THREE DIMENSIONAL EXPERIMENTAL MODAL ANALYSIS ON A WATER FILLED CYLINDRICAL SHELL

Transcription

1 Proceedigs of the IMAC-XXVII February 9-12, 29 Orlado, Florida USA 29 Society for Experimetal Mechaics Ic. THREE DIMENSIONAL EXPERIMENTAL MODAL ANALYSIS ON A WATER FILLED CYLINDRICAL SHELL William K. Boess, Joh B. Fahlie, ad Dea E. Capoe Noise Cotrol ad Hydroacoustics Divisio Applied Research Laboratory / Pe State Uiversity P.O. Box 3 State College, PA ABSTRACT Results from a experimetal modal aalysis o a waterfilled cylidrical shell are compared with a correspodig aalytic model solutio. The resultig modal parameters are used to determie directioal forces from turbulet boudary layer pipe flow excitatio. The thi-walled, 24- ich log, alumium shell has clamped ed coditios imposed withi a sectio of a cotiuous pipe. Mode shapes, dampig, i vacuo ad water-filled resoace frequecies, ad modal masses are evaluated. The three dimesioal shell respose to all three coordiate excitatios is examied ad compared. Directioal couplig exists for all shell modes except the lowest order (=: breathig ) modes. These modes represet a differet class of modes from the higher order bedig ad lobar modes which are coupled i all directios. The lowest order (=) circumferetial ad primarily axial modes ca easily be idetified i the experimetal data, however the lowest order radial modes caot. NOMENCLATURE E Youg s modulus F force H xy FRF betwee x (referece) ad y L cylider half-legth N DFT Sample sie P TBL wall pressure spectrum S structural sesitivity fuctio T excitatio vector surface tractio a, h cylider radius, wall thickess k 1, k 3 streamwise, crossflow waveumbers, m circumferetial, axial mode orders r, φ, cylider coordiate directios s dyamic stiffess matrix u cylider displacemet vector x 1, x 3 streamwise, crossflow spatial locatios α, β cylider locatios η dampig loss factor ρ, ν structural desity, Possio s ratio ψ ormalied structural mode shape fuctio ω radia frequecy 1. INTRODUCTION Pipig systems ad other structures ivolvig fiite cylidrical shells are importat i may practical egieerig applicatios. The mitigatio of uwated vibratio ad soud i these structures ofte begis with a uderstadig of their structural dyamics. Most work o the structural dyamics of fiite shells is focused o radial displacemet. While vibratio i the radial directio is typically the most relevat ad ofte leads directly to radiated soud, vibratio i the other coordiate directios (axial ad circumferetial) ca also be importat. The curvature of shells leads to couplig i all three coordiate directios. Therefore, excitatio ad modal displacemet i oe directio is accompaied by correspodig displacemet i the other two directios for almost all modes of vibratio. This paper compares the results of a three-dimesioal structural aalysis of a water-filled cylidrical shell from a aalytical model ad a experimetal modal aalysis for a specific set of fiite shell parameters. This work is motivated by a desire to determie the fluctuatig forces associated with turbulet boudary layer (TBL) excitatio from fully developed pipe flow usig water. The forces are geerated by fluctuatig ormal wall pressure as well as fluctuatig wall shear stress i the streamwise ad crossflow directios. The procedure preseted for idirectly determiig forces by measurig the structural respose ca be applied to other structures ad broadbad forces. a h F 2L Y Alumium cylider ρ = 27 Kg/m 3 E = 6.9x1 1 N/m 2 ν =.3 a = 3 i h =.125 i 2L = 24 i η =.1 Figure 1. Diagram ad Parameters of Aalytical Cylidrical Shell

2 2. ANALYSIS APPROACH 2.1 Structural Vibratio ad TBL Force Measurig TBL forces from a structure s respose to TBL excitatio requires kowig the dyamic respose of the structure. The modal force spectrum, F (ω), for a give mode,, exerted o a structure from TBL geerated wall pressures is give by Hwag ad Maidaik [1] as ( ω) (,, ω) (, ) F P k k S k k dk dk =, (1) where, P(k 1,k 3,ω) is the TBL wall pressure spectrum which is a fuctio of radia frequecy, ω, ad streamwise ad cross-flow waveumbers, k 1,k 3, respectively. The sesitivity fuctio, S(k 1,k 3), is a spatial Fourier trasform of the ormalied structural mode shape fuctio, ψ (x 1,x 3), ik1x1 ik3x3 (, ) ψ (, ) S k k x x e e dxdx =, (2) A where ψ has a peak value of oe, ad x 1,x 3 are streamwise ad cross-flow structural locatios respectively. The relatioship betwee TBL forces ad TBL iduced vibratio ca be established usig the Frequecy Respose Fuctio (FRF) as give by Ewigs [2]. The displacemet u (for a idividual mode ) at a poit α ad a force F at a poit β is give as u ( αω, ) 1 = ψ α ψ β 2 2 F( βω, ) m ω + ω + iηωω, (3) where ω is the resoace frequecy, η is the loss factor, ad m is the modal mass. Combig Equatio 3 with the relatioship betwee the modal force spectrum, F (ω), ad the spectral force at poit, β, F yields the relatioship, ( ω) ψ ( β) F( β, ω) =, (4) 1 ψ ( α) F( ω) u( αω, ) = 2 2 m ω + ω + iηωω. (5) Equatio 5 requires a assessmet of the modal parameters of resoace frequecy, dampig, mode shape, ad modal mass to determie the TBL forces from vibratio measuremets o a TBL excited structure. 2.2 Aalytical Cylidrical Shell A aalytical shell model is examied to provide a basis for compariso with experimetal modal aalysis results for cylidrical shell dyamics. The aalytical model from Skelto ad James [3] is a fluid-loaded, simply-supported, cylidrical-shell usig equatios of motio from Goldeveier ad Novohilov. These equatios apply for a isotropic, thi-walled (h<<a) cylider i which rotatioal iertia ad trasverse shear effects are eglected. The time-harmoic, spectral form of the equatios of motio become, s s s u, m T, m s21 s22 s 23 uφ, m = Tφ, m s31 s32 s33 + f m ur, m Tr, m The terms of the dyamic stiffess matrix, s, are give by Skelto ad James [3]. The excitatio vector, T (surface tractio), ca be specified as a poit force or distributed radom excitatio. Iteral or exteral fluid loadig (or both) ca be icluded usig the term, f m, defied i terms of Bessel fuctios. Kowig s ad T, oe ca solve for the modal displacemet, u(,m). From the modal displacemet, the spatial displacemet, u(φ,), ca be obtaied from the Fourier Series expasios, jφ u ( φ, ) = u (, m)si( mπ( + L)/2 L) e, r m, jφ u ( φ, ) = u (, m)si( mπ( + L)/2 L) e, φ m, jφ u ( φ, ) = u (, m)cos( mπ( + L)/2 L) e. m, r φ The cylidrical shell respose fuctios or mode shapes iclude displacemet i all three coordiate directios: radial - r, circumferetial - φ, ad axial. The displacemets cotai a siusoidal depedece i the circumferetial directio ( represets the umber of full wavelegths) ad axial directio (m represets the umber of half wavelegths). Shell curvature accouts for the couplig of displacemet i all three directios for all modes except the special case of = or breathig modes. These modes represet a differet class of modes which are ot fully-directioally-coupled so three distict mode shapes ad correspodig resoace frequecies exist for each,m combiatio. I cotrast, the higher order bedig (=1) ad lobar (>1) modes are fully-coupled ad a uique mode shape ad resoace frequecy exists for each,m combiatio. A diagram of the aalytical shell model ad parameters used for the calculatios is show i Figure Experimetal Modal Aalysis A three-dimesioal modal aalysis is coducted i reciprocal maer o the cylider show i Figure 2 with the cylider ad adjacet pipig full of water. The cylidrical shell is machied from a 4 feet log, 6 ich diameter, schedule 4 (.25 i. thick) alumium pipe. A two feet log (6) (7)

3 mid sectio of pipe is machied from the outside dow to a wall thickess of.125 iches to create the actual thi walled shell. Circumferetial V-shaped grooves are machied aroud the pipe at the discotiuity betwee the ucut pipe ad the eds of the test sectio to a depth of.25 iches. The grooves help simulate simply supported coditios at the test sectio boudaries. Alumium blocks are rigidly bolted aroud the ucut portio of pipe adjacet to the grooves ad rigidly attached to the floor which resides ie iches below the pipe ceterlie. The rigid blocks, grooves, ad corer braces show i Figure 2 help limit motio of the shell ed boudaries. separate modes ad idetify resoace frequecies ad mode shapes (Catbas et. al. [4], Fahlie, et. al. [5]). A Ratioal Fractio Polyomial scheme (Ewis [2]) is the used to refie the resoace frequecy ad dampig estimates. Fially, a surface average of measured FRF data is compared to correspodig FRF estimates usig Equatio 3 to determie the modal mass. I additio to extractig modal parameters from the measured data, average vibratio spectra are decomposed ito their circumferetial Fourier waveumber compoets. A N-poit spatial Discrete Fourier Trasform (DFT) is performed aroud each rig of measured data for all frequecies (Boess et. al. [6]): 1 H ( f) H ( f) e N j2 π ( 1)( y 1)/ N =, (8) xy N y= 1 where H represets the waveumber compoets (positive ad egative) ad H xy are the measured FRF data. The resultig Fourier compoets ca be summed to recover the average measured FRF data for each rig. 3. EXPERIMENTAL AND ANALYTICAL RESULTS 4 3 Radial Respose Figure 2. Experimetal Cylidrical Shell ad Modal Aalysis Measuremet Setup To acquire the modal trasfer fuctio data, a evely spaced 156 poit grid (13 poits axially by 12 poits circumferetially) is excited with a rovig force hammer. The measuremets are acquired i reciprocal maer with the hammer hits poits represetig respose poits ad 12 attached accelerometers represetig excitatio poits. Four acceleratios are orieted i each of the three coordiate directios (12 total) ad the etire grid is excited three separate times correspodig to the three coordiate directios. Flat spots are machied ito the cylider wall at each grid poit to allow small alumium cubes holdig the accelerometers (or dummy masses) to be securely adhered to each locatio. The alumium cubes allow excitatio i all coordiate directios. Frequecy Respose Fuctios (FRFs) are simultaeously measured betwee the force hammer (PCB model type, ses ~ 5 mv/lbf) ad the 12 referece accelerometers (PCB W532, ses ~ 1 mv/g). After acquirig the FRF s, the data is post processed usig stadard modal aalysis techiques to extract resoace frequecy, dampig, mode shape, ad modal mass for idetifiable cylider resoaces ad modes. The Complex Mode Idicator Fuctio (CMIF) techique i combiatio with Sigular Value Decompositio (SVD) is iitially used to Figure 3. Aalytical radial vibratio i respose to excitatio i three coordiate directios Radial Respose Figure 4. Experimetal radial vibratio i respose to excitatio i three coordiate directios

4 Because a aalytical solutio for a cylidrical shell with clamped boudary coditios does ot exist, the solutio for simply supported boudaries is used to represet the experimetal setup. While the experimetal structure is more clamped tha simply supported, the aalytical model provides isight ito cylidrical shell dyamics ad a helpful basis for compariso. The loss factor (dampig) used for the aalytical model is set to η = Circumferetial Respose =,m= Figure 5. Aalytical circumferetial vibratio i respose to excitatio i three coordiate directios Axial Respose =,m= Figure 6. Aalytical axial vibratio i respose to excitatio i three coordiate directios Both the aalytical model ad experimetal modal results idicate the cylider respose to is higher tha the respose to excitatio i the other two coordiate directios (axial ad circumferetial) for early all modes. Figures 3 ad 4 show the average radial accelerace i respose to excitatio i all three coordiate directios (four drive poits i each directio) for the aalytical model ad correspodig experimetal modal data respectively. From the aalytical model results i Figure 3, the radial respose to the radial drive is roughly 1 db higher tha the circumferetial drive which is roughly 1 db higher tha the axial drive. The experimetal data, show i Figure 4, provides similar results. The radial respose show i Figures 3 ad 4 also represets the relative respose i the axial ad circumferetial directios as well. I other words, a radial drive geerates the highest vibratio levels i all three directios. Therefore, measurig radial vibratio ca provide a assessmet of the radial iput force, assumig excitatio levels i all three directios are roughly similar, but measurig axial ad circumferetial vibratio does ot provide a assessmet of their respective iput forces. It oly provides aother assessmet of the radial iput force Circumferetial Respose =,m= Figure 7. Experimetal circumferetial vibratio i respose to excitatio i three coordiate directios =,m=1 Axial Respose Figure 8. Experimetal axial vibratio i respose to excitatio i three coordiate directios The loe exceptio to this observatio is provided by oe class of modes ot evidet i Figures 3 ad 4. The = modes, as metioed previously, are ot fully coupled i all directios. Therefore, three distict mode shapes ad correspodig resoace frequecies exist for each,m combiatio ad represet each coordiate directio. Aalytical model results i Figures 5 ad 6 show the average circumferetial ad axial respose respectively to excitatio i all three coordiate directios up to 5 kh. A sigle resoace peak domiates the spectra i both figures. The peak at roughly 25 H i Figure 5 represets the lowest order mode (=, m=1) i the circumferetial directio. The peak at roughly 43 H i Figure 6 represets the lowest order mode (=, m=1) i the axial directio. As evidet i these figures, both of these modes

5 are preferetially excited by drives aliged i the correspodig coordiate directio. Experimetal modal aalysis results correspodig to Figures 5 ad 6 are show i Figures 7 ad 8. The resoace peak of the low order circumferetial mode (=, m=1) is clearly visible ear 21 H i Figure 7. Somewhat less obvious i Figure 8 is the resoace peak of the low order axial mode (=, m=1) ear 34 H. Because these two modes are preferetially excited by circumferetial ad axial drives, they represet the best opportuity to evaluate the magitude of the iput force i their respective directios Sum = =1 =2 =3,m 2,1 1,1 3,1 3,2 2, Figure 9. Average aalytical radial vibratio decomposed ito circumferetial Fourier compoets Sum = =1 =2 =3 2,1,m 1,1 3, Figure 1. Average experimetal radial vibratio decomposed ito circumferetial Fourier compoets Circumferetial Fourier compoets of the surface average vibratio field for both the aalytical results ad experimetal data are show i Figures 9 ad 1 respectively. Modal parameters determied for these modes are compared i Table 1. The higher order modes (>1) idetified i Figure 1 form the basis for determiig the radial iput force. The low order = modes also listed i Table 1 form the basis for determiig the axial ad circumferetial iput forces. 2,2,1 3,2 3,3 3,3 The resoace frequecies ad modal masses listed i Table 1 provide isight ito the effect of fluid loadig o differet modes. The modal mass of these modes is a combiatio of some fractio of the alumium shell mass (2.5 Kg) ivolved i modal displacemet ad some fractio of the mass of the water (11 Kg withi the shell boudaries) etraied i the modal displacemet. From the i vacuo aalytical results, the higher order modes (>1) have a modal mass represetig roughly oe-quarter of the mass of the alumium shell. The low order modes (=,1) have a modal mass of half the mass of the alumium shell. Aalytical results idicate the presece of water iside the shell icreases the modal mass by roughly a factor of four for modes where >. This results i a dowward shift i resoace frequecy by roughly a factor of two. Table 1. Cylidrical Shell Modal Parameters Mode,m Res. Freq. I vacuo (H) M. Mass I vacuo (Kg) Res. Freq. fl. load (H) M. Mass fl. load (Kg) Res. Freq. Exp. Data (H) M. Mass Exp. Data (Kg) Dampig Exp. Data (%) 2, , , , , ,1-r 11, , , ,1-φ , The effect of heavy fluid-loadig o the modal mass of the = modes is very differet. Modal mass of the axial ad circumferetial = modes is uchaged with ad without fluid loadig. The resoace frequecy for the circumferetial = mode is uaffected. The resoace frequecy for the axial = mode icreases slightly suggestig a small icrease i stiffess. The radial = mode is dramatically affected by the presece of iteral water. Because the iteral fluid is early icompressible ad has owhere to go (i the case of this radial breathig mode), the effect of mass loadig is much greater tha just the mass of the fluid itself. The modal mass for this mode icreases by more tha two orders of magitude with water iside the cylider. This results i a reductio of the resoace peak by more tha 4 db ad a resoace frequecy shift by more tha a factor of te. Because of the iheretly low respose level ad the likelihood of high dampig for this mode, there is o evidece of the radial = mode i the experimetal data. Because the radial = mode is a particularly efficiet radiator of soud from cylidrical structures, it is ofte helpful to idetify ad structurally alter this mode i attempts to mitigate soud radiatio.

6 4. CONCLUSIONS Modal parameters from a aalytical model solutio ad a correspodig experimetal ivestigatio o a water filled cylidrical shell are evaluated for the purpose of determiig directioal forces from TBL pipe flow excitatio. The procedure preseted for idirectly determiig forces o a structure by measurig the structural respose ca be applied to other structures ad broadbad forces. For the cylider uder ivestigatio, a radially applied force geerates the highest cylider vibratio levels i all three coordiate directios for all higher order modes (>1). Ay of these modes ca be used to evaluate the magitude of. Oly the special case of = modes are ot directioally coupled, therefore excitatio of these modes i oe directio produces a preferetially high respose i the same directio. These modes are best suited to evaluate the magitude of axial ad circumferetial excitatio. Stadard modal aalysis techiques are used to determie modal parameters from experimetal data. Favorable comparisos exist betwee aalytical model results ad experimet give the iheret differeces betwee the two. While the lowest order =, m=1 circumferetial ad axial modes ca easily be idetified i the experimetal data, the uiquely high modal mass of the radial =, m=1 mode prevets experimetal idetificatio of this mode. Idetifyig this mode ca have practical importace because of its particularly high soud radiatio efficiecy. ACKNOWLEDGEMENTS The authors gratefully ackowledge iteral fudig from the Applied Research Laboratory at Pe State Uiversity to support this work. REFERENCES [1] Hwag, Y.F. ad Maidaik, G., A Waveumber Aalysis of the Couplig of a Structural Mode ad Flow Turbulece, Joural of Soud ad Vibratio, 142(1), pp , 199 [2] Ewis, D.J., Modal Testig: Theory, Practice, ad Applicatios, 2d Ed., Research Studies Press, Hertfordshire, Eglad, 21 [3] Skelto, E. A. ad James, J. H., Theoretical Acoustics of Uderwater Structures, Imperial College Press, 1997 [4] Catbas, N., Brow, D.L., ad Akta, A.E., Parameter estimatio for multiple-iput multiple-output modal aalysis of large structures, Joural of Egieerig Mechaics, 13 (8), , 24 [5] Fahlie, J.B., Campbell, R.L., ad Hambric, S.A., Modal Aalysis Usig the Sigular Value Decompositio, Applied Research Lab, Techical Report 4-8, May, 24 [6] Boess, W.K., Fahlie, J.B., ad Jekis, D.M., Circumferetial Waveumber Decompositio of Experimetal Data from Structures Cotaiig Circular Symmetry, Proceedigs of IMAC-XXI, Orlado, FL, February, 23