigates the otat harateristis o beveloid gear pairs with iterseted, rossed ad parallel axes. Tooth otat aalysis (TCA) is perormed to examie the meshig ad bearig otat o the beveloid gear pairs omposed o a piio ad a gear. additio, the priipal diretios ad urvatures o the piio ad gear suraes are ivestigated ad the otat ellipses o the matig tooth suraes are also studied. Numerial illustrative examples are provided to demostrate the omputatioal results. Ó 2002 Elsevier Siee Ltd. All rights reserved. 1. trodutio Beveloid gears, also kow as oial ivolute gears, are ivolute gears with tapered tooth thikess, root ad outside diameters. This type o gears a trasmit rotatioal motio betwee iterseted, rossed ad parallel axes i ay relative positio. Owig to its tapered tooth thikess, the beveloid gear a also be used as the baklash otrol gears. Moreover, uder o-parallel axes meshig, beveloid gears are ot sesitive to assembly errors. The meshig o beveloid gears has seldom bee studied, ad Mitome [1 4] has oduted most o the researhes i this area. The tooth atio o a beveloid gear pair was ivestigated both theoretially ad experimetally by Mitome [1]. His ivestigatio veriied that beveloid gear pairs trasmit a uiorm rotatioal motio betwee iterseted axes eve whe assembly errors exist. Mitome [2] also proposed a desig or the rossed axes beveloid gears ad established their egagemet models. additio, the oept o ieed gridig whih a elarge the otat patters o straight beveloid gears with iterseted axes was proposed []. More reetly, Mitome [4] also proposed a ovel desig or miter beveloid gears, whih eables the tooth bearig otat loated at the middle regio o the tooth width. * Correspodig author. Tel.: ;ax: address: btsay@.tu.edu.tw (C.-B. Tsay) X/02/$ - see rot matter Ó 2002 Elsevier Siee Ltd. All rights reserved. P: S X(01)

2 4 C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) 50 Although the above ivestigatios otributed sigiiatly to the meshig o beveloid gears, they ailed to establish a omplete simulatio model or tooth otat aalysis (TCA). The TCA method was proposed by Litvi [5,6], ad it has bee applied to simulate the meshig o various types o gear drives. The TCA results provide valuable iormatio o the otat poits, otat lies, lies o atio ad trasmissio error (TE) o the matig gear pair durig their meshig proess. This study ivestigates the meshig harateristis o beveloid gear pairs with iterseted, rossed ad parallel axes. Theoretially, the bearig otat o a beveloid gear pair uder oparallel axes meshig is a poit otat. Due to the elastiity, the gear tooth otat is spread over a elliptial area aroud the istataeous otat poit. The diretio ad dimesio o the otat ellipse provide urther otat ad lubriatio harateristis to the gear meshig. this work, methodologies proposed by Litvi [5,6] osiderig the priipal diretios ad urvatures o matig suraes was adopted to obtai the otat ellipses o the o-parallel axes beveloid gear pairs. The umerial illustrative examples aurately relet the otat ature o a beveloid gear pair uder several assembly oditios. 2. Mathematial model o beveloid gear pairs Aordig to Merritt s geeratio oept [7], a beveloid gear a be geerated by a basi rak whose pith plae itersets with the axis o the gear, ad orms a agle that equals the geeratig oe agle. idustrial appliatios, the most ovetioal meas o beveloid gear mauaturig method is the taper hobbig proposed by Mitome [8,9]. Aordig to Mitome s results, the evelope geerated by the hob i the spae a be osidered a rak utter. this work, the rak utter is used to simulate the geeratig proess o beveloid gears Mathematial model o rak utter The beveloid gear pair or the meshig simulatio omprises a piio ad a gear. Assume that the utter surae R F geerates the piio tooth surae R 1, ad the utter surae R G geerates the gear tooth surae R 2. Notably, subsripts i ¼ 1 ad 2, ad j ¼ F ad G represet the suraes o piio R 1 ad gear R 2 ad their orrespodig utters R F ad R G, respetively, i the ollowig derivatio. Aordig to Fig. 1, the ormal setio o the rak utter osists maily o two Fig. 1. The ormal setio o rak utter.

3 straight edges. The straight edge M ðjþ 0 M ðjþ 2 a be represeted i oordiate system ðjþ ðx ; Y ðjþ; ZðjÞÞ by S ðjþ x ðjþ y ðjþ ¼ j os a ðjþ a j ; ¼ j si a ðjþ a j ta a ðjþ b j ; z ðjþ ¼ 0; ƒƒƒƒƒ! where desig parameter j ¼jM ðjþ 0 M ðjþ 1 j represets the distae measured rom the iitial poit M ðjþ ðjþ 0, movig alog the straight lie M0 M ðjþ ðjþ 2, to poit M1 ; aðjþ deotes the ormal pressure agle, ad symbols P ad p represet the gear diametral pith ad irular pith, respetively. To obtai the rak utter surae or beveloid gear geeratio, the above-metioed ormal setio o the rak utter, attahed to plae X ðjþ ðjþ Y, is traslated alog the lie OðjÞ r O ðjþ with ƒƒƒƒ! respet to the oordiate system Sr ðjþ ðxr ðjþ ; Yr ðjþ ; Zr ðjþ Þ, as illustrated i Fig. 2. Herei, u j ¼jO ðjþ r O ðjþ j is also a desig parameter o the rak utter surae. Thus, the proile o the rak utter a be traed out i oordiate system Sr ðjþ, ad plae Yr ðjþ Zr ðjþ a be regarded as the pith plae o the rak utter. The agle b j, whih determies the diretio o tooth trae, is the helix agle o tooth trae o the pith plae o rak utter. To simulate the taper hobbig proess, the pith plae o the rak utter is set to have a iliatio agle d i with respet to the plae axode oordiate system S ðjþ ðjþ ðx ; Y ðjþ ; Z ðjþ Þ, ad the rak utter surae a be represeted i oordiate system SðjÞ as ollows: x ðjþ ¼ j os a ðjþ a j os di þ j si a ðjþ y ðjþ ¼ j si a ðjþ a j ta a ðjþ b j os bj þ u j si b j ; z ðjþ ¼ j os a ðjþ a j si di þ j si a ðjþ a j ta a ðjþ b j si bj þ u j os b j si di ; a j ta a ðjþ b j si bj þ u j os b j os di : Owig to that the surae oordiates o rak utter are j ad u j, the uit ormal to the rak utter surae a be attaied by ðjþ ¼ N ðjþ N ðjþ C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) 50 5 ðj ¼ F ; GÞ; ð1þ ð2þ ðþ Fig. 2. Relatios amog oordiate systems S ðjþ, SðjÞ r, S ðjþ.

4 6 C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) 50 where N ðjþ ¼ orðjþ ou j orðjþ : ð4þ o j Herei, R ðjþ deotes the positio vetor o the rak utter surae represeted i plae axode oordiate system S ðjþ. Eqs. (2) ad () result i the orrespodig uit ormals to the rak utter surae as ollows: ðjþ x ¼ si d i si b j os a ðjþ os d i si a ðjþ ; ðjþ y ðjþ z ¼ os b j os a ðjþ ; ¼ os d i si b j os a ðjþ þ si d i si a ðjþ : 2.2. Mathematial model o beveloid gear Fig. illustrates a shemati geeratio mehaism ad the oordiate relatioship betwee the rak utter ad the geerated gear. Herei, S ðjþ ðjþ b ðxb ; Y ðjþ b ; ZðjÞ b Þ is the ixed oordiate system, S i ðx i ; Y i ; Z i Þ is the oordiate system attahed to the geerated piio R 1 ði ¼ 1Þ or gear R 2 ði ¼ 2Þ, ad S ðjþ is the plae axode oordiate system attahed to the rak utter. the gear geeratio proess, the gear blak rotates with agular veloity x i while the rak utter traslates with veloity V ¼ x i r i. The plae axode ad the gear axode roll over eah other without slidig o the istataeous axis o rotatio. Based o the theory o gearig [5,6], the mathematial model o the geerated beveloid gear tooth suraes a be attaied i oordiate system S i as ollows: x i ¼ x ðjþ os / i y ðjþ si / i þ r i ðos / i þ / i si / i Þ; y i ¼ x ðjþ si / i þ y ðjþ os / i þ r i ðsi / i / i os / i Þ; ð6þ z i ¼ z ðjþ ; where. / i ¼ y ðjþ ðjþ x x ðjþ ðjþ y r i ðjþ x : ð7þ ð5þ Fig.. Coordiate relatioship betwee the rak utter ad geerated gear.

5 C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) 50 7 Herei, i ¼ 1 ad 2, ad j ¼ F ad G represet the suraes o piio R 1 ad gear R 2 ad their orrespodig utters R F ad R G, respetively; r i deotes the pith radius o the geerated gear ad / i represets the gear rotatio agle i the geeratig proess. Eq. (7) is the geeral orm o the equatio o meshig whih a be obtaied by applyig the oept that the ormal vetor to ay poit o the geerated tooth surae passes through the istataeous axis o gear rotatio [5,6]. Substitutig Eqs. (2) ad (5) ito Eqs. (6) ad (7) yields the mathematial models o beveloid piio R 1 (i ¼ 1 ad j ¼ F ) ad gear R 2 (i ¼ 2 ad j ¼ G) expressed i oordiate systems S 1 ðx 1 ; Y 1 ; Z 1 Þ ad S 2 ðx 2 ; Y 2 ; Z 2 Þ, respetively. The surae uit ormal o the geerated beveloid gear tooth suraes a also be obtaied by ix ¼ ðjþ x os / i ðjþ y si / i; iy ¼ ðjþ x si / i þ ðjþ y os / i; iz ¼ ðjþ z : ð8þ. Meshig model ad tooth otat aalysis Adoptig the egagemet model proposed by Mitome [2], Fig. 4 illustrates the shemati meshig model o the beveloid gear pair, iludig the piio R 1, the gear R 2, ad the pith plae o the imagiary egagig rak. Beveloid piio ad gear a be osidered two imagiary oes with oe agles d 1 ad d 2, lyig o opposite sides o the pith plae o the imagiary egagig rak, while S ðx ; Y ; Z Þ ad S g ðx g ; Y g ; Z g Þ are the reeree oordiate systems or the piio oordiate system S 1 ðx 1 ; Y 1 ; Z 1 Þ ad gear oordiate system S 2 ðx 2 ; Y 2 ; Z 2 Þ, respetively. Furthermore, / 0 1 ad /0 2 deote the rotatio agles o the piio ad gear durig meshig. Meawhile, poits O ad O g are the eters o the pith irles o the piio ad gear, where r 1 ad r 2 deote Fig. 4. Shemati relatioship or the meshig o piio, gear ad the pith plae o the imagiary egagig rak.

6 8 C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) 50 their respetive pith radii, ad P ad P g are the apexes o the imagiary oes; P O ad P g O g are the axes o the imagiary oes. The pith irles o the piio ad gear are i tagey at poit Q. additio, the taget lies o the two imagiary oes with respet to the pith plae o the imagiary egagig rak, P Q ad P g Q, orm a agle C ¼ b F þ b G (or C ¼ b F þ b G þ p). Herei, b F ad b G deote the helix agle o the pith plae o the rak utter i the geeratio o the piio R 1 ad the gear R 2, respetively. The preedig meshig model o the beveloid gear pair is desiged to simulate the meshig o a beveloid gear pair with rossed axes. Cosider that whe the agle C ¼ 0orC¼p, the axes o two imagiary oes iterset ad orm a iterseted agle o d 1 þ d 2 (whe C ¼ 0) or jd 1 d 2 j (whe C ¼ p). Thus, the meshig model regresses to the ase whih a simulate the meshig o beveloid gears with iterseted axes. Furthermore, whe the iterseted agle beomes zero, the axes o the two imagiary oes beome parallel, ad the meshig model regresses agai to the ase whih a simulate the meshig o beveloid gears with parallel axes. To ivestigate the meshig o beveloid gear pair with assembly errors, auxiliary oordiate systems S e ðx e ; Y e ; Z e Þ, S h ðx h ; Y h ; Z h Þ ad S v ðx v ; Y v ; Z v Þ have bee set up to simulate the assembly errors o the gear R 2 as depited i Fig. 5. Coordiate system S e is set up ad keeps its orietatio with respet to oordiate system S g. However, the origi O e o oordiate system S e has a oset deviatio rom the origi O g o oordiate system S g. The oset O g O e ¼ Dd ¼ðDx g ; Dy g ; Dz g Þ idiates the moutig positio deviatio o the gear R 2. Moreover, oordiate system S h simulates the gear R 2 havig a horizotal misaliged agle D h with respet to oordiate system S e, while oordiate system S v simulates the gear R 2 havig a vertial misaliged agle D v with respet to oordiate system S h. Hee, oordiate system S v is the reeree oordiate system or the gear oordiate system S 2 whe assembly errors Dd; D h ad D v exist. Applyig the oordiate trasormatio matrix equatio, the positio vetors ad uit ormal vetors o the piio R 1 ad the gear R 2 a be represeted i the oordiate system S. The matig piio ad gear must satisy the ollowig oditios at their istataeous otat poit [5,6]: Fig. 5. Assembly errors o the beveloid gear.

7 C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) 50 9 ad R ð1þ ¼ R ð2þ ; ð9þ ð1þ ¼ ð2þ ; ð10þ where R ð1þ ad R ð2þ are the positio vetors while ð1þ ad ð2þ deote the uit ormal vetors o piio R 1 ad gear R 2, represeted i oordiate system S, respetively. a three-dimesioal spae, Eqs. (9) ad (10) orm a system o ive idepedet o-liear equatios with six ukows / 0 1, F, u F, / 0 2, G ad u G. By hoosig the piio rotatioal agle / 0 1 as a iput variable, all other ukows a be solved i terms o / 0 1. The istataeous otat poits o the piio ad gear suraes a be obtaied by substitutig the solved ukows ito the piio ad gear tooth-surae equatios. The deviatio o the real gear rotatioal agle / 0 2 ð/0 1 Þ rom its ideal rotatioal agle is deied as the trasmissio error (TE) whih a be expressed as ollows: TE ¼ / 0 N 1 2 ð/0 1 Þ /0 1 ; N 2 ð11þ where N 1 ad N 2 deote the umber o teeth o the piio ad gear, respetively. 4. Curvature aalysis ad otat ellipses this study, the methodologies proposed by Litvi [5,6] evaluatig the priipal diretios ad urvatures o the geerated suraes i terms o their orrespodig geeratig tool suraes are adopted to obtai the otat ellipses o the matig beveloid gear pairs with oparallel axes. Notably, oly the derivatio proesses o urvature relatioships betwee the utter surae R F ad the geerated piio suraes R 1 are show hereiater. The priipal diretios ad urvatures o the gear suraes R 2 a also be derived as well by the same proedure Priipal diretios ad urvatures o rak utter surae R F The positio vetor ad uit ormal vetor o the utter surae R F are expressed i Eqs. (2) ad (5), respetively. Aordig to Rodrigues equatio, the priipal urvatures o the utter surae R F a be obtaied as [5,6] j ðf Þ ; V ðf Þ r ¼ _ ðf Þ r ; ð12þ is the relative veloity at the otat poit i its motio over the utter surae R F ; _ ðf r Þ is the veloity o the tip o the surae uit ormal i the above motio;ad j ðf Þ ; represets the where V ðf Þ r two priipal urvatures o the surae at the otat poit. Sie u F ad F are the surae parameters o the rak utter surae R F, Eqs. (2), (5) ad (12) are used to alulate the priipal diretios ad urvatures o the rak utter surae R F as ollows:

8 40 C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) 50 are derived as ol- 1. Whe ðd F =dtþ ¼0, the irst priipal diretio i ðf Þ 2 i ðf Þ ¼ VðF Þ si d 1 os b F r V ðf Þ ¼ 4 si b F r os d 1 os b F 2. Whe ðdu F =dtþ ¼0, the seodary priipal diretio i ðf Þ lows: i ðf Þ ¼ VðF Þ r V ðf Þ r 2 ¼ 4 Herei, uit vetors i ðf Þ os d 1 os a ðf Þ si d 1 os a ðf Þ ad i ðf Þ ad urvature j ðf Þ are derived as ollows: 5; ad j ðf Þ ¼ 0: ð1þ si d 1 si b F si a ðf Þ os b F si a ðf Þ os d 1 si b F si a ðf Þ ad urvature j ðf Þ 5; ad j ðf Þ ¼ 0: ð14þ are represeted i the oordiate system S ðf Þ b ðx ðf Þ b ; Y ðf Þ b ; Z ðf Þ b Þ as are both zero sie the ge- show i Fig.. Noted that the priipal urvatures j ðf Þ eratig surae R F is a plae. ad j ðf Þ 4.2. Priipal diretios ad urvatures o the geerated piio tooth surae R 1 The geerated piio tooth surae R 1 ad the utter surae R F are i lie otat ad i otiuous tagey at every istat durig the geeratio proess. Thereore, the priipal diretios ad urvatures o the geerated piio tooth surae R 1 a be determied by applyig the ollowig equatios [5,6]: ad where ta 2r ðf 1Þ ¼ j ð1þ j ð1þ F ð1þ ¼ G ð1þ ¼ S ð1þ ¼ þ j ð1þ j ð1þ j ðf Þ ¼ j ðf Þ ¼ jðf Þ 2F ð1þ b ð1þ þ V ðf 1Þ i ðf Þ b ð1þ þ V ðf 1Þ i ðf Þ b ð1þ þ V ðf 1Þ i ðf Þ h i 1 ¼ ðf Þ x ðf 1Þ i ðf Þ ; ð15þ j ðf Þ þ Gð1Þ þ j ðf Þ þ S ð1þ ; ð16þ j ðf Þ þ G ð1þ ; ð17þ os 2r ðf 1Þ 1 að1þ 2 1 þ V ðf 1Þ i ðf Þ 2 2 ð1þ 1 a 2 1 þ V ðf 1Þ i ðf Þ 2 2 ð1þ 1 þ a 2 1 þ V ðf 1Þ i ðf Þ j ðf Þ V ðf 1Þ i ðf Þ ; ð18þ ; ð19þ ; ð20þ ; ð21þ

9 ad C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) h i 2 ¼ ðf Þ x ðf 1Þ i ðf Þ j ðf Þ V ðf 1Þ i ðf Þ ; ð22þ h i b ð1þ ¼ ðf Þ x ð1þ V ðf Þ tr h i ðf Þ x ðf Þ V ð1þ tr : ð2þ All the vetors i Eqs. (15) (2) are represeted i the oordiate system S ðf Þ b. Herei, i ð1þ ad i ð1þ are the irst ad seod priipal diretios o the geerated piio tooth surae R 1 ; j ð1þ ad j ð1þ represet the irst ad seod priipal urvatures o R 1 ;ad r ðf 1Þ is the agle ormed by the irst priipal diretios o the utter surae i ðf Þ ad the geerated piio surae i ð1þ. t is realled that the vetor x ðf Þ ¼ 0 due to the traslatioal motio o rak utter R F. Thereore, the relative agular veloity betwee the rak utter surae R F ad the geerated piio tooth surae R 1, a be expressed as 2 x ðf 1Þ ¼ x ðf Þ x ð1þ ¼ x ð1þ ¼ x 1 5: ð24þ Meawhile, the traser veloity o the rak utter R F is 2 0 V ðf Þ tr ¼ 4 x 1 r 1 5: ð25þ 0 additio, the positio vetor o the otat poit o the geerated piio tooth surae R 1, represeted i oordiate system S ðf Þ b, a be expressed as 2 x ðf Þ R ð1þ þ r 1 b ¼ 4 y ðf Þ r 1 / 1 5; ð26þ z ðf Þ where x ðf Þ, y ðf Þ ad z ðf Þ are expressed i Eq. (2). Thereore, the traser veloity o otat poit o R 1 is 2 y ðf Þ V ð1þ tr ¼ x ð1þ R ð1þ r 1 / 1 b ¼ x 1 4 x ðf Þ r 1 5: ð27þ 0 Thus, the relative veloity betwee suraes R F ad R 1, represeted i oordiate system S ðf Þ b, a be obtaied as 2 y ðf Þ V ðf 1Þ ¼ V ðf Þ tr V ð1þ þ r 1 / 1 tr ¼ x 4 1 x ðf Þ 5: ð28þ 0 additio, two ommo terms appeared i Eqs. (18) (20) a be obtaied by applyig Eqs. (1), (14) ad (28): V ðf 1Þ i ðf Þ ¼ x 1 y ðf Þ þ r 1 / 1 si d1 os b F þ x ðf Þ si b F ; ð29þ

10 42 C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) 50 ad V ðf 1Þ i ðf Þ ¼ x 1 y ðf Þ þ x ðf Þ Furthermore, sie j ðf Þ ollows: ad þ r 1 / 1 os d1 os a ðf Þ si d 1 si b F si a ðf Þ os b F si a ðf Þ : ð0þ ¼ j ðf Þ h i 1 ¼ ðf Þ x ðf 1Þ i ðf Þ ¼ x 1 si d 1 os a ðf Þ h i 2 ¼ ðf Þ x ðf 1Þ i ðf Þ ¼ x 1 os d 1 os b F ; h i b ð1þ ¼ ðf Þ x ð1þ V ðf Þ tr ¼ r 1 x 2 1 ¼ 0 ad x ðf Þ ¼ 0, oeiiets 1 ; að1þ 2 ad bð1þ a be attaied as si d 1 si b F os a ðf Þ þ x 1 os d 1 si b F si a ðf Þ ; ð1þ þ os d 1 si a ðf Þ : t is obvious that all terms eeded to express F ð1þ, G ð1þ ad S ð1þ have bee derived ompletely. Cosequetly, Eqs. (15) (17) determie the priipal urvatures ad diretios o the piio tooth surae R 1 represeted i oordiate system S ðf Þ b. Notably, the priipal diretios o R 1 a be represeted i oordiate system S 1 by applyig the oordiate trasormatio matrix equatio. Similarly, the priipal urvatures ad diretios o ay poit o the gear tooth surae R 2 geerated by utter R G a be derived as well ad expressed i oordiate system S 2 by the same proedure. 4.. Cotat ellipses Aordig to the TCA results, the priipal diretios i ð1þ ad i ð2þ systems S 1 ad S 2, respetively, a be desribed i oordiate system S represeted i oordiate by applyig the oor- ad i ð2þ diate trasormatio matrix equatio. Obviously, at ay istataeous otat poit, i ð1þ ð2þ ðþ are loated o the ommo taget plae o the matig suraes R 1 ad R 2, as show i Fig. 6. The orietatio o the otat ellipse is determied by the agle whih a be represeted with the ollowig equatios [5,6]: Fig. 6. Orietatio ad dimesios o otat ellipse.

11 ta 2 ¼ g 2 si 2r g 1 g 2 os 2r ; where ad g 1 ¼ j ð1þ g 2 ¼ j ð2þ C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) 50 4 j ð1þ ; j ð2þ : Meawhile, agle r is ormed by the irst priipal diretios o the gear ad piio tooth suraes i ð2þ ad i ð1þ, ad it a be evaluated by r ¼ os 1 i ð1þ i ð2þ : ð7þ The hal legth o the major ad mior axes o the otat ellipse, a ad b, a be expressed i terms o the elasti approah D by [5,6] a ¼ D 1 2 A ; ð8þ ð4þ ð5þ ð6þ ad b ¼ D 1 2 B ; ð9þ where A ¼ 1 h 4 jð1þ R B ¼ 1 h 4 jð1þ R ad j ð1þ R j ð2þ R ¼ jð1þ ¼ jð2þ j ð2þ R ðg2 1 2g 1g 2 os 2r þ g 2 2 Þ1=2i ; ð40þ j ð2þ R þðg2 1 2g 1g 2 os 2r þ g 2 2 Þ1=2i ; ð41þ þ j ð1þ ; þ j ð2þ : Thus, the orietatio ad dimesio o the otat ellipse a be determied by utilizig Eqs. (4) (9). ð42þ ð4þ 5. Numerial illustrative examples or gear meshig simulatios Fig. 7 illustrates three typial types o gear moutig or beveloid gear pairs with iterseted, rossed ad parallel axes. Applyig the developed omputer simulatio programs, the TCA results a be obtaied ad the otat ellipses a be plotted o the otat tooth suraes. The elasti approah D or the otat ellipse simulatio is hose as mm, idetial with the

12 44 C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) 50 Fig. 7. Three typial moutig types o beveloid gear pairs: (a) iterseted axes;(b) rossed axes;() parallel axes. Table 1 Major desig parameters o the beveloid piio ad gear Piio Gear Number o teeth N 1 ¼ 24 N 2 ¼ 6 Fae width (thikess o gear blak) F 1 ¼ 14 mm F 2 ¼ 14 mm Normal pressure agle a ðf Þ ¼ 20 a ðgþ ¼ 20 Normal module m ¼ 5 (mm/teeth) thikess o the oatig pait used or otat patter tests. Table 1 lists some major desig parameters o the beveloid gear pair hose or the ollowig examples Example 1: Straight beveloid gear pair with iterseted axes this example, the gear pair is omposed o straight beveloid piio ad gear ðb F ¼ b G ¼ 0 Þ with oe agles d 1 ¼ d 2 ¼ 0, mouted with a iterseted agle o 60, as Fig. 7(a) illustrates. Cases 1 simulate the meshig o the gear pair uder the ollowig assembly oditios: Case 1: D h ¼ D v ¼ 0 ad Dx g ¼ Dy g ¼ Dz g ¼ 0 mm (ideal assembly oditio). Case 2: D h ¼ D v ¼ 0 ad Dx g ¼ Dy g ¼ Dz g ¼ 0: mm. Case : D h ¼ 0:5, D v ¼ 0:5 ad Dx g ¼ Dy g ¼ Dz g ¼ 0: mm. Case 1 is the ideal assembly oditio. Case 2 idiates that the gear has a moutig positio deviatio but there is o axial misaligmet. Case idiates that the gear pair has both moutig positio deviatio ad axial misaligmets. The bearig otat o a straight beveloid gear pair with iterseted axes is a poit otat. The TCA results ad the TE were alulated ad summarized i Table 2. Meawhile, the loi o otat poits ad their orrespodig otat ellipses o the piio surae were illustrated i Fig. 8. Eve whe the gear pair meshes uder various assembly errors, the TEs remai zero ad the loi o otat poits remai i the middle regio o the tooth lak. These results idiate that the straight beveloid gear pair mouted with iterseted axes is isesitive to assembly errors. Uder some irumstaes, iterseted straight beveloid gears a replae bevel gears ad take the advatage o isesitivity to the gear assembly errors. However, the otat ellipses o a beveloid gear pair with iterseted axes are relatively small, espeially with large oe agles. Thereore, tooth surae durability is geerally low owig to its high otat stress. Cosiderig the straight beveloid gear pair i this example, oe urve

13 C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) Table 2 TCA results o the straight beveloid gear pair with iterseted axes Case / 0 1 (deg.) F u F / 0 2 (deg.) G u G TE (ar-se.) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Fig. 8. Bearig otat o a straight beveloid gear pair with iterseted axes. ðb F ¼ b G ¼ 0 Þ i Fig. 9 shows how with the irease o oe agles d 1 ad d 2, the ratio betwee the major ad mior axes o the otat ellipse, a=b, derease sigiiatly. Notably, the ratio a=b approahes to iiity whe the oe agles d 1 ad d 2 ted to zero, whih represets a spur gear pair i lie otat.

14 46 C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) 50 Fig. 9. The ratio a/b with respet to the oe agle Example 2: Helial beveloid gear pair with rossed axes Fig. 7(b) illustrates a beveloid gear pair osistig o helial beveloid piio ad gear with rossed axes. The oe agles o the helial beveloid piio ad gear are d 1 ¼ d 2 ¼ 20, ad the helix agles o the pith plaes o the rak utters or the piio ad gear are b F ¼ b G ¼ 15 (right haded). The rossed agle ormed by axes Z ad Z g, as show i Fig. 4, is alulated as by applyig the algorithms proposed by Mitome [2]. This example ivestigates the meshig simulatios o gear pairs uder the ollowig assembly oditios: Case 4: D h ¼ D v ¼ 0 ad Dx g ¼ Dy g ¼ Dz g ¼ 0 mm (ideal assembly oditio). Case 5: D h ¼ D v ¼ 0 ad Dx g ¼ Dy g ¼ Dz g ¼ 0: mm. Case 6: D h ¼ 0:5 ; D v ¼ 0:5 ad Dx g ¼ Dy g ¼ Dz g ¼ 0: mm. Table summarizes the TCA results ad TEs o the gear pair, ad Fig. 10 illustrates the loi o otat poits ad their orrespodig otat ellipses o the piio surae. Eve uder various assembly errors, the TEs remai zero ad the loi o otat poits also remai i the middle regio o the tooth lak. Thereore, the helial beveloid gear pair with rossed axes is also isesitive to assembly errors. Aother urve i Fig. 9 idiates the relatioship betwee the oe agles ad the ratio a=b o this helial beveloid gear pair with b F ¼ b G ¼ 15. t is reasoable to id the ratio a=b approahes to a limited value whe the oe agles ted to zero, whih represets that a rossed axes helial gear pair is i poit otat. 5.. Example : Straight beveloid gear pair with parallel axes Fig. 7() illustrates a straight beveloid gear pair mouted with the parallel axes. The piio ad gear are idetial to those i Example 1 (. Setio 5.1). However, the gear is ow tured over to allow the heel o the gear to egage with the toe o the piio. Thus, the axes o piio ad gear are ow parallel. Cases 7 9 simulate the meshig o the gear pair uder the ollowig assembly oditios:

15 C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) Table TCA results o the helial beveloid gear pair with rossed axes Case / 0 1 (deg.) F u F / 0 2 (deg.) G u G TE (ar-se.) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Fig. 10. Bearig otat o a helial beveloid gear pair with rossed axes. Case 7: D h ¼ D v ¼ 0 ad Dx g ¼ Dy g ¼ Dz g ¼ 0 mm (ideal assembly oditio). Case 8: D h ¼ D v ¼ 0 ad Dx g ¼ Dy g ¼ Dz g ¼ 0:5 mm. Case 9: D h ¼ 0:5, D v ¼ 0 ad Dx g ¼ Dy g ¼ Dz g ¼ 0:5 mm (edge otat oditio).

16 48 C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) 50 Fig. 11. Bearig otat o a straight beveloid gear pair with parallel axes. Fig. 11 illustrates the bearig otats o the gear pair o the piio surae uder three assembly oditios. Meawhile, the TCA results ad TEs are listed i Table 4. Uder the ideal assembly oditio (ase 7) or with a moutig positio deviatio without a axial misaligmet (ase 8), similar to a spur or helial gear pair mouted with parallel axes, the bearig otat is a lie otat ad the TEs equal zero. However, whe a axial misaligmet ours (ase 9), the bearig otat beomes a edge otat ad the TEs are idued as show i Table 4. Notably, whe the edge otat ours, Eqs. (9) ad (10) whih desribe the surae-to-surae otiuous tagey are o loger appliable. Litvi [5,6] proposed the geeral oept o urve-to-surae otiuous tagey to simulate edge otat i gear meshig. Case 9, the edge otat ours o the heel edge o the piio whih is i tagey with the surae o the gear. Thereore, the ollowig equatios must be observed: z 1 ¼ F 1 =2; ð44þ ad R ð1þ ¼ R ð2þ ; ð45þ t ð1þ ð2þ ¼ 0; ð46þ where, F 1 is the ae width o the piio, ad z 1 ¼ F 1 =2 represets the trasverse plae o the heel side o the piio. By ombiig R ð1þ ad Eq. (44), the urve whih represets the edge o the heel o the beveloid piio tooth is expressed i oordiate system S ;i additio, the taget o this edge a be derived ad represeted as t ð1þ. Eqs. (44) (46) a be solved similarly to Eqs. (9) ad (10).

17 C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) Table 4 TCA results o the straight beveloid gear pair with parallel axes Case / 0 1 (deg.) F u F / 0 2 (deg.) G u G TE (ar-se.) Lie otat Lie otat ) ) Lie otat Lie otat ) ) ) ) ) ) Colusios This study simulates the meshig o beveloid gear pairs with iterseted, rossed ad parallel axes. Bearig otat harateristis o the beveloid gear pairs uder various assembly errors, ad the otat ellipses o beveloid gear pairs with o-parallel axes are also ivestigated. Several illustrative umerial examples are preseted as well. Based o the results attaied i this study, we olude the ollowig: 1. Beveloid gear pairs with o-parallel axes exhibit poit otat ad are isesitive to assembly errors. Besides, the orietatio ad dimesio o the otat ellipse a be determied. The results are helpul to urther ivestigatios o the otat harateristis ad lubriatio o this type o gear pairs. 2. The otat ellipses o o-parallel axes beveloid gear pairs are relatively small espeially with large oe agles. Thereore, tooth surae durability is geerally low owig to its high otat stress. Our upomig researh will provide some ovel geeratio methods whih a elarge the otat ellipses o o-parallel axes beveloid gear pairs.. Straight beveloid gear pairs with parallel axes without axial misaligmets a mesh ojugately i lie otat. However, whe axial misaligmets are preset, the edge otat replaes the lie otat ad uavorable TEs, stress oetratios, oises ad vibratios will our. real appliatios, oe o the beveloid gears should be rowed to loalize the bearig otat i the middle area o the tooth rak ad to avoid edge otat. Akowledgemets The authors are grateul to the Natioal Siee Couil o the ROC or the grat. Part o this work was perormed uder otrat No. NSC E

18 50 C.-C. Liu, C.-B. Tsay / Mehaism ad Mahie Theory 7 (2002) 50 Reerees [1] K. Mitome, Coial ivolute gear, Part : tooth atio o a pair o gears, Bulleti o the JSME 28 (245) (1985) [2] K. Mitome, Coial ivolute gear (Desig o oitersetig oparallel-axis oial ivolute gear), JSME teratioal Joural, Series 4 (2) (1991) [] K. Mitome, eed gridig o straight oial ivolute gear, JSME teratioal Joural, Series C 6 (4) (199) [4] K. Mitome, Desig o Miter oial ivolute gears based o tooth bearig, JSME teratioal Joural, Series C 8 (2) (1995) [5] F.L. Litvi, Theory o Gearig, NASA Publiatio RP-1212, Washigto DC, [6] F.L. Litvi, Gear Geometry ad Applied Theory, Pretie-Hall, Eglewood Clis, NJ, [7] H.E. Merritt, Gear, third ed., ssa Pitma ad Sos, Lodo, [8] K. Mitome, Table slidig taper hobbig o oial gear usig ylidrial hob, Part 1: theoretial aalysis o table slidig taper hobbig, ASME Joural o Egieerig or dustry 10 (1981) [9] K. Mitome, liig work-arbor taper hobbig o oial gear usig ylidrial hob, ASME Joural o Mehaial Desig 108 (1986)