Aalborg Universitet

Stochastic Response of Energy Balanced Model for Vortex-Induced Vibration

Nielsen, Søren R. K.; Krenk, S.

Publication date:1997

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Citation for published version (APA):Nielsen, S. R. K., & Krenk, S. (1997). Stochastic Response of Energy Balanced Model for Vortex-InducedVibration. Aalborg: Dept. of Building Technology and Structural Engineering. Structural Reliability Theory, No.166, Vol.. R9710

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INSTITUTTET FOR BYGNINGSTEKNIK DEPT. OF BUILDING TECHNOLOGY AND STRUCTURAL ENGINEERING AALBORG UNIVERSITET • AAU • AALBORG • DANMARK

STRUCTURAL RELIABILITY THEORY PAPER NO. 166

To be presented at ICOSSAR '97, Kyoto, Japan, November 24-28, 1997

S. R. K. NIELSEN & S. KRENK STOCHASTIC RESPONSE OF ENERGY BALANCED MODEL FOR VOR­TEX-INDUCED VIBRATION MAY 1997 ISSN 1395-7953 R9710

The STRUCTURAL RELIABILITY THEORY pnprrs are issurd for early dissemin tion of rC'sC'arch results from the Structural Rdiability Group at the DcpartnH'llt Building Technology and Structural Engineering, UnivC'rsity of Aalborg. These pap<' are gC'nerally submitted to scientific mcetmgs, conf<' rt' ll<"<'!:i or journals and should t.hcr fore' not hC' widPly clistrihutC'd. \'lfhenever possible' rdrrence should be given to the fill' public<ltions {pro<"eedings, journals, etc.) and not to the Structural Reliability Thf'o papers.

I Printed at Aalhorg lfniversity I

INSTITUTTET FOR BYGNINGSTEKNIK DEPT. OF BUILDING TECHNOLOGY AND STRUCTURAL ENGINEERING AALBORG UNIVERSITET • AAU • AALBORG • DANMARK

STRUCTURAL RELIABILITY THEORY PAPER NO. 166

To be presented at ICOSSAR '97, Kyoto, Japan, November 24-28, 1997

S. R. K . NIELSEN & S. KRENK STOCHASTIC RESPONSE OF ENERGY BALANCED MODEL FOR VOR­TEX-INDUCED VIBRATION MAY 1997 ISSN 1395-7953 R9710

STOCHASTIC RESPONSE OF ENERGY BALANCED

MODEL FOR VORTEX-INDUCED VIBRATION

S.R.K. Nielsen Department of Building Technology and Stroctuml Engineering,

Aalbo1"!} University , Sohngaardsholmsvej 57, DK-.9000 Aalborg, Denma1·k

S. Kre nk Division of M echanics, Lund Unive1·sity, S-z21 00 Lund, Sweden

ABSTRACT: A double oscillator model for vortex-induced oscillations of structural elements based on exact power exchange between fluid and structure, recently proposed by the authors, is extended to include t he effect of the turbulent component of the wind. In non-turbulent flow vortex-induced vibrations of lightly damped structures are found on two branches, with the high­est amplification on the low-frequency branch. The effect free wiliCl turbulence is to destabili;~<· t he vibrations on the high amplifi cation branch, thereby reducing the oscillation ampli tude. The effect is most pronounced for very lightly damped structures. The character of the structural vibrations changes with increasing turbulence and damping from nearly regular harmonic oscil­lation to typical narrow-banded stochastic response, closely resembling observed behaviour in experiments and full -scale structu res.

1 INTRODUCTION

\'ort.cx-induced oscillations play an important role in the design of slender structures ex­posed to fluid flow. It is commonly observed that if the natural vortex shedding frequency, determined by the flu id flow velocity via the Strouhals relation and the e igenfrequen cy of the structure arc close, t he structure may develop oscillations that control the vortex shedding frequency, and this syncronization may lead to large osci llation amplitudes in the so-called lock-in interval. The resonance amplitude depends on the structural clamping, the free t urbulence, and for smooth st ructural shapes on the Reynolds number. It has been found experimentally e.g. by van Kotcn (1984) that in turbulent flow very lightly damped cylindrical structures may develop amplitudes up to around half the diameter in nearly harmonic oscillation. With increasing damping the amplitude is decreased and the response changes gradually towards nearly linear Gaussian response.

Vickery & Basu ( 1983) have proposed a model equation for the stochastic response during lock- in. The wind exci tation is assumed in the form of a narrow-band Gaussian process, and the self-limiting amplitude is obtained by assuming t hat the aerodynamic damping is

negative for small osci llations and increases quadratically with amplitude. This model ha been fitted to full scale observtions on chimneys by Daly (1986) and is used in simplifie form in the National Building Code of Canada (1990). An important point is the modt parameters, and thei r dependence on Reynolds number and turbulence. In the study b Dal) (1986) only dependence on the Reynolds number was assumed. However, it appear that structures that have de\·eloped severe vortex- induced vibrations may have experience, ye'l-rs of service without vibration problems. The incidents of severe vortex-induced vi bra tions seem to be associated with atmospheric conditions with low turbulence intensit) Dyrbye & Han~en (1997).

The present paper extends a double osciallator model for vortex-induced vibrations, re cently developed by t he authors fo r non-turbulent free flow, I<renk & Nielsen ( 1996), t • fiow with free turbulence representing e.g. natural wind. The double oscillator model i based on exact energy exchange between fluid and structure, and it predicts two clifferett modes of osci llation in the lock-in regime leading to hysteresis effects when the wind spcec passes up and down through the lock-in interval. This behaviour is also observed in ex periments e.g. by Feng (1968) and Brika & Laneville (1993). The effect of the tu rbuleno is to destabilize the mode with the highest amplification, thereby reducing the respons• amplitude, and for higher turbulence intensity to change the self-excited harmonic respons• to stochastic narrow-banded response with changing amplitude.

2 DOUBLE OSCILLATOR MODEL

Figure 1 shows cylinder of length l and diameter D suspended by linear springs in < fluid flow with mass density p and total fluid velocity Ut = U + u(t), where U is the undisturbed mean-flow velocity a.nd u(t) is the turbulent velocity component.. The cylinde1 can move in the transverse direction with displacement X(t). The sum of the structura: and added fluid mass is m 0 . The structural stiffness is represented via the unclampec circular eigenfrequency of the oscillator w0 . The structural damping is modelcd as a lineat viscous with the damping ratio (o.

x(L)

~D/2S

FIG . 1. Vortex-induced ,·ibrations of cylinder in cross-flow.

The oscillating fluid in the near wake of t he cyl inder is modelled pbenomenologi cally as a

2

single degree of freedom non-linear oscillator of the van der Pol type, where the sign of the damping term, which serves as the power supply to the integrated system, is controlled by the mechanical energy of the fluid as proposed by Hartlen & Currie ( 1970). Several proposals have been made for the form of coupling between the solid an fluid oscillator, e.g. Iwan & Blevins (1974). Recently the authors proposed a model in which the coupling is prescribed such that the energy exchange between the two oscillators is balanced at all times, Krenk & Nielsen (1996). The governing equations for non-turbulent flow were derived from dimensional analysis in the form

mo[ X + 2(owoX + w6X] ( I )

I 2 X --pU Dl- "" 2 [/ I (2)

1 is a non-dimensional coupling parameter, and the quantity 'YW j[' may be considered as a time-dependent lift-coefficient. The velocities X and W are normalized with respect to the wind velocity U. m., is a generalized mass proportional to the fluid mass density p and the volume of the cylinder. The proportionality factor can be absorbed in W and "'f , so t he equivalent mass can be set to m, = pD2 1. The natural frequency of the cylinder is defined as the circular Strouhals frequency w. = 21rSUj D, where S-:::: 0.2 is the Strouhals number , indicating the circular shedding frequency on a fixed cylinder in laminar How. In this case the exact solution to the fluid oscillator equation is W(t) = w 0 sin(w.t + 1/> ), with the ar­bitrary phase 1/J. Hence wo is the amplitude of the fluid oscillations on a fixed cylinder i11 laminar flow. The power supplied to the mechanical oscillator becomes ~pU12 DIW X 'Y j U, which exactly balance the power extracted from the fluid oscillator. Previous double oscil­lator models have not met this power flow condition.

:1.1 Equations for ttt7·bulenl flow

The effect of the turbulence is to modify the right-band side of the equat ions by introduction of the inst<~.nta.neous wind pressure ~pU12 and by evaluating the Strouhals frequency in tlw fluid oscillator stiffness term by use of the instantaneous wind velocity U1 , whereby t he fluid 'stiffness' takes the form w;(l + u(t)/U))2

. The fluid damping term was devised to supply a typical rate of energy and is not known in sufficient detail to warrant more detail. The equations including turbulence then are

.. . 2 mo[ X + 2(awaX + w0 X ] ' U2Dl( u(t))21¥ (3)

2P I+ V [j'Y

3

The• following nOII-dilncnsiona.l variabl!'s it)'(' int.rodue<!U for t.lw st.ruct.mal and rluid di s· pla.('C'Ill!'ll t.

)" X

\t w (r. '

/) .) ,

111o

'l'h<' e•quat.ions (:I)-( If) can t.h<'u lw writf.c·n

)'·· + 2(owo ):· + w~Y Jl 3 CW,,(I + /l(/. )) \f

(7)

wit.h t.lw fo llowing uou-di1ue•usioual qua.ut.it.ie·s

'lOo 711_., {J !Jll 'Po = JL , = /) 1110 111 0

(~) 'IIIo I !l( I) ·> u( I)

(' = = /) 111rS ~ {I

fL,, is t h<' llla.~s rat io, a.ud r· is a. n•sca.l<'d couplinp; rodficic·u t. proport.iou<tl t.o t.h<' <l.lllplit.ud!' or l.h!' lift. rol'fficii'Jlf,. Ju ((i)-(7) t he• I.UJ'hllle•ll('(' int.e'IISit.y is <I.~Slllll<'d sulficic•u!.ly Slllall I.O justify Lh<' o1nissiou of q ua.dmt.i c Lc•m•s in '1L(i), <tnd t.h <' t u rhnlcnn• is t.lwrdon• re• tm•sc•nt.C'd hy t.IIC' non-dilll<'IISiona.l st.orl1a.st.ir proc!'ss /?(1).

> n 1/annonit rcsprmsr· for 71011.-lu1'1mlnl.! .flow

In t.h<' ahsc·n n · of t.urhu l<'un·, i. c•. for 1t{l) = 0, t.lw r<'spons<' eau lw n ·pn·se•ut.C'd by !.lw ha.riii OII ic approxin1a.t.ion

y = !I sin (w/) ·p = /J sin(w/) (!J)

The· frc•epwry or OSC'i llat.ioll w is dct.e•rtlliiiC'd by t.h<' Stronha.ls frc•que·ury w . 'l'hC' illWJ'S(' re·la t. ion giving w., a.~ fuui'Lion o f w is, 1\r<'llk lv. Nic·ls<'u ( I!J!JG),

w ) 1 w ( r· ) 1

w ( - (-)1 + tJ(d( - )1 - JI.,, - (-)1 I Wo Wu '/Ju Wo

'l'h<' a.m pl it.ude•s ra.n t.he11 lw fouud illi

!I

(

• 'l 2) (uwo w.,- w I - - - - - - -

. 'l 2 (,w,, w0 - w

w." w Jl., l' - - /J Wo wu

( 10)

( 11 )

( 12)

2

1.5

0

8 ..._ 8

0.5

~1 =0.<Xl05

{) {) 0.5 1.5 2

Fl<;. 2. H.<'SJH>nsc frcqu<'nry bmnrlws.

H.<'pr<'S('nl.a.Livc values of t.lw mod<· I paramct.<·rs a.re <'::::: 0.10 a.nd (., ::::: 0.0 I 0. Figure 2 shows

t.ypiral frcqtwncy brandws for a. Vl'ry lightly danlJH'd sysl.<'lll wil.h fL_, = 0.0!\ (u = O.OOO!i, and Pu = 0.2fi. Tlw I'OI'I'I'SJH>IHiing a.111plit.nde curv<·s <H<' shown in Fig. :L

Tlw fignn·s show l.wo solutions with disl.incl.i ve r<•sona.nn· when the Stroulmls frequ<·nry

is close t.o l.lw ua.tma.l slnll'tura.l freq<·ncy. Tlw solutions a.re lin1il.ed by tlw condition

/J~ ~ 0. /1. sl.a.hilil.y i\.IHdys is nt.rri<·d out in J..:renk .'\.•. Ni<·lsen ( I!J!J(i ) shows th;il. l.lw low

frcqu<·ncy bmnch is only stahl<· for fr<:<JIH'IICi<'s lowl'l' than 1111 uppn l imit approxi111at<'ly

<'qual l.o l.lw JH'a.k i\.111plitude frcqt11:nry. Si111ila.rly Llw high fn•qu<•nry hranrl1 is only sta.bl<·

for frequ<'lll'i<·s a.hov<' a. frequency ('orn•sponding <tpprox illla.t<·l y to tlw JH'it.k a.111 plitude of

!.his hra.nrh.

3 THE TURBULENCE PROCESS

The non-dinwnsionalturhul<·nn· pro<'<'ss /((/)is IH'I'<' used to n ·p res<·nL t.lw rapidly flncl.ll ­

<t.ting pa.rl. of t.lw na.tura.l (.urhulenre. This is thl' pa.rt of l.h<' l.urbnl<'nl'e !.ha.!. I<'<His l.o a

nlOd ifi<·d s(.al.iona.ry rspolJS<', whil<• low frequ<'nry !'OIIIJH>IH'IJ!.s lllit.y l<·a.d l.o l.ransi<'nts wlwn <·nt<·ring and le<wing lock- i11 intervals of l i111iL('(I length. Tlw l.mbulence pron·ss /((/ ) is

introduced ill th<· forlll or i\.11 Ornstcin- llhlc•niH·rk p1'0('('SS with l.lw sta.!.iona.ry rova.ria.nn·

function

(1:1)

wh<·r<' a11 is tlw standard d<·via.t.ion and r,. the rorrda.Lion tinll' snt.l<·. T lw turhtll<'lln'

iut<•nsity is defin<•d by / 11 = au/11 = 1atl·

The r< 'SJIOII S<' it.llidysis is ba.sc·d on Monte ( :a.rlo si n~t!l a.tion . The broa.d-lmnded turl>tll<·n<·<·

0.2 .----~--~--~------,

0.15

FIG. 3. Harmonic amplitude versus frequency.

process R(t) is simulated from the !to-differential equation

1 {f-dR(t) = --R(t)dt + - aRdW(t) Tr Tc

(14)

where dW(t) is the increment of a unit intensity Wiener process.

The reference time is the 'Strouhals period' T. = 27r /w., representing a typical oscilla­tion period. The correlation time Tc is chosen to be 'short' relative to T., Tc = T,/50. The simulation interval /::,.t of the process dW(t) was .:':!.t = T,/250, corresponding to /::,.t = kTc, and the simulated points connected by a broken line as suggested by Clough & Penzien (1975). The response was obtained by introducing the state space vector [Y(t), Y(t), V(t), V(t),R(t)] and integrating the differential equations (6), (7) and (14) with a 4th order Runge-Kutta scheme using the time-step /::,.l. By this integration scheme the response to rapid Auctutions are represented with good accuracy. The probability den­sity Jy(y) and the standard deviation ay are obtained from ergodic sampling over a t ime interval of length 105T., following an initial interval of length 3 · 1037~ to allow for decay of transients due to initial conditions. The probability density function was sampled with 301 classes of equal length covering the interval (-4ay, 4ay] for an estimated value of av.

4 RESPONSE CHARACTERISTICS

In the absense of turbulence two near-harmonic solutions exist in the frequency interval between the peaks of the amplitude curves in Fig. 3. The effect of turbulence is illust rated in Figs. 4 and 5, showing the simulated probability density function f>-(y) at mean wind speeds corresponding to the Strouhals frequencies w,/w0 = 0.9, 0.95, 1.0, 1.05, l.l , 1.15 for turbulene intensities a,J[J = 0.02 and 0.10, respecively. At low turbulence intensity the

6

40 40 wJw0 = 0.90 wJw0 = 0.95

30

N 30

20 20 u 10 10

0 0 -0. 1 0 0.1 -0.1 0 0.1

40 40

30 WJWo = 1.00

30 w, lw0 = 1.05

20 20 (\) 10 10

0 0 -0.1 0 0.1 -0.1 0 0.1

40 40 wJ w0 = I. I 0

30 w,lw0 = 1.15

30

~ 20 ~ 20

10 10 ·····

0 0 -0.1 0 0.1 -0.1 0 0.1

FIG. 4. Probability densities, (o = 0.0005, <7u/U = 0.02.

response retains a nearly harmonic cha.rader and around the resonance frequecy two so­lutions exist, as in the non-turbulent case. In the imulation study the two solutions arc•

obtained by using the initial conditions Y(O) = V(O) = V(O) = 0 together with either Y(O) = 1 or 0. Segments of the corresponding time histories are shown in Figs. fia and 6b for w,fw0 = 1.05. The mean amplitudes correspond closely to the deterministic val­ues predicted by Fig. 3. The resonant solut.ion remains most regular. This confirms the experienta.l observa.ion of Goswami et al. ( 1993), who found only liit.le effect. of turbu­lence of this low intensity. When the turbulence intenity is increased the upper branch is destabilized and the response changes character from slightly perturbed harmonic to narrow-banded stocha.stic response containing also very small amplitudes as shown in the probability density functions h(y) in Fig. 5 and the time history show in Fig. fie.

7

40 40

30 mJ m0 = 0. 90

30 mJ m0 = 0.95

20 20

10 10

0 0 -0. 1 0 0. 1 -0.1 0 0.1

40 40

30 m, /m0 = 1.00

30 w,lm0 = 1.05

20 20

10 10

0 / '\ 0 - 0.1 0 0.1 -0.1 0 0.1

40 40

30 m, /m0 = 1.10

30 m.fm0 = 1. 15

20 20

lO 10

0 0 - 0.1 0 0. 1 - 0.1 0 0 .1

FIG. 5. Probabi lity densities, (o == 0.0005, au/U == 0.10.

5 CONCLUSIONS

A recently developed energy balanced double osci llator model has been used to investigate the effect of turbulence. Two effects were identified: tu rbulence tends to decrease the re­sponse by destabilising t he most resonant branch, and with increasing tu rbulence intensity t he response changes to typical stochastic narrow-banded with varyi ng a mplitudes.

ACKNOWLEDGMENT: Support from the European Union Programme for Hu man Re­sources and Mobil ity and t he Danish Technical Research Council is gratefully acknowl­edged.

REFERENCES Bri ka, D. and Laneville, A. (1993). Vort.f'x-induced vibrations of a long flexible ci rcular

8

0.1 cr /U = 0.02

0

- 0.1

0 5 10 15 20

0.1 cru/U = 0.02

-0.1

0 5 10 15 20

-0. 1

0 5 10 15 20

F rG. 6. Time histories for (0 = 0.0005.

cylinder. Journal of Fluid Mechanics, 250, 481-508.

('lough, R.W. and Penzi~n, J . (1975) . Dynamics of Slntclures, Mc-Graw- Hill , New York.

Daly, A.F. (1986). Evaluat ion of methods of predicting the across-wind response of chim­neys, in C!CJND Report, Vol. 2, No. l, 9-46.

Dyrbye, C. and Hansen, S.O. (1997). Wind Loads on Stntclures, Wilcy, Chichester, Eng­land.

Feng, C.C. ( 1968). The measurement of vortex-indued effects in flow past stationary and oscillat ing circula r and D-section cylinders. M.Sc. Thesis, Univers ity of British Columbia.

Goswami, I., Scanlan, R.H. and Jones, N.P. (1993). Vortex-induced vibrations of ci rcular cylinders. I: Experimental data, Il: New model. Joumal of Enginetring Mechanics, 119, 2270-2287, 2288-2302.

Hartlen, R.T. and Currie, I.G. (1970). Lift-oscillator model of vortex induced vibration. Journal of Engineering Mechanics, ASCE, 96, 577-591.

!wan, W.D. and Blevins, R.D. (1974). A model for vortex induced oscillation of structures. Journ al of Applied Mechanics, ASME, 41, 581-586.

9

Krenk , S. and Nielsen, S.R.K. (1996). An energy balanced double oscillator model for vortex-induced vibration . (to be published).

van Koten, H. (1984). Wind induced vibrat ions of chimneys: the rules of the CICIND code for steel chimneys. Engineering Sh·uctures, 6, 350-356.

Vickery, B.J. and Basu, R.L (1983). Across-wind vibrations of structures of circular cross­section. Part I. Journal of Wind Engineering and Industrial Aeodynamics, 12, 49-73.

10

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PAPER NO. 154: .J. D. S0rPns<'n, M. H. FabPr, I. B. Kroon. Optimal Reliability-Base Planning of Experiment.• for POD Cttr11c.•. ISSN 1395-7953 R9542.

PAPER NO. 155: .J. D. S(>m·nsm. S. Enp;Plnnd: Stochastic Finite Elemr.nt.• in Rdtability· Based Stntctural Optimtzation. ISSN 1395-7953 R9543.

PAPER NO. 156: C. Pc•c!Prsm, P. Thoft-Christms<'!l: Gmdclincs for lnteractivr. Rcliabil BaM:d Structural Opttmization nsmg Q·ua.•i-Ncwton Algonthm.•. ISSN 1395-7953 R9615

PAPER NO. 157: P Thoft.-Christ.Pnsc•n, F. M. kmPn, C. R. Midd!Pton, A. Blackmorc A.•.•e.•.•mmt of thr. Reliabihty of Concrete Slab Bndge.•. ISSN 1395-7953 R9616.

PAPER NO. 158: P. Thoft-C'hristcnsc•n: Rc-A•M~.•.•ment of Concrete Bridges. ISSN 1395-7953 R9605.

PAPER :NO. 100: H. I. Hatuwn, P. Thoft-ChristPnsc•n: Wmd T1mncl Testing of Acti11 Control Systr:rn for Bridge.•. ISSN 1395-7053 R9662.

PAPER NO 160: C P<'<lPrsPn: lnterachm: Rel?abildy-Ba.H:d Optimization of Strnct11.ra Sy.•tw~.- Ph.D.-Th<'~is. ISSN 1395-7953 R9638.

PAPER NO. 161: S. Engdnnd . .J. D S~t~rf'llsPu: Stocha.•tic M odd.• for Chlorid~;-initiate ' Corro~wn m Reinforced Concretc.ISSN 1305-7953 R9608.

PAPER NO. 165· P. H. KirkPp;aard, F. M . .Jc•nsm. P. Thoft-Christcnsen: Modelling o Surface Shtp.• u.1ing Artificial Ne·ural NctvJOrJ,;,,, ISSN 15!>3-7953 R0625.

PAPER NO 166: S. R K. NidsPn, S. Kr<'nk: Stocha.~iic Rr..•pon.H! of Energy Balanced Model for Wortcx-lnd1tcd Vibration. ISSN 1395-7953 R9710

Pt\.PER NO. 167: S.R.K. Nidscn, R Iwankic·wic)l: Dynamic ,qystem.~ Driven by Non Pomonian Imp1Ll.H~s: Ma1·ko11 Vector Approach. ISSN 1395-7953 R0705.

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