Free Flapping Vibration of Rotating Inclined Euler Beams

Transcription

1 World cdemy of Sciece, Egieerig d Techology Free Flppig Vibrio of Roig Iclied Euler Bems Chih-ig Hug, We-Yi i, d Kuo-Mo Hsio bsrc mehod bsed o he power series soluio is proposed o solve he url frequecy of flppig vibrio for he roig iclied Euler bem wih cos gulr velociy. The vibrio of he roig bem is mesured from he posiio of he correspodig sedy se xil deformio. I his pper he goverig equios for lier vibrio of roig Euler bem re derived by he d'lember priciple, he virul work priciple d he cosise lierizio of he fully geomericlly olier bem heory i roig coordie sysem. The goverig equio for flppig vibrio of he roig iclied Euler bem is lier ordiry differeil equio wih vrible coefficies d is solved by power series wih four idepede coefficies. Subsiuig he power series soluio io he correspodig boudry codiios wo ed odes of he roig bem, se of homogeeous equios c be obied. The url frequecies my be deermied by solvig he homogeeous equios usig he bisecio mehod. Numericl exmples re sudied o ivesige he effec of icliio gle o he url frequecy of flppig vibrio for roig iclied Euler bems wih differe gulr velociy d slederess rio. Keywords Flppig vibrio, Icliio gle, Nurl frequecy, Roig bem. I. INTRODUCTION OTTING bems re ofe used s simple model for Rpropellers, urbie bldes, d sellie booms. Roig bem differs from o-roig bem i hvig ddiiol cerifugl force d Coriolis effecs o is dymics. The free vibrio frequecies of roig bems hve bee exesively sudied [1-1]. However, he vibrio lysis of roig iclied bem is rher rre i he lierure [8, 11, 1]. To he uhors kowledge, he url frequecy for very sleder roig iclied bem high gulr velociy is o repored i he lierure. The objecive of his pper is o derive he correc goverig equios for lier flppig vibrio of roig iclied Euler bem, d ivesige he effecs of icliio gle d slederess rio o he url frequecy of roig Euler bems. The equios of moio for roig C.. Hug is wih Deprme of Mechicl Egieerig, Niol Chio Tug Uiversiy, 1001 T Hsueh Rod, Hsichu, 300, Tiw (phoe: ; e-mil: clhug@roc.edu.w). W. Y. i is wih Deprme of Mechicl Egieerig, De i Isiue of Techology, 1 lley 380, Chig Yu Rod, Tucheg, 36, Tiw (e-mil: wyli@dli.edu.w). K. M. Hsio is wih Deprme of Mechicl Egieerig, Niol Chio Tug Uiversiy, 1001 T Hsueh Rod, Hsichu, 300, Tiw (e-mil: kmhsio@mil.cu.edu.w). Euler bem re derived by he d'lember priciple d he virul work priciple. I order o cpure ll ieri effec d couplig bewee exesiol d flexurl deformio, he cosise lierizio [13, 14] of he fully geomericlly o-lier bem heory [14, 15] is used i he derivio. mehod bsed o he power series soluio is proposed o solve he url frequecy. Numericl exmples re sudied o ivesige he effec of icliio gle d slederess rio o he url frequecy of flppig vibrio for roig iclied Euler bems wih differe gulr velociy. II. FORMUTION. Descripio of Problem Cosider iclied uiform Euler bem of legh rigidly moued wih icliio gle o he periphery of rigid hub wih rdius R roig bou is xis fixed i spce cos gulr velociy s show i Fig. 1. The deformio displcemes of he bem re defied i roig recgulr Cresi coordie sysem which is rigidly ied o he hub. The origi of his coordie sysem is chose o be he iersecio of he periphery of he hub d he ceroid xis of he udeformed bem. The X 1 xis is chose o coicide wih he ceroid xis of he udeformed bem, d he X d X 3 xes re chose o be he pricipl direcios of he bem cross secio he udeformed se. The direcio of he X 3 xis is coicide wih he xis of he roig hub. Thus, he gulr velociy of he hub my be give by { 0 0 } (1) where he symbol { } deoes colum mri which is used hrough he pper. Here i is ssumed h he bem is oly deformed i he X1 X ple. Thus oly xil d flppig vibrios re cosidered. Noe h he xil d flppig vibrios re o coupled d c be lyzed idepedely. I is well kow h he bem susis sedy se deformios (ime-idepede displceme iduced by cos roio [16]. I his sudy, he vibrio (ime-depede displceme of he bem is mesured from he posiio of he sedy se xil deformio, d oly ifiiesiml free vibrio is 790

2 World cdemy of Sciece, Egieerig d Techology cosidered. Here he egieerig sri d sress re used for he mesure of he sri d sress. I is ssumed h he sris re smll d he sress-sri relioships re lier. X X y P s P Q y v Q z X 3 R O X 1 O x u X 1 Fig. Kiemics of deformed bem X 3 () From (3) d he defiiio of egieerig sri, mkig use of he ssumpio of smll sri, d usig he pproximio si v,x d cos 1, he egieerig sri i he Euler bem my be pproximed by X 1 u, x v, x yv, xx (5) X 3 (b) Fig. 1 roig iclied bem () Top view, (b) Side view B. Kiemics of Euler Bem e P (see Fig. ) be rbirry poi i he roig bem, d Q be he poi correspodig o he bem cross-secio of P o he ceroid xis. The posiio vecor of poi P i he udeformed d deformed cofigurios my be expressed s r 0 { R x y z} () r { x u( ysi v( ycos z} r i ei (3) u( us ( x) u( (4) where is ime, u s (x) is he sedy-se xil deformios iduced by cos roio, u ( d v( re he ifiiesiml displcemes of poi Q i he X 1 d X direcios, respecively, cused by he free vibrio, ( is he ifiiesiml gle of roio of he cross secio pssig hrough poi Q bou he X 3 xis, cused by he free vibrio, ei ( i 1,, 3 ) re ui vecors i he X i direcios. O C. Equios of Moio The equios of moio for roig iclied Euler bem re derived by he d'lember priciple, he virul work priciple d he cosise firs order lierizio of he fully geomericlly o-lier bem heory [14]. Fig. 3 shows porio of he deformed ceerlie of he bem. Here he geerlized displcemes re chose o be u, v, d defied i (3). The correspodig geerlized forces re F 1, F, d M, he forces i X1, X direcios, d mome bou X 3 xis. F 1 j, F j, d M j (j =, b) i Fig. 3 deoe he vlues of F1, F, d M secio j. F 1 Fig. 3 Free body of porio of deformed bem For lier elsic meril, he virul work priciple my be wrie s Wex W i (6) b W ex ( F1 u F v M ) b d dx s M F X F1 u F v M dx M b X 1 b F b F 1b (7) 791

3 World cdemy of Sciece, Egieerig d Techology Wi E dv V b V b r rdv (8) correspodig o he sme geerlized virul displcemes, oe my obi where Wex d Wi re he virul work of he exerl b forces d he ierl sresses, respecively, ( ) is he vlue of ( ) secio b mius he vlue of ( ) secio, u, v d re he virul displcemes, is he vriio of give i (5), E is Youg s modulus, V b is he volume of he udeformed bem bewee secio d secio b. The differeil volume dv my be expressed s dv = dd where d is he differeil cross secio re of he bem, is he desiy, r is he vriio of r give i (3), d r d r / d is he bsolue ccelerio. The symbol (. ) deoes differeiio wih respec o ime. The exc expressio of Wi my be very compliced. However, due o he ssumpio of ifiiesiml vibrio, he quiies u, v, d defied i (3) d (4), d heir derivives wih respec o x d re ll ifiiesiml quiies. For lier vibrio lysis oly he erms up o he firs order of ifiiesiml quiies re required. ll erms up o he firs order of ifiiesiml quiies i Wi re reied. Noe h he sedy se xil deformios u s (x) i (4) d is derivives wih respec o x re smll fiie quiies, o ifiiesiml quiies, d re ll reied s zeroh order erms of ifiiesiml quiies. From (3) d (5), usig he pproximio v, x, d reiig ll erms up o he firs order of ifiiesiml quiies, r d my be pproximed by r { u yv, x v yv, x 0} (9) u, x v, xv, x yv, xx (10) The secod ime derivive of r i (3) my be expressed s r ( ro ) ( r) ( r iei ) r i ei (11) r O { Rcos Rsi 0} (1) where i 1,, 3 d is give i (1). From (1), (3), (11) d (1), usig he pproximio v,x, r i (11) my be pproximed by u yv, x ( R cos x u yv, x ) r v (13) ( u yv ), x z R si Subsiuig (5), (7)-(10) d (13) io (6), usig yd 0, d reiig ll erms up o he firs order of ifiiesiml quiies, d he equig he erms i boh sides of (6) F1, x [ u ( Rcos x u)] (14) F, x v (15) F M, x Eu, xv, x I ( v, x v, x ) (16) M EIv, xx (17) F 1 Eu, x (18) where I y d is he mome of ieri of he cross-secio. (14)-(16) re equios of moio d (17) d (18) re cosiuive equios. Subsiuig (18) io (14), d subsiuig (15) d (17) io (16), oe my obi Eu, xx [ u ( Rcos x u)] (19) EIv,xxxx E u xv x x I v (,, ), (,xx v, xx ) v (0) The boudry codiios for roig Euler bem wih fixed ed x = 0 d free ed x = re give by us ( 0) u(0, 0, v(0, 0, v, x(0, 0 (1) F1 (, 0, M (, 0, F (, 0 D. Sedy-Se xil Deformio For he sedy-se xil deformios, u( us ( x), u ( v( 0. Thus (19) d (1) c be reduced o Eus, xx ( Rcos x us ) () u s ( 0) 0, u s, x ( ) 0 (3) e k E (4) where k is dimesioless gulr velociy. If k 1, he sedy-se xil deformio u s (x), which sisfies () d (3), my be my be pproximed by [9] 3 k x Rcos x u s ( x) [ ( Rcos ) x] 6 (5) The mximum vlue of he sedy se xil sri correspodig o he xil deformio give i (5) occurs he roo of he bem d my be expressed s mx u s, x (0) k ( r cos 0.5) (6) 79

4 World cdemy of Sciece, Egieerig d Techology r R / (7) where R is he rdius of he roig hub. E. Free Vibrio The vibrio of he roig bem is mesured from he posiio of he sedy-se xil deformio. From (4), (19), (0) d (), he goverig equios for free vibrio my be expressed s u, xx u u 0 E E (8) v,xxxx us xv x x v (,, ), (,xx v, xx ) v I E EI (9) I c be see from (8) d (9) h he xil vibrio d he flppig vibrio re o coupled d c be solved idepedely. e deoe he url frequecy of roig bem d K / E (30) deoe odimesiol url frequecy. From (8) (30)d boudry codiios i (1), he url frequecy d vibrio mode of roig bem correspodig o he xil vibrio my be expressed s 1 K ( k ) (31) u R si( x / ) (3) where ( 1) /, u R is he xil vibrio mode. For coveiece, he followig odimesiol vribles re used: u x / 0.5, U U u s s s ( ) s /, V V ( ) v / (33) / I (34) where 0 x, is he slederess rio of he bem. From (9), (33) d (34), he dimesioless goverig equios of free vibrio my be expressed s V, ( U s, k ) V, U s, V, V, V 0 E E (35) U s, k [0.5 ( r cos 0.5) 0.5r cos 0.375] (36) We shll seek soluio of (35) i he form V (, V e i R ( ) (37) where i 1, d is he url frequecy o be deermied Iroducig (37) io (35), oe my obi VR, ( b c d) VR, (b c) VR, evr 0 (38) b 0.5 k (39) c k ( r cos 0.5) d K k k (0.5r cos 0.375) e K F. Power Series Soluio The soluio of (38) c be expressed s power series i he idepede vrible : VR( ) C (40) 0 where C re udeermied coefficies. Subsiuig (40) io (38) d equig coefficies of like power of, we obi he recurrece formul 4 j C C j, 4 (41) j1 1 0, d 1 (4) 3 c( 3) 1 4 [ e b( 3)( 4)] 1 3 From (41), I c be see h oly C 0, C 1, C d C3 re idepede coss i (40), d C ( 4 ) c be rewrie s 3 C Yi Ci, 4 (43) i0 4 j j Yi Yi, i 0,1,, 3, 4 (44) j1 if i j Y j 1 i, 0 if i j i, j 0, 1,, 3 (45) Subsiuig (43) io (40), oe my obi 793

5 World cdemy of Sciece, Egieerig d Techology VR ( ) E ( )C (46) E ( ) { E0 E1 E E3} (47) E0 1 Y0, E1 Y E Y, E3 Y3 4 4 C { C0 C1 C C3} (48) From boudry codiios give i (1), d (16), (17), (5), (36), (37) d (46), oe my obi se of homogeeous equios expressed by K(K ) C 0 (49) E ( 1) E ( 1) K ( K) EI, [( U ( ) ) ( ) ( s K k E E )] (50) EI E where K is 4 4 mri K(K) deoes K is fucio of K, E E,, E E,, E E,, d d 0. 5 deoe he vlues of he odimesiol coordies wo ed odes for he roig bem. For orivil C, he deermi of he mrix K mus be equl o zero. The vlues of K which mke he deermi vish re clled eigevlues of mrix K d give he url frequecies of he roig Euler bem hrough K i Eq. (30). The bisecio mehod is used here o fid he eigevlues. III. NUMERIC EXMPES To demosre he ccurcy of he proposed mehod d o ivesige he effec of icliio gle o he url frequecy of roig iclied Euler bems wih differe gulr velociy d slederess rio, severl umericl exmples re sudied. Here he followig cses re cosidered: r 0.5, 1.0, k 0, 0.03, 0. 06, d ( ) 0, 15, 30, 45, 60, 75, 90. From (6), oe my obi mx , he mximum sedy xil sri occurred he roo of he bem for ll cses sudied. The xil d lerl vibrio modes re o coupled here. For coveiece, le K i d K i deoe he ih dimesioless url frequecies of lerl d xil vibrio, respecively. From (31), i is oed h K i re fucios of he dimesioless gulr velociy k oly, d from (35) d (36), i is oed h Ki re fucios of slederess rio, he dimesioless gulr velociy k, he dimesioless rdius of he roig hub r, d he icliio gle. From (31), he firs hree dimesioless url frequecy of xil vibrio correspodig o he dimesioless gulr velociy k 0, 0.03, 0.06 my be obied d give s follows: K , , , K , 4.719, , d K , , s expeced, he vlue of K i decreses slighly wih he icrese of k. Tbles 1-6 prese dimesioless url frequecies Ki ( i 1-5) for he roig iclied bem wih differe slederess rio. I c be see from Tbles 1-6 h he vlues of Ki correspodig o he sme icreses wih icrese of k d r, bu decreses wih icrese of. However, whe 90, he vlues of Ki correspodig o he sme d k re ideicl for differe r. These resuls my be explied by he cerifugl siffeig effec d (36). s c be see from (36) h he cerifugl force icreses wih icrese of k d r, bu decreses wih icrese of. However, r cos90 0 for ll vlues of r. I c be see from Tbles 1-3 d 4-6 h he cerifugl force, which is proporiol o k, hs sroger effec o he lower Ki. For 50, he effec of he cerifugl force o K i ( i 4) is egligible. However, for 500, he effec of he cerifugl force o K 5 is sill remrkble. I idices h he effec of he cerifugl force o K i icreses wih icrese of. IV. CONCUSIONS I his pper, he correc goverig equios for lier vibrio of roig iclied Euler bem re derived. The vibrio of he bem is mesured from he posiio of he sedy-se xil deformio, d oly ifiiesiml free vibrio is cosidered. The equios of moio for roig Euler bem re derived by he d'lember priciple, he virul work priciple d he cosise lierizio of he fully geomericlly o-lier bem heory. The goverig equio for lier flppig vibrio of roig bem is solved by power series wih four idepede coefficies. Subsiuig he power series soluio io he correspodig boudry codiios wo ed odes of he roig bem, se of homogeeous equios c be obied. The url frequecies my be deermied by solvig he homogeeous equios usig he bisecio mehod. The resuls of umericl exmples show h he effec of he cerifugl force o he url frequecies correspodig o lerl vibrio mode decreses wih icrese of he icliio gle, bu icreses wih icrese of he slederess rio for he iclied roig Euler bem. 794

6 World cdemy of Sciece, Egieerig d Techology TBE 1 ( r 0.5, 50) k K1 K TBE 4 ( r 1. 0, 50 ) k K1 K TBE ( r 0. 5, 100 ) k K1 K TBE 5 ( r 1. 0, 100 ) k K1 K TBE 3 ( r 0. 5, 500 ) k K1 K TBE 6 ( r 1. 0, 500 ) k K1 K

7 World cdemy of Sciece, Egieerig d Techology REFERENCES [1] M. J. Schilhsl Bedig frequecy of roig cilever bem SME Jourl of pplied Mechics vol. 5 pp. 8-30, [] J. T. S. Wg, O. Mhreholz d J. Bohm, Exeded Glerki's mehod for roig bem vibrios usig egedre polyomils Solid Mechics rchives, vol. 1, pp , [3]. eiss Vibriol specs of roig urbomchiery bldes SME pplied Mechics Reviews vol. 34 pp , [4] D. H. Hodges d M. J. Rukowski, Free-vibrio lysis of roig bems by vrible-order fiie-eleme mehod, I Jourl, vol. 19, pp , [5] T. Yokoym Free vibrio chrcerisics of roig Timosheko bem Ieriol Jourl of Mechicl Sciece vol. 30 pp , [6] H. H. Yoo d S. H. Shi, Vibrio lysis of roig cilever bems Jourl of Soud d vibrio, vol. 1, pp , [7] S. Y. ee d Y. H. Kuo Bedig frequecy of roig bem wih elsiclly resried roo SME Jourl of pplied Mechics vol. 58 pp , [8] H. P. ee, Vibrio o iclied roig cilever bem wih ip mss, SME Jourl of Vibrio d cousics, vol. 115, pp. 4145, [9] S. C. i d K. M. Hsio Vibrio lysis of roig Timosheko bem Jourl of Soud d Vibrio vol. 40, pp , 001. [10].. l-qisi, No-lier dymics of roig bem clmped wih chme gle d crryig ieri eleme, The rbi Jourl for Sciece d Egieerig, vol. 9, pp , 004. [11] S. Y. ee d J. J. Sheu, Free vibrios of roig iclied bem, SME Jourl of pplied Mechics vol. 74 pp , 007. [1] S. Y. ee d J. J. Sheu, Free vibrio of exesible roig iclied Timosheko bem, Jourl of Soud d Vibrio, vol. 304, pp , 007. [13] J. C. Simo d K. Vu-Quc The role of o-lier heories i rsie dymic lysis of flexible srucures Jourl of Soud d Vibrio vol. 119, pp , [14] K. M. Hsio, Coroiol ol grgi formulio for hree-dimesiol bem eleme, I Jourl, vol. 30, pp , 199. [15] K. M. Hsio, R. T. Yg, d. C. ee, cosise fiie eleme formulio for olier dymic lysis of plr bem, Ieriol Jourl for Numericl Mehods i Egieerig, vol. 37, pp , 1994 [16] P. W. ikis, Mhemicl modelig of spiig elsic bodies for model lysis I Jourl, vol. 11, pp ,