Performance analysis of a novel planetary speed increaser used in single-rotor wind turbines with counter-rotating electric generator

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1 OP Conference Seres: Mterls Scence nd Engneerng PAPE OPEN ACCESS Performnce nlyss of novel plnetry speed ncreser used n sngle-rotor wnd turbnes wth counter-rottng electrc genertor elted content - Fult Prognoss nd Dgnoss of n Automotve er Axle Ger Usng BF- BP Neurl Network Ymn Sho, Je Lng, Fengshou Gu et l. To cte ths rtcle: Sulescu et l 20 OP Conf. Ser.: Mter. Sc. Eng Vew the rtcle onlne for updtes nd enhncements. Ths content ws downloded from P ddress on /08/208 t 22:59

2 7th nterntonl Conference on Advnced Concepts n Mechncl Engneerng OP Publshng OP Conf. Seres: Mterls Scence nd Engneerng 7 (20) do:0.088/ x/7//02090 Performnce nlyss of novel plnetry speed ncreser used n sngle-rotor wnd turbnes wth counter-rottng electrc genertor Sulescu, M Negoe, O Muntenu nd N Cretescu enewble Energy Systems nd ecyclng eserch Centre, Trnslvn Unversty of Brşov, omn E-ml: rsulescu@untbv.ro Abstrct. The pper presents study on the knemtc nd sttc performnces of new type of DOF (Degree Of Freedom) plnetry speed ncreser to be mplemented n wnd turbnes, trnsmsson wth three opertng cses: ) one nput nd one output, b) one nput nd two outputs, n whch the speed of the secondry output s equl to the nput speed, nd c) wth one nput nd two outputs, where the secondry output speed s hgher thn the nput speed. The proposed speed ncreser contns two sun gers nd double stellte, llowng operton wth n output connected to the fxed sttor of clssc genertor (cse ) or wth two counterrottng outputs tht drve counter-rottng genertor (wth moble sttor). A new vrnt of plnetry trnsmsson cpble of provdng the speed ncrese of the genertor sttor nd, thus, the ncrese of the reltve speed between the genertor rotor nd sttor s obtned by the prllel connecton of the speed ncreser wth plnetry ger. The three conceptul vrnts of plnetry trnsmsson re nlytclly modelled nd comprtvely nlysed bsed on set of knemtc nd sttc prmeters. The proposed trnsmsson hs hgher performnces compred to the sme trnsmsson wth one nput nd one output, the ncrese of the knemtc mplfcton rto nd effcency beng cheved smultneously.. ntroducton The use of speed ncresers for hgher performnces of wnd turbnes ws mposed by the ncrese of ther producton cpctes; however, n the lst decde, new types of speed ncresers were dpted to better meet the functonl requrements of the wnd turbnes mplemented both n hgh power systems nd for the bult envronment. The frst step n these pproches hs been to dopt common termnology regrdng ger mechnsms wth moble xes [], n whch the problem of the types of plnetry trnsmssons dependng on the number of nputs nd outputs ws dscussed. Bsed on these pproches, [7] presents methodology usng grphs for the knemtc nlyss of complex ger mechnsms, whch nclude plnetry gers (PU); n exmple of spur ger mechnsms optmton by pplyng mthemtcl model for selectng the optml soluton bsed on the optml speed rto or the optml poston of the shfts xes s presented n [9]. A knemtc nd dynmc model bsed on the trnsmsson rtos nd effcences, whch s derved from generled model of the vgneux type ger mechnsms s descrbed n [0]. Two nlytcl methods for determnng the effcences of the plnetry gers re presented n [5], methods tht use the equtons of speeds nd torques, nmely the reltons between powers nd the speed rto. Ths concept s further developed n [] by descrbng methodology for clcultng the power Content from ths work my be used under the terms of the Cretve Commons Attrbuton 3.0 lcence. Any further dstrbuton of ths work must mntn ttrbuton to the uthor(s) nd the ttle of the work, journl ctton nd DO. Publshed under lcence by OP Publshng Ltd

3 7th nterntonl Conference on Advnced Concepts n Mechncl Engneerng OP Publshng OP Conf. Seres: Mterls Scence nd Engneerng 7 (20) do:0.088/ x/7//02090 through the complex plnetry trnsmssons consstng of plnetry gers nd vrble speed trnsmsson, presentng the equton of power crcultng through brnch by knemtc pproch. A method for estblshng the effcency of plnetry gers bsed on the screw theory s descrbed n [8], whle [2] presents n pproch on modellng the effcences of power trnsmssons used n hybrd vehcles bsed on grph theory. A combned nlyss of effcency nd trnsmsson rto [3] llows dentfyng trnsmssons wth hgh ger rtos nd hgh effcences, useful pproch n the desgn of plnetry gers. Thus, ccordng to the prevous models, the plnetry gers cn be ddressed n two wys: s seprte stnd-lone entty or s subsystem of complex system. A descrpton of the plnetry gers used n wnd turbnes s presented n [7], whch proposes type of 2 Degrees Of Freedom (DOF) plnetry speed ncreser controlled by servomotor n order to obtn constnt speed t the genertor. Strtng from the de of usng 2DOF plnetry gers s speed ncresers, [3] proposes knemtc nd dynmc nlyss of counter-rottng wnd system consstng of bevel ger nd 2DOF plnetry ger wth mproved effcency. Ths pproch s contnued n [] through cse study of kw wnd turbne modelled by dynmcs lgorthm bsed on the property of 2DOF plnetry gers of summng two externl motons for the speed ncrese t the genertor. A model of plnetry speed ncreser tht s smlr to the bove solutons s descrbed n [5, ], whch present knemtc nlyss nd the effcency compred to clsscl solutons; the knemtc nd sttc performnces of the plnetry speed ncreser re compred for two opertng cses: wth n nput nd n output or two nputs nd one output. The desgn nd development of counter-rottng genertor tht s ntegrted n wnd turbnes for urbn res [2], wth hgher performnces thn of clsscl genertor, contnng moble rotor nd sttor, represents new pproch of the 2DOF plnetry trnsmssons tht functon wth n nput nd two outputs. The use of these trnsmssons brngs some dvntges, lke: the possblty of mplementton nto wnd turbnes wth se constrnts, s well s n ncresed effcency due to the prllel connecton of plnetry gers. The pper nlyses the performnces of type of DOF plnetry speed ncreser wth n nput nd n output or n nput nd two outputs, whch cn be used n wnd turbnes. 2. Problem formulton The pper refers to the comprson of knemtc nd sttc performnces of plnetry speed ncresers to be used n sngle-rotor wnd turbnes. The pper proposes new type of DOF plnetry speed ncreser usble n sngle-rotor wnd systems wth counter-rottng genertor, ble to ncrese the energetc performnces versus those of clsscl systems wth one output. The proposed speed ncreser s spur ger trnsmsson composed of two sun gers nd stellte wth two gers n serl connecton, whch cn functon wth n output (fgure ) or two outputs, nd n whch the speed of the secondry output (connected to the moble sttor of the counter-rottng genertor [2]) s equl to the nput speed (fgure b). The counter-rottng motons led to the genertor speed ncrese due to the summng up of rotor speed nd sttor speed (fgure b). Bsed on ths concept, the uthors propose nother vrnt of speed ncreser obtned by prllel connecton of two PU: the frst PU (-2-3--H) s descrbed n fgure b, whle the second PU wth double stellte ger (3--5--H) n fgure 2c. Ths vrnt of speed ncreser ncreses the output moton (secondry) versus the nput moton H, ccordng to the speeds dgrm from fgure 2d. Accordng to fgure, the trnsmsson of the mechncl power from the wnd rotor to the genertor, through the DOF speed ncreser cn be performed n two opertng cses: wth one output (fgure e), whch corresponds to the cse of the genertor wth fxed sttor, nd wth two outputs (fguref) n cse of the counter-rottng genertors (wth moble sttor), n whch the sttor cn rotte wth the sme speed or hgher thn the nput speed. Therefore, three vrnts of plnetry speed ncresers re proposed n the pper: 2

4 7th nterntonl Conference on Advnced Concepts n Mechncl Engneerng OP Publshng OP Conf. Seres: Mterls Scence nd Engneerng 7 (20) do:0.088/ x/7//02090 () (b) (c) (d) (e) Fgure. Knemtc schemes of DOF plnetry speed ncreser wth: () L=2; (b) L=3 nd ; (c) L=3 nd ; (d) speeds dgrm for cse (c); (e) block dgrm ssocted to cse (); (f) block dgrm ssocted to cses (b) nd (c). Cse : speed ncreser wth one nput nd one output (L=2, M=), nd genertor wth fxed sttor, ω = 0 (fgure ); Cse : speed ncreser wth one nput nd two outputs (L=3, M=), nd counter-rottng genertor wth the speed of the sttor equl to the rotor speed, ω = ω (fgure b); Cse : speed ncreser wth one nput nd two outputs (L=3, M=), the sttor of the counterrottng genertor rottng wth speed hgher thn the nput speed: ω > ω (fgure c). The knemtc nd sttc nlyss s further presented for the three vrnts of plnetry speed ncreser, ther performnces n cse of usng them n wnd turbnes beng lso hghlghted. 3. Modellng the plnetry speed ncreser wth one output (cse, fgure ) The DOF plnetry trnsmsson from fgure hs two stellte gers n serl connecton (2 nd 3) nd trnsmts the mechncl power generted by the wnd rotor to the rotor of the genertor wth fxed sttor, multplyng the speed nd chngng ts drecton whle reducng the torque. The trnsmsson functons nd the propertes of the plnetry trnsmsson were obtned n the knemtc nd sttc nlytcl modellng by consderng frcton n the ger prs. (f) 3

5 7th nterntonl Conference on Advnced Concepts n Mechncl Engneerng OP Publshng OP Conf. Seres: Mterls Scence nd Engneerng 7 (20) do:0.088/ x/7// Knemtc nlyss The trnsmsson rto of the plnetry speed ncreser, where H H HH H H H H s determned by relton () (fgure ): 0 s the nteror knemtc rto of the plnetry ger H H H H 0. (2) The speed trnsmsson functon, relton (3), nd the relton between the wnd rotor speed ( ) nd the speed of the genertor rotor ( mplfcton rto: H 0 () ) cn be wrtten bsed on relton (), where s the 0 H (3) H H H (, H ) () where H Sttc nlyss The sttc functon of the speed ncreser s obtned from the trnsmsson energy equlbrum equton consderng frcton []: H TH H T 0, (5) where the effcency of the plnetry speed ncreser nternl effcency []: H H depends on the trnsmsson where T T TH H 0 H, () x H TH H H sgn sgn HT sgn x HT. (7) T 0 The dependence of the nput torque (T H ) on the output torque (T ), or between the wnd rotor (T ) nd genertor rotor torques (T ), respectvely s obtned from relton () s follows, relton (8): HHTH TH or T T T. (8). Modellng the plnetry speed ncreser wth two outputs nd ω = ω (cse, fgureb) The plnetry trnsmsson from fgure cn functon wth two counter-rottng outputs, ccordng to fgure b, the mn output beng the genertor rotor, s n the prevous cse; moreover, secondry output s obtned by drect couplng of the nput body H (the crrer) to the moble sttor of the electrc genertor. Thus, the genertor speed s ncresed by ncresng the reltve speed of the rotor gnst the sttor, the genertor functonng n more dvntgeous wy, mplctly. n ths

6 7th nterntonl Conference on Advnced Concepts n Mechncl Engneerng OP Publshng OP Conf. Seres: Mterls Scence nd Engneerng 7 (20) do:0.088/ x/7//02090 opertng cse, the knemtc nd sttc nlyss of the plnetry speed ncreser s bsed on the two power flows modellng: one from the wnd rotor to the genertor rotor (-), nd the second one from the wnd rotor to the moble sttor of the genertor (-), by consderng the propertes of DOF mechnsms wth 3 nputs nd outputs regrdng the dstrbuton of speeds nd summng of externl torques, s well s the property of counter-rottng genertor of hvng equl nd opposte torques t ts rotor nd sttor ( T T )... Knemtc nlyss The speed of the counter-rottng genertor, relton (0), defned s the reltve speed of the rotor gnst the sttor s obtned strtng from the prevous relton ()-() estblshed for the trnsmsson wth M= nd L=2, whch chrctere the mn power flow -, nd lso consderng relton (9): S H,, S, (9) ) H H ( 0 ) H 0H (. (0).2. Sttc nlyss The energy equlbrum equton for the trnsmsson wth two outputs nd consderng frcton cn be expressed s follows: T T T 0 T T ( ) 0. () As the plnetry speed ncreser s DOF trnsmsson wth L=3, ts effcency cn be gven by relton (2) []: 0 (2) where H (relton ()) nd ( H ). The genertor torque T G, whch conventonlly s the torque on the genertor rotor T G = T s obtned from relton () nd (2): T 0 T T. (3) 0 5. Modellng the plnetry speed ncreser wth two outputs nd ω > ω (cse, fgure c) The plnetry speed ncreser from fgure c llows hgher speed t the secondry output (fgure d, v > v H ) by ntegrtng the plnetry ger 5- nto the vrnt from fgure b. Ths soluton trnsmts the mechncl power generted by the wnd rotor to the electrc genertor through two power flows (smlr to the prevous cse); the complex trnsmsson conssts of two plnetry gers n prllel connecton wth common nput (the block dgrm from fgure 2), beng chrctered by wnd rotor ssembled on the crrer (H), the genertor rotor connected to the shft, nd the moble sttor to shft. The complex plnetry speed ncreser (fgure c) hs three sun gers (, nd ) nd s composed of two DOF plnetry gers (PU- nd PU-) wth common nput (H). Accordng to the block dgrm from fgure 2, the two solted plnetry gers from the complex trnsmsson re chrctered by the followng knemtc nd sttc equtons: 5

7 7th nterntonl Conference on Advnced Concepts n Mechncl Engneerng OP Publshng OP Conf. Seres: Mterls Scence nd Engneerng 7 (20) do:0.088/ x/7//02090 Fgure 2. The block dgrm of the plnetry speed ncreser obtned by prllel connecton of two DOF plnetry gers. H H ' 0 ' H PU : ' 0 H 0 T 0 T' TH x T 0 0 T' 0 H H " 0 " H PU : " 0 H 0 T 0 T" TH w T 0 0 T" () 0. (5) The plnetry speed ncreser s chrctered by the followng correltons for the nternl lnks between the plnetry gers, for the externl lnks, relton () (9), nd between the rotor nd sttor of the electrc genertor, relton (20), whch re obtned bsed on the power nd coxl torques equlbrum equtons: H H H H H () T TH T 0 H T T T T 0 0 ' " 0 ' " T T' T" 0 (7) (8) (9)

8 7th nterntonl Conference on Advnced Concepts n Mechncl Engneerng OP Publshng OP Conf. Seres: Mterls Scence nd Engneerng 7 (20) do:0.088/ x/7//02090 T T S T T T T 0 T T TS T 0. (20) These correltons llow estblshng the knemtc nd sttc trnsmsson functons of the complex speed ncreser. 5.. Knemtc nlyss The mplfcton rtos for the two PU, relton (2) nd (22), to be used n estblshng the dependence between the speeds of the wnd rotor nd counter-rottng genertor, relton (23), re obtned bsed on the knemtc equtons for the two PU nd on the nternl nd externl correltons: 5.2. Sttc nlyss ' ' H ' H H 0 H ' ' H (2) " " H " H H 0 H " " H (22) ) S ' " ( 0 0. (23) The effcency of the plnetry trnsmsson s determned on the bss of the effcences of the two PU: where ' T TH ' T ' H 0 0 H (2) ' x H ' TH H ' ' H 0 00 " T TH " T " H 0 0 H (25) " w H " TH H " " H 0 00 x sgn H T 0 sgn sgn H sgn T H ' ' 0 T H H (2) H 0 " 2233 (27) w sgn H T 0 sgn sgn H sgn T H " " 0 T H H (28) H 0 " 53. (29) The trnsmsson effcency, relton (3) nd the correlton between the torques on the wnd rotor nd genertor, relton (32) re obtned smlr to cse nd tkng nto ccount the power equlbrum equton, relton (30): T T T T T ( ) 0 (30)

9 7th nterntonl Conference on Advnced Concepts n Mechncl Engneerng OP Publshng OP Conf. Seres: Mterls Scence nd Engneerng 7 (20) do:0.088/ x/7// (3) T 0 0 S T T T T. (32) Comprtve nlyss. Dscussons The reltons for the knemtc nd sttc prmeters for the three vrnts from fgure re systemted n tble, followed by numercl exmple of wnd turbne contnng speed ncreser wth n mplfcton rto = 80 nd n effcency of the spur ger of Tble. The mn knemtc nd sttc prmeters for the vrnts from fgure Plnetry ncreser type Crteron nternl knemtc rto 0 Numercl soluton for = 80 nd = 0.95 Fgure Fgure b Fgure c (=8) (=80) (=57) 0 5 (= 23) 3 nternl effcency 0 0 Knemtc rto S 2233 (=0.8573) (= 80) 0 (=0.8573) 2233 (=0.8573) 5 3 (=0.9025) (= 80) 0 0 (= 80) 0 (= 80) 0 (= 79) 0 (= 5) 0 0 (= 2) Sttc rto T T Effcency (=0.00) 0 (=0.007) 0 0 (=0.008)

10 7th nterntonl Conference on Advnced Concepts n Mechncl Engneerng OP Publshng OP Conf. Seres: Mterls Scence nd Engneerng 7 (20) do:0.088/ x/7// (=0.8558) 0 (=0.8573) (=0.898) Accordng to tble, the vrnts wth counter-rottng genertor (L=3) hve superor performnces versus those of the vrnt wth clsscl genertor (L=2): the nteror knemtc rtos re reduced nd the effcences re hgher t the sme vlue of the mplfcton knemtc rto. To lmt the nertl effects, the speed of the moble sttor s recommended to be lower thn the speed of the genertor rotor; n the cse of the trnsmsson from fgure c nd lmtng the speed of the sttor t the thrd prt of the rotor speed ( ), the dgrm from fgure 3 llows estblshng the nteror knemtc rto 0 s functon of 0. Fgure 3. The vrton of the rto 0 / 0 s functon of 0 for the plnetry speed ncreser from fgure c (cse ) wth The knemtc mplfcton rto (fgure ), the sttc rto (fgure b) nd the effcency (fgure c) for the three opertng cses re expressed s functons of 0, strtng from the prevous mthemtcl model, nd from the dgrm n fgure 3. () (b) 9

11 7th nterntonl Conference on Advnced Concepts n Mechncl Engneerng OP Publshng OP Conf. Seres: Mterls Scence nd Engneerng 7 (20) do:0.088/ x/7//02090 (c) Fgure. The vrton of knemtc nd sttc prmeters of the plnetry speed ncreser for the three opertng cses s functon of the nteror knemtc rto of PU-: () the mplfcton rto ; (b) the sttc rto ; (c) the effcency. Smlrly, the vrton of the knemtc mplfcton rto for the two power brnches re represented n fgure 5 s functon of 0. () (b) (c) Fgure 5. The knemtc trnsmsson rtos from the wnd rotor to the genertor rotor ( ), nd to the genertor sttor ( ) for the plnetry trnsmssons: ) cse ; b) cse ; c) cse ; d) the prevous cses superposed. The comprtve nlyss of the dgrms from fgure nd 5 outlnes tht the most fvorble soluton s the trnsmsson wth two PU n prllel connecton, due to the followng dvntges: (d) 0

12 7th nterntonl Conference on Advnced Concepts n Mechncl Engneerng OP Publshng OP Conf. Seres: Mterls Scence nd Engneerng 7 (20) do:0.088/ x/7// the nteror knemtc rto of the second PU tends to be stbled t lmted vlue wth the ncrese of the PU- nteror knemtc rto, e.g. 0 / 0 0. for 0. 29, fgure 3; - the bsolute ncrese of the mplfcton rto wth 0 s hgher thn for the cses nd (fgure ), whle the sttc rto hs more pronounced decrese compred to the other cses, decrese tht tends to stble t hgher vlues of the mplfcton rto, fgure b; - t the sme totl mplfcton rto, the vrnt wth two PU n prllel connecton hs smller dmensons, the trnsmsson rtos beng lower compred to the other cses, tble ; - the effcency of the speed ncreser from cse s hgher thn the effcences of cses nd, nd ncreses wth the ncrese of the PU- nteror knemtc rto, fgure c; t s hghlghted tht the trnsmsson from cse hs constnt effcency, equl to the nternl effcency of PU-; - the use of genertor wth moble sttor ncreses the knemtc mplfcton rto, beng dependent on the reltve speed between the genertor rotor nd sttor : the sttor speed s null n cse ; n the second cse, the sttor speed s equl to the nput speed, whle n cse, the sttor speed s hgher dependng on the trnsmsson rto of the second PU, 0, tble nd fgure 5; - for the sme sttc rto, the trnsmsson wth two PU n prllel connecton nd wth one nput nd two outputs hs hgher effcency nd smller overll dmenson compred to the other two cses; e.g. the results for re: cse : 33, 5, compred to cse : 0, 5, The speed ncresers contnng plnetry trnsmssons n prllel connecton re chrctered by hgher performnces compred to those of trnsmssons n seres connecton. The use of plnetry trnsmssons n prllel wth counter-rottng output llows hgher effcences nd smller overll dmensons. For nstnce, ccordng to Tble, n ncrese wth pprox. % of the effcency nd decrese of the rdl dmenson wth pprox. 0.3% s obtned for n mplfcton rto = 80: cse - 0 = 57 nd η = 0.898, compred to cse : 0 = 80 nd η = The vrnt from cse brngs more dvntges for smll mplfcton rtos, whch mkes t recommendble for low-power systems, ncludng those nstlled n the bult envronment. For nstnce, for 0 = the followng results re obtned n cse : 8, η = 0.82; n cse : 9 nd η = 0.857; n cse :. 3, η = 0.8; thus, for the sme rdl dmenson, both the mplfcton rto nd the effcency re ncresed (ncrese of pprox.. 2 % for the mplfcton rto nd of pprox % for effcency) n cse compred to the cse of clsscl genertor (cse ). 7. Conclusons The pper presents n lgorthm for the nlyss of new type of DOF plnetry trnsmsson used s speed ncresers n wnd turbnes, consderng three of opertng cses: wth one nput nd one output (L = 2, M = ), wth one nput nd two counter-rottng outputs, the nput beng fxed to the secondry output (L = 3, M = ), nd wth two counter-rottng outputs n whch the speed of the secondry output s mplfed reltve to the nput speed (L = 3, M = ). Dependng on the requred power on the genertor shft, the proposed lgorthm determnes the prmeters of mechncl power on ech brnch seprtely, correspondng to the trnsmsson outputs. The lgorthm llows dmensonng plnetry speed ncreser by tkng nto ccount the possblty of estblshng the nteror knemtc rtos nd the mplfcton rto, mplctly, consderng rtos tht re recommended between the genertor sttor nd the rotor speeds. A speed ncrese of the genertor rotor reltve to the sttor nd, thus, more dvntgeous operton of the genertor re obtned due to the use of genertor wth moble sttor; the vrnt of plnetry trnsmsson obtned by couplng two prllel PU (cse ) cheves hgher trnsmsson rtos nd hgher effcency compred to the cses wth one PU nd one 0

13 7th nterntonl Conference on Advnced Concepts n Mechncl Engneerng OP Publshng OP Conf. Seres: Mterls Scence nd Engneerng 7 (20) do:0.088/ x/7//02090 output (cse ) or two outputs (cse ). The results re useful n the synthess of speed ncresers wth hgh performnces to be ntegrted n wnd turbnes, provdng speed mplfcton, hgher effcences nd reduced dmensons. 8. eferences [] Arnudov K, Genov P, Dmtrov L 2005 For n unfed nd correct FToMM termnology n the re of gerng Mechnsm nd Mchne Theory 0 pp [2] Booker J D, Mellor P H, Wrobel, Drury D 200 A compct, hgh effcency contr-rottng genertor sutble for wnd turbnes n the urbn envronment enewble Energy 35 pp [3] Clmescu O, Săulescu, Jlu C 20 Specfc fetures of counter-rottng trnsmsson for renewble energy systems Envronmentl Engneerng nd Mngement Journl 0 pp 05-3 [] Dconescu D V, Dudt Fl 99 Wrkungsgrdberechnung von wngläufgen Plnetengetreben. Tel : Wetere Bespelrechnungen und Vortele Antrebstechnk 33 pp -3 [5] del Cstllo J M 2002 The nlytcl expresson of the effcency of plnetry ger trns Mechnsm nd Mchne Theory 37 pp 97-2 [] Gomà Ayts J et l. 20 Power trnsmtted through prtculr brnch n mechnsms comprsng plnetry ger trns nd other fxed or vrble trnsmssons Mechnsm nd Mchne Theory pp 7-75 [7] Gomà Ayts J, Dego-Ayl U, Mnguell Cnel J, Fenollos F, Vvncos J 202 Hypergrphs for the nlyss of complex mechnsms comprsng plnetry ger trns nd other vrble or fxed trnsmssons Mechnsm nd Mchne Theory 5 pp [8] Lus L P, Sms H, Mrtns D 202 Effcency of ger trns determned usng grph nd screw theores Mechnsm nd Mchne Theory 52 pp [9] Mrjnovc N et l 202 A prctcl pproch to the optmton of ger trns wth spur gers Mechnsm nd Mchne Theory 53 pp [0] Mths, emond Y 2009 Knemtc nd dynmc smulton of epcyclc ger trns Mechnsm nd Mchne Theory pp 2 2 [] Mlou Gh, Dudt Fl, Dconescu DV 980 Modern Mechncl Trnsmssons (n omnn) (Bucurest: Ed. Tehnc) [2] Pennestrì E, Mrt L, Vlentn P P, Mucno V H 202 Effcency evluton of gerboxes for prllel hybrd vehcles: Theory nd pplctons Mechnsm nd Mchne Theory 9 pp 57-7 [3] Slgdo D, del Cstllo J M 20 Anlyss of the trnsmsson rto nd effcency rnges of the four-, fve-, nd sx-lnk plnetry ger trns Mechnsm nd Mchne Theory 73 pp [] Săulescu, Jlu C, Muntenu O, Clmescu O 20 Plnetry Ger for Counter-rottng Wnd Turbnes Appled Mechncs nd Mterls 58 pp 35-0 [5] Sulescu et l. 20 Structurl nd Knemtc Fetures of 2 DOF Speed ncreser for enewble Energy Systems Appled Mechncs nd Mterls 823 pp [] Sulescu et l. 20 Comprtve Anlyss of Two Wnd Turbnes wth Plnetry Speed ncreser n Stedy-Stte Appled Mechncs nd Mterls 823 pp [7] Zho X, Mßer P 2003 A novel power splttng drve trn for vrble speed wnd power genertors enewble Energy 28 pp Acknowledgments Ths work ws supported by UEFSCD nd the PN Prtnershp progrm through the project no PN- -PT-PCCA Sm-PhotoAd. 2