Journal of Chemical and Pharmaceutical Research, 2014, 6(5): Research Article

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1 Available online Jornal of heial and Pharaetial Researh, 4, 6(5): Researh Artile ISSN : OEN(USA) : JPR5 Fltter and response of oposite wind trbine blade with bending-torsional opling ingri i ollege of Mehanial & Eletroni Engineering, Shandong Universit of Siene & ehnolog, Qingdao, hina ABSRA In this std, fltter and aeroelasti responses of rotor blade with bending-torsional opling have been investigated for oposite thin-walled strtre alled integration line eqation losed-setion. he aerodnai odels adopted here are a siple aerodnai odel sed for lassial fltter and a reded nonlinear aerodnai odel sitable for onstant pith otion for stall fltter, whih are sed in order to investigate qikl paraeters that inflene the behavior of the aeroelasti sste. he opled partial differential strtral eqation are solved b disretiation, with tie responses investigated. ewords: lassial fltter; Stall fltter; Bending-torsional opling; isretiation INROUION Althogh stabilit of lassial fltter has generall not been a driving isse in tilit-sale design, one ase in whih lassial fltter was observed involved a high-speed rotating blade in pith exitation proess. However, with the advent of large wind trbine fitted with relativel slender blades, lassial fltter a beoe a ore iportant design onsideration. In addition, innovative blade designs involving the se of aeroelasti tailoring, wherein the blade twists as it bends nder the ation of aerodnai loads to shed load reslting fro wind trblene, a inrease the blade s prolivit for fltter[]. he analsis of lassial fltter in wind trbines neessitates the se of nstead aerodnais []. For horiontal axis wind trbines (HAW) there are two interonneted sores of nstead aerodnais. he tehniqe odified for the aeroelasti analsis of a HAW blade was developed b heodorsen[3]. Most of the literatre of lassial fltter fosed on the heliopter blades. haviaropolos[4] investigated lassial fltter of - setion of wind trbine blade on the non-opling ondition. However the aerodnais of a HAW sste, dring atastrophi aeroelasti phenoena, a involve highl separated flows. he nonlinear blade theor for rotating sstes has far srpassed what the lassial fltter has revealed. So stall fltter shold be investigated. Ren[5]adopts a strtral eqation based on a oposite thin-walled bea odel with bending-twist opling, and anale nonlinear aeroelasti stabilit for stall fltter. he athor has ade a literatre srve in the sbjet atter on rotating thin-walled strtre (not blade setion) sstes withot strtral daping and investigated the nonlinear aeroelasti stabilit based on lineariation in another paper [6]. As for the partial differential eqations, whih represent the dnai behavior of wind trbine blade, a radial basis fntion proedre [7] and Galerkin ethod [5] are sed to transfor partial differential eqations into a disrete eigenvale proble. In this paper, fltter and responses nder the irferentiall asetri stiffness (AS) sitation of slender oposite blade with thin-walled losed ross setion are investigated. A nber of researhers have developed the neros soltion ethods to anale the vibration probles of oposite beas in reent 3 ears, espeiall in the developent of analtial expressions for extension-twist opling and bending-twist opling. Here a blade 544

2 an i and Xejn Wang J. he. Phar. Res., 4, 6(5): setion strtral odel of opled partial differential eqations is applied based on oposite bea theories [8-9].his odel indes the variation of shear odls in different walls of the ross setion in the warping fntion and transverse shear related oplings. However, restrained warping effets are not inlded in the odel. Fltter and responses of bending-torsional opling blade integrated with strtral daping are analed, with inflenes of pl angles investigated. In this paper, a siple aerodnai odel sed for lassial fltter and a reded nonlinear aerodnai odel sitable for onstant pith otion for stall fltter are sed in order to investigate qikl a wide range of paraeters that ontrol the behavior of the aeroelasti sste.. Analtial odel. Strtral odel onsidering the losed-setions of setri slender thin-walled oposite blade shown in figre (a)-(b), the length of the blade is in x diretion. he ross-setional diension, hord length, is. he thikness of blade setion is denoted b h, and the radis of rvatre of the iddle srfae b r. It is assed that, <<, h<<, h<<r. φ represents twist angle of the losed-setion, the pith angle is, and the wind veloit is denoted b U. he iddle-line of the thin-walled strtre alled integration line eqation losed-setion is as follows []: = e[exp(.5(( os θ ) + (sin θ ) )] + exp[.5(( os θ ) + (sin θ ) ))os θ ], (a) = e[exp(.5(( os θ ) + (sin θ ) )] exp[.5(( os θ ) + (sin θ ) ))sin θ ], (b) where e denotes /4 hord length, and paraeter θ φ hanges fro to π. F α r M U e x Rotor Plαne α M U e x Rotor Plαne (a) (b) Figre. Aerodnai setional fores of lassial fltter (a) and stall fltter (b). Eqations of otion for lassial fltter he linear fore and torqe, illstrated in figre (a) and deded in ters of the lift and oent oeffiients for sall displaeents with the air densit ρ a, are expressed respetivel as follows[4]: F = ρ a π( a + ), M e( = ρa π a + ). () he eqations of free vibration with bending-torsional opling of anisotropi thin-walled losed-setion blade setion are derived sing a variational asptoti approah and Hailton s priniple [8-9]. onsidering rotation with rotate speed Ω, nder variable paraeters, sh as pl angle and wind veloit U, the following governing eqations of otion for blade tip are obtained: [ + Ω ( x )] + 3 Ω x I Ω ( ) A A, (3a) = M + Fe os( ψ ) Ω [ ( x ) + ( x )] = F os( ψ ), (3b) where, ρ, ρ, = +, A = dd, A A = ( + ) dd. herein, is the ass per nit length, I is the polar ass oent of inertia per nit length abot the x axis., φ denote flap displaeent in diretion and torsional angle, respetivel. ij are the ross setional stiffness. he AS sed here onsists of [θ] n in the top side, and [ θ ] n in the botto side, with pl nber being 6, pl thikness being.7 3.

3 ingri i J. he. Phar. Res., 4, 6(5): Eqations of otion for stall fltter onsidering ostant pith otion, the nonlinear aerodnai aerodnai fores[], lift (), oent (M ) and drag (), of the ross-setion illstrated in figre (b), are siplified and reded to: = ρ S + ), M = ρ S + ), = ρ.4 + ) (4) ( M( M M ( where the nonlinear ites are: + λ = λa, + a + r = r, M b b b =, (5a) M + am M + rm M = rm, M + a + r = r ; b b b b b b (5b) herein, nonlinear parts in nonlinear ites an be expressed as:, ψ = a( ψ.396), < ψ.34 + ( a + a )( ψ ).396a.34a, ψ > = a ( ψ ) a ( ψ ) a ( ψ ). 3 where = denotes lift, and =M denotes oent respetivel. Other related oeffiients an be fond in referene [6]. So the eqations of the stall fltter sste are depited as: [ + Ω ( x )] + 3 Ω x I Ω ( ) = A A (6a) M + e sin( ψ ) + e os( ψ ), + + Ω [ ( x ) + ( x )] = os( ψ ) + sin( ). (6b) 3 33 ψ 3. Reded eqations and strtal daping 3. Reded eqations of otion owards the goal of solving the eqations as given b Eq. (3), the following steps of disretiation will be ipleented. he first step onsists of representation of displaeent fntions in the for: ( x, t) = Φ ( x ) q ( t), U( x, t) = U ( x ) q ( t) ; (7) herein, shape fntions are reqired to satisf the kineatis and fore bondar onditions of the antilever blade written as: ( x ) [,, 3, 4,, Φ = 3 N ], U ( x ) = [,, 3, 4, 3, N ], where bea fntions φ j and j, for a antilevered bea, an be fond in referene[]. Sbstitting Eq. (7) into Eq.(3) and arring ot the indiated variations and the reqired integration, reslt in the eqation governing the otion of the sste: M q + q = Q, (8) where M,, and Q are the N N oeffiient atries, and =. q [ q, ] q 3. allation and integrating of strtral daping Modal daping ratio of oposite ross-setion bea is defined as ratio of dissipated energ to axi strain energ in the le of vibration. Referene [3] stdied the loal (plate) /global (bea) vibration and daping behavior of oposite thin-walled box eber sbjeted to vibrator environent. Based on above theories, referene [5] gave a strtral analsis odel whih an deterine the inflenes of different order odal daping fators for different strtral and aterial paraeters. o std the flexral vibration of the slender blade, the paper adopts another ethod to opte strtral daping as depited in referene [4]. Strtral daping an be expressed in ters of an eqivalent visos daping atrix as follows: a s [ Φ ( x )] as 3[ U ( x )] = a =, a = η s s s / ω. (9) s a [ Φ ( )] [ ( )] s 3 x as 33 U x where ω, and η are the first order natral freqen and the orresponding first order odal daping ratios respetivel that an be fond in referene[5]. For strtral daping optation, a set of oposite aterial paraeters are given as, aterial densit ρ=67 kg/ 3, E =E 33 =8.7 GPa, E =5.8 GPa, v =v 3 =.34, G = G 3 =3.5 Gpa. 546

4 an i and Xejn Wang J. he. Phar. Res., 4, 6(5): Integrating strtral daping s into Eq.(8) and arring ot the indiated variations and the reqired integration reslting in the atrix eqations governing the sste otion for blade tip, with N N daping atrix SN, as follows: M q + SN q + q = Q, () where the added daping atrix SN is given as: Φ = as [ ( x )] jjdx as 3[ U ( x )] jjdx. SN Φ as 3[ ( x )] jdx as 33[ U ( x )] jdx As for soltion of the eqations for stall fltter as given b Eq. (6), dring the integration in the proess of disretiation, a strip theor [6], sbstitting s operation for integral operation, is sed at the right-hand-side ters in Eq. (6). onsidering in onjntion with strtral disretiation eqations and aerodnai eqations, and integration with strtral daping entioned above, reslt in the eqations governing the otion of stall fltter: M X + X + X = Q, () where X is the overall vetor of generalied o-ordinates as defined in referene [6]. 4. Analsis and reslts For this std, soe paraeters are defined: pith angle is= o, the length of the blade =8.4455, wind speed U= /s, rotate speed Ω=6rp. he ross setional stiffness ij, are opted and listed in able. he inflenes of pl angles are investigated and deonstrated. able. he ross setional stiffness (N ) against pl angles Pl angles( o ) e e e5 9.59e e e e e- -.65e -3.4e -.599e e e e- 6.87e e- 3.86e e e- o deterine the responses of the aeroelasti sstes in Eqs. (-), Silink an be diretl applied. As for eigenvale proble, take Eq.() for exaple, it will be expressed in state-spae for of first order sste [6], then nerial silation for eigenvale and freqen an be perfored. 4. alidit of ethodolog and silation o testif validit of the ethodolog of reded eqations, onsidering free vibration in Eq.(3) of a antilever strtre, and let Ω=rp and a s =, the first five order harateristi natral freqenies of free vibration are opted b oparison with the reslts of an approxiate allation (A) ethod[9]. he approxiate soltion old be ast as the prodt of deopled bending and twisting behavior. he anglar freqen an be solved as: λ ( I ( I ) 4I( ) λ 33 λ ) λ 33 λ ω = ± (3) I I where λ = ik and λ = k ; k = ( + ) π, and k is the soltion of the eqation os( k )osh( ) =. able shows the first five order harateristi freqenies against pl angles fro o to 9 o at interval of 5 o solved b ethodolog in the paper, b oparison with that of A ethod. It is obvios that the reslts both are qite idential. able. he first five order harateristi freqenies(h) against pl angles Pl angles( o ) st freqen Present A nd freqen Present A rd freqen Present A th freqen Present A th freqen Present A

5 ingri i J. he. Phar. Res., 4, 6(5): o testif silation validit of the Silink proess of nerial silation in stall fltter, reslts of tie response of Eq.(), representing all the responses of all variables inlding strtal displaeent variables and aerodnai variables, are displaed in figre (a). It is lear that all variables are stable, whih an be affired b analsis of sste eigenvales. Figre (b) deonstrates the first ten axi eigenvales, all vales are far less than, whih obviosl shows the sste onvergene. (a) (b) Figre. Responses(a) and eigenvales(b) of stall fltter sste. 4. he effets of pl angles on stabilit Firstl, the inflenes of pl angles are investigated b responses. Figre 3(a)-(b) show blade tip tie responses of the displaeents of torsion for lassial fltter and stall fltter, respetivel. As a rle aplitdes of lassial fltter are higher than that of stall fltter, whih indiates destrtive fltter. For the pl angle of 45 o, the trend shows less stabilit than others. On both sides, stabilit gradall inreases. In pl angle 5 o, it is the ost stable sitation. However in figre 4(a)-(b), it deonstrates the ost axi stabilit in 45 o, and on both sides, stabilit gradall dereases. In pl angle 5 o, it is the ost instable sitation. It an be deonstrated that for the given ases, tie response aplitdes of the fltter vibration, either for lassial fltter or for stall fltter, the hange trends of both torsion and flap are onsistent, whih indiates fltter dereasing. For the given paraeters, the sste is onvergent. It shold be stated that the stabilit rles ahieved here are eant to be generalied onlsions for all oposite thin-walled blade with different ross-setions and different paraeters, in addition to a soewhat data offset of the speifi nerial point. Seondl, the inflenes of pl angles an be testified b eigenvale analsis. he aeroelasti sste integrated with strtral daping has an of 4N eigenvales or ore. It is not eas to distingish the strtral eigenvales and aerodnai eigenvales. So it is nrealisti to prove the strtral stabilit onl b eigenvales. However, for fltter sste, real parts of soe strtral eigenvales st be eros, and for the whole aeroelasti sste soe real parts of eigenvales st be negative. Frtherore, if the eigenvales are arranged in order (denotes real parts) fro large to sall and naed in order as the st eigenvale, the nd eigenvale, the 3 rd eigenvale, et., eigenvale rles against pl angles ight be larified nder onditions of different pl angles for lassial fltter and stall fltter. (a) lassial fltter (b)stall fltter Figre 3. ie responses of torsion against different pl angles. 548

6 an i and Xejn Wang J. he. Phar. Res., 4, 6(5): (a) lassial fltter (b)stall fltter Figre 4. ie responses of flap bending of different pl angles, respetivel Figre 5 show the first for eigenvales, hanging with pl angles fro 5 o ~75 o at interval of 5 o. he reslts deonstrate that a signifiant variation in the aeroelasti stabilit an be ahieved against pl angles in general. In addition, the reslts show that fltter effets are obviosl different in pl angle of 45 o opared with the other pl angles, whih are onsistent with distribtion harateristis of responses shown in figre 3-4. For lassial fltter in figre 5(a), the for eigenvales in pl angles 5 o are ini,whih shows the ost stable state, and in in pl angles 45 o are the the largest with the ost nstable state. For stall fltter in figre 5(b), the for eigenvales in pl angles 45 o are alost ini,whih shows the ost stable state. (a) lassial fltter (b)stall fltter Figre 5. Eigenvales hange with pl angles fro 5 o ~75 o at interval of 5 o ONUSION ibration and responses of lassial fltter and stall fltter of thin-walled rotor blade with bending-torsional opling are investigated. wo onlding rearks an be drawn: () he opled partial differential eqations are reded to ordinar eqations b disretiation, and espeiall strtral daping is integrated. ie doain response based on Silink an be perfored to anale sste stabilit and vibration harateristis of fltter, aopanied b eigenvale analsis. () he inflenes of pl angles of oposite blade on stabilit are investigated. It is obviosl deonstrated that different pl angles have different effets on lassial fltter and stall fltter. Aknowledgent his work is spported b the Natral Siene Fondation of Shandong Provinial of hina (ZR3AM6), and Qnxing Projet of Shandong Universit of Siene & ehnolog (qx3). REFERENES [] on W. obit, Wind Energ, ol.7,pp. 4, 4. [] eishan J G, Wind Energ,ol.5,pp. 85 3,. [3] owell E E, A Modern orse in Aeroelastiit, lwer: ordreht, 995. [4] haviaropolos P, Soerensen N N, Hansen M O, et al, Wind Energ, ol.6,pp ,3.

7 ingri i J. he. Phar. Res., 4, 6(5): [5]Ren Y S, i R, Jornal of ibration and Shok,ol.3, no.8,46 5, 3. [6]ingri, Yongsheng R, Ata Energiae Solaris Sinia,ol.3, no.,pp.5,. [7]Ming-Hng Hs, Advanes in Aostis and ibration,ol.,pp.,. [8] Aranios E A, Badir A M, AIAA jornal, ol.,pp. 95 9,995. [9] anila S, Aranios E A, International Jornal of Solids and Strtres, ol.35, no.3,pp.35 39,998. [] Xdon W, Jin, Wenhong S, hinese Jornal of Mehanial Engineering,ol., no.,pp. 3,9. [] ingri, Yongsheng R, Xingha Y, Jornal of Flid and Strtre,ol.4,pp.488 5, 3. [] Raesh, Inderjit, AIAA Jornal,ol. 9, pp ,99. [3] Rakar, Ganesan N, annan R, Eropean Jornal of Mehanis-A/Solids,ol.9,pp.53 65,. [4] ar R, atta P, Prabhakara, International Jornal of Mehanial Sienes, ol.45,pp ,3. 55