pure VIF on bluff bodie, almot all the exiting parameter identification procee have baed on the diplacement repone of VIV, intead of the force hitory

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1 The Seventh International Colloquium on Bluff Body Aerodynamic and it Application (BBAA7) Shanghai, China; September -6, A novel empirical non-linear model for vortex-induced force on a cloed box bridge deck Le-Dong Zhu a,b,c, Xiao-Liang Meng a,b, Zhen-Shan Guo b,c a State Key Laboratory for Diater Reduction in Civil Engineering, Tongji Univerity, Shanghai, China b Department of Bridge Engineering, Tongji Univerity, Shanghai, China c Key Laboratory for Wind Reitance Technology of Bridge of Minitry of Tranport, Tongji Univerity, Shanghai, China ABSTRACT: By taking a cable-tayed bridge a background, the non-linear vortex-induced force on it flat fully-cloed box girder wa invetigated via a erie of wind tunnel tet of pring-upended ectional model at a large cale of /5, in which the dynamic diplacement of the model and the dynamic force acting on the outer cladding of the model were imultaneouly meaured. A non-linear leat quare fitting method baed on the meaured hitory of vortexinduced force wa etablihed at firt to identify the relevant parameter of variou empirical model of vortex-induced force. The well known Scanlan' nonlinear empirical model for the vortex-induced force wa then examined by comparing the calculated diplacement repone with the meaured one, and wa demontrated to be not uitable for the vortex-induced force of the flat box girder. A novel empirical model wa finally propoed and proved to be proper to the vortex-induced force of the flat box girder. By uing thi novel model, not only the meaured hitory of vortex-induced force can be well fitted, but alo the meaured hitory of dynamic diplacement of the ectional model can be well reproduced via numerical calculation. KEYWORDS: Cloed box girder; vortex-induced vibration; nonlinear vortex-induced force; parameter identification; nonlinear leat quare fitting. INTRODUCTION The flexible teel bridge with low damping often uffer vortex-induced vibration (VIV) at relatively low wind peed. However, the VIV with mall amplitude i ometime allowable, unle it could poibly caue fatigue problem to the tructure or dicomfort problem to it uer. Therefore, a careful prediction of the VIV performance via wind tunnel tet or theoretical analyi i very neceary for a bridge upected of VIV problem. In thi connection, a reliable model of vortex-induced force (VIF) hould be firtly etablihed to decribe the trongly-nonlinear interaction between fluid and tructure. Neverthele, no completely ucceful analytical method ha yet been developed, tarting from baic flow principle, to repreent the repone behavior of an elatic bluff body under the action of vortex hedding (Scanlan, 986). Intead, it ha been relatively fruitful in building empirical model of uncoupled VIF baed on wind tunnel tet, uch a propoed by Vickery & Bau(983), Staubli(983), Scanlan(986) and Laren(995), and the fluid ocillator model by Hartlen & Currie(97), Landl(975), Tamura & Matui(979), Berger(987) and Diana & Cheli(995). Among thee propoed VIF model, only a few, uch like the nonlinear model propoed by Scanlan(986), a well a it generalized model propoed by Laren(995), are applicable to bridge VIV becaue of the pecial characteritic of vortex hedding mechanim of bluff bridge deck. Secondly, the identification of the parameter in the empirical model i to ome extent decided whether a model i applicable or not. Since the difficulty on the technique of meauring 49

2 pure VIF on bluff bodie, almot all the exiting parameter identification procee have baed on the diplacement repone of VIV, intead of the force hitory excited by vortex hedding. Several aumption have been induced to deduce the approximate olution of the nonlinear differential equation of motion (Ehan et al. 99; Laren 995). In pite of that, it i till unknown how thee aumption will affect the analyi reult. In thi tudy, a new technique of ectional model wind tunnel tet wa developed to invetigate the VIF directly. The pecial technique could imultaneouly meaure the total force (coniting of vortex-induced force and the inertial force), the VIV diplacement and acceleration of an ocillating deck model, in free vibration tet, via force balance along with laer diplacement enor and acceleration enor repectively. The reliability of exiting Scanlan and Laren model and their parameter identification method were judged by the vortex-induced force hitory. Moreover, a novel promiing empirical model for VIF on a cloed box bridge girder i propoed and it parameter are identified uing nonlinear leat quare method. MODEL, WIND TUNNEL AND MEASUREMENT SYSTEM. Sectional model deign The Xianghan Harbor Bridge, a typical long pan highway bridge in China, wa taken a an example to invetigate the vortex-induced force on cloed box bridge deck. The Xianghan Harbor Bridge i a cable-tayed bridge with a main pan of 688m over the Xianghan Harbor in Ningbo of China. The deck of the bridge i of flat teel trapezoidal box girder, which i 3.m wide, 3.5m high and with two et of ide fairing of about 56 degree. Conidering that the VIV performance i alway enitive to the aerodynamic configuration of bluff bodie, the deck exterior appearance, including the ide fairing, hand rail, protection rail and maintenance trace, were modeled in the light of the principle of geometric imilarity eriouly. The geometric cale of the ectional model wa et to be : while it length i 3.6m after a careful conideration of many factor. Thee factor include the ize of working ection of the wind tunnel, the ufficiently large apect ratio (length-over-width) requirement for the model, the lock-in wind peed, the frequency of the ma-pring ytem, and the enitivity and capacity of the force balance. After caled, the cro ection of the model i.6m wide (noted a B) and.75m high (noted a D). The configuration of the model cro ection i illutrated in Fig.. 8 Hand Rail Clothe Part Force Balance 8 Protection Rail Rigid Frame Part D = 75 Maintenance Rail B = 6 Figure. Cro Section of the ectional model (Unit: mm). The force balance technique wa adopted in thi tudy for it i the mot reliable way of meauring fluctuating lift in wind engineering. To get the pure vortex-induced force acting on an ocillating bridge deck, the inertial force, which could obtained from the product of acceleration 43

3 The Seventh International Colloquium on Bluff Body Aerodynamic and it Application (BBAA7) Shanghai, China; September -6, and ma, along with the aerodynamic force in till air, hould be ubtracted from the total force meaured by the force balance. In factor, the aerodynamic force i alway very mall regarding to the inertial force caued by the motion of the model. The numerical error will be inevitably induced in thi ubtraction when the ma connected to the balance i too large. While, low ma of ectional model will reult in wake of tiffne and large deformation of the elatically upported rigid body. To avoid thi confliction, an innovative improvement wa made in thi tudy. The model wa deigned into two part. One of which wa the clothe part made of wood plate and thin-walled tainle teel tube tiffener to hape the bridge deck a light a poible, while the other wa the frame part made of teel tube to provide the tiffne of the model (See Fig. ). The force balance were et interiorly between the clothe part and the rigid frame part while the frame part wa elatically mounted on an eight pring ytem (See Fig. ). Thu, the ampled force ignal could contain le inertial component rather than the whole inertial force of the cumberome ectional model. In other word, the demand of the preciion on force balance could not be too rigorou any more. Furthermore, in order to promote the veracity of the aerodynamic force obtained, the clothe part wa divided into three egment. The main egment et in middle of the model i.4m long, and the two ide one are.598m long each, with mm wide gap between the main egment. With the force balance meauring only the force come from the mid egment, the effect caued by the three dimenional flow around the end of the model could be ignificantly eliminated.. TJ-3 wind tunnel and meaurement ytem Wind tunnel tet were carried out in TJ-3 boundary layer wind tunnel of the State Key Laboratory for Diater Reduction in Civil Engineering at Tongji Univerity, Shanghai, China. The facility i a cloed-circuit wind tunnel which ha a tet ection of 5m wide, m high and 4m long. The achievable mean wind peed in the tunnel ranged from. to 7.6m/. The whole pring-ma ytem of the bridge deck model wa fixed on two temporarily treamline wall built in the interior of TJ-3 wind tunnel (See Fig. 3). All the tet were carried out in mooth flow. The flow field between the wall wa verified by hot wire at the cro ection parallel to the central axe of the deck. The maximum non-uniformity of mean wind within the cro ection i le than % and the turbulence intenity within the ame cro ection i le than.5%. Force Balance Figure. Set up of the ectional model Figure 3. Sectional model in TJ-3 wind tunnel To obtain the time-hitorie of ocillation curve of the model at a given poition and a given wind peed, four accelerometer along with four laer diplacement enor were mounted on each end of the two upport arm. Four electric bridge of the train force balance were powered by a high-accuracy potentiontatic DC power upply and connected to an eight channel DC 43

4 voltage amplifier. A 6-channel data acquiition unit and a computer were ued to digitize and record the meaurement data. Calibration of the force meaurement ytem wa performed before the tet to obtain the calibration factor of each force balance, reflecting the relationhip between the applied force and the normalized output voltage. In factor, there were nonneglectable phae delay between the different type of enor, for the electrical impedance are not the ame in each electric circuit. A imple ma-pring ocillator wa deigned to modify the ampled time hitorie of force, diplacement and acceleration (See Fig. 4). Fig. 5 and Fig. 6 illutrated the normalized time hitorie before and after modification repectively. Balance Acce. Senor Balance Dip. Senor Figure 4. Phae calibration ytem Normalized Signal Dip. Acce Figure 5. Signal before modified Force Force Normalized Signal Dip. Acce Figure 6. Signal after modified Force Force 3 WIND TUNNEL TEST RESULTS AND DISCUSSIONS 3. The added ma parameter m and added damping parameter c in till air Air will certainly reult in force on ocillating deck model regardle of whether the air i flowing or not. But, in the differential equation of motion, the aerodynamic force acting on the model in till air i ued to be idealized to zero. Thi idealization will caue problem when the ytem parameter are identified through free vibration repone in till air while the aerodynamic force i meaured by experiment. Becaue the ytem parameter ued in the equation of motion are identified in till air. That ha automatically taken into account the till air force effect. But the pecial force meaured in flowing air till contain the component repreent the force effect in tatic air. In thi tudy, thi till air aerodynamic force effect mentioned above i conidered by introducing added ma parameter m and added damping parameter c. Therefore, the total air force in lock-in wa the ummation of inertial force caued by added ma, damping force caued by added damping, and the pure aeroelatic force regarded a the pecial vortex-induced force caued by the flowing air. The referred parameter m and c can be identified via the free decay repone and the force hitory by the force enor. The total force meaured by the force balance can be regard a the force come from equivalent ma m e and damping c e, a decribed by Equation : f = m y '' + c y ' = ( m + m ) y '' + c y' = f + f () meaured e e inertial tatic where f meaured = total force meaured per unit length; m = ma of the meaured mid egment per unit length; m = added ma per unit length; and c = added damping per unit length; 43

5 The Seventh International Colloquium on Bluff Body Aerodynamic and it Application (BBAA7) Shanghai, China; September -6, f inertial i inertial force caued by the ma of the mid egment per unit length, which could be expre a: f tatic = m y '' () f tatic i the till air force per unit length: f = m y '' + c y ' (3) tatic The velocity repone can be indirectly obtained from diplacement ignal by central difference method in time domain. Fig. 7a to Fig. 7d how the ampled diplacement, acceleration, total meaured force along with the computed velocity hitorie of free decay vibration repone in till air. With the meaured total force hitory along with the repone hitory of acceleration and velocity, the identification of the parameter of equivalent ma m e and damping c e wa operated by time domain leat quare fitting method (See Fig. 7e). The added ma m and added damping c can be obtained by Equation, for the ma of the mid egment m e could directly weighted by the force balance. Then the till aerodynamic force hitory could be reproduced by the parameter m and c (See Fig. 7f).. 6. y (m).. -. y'' (m/ ) f meaured (N/m) (a) Diplacement hitory (b) Acceleration hitory (c) Total meaured force f meaured.4 Fitting. 6 6 y' (m/) f meaured (N/m) (d) Velocity hitory (e) Leat quare fitting of f meaured (f) Still air force f tatic Figure 7. Force and repone hitory of free decay vibration of ectional model in till air f tatic =m y''+c y' (N/m) The pure vortex-induced force at lock-in With the etimated parameter m and c (or m e and c e ), it i eay to obtain the pure vortexinduced force at lock-in. Jut a the operation conducted in the till air, the total force along with the VIV repone hitorie were recorded via the enor equipped on the model. Then the pure vortex-induced force could be achieved by ubtract the inertial force of the mid egment and the till air force caued by the added ma and damping from the meaured force. Taking the VIV performance at the wind peed of 9.m/ a example, the time hitorie of repone and force were illutrated in Fig. 8. To verify the reliability of the indirectly meaured vortex- 433

6 induced force, a time domain analyi on the repone of the ectional model wa conducted by Newmark-β method, inputting the pecial vortex-induced force hitory obtained above. y (m) y'' (m/ ) (a) Diplacement hitory (b) Acceleration hitory (c) Total meaured force f meaured y' (m/) m e y''+c e y' (N/m) (d) Velocity hitory (e) Equivalent inertial and damping force (f) Pure vortex-induced force Figure 8. Force and repone hitory of vortex-induced vibration at lock-in f meaured (N/m) f VIV (N/m) f VIV (N/m) (a) Detail view of pure vortex-induced force y (m) Computed via Newmark-β method (c) The teted diplacement repone and the computed repone Magnitude of f VIV (N/m).. E Frequency (Hz) (b) f VIV in frequency domain Magnitude of y (m).. E-3 E-4 E-5 Computed E Frequency (Hz) (d) y in frequency domain Figure 9. The vortex-induced force and repone in time and frequency domain 434

7 The Seventh International Colloquium on Bluff Body Aerodynamic and it Application (BBAA7) Shanghai, China; September -6, Fig. 9a how the vortex-induced force time hitory in detail and Fig. 9b how the amplitude pectrum of the vortex-induced force. Fig. 9c and Fig. 9d illutrate the repone and force ignal in frequency domain, with the ordinate to be the magnitude of the ignal. It can be oberved that while the repone hitorie. It can be oberved form the figure that, while the motion repone are harmonic-like ignal, the vortex-induced force hitory i much complicate. The vortexinduced force hitory ha two negative peak and two poitive peak during one period. It reult in the ignificant double and triple frequency component, which could be the main reaon for the nonlinear phenomena in VIV. A the computed repone i very cloe to the teted, both in time and frequency domain, it could be concluded that the teted force i reliable and with high preciion. 4 SCANLAN S NONLINEAR MODEL AND PARAMETER IDENTIFICATION METHOD 4. Scanlan nonlinear model Empirical model decribing the acro-flow repone of a rigid bluff body mounting on linear pring ubmerged in a fluid tream with uniform velocity can be written a following m( y + ξω y + ω y) = f ( y, y, y, Ut, ) (4) VIV where m = ma per unit length; ω = mechanical tructure circular frequency; ξ = mechanical tructure damping ratio; y = diplacement repone in acro-flow direction; U = the mean follow peed. Scanlan (986) propoed a emi-empirical nonlinear model yield the general form mentioned above. In which, the vortex-induced forcing function wa defined a y y y fviv ( y, y, y, U, t) = ρu ( D) Y( K) ε Y + ( K) + C % L in( ωt+ ϕ) D U D (5) where D = characteritic length of the bluff body, in thi tudy, it i the depth of the bridge deck; ρ = air denity; the reduced frequency K = ωd/u, where ω i the vortex frequency; and the phae φ. The non-dimenional form of Equation 4 with the ubtitution of f VIV in Equation 5 i η () + ξkη () + K η() = my r ( K)[ εη ( )] η ( ) + my r ( K) η( ) + mc r LV in( K+ θ) where η() = y(t)/d; the ma ratio m r = ρd /m. (6) 4. Parameter identification method via GTR or DTR method Ehan and Scanlan (99) ha developed a parameter identification procedure via the Decay to Reonance (DTR) or Grow to Reonance (GTR) repone of the ectional model in wind tun- 435

8 nel tet. The approximate analyi olution of the Van der Pol type function decribed by Equation 6 wa firtly deduced by introducing the low variation and Van der Pol aumption. The lat term of the direct forcing wa dropped for the value of C LV wa thought to be very mall. Then the approximate olution could be decribed like following β η() = co K my r ( K) ( K K ) A K + ψ β ( αβ /4) e A (7) where A and ψ are the initial amplitude and phae of the Decay to Reonance (DTR) or Grow to Reonance (GTR), repectively. α and β are variable containing parameter ε and Y α = my ( ) r K ε (8) β = ξ K ε my r ( K) / (9) The variable α and β could be obtained from the reonance amplitude and the envelope of the decay or grow tage of the repone. Thu, the parameter ε and Y could be obtained by equation (8) and (9). Furthermore, the parameter Y, which wa thought to be very mall, could alo be obtained from the difference between the mechanical frequency and the total frequency during VIV. When the parameter were identified, not only the vortex-induced ocillating repone, but alo the vortex-induced fore can be numerically achieved. In thi ection, the GTR method wa adopted to identify the parameter. The etimated parameter are hown in Table. Fig. how the non-dimenional vortex-induced forcing f nonviv hitorie achieved by wind tunnel tet and computation. Fig. how the teted and computed grow to reonance non-dimenional hitorie. It can be oberved from the figure that, although the repone of the ectional model calculated from Scanlan nonlinear empirical model, which parameter are identified via the motion repone, i quite cloe to the tet repone hitory, the parameter decribed vortexinduced force hitory i remarkably diimilar to the teted force. The parameter decribed force ha three negative peak a well a three poitive peak in one period, while, the teted force hiroy ha only two negative peak and two poitive peak. x -3 5x -4 Numerical f nonviv -5x -4 -x Figure. Non-dimenional vortex-induced force hitory (Scanlan model, parameter identified via GTR method) 436

9 The Seventh International Colloquium on Bluff Body Aerodynamic and it Application (BBAA7) Shanghai, China; September -6,.. Numerical η Figure. Non-dimenional diplacement repone (Scanlan model, parameter identified via GTR method) 4.3 Parameter identification method via LSQ method Jut a decribed above, with the parameter identification method propoed by Ehan and Scanlan, the Scanlan nonlinear empirical model could not perfectly rebuild the teted vortexinduced force. That could reult from two poible reaon, firt, the parameter identification procedure i not uitable for it improper aumption induced in the approximate olution, and econd, the model itelf i not able to decribe the vortex-induced force. In thi ection, a direct curve fitting method baed on the nonlinear leat quare (LSQ) technique wa developed to identify the parameter in Scanlan nonlinear empirical model. The um quared error wa regarded a the objective function, and the numerical optimization function can be defined a ( ˆ ) R( Y, Y, ε, C, K, ϕ ) = f ( ) f ( ) () L nonviv i nonviv i where ˆf nonviv ( i ) i the etimated non-dimenional vortex-induced force decribed by Scanlan model at time i,and f nonviv i the teted value at the ame time. The objective i to find Y,Y, ε, C L, K and φ to make the value of function R(Y,Y,ε,C L, K, φ) to be minimum. The Levenberg- Marquardt method wa adopted to olve the nonlinear Equation. η ( ( ) ( ))( ) = n ˆ i fnonviv i fnonviv i ε η i i= D n ˆ ( fnonviv ( i ) fnonviv ( i )) ηη i i = i= n i= n i= ( f ( ) fˆ ( )) η = nonviv i nonviv i i i= n ( f ( ) ˆ nonviv i fnonviv ( i ))in( Ki + ϕ) = i= n ( f ( ) fˆ ( )) co( K + ϕ) = nonviv i nonviv i i i ( f ( ) fˆ ( ))co( K + ϕ) = nonviv i nonviv i i () 437

10 Table. The parameter identified in the empirical model Model Method Y Y Y 3 ε ε N C LV K θ Scanlan GTR Scanlan LSQ New model LSQ The identified parameter for Scanlan nonlinear model by LSQ method wa given in Table. Fig. and Fig. 3 how the rebuilt force and the numerical olution of the differential equation of motion, with the vortex-induced force decribed by Scanlan model and the parameter by LSQ method. The reult how that, the coefficient of the nonlinear aerodynamic term, ε, wa a negative value. That will certainly reult in nonlinear negative aerodynamic damping in the numerical ytem. In other word, the negative aerodynamic damping will grow if the ocillation amplitude grow, and the ytem will how divergence finally, jut a the repone hitory illutrated in Fig. 3..x -3 5.x -4 Fitting f nonviv. -5.x -4 -.x Figure. Non-dimenional vortex-induced force hitory (Scanlan model, parameter identified via LSQ method)..5 Numerical η Figure 3. Non-dimenional diplacement repone (Scanlan model, parameter identified via LSQ method) 5 NEW VORTEX-INDUCED FORCE MODEL The evidence above how overwhelmingly that the Scanlan nonlinear empirical model i not proper to predict the vortex-induced force on cloed bridge deck. A new empirical model wa uggeted, after a really hard try of poible model with the general form of Equation 4, to properly decribe the vortex-induced force on the bridge deck. The new model can be defined a y y y yy fviv ( y, y, y, U, t) = ρu ( D) Y( K) εn Y + ( K) + Y3 + C % L in( ωt+ ϕ) D U D UD () 438

11 The Seventh International Colloquium on Bluff Body Aerodynamic and it Application (BBAA7) Shanghai, China; September -6, Thi new model alo induced a cubical term to enure that the ocillation i elf-limiting, but the pecial nonlinear aerodynamic damping term wa in phae with the cube of velocity, intead of with the product of quare diplacement and velocity ued in Scanlan model. Furthermore, a econd order nonlinear term, which wa in phae the product of diplacement and velocity, wa induced to the model a well, for the remarkable force component of double frequency oberved in Fig. 9b. The LSQ method wa alo adopted to identify the parameter in the new model. The value of the parameter in the new model are given in Table. Fig. 4 and Fig. 5 how the vortex-induced force and the repone calculated by the new model. The reult how that, with the parameter etimated by LSQ method, the new model could not only perfectly rebuild the vortex-induced force, but alo reliably predict the vortex repone by numerically olving the nonlinear differential equation of motion. However, the computed repone i a little larger than the teted repone. It could be reult from the lack of panwie correlation on the vortexinduced force and the incoming flow, for the force teted wa only the aerodynamic force acting on the mid egment of the model, while the ocillation ytem wa driven by the force acting on the whole model..x -3 5.x -4 Fitting f nonviv. -5.x -4 -.x Figure 4. Non-dimenional vortex-induced force hitory (New model, parameter identified via LSQ method).. Numerical η Figure 4. Non-dimenional diplacement repone (New model, parameter identified via LSQ method) 6 CONCLUSIONS By taking a cloed box bridge deck a an example, the vortex-induced force wa teted by ectional wind tunnel tet. The tet reult ha hown that the vortex-induced force i evidently nonlinear, for the ignificant double and triple frequency component oberved in the amplitude pectrum of the teted force. The reliability of the well known Scanlan nonlinear empirical model and the parameter identification procedure uggeted by Ehan and Scanlan wa dicued. A new parameter etimate procedure, which wa baed on leat quare technique to directly fitting the vortex-induced force hitory, wa alo developed. By comparing the vortex-induced force and tructure repone rebuilt by Scanlan model with the meaured reult, it can be concluded that the Scanlan 439

12 nonlinear empirical model could not properly decribe the pecial vortex-induced vibration phenomenon. A new promiing nonlinear model wa propoed. The model till introduced a three order term to enure the elf-limiting property a well a a econd order term to adapt the double frequency component of the vortex-induced force. Aided with the parameter identified by the nonlinear leat quare fitting method, the new model could either perfectly fit the teted force hitory, or exactly reproduce the repone of VIV. 7 ACKNOWLEDGEMENTS The work decribed in thi paper wa jointly upported by the National Nature Science Foundation of China (Grant 59784), and the Fundamental Reearch Fund for State Key Laboratorie from the Minitry of Science and Technology of China (Grant No. SLDRCE8-A-). Any opinion and concluding remark preented in thi paper are entirely thoe of the writer. 8 REFERENCES A. Laren, A generalized model for aement of vortex-induced vibration of flexible tructure, Journal of Wind Engineering and Indutrial Aerodynamic, 57(995) B. Vickery, R. Bau, Acro-wind vibration of tructure of circular cro-ection, Part I. Development of a mathematical model for two-dimenional condition, Journal of Wind Engineering and Indutrial Aerodynamic, (983) E. Berger, On a mechanim of vortex excited ocillation of a cylinder, Journal of Wind Engineering and Indutrial Aerodynamic, 8(988) E. Simiu and R. H. Scanlan, Wind effect on tructure: Fundamental and application to Deign, nd edition, New York, F. Ehan, R. H. Scanlan, Vortex-induced vibration of flexible bridge, Journal of Engineering Mechanic, 6(99) G. Diana, F. Cheli and F. Reta, Time domain aeroelatic force identification on bridge deck, Proc. 9th International Conference on Wind Engineering, New Delhi, India, 995, pp R.T. Hartlen, I.G. Currie, Lift-ocillator model of vortex-induced vibration, Journal of Engineering Mechanic Diviion, ASCE, 96(97) R. Landl, A mathematical model for vortex-excited vibration of bluff bodie, Journal of Sound and Vibration, 4(975) T. Staubli, Calculation of vibration of an elatically mounted cylinder uing experimental data from a forced ocillation, Journal of Fluid Engineering, 5(983) 5 9. Y. Tamura, G. Matui, Wake-ocillator model of vortex-induced ocillation of circular cylinder, Proc. 5th International Conference on Wind Engineering, Fort Collin, Colorado, USA, 979, pp