Analytical synthesis for four bar mechanisms used in a pseudo equatorial solar tracker

INGENIERÍA E INVESTIGACIÓN VOL. No., DECEMBER - 0 (55-60) Analytal synthess fo fou ba mehansms used n a pseudo equatoal sola take Síntess analíta de los meansmos de uato baas mplementados en un segudo sola pseudo euatoal J. M. González Mendoza, C. Palaos Montúfa and J. A. Floes Campos ABSTRACT Photovolta enegy poduton systems geneate eletty wthout emttng pollutants nto the atmosphee and do so fom a fee, unlmted esoue. The hghest level of enegy onveson fom the photovolta panels an be obtaned by plang them pependula to the sun s ays fallng on the sufae; ths s done by nstallng sola takng systems. Ths wok poposes the use of two fouba mehansms as the dvng foe fo a sola take; we popose the use of analytal synthess fo suh mehansms. Ths poedue s amed at optmsng the tansmsson angle, neasng mehanal advantage and deeasng dvng toque. A mathematal model was used to pove synthess esults and a pototype of the sola take was bult. Keywods: Tansmsson angle, fou-ba mehansms, sola take, synthess. RESUMEN Los sstemas de poduón de enegía fotovoltaa genean eletdad sn la emsón de ontaantes a la atmósfea, a pat de un euso lbe e lmtado. Con el fn de obtene el mayo nvel de onvesón de enegía de los paneles fotovoltaos, éstos deben oloase de manea pependula a los ayos del sol; esto se loga medante la nstalaón de sstemas de segumento sola. En éste tabajo, se popone el uso de dos meansmos de uato baas omo mpulsoes de un segudo sola, y el uso de un poedmento analíto de síntess paa tales meansmos. Dho poedmento tene po objetvo optmza el ángulo de tansmsón, nementa la ventaja meána y dsu el toque mpulso. Se obtuvo,un modelo matemáto paa poba los esultados de dha síntess y se onstuyó un pototpo del segudo sola. Palabas lave: ángulo de tansmsón, meansmos de uato baas, segudo sola y síntess óptma. Reeved: Deembe th 0 Aepted: Novembe 5th 0 Intoduton Thee s a need to geneate eletty fom enewable esoues nowadays; photovolta enegy poduton systems use the photoelet effet to tansfom sola enegy nto eletty, usng photovolta ells plaed on panels. Suh enegy onveson takes plae wthout emttng pollutants nto the atmosphee and also epesents a fee, unlmted esoue (Tatu, 00). Howeve, photovolta systems do not ompete wth standad enegy poduton systems yet, manly due to thee poblems (Oheng, 00): Low effeny,.e. low enegy poduton densty pe aea nstalled; The elatvely hgh ost of the mateals nvolved n the onstuton; and Vaablty dung poduton, manly elated to hanges n the weathe. One way of neasng enegy onveson by 0% to 50% n photovolta enegy poduton systems s the nstallaton of sola takng systems. Ths s beause they must be plaed pependula to the sun s ays fallng on the sufae to obtan the hghest level of enegy onveson fom the photovolta panels; ths s aheved though the use of sola takes (Comst, 007). Howeve, many photovolta enegy poduton systems do not have takng systems, manly beause of the ost and omplexty (Duate, 00; Comst, 007). Thee s thus a need fo developng effent, low-ost, sola takng systems. It has been demonstated that the best way to follow the path of the sun thoughout the yea s by usng plana mehansms, wth the otaton axes aanged pependulaly. Ths also smplfes Juan Manuel González Mendoza. Mehanal Engnee, ESIME-IPN, Méxo. MS en Ingeneía Meána, SEPI-ESIME-IPN, Méxo. Afflaton: Canddato a Doto en Cenas, SEPI-ESIME-IPN, Méxo. E-mal: johann009@yahoo.om Cánddo Palaos Montufa. Mehanal Engnee, ESIME-IPN, Méxo. M. S., UAP, Mosú-Rusa. Aflaón: SEPI-ESIME-IPN, Méxo. E-mal: pmontufa@yahoo.om.mx Juan Alejando Floes Campos. Mehanal Engnee y M. S. FI-UNAM, Méxo. Ph. D. en Ingeneía Meána, SEPI-ESIME-IPN. Aflaón: UPIITA-Insttuto Polténo Naonal, Méxo. E-mal: jafloes@pn.mx How to te: González Mendoza, J. M., Palaos Montúfa, C., Floes Campos, J. A., Analytal synthess fo fou ba mehansms used n a pseudo equatoal sola take., Ingeneía e Investgaón, Vol., No., Deembe 0, pp. 55 60. 55

ANALYTICAL SYNTHESIS FOR FOUR BAR MECHANISMS USED IN A PSEUDO EQUATORIAL SOLAR TRACKER the knemat and dynam analyss of the poblem and has advantages egadng a ontol sheme and ts mplementaton. As fo sola takes, eseah has been amed at synthessng fouba lnkages wth lnea atuatos (Vsa, 0), geaed lnkages, o developng new methods fo evaluatng the effeny (Popa, 008). Although n seveal efeened papes, dmensonal synthess fo fou-ba lnkage (Comst, 007) and geaed lnkage (Ceanga, 00) has been aed out, analyt dmensonal synthess has not been used to mpove tansmsson angle and deease the moto s dvng toque dung desgn and onstuton phases. Spef methods fo alulatng the dvng toque n suh lnkages (whose ultmate goal should be the onstuton and assembly) have not been pesented n these soues. Ou man objetves wee thus to use analytal synthess fo two fou-ba mehansms and mplement them n a pseudo-equatoal sola take. Ou seond am was to obtan a mathematal model whh would be used fo povng that by optmsng tansmsson angle, the peak toque equed by these mehansms dvng motos would beome deeased, thus loweng the enegy onsumpton of the whole takng system and makng t moe effent. Fgue : Pseudo equatoal sola take: somet vew The bas dffeene between equatoal and pseudo equatoal takes s the ode of otaton of the axes (Buduhos et al., 008). Two fou-ba mehansms wee desgned, bult, assembled and oupled n sees to dve the axes of movement of the pseudo equatoal sola take, and hene the photovolta panels. The motos woked n a stepwse manne (Fgues and ). Synthess of fou ba mehansms We poposed an analytal synthess poedue fo the fou-ba mehansms, known as Bodell and Son poedue (Shgley, 998). Ths poedue was amed at optmsng the mehansms tansmsson angle (angle γ, between ouple and oke, whh s a measue of the qualty of tansmsson foe and speed at the jont, omplement angle beng alled the pessue angle). Seveal authos have suggested keepng ths angle s values geate than 0 to ensue the hghest mehanal advantage and edue dvng toque (Shgley, 998; Noton, 999). The poedue s desbed as efeene n the followng lnes. Fgue shows the geomet elatonshps fo the fou ba mehansm: they meet Gashof s law, have a ank oke onfguaton and Q= speed ato,.e. when the ank goes fom pont A to pont A, 80º angula dsplaement ous and when the oke goes fom pont B to pont B ths podues total angula dsplaement ϕ (Fgue a). Usng Fgues a) and b), and the osne law, equatons () () wee fomed. These equatons wee n tems of γ and ϕ, and fomed a system of lnea equatons, havng unknowns,, and. Bodell and Son poposed equaton (5) as a ondton to be met: os θ () os () os () os () 80 (5) Fgue : Detal of dunal axs dvng mehansm Despton of the takng system The sola take had 0 photovolta panels (ERDM Sola 0TP6); they had a maxmum powe of 0 watts eah. The pseudo-equatoal sola take had two axes of movement (see Fgue ): a dunal axs (movng a 55 kg mass) and an elevaton axs (movng an estmated 0 kg mass). It was nstalled n Tejalpa, Jutepe, Moelos, Mexo. Pseudo equatoal takng systems use a loal-obseve angula system (wth elevaton and dunal angles), whle equatoal takng systems use a global angula system. Fgue : a) Osllaton angle φ on lnk, b) mum tansmsson angle and maxmum angle fomed at the nteseton of ouple and oke 56 INGENIERÍA E INVESTIGACIÓN VOL. No., DECEMBER - 0 (55-60)

GONZÁLEZ, PALACIOS AND FLORES Solvng equatons () - (5) gave the atos between lnks, gven by equatons (6)-(8): whee: os os os (6) (7) (8) oke osllaton angle mum allowed tansmsson angle maxmum angle fomed at the nteseton of ouple and oke length [m] of lnk, fxed length [m] of lnk, ank length [m] of lnk, ouple length [m] of lnk, oke Values fo the osllaton angles of lnk (oke) on both dunal and elevaton axes wee poposed. These values an be seen n Fgue. They ae maked by Geek lette ϕ, beause ths lette denotes the angula dsplaement of the oke n both mehansms. S S S J S S q S P C S q SS SS S C q S C q G S G T q G q S (9) As a means of showng how dvng toque deeased as tansmsson angle was optmsed, equaton (9) was evaluated fo vaous values of,, and (esultng n vaous values fo tansmsson angle, γ, as explaned n the pevous seton), opeatng at onstant speed. The equed dynam paametes wee obtaned afte pelay modellng of the mehansms usng Autodesk Invento 00. Fgue 5: Compang dvng toque fo vaous tansmsson angle values fo the sola take s dunal mehansm Fgue : Poposed angles fo lnk (oke), on the pseudo equatoal sola take: a) dunal axs, b) elevaton axs Osllaton angle, dunal axs: 50 50 Osllaton angle, elevaton axs: 0 0 Equatons (6) - (8) wee evaluated usng Matlab R00, lettng ϕ be a fxed value, and vayng atos [], [], [] untl γ beame optmsed,.e. untl tansmsson angle γ values wee always geate than 0º and maxmum values aheved fo a full evoluton of the ank, on both axes. These values wee 9º γ 90º fo the dunal axs and 59º γ 90º fo the elevaton axs, povng the usefulness of ths poedue as a means of synthessng fou-ba mehansms to be used n sola takes. Dynams: dvng toque Although a knet-stat appoah would be enough fo alulatng shape, sze and hoosng mateals fo the pototype s stutual membes at low speed and aeleaton, the dynams of both mehansms wee alulated to evaluate the dvng toque equed by the motos. A dynam model of the fou-ba mehansms was obtaned usng Lagange s fomulaton (Wu, 00) (detaled n appendx A). Dvng toque ould thus be wtten as n equaton (9): T J J S J S PC S q S S S J S S q S Fgue 6: Compang dvng toque fo vaous tansmsson angle values fo the elevaton mehansm of the sola take Fgue 7: Tansmsson angle values fo a full evoluton of the ank INGENIERÍA E INVESTIGACIÓN VOL. No., DECEMBER - 0 (55-60) 57

ANALYTICAL SYNTHESIS FOR FOUR BAR MECHANISMS USED IN A PSEUDO EQUATORIAL SOLAR TRACKER Fgue 5 shows how the peak value of the dvng toque equed fo the dunal mehansm deeased as tansmsson angle value mpoved, untl eahng γ=9º. Howeve, some dynam paametes egaded as onstant values (suh as mass, neta moment, and loaton of the ente of gavty of the lnks) undewent some vaaton as the synthess esults also vaed; futhe studes wll help detee whethe the effet of suh vaaton aounted fo the elatonshp lealy seen n Fgue 5 (dunal mehansm), but not lealy enough n Fgue 6 fo the elevaton mehansm. Fgue 7 shows the tansmsson angle values fo a full evoluton of the ank n both mehansms (dunal and elevaton). Mehanal desgn: buldng a pototype The wnd load the sola take had to wthstand was alulated, followed by the eaton foes on the jonts and stess on lnks. Stutual membe shape, sze and mateals wee assgned and a pototype was bult. Wnd load Regadng the loads a sola take must stand (weght of ts own stutue, weght of the photovolta panels, neta load, wnd load), tal load s due to wnd. Ths s manly beause of the takes onstuton haatests,.e. the sufae, heght of nstallaton, oentaton egads wnd deton (Butu, 00). Equatons (0) - () wee used to detee wnd load (Messenge, 00; Velu, 00). whee: F N N P f vw F A (0) f () Nd 0 p p T F () nomal foe of wnd on the sufae of photovolta panels [N] a densty [kgm ] A aea of photovolta panels [m ] P v wnd speed [ms] W f foe oeffent [untless] p pessue oeffent [untless] p pessue oeffent [untless] d0 T dstane fom photovolta panel axs to the pont of applaton of nomal foe, as a esult of pessue dstbuton on the panel [m] extenal toque on lnk, due to wnd load [N-m] Fgue 8 shows the pont of nomal foe beng appled to the photovolta panels. The effet of wnd load on the photovolta panels was alulated, and t was assumed that suh a foe would make lnk otate on ts own axs, thus equng geate moto dvng toque to ompensate fo suh extenal dstubane. It was poposed that the sola take must be able to move to ts defene poston, namely, havng photovolta panels oentated hozontally, at wnd speeds of up to 0 kmh (a wnd speed of 6 on Beaufot sale, aodng to (http:www.sp.noaa.gov)). Ths teon was used fo alulatng dvng toque (Eq. 9). Fgue 8: Applyng wnd load to photovolta panel sufae It was statally assumed that the sola take, n one of ts lmt postons, must be able to wthstand wnd speeds of up to 80 kmh (a wnd speed of 9 on Beaufot sale). Ths teon was used fo alulatng stesses and defletons on lnks and seletng mateals fo the model. Reaton foes on the jonts: stat ase The appoah used fo alulatng stesses on the lnks of the mehansm was as follows. The eaton foes on the lnks and jonts wee found by onsdeng eah mehansm n one of ts lmt postons (statally), as they had to stand wnd speeds of up to 80 kmh (wnds apable of ausng slght stutual damage, aodng to (http:www.sp.noaa.gov)). Fgues 9 and 0 show the mehansms n the lmt postons as they wee used fo alulatng eaton foes on the jonts. Fgue 9: Shemat vew of lmt poston fo stat analyss of dunal mehansm Fgue 0: Shemat vew of lmt poston fo stat analyss of elevaton mehansm 58 INGENIERÍA E INVESTIGACIÓN VOL. No., DECEMBER - 0 (55-60)

GONZÁLEZ, PALACIOS AND FLORES Stesses and defletons on lnks Stesses and defletons on the lnks wee alulated, as a fnal step n the desgn stage. Fgue shows some of the esults of these alulatons, as vefed usng Autodesk Invento 00. Fgue : FEM model fo ouple, dunal mehansm Expemental setup Fgue shows the dunal mehansm assembly, oupled to a wom gea mehansm (also used as a beak fo the whole system) and to the atuatng moto. As a esult of ths expement, the angles theoetally pedted by the analytal synthess (Fgue 7) wee poved. Vaatons angng fom % to % of theoetal values wee obseved, whh ould have been attbuted to the manufatung poess. Fgue : Dunal mehansm, whole assembly Conlusons Aodng to Fgue 5, the mathematal model fo alulatng the dvng toque (gven by equaton (9)), poved that analytal synthess fo optmsng the tansmsson angle of fou-ba mehansms, known as Bodell and Son s poedue, ould be suessfully used n the synthess of a pseudo-equatoal sola take s dvng mehansms. Fgue 5 shows a 0 N-m eduton fo evey egadng peak value fo the dvng toque equed fo the dunal mehansm; tansmsson angle γ value beame mpoved. Sne dvng toque beame edued, the poposed desgn led to a eduton n the enegy onsumpton of the takng system tself, theeby makng the pseudo-equatoal sola take moe effent. The poedue tself an be used fo the synthess and desgn of fou-ba mehansms to be used n any szed, vey heavy, pseudo equatoal sola takes and fo vaous knds of faltes. Aknowledgements Ths wok was fnaned by the ConsejoNaonal de Cena y Tenología (CONACYT) and the Insttuto Polténo Naonal, IPN, Seetaía de Investgaón y Posgado, SIP-IPN. Refeenes Beaufot Wnd Sale., Consulted n: http:www.sp.noaa.gov faqtonadobeaufot.html Buduhos, B., Saulesu, R., et al., On the dependene of the eeved det sola adaton on the pv pseudo equatoal takng steps numbe., Annals of the Oadea Unvesty. Fasle of management and tehnologal engneeng, Vol. 7, No. 7, 008. Butu, B., Lates, M., et al., FEM analyss of the bevel gea housng of an azmuthal taked PV platfom., Annals of the Oadea Unvesty. Fasle of management and tehnologal engneeng, Vol. 9, No. 9, 00. Comst, M., Vsa, I., Desgn of the lnkages type takng mehansms of the sola enegy onveson systems by usng Mult Body Systems Method. th IFToMM Wold Congess, Besançon, Fane, 007. Ceanga, N., Vsa, I., et al., Geaed lnkage wth lage angula stoke used n taked PV systems., Bulletn of the Tanslvana Unvesty of Başov, Vol., No. 5, 00. Duate, F., Gaspa, P., et al., Two axs sola take based on sola maps, ontolled by a low-powe moontolle. Intenatonal Confeene on Renewable Eneges and Powe Qualty (ICREPQ 0), Ganada (Span), 00. Messenge, R., Vente, J., Photovolta systems engneeng., London, CRC Pess, 00. Noton, R. L., An ntoduton to the synthess and analyss of mehansms and mahnes., USA, MGaw Hll, 999. Oheng, R., Chong, K. K., Wong, C. W., et al., Sola olletos and panels, theoy and applatons., Syo. 00. Popa, V., Daonesu, D., et al., The takng lnkage synthess destned to dve the azmuthal moton fom a PV take., Annals of the Oadea Unvesty, Fasle of management and tehnologal engneeng, Vol. 7, No. 7, 008. Shgley, J. E., Uke, J. J., Teoía de máqunas y meansmos., Méxo, M Gaw Hll, 998. Tatu, I., Alexandu, C., A study on the takng mehansms of the photovolta modules., Annals of the Oadea Unvesty, Fasle of management and tehnologal engneeng, Vol. 9, No. 9, 00. Velu, R., Moldovean, G., Wnd loads on an azmuthal photovolta platfom. Expemental study., In: Intenatonal Confeene on Renewable Eneges and Powe Qualty (ICREPQ 0), Ganada, Span, 00. Vşa, I., Saulesu, R., et al., New lnkage wth lnea atuato fo takng PV systems wth lage angula stoke., Chnese jounal of mehanal engneeng, Vol., No. 5, 0. Wu, F., Zhang, W., Integated desgn and PD ontol of hgh speed losed loop mehansms., ASME, Vol., 00. INGENIERÍA E INVESTIGACIÓN VOL. No., DECEMBER - 0 (55-60) 59

ANALYTICAL SYNTHESIS FOR FOUR BAR MECHANISMS USED IN A PSEUDO EQUATORIAL SOLAR TRACKER Appendx A The dynam model of the fou ba mehansms was obtaned usng Lagange s fomulaton (Fgue ). Fgue : Nomenlatue fo the fou ba mehansms, usng Lagange s fomulaton q length of the lnk genealsed oodnate fo the mehansm angles defnng lnk poston m mass of the lnks I neta moment of lnks dstane fom ente of gavty to pvot CG Equatons fo poston analyss wee gven by () (7) q atan k k k k k k () q, sen senq sen os os q atan os q sen senq k k () (5) q os os q kq sen (6) senq os os q (7) Equatons fo speed analyss ae gven by (8) (0) S q,, S q q (8),, S q,, q sen q sen S q,, q sen q sen (9) (0) The total knet enegy, potental enegy and Lagangan fomulaton of the system ae gven by equatons () (9). T m v CG I q m vcg I m vcg I CG CG CG () V m gy m gy m gy () L q,,, q,, PC q, q Gq,, J q m I m m m J J () J CG () J (5) CG I J (6) CG I P m (7) CG q, q C (8) os,, m gcg mg m g sn m g sn G q CG sn q CG sn CG (9) Afte fndng the Lagangan fomulaton, the equaton of movement fo the system (equaton (9)) was found usng equaton (0): V m gy m gy m gy (0) CG CG CG 60 INGENIERÍA E INVESTIGACIÓN VOL. No., DECEMBER - 0 (55-60)