EXACT STOCHASTIC ANALYSIS OFCOUPLED BENDING-TORSION BEAMS WITH IN-SPAN SUPPORTS AND MASSES

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1 Meccc de Mterl e delle Strutture Vol. VI 06), o., pp. 5-3 ISS: X Dprtmeto d Igeger Cvle, Ambetle, Aerospzle, De Mterl DICAM EXACT STOCHASTIC AALYSIS OFCOUPLED BEDIG-TORSIO BEAMS WITH I-SPA SUPPORTS AD MASSES AdreBurlo, GuseppeFll dfelceare * Dprtmeto d Igeger Cvle, dell Ambete, dell Eerg e de Mterl DICEAM) Uverstà d Reggo Clbr 894 Reggo Clbr, Itly e-ml:.burlo9@gml.com guseppe.fll@urc.t re@urc.t Key words: Rdom lods, Coupled bedg-torsol vbrtos, Geerlzed fuctos, Elstc supports, Attched msses. Prole chve:crch letor, Vbrzo ccoppte torsol-flessol, Fuzo geerlzzte, Support elstc, Msse ppese. Abstrct. The stochstc respose of coupled bedg-torso bem, crryg rbtrry umber of supports/msses, s vestgted. Usgthe theory of geerlzed fuctos coucto wth the elemetry coupled bedg-torso bem theory, exct lytcl solutos uder sttory puts re obted bsed o frequecy respose fuctos derved by two dfferet closed-form expressos. The lytcl solutos re obted for ll respose vrbles, cosderg y umber of supports/msses log the bem d rbtrry sptl lod dstrbutos. A pertetumerclexmplesreported. Sommro. Vee dgt l rspost stocstc d u trve co u umero rbtrro d support elstc e msse ppese,che preset u comportmeto flessole e torsole ccoppto. Utlzzdo l teor delle fuzo geerlzzte e l teor elemetre d ccoppmeto flessole-torsole, soluzo ltche estte soo otteute, per ecctzo stzore, ttrverso fuzo d rspost frequez dervte ttrverso due dverse espresso form chus. Le soluzo ltche soo otteute per tutte le vrbl dell rspost, cosderdo u umero rbtrro d support e msse lugo l trve e u qulss dstrbuzoe spzle del crco. S rportfe uesempoumerco. ITRODUCTIO The respose of bems to rdom lods hs bee wdely vestgted lterture, d some exmples c be foud ref. [-]. Most works coceredbedg vbrto lyss bsed o Euler-Beroull or Tmosheko bem theory, uder the fudmetl ssumpto thtthe bem cross-secto s doubly-symmetrc, so tht the sher ceter SC) cocdes wth the grvty ceter GC). However, bems wth moo-symmetrc cross sectos re frequetly employed dfferet egeerg pplctos such s wgs, turbe bldes d propellers. Becuse, ths cse, the SC does cocde wth the GC, the dymcs of these bems exhbts coupled bedg- Meccc de Mterl e delle Strutture VI 06),, PP

2 Adre Burlo, Guseppe Fll e FelceAre. torso pheome. Ideed, bedg-duced ertl forces, whch rse log the mss xs.e. the locus of the grvty ceters of the bem cross sectos), re eccetrc wth respect to the elstc xs.e. the locus of the sher ceters of the bem cross sectos) d, s result, bedg s heretly coupled wth twstg. The couplg effects hve geerlly bee descrbed by the so-clled elemetry coupled bedg-torso bem theory [3-4],.e. eglectg wrpg effects d ot cosderg rottory ert d sher deformto of the bem. These effects were lso cosdered other studes [5-6]. As for forced vbrto lyss, usg the orml mode method Eslmy et l. [7-8] obted the respose of coupled bedg-torso bem whe subected to determstc s well ssttory, Guss lods. Specfclly, they ppled the method to ctlever rcrft wg for whch there s substtl couplg betwee bedg d torsol vbrtos. It must be otced, however, tht ll works ref.[] through ref.[8] geerlly ddressed uform bems, wth o ttchmets or -sp supports. The ltter re of gret terest for egeerg pplctos but hve rrely bee cosdered studes o coupled bedgtorso pheome. Ths pper proposes exct method for bems wth moo-symmetrc cross secto, crryg rbtrry umber of elstc supports d ttched msses, subected to sttory lods. The elemetry bedg-torso theory s cosdered s ref.[3-4], coucto wth the theory of geerlzed fuctos to hdle the dscotutes of the respose vrbles t the pplcto pots of supports d msses. Exct lytcl solutos for the respose re bult bsed o frequecy respose fuctos of the bem obted, ths pper, by two dfferet closed-form expressos. The key step to buld the frequecy respose fuctos s ovel closed-form lytcl expresso derved, ths pper, forthe respose of the bem wthout respose supports/msses, subected to rbtrrly-plced hrmoc ut force d ut twstg momet. A umercl exmple wll show the respose power spectrl desty fuctoof bem wth gulr secto, crryg trsltol d torsol-rottol elstc supports, uder trsverse sttory lod. PROBLEM STATEMET Fgure shows the cse uder study,.e. uform strght bem of legthl, referred to rght hded coordte system, crryg rbtrry umber of elstc supports d ttched msses d subected to trsverse dstrbuted lod. Bem cross secto s ssumed to be moo-symmetrc, beg x ts symmetrc xs. The loc of the sher ceters d mss ceters of the bem cross secto re respectvely the elstc xs d the mss xs; the frst cocdes wth the y-xs, whle the ltter s t dstce x from y-xs. The bedg deflecto the z- drecto, the bedg rotto bout the x-xs d the torsol rotto bout the y-xs of the sher ceters re deoted respectvely by h y, t), θ y, t) d ψ y, t). The trsverse dstrbuted lod, ctg prllel to O z, s ppled t dstce x c from y - xs; t c be seprted to trsverse lod ppled log the elstc xs, d dstrbuted torque bout Oy, s show Fgure. The elstc support d ttched msses re ppled t y, wth 0 < y <... < y < L d stffess prmeters of the -th support d propertes of the -th mss re deoted s follows: k for trsltol support, k Ψ for torsol-rottol support d for bedg H rottol support. M s the mss, I xx d Iyy the compoets of the mss ert tesor bout xes x Meccc de Mterl e delle Strutture VI 06),, PP k θ

3 Adre Burlo, Guseppe Fll e FelceAre. d y Fgure. Fgure : Bem wth moo-symmetrc cross-secto crryg rbtrry umber of elstc supports d ttched msses d subected to trsverse dstrbuted lod d dstrbuted torque. Equtos of moto Pursug the prmry purpose of obtg the frequecy respose fuctos order to determe the stochstc respose, ssume tht the bem, crryg supports/msses s show Fgure, s loded by hrmoc dstrbuted lod f y) e t, d hrmoc dstrbuted torque ) g y e t f y) x e = t, o the tervl, b ), wth 0 b L, where s the c t t frequecy. Let be h y,, t) = H y, ) e, ψ y,, t) = Ψ y, ) e, the stedy stte respose vrbles. By usg the elemetry coupled bedg-torso theory d mkg use of the geerlzed fuctos, the followg stedy stte coupled equtos of moto re derved: 4 d H EI m c H + mx c x Ψ P y y + Mf y y + f y = 4 dy ) ) ) δ ) δ ) ) 0 ) = = d Ψ GJ m x c x H I c P x y y Mt y y g y + Ψ + + = dy ) ) ) δ ) δ ) ) 0 α ) = = where EI d GJ re respectvely bedg d torsol rgdtes, m s the mss per ut legth, I α s the polr momet of ert per ut legth bout the elstc xs, whle c d c re vscous dmpg coeffcet per ut legth respectvely bedg d torso, to be ssged so tht dmpg s proportol [8]; Eqs.)-) δ k ) y y ) s the k-th forml dervtve of Drc s delt δ y y ), whle P, Mt d Mf re cocetrted force, twstg momet d bedg momet ssocted wth supports d ttched msses t y, d they re gve s: P = κ P ) H y ) x Ψ y ) ; Mt ) ) = κ T Ψ y ; Mf = κ M ) Θ y ) 3) beg H y ), Ψ y ) d Θ y ) the deflecto, torsol d bedg rotto t y = y, d κ P ), κ T ), κ M ) frequecy depedet terms gve s: P ) = H M ; κ κ κ = k I M x ) ; T Ψ yy M ) = k Θ I 4) xx κ Eqs.)-)c be combed, by elmtg ether Hor Ψ, to obt two ucoupled 6-th order dfferetl equtos for deflecto d torsol rotto: Meccc de Mterl e delle Strutture VI 06),, PP

4 Adre Burlo, Guseppe Fll e FelceAre. 6 4 d H d H d H I GJ ) H x ) f y) f y) g y) R y) H ext α + β γ + η + = 5) dy dy dy mx mx 6 4 Ψ Ψ Ψ EIx x x 4) 4) + + Ψ + f y) f y) + g y) g y) + R y) = 0 6) 6 4 Ψext d d d EI m α β γ η dy dy dy mx x mx mx where I GJ ) α R y) = ) ) ) ) H P ext x δ y y + δ y y Mt δ y y + = mx mx = I + ) ) ) GJ α 3) Mf δ y y + y y δ = mx mx 7) EIx x x R y = P y y y y + 4) ) ) ) H ext δ δ = mx x EI 4) m ) + Mt δ y y ) δ y y ) + Mf δ y y ) = mx mx = 8). Solutos of the equtos of moto Cosder frst the followg equto: 6 4 X y y 6 4 0) 0 d X d X d X α + β γ + η dx dx dx δ = 9) The soluto Xtkes the form: 6 *) 0 = Ω + 0 = X y, y ) c J y, y ) 0) where Ω = cosh r X ); Ω = sh r X ); Ω 3 = cos r X ); ) Ω 4 = s r X ); Ω 5 = cos r3 X ); Ω 6 = s r X ); whle the prtculr tegrl s obted by Mthemtc, fter few mpultos the followg form: *) J y, y0) = D sh r y y0)) r r3 r3 r ) s r y y0)) r r3 r + r3 ) + ) + s r3 y y0)) r r r + r ) U y y0) beg U ) the Ut-Step fucto, d = α / 4ηβ + 4γ α + 7α η γ β 8 αβγη) d D = d r + r ) r + r ) r r ) r r r Strtg from Eq.9) closed-form solutos c redly be derved for Eqs.5)-6), s deed the prtculr tegrls ssocted wth forml dervtves of the Drc s delt, whch re relted to ut force d momets Eqs.5)-6), c redly be obted from Eq.9) by successve dervto. Meccc de Mterl e delle Strutture VI 06),, PP

5 Adre Burlo, Guseppe Fll e FelceAre. 3 FREQUECY RESPOSE FUCTIO OF BEAMS WITH SUPPORTS/MASSES VIA GEERALIZED FUCTIOS I ths Secto exct method to buld the frequecy respose fucto of the bem wth supports/msses Fgure wll be show. Frstly cosderthe bem Fgure subected oly to the trsverse dstrbuted lod f y ) the tervl, b ) wth 0 b L. By pplyg the ler superposto prcple, the vector the bem wth supports/msses c be wrtte s: Y f ) f ) f ) f ) f ) f ) f ) y) = H Θ M S Ψ T collectg ll respose vrbles of f ) = f ) f ) + + f ) = Y y) Ω y) c J y, y ) Λ Y y) 3) T f ) I Eq.3), vector Λ = P Mt Mf collects the ukow recto force P ), twstg momet Mt ) d bedg momet Mf ) t locto y ; J y, y ) s 6 3 mtrx collectg the prtculr tegrls relted to the respose dscotutes t locto y, bult s expled Secto.; Y ) ) y re the prtculr tegrls relted to lod f y ), f = ) b P), ) ) f ) Y y J y ξ f ξ dξ 4) where J P) y, ξ) s the vector of the prtculr tegrls relted to ut trsverse lod. Through recursve procedure, the ukows Λ Eq.3)c be obted s fuctos f ) the vector of tegrto costts c oly, to flly derve the followg expresso f for Y ) y) f ) f ) Y y) = Y y) c + Y f ) y) 5) where Y y) s vector depedg o the bem prmeters oly, d Y f ) y) depeds o the bem prmeters d the ppled lod expressos re omtted for brevty). The vector of f ) tegrto costts c c be determed o settg the bem boudry codtos. Specfclly, t wll be obted by vertg 6 6 mtrx closed form usg Mthemtc. A ltertve exct expresso of the frequecy respose fuctos c be bult by modl superposto. For ths, exct turl frequecesc be obted from the egeproblem B c = 0 6) ) where B s 6 6 mtrx bult bsed o Eq.3),where the lod depedet term Y f ) y ) = 0 d c = c = 0 Eqs.)-). Upo computg the turl frequecesfrom Eq.6), correspodg exct egefuctos re derved, closed lytcl form, from Eq.3)for Y f ) y ) wth, g, Y f ) y ) = 0 d c = c = 0 Eqs.)-).Deotg H d Ψ s the egefuctos of deflecto d torsol rotto, respectvely,d by cosderg the orthogolty codto,derved s smlrly ref.[8], the followg expresso c be derved for the frequecy respose fucto for bedg deflecto d torsol rotto where f ) f ) f ) f ) = Ψ = Ψ = = H y, ) q ) H y) ; y, ) q ) y) 7) f q ) ) deotes the modl frequecy respose fucto of the th mode: Meccc de Mterl e delle Strutture VI 06),, PP

6 Adre Burlo, Guseppe Fll e FelceAre. L [ ] f y) H y) dy f ) 0 ) = µ + ξ ) q Eq.8)provdes sght to the cotrbutos of every mode to the frequecy respose of the bem, very useful for desg purposes. Frequecy respose fucto for the other respose vrbles c be obted from Eq.7) by cosderg St-Vet d Euler-Beroull bem theores. Smlrly the frequecy respose fuctos Y g ) y) due tothe torque g y) = f y) x c, c be obted replcg Eq.3) Y ) ) y wththe prtculr tegrl Y ) ) y,gve s f = ) b Mt), ) ) g ) g q g 8) Y y J y ξ g ξ dξ 9) d replcg Eq.7) q f ) ) wth ) ), whose expresso s: L [ Ψ ] g y) y) dy g ) 0 ) = µ + ξ ) q 0) 4 RESPOSE TO RADOM LOADS ext, the stochstc respose of the coupled bedg-torso bem wth supports d ttched msses, subected to sttory lods,wll be obtedusgthe frequecy respose fuctos derved Secto Respose to cocetrted lods Cosder the bem Fgure subected to fte umber K of sttory cocetrtedtrsverse forcesp r ppled t dstce x from the elstc xs,ssumedto be sttstclly depedet. The power spectrl desty fuctos of the deflecto, SHH y, ), Y y) = H Θ M S Ψ T, c be obted by the dll respose vrbles vector [ ] followg expressos volvg the cross spectrl desty fuctos of the forces S ) { K K f )* f ) g )* g) Y, ),, ),, ),, ),, ) Y r s r s r= s= S y = Y y y Y y y + Y y y Y y y + ) f )* g ) g )* f ) Y y, yr, ) Y y, ys, ) + Y y, yr, ) Y y, ys, ) SP )} r P s f ) g ) I Eq.), the stersk deotes complex cougte d Y y, ξ, ), Y y, ξ, ) re gve f s Eq.5) d ts logous for gy),where Y ) y) Eq.4) d rereplced respectvely by P J ) y, ξ) d Mt J ) y, ξ ). 4. Respose to dstrbuted lods Meccc de Mterl e delle Strutture VI 06),, PP c Pr Ps Y g ) y) Eq.9) Assume ow tht the bem Fgure s cted upo by sttory dstrbuted trsverse lodft), ppled t dstce x c from the elstc xs. The lod srdomly vryg wth respect to tme, but ot sptlly.if S ) deotes the power spectrl desty fucto of ft), the ff power spectrl desty fuctos of ll respose vrbles c be obted s follows: f ) g ) f )* g ) g )* f ) S y, ) = Y y, ) + Y y, ) + Y y, ) Y y, ) + Y y, ) Y y, ) S ) YY ff )

7 Adre Burlo, Guseppe Fll e FelceAre. where f ) Y y, ), Y g ) y, ) re gve respectvely s Eq.5)d ts logous for g y ). 5 UMERICAL EXAMPLE Fgure : Clmped-clmped bem wth gulr secto crryg elstc supports Cosder the clmped bem wth gulr cross secto show Fgure. Propertes re chose s follows: I = m, J = , I = kg m, x = 0.05m, L = 3m, m = 7.83kg m, E = 70 0 m, G = m, c = 5 0 m s, c = 3.5 s, b = 0.5m, t = 0.0m. 9 9 α The bem crres two trsltol elstc supports t loctos y = 0.5L d y3 = 0.75L, both ppled t dstce x = x 3 = 0.05 from the SC of the bem crosssecto, d torsol-rottol elstc support t the locto y = 0.5L s show Fgure. The bem s cted upo by sttory trsverse lod P wth S ) =, ppled t y = 0.35L log the elstc xs prllel to O z, t dstce xc = x from y xs. Sce the SC s eccetrc wth the respect to the MC log the x xs, s show Fgure, the bem rdom respose the y z ple shll be vestgted cosdergthe couplg betwee bedg d torsol vbrtos. Fgure 3 shows the power spectrl desty fuctos of the bedg deflecto, Hy), s well sthe deflecto of the MC due to torsol rotto,ψy)x, clculted t y = 3L 7 through the exct frequecy respose fuctosbult Secto 3. Results re excellet greemet. Fgure 3 shows tht twstg cotrbutes sgfctly to the deflecto of the MC, thus cofrmg the mportce of ccurte methods to cpture bedg-torsol couplg effects the bem respose. PP 6 COCLUSIOS The pper hs preseted exct lytcl solutos for the respose power spectrl desty fuctos of bems wth moo-symmetrc cross secto, crryg rbtrry umber of sp supports d ttched msses, subected to sttory lods. The solutos re obted Meccc de Mterl e delle Strutture VI 06),, PP

8 Adre Burlo, Guseppe Fll e FelceAre. through two dfferet closed-form expressos of the frequecy respose fuctos, usg the theory of geerlzed fuctos d elemetry coupled bedg-torso bem theory.umercl results demostrte the mportce of bedg-torsol couplg effects bems uder sttory lods, d the ccurcy of the proposed pproch. Fgure 3:Power spectrl destes of pure bedg deflecto S, ) to torsol rotto ) HH ΨΨ, gry), computed t y 3L 7 S y x y blck), ddeflecto of the MC due =, wth frequecy respose fuctos gve v geerlzed fuctos cotuous le) d orml mode method dotted le). BIBLIOGRAFY [] S. D Lorezo, M. D Pol, F.P. Pol, A. Prrott, Stochstc respose of Frctolly dmped bems., Prob. Eg. Mech., 35, ). [] J.R. Beree d D.Keedy, Respose of xlly loded Tmosheko bems to rdom lods., J. Soud Vb., 0 4), ). [3] J.R. Beree, Coupled bedg-torsol dymc stffess mtrx for bem elemets., It. J. um. Method. Eg., 8, ). [4] J.R. Beree, Explct frequecy equto d mode shpes of ctlever bem coupled bedg d torso., It. J. um. Method. Eg., 8, ). [5] S. Tmosheko, D.H. Youg, W.J.R. Wever, Vbrto Problems Egeerg, Wley,ew York 974). [6] A.. Berc d M. Tk, Coupled flexurl-torsol vbrtos of Tmosheko bems., J. Soud Vb., 07 ), ). [7] S.H.R. Eslmy d J.R. Beree, Use of geerlzed mss the terpretto of dymc respose of bedg-torso coupled bems., J. Soud Vb., 38 ), ). [8] S.H.R. Eslmy d J.R. Beree, Respose of bedg torso coupled bem to determstc d rdom lods., J. Soud Vb., 95 ), ). Meccc de Mterl e delle Strutture VI 06),, PP