Dynamic behaviour of a rotating cracked beam

Transcription

1 Journal of Physics: Confrnc Sris PAPER OPEN ACCESS Dynamic bhaviour of a rotating crackd bam To cit this articl: Ahmd Yashar t al 6 J. Phys.: Conf. Sr Viw th articl onlin for updats and nhancmnts. Rlatd contnt - Static and dynamic bhaviours of hlical spring in MR fluid S Kawunrun, O Akintoy and M Papalias - A hybrid approach for modlling dynamic bhaviours of a rotor-foundation systm Z G Zhang, Z Y Zhang, B Jing t al. - Insight into th dynamic bhaviour of th Van dr Pol/Raligh oscillator using th intrnal stiffnss and damping forcs M J Brnnan, B Tang and J C Carranza Rcnt citations - Simplifid modlling and analysis of a rotating Eulr-Brnoulli bam with a singl crackd dg Ahmd Yashar t al This contnt was downloadd from IP addrss on /9/8 at 7:

2 MOVIC6 & RASD6 Journal of Physics: Confrnc Sris 744 (6 57 doi:.88/ /744//57 Dynamic bhaviour of a rotating crackd bam Ahmd Yashar, Maryam Ghandchi-Thrani and Nil Frguson Institut of Sound and Vibration Rsarch, Univrsity of Southampton, Southampton SO7 BJ, UK amiyg4@soton.ac.uk Abstract. This papr prsnts a nw approach to invstigat and analys th vibrational bhaviour of crackd rotating cantilvr bams, which can for xampl rprsnt hlicoptr or wind turbin blads. Th analytical Hamiltonian mthod is usd in modlling th rotating bam and two numrical mthods, th Rayligh-Ritz and FEM, ar usd to study th natural frquncis and th mod shaps of th intact rotating bams. Subsquntly, a crack is introducd into th FE modl and simulations ar prformd to idntify th modal charactristics for an opn crackd rotating bam. Th ffct of various paramtrs such as non-dimnsional rotating spd, hub ratio and slndrnss ratio ar invstigatd for both th intact and th crackd rotating bam, and in both dirctions of chordwis and flapwis motion. Th vring phnomna in th natural frquncis as a function of th rotational spd and th buckling spd ar considrd with rspct to th slndrnss ratio. In addition, th mod shaps obtaind for th flapwis vibration ar compard using th modal assuranc critrion (MAC. Finally, a nw thr dimnsional dsign chart is producd, showing th ffct of crack location and dpth on th natural frquncis of th rotating bam. This chart will b subsquntly important in idntifying crack dfcts in rotating blads.. Introduction Rotating bams play a substantial part in various tchnological applications such as gas turbin blads, hlicoptr propllrs and wind turbins. Fracturs ar most common faults in ths structurs that might contribut to incrasd vibration, which can ultimatly caus complt failur. Invstigations ar mad rgarding th vibrational bhaviour of a crackd rotating blad, which includ crack idntification and dtction mthods so that subsqunt damag could b prvntd. Numrous studis hav invstigatd th ffct of th rotational spd on th natural vibration of cantilvr bams. Ths studis concludd that th natural frquncis of flxural vibration tndd to incras abov thos for th non-rotating bams bcaus of th cntrifugal forc and rsulting stiffning ffcts []. Th fr vibration analysis of rotating bams hav bn widly studid [ ]. Howvr, th analysis of rotating crackd bams ar rlativly scarc in th litratur [, 3]. Th objctiv of this papr is to dvlop a mthod to modl a crackd rotating cantilvr bam. Initially, th uncrackd rotating bam is modlld using a simpl continuous bam (th quilibrium of forcs basd on Hamiltonian principl to find th govrning diffrntial quations [6]. Thn, a finit lmnt mthod and Rayligh Ritz mthod ar usd to obtain th numrical rsults for a rotating intact bam. Subsquntly, a crack lmnt is introducd using fractur mchanics as prsntd by Qiau [4] and Bhra [5] to th finit lmnt modl to calculat th ffct of th crack on th rotating systm. Finally, a paramtric study is carrid out for a crackd rotating bam using paramtrs such as th rotational spd, Contnt from this work may b usd undr th trms of th Crativ Commons Attribution 3. licnc. Any furthr distribution of this work must maintain attribution to th author(s and th titl of th work, journal citation and DOI. Publishd undr licnc by Ltd

3 MOVIC6 & RASD6 Journal of Physics: Confrnc Sris 744 (6 57 doi:.88/ /744//57 th crack location, th hub ratio and th slndrnss ratio.. Thortical modlling of a rotating crackd bam.. Modlling of a rotating bam Th quation of motion for th latral vibration of a uniform thin homognous bam subjctd to an axial forc can b writtn as; ρa w t ( + x EI w x ( P w = f(x, t ( x x whr ρ, A, E, I, w, P and f rprsnt th mass dnsity, cross sctional ara, modulus of lasticity, scond momnt of ara, transvrs displacmnt, longitudinal (axial forc and transvrs latral distributd forc, rspctivly. For fr vibration, uniform cross sction and constant P with rspct to tim, ( can b rwrittn as, ρa w t + EI 4 w x 4 ( P w = ( x x whr r is th radius of th hub and x is th distanc along th bam masurd from th hub. Figur. Latral vibration of th bam subjctd to an axial forc. P (x For a rotating bam, th axial forc P is du to and qual to th cntrifugal forc in Figur and can b obtaind as a function of position x along th bam from, P (x = L x ρaω (x + rdx = ρaω {r(l x (L x } (3 Th quation of motion for th transvrs vibration of th rotating bam thn can b obtaind by substituting (3 into (; ρa w t + EI 4 w x 4 { ρaω [r(l x x (L x ] w } = (4 x For th fr vibration of th rotating bam, a harmonic solution with th frquncy of ω and amplitud of W of th sparabl following form is assumd. w(x, t = W (x cos(ωt (5

4 MOVIC6 & RASD6 Journal of Physics: Confrnc Sris 744 (6 57 doi:.88/ /744//57 Substituting th solution, Eq.(5, into Eq.(4 yilds, ρaw ω + EI d4 W dx 4 d { ρaω [r(l x dx (L x ] dw } = (6 dx and th boundary conditions ar assumd to b as follows for a cantilvr ; W (x = d W (x dx dw (x dx =, x = = d3 W (x dx 3 =, x = L (7 Th xact solution of Eq.(6 is difficult to obtain analytically du to th inclusion of th cntrifugal forc and th trm d dw dw dx (x dx, which is qual to x dx + x d w. Nvrthlss, an dx approximat mthod using a numrical mthod such as th finit lmnt mthod or Rayligh- Ritz mthod can b usd to solv this quation. Figur shows a rotating cantilvr bam with a rigid hub rotating with angular vlocity Ω about th vrtical axis Z. Th cantilvr bam s lngth is along th X axis. Th chordwis vibration occurs in th XY plan, paralll to th plan of rotation and prpndicular to th flapwis vibration which occur in th XZ plan. Th coordinat systm XY Z is fixd to th hub as shown in figur. Th displacmnt of point A to A can b xprssd in trms of th componnt displacmnts u, v and w whr thy corrspond to th axial, chordwis and flapwis dformation, rspctivly. Thr is a gomtrical rlation btwn th lngth of th bam bfor and aftr th dformation as givn in [6] by x + s = x+ux [ + ( uy + η ( ] uz dη (8 η whr s rprsnts th chang in bam s lngth (strtch and η is a dummy variabl. Th vlocity of th point A in figur 3, can also b found from, v A = ( u Ωu y i + [ v + Ω(r + x + u x ]j + ẇk (9 Z Y u = u x + u y + u z A A u u y u z r x L Figur. Configration of a rotating cantilvr bam. 3

5 MOVIC6 & RASD6 Journal of Physics: Confrnc Sris 744 (6 57 doi:.88/ /744//57 Y Ω r + x + u x V A Ω u y A' u y Ω X A u x r + x + u x Figur 3. Top viw of th systm shown th rlation btwn th angular and linar vlocity. Following th rsult by Kim t al [], lads to a st of non-linar partial diffrntial quations. Aftr simplifying th diffrntial quations and linarisation by nglcting highr ordr trms in th displacmnts, th following partial diffrntial quations can b drivd. ( s ρa ρa ρa w t t Ω s Ωv t Ω v EA s x = ρaω (a + x + p s ( ( v t Ω s t Ω v Ωs 4 v + EI z x 4 ( ρaω r(l x + x (L x ] v = p v x ( + EI 4 w y x 4 ( ρaω [r(l x + x (L x ] w = p w ( x whr p s, p v and p w ar th applid forcs pr unit lngth along th bam and in th y and z dirctions and I y, I z ar th scond momnt of ara about th y and z dirctions. It is intrsting that Eq.( and Eq.( ar coupld togthr, whil Eq.( is indpndnt of th othr two. Th boundary conditions from Eq.(7 ar givn by s = v = w = v x = w x =, x =, s x = v x = v x = 3 v x 3 = 3 w = x3, x = L. (3 A solution which satisfis th govrning quation of motion togthr with th boundary conditions at vry point ovr th domain is known as a strong form of solution. Convrsly, a wak form of solution satisfis th conditions in an intgral sns. Following this stp, th wak forms nd to b obtaind from th strong forms that ar givn by th partial diffrntial quations and th corrsponding boundary conditions. To driv th wak forms for th quation of motions for th Eulr-Brnoulli bam, Eqs.(-( ar multiplid by th wighting functions s, v and w rspctivly, summd and intgratd ovr th lngth L as follows. For th chordwis 4

6 MOVIC6 & RASD6 Journal of Physics: Confrnc Sris 744 (6 57 doi:.88/ /744//57 ρa L [ ( s s t Ω v and for th flapwis t Ω s Ωv + L + ρaω L ( EA s x ( v + v t Ω s s v v + EI x x x [ r(l x + (L x = L ] t Ω v Ωs dx ] v v x x dx {ρaω (r + x s + [p v ρa Ω(r + x] v}dx. (4 L w L ρa t dx + EI w w y x x dx L [ + ρaω r(l x + (L x ] w x w x dx = L wp w dx. (5 Th displacmnts and wighting functions ar now approximatd by th shap functions as sv s = ( d T Ns T ; s = N s d sv sv v = ( d T Nv T ; v = N v d sv w = ( d w T Nw T ; w = N w d w (6 whr d sv and d w ar th lmnt displacmnt vctor for chordwis and flapwis, rspctivly. d sv and d w ar arbitrary vctors with th sam dimnsions of d sv and d w corrspondingly. Finally N s,n v and N w rprsnt th trial shap functions. Introducing ths approximat solutions givn by quations (6 in th wak quations givn by Eq.(4 and Eq.(5, th quations can also b writtn in a matrix form, yilding th discrtizd quation for th chordwis and flapwis motion as, = (d sv T {m sv d sv + Ωg sv d sv + [k sv + Ω (s sv m sv + (d w T [m w d w + (k w + Ω s w d w ] = = Ωg sv ]d sv } = = (d sv T f sv (7 (d w T f w (8 whr m sv, g sv, k sv and s sv ar trms in th lmnt mass, th lmnt gyroscopic, th lmnt stiffnss and motion-inducd stiffnss matrics for th chordwis motion. Manwhil, m w, k w and s w ar th corrsponding trms for flapwis motion. f sv and f w ar th lmnt load vctors for th chordwis and flapwis motions. Th wak form solution and approach convrts th continuous systm into discrt lmnts, which can hav a finit numbr of dgrs of frdom and can b solvd by a finit lmnt = 5

7 MOVIC6 & RASD6 Journal of Physics: Confrnc Sris 744 (6 57 doi:.88/ /744//57 mthod or by givn shap functions using th Rayligh-Ritz mthod. Howvr, th nxt stp is to rprsnt th systm by matrics and convrt continuous quations of motion by th wak form into th quation of motion basd on th mass and stiffnss matrics. This can b obtaind by introducing s, v and w into th abov quations. Ths thr trms rprsnt th lmnts displacmnts in th strtch, chordwis and flapwis dirctions, rspctivly, and thy rfr to th local dformation of th discrt lmnt... Modlling of a crack According to fractur mchanics thory, an opn crack in a structur can b considrd as a sourc of additional local flxibility bcaus of th incras in th strain nrgy in th ara surrounding th crack tip. Th ida of substituting masslss springs instad of a crack is to crat th rlation btwn th strain nrgy and th applid loads as shown in figur 4. Th flxibility cofficints ar statd in trms of strss intnsity factors (SIF. Applying Castiglions thorm [7, 8], figur 5 shows th gnralizd loading conditions for a bam of rctangular cross sction with an opn dg surfac crack. Th bam is loadd statically with an axial forc, P, shar forcs, P and P 3, bnding momnts, P 4 and P 5, and torsional torqu, P 6. Howvr, dtrmining th additional flxibility can b achivd using Hook s law, which rlats th stiffnss to th acting forc and th dflction displacmnt. Th flxibility can also b calculatd from Eq.(9. c ij = δ i P j, (i, j =,, 3. (9 whr c ij, δ, Π c and P i rprsnt th additional flxibility, th displacmnt, th additional strain nrgy and th applid loads rspctivly, and can b obtaind from, δ = Π c P i ( Π c = GdA A c ( whr A c is th ffctiv crack ara and G is th strain nrgy rlas rat function xprssd as [9]. ( n ( n ( n G = E K Ii + K IIi + K IIIi. ( i= i= For plan strss problm [8] E = E and for th plan strain problm E = E/( µ, Ks ar th strss intnsity factors (SIF du to forcs. I, II, III rprsnt th mods of th crack namly opning, sliding and taring rspctivly, and i =,, 3,..., n rlatd to th applid loads on th bam P, P, P 3,..., P n. In this study, th in-plan vibration bam modl is dvlopd. Th applid forcs ar shown in figur 6. Rarranging Eq.( to match in-plan strain nrgy rlas rat function [], yilds G = E [(K I + K I + K I3 + (K II ]. (3 i= 6

8 MOVIC6 & RASD6 Journal of Physics: Confrnc Sris 744 (6 57 doi:.88/ /744//57 According to Ozturk t al [], th SIF for ach forc and mod bcoms, K I = P πξf (4 bh K I = 3P L c bh πξfi (5 K I3 = 6P 3 πξfi bh (6 K II = P πξfii bh (7 whr ξ is th crack dpth, F i is th corrction factors of th SIF. Notic that a is th final crack dpth, whras ξ is th dpth within procss of ongoing from zro crack dpth to distanc a. F = F I = πξ tan πξ πξ tan πξ [.75 +.ξ +.37 [ cos ( πξ ( ] 3 sin πξ (8 ( ] 4 sin πξ (9 cos ( πξ F II = (3ξ ξ..56ξ +.85ξ +.8ξ 3 ξ (3 Now substituting quations (-(8 into Eq.(9, w hav, c ij = E P i P j a o [ P πξf + 3P L c πξfi bh bh + 6P ] 3 πξfi bh + [ ] P πξfii dξ (3 bh c ij is th additional flxibility that rsults from th xistnc of a crack. On considrs th additional local flxibility du to th xistnc of th crack by updating th global stiffnss matrix of th uncrackd rotating bam. Th updatd stiffnss matrix is calculatd by rplacing th corrsponding trms in th uncrackd bam stiffnss lmnt whr th crack is introducd. Th stiffnss of th crackd lmnt is obtaind by adding th local flxibility cij du to th crack with th flxibility of th intact bam lmnt. This is thn usd to find th local stiffnss of th crackd lmnt and substitutd into th original stiffnss matrix. = (d sv T {m sv d sv + Ωg sv d sv + [kc sv + Ω (s sv m sv + (d w T [m w d w + (kc w + Ω s w d w ] = = whr kc rprsnt th stiffnss of crackd bam. Ωg sv ]d sv } = = (d sv T f sv (3 (d w T f w (33 = 7

9 MOVIC6 & RASD6 Journal of Physics: Confrnc Sris 744 (6 57 doi:.88/ /744//57 Figur 4. Modlling th crack as a masslss torsional spring Figur 5. Bam with an opn crack undr gnralizd loading condition Figur 6. Bam with an opn crack in-plan loads condition 3. Numrical rsults For th convninc of discussion and for th sak of comparison with rsults in th litratur, th following dimnsionlss paramtrs will b introducd. δ = r L, α = AL, γ = Ω, ϖ = ω, (34 I z ω n ω n whr δ, α, γ, ϖ and ω n rprsnt th hub ratio, th slndrnss ratio, th angular spd ratio, th frquncy ratio and fundamntal natural frquncy of non rotational cantilvr bam. 3.. Bam Without crack Th numrical simulation for th rotating intact cantilvr bam ar xamind first. Th rsults for th approachs (FE and th Rayligh Ritz for th dimnsionlss natural frquncis ar compard in tabl. Th mod shaps ar also compard using th MAC as illustratd in Figurs 7 and 8. Th frquncis obtaind ar in good agrmnt with th prviously publishd rsults [6]. In th MAC plot, in figur 7, th abscissa axis rprsnts th mod shaps calculatd by th Rayligh-Ritz Mthod, and th ordinat axis rprsnts th mod shaps calculatd by th FEM. Th rd squars show th matching mods. Whn th cantilvr bam is non-rotating, th MAC for th chordwis vibration is shown in figur 8. Th 4th and 8th mods rprsnt th longitudinal vibration likwis whn th γ = 7. th 4th and 7th similar ffct can b obsrvd at γ = 4. and γ =

10 MOVIC6 & RASD6 Journal of Physics: Confrnc Sris 744 (6 57 doi:.88/ /744//57 Rgarding th rotating spd, as in flapwis vibration, th natural frquncis ar changing with th variation of th rotating vlocity. Smingly in figur, th xistnc of th gyroscopic coupling influncs th bnding and strtching mods cratd by a rotating movmnt, that rsults in th vring phnomna in th frquncy vrsus spd plot. For xampl, whn approximatly th non-dimnsional spd is γ = 7.39 th 3rd and 4th natural frquncis vr togthr whilst th 5th and 6th natural frquncis vr at about γ = 7.6. Likwis, vring occurs btwn th frquncis of th 6th & 7th mods, 7th & 8th mods for γ =.5 and γ = 4.7, rspctivly. Figur shows in dtail th vring points for th lowr dimnsionlss natural frquncis. Th circls V, V and V3 indicat th spd whr pairs of natural frquncis xhibit th vring phnomna. Th 3rd, 5th and 6th dimnsionlss natural frquncis corrspond to th mods that ar vring from bing primarily bnding vibration to longitudinal vibration at points V and V3. Howvr, th 4th and th 6th ar vring from prdominating longitudinal vibration bhaviour to bnding vibration at points V and V. frquncy ω γ Chung and Yoo ANSYS RRM Error % Tabl. Comparison btwn diffrnt mthods for flapwis dimnsionlss natural frquncy calculatd for a rotating cantilvr bam In gnral, incrasing th rotating spd lads to an incras in th natural frquncis as shown in prvious figurs. Thr ar additional factors that affct th natural frquncis of a rotating bam, which ar not rlatd to a crack such as hub ratio and slndrnss as can b sn in figur 9 and figur, rspctivly. An incras in th hub ratio lads to an incras in th natural frquncis as shown in figur 9, du to a proportional rlationship btwn th stiffnss and th cntrifugal forc. An incras in th slndrnss lads to a dcrasd chanc of vring at low frquncis, sinc th bam bcoms thinnr and mor capabl of bnding rathr than strtch. Dcrasing slndrnss ratio mans a thickr cross sction and th highr probability of vring at low frquncis. Anothr noticabl fact is th xistnc of instability whn th fundamntal natural frquncy of th chordwis vibration rducs to zro. This spd can b dfind as th buckling spd,.g. s [] and [6]. 3.. Bam with a crack A finit lmnt modl of a crackd bam can, consquntly, b simulatd using th formulation in quations (3 and (33. Th crack modl utilizs a cantilvr bam with a crack on a singl 9

11 MOVIC6 & RASD6 Journal of Physics: Confrnc Sris 744 (6 57 doi:.88/ /744//57 Prvious work (FEM Prvious work (FEM Prvious work (FEM Prvious work (FEM This work (Rayligh Ritz (a MAC in flapwis vibration at γ= This work (Rayligh Ritz (b MAC in flapwis vibration at γ=7. This work (Rayligh Ritz (c MAC in flapwis vibration at γ=4. Figur 7. MAC for th flapwis vibration This work (Rayligh Ritz (d MAC in flapwis vibration at γ=8.44 Prvious work (FEM Prvious work (FEM Prvious work (FEM Prvious work (FEM This work (Rayligh Ritz (a MAC for th chordwis vibration at γ= This work (Rayligh Ritz (b MAC in chordwis vibration at γ=7. This work (Rayligh Ritz (c MAC in chordwis vibration at γ=4. Figur 8. MAC in chordwis vibration This work (Rayligh Ritz (d MAC in chordwis vibration at γ=8.44 frq. ratio (ω/ω n δ= δ= δ=3 δ=4 δ=5 Ω=ω frq. ratio (ω/ω n st nd 3rd 4th 5th 6th V V3 V spd ratio (Ω/ω n spd ratio (Ω/ω n Figur 9. Th ffct of th hub ratio (δ on th fundamntal dimnsionlss natural frquncy (ϖ for diffrnt dimnsionlss rotating spd (γ Figur. Dimnsionlss natural frquncis (ϖ vrsus dimnsionlss rotation spd (γ. Th rd circls indicat th vring of mods. dg, in ordr to compar th rsults with a solid finit lmnt modl assmbld in a commrcial FE cod ANSYS, as prsntd in figur 3. In figur 4 th numrical simulations ar shown for th bam with a crack at its midpoint for two diffrnt crack dpths. Th two rsults ar in xcllnt agrmnt. Thr ar numrous factors that affct th natural frquncis rlatd to cracks, for xampl, th dpth of th crack, its location and th orintation of th crack according to th dirction of th loads []. Figur 4, shows that whn th dpth is qual to half th bam thicknss i.. a = h/, th dimnsionlss natural frquncis for a crack at th root of th bam drops to.78, whil for a crack dpth lss than a = h/3 th frquncy ratio is about.9. A dpr crack lads to a

12 MOVIC6 & RASD6 Journal of Physics: Confrnc Sris 744 (6 57 doi:.88/ /744//57 frq. ratio (ω/ω n st nd 3rd 4th 5th 6th α =7 frq. ratio (ω/ω n st nd 3rd 4th 5th 6th 7th α = spd ratio (Ω/ω n spd ratio (Ω/ω n Figur. Chordwis vibration natural frquncis of a rotating cantilvr bam at α = 7 Figur. Chordwis vibration natural frquncis of a rotating cantilvr bam at α = 7.5 gratr rduction in th frquncy than a shallow on bcaus of th additional flxibility, as xpctd. Examining th crack location, it appars that whn th crack is clos to th root of th bam, i.. clos to th nd which is attachd to th hub, th ffct of th crack bcoms mor significant and convrsly this ffct rducs whn th crack is closr to th fr nd. This rsults from th bnding momnt ffct at th fixd nd. In addition, th crack ffct vanishs or is ngligibl at som frquncis whn it is locatd at a nod of th highr ordr mods. This bcoms vidnt in th rsults for th scond and highr mods. Tabl shows th agrmnt btwn th prsnt work and an ANSYS FE modl using solid lmnts. Th maximum prcntag rror is.33% for th fundamntal natural frquncy at γ = and.66% in th scond natural frquncy for th non-rotating bam..5 a=h/3 Prsnt work a=h/3 ANSYS a=h/ Prsnt work a=h/ ANSYS frq. ratio (ω c /ω n Crack location (x/l Figur 3. crackd bam Figur 4. Comparison btwn th prsnt work and an ANSYS FE modl for th rlativ fundamntal frquncy of th crackd stationary cantilvr bam Considring th influnc of th two main paramtrs, th crack location and th rotating spd, in on graph is figur 5, on can obsrv a clar pictur of how th crack rducs th natural frquncy. In addition, it can giv a pr-stimation of th possibl crack location.to assist th idntification of th crack location th rsults ar shown using th sam two paramtrs in a 3D plot of th scond rlativ natural frquncy as shown in figur 6. Utilizing th fundamntal natural frquncy chart in figur 5 sid by sid with th scond

13 MOVIC6 & RASD6 Journal of Physics: Confrnc Sris 744 (6 57 doi:.88/ /744//57 st ϖ nd ϖ γ prsnt work ANSYS Error % prsnt work ANSYS Error % Tabl. Comparison btwn th prsnt work and an ANSYS FE modl for a singl dg crackd rotating bam natural frquncy in figur 6, a bttr stimation of a crack location can b providd whn th crack occurs in th scond half of th bam, whr th fundamntal frquncy has a low snsitivity to th crack location. Figur 7 and 8 clarly show th snsitivity of th two lowst natural frquncis of a rotating bam with rspct to th crack location for a fixd rotating spd. 3.5 frq. ratio (ω / ω n frq. ratio (ω / ω n spd ratio (Ω/ω n.5 Crack location x/l spd ratio (Ω/ω n.5 Crack location x/l Figur 5. Th first rlativ natural frquncy for variabl crack location and rotating spd. Crack dpth a = h/ Figur 6. Th scond rlativ natural frquncy for variabl crack location and rotating spd. Crack dpth a = h/

14 MOVIC6 & RASD6 Journal of Physics: Confrnc Sris 744 (6 57 doi:.88/ /744//57 3. a=h/ a=h/ a=h/ a=h/ frq. ratio (ω /ω n frq. ratio (ω /ω n Crack location (x/l Crack location (x/l Figur 7. Th ffct of th crack s dpth and location on th st rlativ natural frquncy. Rlativ spd γ =.844 Figur 8. Th ffct of crack s dpth and location on th nd rlativ natural frquncy. Rlativ spd γ = Conclusions This study intndd to modl and invstigat th dynamics of rotating bams with a crack, considring th ffct of th rotational spd on th dynamic charactristics for such bams. Two approachs wr usd in modlling th rotating bam, both ar basd on Eulr-Brnoulli bam thory. Firstly, a continuous systm approach for flapwis vibration is dvlopd. This mthod basd on forc quilibrium ld to th drivation of th quation of motion. Rotational ffcts ar includd by simply adding a cntrifugal forc to th diffrntial quation. Scondly, an nrgy mthod using Hamilton s principl providd th drivation of th quations of motion from th kintic nrgy, potntial nrgy and work don on th systm, for th flapwis and chordwis quations of motion. Furthrmor, a crack was introducd and modlld as a masslss spring by using fractur mchanics and th Paris quation. An outcom of th nrgy mthod is that it shows th coupling ffct btwn th strtch and chordwis motion, whil th flapwis quation is not coupld. Rlating to th crack problm, it is dividd into two modls according to th dirction of motion i.. flapwis or chordwis. For th flapwis situation it nds two dgrs of frdom for ach lmnt nod (latral dirction and slop, whilst in th chordwis dirction it nds thr dgrs of frdom for ach nod (latral dirction, longitudinal dirction and slop bcaus of th combind bnding and strtch motion. Two diffrnt numrical mthods wr usd to solv th quations of motion; th Rayligh- Ritz mthod for th intact rotating bam whr th finit lmnt mthod ar usd for both th intact and crackd rotating bams. Th numrical rsults wr prsntd and discussd in dtail for rotating bams with and without cracks. For th flapwis motion, th natural frquncis monotonically incras with incrasing rotational spd. No critical angular spd, i.. whr th critical spd mans any natural frquncy quals th rotational spd, ariss. Th incras in th hub ratio also incrass ths natural frquncis. Various paramtrs wr considrd and thir ffcts invstigatd with changs in th rotational spd, hub ratio, slndrnss ratio, critical rotating spd, and inspction of th vring phnomna and buckling spd wr xplord. In addition, th ffct of th crack location and crack dpth ar xplord and illustratd in thr dimnsional charts for th crackd rotating bam. Rgarding chordwis motion, th natural frquncis do not incras monotonically du to th gyroscopic coupling. A critical angular spd can occur though. Thr is th xistnc of a buckling spd, i.. whr th buckling spd causs th bam natural frquncy to rduc to zro as th rotational spd incrass. Slndrnss and hub ratio hav a significant ffct on th critical angular spd, sinc th thinnr bams tnds to bnd mor than strtch. 3

15 MOVIC6 & RASD6 Journal of Physics: Confrnc Sris 744 (6 57 doi:.88/ /744//57 For th crackd bams flapwis motion, th crack location has a significant ffct on th natural frquncis. Th closr th crack is to th hub, th lowr ar th natural frquncis. Gnrally thr is an invrs rlationship btwn th crack dpth and th natural frquncy. Whn a crack is locatd at on of th nods of vibration of a particular mod thn th frquncy of that mod is unaffctd. Acknowldgmnts Th lad author gratfully acknowldgs th Ministry of Highr Education and Scintific Rsarch in Iraq for providing full PhD scholarship. Rfrncs [] Rubinstin N and Stadtr J T 97 Journal of th Franklin Institut ISSN 63 [] Bhat R 986 Journal of Sound and Vibration 5 99 [3] Yoo H and Shin S 998 Journal of Sound and Vibration ISSN 46X [4] Rao S and Gupta R Journal of Sound and Vibration ISSN 46X [5] Cai G P, Hong J Z and Yang S X 4 Intrnational Journal of Mchanical Scincs ISSN 743 [6] Chung J and Yoo H Journal of Sound and Vibration ISSN 46X [7] Yang J, Jiang L and Chn D 4 Journal of Sound and Vibration ISSN 46X [8] Banrj J, Su H and Jackson D 6 Journal of Sound and Vibration ISSN 46X [9] Yoo H H, Cho J E and Chung J 6 Journal of Sound and Vibration ISSN 46X [] Özdmir Ö and Kaya M 6 Journal of Sound and Vibration ISSN 46X [] Kim H, H Yoo H and Chung J 3 Journal of Sound and Vibration ISSN 46X [] Banrj A and Pohit G 4 Procdia Tchnology ISSN 73 [3] Chng Y, Yu Z, Wu X and Yuan Y Finit Elmnts in Analysis and Dsign ISSN 68874X [4] Qian G L, Gu S N and Jiang J S 99 Journal of Sound and Vibration ISSN 46X [5] Bhra K and K V 4 Procdia Enginring ISSN [6] Yoo H, Ryan R and Scott R 995 Journal of Sound and Vibration ISSN 46X [7] Silani M, Ziai-Rad S and Talbi H 3 Applid Mathmatical Modlling ISSN 3794X [8] Zhng D and Kssissoglou N 4 Journal of Sound and Vibration ISSN 46X [9] Tada H, Paris P C and Irwin G R Th Strss Analysis of Cracks Handbook (ASME Prss ISBN [] Ibrahim A M, Ozturk H and Sabuncu M 3 Structural Enginring and Mchanics [] Ozturk H, Yashar A and Sabuncu M 6 Mchanics of Advancd Matrials and Structurs [] Lima M A C F Rotating Cantilvr Bams : Finit Elmnt Modling and Vibration Analysis Ph.D. thsis Faculdad d Engnharia da Univrsidad do Porto 4