Parametric Instability Investigation and Stability Based Design for Transmission Systems Containing Face-gear Drives

Transcription

1 Unvey of Tenneee, Knoxvlle Tace: Tenneee Reeach and Ceave Exchange Docoal Deaon Gaduae School 8- Paamec Inably Invegaon and Sably Baed Degn fo Tanmon Syem Conanng Face-gea Dve Meng Peng Recommended Caon Peng, Meng, "Paamec Inably Invegaon and Sably Baed Degn fo Tanmon Syem Conanng Face-gea Dve. " PhD d., Unvey of Tenneee,. hp://ace.enneee.edu/u_gadd/434 Th Deaon bough o you fo fee and open acce by he Gaduae School a Tace: Tenneee Reeach and Ceave Exchange. I ha been acceped fo ncluon n Docoal Deaon by an auhozed admnao of Tace: Tenneee Reeach and Ceave Exchange. Fo moe nfomaon, pleae conac ace@u.edu.

2 To he Gaduae Councl: I am ubmng heewh a deaon wen by Meng Peng enled "Paamec Inably Invegaon and Sably Baed Degn fo Tanmon Syem Conanng Face-gea Dve." I have examned he fnal eleconc copy of h deaon fo fom and conen and ecommend ha be acceped n paal fulfllmen of he equemen fo he degee of Doco of Phloophy, wh a majo n Mechancal Engneeng. We have ead h deaon and ecommend accepance: J. A. M. Boule, Sedd M. Djouad, Xaopeng Zhao Ognal gnaue ae on fle wh offcal uden ecod. Han A. DeSmd, Majo Pofeo Acceped fo he Councl: Dxe L. Thompon Vce Povo and Dean of he Gaduae School

3 Paamec Inably Invegaon and Sably Baed Degn fo Tanmon Syem Conanng Face-gea Dve A Deaon Peened fo he Doco of Phloophy Degee The Unvey of Tenneee, Knoxvlle Meng Peng Augu

4 ACKNOWLEDGEMENTS I would le o expe my deepe gaude o my pmay advo, D. Han A. DeSmd, fo h excellen gudance and ong uppo n all my eeach owad h deaon. He eablhed an odeed and open leanng envonmen o nuue my gowh n academc feld and nped my eeach nee wh h pecal cenfc vew and gea paence. Appecaon alo exended o he membe of my Ph.D. commee, D. J. A. M. Boule, D. Sedd M. Djouad and D. Xaopeng Zhao, fo he helpful dcuon and uggeon. I alo wan o han D. Zhn B. Sabay fom he Pennylvana Sae Unvey fo h echncal advce and aance n he face-gea modelng. Th eeach wa uppoed by a gan fom he Naonal Scence Foundaon Gan Numbe: NSF-CMMI-748 unde he Dynamcal Syem Pogam deced by D. Eduado A. Mawa.. Fnally, I wan o han my paen and my wfe fo he uncondonal love and cae.

5 ABSTRACT The objecve of h deaon o povde a novel degn mehodology fo face-gea anmon baed on yem ably - a dynamc vewpon. The ucual dynamc model of anvee and oonal vbaon ae developed fo face-gea dve wh pu pnon o nvegae he paamec nably behavo n gea deph. The unque face-gea mehng nemac and he flucuaon of meh ffne due o a nonuny conac-ao ae condeed n hee model. Snce he yem peodcally mevayng, Floque heoy ulzed o olve he Maheu-Hll yem equaon and deemne he yem ably numecally. To avod complex numecal compuaon, Teglod appoxmaon employed o calculae face-gea conac-ao. Fo anvee vbaon, he model of face-gea wh one pu pnon and n-plane ymmec cenfugal e feld nvegaed f, and nex he face-gea mehng wh mulple pnon exploed, fnally, he one pnon cae ecalculaed by ang no accoun he n-plane aymmec e feld eulng fom n-plane dvng foce. The eul how ha he yem ably depend on oaon peed, geomecal dmenon and meh load. In ably baed degn, he yem ably one degn conan; he ohe conan npu powe. The powe level deemne he maxmum e a pnon ooh oo and he n-plane dvng foce on face-gea body. Baed on paamec nably nvegaon, he macocopc degn mehodology of he facegea body exploed by condeng he npu powe and ably conan. Moeove, he elaonhp of yem ably o paal confguaon of npu pnon alo exploed fo he mulple pnon cae.

6 Fo oonal vbaon, he yem ably nvegaed numecally wh epec o oaon peed, oaonal nea, meh ffne, and chaacec of anmon haf. Fuhemoe, a peubaon mehod appled o he ably bounday acng fo degn pupoe. The ably eul povde he neceay nfomaon fo vbaon uppeon. Heeno, he effec of he yem nea dbuon on he yem ably exploed o develop pave vbaon uppeon mehod and o fnd an opmal degn wh lea wegh; he dampng and ffne of haf can alo be adjued ndvdually o a o acheve pave and acve vbaon conol. v

7 TABLE OF CONTENTS LIST OF TABLES... x LIST OF FIGURES... x Chape... INTRODUCTION.... Bacgound.... Helcope Man Tanmon Bevel gea / planeay gea ype helcope man geabox Bevel gea / cylndcal gea ype helcope man geabox Face-gea / planeay gea ype helcope man geabox Face-gea / face-gea ype helcope man geabox Developmen of Face-gea Reeach Dynamc of Geabox Objecve of Deaon... 9 Chape... MATHEMATICAL METHODS FOR STABILITY ANALYSIS.... Syem Inoducon.... Sably Analy Mehod..... Floque Theoy and Appoxmae Algohm Hu Mehod and Combnaon Reonance Summay... 3 Chape GEOMETRY AND MESHING KINEMATICS OF FACE-GEAR DRIVES Ovevew Shape Pnon Tooh Suface Coodnae Tanfomaon v

8 3.4 Equaon of Mehng Face-gea Tooh Suface Geneaon Conac Pon and Conac Cenod Appoxmae Movemen Pah of Conac Pon Conac-Rao Appoxmaon Summay... 5 Chape STRUCTURAL DYNAMICS MODEL AND MACROSCOPIC DESIGN FOR FACE- GEAR DRIVES WITH A SPUR PINION Ovevew Sucual Dynamc Model of Face-gea Dve wh a Spu Pnon Syem Dcezaon Pnon Paamee Bendng Se of Pnon Tooh Sably Reul Sably Baed Face-Gea Macocopc Degn Degn wh conan dameal pch Degn wh conan pnon pch adu Summay Chape PARAMETRIC STABILITY ANALYSIS AND SPATIAL CONFIGURATION DESIGN FOR FACE-GEAR DRIVES WITH MULTIPLE SPUR PINIONS Ovevew Phae Dffeence beween Pnon Sucual Dynamc Model of Face-gea Dve wh Mulple Spu Pnon Syem Dcezaon Sably Analy Sably degn baed on oal angle and oaon peed Sably degn baed on poonng angle and oaon peed v

9 5.5.3 Sably degn baed on phang angle and oaon peed Sably degn baed on pnon paamee and oaon peed Sably degn baed on wo pace-fxed pnon Summay... 8 Chape 6... FACE-GEAR STRUCTURAL DYNAMICS MODEL CONSIDERING THE EFFECT OF IN-PLANE DRVING FORCE Ovevew In-Plane Se Feld Face-gea Model Condeng In-Plane Dvng Load Sably Reul fo Vaou Powe Level Summay... 3 Chape TORSIONAL VIBRATION OF FACE-GEAR DRIVE SYSTEMS Ovevew Meh Sffne Evaluaon Sucual Dynamc Model fo Toonal Vbaon of Face-Gea Dve Syem Toonal Sably Reul Paamec Inably due o Ma Momen of Inea Paamec Inably due o Shaf Sffne Paamec nably due o haf dampng Summay... 5 Chape CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK Summay and Concluon Recommendaon fo Fuue Wo v

10 LIST OF REFERENCES... 6 VITA v

11 LIST OF TABLES Table 4. : Face-gea dve paamee Table 4. : Face-gea model paamee Table 4. 3: Maeal chaacec Table 4. 4: Degn equemen Table 7. : Requemen and gea paamee x

12 LIST OF FIGURES Fg.. : Example of face-gea dve mehng wh pu pnon.... Fg.. : OH-58 man geabox Fg.. 3: CH-47 fowad geabox... 6 Fg.. 4: Schemac of UH-6 Blac Haw man geabox... 7 Fg.. 5: Spl-oque upgade geabox compable wh he UH-6A helcope Fg.. 6: Spl-pah helcope anmon fo wo engne Fg.. 7: Example of pl-oque degn baed on cylndcal gea... Fg.. 8: Example of face-gea pl-oque / planeay hybd anmon... Fg.. 9: Example of pl-oque degn baed on bevel and face-gea dve... Fg.. : Face-gea yem n he Apache Bloc III Impoved Dve Syem Fg.. : 3D CAD model of Soy RDS- dvean yem Fg.. : An example of gea dynamc model... 7 Fg. 3. : Dagam of face-gea geneaon by a hape... 3 Fg. 3. : Shape pnon wh nvolue ooh pofle Fg Coodnae yem fo face-gea and hape Fg. 3. 4: Flowcha of face-gea geneaon Fg. 3. 5: Face-gea ooh and conac lne Fg. 3. 6: Inne and oue lmng end of face-gea ooh Fg. 3. 7: Schemac of ponng poon and co econ of plane and... 4 Fg. 3. 8: Conac pon on face-gea ooh Fg. 3. 9: An example of effecve adal and ccumfeenal poon of conac pon and phae lag beween dffeen mehng ooh pa, fo c = Fg. 3. : Conac cenod, fo < c < Fg. 3. : Sech of conac pon and coepondng conac cenod, fo c = Fg. 3. : Poon of conac pon and coepondng conac cenod, fo c = Fg. 3. 3: Appoxmae loc fo he poon of conac pon, fo c = Fg. 3. 4: Tegold appoxmaon Fg. 4. : Face-gea ucual dynamc model: pnnng elac d and a concenaed mehng load un ha move a he unque face-gea mehng nemac. 56 x

13 Fg. 4. : Pnon ooh geomey and oo e Fg. 4. 3: Appoxmae conac-ao of face-gea dve, m = 5, =9 and P d = 8 /n Fg. 4. 4: Face-gea/pnon yem paamec nably, Cae I: N = 6, N = 3, a = 3.94 mm, h =.3a, c = Fg. 4. 5: Face-gea/pnon yem paamec nably, Cae II: N = 3, N = 55, a = mm, h =.5a, c = Fg. 4. 6: Flowcha of he ably baed face-gea degn... 7 Fg. 4. 7: Tooh face wdh fo dffeen numbe of pnon eeh, P d = 8 /n Fg. 4. 8: Maxmum ee a he oo of pnon ooh fo dffeen numbe of pnon eeh, and 6 5 RPM, P d = 8 /n Fg. 4. 9: Face-gea/pnon yem paamec nably baed on face-gea body hcne and oaon peed, =, =.8, = -5 ec, P d = 8 /n Fg. 4. : Maxmum pnon ooh oo e fo 5: educon degn VS face-gea ccal hcne fo vaou pnon ooh coun and opeang peed, P d = 8 /n Fg. 4. : Pnon ooh face wdh fo dffeen numbe of pnon eeh, p =.9375 nch o p = 49. mm Fg. 4. : Maxmum ee a he oo of he pnon ooh fo dffeen numbe of pnon eeh, and 6 5 RPM, p =.9375 nch o p = 49. mm Fg. 4. 3: Face-gea/pnon yem paamec nably baed on face-gea body hcne and oaon peed, =, =.8, = -5 ec, ρ p = 784 g/m 3, p =.9375 nch o p = 49. mm Fg. 4. 4: Maxmum pnon ooh oo e fo 5: educon degn VS face-gea ccal hcne fo vaou pnon ooh coun and opeang peed, wh conan pnon pch adu p =.9375 nch o p = 49. mm Fg. 5. : Sech of one face-gea mehng wh wo pu pnon Fg. 5. : Illuaon fo oal angle, poonng angle and phang angle Fg. 5. 3: Relaonhp of oal angle o dmenonle phae dffeence fo N = x

14 Fg. 5. 4: Face-gea ucual dynamc model: pnnng elac d and mulple mehng load un ha move a he unque face-gea mehng nemac. 9 Fg. 5. 5: Face-gea/wo pnon yem paamec nably wh epec o oal angle and oaon peed Fg. 5. 6: Face-gea/wo pnon yem paamec nably wh epec o poonng angle and oaon peed fo Example B Fg. 5. 7: Face-gea/wo pnon yem paamec nably wh epec o phang angle and oaon peed fo Example B... Fg. 5. 8: Syem paamec nably due o pnon ma, N = 3, N = 55, h =.7a, =.5, = -5 ec, c =.... Fg. 5. 9: Syem paamec nably due o pnon beang ffne, N = 3, N = 55, h =.7a, = -5 ec, c = Fg. 5. : Syem paamec nably due o pnon beang dampng, N = 3, N = 55, h =.7a, =, =.5, c = Fg. 5. : Sably plo fo he face-gea mehng wh wo pace-fxed pnon: conan gea ao m = 5 and conan module m d =3.75mm; h =.7a, =, =.5, = -5 ec... 7 Fg. 6. : Spnnng annula d ubjeced o a pace-fxed concenaed n-plane angenal edge load... Fg. 6. : Face-gea dynamc model: pnnng elac d wh pace-fxed n-plane dvng foce and movable ou-of-plane meh load un Fg. 6. 3: Aymmec n-plane adal e feld,, eulng fom n-plane edge load: N =3, N =55, a=74.54mm, h=.5a, P=8hp@ =5pm.... Fg. 6. 4: Aymmec n-plane hea e feld,, eulng fom n-plane edge load: N =3, N =55, a=74.54mm, h=.5a, P=8hp@ =5pm Fg. 6. 5: Aymmec n-plane ccumfeenal e feld,, eulng fom n-plane edge load: N =3, N =55, a=74.54mm, h=.5a, P=8hp@ =5pm x

15 Fg. 6. 6: Face-gea/pnon yem paamec nably condeng n-plane dve load, Cae I: N = 6, N = 3, a = 3.94 mm, h =.3a, c =.553, =, = 3-6 ec Fg. 6. 7: Face-gea/pnon yem paamec nably condeng n-plane dve load, Cae II: N = 3, N = 55, a = mm, h =.5a, c =.5668, =, = -6 ec Fg. 7. : Tooh ffne model; a p loaded canleve beam wh aveage ooh dmenon... 3 Fg. 7. : Meh ffne model Fg. 7. 3: Meh ffne and meh defomaon Fg. 7. 4: Top vew of he face-gea end of he meh ffne Fg. 7. 5: Meh defomaon due o elac devaon angle Fg. 7. 6: Schemac dagam of a geabox conanng a face-gea dve Fg. 7. 7: Sucual dynamc model of a face-gea dve yem Fg. 7. 8: Syem naual fequence v. face-gea and load ma momen of nea: =, h=.5r, cf=.5, J =.6-3 gm, m =.58 8 N/m, =3.5 8 Nm/ad, = Nm/ad Fg. 7. 9: Face-gea yem oonal nably due o face-gea and load ma momen of nea and oaon peed: h=.5r, cf=.5, J =.6-3 gm, m =.58 8 N/m, c m =5-6 ec m, =3.5 8 Nm/ad, c = 5-6 ec, = Nm/ad, c = 5-6 ec Fg. 7. : Syem naual fequence v. haf ffne: =, h=.5r, cf=.5, J =.6-3 gm, J =9.97J, J o =.7J, m =.58 8 N/m Fg. 7. : Face-gea yem oonal nably due o haf ffnee and oaon peed: h=.5r, cf=.5, J =.6-3 gm, J =9.97J, J o =.7J, m =.58 8 N/m Fg. 7. : Face-gea dve-yem oonal nably due o npu haf dampng and oaon peed: J =.6-3 gm, J =9.97J, J o =.7J, h=.5r.. 5 Fg. 7. 3: Face-gea dve-yem oonal nably due o oupu haf dampng and oaon peed: J =.6-3 gm, J =9.97J, J o =.7J, h=.5r.5 x

16 Fg. 8. : Tanvee ably baed degn dagam Fg. 8. : Toonal ably baed degn dagam Fg. 8. 3: Face-gea anvee vbaon model condeng ooh flexbly: pnnng d wh a meh load ocllao Fg. 8. 4: Nonlnea meh defomaon wh baclah gap cleaance Fg. 8. 5: Double-de ooh conac and baclah effec... 6 xv

17 Chape INTRODUCTION. Bacgound Face-gea dve an mpoan mechancal componen whch anfe oaon and powe beween neecng haf. I ha exed fo a long me a an alenave o bevel gea, bu he applcaon of face-gea ae lmed due o complex ooh geomey and elavely hgh co. Face-gea dve wee moly ued fo low powe applcaon unl helcope nduy howed eeach nee n hem abou wo decade ago []. Fo helcope anmon applcaon, he bac face-gea dve con of a face-gea and an nvolue pu pnon. Some example of face-gea dve wh pu pnon ae dplayed n Fgue.. Dffeen fom bevel gea, hee ae no c nallaon equemen fo facegea dve nce he pu pnon can move along he axal decon feely and alo ha favoable malgnmen oleance n ohe decon. The mehng of pu pnon and face-gea doe no yeld any axal foce, whch ave lage hu beang. Th componen alway equed fo bevel gea. The wegh benef ae obaned fom he low envy of face-gea o pnon malgnmen by educng beang uppo ffne equemen. Moeove, he low malgnmen envy of face-gea dve poduce

18 le anmon eo, alo le vbaon and noe, han n ohe ype of gea dve. Face-gea dve ha conan oaon veloce heoecally due o conjugaed geneaon by a hape. In addon, he axal feedom of pu pnon mae face-gea dve moe favoable han bevel one n pl-oque anmon degn []. a Face-gea / pu pnon wh dffeen haf angle [3-4]. b A e face-gea dve n RDS- pogam [5]. Fg.. : Example of face-gea dve mehng wh pu pnon.

19 Lvn e al. [6] ummazed he heoecal advanage of face-gea dve: lowe malgnmen envy, educed level of noe owng o mall anmon eo 3 moe favoable load anfe beween adjacen ooh pa; and 4 no accuae axal locaon equemen fo he pu pnon. Afe weny yea developmen unde he jon Amy/NASA Advanced Roocaf Tanmon ART pogam [], helcope man anmon conanng face-gea dve wee fnally mplemened. The hghlgh he Boeng 34 HP Apache Bloc III Impoved Dve Syem, baed on a pl-oque face-gea yem, whch can povde a % moe powe houghpu han he pevou Apache Longbow Dve Syem [7-8]. A he ame me, Soy Acaf ha alo been developng and eng own helcope face-gea anmon yem [9]. A one of ey echnologe fo he U.S. Amy Roocaf Dve Syem fo he Cenuy RDS- Pogam, he fuhe advancemen of face-gea echnology ae n developmen [5, 8-9]. Howeve, he dynamc conequence of face-gea dve have no been fully nvegaed. Thu, h deaon o advance he cuen ae-of-he-a.. Helcope Man Tanmon Helcope man anmon anfe powe fom engne o oo wh educed peed and enhanced oque. Tanmon echnology deemne, o a gea exen, helcope opeaon pefomance, noe and vbaon, manenance equemen, evce lfe, and manufacung co. A epoed n pevou leaue, anmon yem accoun fo abou ~% of helcope co, 3% of manenance expene, ~5% of helcope wegh, and 6% of he mechancally elaed malfuncon ha ofen eul n 3

20 he lo of helcope [-]. Boh ART pogam and RDS- pogam am a he anmon yem wh hghe powe-o-wegh ao, lowe noe level, longe evce lfe and economc co [, 5, 8-9]. The ypcal man geaboxe appled n helcope anmon can be denfed a bevel gea / planeay gea ype, bevel gea / cylndcal gea ype, face-gea / planeay gea ype, and face-gea / face-gea ype accodng o he componen; o ngle powe pah degn and pl powe pah degn accodng o he dvean confguaon. In ode o educe he wegh of dve yem, helcope man geabox demand fewe age, moe paallel powe pah and lage educon ao a he fnal age. The volume of gea popoonal o he quae of gea adu, whle he oque-cayng capacy popoonal o lowe ode of gea adu. So, he gea ze can be educed ubanally when he npu oque pl equally no mulple pah. Alo, he geabox wegh can be educed by aangng a lage educon ao a he fnal age becaue of he lowe oque a pecedng age [, ]. Face-gea dve poee hgh educon ao capacy and he unque advanage n pl-oque degn; hence he mo favoable canddae fo helcope anmon applcaon now... Bevel gea / planeay gea ype helcope man geabox The helcope man geaboxe, conng of bevel gea angle un age and planeay gea deceleaon oupu age, had been appled o Bell OH-58 Kowa, Boeng CH-47 Chnoo, and Soy UH-6 Blac Haw befoe 98 [3]. 4

21 a Schemac b Deal Fg.. : OH-58 man geabox [3]. The OH-58 man geabox compe an npu bevel gea age evng a angle unng funcon and a planeay gea age a educon oupu, hown n Fgue.. I a ngle man oo anmon wh only one powe npu pah, uually ued n lgh helcope. Th geabox povde a peed educon fom 66 pm o pm. 5

22 a Schemac b Deal Fg.. 3: CH-47 fowad geabox [3]. The CH-47 anmon ae a ypcal andem oo degn. I fowad geabox vey mla o OH-58, dplayed n Fgue.3. The dffeence ae: CH-47 conan wo planeay gea age n ee fo a hghe educon ao whle OH-58 ha only one; OH-58 oupu haf goe downwad hough he bevel gea whle CH-47 doe no. 6

23 Fg.. 4: Schemac of UH-6 Blac Haw man geabox [3]. The chemac of UH-6 man geabox lluaed n Fgue.4. Th geabox con of one man module, wo engne npu module, and wo acceoy module. The powe anmed no he man module fom wo engne va wo epaaed npu module. The powe pah change he decon hough bevel gea dve, and hen, a combnng bevel gea ued o anfe powe o man planeay gea yem and al oo geabox. The planeay gea yem conan fve plane pnon wh a educon ao of 4.67: [3]. The powe anmed fom he cenal un gea o he man oo haf va he fve plane pnon and he plane cae uccevely [4]. 7

24 .. Bevel gea / cylndcal gea ype helcope man geabox The pl-oque helcope man geabox wh cylndcal gea a he fnal age a pomng alenave fo convenonal planeay gea. The educon ao of he fnal age can each up o 4: unde he pl-oque confguaon wh wo paallel pah, whle he maxmum educon ao abou 7: fo he uual planeay gea e wh 3 o 8 paallel powe pah []. In pl-oque degn, boh powe plng and lage educon ao a he fnal age conbue o wegh avng. The ypcal applcaon ae he popoed pl-oque upgade geabox fo Soy UH-6A Blac Haw helcope and he man geabox of Boeng-Soy RAH-66 Comanche helcope, hown n Fgue.5 and Fgue.6 epecvely. The UH-6A upgade geabox anfe powe fom wo engne o he man oo va one bevel gea angle unng age, one bevel gea oque plng age, and one pu gea oque ecombnaon age. The powe fom each engne dvded no wo paallel pah va he bevel gea oque plng age [5]. Fg.. 5: Spl-oque upgade geabox compable wh he UH-6A helcope [5]. 8

25 a Layou of Comanche helcope man geabox [] b A popoed degn by G. Whe [6] Fg.. 6: Spl-pah helcope anmon fo wo engne. RAH-66 Comanche helcope alo equpped wh a pl-oque man oo geabox fo wo engne. A dual-helcal combnng gea ued o cancel axal hu. Dffeen fom UH-6A upgade geabox, RAH-66 employ cylndcal pu gea o pl 9

26 oque. The deal of he pu gea age fo pl-oque degn lluaed n Fgue.7. The npu pnon mehe wh wo paallel haf gea mulaneouly o pl he powe no wo pah evenly. Fg.. 7: Example of pl-oque degn baed on cylndcal gea []...3 Face-gea / planeay gea ype helcope man geabox The ypcal example of face-gea / planeay gea helcope man geabox a pl-oque / planeay hybd confguaon popoed by McDonnell Dougla Helcope Company fo he ART pogam []. See Fgue.8.

27 Fg.. 8: Example of face-gea pl-oque / planeay hybd anmon [-]. Th concepual baelne wa deved decly fom he AH-64A helcope []. Each npu pnon pl he oque no wo face-gea and change he dvean decon mulaneouly. Nex, a pu gea age ued o ecombne all oque and povde he npu powe fo he fnal planeay age and he al oo. Th degn employed face-gea/pu-pnon dve, nead of pal bevel gea, evng a he dvean angle unng age becaue of he favoable nallaon malgnmen oleance and no axal poonng equemen. The compae beween bevel and facegea dve n pl-oque degn lluaed n Fgue.9. The bevel pnon ha no axal feedom, whle he pu pnon ha.

28 Fg.. 9: Example of pl-oque degn baed on bevel and face-gea dve []. In addon, he lae Apache Bloc III Impoved Dve Syem alo ulze facegea o pl he oque and planeay gea o dve he man oo yem. The pl-oque face-gea yem con of wo npu apeed pnon, uppe face-gea, lowe face-gea, wo apeed dle, and one apeed dle fo ubday powe oupu, hown n Fgue.. The oque on each npu pnon pl beween dle and pnon va lowe facegea, and hen, he uppe face-gea ecombne all oque fo planeay gea age npu. The Boeng 34 HP Apache Bloc III Impoved Dve Syem conanng h ploque face-gea yem can povde a % moe powe houghpu han he pevou Apache Longbow Dve Syem [7-8].

29 Fg.. : Face-gea yem n he Apache Bloc III Impoved Dve Syem [4, 7]...4 Face-gea / face-gea ype helcope man geabox A one of ey echnologe fo he RDS- pogam, Soy ha been wong on a novel anmon yem wh wo face-gea age n ee [9]. Fgue. how he 3D CAD model of h dvean yem. Fg.. : 3D CAD model of Soy RDS- dvean yem [4, 9]. The RDS- dvean yem degned fo a dual engne confguaon and conan wo face-gea educon age. The powe anmed o he f age va he 3

30 npu pnon ha pl oque no wo face-gea; he econd age ue a face-gea o ecombne all oque and dve he man oo. Thee wo age change he powe decon connuouly. A ade udy on dffeen educon ao aangemen beween wo age wa conduced fo a 5 HP RDS- demonao geabox by Soy. The eul ndcaed ha he degn wh he econd age gea educon ao of.4 and he oal gea educon ao of 85.3 acheve he maxmum powe-o-wegh ao [9]..3 Developmen of Face-gea Reeach The bac heoy and poce echnology fo face-gea appeaed a ealy a he 93 [3]. The eeach on face-gea pogeed lowly unl poble applcaon n helcope anmon wa popoed n he ealy 99. Lvn e al. [7] developed he dealed analycal geomey of face-gea dve wh pu nvolue pnon va mulang he mehng of he face-gea and hape. In ode o educe he ee and anmon eo, ome face-gea degn wh modfed ooh pofle wee peened n [8-9]. The e expecaon fo hee modfed veon wa alo valdaed baed on he FEA model of ooh pa. Fuhemoe, Lvn e al. f noduced he analycal devaon fo he face-gea geneaon by a gndng wom n [9-], and he e analye on h ype of face-gea wee conduced by FEA ool oo. The gndng echnc help haden ooh uface. The dynamc nably of face-gea dve wh pu pnon, nduced by mehng nemac, wa nvegaed baed on a model of pnnng d / movng load n []. The applcaon of helcal pnon educe conac e and nceae conac ao. The heoy of face-gea dve wh a helcal pnon wa eablhed by Lvn and 4

31 cowoe n [-3]. They nvegaed wo ype of helcal pnon: one cew nvolue helcod, he ohe deemned a he envelope o a ew paabolc ac-cue; he geneaon of he conjugaed face-gea wa accomplhed by hape and wom epecvely. The eeach on wom face-gea dve wh concal and cylndcal wom wa peened n [4]. In addon, Lvn e al. [] and He e al. [9] exploed offe facegea dve, namely, he npu and oupu haf ae coed bu no neeced. Handchuh e al. [5] conduced expemenal evaluaon of face-gea a NASA Lew facly wh he focu of falue mode and load capacy e. Lewc e al. [5, 6] and Heah e al. [8] exploed face-gea fague chaacec. The nananeou load hang on all eeh n meh and he coepondng expemenal valdaon wee nvegaed fo face-gea n [7]. To mplfy he complexy n face-gea degn, Gungand e al. [8] povded wo mple analycal fomulae baed on acal mehod o calculae he nne and oue ad of face-gea ooh epecvely, whch ave he nenve compuaon equed fo plong Lvn adu faco cha [7]. FEA echnque wee developed fo face-gea dve o pefom a 3-D ooh conac analy n [9], o mulae he mehng wh pofle modfed pnon o malgnmen n [3-3], and o analyze he bendng e baed on dffeen paamee n [3]. Fnally, ome lghwegh pl-oque helcope anmon degn baed on face-gea dve wee popoed, eaoned fo feably, eed unde hgh-peed hgh-load condon, and adjued o acheve evenly oque plng [6, 33-34]. The ooh conac mode an mpoan faco o deemne he opeaon chaacec of face-gea dve. When he pnon an dencal copy of he hape ha geneae he face-gea, he mehng ooh pa ae n lne conac mode; when he pnon 5

32 ha le numbe of eeh han he hape, hey ae n pon conac mode. The pon conac mode ha advanage n malgnmen oleance, whle he lne conac mode gve bee pefomance n conac ao, maxmum load, conac peue, and anmon eo [7, 35]. To opmze he lne conac mode, a lgh pnon ooh pofle modfcaon cownng wa popoed o educe he envy o malgnmen n [35]. Theefoe, he lne conac mode choen a he objec of h udy n he followng chape..4 Dynamc of Geabox In he pa half cenuy, dynamc chaacec of gea/geabox have been nenvely exploed by lo of chola. Ozguven and Houe [36] peened an excellen ummay on he mahemacal model of gea dynamc publhed befoe 987 and clafed hem accodng o he flexble elemen ha wee condeed n model. Thee model evolved fom eang all componen a gd o ncopoang a many a flexble degee-of-feedom. Cuenly, hey ae beng fuhe developed baed on newly gea geomey and maeal chaacec. Uagawa and Haada [37-38] conduced boh expemenal and heoecal ude on he dynamc load ove he pu gea eeh wh peue angle and pch eo epecvely unde hgh peed condon. In he heoecal analye, an equvalen lnea png-ma model wa employed. Kauba and Evan [39] popoed a concep of vaable-vaable meh ffne fo he dynamc load calculaon by condeng he effec of anmed load, ooh pofle modfcaon and eo, hub oonal defomaon, poon of conac aea, and o on. Kubo [4] eeached he oaonal 6

33 vbaon of helcal gea wh manufacung and algnmen eo baed on he gd d / flexble eeh model. Th ype of model wdely ued n gea dynamc analye and a ypcal example, ncludng meh ffne, meh dampng and he excaon due o gea eo, fo one pa of gea hown n Fgue.. Fuhemoe, he gea dynamc model wh nonlneae, uch a gea eo and baclah, wee alo developed n he mla way [4-4]. Fg.. : An example of gea dynamc model [4]. The dynamc chaacec of haf and beang ae able o affec he pefomance of gea yem and can alo be adjued o mpove he pefomance. Kubo and Kyono [43] ncopoaed haf and beang ffnee no he vbaon model of gea wh ooh eo. Bahga e al. [44] condeed he ma of haf bede he ffne when calculang he nananeou dynamc load on pu gea eeh. Howeve, he ma of haf uually much le han gea, o he accuacy of he model whou haf ma enough n mo cae. By ang no accoun he chaacec of haf and beang, boh lnea and nonlnea model wee developed o analyze yem 7

34 ably, dynamc ooh load, foced epone, pofle modfcaon, baclah effec, and o on n [45-47]. Va he mul-cale mehod, he paamec nably wa exploed fo a wo-age gea yem wh me-vayng mehng ffne eulng fom a non-uny conac ao n [48]. Addonal, a dual geabox yem, conneced by a flexble haf, wa analyzed n [49]. Nonlnea baclah, me-vayng mehng ffne and dynamc chaacec of couplng haf wee condeed, and he peodc eady-ae epone wa alo compued by ung a hamonc balance and numecal ac-lengh connuaon appoach. By condeng coupled oonal-laeal moon of haf, he gea model become oo model wh he unque gea chaacec eenally. Geabox wa modeled a he oo yem coupled by gea meh load and alo analyzed fo ably and epone unde dffeen ype of meh aumpon n [5-5]. When helcal o bevel gea ae ued, he axal hu beang ae needed. Conequenly, he coupled axaloonal-laeal dynamc model wee ulzed o eeach uch geaed oo yem [53-56]. Moeove, ome nvegao conde he whole gea a gd by neglecng he flexbly of meh ooh pa. Th aumpon mae he gea dynamc cloe o pue oonal vbaon poblem. Ida e al. [57-58] focued on dynamc of haf by ung mehng foce o connec gea nead of ffne. The connuou yem model of haf wa alo ncopoaed no gea yem n ode o conan haf nea [49, 59]. In addon, he plac defomaon of gea eeh unde heavy load wa nvegaed expemenally and heoecally o mnmze he dynamc load [6]. 8

35 .5 Objecve of Deaon A ced above, he heoy of face-gea degn, geneaon and e analy have been well eablhed; he feably and duably wee alo valdaed by expemen. Howeve, he pape abou he dynamc model of face-gea dve ae cace and ncompehenve. The objecve of h deaon o develop he ucual dynamc model of anvee and oonal vbaon fo face-gea dve wh pu pnon o fully nvegae he paamec nably behavo. Subequenly, he degn gudelne fo face-gea anmon ae popoed baed on yem ably - a dynamc vewpon. The unque face-gea mehng nemac and he flucuaon of meh ffne due o a non-uny conac-ao nemac ae condeed n hee model. Snce he yem peodcally me-vayng, Floque heoy ulzed o olve he Maheu-Hll yem equaon and deemne he yem ably numecally. To avod complex numecal compuaon, Teglod appoxmaon employed o calculae face-gea conac-ao. Fo anvee vbaon, he ably of a face-gea mehng wh one pu pnon and ubjeced o an n-plane ymmec cenfugal e feld nvegaed f, and nex he face-gea mehng wh mulple pnon exploed, fnally, he one-pnon cae ecalculaed by ncopoang n-plane aymmec e feld eulng fom he nplane angenal dvng foce. The yem ably analyzed wh epec o vaou oaon peed, geomec dmenon and meh load. The anvee ably baed face-gea degn compe wo man conan, one yem ably; he ohe npu powe. Th powe deemne he maxmum 9

36 e a pnon ooh oo and he n-plane angenal dvng foce on face-gea body. All paamee ha ae able o affec yem ably can be degned o avod nably and afy wh pnon ooh e econ. Baed on paamec nably nvegaon, he macocopc dmenon of face-gea body and he npu powe level wll be degned fo pecfc opeaon peed ange. Moeove, he mulple pnon cae wll alo exploe he elaonhp of yem ably o paal confguaon of npu pah. Fo oonal vbaon, he yem ably nvegaed baed on meh ffne, yem nea dbuon, and chaacec of npu and oupu haf. I poble o develop pave and acve vbaon uppeon aege va adjung he dampng and ffne of haf o he ma momen of nea dbuon among pnon, face-gea and load. Alo, a peubaon echnque employed o oban he appoxmae ably bounday analycally fo degn pupoe.

37 Chape MATHEMATICAL METHODS FOR STABILITY ANALYSIS. Syem Inoducon Th deaon o eablh ucual dynamc model fo anmon yem conanng face-gea dve, ncopoang he unque face-gea mehng nemac and flucuaon of meh load eulng fom non-uny conac-ao nemac. Due o he nemacally nduced vaaon of he face-gea/pnon conac locu dung gea oaon, h yem ha peodcally me-vayng dynamcal behavo. Boh mehng nemac and meh load flucuaon may exce dynamc nably. Th chape o noduce he chaacec of h ype of yem and he compuaon mehod fo he yem ably cea. F of all, le u conde a geneal peodcally lnea me-vayng yem whou foce npu n mechancal engneeng. The equaon-of-moon ae gven a M q C q K q - and M T M, C T C, K T K - whee M, C, K ae epecvely genealzed yem ma, dampng, ffne mace wh a peod of T, q n-dmenonal genealzed coodnao, and

38 epeen dffeenaon wh epec o me. The Eq. - clafed a Hll equaon n geneal o a Maheu equaon when he me-vayng coeffcen mace ae nuodal. The coepondng ae-pace fom and T T X wh X q q T X A -3 I A and A T A M K M C -4 nn Hee, X a column veco ncludng n yem ae vaable; A a n n peodc yem max wh he peod of T ; and I ae nn zeo and deny max epecvely.. Sably Analy Mehod To pedc and deemne he yem dynamcal behavo, he ably analye mu be mplemened fo Hll equaon, ehe n Eq. - fom o n Eq. -3 fom. Founaely, pevou chola have uccefully developed a lo of mahemacal mehod. The mo common one nclude Floque mehod, Bolon appoach, Lyapunov heoy, and peubaon echnque [6]. The eence of Floque mehod o examne he ably of he ae anon max ha map an nal ae o he ae afe one peod [6-63]. The ae anon max can be fomed fom Wonan max, bu he numecal negal eque meconumng compuaon epecally fo hgh-dmenonal yem. An appoxmae aegy ha ea he ae anon max a epwe conan o dcee wa ulzed o evaluae he anon max [6-65], and convegence and ably chaacec wee poved o be he ame a he ognal yem heoecally by Hu [66]. Bauchau and

39 Nhov [67-68] conduced an mplc Floque analy whch examne only he domnan egenvalue of he anon max. Bolon [69] popoed a mehod fo ably bounday acng: wo ype of oluon wh peod of T and T ae gven a Foue ee epecvely; he aea uounded by wo oluon wh dencal peod ae unable and by wo oluon wh dffeen peod ae able. Th mehod wa exended o fnd combnaon eonance boundae n [7-7]. Jang and Jeong [7], and Pe [73] exploed oang yem wh gyocopc effec by Bolon appoach. Pe alo found he unable aea may be enlaged fo gyocopc yem va compang wh he eul fom Floque mehod. Lyapunov heoy wa employed o deemne he ably of me-vayng yem a a vewpon of enegy vaaon [74-77]. Hu [78-79] developed a peubaon echnque o deemne he yem combnaon eonance. By compang he ably eul, evealed ha Hu echnque wo much fae han Floque mehod due o no numecal compuaon, bu Hu echnque canno cach all nable nce only eonance em ae condeed [8]. In addon, Xu and Gach [8] peened a hamonc balance mehod o anfe he peodcally me-vayng yem max no a conan hype-dmenonal one. The egenvalue of he hype-dmenonal yem max goven he ably of he ognal yem. Th deaon employ Floque heoy and Hu mehod fo ably analye, and hey ae epecvely explaned below n deal. 3

40 .. Floque Theoy and Appoxmae Algohm Accodng o Floque heoy, he ably of he peodcally lnea me-vayng yem, uch a Eq. -3 and Eq. -4, can be epeened by he ably au ove only one peod. The yem ae anfeed fom one me pon o anohe hough a anon max,,, and expeed a X Φ, X -5 Thu, when he nal ae deny, he ae afe one peod, T, X I -6 T Φ T, X Φ T, when X n Hee, I n a nn deny max. In geneal, he ae afe -negal peod [6-63] X T Φ T, X -7 The ae anon max, T,, can be obaned by numecally negang Eq. -3 fom o T wh deny nal condon. Th anon max called Floque Tanon Max FTM, and egenvalue,, ae Floque mulple whch goven he oveall ably chaacec of he yem. The ably ceon gven ln α able and -8 T α unable Th mehod can cach all nable bu pay bac nenve compuaon due o numecal negal. A menoned above, an appoxmae aegy ha ea he yem max a epwe conan ued o evaluae FTM n h deaon [6-65]. By dvdng one peod no Q me ub-neval wh nal value [,,, Q- ], he yem max A aumed a conan n each ub-neval and evaluaed a he mddle me pon of he ub-neval. Thu, he oluon fo each ub-neval ae obaned 4

41 5 Q T T e T e e Q Q X X X X X X A A A -9 By ung he oluon of he la ub-neval a he nal ae fo he nex ubneval, cacadng all equaon n Eq. -9 yeld T wh e T Q Q p p p p p X X A - and he appoxmae FTM obaned Q p p p p p e T A Φ - Hee, he compuaonal accuacy and effcency can be ealy conolled and balanced by adjung he numbe of ub-neval, Q, whn one peod... Hu Mehod and Combnaon Reonance Compaed wh he numecal negal baed Floque mehod, peubaon echnque povde moe favoable compuaon effcency becaue hey ae baed on analycal appoxmaon. A geneal and mple peubaon algohm, developed by Hu [78-79], employed hee o ace he ably bounday appoxmaely. Th mehod explaned below: Eq. - anfomed a a andad fom hough nomalzaon and dagonalzaon pocee ] [ q q q U U P -

42 6 whee a mall eal numbe. U a dagonal max wh pove eal numbe quae of naual fequence on dagonal lne and can be expeed a n n wh U -3 Hee,,, n ae yem naual fequence. P and U ae peodcally mevayng and expanded a Foue ee S S n co n co G F F P E D U -4 Subung Eq. -3 and Eq. -4 no Eq. - yeld, n co n co q q q q S S E D G F F U -5 and componen fom, n q e d q g f q f q q S n j j j j S n j j j j n j j j,,,, n co n co -6 whee lowecae lee epeen he elemen of coepondng capal lee max. The f ode fom of Eq. -6 can be wen a

43 7 S n j j j j S n j j j j n j j j q e d y g f y f q y y q n co n co -7 One poble peubaon oluon aen a he fom of co n n co q B A y q B A q -8 On he gh hand de of Eq. -8, he em nvolvng A and B ae he vaaonal pa and he e he peubaon pa. Replacng he y and q n Eq. -7 by he expeon n Eq. -8 and uncang o he f ode of, he eulng equaon ae n j j j j j j j S n j j j j j A B f H H H H q q B A B A 4 3 n co n n co co co n n co -9 whee j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j B g A f A e B d H B g A f A e B d H A g B f B e A d H A g B f B e A d H Fo he econd equaon n Eq. -9, all poble unable em on he gh hand de ae colleced wh he vaaonal pa and he e aocaed wh he

44 peubaon pa. By eang he A and B a conan, he peubaon equaon ae alway able nce >. Thu, he vaaonal equaon goven he yem ably. When he excaon fequency of he coeffcen mace n Eq. - equal o cloe o ome combnaon of he yem naual fequence, he paamec eonance occu. F of all, he cae of he excaon fequency cloe o ome ummaon combnaon of wo naual fequence analyzed below, j - Hee, a fne eal numbe and a mall quany nce mall. The and j em ae ncluded n he vaaonal equaon a A co B n A n B co H f A co B n j j j A j n j B j co j H j f j jj B H n B co A n j co co co j H 4 n j A n j j j 4 - whee H and H 4 ae he H and H 4 wh he ubcp and j exchanged epecvely. Afe decouplng A, B, A j, and B j fom Eq. -, all eulng equaon ae negaed wh epec o and j ove [, ] and hen ubued by he negal aveage. The vaaonal equaon become 8

45 A B A j B j f f f f jj jj A B 4 4 j j H H Aj H 4 B j H 4 n H co H n H co H co n co n -3 Accodng o Eq. -, he expeon of H, H 4, H, and H 4 ae ubued no Eq. -3, and hen he equaon ae fuhe mplfed baed on he followng anfomaon, X X A B, A B, Y A B Y j A B j j j -4 whee =-. The eul ae X X Y f f Y f f jj jj X X 4 4 Y 4 Y 4 j d g e f d g e f d g e f j j j d g e f e X j j j j j j j j j j j j j j j j j e e j e Y X Y -5 The yem ably deemned by Eq. -5 whch ae lghly coupled f ode equaon. Theefoe, he ably ceon fo he excaon fequency cloe o he ummaon of h and j h naual fequence, + j /, f f f f jj jj able unable -6 wh 9

46 f 4 f f jj jj f 4 f jj f d j j e ej fj j d j j dj gj j e g j j g f f j j d j e j j j j j g f j j -7 Baed on he ame appoach, he ably ceon fo he excaon fequency cloe o he ubacon of h and j h j> naual fequence, j - /, mla o he ummaon cae expec dffeen and, f 4 f f jj jj f 4 f jj f dj j e j d j j e g j f e j j j d j j j f g g f j j d j e j j j j j g f j j -8 When he cae of mulple dencal naual fequence appea, all coepondng em mu be ncluded n he analy f he dencal naual fequency nvolved n combnaon eonance..3 Summay Th chape gve he equaon-of-moon and he coepondng ae-pace fom fo a geneal peodcally lnea me-vayng yem. The equaon ae clafed a Maheu-Hll equaon. Accodng o Floque heoy, he ae anon max ha map an nal ae o he nex ae afe one peod deemne he ably of he ae mappng, alo he ably of he yem. Th anon max Floque Tanon Max and an nenve numecal negal equed o calculae FTM. To nceae he compuaon 3

47 effcency, a valdaed appoxmaon aegy employed o evaluae FTM by eang he yem max a a epwe conan funcon. A an analycal mehod, he peubaon echnque ued o pedc he yem ably bounday appoxmaely. The hgh-accuacy eul can be acheved when he yem dampng and he me-vayng componen of he yem ffne ae much malle han he conan componen of he yem ffne. 3

48 Chape 3 GEOMETRY AND MESHING KINEMATICS OF FACE-GEAR DRIVES 3. Ovevew The face-gea geneaed by hape o wom and he neceay mahemacal heoy ha been well eablhed. Th chape o noduce he mahemacal decpon, developed by Lvn e al. [, 7, 8-83], fo he analycal geomey of facegea dve and he mehng mulaon. The face-gea dve, dcued n h deaon, compe one mulple pu nvolue pnon and one face-gea geneaed by a hape ha dencal o he pnon. Theefoe, all paamee of he hape ae he ame a hoe of he pnon. The face-gea geneaon by a hape lluaed n Fgue 3.. Fg. 3. : Dagam of face-gea geneaon by a hape. 3

49 3. Shape Pnon Tooh Suface The hape pnon a andad pu nvolue gea and he pojecon of ooh pofle on x -y plane hown n Fgue 3.. The angle coepondng o half of he ccula wdh of he pace beween wo adjacen eeh,, deemned by [8], nv 3- N c whee N numbe of hape eeh, c hape peue angle and nv epeen he equaon of nvolue cuve. Fg. 3. : Shape pnon wh nvolue ooh pofle [, 8]. 33

50 The poon veco of a pon on hape ooh uface expeed n hape body fxed coodnae x, y, z a [8] x b[n co ] u, y b[co n ] 3- z u Hee, u,θ ae he Gauan coodnae of he nvolue hape uface, and b he adu of hape bae ccle. Fuhemoe, he un nomal o he hape ooh uface alo obaned baed on he poon veco [8], co u n n 3-3 u 3.3 Coodnae Tanfomaon The hape body fxed coodnae x, y, z anfomed o he face-gea body fxed coodnae x, y, z hough wo neal coodnae yem: one m x m, y m, z m, algnng wh he nal hape poon; he ohe p x p, y p, z p, algnng wh he nal face-gea poon. The coodnae fo he face-gea geneaon by a hape eched n Fgue 3.3a; he hape coodnae and coepondng neal coodnae m ae hown n Fgue 3.3b; he face-gea coodnae and coepondng neal coodnae p ae hown n Fgue 3.3c; Th pocedue compe hee coodnae oaon: fly, oae clocwe o neal coodnae m along z m ax; nex, m oae clocwe o p along x m ax; and fnally, p oae coun-clocwe o along z p ax. 34

51 a Sech of he face-gea geneaon by a hape b Coodnae fo hape c Coodnae fo face-gea Fg Coodnae yem fo face-gea and hape [, 8]. 35

52 The anfom max fom o he poduc of hee coodnae oaon mace n ode [8], M M p M pm m M m co co co mn n n co co m co n n m n and co n co n co n n co co co m m m n co n m n n m co co m 3-4 N u,, M u, wh 3-5 N whee m he haf angle of face-gea dve m 9, he poon veco n, N he numbe of face-gea eeh, and, ae oaon angle of hape and face-gea epecvely. 3.4 Equaon of Mehng The equaon of mehng deved fom he condon ha he elave velocy of a conac pon on ooh uface vecal o nomal [8-83]. The elave velocy of hape o face-gea expeed n a [8] v v v v v v x y z Ω Ω y m co m z m n m co x m co m zm n m n m n co n m x y 3-6 whee gea ao m = N /N, v and v ae he velocy veco of he hape and he face-gea n epecvely, hape oaon peed, and and ae he 36

53 angula velocy veco of he hape and he face-gea n epecvely. Combnng wh Eq. 3-3, he equaon of mehng gven a [8] f b m m u,, n v m co u m n co Face-gea Tooh Suface Geneaon Snce face-gea ooh geneaed by mehng wh a hape, he ooh uface mu afy wh he equaon of mehng, Eq The mahemacal pocedue fo fomng he face-gea ooh uface ae: defne he poon veco of he hape ooh uface; olve he equaon of mehng o locae ooh conac aea baed on u,, paamee can; 3 expe he conac aea n face-gea body fxed coodnae yem va coodnae anfom max Eq Fo each, he value of u, defne an nananeou conac lne beween he eeh of face-gea and hape pnon. The conac lne conue he face-gea ooh uface. The flowcha of face-gea geneaon lluaed n Fgue 3.4. Fg. 3. 4: Flowcha of face-gea geneaon [7]. 37

54 Fg. 3. 5: Face-gea ooh and conac lne []. Fgue 3.5 how a ample geneaed face-gea ooh and conac lne. Each conac lne epeen he locaon whee face-gea and hape pnon eeh neac wh each ohe n an nan. The meh cycle a fom econ B a he oue edge of face-gea ooh and end n econ A a he nne edge. I evealed n Fgue 3.5 ha conac lne connuouly change he lengh and locaon n one meh cycle. Th vaaon ndcae he meh load move effecve poon peodcally dung oaon, whch poble o exce dynamc nably. The co econ of a face-gea dve hown n Fgue 3.6 o lluae he facegea ooh adu calculaon. The nne edge of face-gea ooh Secon A lmed by nonundecung condon. I avod he undecung on he flle uface ha unde he bounday cuve L p n Fgue

55 39 Fg. 3. 6: Inne and oue lmng end of face-gea ooh [, 8]. The equaon fo nonundecung condon ae gven [8] whee y x f f u f v y u y v x u x 3-9 z x f f u f v z u z v x u x 3-

56 y u z y 3 vz u f u y z f v f 3- Hee, f he equaon of mehng n Eq. 3-7, x, y, z he poon veco of he hape ooh uface n Eq. 3-, and v x, v y, v z he elave velocy veco n Eq The ccal pon of undecung locae on he neecon of he hape addendum and he bounday cuve L p on he face-gea ooh. The pola angle of he hape addendum n Gauan coodnae, denoed by, [8] a b 3- b wh he adu of he hape addendum ccle, a. The upecp ae ndcae he paamee fo he ccal pon of undecung. The nne lmng end of he face-gea ooh found by followng pocedue: npu no any equaon of Eq. 3-9, Eq. 3-, and Eq. 3- o olve he paamee ; ubue and no he equaon of mehng Eq. 3-7 o deemne he paamee u ; 3 locae he coodnae x, y, z of he ccal pon of undecung by Eq. 3- and paamee u,. The nne lmng value, L, obaned by [7] L d z and an whee d he adu of he hape dedendum and m + =8. 4

57 Fg. 3. 7: Schemac of ponng poon and co econ of plane and [, 8]. 4

58 The oue edge of face-gea ooh Secon B n Fgue 3.6 lmed by ponng condon. Ponng mean he plane of boh ooh de mee a a pon whee he ooh hcne equal o zeo maed n Fgue 3.5. The plane conanng ponng poon hown n Fgue 3.7. The oue lmng value, L, gven by [8] N a S g NS coc co L 3-4 Pd an an an Pd an co whee P d dameal pch, a g he hape ooh addendum, and, ae olved va Eq. 3-5 and Eq. 3-6 epecvely [8] N n 3-5 N coc co m co 3-6 m n Conequenly, he coepondng nne and oue lmng ad of face-gea ooh, meaued on he plane of he face-gea body d, ae epecvely Fgue 3.6 R L n 3-7 R L n Conac Pon and Conac Cenod The ooh ze much malle han he face-gea dmenon, o doe he lengh of conac lne. Th deaon aume he meh load ha dbue along he ooh uface a a lumped one acng on he mdpon of each conac lne n un. Th mdpon called Conac Pon and maed n Fgue 3.8 wh x ymbol. 4

59 a 3D vew b Sde vew [] Fg. 3. 8: Conac pon on face-gea ooh. The conac pon move fom he oue end of he face-gea ooh o he nne end peodcally a gea oaon. The oal me of one engaged ooh pa unnng fom nal conac o ou of mehng, T e, T e c 3-9 T and T m and m N 3-43

60 whee c conac-ao, T meh peod, m meh fequency and face-gea oaon peed. The adal and ccumfeenal poon of conac pon, and, fo fou ucceve mehng ooh pa on an example face-gea dve ae ploed n he pace fxed coodnae fame ovelayng he face-gea uface, hown n Fgue 3.9. Fo clocwe oaon decon fom he op vew of face-gea, boh adal and ccumfeenal poon of he conac pon connuouly vay fom he maxmum o mnmum dung one meh cycle, and hen he mehng ooh pa un ou of conac. Fg. 3. 9: An example of effecve adal and ccumfeenal poon of conac pon and phae lag beween dffeen mehng ooh pa, fo c =.5. Fo he conac-ao geae han one, mulple pa of eeh may ay n meh mulaneouly bu wh dffeen phae lag. Fo example, a.7 conac-ao mean one pa of eeh ae alway n conac and 7% of he meh me wo pa ae n conac []. 44

61 The conac-ao n Fgue I ndcae ha he f and econd pa ay n meh mulaneouly fo half meh peod, fom me A o me B. Afewad, he f pa goe ou of conac and he econd pa eep n engagemen alone fo he e of he meh peod unl he hd pa begn o meh, fom me B o me C. The cycle epea n he ame paen a gea oaon. Fo a non-uny conac-ao, he numbe of mehng pa flucuae dung one peod, whch caue he vaaon of he oal meh load. Moeove, fgue 3.9 alo lluae he phae dffeence beween adjacen mehng pa T. Fo he anvee vbaon analy, one-pon-conac equed n ode o ncopoae he effec of he pnon beang on he face-gea dynamc. Theefoe, a mplfed aegy employed o combne all conac pon n meh by an equvalen conac pon, called Conac Cenod and lluaed n Fgue 3.. Fg. 3. : Conac cenod, fo < c <. 45

62 The conac cenod poon c, θ c calculaed n pace fxed fame, θ by he followng equaon: c c cel j j cel j c cel c c cel c j [ j l [ l [ l j j [ l ] j j ] j ] ] 3- whee j, θ j he conac pon poon of he j h mehng pa, l j he coepondng conac lne lengh, and cel he funcon ha ound a numbe upwad owad neae nege. Hee, he conac lne lengh ued a a wegh paamee o evaluae he conac cenod poon. Boh conac pon poon and conac lne lengh depend on gea oaon angle, alo on me. Fgue 3. lluae he poon of conac pon and coepondng conac cenod on face-gea plane fo.5 conac-ao. When he face-gea oaon angle me beween angle A and angle B ndcaed n Fgue 3.9, wo mehng pa ae n conac mulaneouly and he conac cenod locae beween wo conac pon ee Fgue 3.a. When he face-gea oaon angle beween angle B and angle C, only one mehng pa engage and he conac cenod ovelap wh h conac pon ee Fgue 3.b. The poon of conac pon and coepondng conac cenod dung wo adjacen meh peod fo a ample face-gea dve ae ploed n Fgue 3.. The adal and ccumfeenal poon ae hown n Fgue 3.a and Fgue 3. b epecvely. 46

63 a Beween me A and me B b Beween me B and me C Fg. 3. : Sech of conac pon and coepondng conac cenod, fo c =.5. 47

64 a Radal poon b Ccumfeenal poon Fg. 3. : Poon of conac pon and coepondng conac cenod, fo c =.5. 48

65 3.7 Appoxmae Movemen Pah of Conac Pon To ealy expand he mehng nemac a Foue ee fo peubaon analye, a mplfcaon aegy employed n he oonal vbaon nvegaon. Th aegy o appoxmae he movemen pah of conac pon a a ew lne fo each mehng ooh pa ndvdually, and hen upepoe he meh load fom all mehng pa bu wh he phae lag, hown n Fgue 3.3. I hould be noed ha he lope of he ccumfeenal vaaon appoxmae o one. Th ndcae he conac pon oae wh he face-gea a almo he ame peed o ha ccumfeenal vaaon elave o he face-gea body can be negleced. Fg. 3. 3: Appoxmae loc fo he poon of conac pon, fo c =.5 []. 49

66 Baed on h mplfcaon aegy, he appoxmae adal and ccumfeenal poon of conac pon fo a mehng ooh pa,, θ, ae expeed n pace fxed fame, θ a: R R R ct no conac no conac ct ct T cel c ct ct T cel c 3- whee θ nal angle. Whou lo of genealy, he nal angle e a zeo fo mplcy. 3.8 Conac-Rao Appoxmaon Conac-ao an mpoan paamee fo gea dynamc nvegaon, bu no exac cloe-fom fomula ex fo pu pnon/face-gea dve. Snce he numecal mehod, baed on dffeenal geomey and heoy of geang, ae me-conumng [7, 8], h deaon employ o-called Tegold appoxmaon [84] fo face-gea conac-ao calculaon. The dea o mage a pu gea nead of he face-gea o meh wh he pu pnon, whch anfom he face-gea dve no a fomave pu gea dve. Th mehod a common pacce ued n bevel geang and lluaed n Fgue 3.4. The apex emangle of pnon and face-gea pch cone, and, ae gven n [8]: m co m co ac co and co ac 3-3 n m n 5

67 whee he gea ao m equal o N /N, N he numbe of pnon eeh and N he numbe of face-gea eeh. a Fomave pu gea [8, 84] R vg I R g A P R vp O R p b Mappng fom face-gea dve o pu gea dve [, 84] Fg. 3. 4: Tegold appoxmaon. 5

68 by [84]: The numbe of eeh fo he fomave pu gea dve, N v and N v, ae calculaed N N v v N / co N / co 3-4 The conac-ao of face-gea dve appoxmaed by he fomave pu gea conacao. The analycal fomula fo he conac-ao of pu gea [8]: N v an va an c N v an va an c c 3-5 co, 3-6 va vb whee α c, α va, vb, va ae epecvely he peue angle, he addendum peue angle, he bae ccle adu, and he addendum ccle adu of he fomave pu gea. Hee, va obaned n he uual way by addng ooh addendum o he pch adu. va 3.9 Summay Th chape noduce he analycal geomey of face-gea dve, developed by Lvn e al. The face gea ooh uface geneaed hough a conjugaed mehng wh he hape. The conac lne beween face-gea and hape fom he face-gea ooh uface. Eenally he ooh geneaon o fnd he mahemacal expeon of he conac lne and he lmng value of he ooh. Baed on he nown paamee of he hape, he poon veco of conac lne on hape ooh uface ae olved va he equaon of mehng. Nex, a coodnae anfomaon appled o oban he poon veco of conac lne on face-gea ooh uface. Though he paamee can, he conac lne whn one meh cycle conue he face-gea ooh uface. The nne and 5

69 oue end of he face-gea ooh ae epecvely lmed by nonundecung and ponng condon. Snce he pu pnon an dencal copy of he hape, all paamee of he hape ae he ame a hoe of he pnon. The conac pon defned a he mdpon of each conac lne. The movemen pah of he conac pon can be decly obaned hough he mahemacal expeon of he conac lne. Fo he anvee vbaon of face-gea, he concep of conac cenod ulzed o collec he effec of mulple conac pon ha ae n meh a he ame me, when he conac-ao geae han one. Fo he oonal vbaon, he movemen pah of he conac pon appoxmaed a a ew lne fo each mehng ooh pa. The meh load fom mulple mehng ooh pa ae upepoed afe calculang each meh load fom a ngle pa ndvdually. Fnally, Tegold appoxmaon employed o calculae face-gea conac-ao. Th mehod anfom a pu pnon/face-gea dve o a fomave pu gea dve va aumng a pu gea nead of he face-gea o meh wh he pu pnon. Snce no exac cloed-fom fomula ex fo he face-gea conac-ao calculaon, h appoxmaon povde an effcen mehod o ave he complex numecal compuaon baed on he face-gea dffeenal geomey. 53

70 Chape 4 STRUCTURAL DYNAMICS MODEL AND MACROSCOPIC DESIGN FOR FACE-GEAR DRIVES WITH A SPUR PINION 4. Ovevew Th chape o eablh a ucual dynamc model fo anvee vbaon of face-gea dve wh a pu pnon and povde nfomaon fo he macocopc degn of face-gea body baed on ably and e conan. The face-gea modeled a a pnnng annula d accodng o Kchhoff plae heoy and he meh load modeled a a un wh he pecbed unque face-gea nemac. Th meh load un compe pnon ma, pnon beang ffne, and pnon beang dampng. The ou-of-plane movemen of he un couple wh he anvee vbaon of he pnnng d. The eeach on he anvee vbaon of pnnng d ha a long hoy owng o wde applcaon n ccula aw, compue oage devce, vehcle bae, and gea anmon. The naual fequence of anvee vbaon wee nvegaed heoecally and expemenally fo annula d wh pe-ee eulng fom oaonal cenfugal foce by [85-86]. Adam [87] and Renhaw [88] deved he ccal peed fo floppy d on elac foundaon and n gd encloue epecvely. Pe and Tan [89] conduced a paamec udy on he laeal nably of a d unde 54

71 peodcally vayng angula peed. The ably and dynamc epone analye of he pnnng d ubjeced o aonay anvee load, ffne, dampng, ma, foce o he combnaon, wee nenvely mplemened n [9-93]. Moeove, Chen and Bogy [94-95] oo no accoun pchng load, mang he d vecal plane lng, bede anvee load. Exa ngle degee-of-feedom wa added o he connuou d yem fo couplng an ocllang load n [96-97]. The epone of pnnng d o a movng foce wa calculaed by [98-99] and he ably of pnnng o aonay d unde oang load wa analyzed by [-3]. Pe e al. [4] developed a geneal model o udy he paamec nably nduced by magnec head ecpocaon n had d dve, whch mla o he face-gea model popoed n h deaon. In addon, he pnnng d flue wa wdely nvegaed by [5-9]. Manzone and Hayfeh [- ] exploed he nably mechanm of oang d unde aonay load baed on nonlnea plae heoy. Alo, he conol aege wee developed o uppe he vbaon of pnnng d yem wh vaou nable n [-3]. 4. Sucual Dynamc Model of Face-gea Dve wh a Spu Pnon To mplfy he face-gea model and he ably analy, majo aumpon made n h deaon fo he ucual dynamc model of anvee vbaon of face-gea dve ae a follow: The pu pnon an dencal copy of he hape geneang he face-gea. So, he pnon ha a lne conac mode wh he face-gea and all paamee of he pnon ae he ame a hoe of he hape. The face-gea beang, pnon body and gea eeh ae gd. 55

72 3 The face-gea body an elac annula d. I nne edge clamped and oue edge fee. 4 The meh load ha pecbed n-plane movemen and made up of he pnon ma and he pnon beang. 5 The whole yem well manufacued and nalled whou any malgnmen. Fg. 4. : Face-gea ucual dynamc model: pnnng elac d and a concenaed mehng load un ha move a he unque face-gea mehng nemac. 56

73 The ucual dynamc model of he face-gea dve wh a pu pnon lluaed n Fgue 4.. Baed on aumpon above and he geomecal analye n chape 3, he face-gea modeled a oopc, homogeneou, elac annula d wh unfom hcne baed on Kchhoff lnea plae heoy. I nne bounday gdly clamped whle he oue one fee. The meh load of all ooh pa n meh ae accumulaed and modeled a an equvalen concenaed load un conacng he facegea uface vecally a he conac cenod. The n-plane movemen of he conac cenod eul fom he unque face-gea mehng nemac decbed n econ 3.6. Th concenaed load un compe pnon ma, pnon beang ffne, and pnon beang dampng. Snce he pnon gd, he ou-of-plane dplacemen and velocy of he pnon beang ffne and dampng ae he ame a hoe of he d a he conac cenod. The face-gea/pnon model clafed a a pnnng d / movng load yem. Some mla yem, fo example had d/magnec head yem [4], have been nvegaed. Followng he wo [85-4, -3], he govenng equaon-of-moon fo he face-gea model expeed n pace fxed pola coodnae, θ a: w h w h w w D w 4 w 57 c w 4 D w c m p w c w w b b 4- whee w, θ, he face-gea d anvee deflecon and 4 = he bhamonc opeao. Hee maeal deny of face-gea, h face-gea d hcne, he vcou lo faco, D=Eh 3 /- plae bendng ffne wh Young Modulu E and Poon ao. Alo, m p pnon ma, b pnon beang ffne, and c b

74 58 pnon beang dampng. Fnally, Dac dela funcon,, ued o ndcae ha he concenaed meh load un ac on he conac cenod, c, θ c, defned n Eq. 3-. The cenfugal membane pe-e due o oaon a ymmec e feld, gven by [3-4] a b a b a a b a b a 4- wh ] / 8[ ] / 3 [ ] / 8[ ] / 3 [ / 4 a b a b b a b a b a b b 4-3 hee, a and b ae he oue and he nne ad of he d epecvely. The clamped bounday condon a =b and he fee bounday condon a =a ae [86, 3] / a a b b w w w w w w w w Syem Dcezaon The egenfuncon of he mple elaed poblem can be ued a admble funcon fo he ognal poblem [5]. The aonay d poblem he aocaed

75 59 mple poblem fo he face-gea model n Eq Eq A he aonay d egenfuncon, Beel funcon have been ulzed a he adal mode funcon o olve vaou poblem conanng annula d [89, 93, 97-98, -, 4, 6-9]. Hence he oluon of Eq. 4- aumed o be n he fom: M m N N n mn n mn e R w,, 4-5 wh K D I C Y B J A R mn n mn mn n mn mn n mn mn n mn mn 4-6 whee m and n ae epecvely he numbe of nodal ccle and nodal damee of vbaon mode m, n, mn he coepondng genealzed coodnae and =-; J n, Y n ae Beel funcon and I n, K n ae modfed Beel funcon of ode n. The conan coeffcen, A mn, B mn, C mn, D mn and paamee, β mn, ae deemned by bounday condon, Eq. 4-4, and he ohonomaly condon below a b j jn n j j d R R, 4-7 By applyng Galen mehod, he yem equaon-of-moon can be paally decoupled, va ubung he aumed oluon Eq. 4-5 no Eq. 4-, mulplyng he eulng equaon by R mn e -nθ, and hen negang ove he doman {b a, θ π}, a, q N N q b q b q p c q n q n n n n n n n n η η c η m h e η n n η n η c H I L β β I β 4-8 wh

76 6 ] full[ ] full[ ] dag[,,,,,,,,,,, n 4 R R h wh h d R R n R R wh of ame ze an deny max wh he h D c q m c n m c m m nq c m m nq c q n a b n m n m n m n m n m m n m m n mn n H L β I β 4-9 whee η n =[ η mn ] T fo m =,,, M, epeen dffeenaon wh epec o me, dag[] epeen dagonal max o dagonal bloc max, and full[] epeen full max. By collecng all geneal coodnae, q=[ η nt ] T fo n = -N,, -,,,, N, he eulng equaon-of-moon ae obaned a: ] [ ] [ ] [ q q q K K C C M M 4- wh h e h c e h m e n n n b c q n q n b c q n q n p c q n q n n n n n c c c ] full[ ] full[ ] full[ ] dag[ ] dag[ ] dag[,,, H K H C H M I β L β K I β C I M 4- In ode o ue Floque heoy o deemne he ably of he peodcally mevayng yem decbed by Eq. 4-, f eca no he andad ae-pace fom a: T T T q q wh X X A X 4-

77 6 and ] [ ] [ ] [ ] [ C C M M K K M M I A 4-3 The e o deemne he ably by followng he mehod gven n econ Pnon Paamee Pnon paamee ae ecaled o oban a bee undeandng n he phycal meanng of he effec of pnon paamee on he yem ably. Fly, he ffne of face-gea body choen a he efeence quany o cale he pnon beang ffne. In he face-gea model, h body ffne value can be numecally evaluaed va dvdng a anvee foce by he d eady-ae dplacemen poduced by he foce. Th poce ummazed below a: X C B X A d d d d d y F 4-4 whee F d he anvee foce acng on he face-gea d and ], [ ], [ - d d n d mn T n d mn e R e R h d d d C B C M K M I A 4-5 By olvng Eq. 4-4, he face-gea body ffne, d, obaned a d d d B A C,, d d d d y F h 4-6 The ffne of face-gea body depend on oaon peed due o he cenfugal e feld, effecve poon, and hcne. Hee, he efeence he aveage value of

78 he body ffne a he face-gea ooh cene ove he ene peed ange. The nondmenonal N.D. beang ffne,, defned a he ao of pnon beang ffne o he efeence b 4-7 R R avg[ d,, h] whee avg[] epeen he aveage opeao ove he ene oaon peed ange, and R and R ae gven epecvely n Eq. 3-7 and Eq The popoonal beang dampng,, he ao of pnon beang dampng o he pnon beang ffne, c b 4-8 b The N.D. pnon ma,, he ao of eal pnon ma o he nomnal pnon ma whch calculaed by eang pnon a a old cylnde wh he pnon dmenon, mp L L 4-9 p Hee, p he pch ccle adu of he pnon, and L and L ae gven epecvely n Eq. 3-3 and Eq Bendng Se of Pnon Tooh The maxmum bendng e occu a he oo of pu pnon ooh. Th e cloely elaed o he pnon geomey, npu powe, and oaon peed. Fo mplfcaon, he e value emaed by eang he ooh a a canleve beam wh a dbued load on he pch ccle [, ], lluaed n Fgue 4.. 6

79 Fg. 4. : Pnon ooh geomey and oo e []. The maxmum nomal e, max, a he ooh oo calculaed appoxmaely by he followng fomula: 6F H max 4- LW wh F P and L L L 4- p whee H dedendum, L ooh face wdh, W ooh oo hcne, F angenal load on pch ccle, P npu powe, and he pnon oaon peed. The maxmum bendng e can no be geae han he yeld engh, σ Y, wh ome afey faco, SF, namely Y max 4- SF 63

80 4.6 Sably Reul Two ample face-gea dve ae employed o demonae he ably analy mehod developed above. The paamee of he face-gea dve ae gven n Table 4.; he coepondng paamee n he face-gea model ae hown n Table 4.; and he maeal chaacec ae led n Table 4.3. Fuhemoe, he convegence e ae caed ou by examnng he lage eal pa of he yem egenvalue a he nceae of mode M and N. I eveal ha M= and N=3 can yeld good eul fo he model of he wo ample face-gea dve. The paamec nably numecally deemned fo vaou meh load paamee and oaon peed. Table 4. : Face-gea dve paamee Paamee Cae I Value Cae II Pnon numbe of eeh, N 6 3 Gea ao, m 5 Module, m d 3.75 mm [P d = 8 /n] Shaf angle, γ 9 Peue angle, c 5 64

81 Table 4. : Face-gea model paamee Paamee Cae I Value Cae II Face-gea body hcne, h.3a.5a Oue adu of d, a Inne adu of d, b Inpu peed ange, Ω R nγ 5 mm ~ 5 pm Mode numbe paamee, M Mode numbe paamee, N 3 Table 4. 3: Maeal chaacec Paamee Cae I Value Cae II Gea deny, ρ 784 g/m 3 Yeld engh, σ Y Young modulu, E Mpa Gpa Poon ao,.3 Vcou lo faco, ξ 5e-6 ec 65

82 F of all, he appoxmae conac-ao of face-gea dve calculaed by Tegold appoxmaon decbed n econ 3.8. The conac-ao of face-gea dve wh pependcula neecng haf angle hown n Fgue 4.3. Fg. 4. 3: Appoxmae conac-ao of face-gea dve, m = 5, =9 and P d = 8 /n. The yem ably eul fo wo dffeen face-gea dve ae dplayed below. Fo Cae I, he numbe of pnon eeh 6 and hcne 3 pecen of face-gea ooh oue adu, hown n Fgue 4.4; fo Cae II, he numbe of pnon eeh 3 and hcne 5 pecen of face-gea ooh oue adu, hown n Fgue 4.5. The blac egon ndcae he paamec nably of anvee vbaon of face-gea dve n h deaon. The effec of pnon beang ffne on yem able ae exploed n Fgue 4.4a and Fgue 4.5a. The vecal ax N.D. beang ffne and he hozonal 66

83 ax face-gea oaon peed. I ndcaed ha he yem unable fo ome lowe beang ffne level bu able fo ohe level. The afey zone of he yem alo maed. Fgue 4.4b and Fgue 4.5b how he elaonhp beween pnon beang dampng and yem nable. The vecal ax popoonal pnon beang dampng and he hozonal ax face-gea oaon peed. The ably eul baed on pnon ma ae dawn n Fgue 4.4c and Fgue 4.5c. The vecal ax N.D. pnon ma and he hozonal ax face-gea oaon peed. Paamec nable can be clealy obeved fom Fgue and depend on oaon peed, pnon ma, beang dampng, and beang ffne. 67

84 a Inable due o pnon beang ffne and oaon peed, =, = 3-6 ec. b Inable due o pnon beang dampng and oaon peed, =.6, =. Fg. 4. 4: Face-gea/pnon yem paamec nably, Cae I: N = 6, N = 3, a = 3.94 mm, h =.3a, c =

85 c Inable due o pnon ma and oaon peed, =.6, = 3-6 ec. Fg. 4. 4: Connued 69

86 a Inable due o pnon beang ffne and oaon peed, =, = -6 ec. b Inable due o pnon beang dampng and oaon peed, =.5, =. Fg. 4. 5: Face-gea/pnon yem paamec nably, Cae II: N = 3, N = 55, a = mm, h =.5a, c =

87 c Inable due o pnon ma and oaon peed, =.5, = -6 ec. Fg. 4. 5: Connued 7

88 4.7 Sably Baed Face-Gea Macocopc Degn In h chape, he macocopc degn fo face-gea body dcued. The ably baed degn nclude wo conan: maxmum bendng e and yem ably. The man pocee ae ummazed n Fgue 4.6. Fg. 4. 6: Flowcha of he ably baed face-gea degn []. 7

89 The maxmum bendng e a he oo of pnon eeh mu be le han yeld engh wh ome afey faco. The e conan able o affec face-gea ze becaue he face-gea and pnon have he ame ooh face wdh. The yem ably cloely elaed o face-gea body hcne. Baed on he dynamc vewpon, he mnmum hcne neceay fo uanng ably ove ene opeang peed ange obaned. Boh conan can help decde face-gea dmenonal lm n macocopc degn, and alo deemne he face-gea wegh. Hee, a pooypcal 6HP cla facegea dve uded. Two degn ae exploed n h analy: one baed on conan pnon dameal pch; he ohe baed on conan pnon pch adu. The degn equemen ae led n Table 4.4, and ohe paamee ae gven n Table f no ohe pecal agnmen. Table 4. 4: Degn equemen Paamee Maxmum npu powe, P Value 6 hp@ =5 pm Pnon numbe of eeh, N ~ 35 Se afey faco, SF Degn wh conan dameal pch The face-gea ooh face wdh deemned by ponng and nonundecung condon and calculaed by he econd equaon of Eq. 4-. In h degn, he pnon and he face-gea have he ame ooh face wdh, and he bendng e mu be lowe han he yeld engh wh a afey faco of.5. The ooh face wdh and he 73

90 maxmum e a he oo of he pnon ooh ae epecvely ploed n Fgue 4.7 and Fgue 4.8 fo dffeen numbe of pnon eeh unde a conan dameal pch. Fg. 4. 7: Tooh face wdh fo dffeen numbe of pnon eeh, P d = 8 /n. Fg. 4. 8: Maxmum ee a he oo of pnon ooh fo dffeen numbe of pnon eeh, and 6 5 RPM, P d = 8 /n. 74

91 The gea dmenon hee ae baed on he conan gea ao and conan dameal pch aumpon. Conequenly, he gea wh hghe numbe of eeh bea lowe ooh bendng ee bu n eun have bgge ze []. The aea below he dah lne n Fgue 4.8 e afe. The objec of h degn a 5: educon ao face-gea dve. Unde he ame dameal pch, hee cae fo dffeen choce of numbe of pnon eeh ae analyzed. The face-gea oue adu nceae wh he numbe of eeh. The paamec nably zone, wh epec o face-gea body hcne and oaon peed, fo each of hee cae ae dplayed n Fgue The vecal axal he ao of face-gea body hcne o oue adu and he hozonal ax face-gea oaon peed. 75

92 a N = 5, N = 5, a = 3.4 mm, p = mm, c =.55 b N = 8, N = 4, a = 49 mm, p = mm, c =.559 Fg. 4. 9: Face-gea/pnon yem paamec nably baed on face-gea body hcne and oaon peed, =, =.8, = -5 ec, P d = 8 /n.

93 c N = 3, N = 6, a = 83 mm, p = 5.8 mm, c =.569 Fg. 4. 9: Connued 77

94 The mnmum face-gea body hcne ao h/a neceay fo uanng he yem ably ove he ene opeang peed egon deemned fom he ably plo, called ccal hcne ao. The value fo dffeen opeaon peed egon can be ead decly fom Fgue 4.9a-c. Fo example, h/a.375 fo he opeang peed [~3] pm, whch decded by he unable lobe A and h/a.394 fo [~] pm, whch decded by he unable lobe B, a hown n Fgue 4.9a. I hould be noed ha he ccal hcne ao choen above unable lobe C ahe han below becaue lowe peed ange he pmay opeang zone n many applcaon; he ohe eaon he aea below he unable lobe C have mall hcne o ha he dplacemen epone of he face-gea may each an unaccepable hgh level. Hence, he aea a boom called epone unafe zone and alo maed n Fgue 4.9. Once he ccal hcne ao gven, he mnmum allowable hcne, h c, obaned and called ccal hcne. The annula cylnde volume fomula ued o evaluae a nomnal volume of he face-gea body, V f =πa -b h c. Conequenly, he poduc of he nomnal volume and he maeal deny emae he coepondng nomnal wegh. The degn baed on he yem ably and e conan ae exploed fo dffeen ooh coun, wo opeaon peed egon, a 5: educon ao, and conan dameal pch P d =8 /n. The eul ae hown n Fgue 4.. I eveal ha he degn equemen fo wegh and e conflc wh each ohe becaue he lowe e level degn end o be le able and heefoe eque lage value of face-gea body hcne o uan ably []. Alo, h ade off degn depend on he deed opeang peed ange. 78

95 Fg. 4. : Maxmum pnon ooh oo e fo 5: educon degn VS face-gea ccal hcne fo vaou pnon ooh coun and opeang peed, P d = 8 /n Degn wh conan pnon pch adu Fo he degn wh conan pnon pch adu, he ooh face wdh and he maxmum ee a he oo of he pnon ooh ae epecvely ploed n Fgue 4. and Fgue 4. fo dffeen numbe of pnon eeh. The face wdh nveely popoonal o he numbe of pnon eeh. The e nceae a he pnon ooh coun gowh, becaue lage ooh coun lead o bgge dameal pch unde a conan pch adu, whch educe he ooh oo hcne and alo enhance he e level. Howeve, he nceae of he e mall. Boh elaonhp of face wdh / ooh coun and e / ooh coun fo conan pnon pch adu ae conay o hoe n he degn wh conan dameal pch Fgue 4.7 and Fgue

96 Fg. 4. : Pnon ooh face wdh fo dffeen numbe of pnon eeh, p =.9375 nch o p = 49. mm. Fg. 4. : Maxmum ee a he oo of he pnon ooh fo dffeen numbe of pnon eeh, and 6 5 RPM, p =.9375 nch o p = 49. mm. 8

97 Unde he ame degn equemen, a 5: educon ao face-gea dve wh conan pnon pch adu ae alo analyzed. The hee cae n he conan dameal pch degn above ae e-evaluaed fo he conan pnon pch adu degn, hown n Fgue 4.3a-c. 8

98 a N = 5, N = 5, a = 77 mm, P d = 6.45 /n, c =.55 b N = 8, N = 4, a = 75.7 mm, P d = 7.3 /n, c =.559 Fg. 4. 3: Face-gea/pnon yem paamec nably baed on face-gea body hcne and oaon peed, =, =.8, = -5 ec, ρ p = 784 g/m 3, p =.9375 nch o p = 49. mm. 8

99 c N = 3, N = 6, a = 74. mm, P d = 8.6 /n, c =.569 Fg. 4. 3: Connued 83

100 The eul ndcae ha he dbuon of he paamec nably zone mla among hee cae wh dffeen ooh coun. The ccal hcne ao value fo he opeang peed ange [~3] pm and [~] pm ae alo maed n Fgue 4.3a-c. Fgue 4.4 ummaze he eul fo many dffeen ooh coun fo a 5: educon ao degn wh a conan pnon pch adu p =.9375 nch o p = 49. mm. The maxmum bendng e a he pnon ooh oo and he ccal hcne ae dplayed fo dffeen numbe of pnon eeh. Th fgue how ha he ccal hcne value vae a lle fo dffeen pnon ooh coun, o he e he man conan o be condeed. In h cae udy, he conan pnon pch adu ha a good value o eep all degn away fom he yeld engh. Hence, he pnon pch adu he ey paamee n h degn. 84

101 Fg. 4. 4: Maxmum pnon ooh oo e fo 5: educon degn VS face-gea ccal hcne fo vaou pnon ooh coun and opeang peed, wh conan pnon pch adu p =.9375 nch o p = 49. mm. 4.8 Summay Th chape eablhe a ucual dynamc model fo he face-gea dve wh one pu pnon. The face-gea anvee vbaon and he unque mehng nemac ae condeed n h model. A concep of conac cenod popoed o ang no accoun he effec of boh mehng nemac and non-uny conac-ao nemac, and he conac lne lengh eve a he wegh funcon ha cale he conbuon eulng fom dffeen mehng ooh pa a each momen of oaon angle. The face- 85

102 gea conac-ao calculaed by Tegold appoxmaon ploed fo vaou numbe of pnon eeh. Fnally, he paamec ude ae conduced on pnon paamee. The eul clealy how ha he yem nably depend on pnon ma, pnon beang ffne, pnon beang dampng, and oaon peed. The ably baed degn fo face-gea macocopc dmenon exploed baed on ooh e and yem ably conan. The face-gea and pnon have he ame ooh face wdh n degn. The e level a he oo of pnon ooh nveely popoonal o he face wdh whch deemned by he nne and oue lm of he face-gea ooh. Tha o ay nealy all face-gea dve paamee can affec he face wdh. In he meanme, he macocopc dmenon and ooh lmng ad of he facegea ae cloely elaed o he yem ably. Two degn ae exploed epecvely baed on conan dameal pch and conan pnon pch adu. Unde he conan dameal pch, he degn equemen fo wegh and e conflc wh each ohe becaue he lowe e level degn end o be le able and heefoe eque lage value of face-gea body hcne o uan ably. Alo, h ade off degn depend on he deed opeang peed ange. Fo a conan pnon pch adu, he ccal hcnee fo he yem ably ae almo he ame fo vaou ooh coun. So, he ey o degn he pnon pch adu o afy wh he e equemen. 86

103 Chape 5 PARAMETRIC STABILITY ANALYSIS AND SPATIAL CONFIGURATION DESIGN FOR FACE-GEAR DRIVES WITH MULTIPLE SPUR PINIONS 5. Ovevew A lo of helcope equp wh wo o moe engne ee econ., o moe han one npu pnon equed o wo wh mulple powe npu pah n he helcope anmon yem. A a ey degn me o educe oveall anmon wegh, he oque plng echnque alo mplemened hough he face-gea dve wh mulple pnon. Fgue. and Fgue. dplay wo example of facegea/mul-pnon dve and he ech of one face-gea mehng wh wo pnon hown n Fgue 5.. The ably udy on h ype of dve mpoan n ode o lowe noe, mpove opeaon elably and exend evce lfe. The objecve of h chape o develop a ucual dynamc model fo one face-gea mehng wh mulple pu pnon by followng he pevou wo n chape 4. Th model conde peodcally me-vayng meh load, face-gea body flexbly and pnon paamee pnon ma, pnon beang ffne and dampng. The focu o exploe he elaonhp beween he paal confguaon of pnon and he yem dynamc nably unde vaou pnon paamee combnaon and dffeen ooh coun. 87

104 Fg. 5. : Sech of one face-gea mehng wh wo pu pnon. All aumpon fo he mulple pnon cae ae he ame a hoe fo he onepnon cae n econ 4.. Fuhemoe, all pnon mehng wh face-gea ae dencal o he hape ha geneae he face-gea n h chape. Hence he mehng nemac analy and he govenng equaon-of-moon n pevou chape ae ll applcable fo h mulple pnon cae. 5. Phae Dffeence beween Pnon Toal angle beween pnon, c, compe wo pa: poonng angle, q, and phang angle, p, lluaed n Fgue 5.. The poonng angle defned a he negal mulple of face-gea angula pch beween pnon, whch equvalen o he amoun of face-gea eeh beween pnon fom he pnon o he ynchonou pon of he pnon n Fgue 5.b. I eflec he appoxmae elave ccumfeenal poon of pnon. The phang angle he dffeence of oal angle ubaced by poonng angle, whch coepond o he mco unng whn one angula pch fom he ynchonou pon of he pnon o he nd pnon n Fgue 5.b. 88

105 a Top vew of paal confguaon of wo pnon b Ccumfeenal developng dawng of a face-gea dve wh wo pnon Fg. 5. : Illuaon fo oal angle, poonng angle and phang angle. 89

106 Poonng angle change dceely wh a gade dffeence of one angula pch and vaaon conbue nohng o phae bu may caue nably; phang angle vae connuouly whn one angula pch, whch affec boh phae and ably. The ao of phang angle o angula pch defne a dmenonle phae dffeence beween pnon. Conequenly, he phae dffeence beween he coepondng mehng ooh pa a wo dffeen pnon obaned by T j j p wh and N j j T N N 5- and he poonng angle gven by face-gea angula pch: q N N 5- whee j, j and j, j epeen he conac pon poon of he j h mehng pa a wo dffeen pnon epecvely, T meh peod, dmenonle phae dffeence beween pnon, N face-gea angula pch, N numbe of face-gea eeh, and N he amoun of face-gea eeh beween pnon. In addon, c = q + p, ha o ay oal angle equal o poonng angle when phang angle zeo. The dmenonle phae dffeence e fom zeo o one when he ncemen of oal angle le han one angula pch, namely only phang angle change; h phae vaaon occu ove agan a each me when h ncemen exceed one angula pch, ha o ay poonng angle change. The elaonhp of oal angle o dmenonle phae dffeence dplayed n Fgue 5.3. The poon of conac cenod fo he h pnon, c, c, obaned by addng a ubcp o Eq. 3- a 9

107 cel j c j cel j [ j lj ] [ j lj ] cel c c 5-3 cel [ l ] [ l ] c c c j whee j, θ j epeen he poon of conac pon fo he j h mehng pa a he h pnon, and l j he coepondng conac lne lengh. j j Fg. 5. 3: Relaonhp of oal angle o dmenonle phae dffeence fo N =5. Boh conac pon poon and conac lne lengh depend on gea oaon angle, alo on me. The vaaon have he ame peod and ae n phae. By ubung Eq. 5- no Eq. 5-3, he phae dffeence beween conac cenod a wo dffeen pnon gven a c c cel c cel c [ j l j ] c [ l ] j cel j j [ j l j ] c [ l ] j cel j j cel c [ cel c T l j T ] c [ l T ] j j cel j j T [ j T l j T ] T cel c c [ l T ] j j j c 5-4 9

108 9 5.3 Sucual Dynamc Model of Face-gea Dve wh Mulple Spu Pnon Followng he wo n chape 4, he face-gea dve wh mulple pu pnon modeled a a pnnng d wh mulple mehng load. Heeno, he effec of each pnon condeed an ndependen concenaed mehng load un wh a elave phae dffeence. The ucual dynamc model lluaed n Fgue 5.4. Fg. 5. 4: Face-gea ucual dynamc model: pnnng elac d and mulple mehng load un ha move a he unque face-gea mehng nemac. The equaon-of-moon fo he model of face-gea mehng wh mulple pnon obaned by addng mulple mehng load un o Eq. 4- a 4 4 b b p c c w w c w m w w h w D w w D w w w h 5-5

109 93 whee he cenfugal membane pe-ee, and, ae gven n Eq. 4- and Eq. 4-3, and he bounday condon ae gven n Eq Syem Dcezaon By ung he ame appoach n econ 4.3, he govenng equaon can be dcezed va ubung he oluon fom Eq. 4-5 no Eq. 5-5 and applyng Galen mehod. The eulng equaon ] [ ] [ ] [ q q q K K C C M M 5-6 whee b c q n q n b c q n q n p c q n q n n n n n h e h c e h m e n n n c c c ] full[ ] full[ ] full[ ] dag[ ] dag[ ] dag[,,, H K H C H M I β L β K I β C I M 5-7 Nex, Eq. 5-6 eca no he andad ae-pace fom and Floqeu heoy ulzed o deemne he yem ably by he pocedue noduced n econ...

110 5.5 Sably Analy Th chape ae a face-gea mehng wh wo pnon fo example o exploe he elaonhp beween yem ably and vaou paamee of face-gea dve, ncludng: Toal angle beween pnon, c ; Poonng angle, q ; 3 Phang angle, p ; 4 N.D. pnon ma,, N.D. pnon beang ffne,, and popoonal pnon beang dampng,. Thee paamee above ae equed o be degned fo he yem afey accodng o he ably eul. A la, he ably analy pefomed fo a degn baed on wo pace-fxed pnon bu wh vaable numbe of eeh, conan gea ao and conan module. All degn paamee ae led n Table f no ohe pecal agnmen Sably degn baed on oal angle and oaon peed The value of oal angle beween wo pnon vae connuouly, whch affec boh poonng and phang angle. The oal angle choen a a degn vaable fo a gven opeang peed ange. 94

111 a Example A: N = 6, N = 3, h =.5a, =.8, =.5, and = -6 ec b Example B: N = 3, N = 55, h =.7a, =, =.5, and = -5 ec Fg. 5. 5: Face-gea/wo pnon yem paamec nably wh epec o oal angle and oaon peed. 95

112 The yem ably of wo example face-gea/pnon dve Example A and Example B wh epec o oal angle vaaon ploed f o capue he oveall uaon, hown n Fgue 5.5a and b epecvely. The hozonal ax face-gea oaon peed; he vecal ax oal angle beween pnon. The nable can be clealy obeved blac aea and hey ae cu by able aea no awooh hape fom nde. The unable and able aea appea alenaely a he oal angle vaaon. Snce oal angle he ummaon of poonng angle and phang angle, he combned acon caue he yem nably. The vaaon of poonng angle and phang angle epeen he coae unng and he fne unng of paal confguaon of pnon epecvely. The Example B choen fo fuhe analy. Fo convenence, Fgue 5.5b dvded no hee pa: he pa I low peed zone; II medum peed zone; III hgh peed zone. The nably of Example B a low peed zone obcue and occu almo hough he ene oal angle ange. Th may eul fom he eonance of ngle low fequency mode, combnaon of mulple low fequency mode, o facon of hgh fequency mode. A medum peed zone, he nably uaon become clea and ome cloud fom unable aea a ame peed appea nemenly a he nceae of oal angle, e.g. he aea ccled by gey ellpe n Fgue 5.5b. Fo he hgh peed zone, he yem nably vey clea. The dealed elaonhp of poonng angle and phang angle o he yem nably ae analyzed, baed on Example B, n he followng econ. 96

113 5.5. Sably degn baed on poonng angle and oaon peed A a pa of he vaaon of oal angle, he poonng angle vaaon poble o exce dynamc nable accodng o he eul n Fgue 5.5, bu no clea whch unable aea eul fom he poonng angle vaaon. Theefoe, he effec of he poonng angle on he yem ably exploed hee fo degn pupoe. Poonng angle he nege mulple of face-gea angula pch beween pnon, whch equvalen o he pace whee he face-gea eeh beween pnon ae. I value vae dceely wh a gade dffeence of one angula pch. The vaaon of poonng angle eflec a bg adjumen coae unng fo he paal confguaon of pnon, bu conbue nohng o he phae dffeence beween pnon. Fg. 5. 6: Face-gea/wo pnon yem paamec nably wh epec o poonng angle and oaon peed fo Example B. 97

114 The elaonhp of poonng angle o yem ably fo Example B dplayed n Fgue 5.6. The hozonal ax face-gea oaon peed; he vecal ax poonng angle. In ode o ndcae dcee popey, he hozonal gey lne ae dawn n Fgue 5.6 o epeen each value of poonng angle, alo an amoun of facegea eeh beween pnon. The gade dffeence beween hee lne 6.97, equvalen o hee face-gea eeh. The blac aea on hee hozonal gey lne ndcae nably. Th fgue how he poonng angle eponble fo he nable a lowe peed and hgh peed zone I and III. The nably dbuon dependen on oal angle and poonng angle ae hghly mla o each ohe a hgh peed zone III Sably degn baed on phang angle and oaon peed Snce he poonng angle vaaon canno explan he nably a he medum peed zone II n Example B, he unable aea elaed o he phang angle mu be examned. Alo, he awooh hape unable aea n Fgue 5.5 ndcae he yem ably enve o mall angle vaaon. The phang angle vaaon a mco adjumen fne unng o he paal confguaon of pnon, whch vey ueful o mpove he degn whou much paal flexbly. The yem ably may be affeced va changng phang angle o poduce he phae dffeence beween pnon. Baed on Example B, he ably plo wh epec o phang angle and oaon peed fo hee dffeen poonng angle [ q = 9.6, 8.5, 8.] ae dplayed n Fgue 5.7a-c epecvely. The hozonal ax face-gea oaon peed; he vecal ax dmenonle phae dffeence beween pnon defned n Eq. 5-. The dmenonle phae dffeence popoonal o phang angle. The poonng angle 98

115 equal o oal angle when he phang angle zeo o one, o he ably a = o = dencal o he ably along he coepondng gey lne n Fgue 5.6. A maed n Fgue 5.7, he unable aea appea only a low peed zone I fo q =9.6, bu a boh low peed and hgh peed zone I and III fo q =8.5 and q =8.. The gey ellpe and 3 n Fgue 5.7 ndcae he nable a low peed and hgh peed zone fo = epecvely. The nable of Example B a lowe peed zone can no be uppeed compleely by adjung phang angle, wheea hoe a hgh peed zone ae uccefully ablzed va nceang phang angle. Moeove, he unable aea a medum peed zone n Fgue 5.7 do no appea n Fgue 5.6, whch ell he dynamc nably a medum peed zone eul manly fom phang angle fo Example B. 99

116 a q = 9.6, N = 39 b q = 8.5, N = 5 Fg. 5. 7: Face-gea/wo pnon yem paamec nably wh epec o phang angle and oaon peed fo Example B.

117 c q = 8., N = 78 Fg. 5. 7: Connued

118 5.5.4 Sably degn baed on pnon paamee and oaon peed The pnon ma, pnon beang ffne and pnon beang dampng change yem ma, ffne and dampng mace epecvely, o he yem ably cloely elaed o he pnon paamee. Afe he falue of he nably uppeon by angle adjumen n ome degn, he vaaon of pnon paamee povde anohe way o ablze he yem. The yem paamec nably ploed baed on N.D. pnon ma and oaon peed n Fgue 5.8; N.D. beang ffne and oaon peed n Fgue 5.9; popoonal beang dampng and oaon peed n Fgue 5.. I can be found fom Fgue Fgue 5. ha ome paamee combnaon, fo nance [ =.4, =.5, = -5 ec, c = ], ablze he example face-gea dve ove he ene peed ange. Fg. 5. 8: Syem paamec nably due o pnon ma, N = 3, N = 55, h =.7a, =.5, = -5 ec, c =.

119 a = b =.4 Fg. 5. 9: Syem paamec nably due o pnon beang ffne, N = 3, N = 55, h =.7a, = -5 ec, c =. 3

120 Fg. 5. : Syem paamec nably due o pnon beang dampng, N = 3, N = 55, h =.7a, =, =.5, c = Sably degn baed on wo pace-fxed pnon A poble degn baed on wo pace-fxed pnon wh conan gea ao and module uded n h econ. In h cae, he oal angle fxed bu he numbe of eeh vaable. Boh poonng and phang angle ae changed fo dffeen ooh coun o ha he yem nably may be uppeed. Th exploaon can help upgade pe-nalled yem whou any change excep gea. The ably plo fo he face-gea mehng wh wo pace-fxed pnon mouned n hee dffeen oal angle, [ c = 9, c =, c = 8], ae dplayed n Fgue 5.a-c epecvely. The hozonal gey lne n Fgue 5. epeen each numbe of pnon eeh. The blac aea on hee hozonal gey lne ndcae nably. I ha been hown n Fgue 5. ha he degn wh 4 pnon eeh able ove he 4

121 ene peed ange fo hee oal angle and he degn wh pnon eeh alo able fo he damecally oppoed pnon c = 8. Moeove, unde he ame module, he lage numbe of eeh choen, he bgge ma he yem poee. Th educe he yem naual fequence, o he dynamc nably dbuon move owad lowe peed zone. Th concluon can be obeved n Fgue 5.: he blac aea hf along he negave decon of he oaon peed ax a he nceae of ooh coun. 5

122 a c = 9 b c = Fg. 5. : Sably plo fo he face-gea mehng wh wo pace-fxed pnon: conan gea ao m = 5 and conan module m d =3.75mm; h =.7a, =, =.5, = -5 ec. 6

123 c c = 8 Fg. 5. : Connued 7

124 5.6 Summay Th chape exploe he paamec nably of one face-gea mehng wh mulple pu pnon by followng he wo on one face-gea wh one pnon n chape 4. The phae dffeence beween pnon dealedly analyzed baed on he unque facegea mehng nemac and he paal angle beween pnon. The face-gea/mulple pnon yem modeled a a pnnng d wh mulple movng load accodng o lnea plae heoy. A helcope qualy face-gea mehng wh wo pu pnon aen fo example o udy he yem ably wh epec o vaou paamee: oal angle, poonng angle and phang angle beween pnon; pnon ma, pnon beang ffne and dampng. I ha been hown n he eul ha hee paamee can exce dynamc nably. The paal confguaon of pnon affec he yem ably va changng poonng angle coae unng and phang angle fne unng. The nable can be uppeed by adjung paal confguaon of pnon n ome oaon peed zone. When he angle unng fal o ablze he yem o hee no enough paal flexbly o change he pnon eng, he adjumen of pnon paamee povde anohe way o uppe he yem nably. Fo he degn pacce on wo pace-fxed pnon wh vaable numbe of eeh, conan gea ao, and conan module, a uable choce on he numbe of eeh poble o mae he yem able ove he ene peed ange by ulzng he vaaon of ooh coun o poduce dffeen phae beween pnon. In addon, he yem nably move owad lowe peed zone a he nceae of ooh coun unde he ame module. 8

125 In ummay, he dynamc nably of face-gea/mulple pnon yem cloely elaed o pnon paamee, paal confguaon of pnon and oaon peed. To avod nably, he yem may be degned n a gven opeaon peed ange by adjung oal angle beween pnon, pnon ma, pnon beang ffne, o pnon beang dampng. 9

126 Chape 6 FACE-GEAR STRUCTURAL DYNAMICS MODEL CONSIDERING THE EFFECT OF IN-PLANE DRVING FORCE 6. Ovevew The ooh conac uface nehe vecal no paallel o face-gea body, o he meh load affec he face-gea dynamc hough boh ou-of-plane and n-plane componen. In chape 3 and 4, he ou-of-plane componen ha been condeed by modelng he meh load a a vecal load un wh pecbed movemen, bu he nplane effec negleced. Th chape o add h omed pa o he pevou model. The n-plane componen of he meh load epeened by a pace-fxed n-plane dvng foce whch poduce an aymmec e feld n face-gea body o affec he yem ably. The connecon beween yem ably and npu powe eablhed va he n-plane dvng foce. Conequenly, he npu powe become a degn paamee fo he yem ably bede evng a a e conan n pevou degn. The n-plane e feld eulng fom n-plane edge load wee deved fo aonay ccula plae n [-3], and he n-plane fee vbaon of annula d wee analyzed by [4-5]. The aonay d e feld, obaned by Ay e funcon, wa employed o eeach he anvee vbaon fo aonay d n [6-

127 7] and alo fo oang d n [8-9]. In addon, Roen and Lba [3] nvegaed he anvee vbaon of a aonay d unde unfom adal peue expemenally, and Zajaczow [] alo analyzed he anvee ably fo a aonay d ubjeced o peodcally vayng angenal edge load. Snvaan and Ramamu [3] calculaed he eady ae n-plane e epone due o an n-plane edge load analycally by anfomng he poblem of a oang d unde aonay load o he one of a movng load oang aound aonay d. Howeve, h anfomaon gnoe he cenfugal nal e and he Cool effec eulng fom he oaon of he d, whch mae he concluon only vald fo low peed cae. The ymmec cenfugal e feld well nown and ha been gven n Eq Eq The aymmec e feld due o n-plane edge load, condeng he Cool effec, wa analycally olved by Chen and Jhu [3-34]. The eeach nclude naual fequency, ccal peed and eady ae epone. Koh e al. [35] employed a novel movng elemen mehod o numecally calculae he n-plane dynamc epone of annula d, and he eul wee compaed wh he analycal oluon baed on Chen and Jhu mehod [34] bu ung complex Foue-Hanel ee nead of eal Foue-Beel ee. Radclffe and Moe [36] conduced an expemen o nvegae he effec of nplane concenaed edge load on he oang d anvee ably. Ion [37] analyzed he laeal vbaon of a hc d ubjeced o ymmec n-plane e ang fom oaonal o hemal effec. The anvee nably of a oang d wa exploed baed on he aymmec n-plane e feld of aonay d n [8,38]. Chen [39-4] appled h n-plane e eul o he pnnng d anvee

128 vbaon by expandng he concenaed n-plane edge foce a Foue ee. The cae wh unfom n-plane edge load wee nvegaed n [4-43]. Moeove, Chen [44] alo exploed he nonlnea paamec eonance fo he mla poblem. 6. In-Plane Se Feld Th econ o noduce Chen and Jhu wo n [3-34] fo olvng he aymmec n-plane e feld,,,, n pnnng annula d eulng fom a pace-fxed concenaed n-plane angenal edge load. The poblem lluaed n Fgue 6.. Fo he face-gea model, he edge load equal o he dvng foce on he pnon pch ccle, F, bu n he oppoe decon. F calculaed by Eq. 4-. u and u ae n-plane adal and ccumfeenal dplacemen epecvely. Fg. 6. : Spnnng annula d ubjeced o a pace-fxed concenaed n-plane angenal edge load.

129 3 F of all, ome dmenonle vaable ae noduced [33-34] and denoed wh ae, E E E E a E a a u u a u u a b a b 6- The complee n-plane equaon-of-moon ae gven n he dmenonle pace-fxed pola coodnae, a [33-34] 3 u u u u u u u u u u u u 6- and 3 u u u u u u u u u u u u 6-3 The la em n Eq. 6-,, eul fom he cenfugal effec ha ha been condeed n Eq. 4-, o h em negleced n he followng calculaon fo he aymmec e feld due o n-plane angenal load. The dmenonle bounday condon fo n-plane vbaon ae gven [7, 9, 3]: fo clamped edge a = b, b b u u 6-4 and fo he fee edge a =, Ohewe θ ahe F f f f 6-5

130 4 I hould be noed ha he econd equaon of Eq. 6-5 obaned by aumng he concenaed edge load unfomly dbue ove a mall ccumfeenal aea, - f, f and f Ø, a he oue edge. Moeove, can be fuhe expanded a Foue ee [9], n co n nc 6-6 whee n n n co co d ahe F d ahe F ahe F d ahe F d ahe F d ahe F d f f f f f f f f a n f f f a nc f a 6-7 By ulzng Lame poenal L and L [3, 33-34], he coupled equaon Eq. 6- and Eq. 6-3 ae mplfed. Nex, he eady ae oluon ae obaned baed on all em ndependen of empoal devave. Hee, Chen and Jhu wo n [33-34] ae ummazed a followng u u L L L L 6-8 and he eulng equaon ae H G H G L L L L 6-9

131 5 whee L L L L L L L L L L H G 6- and he conan elaed o Poon ao,, ae expeed by 6- The geneal oluon of G L and H L n Eq. 6-9 ae gven a, n co n co g g h h h L h h g g g L B A B A A H B A B A A G 6- Hee A g, B g, A h, and B h ae conan coeffcen. Accodngly, he oluon of L and L ae n he mla fom a Eq. 6-, n co, n co, c c L c c L 6-3 whee he adal funcon, c, c,, c, c,, ae olved by ubung Eq. 6- and Eq. 6-3 no Eq. 6- and hen applyng hamonc balance mehod. Fo =, he nonval oluon eep only em ha affec n-plane dplacemen and ee ae gven Y f J d Y e J c h h c g g c 6-4

132 6 Hee, J and Y ae Beel funcon; c g, e g, d h, and f h ae conan o be deemned by bounday condon; and ae: 6-5 Fo =, he nonval oluon ae ln ln ln ln B Y f J d B A Y f J d B Y f J d B A Y f J d h h h c h h h h g g g g g g g c 6-6 Hee, d g, f g, d h, and f h ae conan o be deemned by bounday condon; and ae epecvely: Fo, he nonval oluon ae Y f Y e J d J c Y f Y e J d J c Y f Y e J d J c Y f Y e J d J c h h h h c h h h h g g g g g g g g c 6-8

133 7 Hee, c g, d g, e g, f g, c h, d h, e h, and f h, ae conan o be deemned by bounday condon; and ae pove eal oo of he followng equaon: and ae epecvely: 6- Fnally, he n-plane e feld ae expeed baed on Lame poenal a L L L L L L L L L L L L L L 6- By applyng he bounday condon n Eq Eq. 6-7 fo each hamonc componen, all conan coeffcen can be deemned. I hould be noed ha he hea e along he oue edge of he d conan only cone componen fo n-plane angenal edge load. Conequenly, he dmenonal e feld ae epeened n a mplfed fom a [4] n, co, n, 6-

134 8 whee,, ae he h hamonc componen of he aymmec n-plane ee. They ae acually he Foue coeffcen of Eq. 6- mulpled by Young modulu E. 6.3 Face-gea Model Condeng In-Plane Dvng Load The n-plane componen of he meh load modeled a a pace-fxed concenaed n-plane angenal dvng foce, acng on he oue edge of he face-gea d. Th foce aumed a followe ype, whch mean decon alway angenal o he ccumfeenal lope of he laeally vbang d a he pon of applcaon [45-47]. The aymmec n-plane e feld due o he n-plane foce have been gven n Eq. 6- a Foue ee and hey ae ncopoaed no he d anvee vbaon by he followng opeao [8, 4]: n co co n h h h 6-3 Snce he e feld n Eq. 6- deved fo he n-plane edge foce a he zeo ccumfeenal poon, =, he nal ccumfeenal poon of he coepondng ou-of-plane load alo anfomed o zeo, namely c =. The facegea model wh boh n- and ou-of-plane meh load lluaed n Fgue 6..

135 9 Fg. 6. : Face-gea dynamc model: pnnng elac d wh pace-fxed n-plane dvng foce and movable ou-of-plane meh load un. By addng Eq. 6-3 o Eq. 4-, he govenng equaon-of-moon fo he facegea model condeng n- and ou-of-plane componen of he meh load obaned 4 4 w w c w m w w h w w D w w D w w w h b b p c c 6-4 I hould be noed ha he ymmec e feld, eul fom cenfugal effec whle he aymmec e feld,, ae poduced by n-plane dvng foce. The followe ype n-plane edge load yeld nohng on he bounday condon of anvee vbaon, o hey ae he ame a Eq Baed on he ame yem dcezaon pocedue n econ 4.3, he max fom equaon-of-moon can be

136 obaned by addng o Eq. 4- a ffne max, K f, eulng fom he n-plane aymmec e feld, ] [ ] [ ] [ q q q f K K K C C M M 6-5 whee, ] full[ q K f K n 6-6 wh Ohewe n q fo d R R nq n q R R n R R q R R n q a R a R a q n q fo d R R n R R q a R a R a q K K a b q m n m q m n m q m n m q m n m q m n m a b q m n m q m n m q m n m q n m m q n m m q n ] full[,,,,,,,,,,,,,,,,,,,,,,, K 6-7 and all ohe coeffcen mace ae peened n Eq. 4-. Nex, Eq. 6-5 ewen n ae-pace fom o deemne he yem ably va Floque heoy: T T T q q wh X X A X 6-8

137 and I A 6-9 [ M M ] [ K K f K ] [ M M ] [ C C ] 6.4 Sably Reul fo Vaou Powe Level The Cae I and II n econ 4.6 ae ecalculaed hee o exploe he effec of he npu powe level on he yem ably. All paamee ae led n Table 4.-Table 4.3. The f fve x fo Foue componen of he aymmec n-plane e feld fo he Cae II,,,, ae epecvely hown n Fgue Fgue 6.5. The npu powe level 8 hp a he npu peed 5 pm.

138 = = =3 =4 =5 Fg. 6. 3: Aymmec n-plane adal e feld,, eulng fom n-plane edge load: N =3, N =55, a=74.54mm, h=.5a, P=8hp@ =5pm.

139 = = = =3 =4 =5 Fg. 6. 4: Aymmec n-plane hea e feld,, eulng fom n-plane edge load: N =3, N =55, a=74.54mm, h=.5a, P=8hp@ =5pm. 3

140 = = =3 =4 =5 Fg. 6. 5: Aymmec n-plane ccumfeenal e feld,, eulng fom n-plane edge load: N =3, N =55, a=74.54mm, h=.5a, P=8hp@ =5pm. 4

141 Condeng he compuaon effcency, only f x Foue componen of he aymmec n-plane e feld ae ued n he anvee ably analy, =~5. The powe level lnealy popoonal o he oaon peed wh a conan lope, ha o ay, conan n-plane dvng foce. The ably eul dependen on he oaon peed and N.D. pnon beang ffne fo he Cae I and Cae II ae ploed n Fgue 6.6 and Fgue 6.7 epecvely a vaou powe level. The maxmum powe level ae e a [ hp, 3 hp, 6 hp]@ =5 pm fo Cae I Fgue 6.6 and [6 hp, 8 hp, hp]@ =5 pm fo Cae II Fgue 6.7. Compang wh he ably eul whou condeng he n-plane dvng load n Fgue 4.4a and Fgue 4.5a, ndcaed ha hgh powe level enlage he unable aea. The eul alo how ha he Cae II able o bea lage n-plane load han he Cae I baed on he ably deeoaon. Th becaue: he gea module he ame n boh cae, o he pnon wh lage ooh coun ha bgge adu; accodng o Eq. 4-, he n-plane foce nveely elaed o pnon adu, conequenly, he Cae I uffe a hghe n-plane foce han he Cae II fo he ame powe level and oaon peed; econdly, he face-gea body hcne n Cae I malle han he one n Cae II, whch yeld a hghe n-plane bounday e level n Cae I unde he ame nplane foce. 5

142 a P= =5 pm b P=3 hp@ =5 pm Fg. 6. 6: Face-gea/pnon yem paamec nably condeng n-plane dve load, Cae I: N = 6, N = 3, a = 3.94 mm, h =.3a, c =.553, =, = 3-6 ec. 6

143 c P=6 =5 pm Fg. 6. 6: Connued 7

144 a P=6 =5 pm b P=8 hp@ =5 pm Fg. 6. 7: Face-gea/pnon yem paamec nably condeng n-plane dve load, Cae II: N = 3, N = 55, a = mm, h =.5a, c =.5668, =, = -6 ec. 8

145 c P= =5 pm Fg. 6. 7: Connued 9