A Mathematical Model of Tooth Geometry and Undercutting Condition. and Contact Ratio Analysis of the Cosine Gear Drive Generated by a

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1 A Matheatical Model o Tooth Geoety and Undecutting Condition and Contact Ratio Analysis o the Cosine Gea Dive Geneated by a Cosine Rack Cutte Depatent o Industial Engineeing and Manageent, Cheng Shiu Univesity, Kaohsiung, 833, Taiwan, chengkanglee@yahoo.co.tw Abstact This pape poposes a cosine gea dive povided with a pai o cosine geas which ae geneated by a cosine ack cutte. Based on the coodinate tansoation theoy and the theoy o geaing, a atheatical odel o tooth geoety o the cosine gea dive is developed. The condition o undecutting o the cosine gea geneated by a cosine ack cutte is established to deteine the iniu nube o teeth o non-undecutting. An analytic atheatical odel o evaluating contact atio o the cosine gea dive is built. A cause-and-eect analysis o contact atio is executed unde consideing gea odule, gea atio, addendu coeicient, and dedendu coeicient as causes. Results o the analysis show that gea odule does not aect contact atio at all but the othe causes do aect contact atio. The poposed cosine gea dive is able to educe the size o gea box since the iniu nube o teeth o non-undecutting is only 6. Futheoe, the poduction cost and poduction lead tie o the poposed cosine gea dive ae low because it can be anuactued eiciently and econoically by egula hobbing achines. The use o CC achines is not necessay. Keywods: Cosine Gea Dive, Contact Ratio, Rack Cutte, Tooth Geoety, Undecutting. Intoduction It goes without saying that involute geas ae the ost ipotant and popula type o geas in oden industy. Involute geas can be anuactued easily and eiciently by hob cuttes on hobbing achines. Theeoe, the poduction cost o involute geas is low. Moeove involute geas ae not sensitive to cente distance vaiations. Although involute geas have any advantages, a seious liitation o involute geas is that the iniu nube o teeth o non-undecutting cannot be too sall. Fo exaple, i a ack cutte povided with a standad pessue angle o 0 degee and a standad tooth height is applied to geneate an involute gea without any odiication, the iniu nube o teeth o non-undecutting is 7. Theeoe, when acing a ission to educe the size o gea box as uch as possible, designes ay conside to eplace the taditional involute gea dive by the cosine gea dive poposed in this pape since the iniu nube o teeth o non-undecutting o the poposed cosine gea is uch salle than that o involute gea. This pape, inspied by the anuactuing way o involute geas, poposes a new vesion o cosine gea dive which is geneated by a cosine ack cutte. The poposed cosine gea dive can also be anuactued eiciently by using hobbing ethod. Theeoe, the cost o poduction and the lead tie o poduction ae low. Although Luo et al. [] have aleady poposed anothe vesion o cosine gea dive, thei cosine gea dive is not geneated by a ack cutte. The application o CC achines is equied to anuactue thei cosine gea dive. Yu et al. [] conducted a contact atio analysis o the vesion o cosine gea dive poposed by Luo et al. Wang et al. [3] studied the eect o assebly eos on contact stess and bending stess o the cosine gea dive poposed by Luo et al. Tsai and Tsai [4] applied quadatic paaetic poile to design high contact atio spu geas. Fong et al. [5] poposed using the line o action to invesely deteine the paaetic tooth poile o spu gea. Tsay and Fong [6] studied a helical gea dive which is coposed o an involute gea and a cicula-ac pinion. Kooi et al. [7] ceated a logix tooth poile which has zeo elative cuvatue at any contact points. Zhang et al. [8] poposed a double involute gea, the poile o which is coposed o two involute cuves. Aiga and agata [9] poposed a new Wildhabe-ovikov gea dive which is not sensitive to cente distance eos by applying a basic ack cutte povided with cicula-ac and involute poiles. Lee [0] poposed a anuactuing pocess o a cylindical cowned gea dive to obtain a contollable outh ode polynoial unction o tansission eos. Lee et al. [] poposed a systeatic ethod Advances in inoation Sciences and Sevice Sciences(AISS) Volue4, ube5, Aug 0 doi: 0.456/AISS.vol4.issue5.33 7

2 to contol intelligently and pecisely the agnitude o paabolic-like tansission eo o a pai o geas by solving a syste o nonlinea equations. Zhang and Wang [] used tooth contact analysis ethod to study the inluences o installation eos on double cicula ac tooth spial bevel gea. This pape is oganized as ollows. Fistly, a atheatical odel o tooth geoety o the cosine gea dive which is geneated by a cosine ack cutte is established based on the coodinate tansoation theoy and the theoy o geaing. Secondly, the condition o undecutting o the poposed cosine gea is developed and the iniu nube o teeth o non-undecutting is solved. Thidly, an analytic atheatical odel o the analysis o contact atio is poposed. Fouthly, a nueical exaple o the poposed cosine gea dive is pesented and a cause-and-eect analysis o contact atio is executed. Finally, discussions on how the causes aect contact atio ae ade.. Matheatical Model o Tooth Geoety o Cosine Geas Accoding to the coodinate tansoation theoy and the theoy o geaing [3], the atheatical odel o tooth geoety o cosine geas can be obtained by using the tooth geoety o geneating tool and the elative otion between the geneating tool and the geneated geas. Shown in Figue is the cosine ack cutte used to be the geneating tool o cosine geas. Since the tooth poile o the ack cutte is a cosine unction, the ack cutte is given the nae o cosine ack cutte. To epesent the atheatical odel o tooth geoety o the cosine ack cutte, a coodinate syste S( x, y, z ) is applied to connect igidly to the ack cutte. The tooth geoety o the cosine ack cutte can be epesented in S( x, y, z ) by the ollowing equations: ( uv, ) ui h cos( u/ ) j vk () oal vecto o the tooth geoety is deteined by ( u) ( h / )sin( u/ ) 0 u v i j k () Hee, h denotes dedendu tooth depth and denotes gea odule. The elative otion between the geneating ack cutte and the geneated cosine geas ae shown in Figue. Two coodinate syste S( x, y, z ) and S( x, y, z ) ae applied to connect igidly to the lowe cosine gea G and the uppe cosine gea G, espectively. In the pocess o geneation, the cosine ack cutte tanslates hoizontally to the let, the lowe cosine gea G otates counteclockwise, and the uppe cosine gea G otates clockwise. Centode o the ack cutte and centodes o the geas oll without sliding with each othe. Thus, the ollowing condition is obseved: s R R (3) whee s is tanslational paaete o the cosine ack cutte; R is adius o centode o gea G; R is adius o centode o gea G; is otational paaete o gea G; is otational paaete o gea G. The tooth geoety o ack cutte will o a aily o suaces in coodinate syste S( x, y, z ) and the aily o suaces is epesented in S( x, y, z ) by x ( ur )cos [ Rh cos( u/ )]sin y R h cos( u/ )]cos ( R u)sin (4) z v The elative velocity between the geneating ack cutte and the geneated gea G is epesented by () () () () V ( u, ) V i V j V k h cos( u/ ) i ( R u) j 0k (5) x y z whee is angula velocity o gea G. Based on the theoy o geaing [3], the necessay condition o the existence o envelope to the aily o suaces epesented by Eq. (4) is the ollowing equation o eshing: h () ( u, ) ( u) V ( u, ) ur 0 (6) Since is not equal to zeo, the paaete in Eq. (6) can be solved out and can be epesented by a unction o u as ollows: 7

3 u h R (7) The atheatical odel o tooth geoety o cosine gea G can then be obtained as ollows: x ( ur )cos [ Rh cos( u/ )]sin y R h cos( u/ )]cos ( R u)sin z v (8) u h R Siilaly, the suace o the cosine ack cutte will also o a aily o suaces in S( x, y, z ) and the aily o suaces is epesented in S( x, y, z ) by x ( ur)cos [ Rh cos( u/ )]sin y [ R h cos( u/ )]cos ( u R)sin (9) z v The elative velocity between the geneating ack cutte and the geneated gea G is as ollows: ( ) ( ) ( ) ( ) V ( u, ) V i V j V k h cos( u/ ) i ( ur ) j 0 k. (0) x x x whee is angula velocity o gea G. Accoding to the theoy o geaing, the necessay condition o the existence o envelope to the aily o suaces epesented by Eq. (0) is the ollowing equation o eshing: h ( ) ( u, ) ( u) V ( u, ) R u 0 () As is not equal to zeo, the paaete in Eq. () can be solved out as a unction o u as ollows: u h R () The atheatical odel o tooth geoety o cosine gea G can then be epesented by x ( ur)cos [ Rh cos( u/ )]sin y [ R h cos( u/ )]cos ( u R)sin z v (3) u h R 3. Condition o Undecutting Accoding to the eshing theoy o geaing [4], the poposed cosine geas will be undecut by the cosine ack cutte i the ack cutte has undecutting liit points on its tooth suace. In othe wods, i the ack cutte does not have any undecutting liit point on its tooth suace, the geneated suaces o cosine geas will not be undecut. Undecutting liit points ae also eeed to as the liit points o the ist kind. Since the two cosine geas ae geneated by the sae ack cutte, they have the sae condition o undecutting. Hee, the cosine gea G is chosen to deive the condition o undecutting. Accoding to the geneal o o condition o undecutting pesented in [4], the undecutting liit points on the suace o the cosine ack cutte ae deteined by the ollowing condition: 73

4 ( ) E F u V F G 0 EG F v V \(4) d u v dt ( ) whee E ; F ; G u u u v v v Ate substituting Eq. () and Eqs. (5)-(7) into Eq. (4) and eplacing R and h by the ollowing equations: R h h (5) /; the condition o undecutting epesented in Eq. (4) can yield the ollowing equations: h cos( u/ ) 4( h ) cos(4 u/ ) (6) Thee is an extee value o when u satisies the ollowing condition: 4 h u cos 4 6h (7) Ate substituting Eq. (7) into Eq. (6), the lowe bound o the nube o teeth o non-undecutting cosine gea is obtained as ollows: 3/ LB 4 h (8) 3 3 Accoding to Eq. (8), i the geneating cosine ack cutte is povided with a standad tooth height, which eans the dedendu coeicient h is.5, then the lowe bound o the nube o teeth LB will be In othe wods, as long as the nube o teeth o the cosine gea is lage than 6, the cosine gea will not be undecut by the cosine ack cutte. Table shows the elationship o h,, and in. When h is inceased o.5 to.4, say, i h is.5, the iniu nube o teeth o non-undecutting iniu nube o teeth o non-undecutting non-undecutting in is 8. LB will incease o 5.33 to That is to in is 7. I 4. Analytic Matheatical Model o Evaluating Contact Ratio LB in is 6. I h is.34, the h is.4, the iniu nube o teeth o Contact Ratio is a vey ipotant peoance easue o the easueent o tansission quality o a pai o geas. Contact atio also deteines the shae o load between two adjacent teeth. To evaluate contact atio, the ist step is to deteine the line o action. As shown in Figue, two coodinate systes S( x, y, z ) and S ( x, y, z ) ae applied to connect igidly to the gea housing. By tansoing the atheatical odel o tooth geoety o the cosine ack cutte o S( x, y, z ) to S( x, y, z ) and eplacing the paaete by Eq. (7), the atheatical odel o the line o action can be epesented in S as ollows: x h ; y R h cos( u/ ) (9) As shown in Figue 3, the line o action intesects the addendu cicle o G at point P. At point P, the ollowing condition is tue: 74

5 whee ha x y R ha (0) 0 is addendu tooth height. Equation (0) is a nonlinea equation with one unknown paaete u. Based on Eq. (0), the analytic solution o u, denoted by u P, can be expessed as ollows: K C up cos Q AK 4 A (4 h / ); B h A; C hr; D harha; /3 3 3 T B 7AC 7 ABD ; M T 4( B AD ) T ; () /3 B ( B AD ) M K ; /3 3A 3AM 3( ) A /3 4B ( B AD ) M Q /3 3A 3AM 3( ) A Siilaly, the atheatical odel o tooth geoety o the cosine ack cutte is tansoed o S( x, y, z ) to S ( x, y, z ) and the paaete is eplaced by Eq. (7). Then the line o action can be epesented in S by x h ; y R h cos( u/ ) () The line o action intesects the addendu cicle o G at point P. At point P, the ollowing condition is tue: x y R ha 0 (3) Equation (3) is also a nonlinea equation with one unknown paaete u. The analytic solution o u is solved based on Eq. (3) and is denoted by up as ollows: K C u cos P Q AK 4 A (4 h / ); B h A; C hr; D har ha; /3 3 3 T B 7AC 7 ABD ; M T 4( B AD ) T ; (4) /3 B ( B AD) M K ; /3 3A 3AM 3( ) A /3 4B ( B AD) M Q /3 3A 3AM 3( ) A With u and P u P, contact atio, denoted by CR, can then be evaluated by the ollowing equations: h sin(4 u / ) h sin(4 u / ) (5) ( P) ( P) ( P) up P ( P) up P ; ; CR R R R R / 5. ueical Exaple A nueical exaple o a cosine gea dive coposed o a pai o cosine geas is pesented based on the atheatical odel o tooth geoety developed in section o this pape. Figue 4 shows the cosine gea dive which uses the ollowing settings: gea odule is 5, the nube o teeth o 75

6 pinion gea G is 6, the nube o teeth o gea G is 9, the addendu coeicient h a is, the dedendu coeicient h is.5. Accoding to Table, the iniu nube o teeth o a nonundecutting cosine gea is 6 i h =.5. Theeoe, it can be obseved that the shape o teeth o the pinion gea is sooth. Since the nube o teeth o cosine gea dive can be vey sall, the volue o geabox can becoe sall as well. When acing the ission to educe the size o geabox, designes ay conside to eplace involute geas by cosine geas since the iniu nube o teeth o nonundecutting o cosine geas is salle than that o involute geas. 6. Cause-and-Eect Analysis o Contact Ratio Accoding to the analytic atheatical odel o evaluating contact atio poposed in section 4 o this pape, causes that ight aect contact atio include gea odule, gea atio, addendu coeicient, and dedendu coeicient. In this section, the nueical exaple pesented in section 5 o this pape is continued to be used to execute the cause-and-eect analysis o contact atio. 6. Eect o Gea Module The pupose o studying the eect o gea odule on contact atio heein is to deonstate that the vesion o cosine gea dive poposed in this pape is dieent o that poposed by Luo et al. []. Accoding to Yu et al. [], the contact atio o the vesion poposed by Luo et al. is inluenced by gea odule. Howeve, the contact atio o the vesion poposed in this pape is not inluenced by gea odule. Table shows that when gea odule is inceased o 5 to 0, contact atio (CR) eains constant all the tie. Contact atio is not inluenced by gea odule at all. It should be noted that although gea odule does not aect contact atio, gea odule is still an ipotant acto which can aect tooth size and tooth stength o gea. 6. Eect o Gea Ratio Gea atio is deined as the atio o to, whee is the nube o teeth o gea G and is the nube o teeth o gea G. A highe gea atio iplies a highe capability o speed eduction. As shown in Figue 5, the gea atio is gadually inceased o to 5 while the nube o teeth is povided with thee ixed levels: 6, 9, and. When is 6 and gea atio is inceased o to 5, contact atio will ise o.38 to.935 with an inceent o 5.45%. When is 9 and gea atio is inceased o to 5, contact atio will ise o.976 to.5 with an inceent o.94%. When is and gea atio is also inceased o to 5, contact atio will ise o.66 to.369 with an inceent o.67%. It can be ound that the eect o gea atio is signiicant when is sall and the eect o gea atio is not signiicant when is lage. 6.3 Eect o Addendu Coeicient To study the eect o addendu coeicient on contact atio, addendu coeicient is vaied o 0.8 to. and the othe paaetes eain unchanged. The eect o addendu coeicient is shown in Figue 6. Contact atio will ise initially and then all as addendu coeicient is inceased. Contact atio has an extee value o.50 when addendu coeicient is Eect o Dedendu Coeicient To study the eect o deddendu coeicient on contact atio, dedendu coeicient is vaied o.5 to.35 and the othe paaetes ae kept invaiant. The eect o dedendu coeicient on contact atio is shown in Table 3. Contact atio will ise as dedendu coeicient is inceased and will go highe as dedendu coeicient is inceased continuously. 76

7 7. Conclusion This pape has poposed a new vesion o cosine gea dive, which is coposed o a pai o cosine geas geneated by a cosine ack cutte. A atheatical odel o tooth geoety o the cosine geas is developed based on the coodinate tansoation theoy and the theoy o geaing. The condition o undecutting is established based on the eshing theoy o geaing is applied to deteine the lowe bound o nube o teeth o non-undecutting cosine gea. The iniu nube o teeth o nonundecutting o the poposed cosine gea is only 6. An analytic atheatical odel o the analysis o contact atio is poposed. A cause-and-eect analysis o contact atio is peoed. Accoding to the esults o analysis, the ollowing indings can be dawn. Fist, gea odule does not aect contact atio at all. Second, inceasing gea atio can incease contact atio. Thid, to incease contact atio, addendu coeicient should be optiized. Foth, inceasing dedendu coeicient can incease contact atio. In suay, to obtain a highe contact atio, it is suggested to incease the nube o teeth, incease gea atio, optiize addendu coeicient, and incease dedendu coeicient. 8. Reeences [] Luo, S., Y. Wu and J. Wang, The geneation pinciple and atheatical odels o a novel cosine gea dive, Mechanis and Machine Theoy, Vol. 43, o., pp , 008. [] Yu, Y.D, J. Wang, S.M. Luo, and L.F. Chen, Study on the contact atio o a cosine gea, Jounal o Hunan Univesity o Science & Technology, Vol., o. 38, 007. [3] Wang, J., S. Luo, S. Zhu and L. Chen, Analysis o the chaacteistics o a cosine gea dive with assebly eos consideed, Mechanical Science and Technology o Aeospace Engineeing, Vol. 8, o. 5, pp , 009. [4] Tsai, M.H. and Y.C. Tsai, Design o high-contact-atio spu geas using quadatic paaetic tooth poile, Mechanis and Machine Theoy, Vol. 33, o. 5, pp , 998. [5] Fong, Z.H., T.W. Chiang and C.W. Tsai, Matheatical odel o paaetic tooth poile o spu gea using line o action, Matheatical and Copute Modelling, Vol. 36, o. 4-5, pp , 00. [6] Tsay, C.B. and Z.H. Fong, Teeth contact analysis o helical geas with pinion cicula-ac teeth and gea involute shaped teeth, Jounal o Mechanis, Tansission, and Autoation in Design, Vol., o., pp.78-84, 989. [7] Kooi, T., Y. Aiga and S. agata, ew geas poile having zeo elative cuvatue at any contact points (logix tooth poile), Jounal o Mechaniss, Tansissions, and Autoation in Design, Vol., o. 3, pp , 990. [8] Zhang, G.H., H.B. Xu and H. Long, Double involute gea with ladde-shape teeth, Chinese Jounal o Mechanical Engineeing, Vol. 3, o. 6, pp.47-5, 995. [9] Aiga, Y. and S. agata, Load capacity o a new W gea with basic ack o cobined cicula and involute poile, Jounal o Mechaniss, Tansissions, and Autoation in Design, Vol. 07, o. 4, pp , 985. [0] Lee, C.K., Manuactuing pocess o a cylindical cown gea dive with a contollable outh ode polynoial unction o tansission eo, Jounal o Mateials Pocessing Technology, Vol. 09, o., pp.3-3, 009. [] Lee, C.K., C.Y. Wang and Y.C. Cheng, "An intelligent ethod o the contol o agnitude o paabolic-like tansission eo o a pai o geas," Intenational Coneence on Mechanical and Electonics Engineeing, Vol., pp.6-3, 00. [] Zhang, R. and T. Wang, "The inluences o installation eos on double cicula ac tooth spial bevel gea using TCA ethod ", Jounal o Convegence Inoation Technology, Vol. 7, o., pp.-0, 0. [3] Litvin, F.L., Theoy o Geaing, ASA Publication, Washington, 989. [4] Wang, S.R., Concise Couse o the Meshing Theoy o Geaing, Tianjin Univesity Pess, Tianjin, 005. [5] Yunlu Cai, Guodong He, Yuguang Chen, "MATLAB Web Application and Method Discussion", JCIT, Vol. 7, o. 4, pp. 75 ~ 8, 0. [6] Saeed Pasa, Reza Entezai-Maleki, "RASA: A ew Gid Task Scheduling Algoith ", JDCTA, Vol. 3, o. 4, pp. 9 ~ 99,

8 Table. Relationship o h, LB, and in h LB in Table. Gea odule vesus contact atio CR Table 3. Dedendu coeicient vesus contact atio. h CR Figue. Tooth poile o cosine ack cutte Figue 3. The line o action intesecting addendu cicles at points P and P Figue. Relative otion o coodinate systes in the pocess o geneation o cosine geas Figue 4 A cosine gea dive coposed o a pai o cosine geas 78

9 Figue 5. Eect o gea atio on contact atio Figue 6. Eect o addendu coeicient on contact atio 79

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