AIRCRAFT ENGINE RESPONSE DUE TO FAN UNBALANCE AND TO THE PRESENCE OF CONSUMED GAPS IN THE ENGINE DURING THE PHASE OF WINDMILLING

Transcription

1 ICAS CONGRESS AIRCRAF ENGINE RESPONSE DUE O FAN UNBALANCE AND O HE PRESENCE OF CONSUMED GAPS IN HE ENGINE DURING HE PHASE OF WINDMILLING B. Benay AEROSPAIALE MARA AIRBUS 316 ouloue Cedex 3 Fance Abtact When fan blade off and gap in a ubomachine occu non-linea load take place. Afte a tanient phae the main oto eache a windmilling motion inide ome fequencie ange. Jump and hyteei phenomena ae obeved in thi ange. he pape peent two model fo the dynamical imulation of the windmilling motion. Fit a implified model i ued fo which analytic olution ae obtained. he eult ae compaed to the numeical integation of a finite element model of the aicaft. Baed on thi analyi an hamonic method ha been developed to compute the jump fequency; a enitivity tudy of the diffeent paamete elated to the jump phenomenon i pefomed. 1 Intoduction he deign of lage commecial jet aicaft ha equied the ue of high by-pa engine with lage fan. he ie of thee fan ha teadily inceaed to accommodate the thut equied of ucceively lage aicaft and imultaneouly to impove opeating efficiency. It i then neceay to invetigate the load due to fan blade off. Afte the fan blade off event a tanient phae occu. Duing thi phae the main oto ( LP oto o Fan haft peed deceae apidly. hen a tabilied phae occu in which the main oto eache a windmilling motion inide ome fequencie ange depending on the velocity and on the altitude of the aicaft. Duing thi motion eveal contact may occu. hee contact come fom the unbalance poduced by the fan blade off. hi unbalance poduce the bending of the haft. (Lalanne and Feai Moeove duing the tanient phae whee tong vibation occu ome beaing ( fued beaing which ae intended to uppot the LP oto ae boken. It eult a eduction of the tong vibation of the oto induced by the fan blade off but on the othe hand the lo of thee beaing lead to a lage peceional motion of the oto and fo ome value of the otational fequency eveal impact may occu. A phyical model with two degee of feedom which epeent the oto ytem with a beaing cleaance ha been popoed by Ehich and O Conno (1967. hey howed the occuence of jump and hyteei phenomena due to the dynamic coupling between the oto and the tato. Black epoted hi expeimental obevation of thee phenomena (1967 and developed a model (1968 in which pola eceptance ae ued to etablih equilibium condition. In thi pape we ue the implified model of Ehich and O Conno in which a nubbing tiffne i added in ode to take into account the defomation occuing nea the contact one. In a econd tep the eult of the time imulation baed on a finite element model of the ytem {engine+aicaft} ae peented. A paallel i made with the implified model to undetand the mechanim with multigap which lead to the jump phenomenon. Finally a fequency method baed on thi analyi i developed to detemine the fequency at which the jump phenomenon occu ; thi method allow enitivity tudie on the jump fequency to be pefomed. 53.1

2 B. Benay. O. A... I G B x Fig 1 : Phyical model of a oto opeating in beaing cleaance with eal tato dynamic Simplified model A implified model fo the dynamic behavio of a tubomachiney i hown in Fig. 1. (Ehich and O Conno A oto of ma m i ovehung by a k and fixed at one m i ovehung by a k and alo fixed at ymmetic haft of tiffne end. A tato of ma ymmetic haft of tiffne one end. he oto i modeled by a beam of cente B of adiu and of cente of ma G offet fom B by a ditance ε : thi unbalance i poduced by the lo of fan blade. he tato i modeled by a dik of cente A of adiu 1 + δ (δ i the cleaance whoe cente of ma i aumed to be located at A (Fig. 1. he fiction foce ae neglected and the motion i aumed to be ynchonou and axiymmetic. heefoe only two degee of feedom ae ufficient to decibe the motion of the oto and the tato : the adial diplacement of the oto and of the tato. he thee point A B and G ae on the ame line A x (A x oiented in uch a way that BG ε x with ε >. When a contact occu between the oto and the tato (i.e. when > δ the eaction foce R exeted at the contact point I i diected along the line A x if no fiction effect occu. In contat with the tudy made in Ehich and O Conno we aume that mall penetation can occu nea the contact point and a concentated tiffne K i intoduced. hu the model ued in Ehich and al. i obtained fo K infinite. We will fit aume that enegy diipation on the oto and tato tuctue i neglected : c c ( 1 wo poible cae ae then conideed: I. Cae without contact : In thi cae we aume that and < δ. he eaction foce R i abent. he eulting equation ae a follow : m ω ( + ε + k ( ( 3 ( 4 < δ Equation ( 3 give : ω ε ω ω ( 5 with: ω ( 1 k m ( 6 II. Cae with contact : In thi cae δ. Among the two poible cae δ o δ only the fit one i conideed. Ehich and al. (1967 ha hown that the econd poibility coeponding to a motion of the tato with a lage amplitude than the oto i untable. he x pojection R of the eaction foce R exeted at the contact point I on the oto i given by : R K( δ ( 7 he eulting equation ae then : 53.

3 AIRCRAF ENGINE RESPONSE DUE O FAN UNBALANCE AND O HE PRESENCE OF CONSUMED GAPS IN HE ENGINE DURING HE PHASE OF WINDMILLING K( δ mω ( + ε k mω + k δ ( 8 ( 9 Fom equation (8 we obtain the following expeion unde condition (9 : ( µδ ε k K ω ω + ω ( µδ ε 1+ k K ω ω K ( ω ω + ω ( 1+ µ k K ( ω ω ω 1 µ ω ( 1 ε k ω δ ε + ω ( δ ε 1 ( 11 ε k K ( ω ω + ω ( 1+ µ k K ( ω ω ω 1 µ µ m m ( 1 he behavio of the obtained olution depend on eveal paamete namely δ K ω ω µ whee : ε ω ω k n n 1 k + k ω ( 13 n m + m A it i pointed in Ehich and al. (1967 the cae of a tato with a geat tiffne i moe inteeting leading to the aumption ω < ωn < ω. Moeove when δ ε ( 1 ( ω 1 ω paticula n inteeting behavio like jump and hyteei phenomena occu. In (Fig. the vibation epone of the ytem ae hown auming the following numeical data : ω. 5 µ 1. ω n δ ε k (a follow ω ω n K taking diffeent value between (1 and 1. Fom thee cuve we can make the following concluion : -when ω inceae fom.5 to 1 the total banch (cae with contact i decibed followed by a jump down (vetical aymptote to join the banch 1 (cae without contact. -when the ω deceae fom 1 to.5 banch 1 then i followed the latte being joined afte a jump up. hi jump occu when the cleaance i eached (a δ : it i thu the econd inteection between the epone cuve of the oto without contact and the cuve of equation δ that i : ω jump ω δ ε δ ε 1 ( 14 - When K k deceae the amplitude of the jump up inceae and banch move to the left and thu the jump down follow. he cae K k 1 (infinite coepond to the Ehich cae. We will now conide the cae in which damping coefficient c and c ae not neglected. he ynchoniation aumption i then no longe valid and an analytical olution fo banch i thu no longe available. On the othe hand we find the following expeion fo the fit cae without contact : ε ( ω ω + ω ω ξ c ξ m ω ω 4 c k m ( 15 ( 16 Following the ame method a peented befoe (the jump up fequency coepond to the econd inteection between the epone cuve of the oto without contact and the cuve of equation δ the expeion of ω jump i found a : ω jump ω 1 ξ + 4ξ ( ξ 1 ε δ 1 + ε δ ( 17 Banch i obtained by olving dynamic equation of the ytem {oto+tato+beaing}. hi eolution will not be peented hee. It can imply be noted hee that the effect of the damping i a uually obeved to mooth the epone and in paticula to deceae the jump down fequency. 3 Mechanim leading to jump phenomena In the following a finite element model of the aicaft+engine i invetigated in ode to compae the eult obtained fom the implified model. he engine i epeented by a condened finite element model (Guyan condenation of 5 dof. he condened aicaft model (ame condenation i epeented by dof. he modal analyi i made with 5 mode. In thi analyi the 5 th mode ha a fequency aound 6 H which allow coect analyi to be made up to 4 H. In fact the windmilling fequency ange i [8 H ; H]. In contat with the implified model in which only one contact may occu fou conumed cleaance have to be conideed hee : 1. wo between the blade and the caing : between the fan and the fan caing and between the compeo blade (R which ae attached to the intemediate oto and the compeo caing (C. 53.3

4 B. Benay. A cleaance between the oto and the beaing caing (FBH. 3. A cleaance between the LP and oto. Radial nubbing tiffne elated to the eaction foce occuing when ome of the fou poible contact ae peent ae hown fig. 3. he fan-fan caing nubbing foce i moe complex than the othe one a the nubbing tiffne change when the adial diplacement gow (blade elatic buckling. Finally ubbing i not taken into account. Fo a otation fequency of the fan (o LP oto ω the equation govening the aicaft+engine tuctue dynamic expeed in the modal bai i : [ µ ]{} q! + ([ β ] + [ Ψ] [ G( ω ][ Ψ] { q! } + []{ γ q} [ Ψ] ({ Funbalance } + { Fcontact } ( 18 [ µ ] [ β ] and [ γ ] diagonal matice 5 5 epeent the modal ma damping and tiffne. [ Ψ ] i the mode hape matix. [ G ] i the antiymmetic matix of the gyocopic effect. It contain n block if n point epeent the LP oto (the othe oto do not otate duing the windmilling phae each one affected by a pola inetia Ip. he i th bloc i : i ω Ip ω ( 19 ωipi i [ G ( ] i F unbalance ha only two component in the tanvee (y and vetical ( diection with epect to the LP oto axi : F t m.. ω co( ω. ( { (t} unbalance y ( t F t m.. ω in( ω. ( 1 unbalance ( t m epeent the blade off ma and the eccenticity of the oto cg fom the oto geometic centeline due to thi blade lo. he calculation of the tem { F contact (t} i obtained by a loop : fom the computed elative adial diplacement the contact foce ae obtained (fom the nubbing tiffne of the figue 3 and ae then conideed a excitation foce with the unbalance foce. hen equation (18 i olved by a Runge-Kutta integation cheme to obtain the elative adial diplacement. Seveal imulation coeponding to a contant otation fequency of the fan in the windmilling ange [8 H ; H] ae pefomed. Fo each imulation the local peak amplitude (Poincaé point ae hown togethe with the Fouie tanfom of the diplacement fo watefall chat (Ehich 199. Fom the Poincae point method a compaion with the epone without contact i made. o undetand the mechanim which lead to the jump phenomenon the tanvee 1 elative diplacement of the contact point due to the unbalance foce only ae calculated in the ange [8 H ; 3 H] which i a lage fequency ange than the windmilling fequency ange to have an enlaged view of the phenomenon. Compaing thee diplacement and thei cleaance (fo example dy fan dy ca in g and the fit cleaance allow u to know if nubbing occu. A the unbalance foce i hamonic thee diplacement ae obtained by a fequency calculation. Fo example fo the fan-fan caing elative diplacement : dy fan dy ca in g ([ Ψ fan y ] [ Ψca in g y ][. D( ω ][. Ψ] { Funbalance } ( [ D( ω ] ( ω.[ µ ] + jω.([ β ] + [ Ψ] [ G( ω ] [ Ψ] + [] γ 1 ( 3 Funbalance y m.. ω δ ( ω ω ( 4 F ω m.. ω δ ( ω ω.( ( 5 unbalance ( j [ Ψ fan y ] and [ Ψ ca in g y ] epeent the line of the matix [ Ψ ] which coepond to the degee of feedom in the y diection at the fan and fan caing point that can be in contact. In figue 4 the amplitude of thee diplacement i compaed with the Poincaé point of the coeponding diplacement in epone to the unbalance foce and to the nubbing foce when they take place and with thei epective cleaance. wo eonance peak appea coeponding to : at 8.5 H : LP oto fit bending mode. at 5 H : oto fit bending mode. An antieonance alo appea fo the LP oto oto elative diplacement amplitude : the 1 he imulation how that the amplitude of the vetical and tanveal motion ae quai equal fo each otation fequency in the windmilling ange ; that mean that the dynamic tiffne of the ytem i ymmetic. 53.4

5 AIRCRAF ENGINE RESPONSE DUE O FAN UNBALANCE AND O HE PRESENCE OF CONSUMED GAPS IN HE ENGINE DURING HE PHASE OF WINDMILLING diplacement i le than the cleaance in the ange [17 H ; 1 H]. he oto -FBH and R-C diplacement ae imila : oto and R coepond to two nea point on the ame haft and the caing (FBH and C diplacement amplitude ae mall in compaion with oto diplacement amplitude. Fo a otational fequency ω between 17 and 18. H the fou elative diplacement amplitude ae le than thei epective cleaance : the Poincaé point ae then on the cuve ( C coeponding to the diplacement amplitude due to the unbalance foce. Unde 17 H the LP oto- oto diplacement amplitude i le than it cleaance δ : the two otating pat ae in contact and the coeponding Poincaé point epaate fom the cuve ( C. At 15 H the C-R diplacement amplitude eache it cleaance δ 4 ; the jump phenomenon then take place. A in the cae of the implified model aleady conideed the jump phenomenon take place when the cleaance between fixed (C and otating (R pat i conumed. he diffeence between thi indutial cae and the model lie in the peliminay phae when the two oto come into contact. he jump fequency i then obtained at the inteection of the following two cuve : - he C-R diplacement amplitude that i the epone to the unbalance foce and the nubbing foce of the two oto ( LP-. - he cuve of equation : dy 4 δ 4. At 18. H the C-R diplacement amplitude eache it cleaance : the two pat ae in contact and the Poincaé point of thi diplacement epaate fom the cuve ( C. hi diminihe the LP oto- oto diplacement amplitude. he influence on the fan-fan caing diplacement amplitude i malle. 4 Hamonic method to detemine the jump fequency he eult obtained fom the integation of the finite element model how that the jump phenomenon occu when the 4 th cleaance δ 4 (R-C contact i eached while the nd cleaance δ (LP- contact wa aleady eached. he time integation of the finite element model fo eveal otational fequencie of the fan need high compute time. In ode to pefom a enitivity tudy of the jump phenomenon when the value of the two cleaance (LP- and R-C ae changed the finite element model i olved in fequency domain auming that the epone i hamonic with the ame fequency ω a the otational fequency of the fan. Fom fomula (4 (5 it i clea that the unbalance foce i hamonic with the fequency ω. Auming that the mechanim leading to the jump i the ame (fit the LP- cleaance i eached and then jump phenomenon occu when the R-C cleaance i eached even if the value of thee cleaance ae lightly changed we have to take into account in the finite element model the LP- contact foce. he expeimental data give only the value of the adial tiffne K coeponding to the LP- contact. It eult that the contact foce F i given by it tanvee F y and vetical F component in a plane pependicula to the diection of the LP oto axi by the following expeion : F y t ( K.( ( dy LP dy + ( d LP d δ. coθ ( 6 F K.( ( dy LP dy + ( d LP d δ. inθ ( 7 d dlp θ Ac tan ( 8 dy dylp dy LP dy and d LP d coepond to the diplacement in the tanvee and vetical diection of the LP and oto point in contact. Howeve the eult obtained fom the time imulation at contant otation fequency how that when the two haft ae in contact the adial elative diplacement between the two contact point of the LP and oto i nealy contant leading to the following aumption : dy d dy d LP LP d d d d LP LP the peceion angle θ i :.co( ω. t + ϕ( ω.in( ω. t + ϕ( ω.co( ω. t + ϕ( ω.in( ω. t + ϕ( ω ( 9 θ ω t + ϕ( ( 3. ω whee ϕ i the angle between the poition vecto of the two point and the unbalance foce. 53.5

6 B. Benay he tanvee and vetical component of the LP- nubbing foce then become : F t K.[ dy dy δ.co( ω. t + ϕ( ] ( LP ω y ( 31 F K. [ dlp d δ.in( ω. t + ϕ( ω ]( 3 With thee aumption thee component ae alo expeed linealy a a function of the nubbing haft point. he Fouie tanfom of thee component i then eaily detemined : jϕ ( ω F ω K.( dy dy δ. e. δ ( ω ω ( 33 y ( LP j ( ω F ω.( δ. ϕ K dlp d e.( j. δ ( ω ω ( 34 ( he fequency expeion obtained fo the unbalance effot and the LP- nubbing foce ae then put into the Fouie tanfom of the dynamic equation expeed in the modal bai : ( [ ] ([ ] [ ] [ ] [ ] ω. µ i + jω. β i + Ψ G( ω Ψ + [ γ i ] { q( ω } [ Ψ ] ({ F unbalance } + { F } ( 35 he detailed calculation ae peented in the appendix. hi method i applied on the enitivity analyi of the nd and 4 th cleaance on the jump fequency. In fact in compaion with the initial time method thi one allow u to undetand bette the ole of each gap. he vaiation of the R-C cleaance imply coepond to the tanlation of the cuve of equation dy 4 δ. On 4 the othe hand a we calculate the elative R- C diplacement amplitude that i the epone to the unbalance foce and nubbing foce of the two oto ( LP- the numbe of fequency imulation i equal to the numbe of choen LP- cleaance value. hee value ae 75% 8% 135% of the initial gap o 13 imulation (figue 5 have to be made. he limit value ae choen in ode to epect the mechanim decibed befoe : a fequency ange whee no contact occu exit ; the nd cleaance then the 4 th one i eached duing the deceleation. So fo each LP- cleaance value the coeponding value fo the R-C cleaance ae detemined (fo intance fo 8 % of δ δ 4 can vay fom 1 % to 135 %. We obeved that a the LP- cleaance value diminihe the cuve of the R-C diplacement amplitude move to the ight ide and the eonance peak inceae: it i due to the augmentation of the equivalent nubbing tiffne. he eult of thi enitivity tudy can be een in the figue 6. hi figue pove the enitivitie of the two cleaance : fo 16% of the two gap the jump fequency deceae to 11 H. 5 Concluion and popect A paallel ha been found between the dynamic behavio of the implified model and that of the FE integated engine+aicaft model. It allow a fequency method of jump fequency detemination to be developed and then a enitivity tudy on two conumed gap to be made. Concening the implified model an analytical olution of the teady motion with damping will be developed baed on Yamamoto (1954. he blade to caing contact model will be efined a Padovan and Choy (1987 and the ubbing will be taken into account. he utiliation of a FE model of the blade i tudied. he modal bai i fomed by failed engine mode afte the fued beaing have boken. A component mode ynthei will be tudied with component bounday coodinate at the contact point (Bethillie and al An analyi of the hyteei phenomenon will be made : we will how that a ditubance on the engine+aicaft ytem (fo example a gut which could occu duing the flight can caue the hang of the econd tability banch a in the implified model. Finally a bette undetanding of dynamic phenomena i expected with tet on a implified moto ig. Refeence [1] Black H. F. Synchonou Whiling of a Shaft Within a Radially Flexible Annulu Having Small Radial Cleaance. IMechE Pape 4 Vol. 191 pp [] Black H. F. Inteaction of a Whiling Roto With a Vibation Stato Aco a Cleaance Annulu. Jounal Mechanical Engineeing Science Vol. 1 No 1 pp [3] Bethillie M. Dupont C. Mondal R. Blade Foced Repone Analyi with Fiction Dampe. ASME Jounal of Vibation and Acoutic Vol. 1 No. pp Apil [4] Ehich F. F. and O Conno J. J. Stato Whil With Roto in Beaing Cleaance. ASME Jounal of Engineeing fo Induty pp [5] Lalanne M. and Feai G. Rotodynamic Pediction in Engineeing. John Whiley and Son

7 AIRCRAF ENGINE RESPONSE DUE O FAN UNBALANCE AND O HE PRESENCE OF CONSUMED GAPS IN HE ENGINE DURING HE PHASE OF WINDMILLING [6] Padovan J. and Choy F. K. Nonlinea Dynamic of Roto/Blade/Caing Rub Inteaction. ASME Jounal of ubomachiney Vol. 19 pp [7] Yamamoto.. On Citical Speed of a Shaft Memoi of the Faculty of Engineeing Nagoya (Japan Univeity Vol. 6 No. pp Appendix In equation (35 the tem that ha to be calculated i [ Ψ ] { FLP (ω}. With expeion (33 and (34 and a : dylp [ ΨLP y ]{. q( ω } dlp [ ΨLP ]{. q( ω } ( 36 dy [ Ψ y ]{. q( ω } d [ Ψ ]{. q( ω } thi tem become : [ ] { FLP } [ γ ]{ q( ω } δ ( ω ω + [ Ψ] { FLP ( ω ϕ } [ γ ] LP ( [ ΨLP y ] [ Ψ y ].([ ΨLP y ] [ Ψ y ] + ([ ΨLP ] [ Ψ ] ([ ΨLP ] [ Ψ ] Ψ ( 37 K. and : [ ] { F ( ω ϕ } Ψ LP ( 38 ([ Ψ ] [ ] [ ] [ ] LP y j ΨLP Ψ y + j j ( K LP. δ LP. e ϕ ω. δ ( ω ω. Ψ ( 39 and ( 4 he tem ϕ i obtained by the inteection of two cuve whoe equation ae : ϕ ϕ ( 41 ϕ ag( [ H( ω ].({ Funbalance( ω } + { FLP ( ω ϕ } contact [ ] [ ]( [ ] ([ ] [ ] [ ] [ ] [] [ ] 1 H ( ω j G [ ] contact Ψ ω. µ + ω. β + Ψ ( ω Ψ + γ + γ Ψ (

8 B. Benay Fig : Nomalied tato oto and elative amplitude fo K k [1151] ( K k 1 in olid line. Banch 1 : cae without contact. Banch : cae with contact. Fig 3 : Radial nubbing tiffne 53.8

9 AIRCRAF ENGINE RESPONSE DUE O FAN UNBALANCE AND O HE PRESENCE OF CONSUMED GAPS IN HE ENGINE DURING HE PHASE OF WINDMILLING Fig 4 : anvee elative diplacement of the contact point (plu thei cleaance : 1. Due to the unbalance foce (in blue. Due to the unbalance and nubbing foce (Poincaé point in ed Fig 5 : anvee elative R-C diplacement when the LP- cleaance vaie fom 75 % to 135 % and the R-C cleaance. 53.9

10 B. Benay Fig 6 : Senitivity analyi of LP- and R-C cleaance on the jump fequency 53.1