Velocity and Acceleration Simulation of a Vehicle with a Continuously Variable Power Split Transmission

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1 Wold Aademy of Siene, Engineeing and Tehnology eloiy and Aeleaion Simulaion of a ehile wih a Coninuously aiable Powe Spli Tansmission A. Babaei, N. Choupani Absa A oninuously vaiable ansmission sysem has ofen been onsideed fo auomobiles. I offes he poenial o allow he engine o opeae a peak effiieny wihou disubing he dive wih disee shifs. The shaf-o-shaf bel CT is now he mos ommonly used CT podu in he auomobiles. The main dawbak of his kind of CT is limied oque apaiy and he modes powe effiieny. This pevens he bel CT fom being used in he vehiles wih lage displaemen engines. A new onep involves a powe spli funion. A oninuously vaiable powe spli ansmission (CPST) is eaed by ombining a -bel CT wih a planeay gea ain. A sep-up geabox is used o expand he CPST aio fo he appliaions in auomobiles. The CPST enhanes he ansmission oque apaiy and impoves he oveall ansmission effiieny. In his pape he CPST sysem was sudied. Based on he analysis of he vehile dynamis and he CPST sysem, a ompue ode was developed and by using his, he aeleaion of vehiles equipped wih he CPST sysem, sandad (manual) and auomai ansmissions was simulaed. Keywods CT, ehile Simulaion, Tansmission. I. INTRODUCTION LA Aaen is widely onsideed o be he leading expe Oin he field of CT design and uning. He has published nine ediions of he Cluh Tuning Handbook []. Mak Allen and Robe LeMase fom he Univesiy of Tennessee a Main desibe he design of a ansmission involving a CT and a fixed aio gea box in he SAE pape A Hybid Tansmission fo SAE Mini Baja ehiles []. In his pape hey disuss he design of a manual ansmission o be used wih a Salsbuy syle CT. Allen and LeMase used Lagange s equaions o deemine he foes aing in he pulleys. The CT was analyzed as-is wihou onsideing possible modifiaions. The sofwae desibed is A. babaei is wih he Sahand Unvesiy of Tehnology, Depamen of Mehanial Engineeing,P.O.Box , Tabiz, IRAN (oesponding auho o povide phone: ; fax: ; a_babaei@su.a.i). N.Choupani,, wih he Sahand Unvesiy of Tehnology, Depamen of Mehanial Engineeing,P.O.Box , Tabiz, IRAN (oesponding auho o povide phone: ; fax: ; houpani@su.a.i). used as an esimae fo he oveall op speed ahievable based on he gea aio used in he manual ansmission. Mihael A. Kluge and Denis M. Long fom he Souhwes Reseah Insiue desibe seveal ansmission ypes and hei popeies in An Oveview of Cuen Auomai, Manual and Coninuously aiable Tansmission Effiienies and Thei Pojeed Fuue Impovemens [3]. In his Kluge and Long disuss he vaious ypes of ansmissions inluding: auomai, manual and oninuously vaiable. Fo eah ype hey disuss he effiieny as well as he aeas fo impovemen fo eah. They disuss he CT s vaious advanages and disadvanages, and help o deemine he oveall CT effiieny a diffeen opeaing speeds and loading ondiions. A SAE publiaion eniled A Theoeial and Expeimenal Poedue fo Design Opimizaion of CT Bels [4] by Segio Lolli desibes a heoeial and expeimenal mehod fo designing CT bels. In his pape Lolli develops equaions ha deemine he sess wihin he bel and povides a mehod fo deemining he ensile foes seen wihin he bel. Lolli uses a mehod of finie elemen modeling o deemine sesses and uses his infomaion o pedi he lifespan of a CT bel. In Mehanisms and Dynamis of Mahiney [5] Hamilon Mabie and Chales Reinholz desibe geneal mehods fo designing and analyzing am mehanisms. In a CT, he sufae of he flyweigh seves as a am o auae a olle aahed o he moving sheave. As he engine speed ineases, he flyweigh oaes ou, foing he sheaves ogehe and hanging he dive aio. The shape of he am sufae of he flyweigh mus be deemined o pefom a omplee analysis. This was done using a dial indiao as he anslaing olle followe and hen using he esuls found in Mabie and Reinholz o alulae he shape of he am sufae. One his sufae shape is known, i is possible o ompue he oupu displaemen of he olle followe in he CT. Howeve, his poess is ompliaed, anohe aepable mehod is o measue he sheave displaemen based on he oaion of he flyweigh wih a dial indiao. The measued daa an hen be ploed wih a uve fi eaed fo he daa. One he displaemen funion of he followe has been developed, a veloiy plo is found by aking he deivaive of he iniial uve. An aeleaion uve is deemined by pefoming a deivaive on he veloiy. Using his mehod, i is possible o deemine 399

2 Wold Aademy of Siene, Engineeing and Tehnology he enegy developed by he movemen of he pimay sheave. ode em, he quadai oque uve is epesened wih seveal saigh lines desibed as follows[7]: II. SIMULATION EQUATIONS T e q (3) Simulaion equaions fo onsan ansmission aio may ake he following fom : f kma x ( p q ) ( f C D A( w w ) R W sin hx o 3.4 f s ( ) 60.5 ) W Obviously i is vey diffiul o find a heoeial soluion fo Eq. (). If he olling esisane oeffiien is defined as a linea funion of he vehile veloiy fo a fixed gea aio and mehanial effiieny, Eq. () may ake he following fom[6]: d d () P Q C () whee P, Q, and C ae onsan fo eah gea se. Eq. () is vaiable-sepaable, hus, a heoeial soluion is available fo a given se of values of P, Q, and C. Unfounaely, wha kind of fom he soluion will ake depends upon he values of P, Q, and C. Thee is no a unique soluion funion. Fo example, if he igh-hand side of Eq. () is 4 4, he soluion involves /; If i is, he soluion is elaed o an(); while 3 may lead he soluion o ake he fom of an exponenial expession. The uneain soluion of he funion makes i vey diffiul o pogam if i is even possible. To solve his equaion fo vehile veloiy and aeleaion as funions of ime, Eq. () mus be simplified based on some easonable ondiions. The oupu oque of an engine is a funion of he engine speed and he hole valve opening. Fo a vehile equipped wih a sandad ansmission, he hole valve opening is diely onolled by he dive. Diffeen dives will podue diffeen vehile pefomane. Fo a vehile equipped wih an auomai ansmission, i migh now be onolled by an eleoni onol sysem. This allows he dive o sele a pefeed shif shedule mode by means of a onol swih opeaion. The ommon hing fo suh muli-aio ansmissions is ha he engine speed keeps vaying in he vehile aeleaion poess due o shifing. Diffeen shif shedules will podue diffeen engine opeaing pahs, hus poduing diffeen vehile veloiy and aeleaion haaeisis. To simulae he vehile aeleaion pefomane, he fis sep is o define he engine oque funion. Fo a given engine map, he engine oque an be defined as a funion of vehile speed. To avoid he seond Fo diffeen gea aios and segmens of oque uve, q and should be alulaed by using diffeen f values and oque uve paamees. Dag em is elaed o he elaive veloiy of he vehile. The wind speed is assumed o be a onsan fo he simulaion. Hene, he dag will vay wih he vehile veloiy. Theoeially, he dag em should be ombined wih he aion foe em. If he ime ineval of he simulaion is small enough, he veloiy hange fom sep o sep is no lage. I is heefoe possible o use an esimaed veloiy o evaluae he dag. To minimize he eo, he vehile veloiy a he las simulaion sep plus he veloiy diffeene of he las wo seps is used as he esimaed vehile veloiy. This enables he dag o be eaed as a onsan em a eah sep. In suh a way he auay is suffiien. By epesening he oque uve wih seveal linea funions, using he linea equaion fo olling esisane, and eaing dag as a onsan em, Eq. () beomes[8]: d f km ( q ) 0.0( ) sin W d 60 CD A( e w) Rhx e is he esimaed veloiy. When = 0 e = 0; when = 0. seond (simulaion ime ineval), e is assumed o be 0.. The vehile veloiy as a funion of he aeleaion ime an be found by solving Eq. (4). I gives: C Q0 / 3.6 Q ( C 0 ) 3.6 e (5) Q Q The vehile aeleaion, as funions of he aeleaion ime, an hen be obained by finding he fis deivaive of Eq. (5) o ime. I is: a Q Q( 0 ) x C e (6) Eqs. (5) and (6) an be used fo he simulaion of vehiles equipped wih sandad and auomai ansmissions, as well as he CPST when he ansmission aio is a onsan value a low vehile speeds. ehiles equipped wih CPST pesen a diffeen aeleaion pefomane. Due o he sepless aio hange, i is possible fo he engine o wok a he opimum poin a whih i eahes is maximum effiieny (Guo e. al., 988). Fo he aeleaion simulaion of a vehile wih CPST (4) 400

3 Wold Aademy of Siene, Engineeing and Tehnology ombined wih a sep-up geabox, i is assumed ha he vehile aeleaion fom he sai sae o a age veloiy involves hee sages. In he fis sage, he vehile sas o aeleae wih he engine speed ineasing fom he idling speed o he maximum opeaion speed. Afe he engine eahes he maximum opeaion speed, i is possible o say a ha speed, while he ansmission aio will deease wih he inease in vehile veloiy. A he fis sage, a lage oal aio is neessay o ensue ha he vehile gains suffiien aion foe fo aeleaion. The CPST should also disibue he majoiy of powe flowing hough he gea. A highe vaiao aio will lead o a highe CPST aio. A high CPST aio ombined wih a sep-up gea aio should podue a lage ansmission aio. This mahes he lage aio equiemen. I is noed ha he powe flowing hough he bel deeases wih he inease in he vaiao aio. Hene, o use as high a vaiao aio as possible also mahes he equiemen of eduing he powe flowing hough he bel a low speeds. I is heefoe a sma hoie fo he CPST aio o be fixed a a high value a he low speed sage. One he engine eahes is maximum opeaing speed, he fis sage is ove. In he seond sage, he CPST aio sas deeasing wih he inease in he vehile veloiy. When he CPST aio is deeased o is minimum value (o some designed value), he fis sep-up gea is sepaaed and he seond sep-up gea sas o mesh. The seond sage is ompleed and he hid sage begins. The CPST aio should hen be adjused o a eain value o keep he oal ansmission aio he same as befoe swihing. Fo example, if he oal ansmission aio is 3.875~0.7 and CPST aio is.7~0.9, wo sep-up gea aios ae.435 and 0.48, and sep-up gea swih ous a he CPST aio of 0.9; he CPST aio should be adjused bak o.7. In he hid sage, he CPST aio keeps deeasing unil he vehile eahes he age veloiy. The opimum ondiion is ha he vaiao aio of he CPST is bak o when he vehile is aeleaed o he age veloiy, beause his enables he vehile o uise a ha speed wih he maximum ansmission and engine effiieny [9]. I is assumed ha he vehile is aeleaed fom a sai sae o a eain veloiy. A sage I, he engine speed ineases fom he idle speed o he maximum opeaing speed. ehile veloiy ineases wih he inease in engine speed. The CPST akes he maximum aio and he fis sepup gea se is meshed. The fis shif veloiy,, is found by: e max 3. 6 (8) f s max A sage II, he CPST aio oninuously deeases fom is maximum value o he minimum value. The fis sep-up gea is sill in use. The vehile veloiy ineases wih he deease in he CPST aio. The seond shif veloiy is easily found by he following equaion: e max 3. 6 (9) f s min A sage III, he seond sep-up gea is meshed and he CPST aio goes bak o some value (oesponding o he oal aio a he end of sage II), hen again deeases oninuously. This value an be alulaed as: min s (0) s When he vehile veloiy eahes he age veloiy, he CPST aio will be adjused o h whih oesponds o he vaiao aio of, he engine speed is: maxf hs e () 3.6 The vehile aeleaion a sage I is he same as ha of a vehile equipped wih a sandad ansmission. A his sage he vehile veloiy and aeleaion an be alulaed wih Eqs. (5) and (6). Diffeen fom he fis sage, a sage II and III he engine keeps woking a is maximum speed wih a onsan hole opening, while he ansmission aio vaies wih a deease in he vaiao aio. Theefoe, he inpu oque will emain a onsan value, bu he aion foe will be a funion of he ansmission aio and effiieny, whih is also a funion of he ansmission aio. A sage II and III, he ansmission aio vaies wih he hange of vaiao aio duing he vehile aeleaion. A any aeleaion ime, he ansmission aio (inluding he sep-up gea aio) has o obey he following elaionship: e max o () f Whee (3) o s A. CPST Effiieny As desibed befoe, he CT effiieny is a funion of he vaiao aio. The powe loss of a CPST as a funion of he vaiao aio is shown in Figue (). This figue shows ha he maximum effiieny is obained a =. Diffeen aangemens of CPST may podue diffeen powe loss uves, howeve, in mos ases, he powe loss uves of he CPST have simila onfiguaions. Fom his powe loss uve, he CPST effiieny as a funion of he vaiao aio an be found as shown in Figue (). 40

4 Wold Aademy of Siene, Engineeing and Tehnology ansmission simulaion. The mass fao, k, is onsideed a onsan sine he oal ineia is almos he same a hese sages, hough he vaiao aio hanges. Mehanial effiieny is evaluaed using he uen vaiao aio befoe eah simulaion sep. I an be eaed as a onsan fo eah sep. The diffeenial equaion fo hese wo sages beomes: Fig.. Powe loss of CT as a funion of vaiao aio (by Philippe Aques)[0] Te max f kmax o FAD ( f sin ) W R hx (7) Consideing ha he oveall ansmission aio, o, is a funion of he vehile veloiy and he fis em of he ighhand side of Eq. (7) an be ewien as: Te max e max (8) Eq.(7) is simply wien as follows: Fig.. CPST effiieny as a funion of vaiao aio[] The oveall effiieny of he CPST also eahes is maximum value a he vaiao aio of.0. This CPST effiieny uve an be bes fied by wo quadai funions. They ae: (4) ( 0.5 ) And (5) ( ) These wo equaions an be ewien in a genei fom as follows: n s u (6) B. Diffeenial Equaion and Soluions A he seond and he hid sages, hee ae wo hoies o evaluae he olling esisane foe. One is o alulae he olling esisane foe using he veloiy obained fom he las sep; he ohe is using a onsan value fo f. The dag is alulaed wih he same mehod as used in he sandad a x Whee Q P (9) Te e Q max max (0) km P FAD ( f sin ) W Rhx () km By sepaaing vaiables and onsideing he iniial ondiion = 0 a = 0, Eq. () is solved as Q Q P ( 0 ) ln 0 P P Q P () 0 The aeleaion ime equied fo a speifi age vehile veloiy is easily found by using his equaion. The vehile aeleaion a any simulaion sep is diely alulaed fom he vehile veloiy. If negleing he oad esisane foe em, Eq. () beomes: d Q (3) d The soluion of his equaion gives he Powe-Limied Aeleaion pefomane of a vehile. The vehile veloiy, aeleaion, and displaemen a his sae ae defined by following equaions: a x Q( 0 ) 0 ) 0 Q Q( 0 (4) 40

5 Wold Aademy of Siene, Engineeing and Tehnology III. DISCUSSION OF RESULTS To ahieve he age veloiy of 00 km/h, 9.7 seonds aeleaion is equied. The vehile veloiy, aeleaion and displaemen as funions of aeleaion ime as well as ansmission aio as a funion of ime ae shown in figues 3 and 4. Figues 5 and 6 illusae he oesponding vehile veloiy and aeleaion of an auomai ansmission as funions of ime. The aeleaion simulaion of a vehile wih a CPST is based on he given paamees of he CPST wih a sep-up geabox. Combined wih he sep-up geabox wih aios of.55 and 0.46, he oveall ansmission aio span is o 0.694, whih is he same as ha of he sandad and auomai ansmissions desibed befoe. Figues 7 and 8 illusae he oesponding vehile veloiy and aeleaion of a CPST ansmission as funions of ime. Fig.. 5. ehile veloiy as a funion of ime (Auomai) Fig. 3. ehile veloiy as a funion of ime (Sandad) Fig. 6. ehile aeleaion as a funion of ime (Auomai) Fig. 7. ehile veloiy as a funion of ime (CPST) Fig. 4. ehile aeleaion as a funion of ime (Sandad) 403

6 Wold Aademy of Siene, Engineeing and Tehnology I. CONCLUSION Fig. 8. ehile aeleaion as a funion of ime (CPST) As shown in he figues above, he CPST pesens a oninuous aio vaiaion. The veloiy uve, ompaed o he sandad ansmission, seems he same. The aeleaion uve of he CPST, exep fo he sa sage a whih he engine speeds up o is maximum opeaing speed, is smooh and oninuous. The use does no expeiene a jek, making he vehile vey omfoable o ide. Fo he oque-limied aeleaion, if he age veloiy is 00 km/h (6.5 mph), unde he same ondiions, he vehile wih he sandad ansmission needs 9.7 seonds; he vehile wih he CPST needs 7.9 seonds. The diffeene beween hem is.8 seonds. If he age veloiy is 0 km/h (75 mph) he diffeene is 3.7 seonds. The maximum effiieny of he CPST is abou 93.5 %; he minimum effiieny is abou 85%. Compaed o he shaf-o-shaf CT, hey ae ineased fo 3.8% and 4.8%, espeively. The aveage mehanial effiieny of he CPST is abou 89.5 %. A low speeds, he bel aies only 44% of he inpu powe. The powe shifs fom he gea o he bel in a oninuous manne. By using he fis oque uve of he engine used fo he bes fuel eonomy aeleaion wih he auomai ansmission, he aeleaion pefomane of he vehile wih he CPST is simulaed. In his ase, he aeleaion ime is muh shoe han he vehile wih he auomai ansmission. If a good dive diving a vehile wih a sandad ansmission ould follow he opimum pah along whih he bes fuel eonomy is obained, he vehile would no obain as good an aeleaion pefomane as he vehile wih a CPST. Fo he bes fuel eonomy aeleaion, he engine of he vehile wih a CPST opeaes a he bes effiieny ange, exep fo he sa sage. Due o he shifs, he engine of a vehile wih a sandad o an auomai ansmission has o wok following a zigzag pah, hus i is impossible o say a he opimum ange. The bee fuel eonomy fo he vehile wih a CPST is obvious. The vehile aeleaion simulaion displays ha he CPST an povide a oninuously vaiable aio hange and hus a smooh and oninuous aeleaion hange. I is moe omfoable fo he dive and he passenges o ide a vehile equipped wih a CPST. The simulaion of he vehile aeleaion pefomane pesens ha boh he divabiliy and he fuel eonomy fo he vehile wih a CPST ae bee han he one wih a sandad o a muli-aio auomai ansmission. A powe spli funion is available fo he CPST whih expands he powe apaiy of he bel CT. A low speeds, he bel only aies pa of he inpu powe. A equied powe spli peenage is obained by using he ompue pogam o aomplish he sysem design. I is possible o shif he powe flow in he sysem fom he gea o he CT uni in a oninuous manne. REFERENCES [] Ashley, Seven, 994, Is CT he a ansmission of he fuue?, Mehanial Engineeing, Nov. 994, pp [] Avamidis, Sanley, A sae of he a hain diven CT design, SAE Tansaions ol. 95 (986), SAE Tehnial Pape No [3] Ben, D. J., 98, Axial foe and effiieny ess of fixed ene vaiable speed bel dive, SAE Pape [4] Bonhon, Andes, CT-effiieny measued unde dynami unning ondiions, SAE Tehnial Pape No [5] Bowling, Bue, 997, ehile aeleaion pogam, hp://sua.jlab.og/~bowling/ auosim_f.hml. [6] Radke, R. R.; Unneweh, L. E.; Feedman, R. J., Opimizaion of a Coninuously aiable Tansmission wih emission onsains, SAE Tansaions ol. 90 (98), SAE Tehnial Pape No [7] Robins, Fanis B., 993, Coninuously vaiable ansmission, U.S Paen No. 5,83,444, Feb., 993. [8] Takayama e. al, 990, Coninuously vaiable ansmission, U.S Paen No. 5,050,457, Sep. 4, 99. [9] Tanaka, Hiohisa and Imanishi, Takashi, 994, Tooidal ype oninuously vaiable ansmission, U.S Paen No. 5,303,796, Apil 9, 994. [0] Phillips, Andew W., Assanis, Deennis N. and Badgley, Pai, 990, Developmen and use of a vehile poweain simulaion fo fuel eonomy and pefomane sudies, SAE Tehnial Pape No [] Kumm, Emeson L., 99, Coninuously vaiable ansmission using planeay geaing wih egeneaive oque ansfe and employing bel slip o measue and onol pulley oque, U.S Paen No. 5,0,458, Apil, 30,

Lecture 9. Transport Properties in Mesoscopic Systems. Over the last 1-2 decades, various techniques have been developed to synthesize

Lecture 9. Transport Properties in Mesoscopic Systems. Over the last 1-2 decades, various techniques have been developed to synthesize eue 9. anspo Popeies in Mesosopi Sysems Ove he las - deades, vaious ehniques have been developed o synhesize nanosuued maeials and o fabiae nanosale devies ha exhibi popeies midway beween he puely quanum