Investigation of the Scruton Number Effects of Wind

Research ArticleInvestigation of the Scruton Number Effects of Wind-InducedUnsteady Galloping Responses of Bridge Suspenders

Shuai Zhou123 Yunfeng Zou 4 Xugang Hua2 Fanrong Xue4 and Xuandong Lu4

1China Construction Fifth Engineering Division Corp Ltd Zhongyi One Road No 158 Changsha Hunan China2College of Civil Engineering Hunan University Lushan No 1 Changsha Hunan China3China State Construction Engineering Corp Ltd Beijing 100013 China4School of Civil Engineering Central South University Changsha 410075 Hunan China

Correspondence should be addressed to Yunfeng Zou yunfengzoucsueducn

Received 1 March 2021 Accepted 6 September 2021 Published 29 September 2021

Academic Editor Arturo Garcia-Perez

Copyright copy 2021 Shuai Zhou et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

When the critical wind speed of vortex-induced resonance is close to that of quasi-steady galloping a type of coupled wind-induced vibration that is different from divergent galloping can easily occur in a rectangular bar It is a type of ldquounsteadygallopingrdquo phenomenon wherein the response amplitude increases linearly with the increase in the wind speed while a limit cycleoscillation is observed at each wind speed whose mechanism is still in research Mass and damping are the key parameters thataffect the coupling degree and amplitude response estimation For a set of rectangular section member models with a width-to-height ratio of 12 by adjusting the equivalent stiffness equivalent mass and damping ratio of the model system and performingcomparative tests on the wind-induced vibration response of the same mass with different damping ratios it is possible to achievethe same damping ratio with different masses and the same Scruton number with different masses and damping combinationsunder the same Reynolds number e results show that the influence of the mass and damping parameters on the ldquounsteadygallopingrdquo amplitude response is independent and the weight is the same in the coupling state e Scruton number ldquolockedintervalrdquo (124ndash306) can be found in the ldquounsteady gallopingrdquo amplitude response and the linear slope of the dimensionless windspeed amplitude response curve does not change with the Scruton number in the ldquolocked intervalrdquo In addition a ldquotransitionintervalrdquo (268ndash306) coexists with the ldquolocked intervalrdquo wherein the coupling state of the wind-induced vibration is converted intothe uncoupled state e empirical formula for estimating the ldquounsteady gallopingrdquo response amplitude is modified and can beused to predict the amplitude within the design wind speed range of similar engineering members

1 Introduction

e classes of blunt bridge members such as rigid bridgesuspenders whose sections are rectangular and feature largeslenderness ratios are light in mass low in damping andprone to the combined instability stemming from a typicalvortex-induced resonance and galloping

is unsteady galloping exhibiting some special char-acteristics such as the beginning of continuous vibration atvortex resonance wind speed for rectangular cylinderswhich are perpendicular to the flow has been extensivelystudied by previous scholars in engineering practice research[1ndash4]

Coupled vibrations can even be triggered when the onsetwind speeds are close to each other e amplitude of wind-induced vibration increases with an increase in the windspeed in the coupling state At a single wind speed point thevibration of the bar takes the form of a ldquolimit cyclerdquo andldquolimiting amplituderdquo at a single wind velocity not a typicaldivergent galloping response such as in iced catenary system[5] In addition the wind-amplitude response curve shows alinear growth trend ere is no vortex resonance lockedinterval and no divergent galloping at the single wind speedpoint e pneumatic mechanism is unclear and has beencalled ldquounsteady gallopingrdquo by previous researchers Sunet al [6] studied the way to harvest the wind energy

HindawiShock and VibrationVolume 2021 Article ID 5514719 11 pageshttpsdoiorg10115520215514719

generating from the synergetic effect of coupled vortex-induced vibration and galloping responses WhileMohamed et al [7] tried to suppress the coupled vibrationwith a downstream square plate by varying side ratios

Niu et al [8] established a mathematical model ofldquounsteady gallopingrdquo by combining the vortex excitationmodel of the wake oscillator with two degrees of freedom(DOF) and a quasi-stationary vibration force model ewind speed amplitude response curve was classifiedaccording to the coupling degree e numerical calculationresults were in agreement with the results obtained byidentifying the relevant aerodynamic parameters e windspeed amplitude curve of the ldquounsteady gallopingrdquo responseis determined by two parameters that is the onset windspeed U0 of the coupled vibration and the linear slope of thewind speed amplitude response curve e European windresistance design code [9] suggests that a coupled vibrationnecessitates an onset wind speed ratio between the vortexresonance and the galloping vibration in the range of 07 to15 e studies in [10] showed that the European code isunsafe and the ratio should be greater than the range from45 to 85 so that the coupled vibration can be avoided In[11] the wind-amplitude response curve morphology of theldquounsteady gallopingrdquo under different Scruton numbers wasstudied by adjusting the stiffness support form of themember model in the wind tunnel tests and changing thedamping ratio indicating that a Scruton number exceedingthe range of 50 to 60 is a necessary condition for eliminatingcoupled vibration Mannini et al and Chen et al [12ndash14]proposed a prediction model that was more consistent withthe experimental results measured by modifying the aero-dynamic parameters based on the 2-DOF Tamura wakeoscillator model Mannini et al [11] investigated the in-fluence of the wind attack angle and turbulence on theldquounsteady gallopingrdquo proposing that low turbulence (ap-proximately 3) would enhance the coupled vibration of thevortex-induced resonance and galloping

Niu et al [8] proposed an empirical formula applicable tothe amplitude estimation of the ldquounsteady gallopingrdquo ofrectangular section members Studies have shown that thelinear slope of the response curve under the coupled vibrationstate is a function of the ratio of the width of amember sectionto its height (BD) An empirical formula for amplitude es-timation was proposed by accumulating a large amount ofdata for regression analysis e studies in [15 16] indicatedthat mass and damping are key parameters that influence theamplitude of vortex-induced resonance responses and thisinfluence is not independent e degrees of influence ofsingle parameter variations (mass or damping) on amplitudeare different Hence the mass and damping parameterscannot be used to form the Scruton number to evaluate theamplitude response e use of the Scruton number as anindependent parameter was investigated in this study eauthors in [17 18] showed that differences in the Reynoldsnumber can result in different amplitude responses which isalso the reason behind the conclusions in [15 16] To evaluatethe amplitude the mass and damping parameters are usuallycombined as the Scruton number

With respect to the coupled vibration between thevortex-induced resonance and galloping vibration (ldquoun-steady gallopingrdquo) a comparative study of the parametersensitivity of the independent mass and damping parame-ters as well as the combined mass damping parameterldquoScruton numberrdquo should be conducted considering theReynolds number effect To further study the ldquounsteadygallopingrdquo response mechanism and to consolidate the re-sults of more reasonable amplitude estimation formulas thisstudy considers a typical rectangular section member modelwith a width-to-height ratio of 12 (BD 12) as a researchobject and adjusts the equivalent mass equivalent stiffnessand damping ratio of the model system in order to conduct aseries of wind tunnel tests on the same mass with differentdamping ratios the same damping ratio with different massvalues and the same Scruton number with different massand damping combinations It also measures and draws thecurve of the wind speed amplitude responses to compara-tively study the influence of the mass and dampingparameters

2 Theoretical Model

21 Wake Oscillator Model A diagram of the cylinder flowwake oscillator is shown in Figure 1 e cylinder tail fol-lowing the cylinder flow can form a nearly stable vortex thatoscillates up and down and provides an oscillator that caninteract with the vibration of the cylinder e oscillatormass is formed by the air quality and the fluid-solid coupledshear stiffness of the vortex oscillating up and down is thevibrator stiffness It can be simplified as a vortex-inducedforce mechanics model with a 2-DOF wake oscillator asshown in Figure 2 In the figures U is the incoming flowwind speed D is the diameter of the cylinder L is the halfwidth of the wake vortex H is the height of the wake vortexG is the center of gravity of the wake vortex still air and α isthe angular displacement of the up-and-down oscillation ofthe wake oscillator Furthermore I K and C are the tor-sional mass torsional stiffness and aerodynamic damping ofthe wake oscillator respectively e vibration equation ofthe established Birkhoff model is given by

I middotd2αdt

2 + C middotdαdt

+ K middot α 0 (1)

For a two-dimensional (2D) cylinder Tamura consid-ered the influence of changes in the wake vibrator length inthe vibration period He also proposed a modified Birkhoff2-DOF vortex-induced resonance mathematical model

euroα minus 2ζ] 1 minus4f

2

C2L0

1113888 1113889α21113896 1113897 _α + ]α minus mlowast euroY minus ]S

lowast _Y

euroY + 2η +n f + CD( 1113857v

Slowast1113896 1113897 _Y + Y minus

fn]2αSlowast2

CL minus fα + Slowast _Y

v1113888 1113889

(2)

2 Shock and Vibration

where Y is the dimensionless displacement response of thestructure α is the displacement response of the wake

oscillator n is the mass ratio η is the mechanical dampingratio of the structure ] is the dimensionless fluid velocityCDis the cylinder resistance coefficient f is the Magnus effectaerodynamic parameter Hr is the dimensionless wake os-cillator width and CL0 is the lift coefficient amplitude of acylinder at rest [19 20]

According to the quasi-constant galloping force theorythe aerodynamic force can be obtained as shown in equation(3) e research conclusion in [8] shows that the expansionterm of the Taylor series for the quasi-constant gallopingforce can essentially meet the calculation requirements withconsideration of the seventh order and equation (4) isdefined as the coefficient of the quasi-constant gallopingforce

Fy minus12ρU

2D middot A1 middot

_y

U1113888 1113889 + A2 middot

_y

U1113888 1113889

2

+ A3 middot_y

U1113888 1113889

3

+ A4 middot_y

U1113888 1113889

4

+ A5 middot_y

U1113888 1113889

5

+ Δ_y

U1113888 1113889

n

⎡⎣ ⎤⎦ (3)

CFY A1 middot_y

U1113888 1113889 + A2 middot

_y

U1113888 1113889

2

+ A3 middot_y

U1113888 1113889

3

+ A4 middot_y

U1113888 1113889

4

+ A5 middot_y

U1113888 1113889

5

+ Δ_y

U1113888 1113889

n

(4)

As shown in equation (5) the dynamic model of thevortex-induced resonance and galloping aerodynamiccoupling is formed by combining the quasi-stationary

galloping self-excited force model with the vortex-inducedforce mechanics model and the 2-DOF wake oscillatorproposed by Tamura


2

C2L0

⎛⎝ ⎞⎠α2⎧⎨

⎩

⎫⎬

⎭ _α + ]α minus mlowast euroY minus ]S

lowast _Y

euroY + 2η +n f minus A1( 1113857v

Slowast minus

nA2Slowast

] _Yminus nA3

Slowast

v1113888 1113889

3_Y2

minus middot middot middot⎧⎨

⎩

⎫⎬

⎭_Y + Y minus

fn]2αSlowast2

CL minus f minus A1( 1113857Slowast _Y

]minus A2

Slowast _Y

v1113888 1113889

2

minus A3Slowast _Y

v1113888 1113889

3

minus middot middot middot + fα⎧⎨

⎩

⎫⎬

⎭

(5)

U

dydt

O

G

y

L L

D

α

HFigure 1 Diagram of the wake oscillator

M

K

C

IG

y

αFigure 2 Mechanical model of the wake oscillator

Shock and Vibration 3

where A1 A2 A3 are the polynomial coefficients of theTaylor series expansion of the quasi-constant gallopingforce and the meanings of the other symbols are the same asmentioned above e coupling model can qualitativelypredict the vortex-induced resonance and galloping andidentify the complete separation of the vortex-inducedresonance locked interval divergent galloping and couplingldquounsteady gallopingrdquo response according to the degree ofsimilarity between the two onset wind speeds e identi-fication such as the recognition accuracy of the onset windspeed point of the dimensionless speed-amplitude responsecurve and amplitude response depends on the accuracy ofthe related aerodynamic parameters It should be pointedout that there is no analytical but numerical solution forequation (5) and the numerical solution could be obtainedby the RungendashKutta mathematical method through nu-merical iteration

For the coupled vortex-induced vibration and gallopingphenomenon based on the previous classical 2-DOF wakeoscillator model Mannini [21] proposed a more useful modelby taking into account the effects of turbulence intensity andintegral length scale and verified by wind tunnel tests Gao et al[22] proposed an empirical model by considering the aero-dynamic nonlinearity with two amplitude-dependent dampingterms and aerodynamic unsteadiness by expressing aerody-namic parameters as functions of reduced frequency Chenet al [23] investigated the unsteady effects from field mea-surement wake-induced vibrations of aligned hangers of long-span suspension bridge

22 Classification of ldquoUnsteady Gallopingrdquo Response CurvesAccording to the degree of coupling between the vortex-in-duced resonance and galloping the wind-amplitude responsecurve of the ldquounsteady gallopingrdquo can be divided into two typesthat is the coupling state and the uncoupled state From thestudy in [19] the response curves measured when the onsetwind speed ratios of the galloping and vortex-induced reso-nance were 106 149 and 215 are shown in Figure 3 eworking conditions (a) and (b) are under the ldquocoupling staterdquoe amplitude increases as the wind speed gradually increasese features of the vortex-induced resonance in the ldquolockedintervalrdquo do not appear in the response curve e time historyof the amplitude curve at a single wind speed indicates a stablestate with a constant amplitude and divergent galloping doesnot appeare amplitude linearly increases with an increase inthe wind speed e features of the response curve can bedescribed by calculating the slope of the curve e ratio of theonset wind speed between the galloping and vortex-inducedresonance further increases to 215 and the working condition(c) shows the ldquouncoupled staterdquo With the increase in the windspeed the separated ldquolocked intervalrdquo of the limiting vortex-induced resonance and the galloping response appear in theamplitude response curve

3 Wind Tunnel Test

31 Model and Setting All tests were performed in a closed-circuit wind tunnel at the Hunan University China e

cross sections of the test section had a width of 30m heightof 25m andmaximumwind velocity of up to 58ms All thetests were designed to be performed under smooth flowconditions e model size of the rectangular membersection for the wind tunnel test is100mmtimes 120mmtimes 1530mm and the ratio of the sectionwidth to its height is BD 12 e installation of the modelin the wind tunnel test is shown in Figure 3 e modelstiffness is provided by eight vertical mounting springs andthe system stiffness can be adjusted by changing the springse mass of the system is composed of the mass of thesection model itself the end plate the connecting rod theweight of the spring reduction and the balance weight eequivalent mass of the model system can be adjusted byincreasing or decreasing the balance weight e naturaldamping of the model system can be effectively increased bywrapping tape around the springe damping of the systemcan be identified before and after the wind tunnel experi-ments for the model vibration measurement e compar-ison results show that the damping is essentially the samewhich indicates that the damping is stable after the modelsystem undergoes substantial wind-induced vibrations

Acceleration sensors are used in the wind tunnel tests tomeasure vibrations e installation diagram is shown inFigure 4 A total of four acceleration sensors (1 2 3 and5) with a sampling frequency of 500Hz were placed at thefront and rear above the horizontal connecting rod at bothends of the sectionmodel on its windward and leeward sidese section model system has six possible degrees-of-free-dom (DOFs) as shown in Figure 5 ere are three trans-lational DOFs and three rotational DOFs along with X-X Y-Y and Z-Z axes respectively e translational DOFs in thedirections of the X-X and Z-Z axes were restricted in thesection modal system as well as the rotational DOF in thedirection of the Y-Y axis e other three available DOFs ofthe section model system were the translational DOFs in thedirections of the Y-Y axis (heaving mode) and Z-Z axis(rolling mode) respectively as well as the rotational DOF inthe direction of the X-X axis (pitching mode) e recog-nition of the vibration form and processing of the amplituderesponse can be performed via signal processing using thefour acceleration sensors rough the adjustment of thetorsional rigidity of the section model system the vibrationmodes with the exception of the heaving mode are

Figure 3 Installation of the sectional model in the wind tunnel


eliminated in all working conditions studied herein the ratioof the torsional frequency to the vertical frequency is from35ndash45 e vibration acceleration response of the sectionmodel was obtained by determining the weighted average ofthe time-history signals of the four sensors According to thefeatures of the simple harmonic single-frequency vibrationdisplacement response signals can be easily obtained usingacceleration vibration signals

32 Configurations and Parameters A comparison of themain parameters of the three-dimensional (3D) prototypeprism with those of the 1 1 2D section model is presented inTable 1 e surface roughness of the prototype prism andsection model differ e steel prototype prism was paintedas smoothly as possible to make it similar to the organic glassmaterial sectionmodel Both the 3D prototype prism and the2D section model have the same cross-sectional dimensionsand roughly the same mass damping and natural fre-quencies e Reynolds number corresponds to the peak ofthe VIV lock-in

e physical quality per linear meter of the sectionmodelsystem is divided into four different levels using counter-weights 827 kgm 1053 kgm 1173 kgm and 1608 kgme equivalent damping of the vertical bending is dividedinto five different levels through the combination of quality

and damping parameters A1ndashA20 0146 0200 02230284 and 0500 A total of 20 test working configura-tions were established by combining the mass and dampingparameters e wind-induced vibration responses underthe combinations of working configurations such as ldquothesame mass with different damping ratiosrdquo and ldquothe samedamping ratio with different massesrdquo can be comparativelystudied

e Scruton number ranges from 124 to 825 When theScruton number is 241 the wind-induced vibration re-sponse can be observed under the configuration that has thesame Scruton number with a combination of different massand damping parameters By co-adjusting the equivalentmass and equivalent stiffness of the model system thebending natural frequencies in the vertical direction ofA1ndashA20 are consistent and the Reynolds number of theonset wind speed corresponding to the vortex-inducedresonance is the same e effect generated by the Reynoldsnumber is eliminated to study the influence of the mass anddamping parameters on the ldquounsteady gallopingrdquo response

4 Results and Discussion

41 Coupling State e time-history curve of the typicalwind-induced vibration was obtained by conducting anexperiment with a rectangular section member model (A1

Rolling Heaving

Spring

Spring

Fixed Boundary

Fixed Boundary

Pitching

Spring

Spring

Fixed Boundary

Fixed Boundary

PitchingSection Model RollingHeaving

X

Y

Zo

Figure 5 Potential vibration mode of the sectional model

1

2

3

5

accelerometer

Figure 4 Indication of a sensor installation on the sectional model


working configuration flow velocity U 37ms) as shownin Figure 6 Under the action of stable wind speed the modelis essentially in a state of constant amplitude vibration witha small-amplitude ldquobeat frequency phenomenonrdquo Fouracceleration sensors were installed at the front rearwindward and leeward sides of the sectional model eymeasured the time-history curve with the same phase whichindicated that the vibration of the model had no torsion orswaying mode and was completely a vertical vibration An

analysis of the Fourier transformation (FFT) spectrumcorresponding to the measured time-history curve is shownin Figure 7 e remarkable vibration frequency is 293Hzwhich is consistent with the measured natural frequency inthe vertical direction under the static wind state and is asingle-frequency vibration state

By adjusting and matching the equivalent mass andequivalent stiffness of the model system the natural fre-quency of the vertical vibration under the static wind state of

0 5 10 15 20 25 30 3500

02

04

06

08

10

12Re=16000 Sc=124~241

A4 ξ=0146 m=1266A8 ξ=0146 m=1611

A12 ξ=0146 m=1795A13 ξ=0146 m=2460

YD

UfD

10

0049

(a)

A3 ξ=0200 m=1266A7 ξ=0200 m=1611

A9 ξ=0200 m=1795

00

02

04

06

08

10

12

YD

0 5 10 15 20 25 30 35UfD

Re=16000 Sc=170~241

100048

(b)

Figure 6 Comparison of different quality responses (a) ξ 0146 (b) ξ 0200

Table 1 Working configurations of research and parameters

Case number m (kgm) ξ () Scruton Spring stiffness (Nm) Measured frequency (Hz) Reynolds VgVv

A1

827

0284 241

616 293 16000

28A2 0223 189 22A3 0200 170 20A4 0146 124 15A17 0500 425 50A5

1053

0223 241

767 293 16000

28A6 0284 306 36A7 0200 216 25A8 0146 158 19A18 0500 540 64A9

1173

0200 241

852 293 16000

28A10 0284 341 40A11 0223 268 32A12 0146 176 21A19 0500 602 71A13

1608

0146 241

1181 293 16000

28A14 0284 468 55A15 0223 368 43A16 0200 330 39A20 0500 825 97Note Sc 4πMξρD2 is the Scruton numberM is the equivalent mass of the structural damping ratio ρ is the air densityD is the model upwind height Vg isthe galloping onset wind speed Vv is the primary wind speed of the vortex-induced resonance and ξ is damping ratio identified from decaying curves ofacceleration e Reynolds number corresponds to the onset wind speed point of the vortex-induced resonance vibration


the configuration A1ndashA20 in all the working configurationsstudied herein is the same that is 293Hz is correspondsto the Reynolds number Re 16000 of the onset wind speedof the vortex-induced resonance e same massm 827 kgm is used in the A1 A2 A3 and A4 workingconfigurations with different damping ratios of ξ 01460200 0223 and 0284 e curves of the dimen-sionless wind-amplitude response corresponding to theScruton numbers ranging from 124 to 241 are shown inFigure 8 In the figure the dimensionless wind speed isdefined as UfD where U is the incoming flow wind speedF 293Hz is the outstanding vibration frequency and

D 100mm is the windward height of the rectangularmembere dimensionless amplitude is YD where Y is thedisplacement amplitude response According to the simpleharmonic features of the single-frequency vibration in thetime-history curve as shown in Figure 1 the amplitude ofthe displacement responses is obtained based on the ac-celeration signal

It can be seen from Figure 9(a) that the curves of thedimensionless wind-amplitude responses in the config-urations A1 to A4 basically coexist e dimensionlessonset wind speed is 75 and the corresponding Strouhalnumber is 013 which is higher than the value

0 5 10 15 20 25 30 35 40 45 50 55 60 6500

01

02

03

04

05

06

07

08

09

10

003110

0025

0027

YD

UfD

Re=16000 Sc=268~825

A14 Sc=468A15 Sc=368A16 Sc=330A20 Sc=825A10 Sc=341

A11 Sc=268A19 Sc=602A18 Sc=540A17 Sc=425

10

10

Figure 8 Amplitude response under dimensionless wind speed in the uncoupled state

A1 m=827 ξ=0284A5 m=1053 ξ=0223

A9 m=1173 ξ=0200A13 m=1608 ξ=0146

0 5 10 15 20 25 30 3500

02

04

06

08

10

12

0048

Re=16000 Sc=241

YD

UfD

10

Figure 7 Comparison of different quality and damping responses (Sc 241)


recommended by the European specifications e curveof the dimensionless wind-amplitude responses shows alinear growth trend and the slope value is 0047 after thestart of the vibration Figure 9(b) shows the correspondingworking configurations A5 A6 A7 and A8 with anequivalent mass of m 1053 kgm e combinations ofdifferent damping ratios of 0146 0200 0223 and0284 correspond to the curve of the dimensionlesswind-amplitude responses with Scruton numbers rangingfrom 158 to 306 e response trend and features areconsistent with those in Figure 8(a) in which the obtainedslope value is 0051 It can be seen from the comparisonresults that under the same mass parameter the dampingratio of the studied rectangular section with a width-to-height ratio of 12 changes from 0146 to 0284 whichhas no influence on the trend of vibration responses andthe slope value of ldquounsteady gallopingrdquo e variationrange of the corresponding Scruton number is from 124to 306

Figure 6(a) shows a comparison of the vibration re-sponses through experiments on the measured models inwind tunnel tests under the combined configuration ldquosamedamping ratio and different massesrdquo e measureddamping ratios of configurations A4 A8 A12 and A13 are0146 with equivalent masses of m 1266 kgm1611 kgm 1795 kgm and 2460 kgm respectively esecorrespond to the curves of the dimensionless wind-am-plitude responses of the Scruton numbers ranging from 124to 241 e response curves of the four working configu-rations tend to coincide e vibration point of the di-mensionless wind speed was 75 and the slope value was0049 As shown in Figure 10 the damping ratio of 0200and equivalent mass m 1266 kgm 1611 kgm and1795 kgm in the working configurations A3A7 and A9 areessentially the same as those in Figure 6(b) e slope valueof the dimensionless response curve was 0048 It can be seenfrom the comparison of the test results that the equivalentmass of the model system changes from 1266 kgm to2460 kgm at the same damping level and the Scrutonnumber ranges from 124 to 241 e obtained curve of thedimensionless wind-amplitude responses has the same onsetwind speed point Furthermore the slope of the amplituderesponse after vibration is the same

Figure 7 shows a comparison of the curve of the di-mensionless wind speed amplitude responses with differentcombinations of mass and damping when the Scrutonnumber (241) is the same e dimensionless wind speedobtained at the starting point is 75 and the slope value of thecurve after vibration is 0048 e response curve features donot change with the different combinations of mass anddamping parameters as shown in Figures 8 and 10 estudy shows that when the width-to-height ratio is 12 of therectangular member section model with a Scruton numberranging from 124 to 306 the coupling of the vortex-

05 10 15 20 25000001002003004005006007008

BD

Slope = -00053(BD)3 + 00505(BD)2 -0147(BD) + 01506

R2=08069

Slop

e

Figure 10 Fitting of response slope value and ratio of width toheight of rectangular section

0 5 10 15 20 25 30 3500

02

04

06

08

10

12Re=16000 Sc=124~241

YD

UfD

A1 m=827 ξ=0284A2 m=827 ξ=0223

A3 m=827 ξ=0200A4 m=827 ξ=0146

10

0047

(a)

00

02

04

06

08

10

12

YD

A5 m=1053 ξ=0223A6 m=1053 ξ=0284

A7 m=1053 ξ=0200A8 m=1053 ξ=0146

0 5 10 15 20 25 30 35

Re=16000 Sc=158~306

UfD

10

0051

(b)

Figure 9 Comparison of different damping responses (a) m 827 kgm (b) m 1053 kgm


induced resonance and galloping produces an ldquounsteadygallopingrdquo response e response curves obtained byconducting experiments on different working configura-tions such as the same mass with different damping ratiosthe same damping ratio with different masses and the sameScruton number with different combinations of mass anddamping ratio are essentially the same e onset windspeed point of the curves of the dimensionless displacementamplitude responses is 75 corresponding to a Strouhalnumber of 013 e linear slopes of the curves of the am-plitude responses are essentially the same with a smallamplitude range of 0047ndash0051

It can be speculated that there exists a Scruton numberldquolocked intervalrdquo for the ldquounsteady gallopingrdquo responsewhich behaved under similar phenomenon as vortex-in-duced vibration lock-in In the ldquolocked intervalrdquo the re-sponse curve of the wind speed amplitude does neitherchange with the change in the value of the single mass anddamping parameter nor with the change in the overallScruton numbere onset wind speed point and slope valueof the amplitude curve are always locked at the same valueFor the bar studied herein the ldquolocked intervalrdquo is 124ndash306

42 Uncoupled State Figure 8 shows a comparison of thecurve of the dimensionless wind speed amplitude responsesunder the uncoupled state of the vortex-induced resonanceand galloping in the working configurations from A1 to A20listed in Table 1 e response curve of the uncoupled statemainly takes on three forms the locked interval and large-amplitude galloping response of the complete-separationvortex-induced resonance the locked interval and large-amplitude galloping response of the semiseparation vortex-induced resonance and the locked interval and large-am-plitude galloping response of the negligible vortex-inducedresonance

e studies in [10 11] show that the critical wind speedratio of the galloping and vortex-induced resonance is inthe interval 45ndash85 and a Scruton number ranging from 50to 60 is a necessary condition to avoid the occurrence ofcoupled vibration e present study shows that the vortex-induced resonance can be separated from galloping toproduce uncoupled vibrations if the critical wind speedratio between the galloping and the vortex-induced reso-nance ranges from 32 to 97 corresponding to the Scrutonnumber from 268 to 825 In the states of complete sep-aration and semiseparation the galloping response can bedivided into two types divergent vibration and limit cyclelimited vibration at a single wind speed point Amongthem the slope values of 0025ndash0031 of the gallopingresponse dimensionless wind amplitude obtained underthe action of high wind speed (UfD gt 15) in the limit cyclevibration state are significantly lower than those in thecoupling state of 0047ndash0051 is is possibly attributableto the disappearance of the coupled action resulting in alack of the contribution of the vortex-induced aerodynamicforce in the state of limit cycle vibration Accordingly thisweakens the energy accumulation and decreases the am-plitude response

It is worth noting that the Scruton number corre-sponding to the ldquounsteady gallopingrdquo response of the cou-pling state in the experimentally studied condition is124ndash306 and it is 268ndash825 in the uncoupled state ere isan overlapping interval of the Scruton number (268ndash306)which can be considered the ldquotransition intervalrdquo of theScruton number from the coupling state to the uncoupledstate

43 Modified Empirical Formula for Amplitude EstimationWith regard to the fact that the onset wind speed of theldquounsteady gallopingrdquo response can be roughly determined bythe Strouhal law and the fact that the dimensionless responseamplitude increases linearly with the dimensionless windspeed after vibration Niu et al [8] identify two key pa-rameters that is the onset wind speed point U0 and thelinear growth slope A numerical study and regressionanalysis were then carried out and an empirical formula foramplitude estimation was presented e research showsthat the slope of the key parameters of the empirical formulais a function of the ratio of the width to the height (BD) ofthe rectangular section bar and the relevant mathematicalexpressions are obtained through polynomial fitting basedon a large amount of measured data

Based on the measured ldquosoft vibrationrdquo response data ofthe rectangular section bar in the wind tunnel test with awidth-to-height ratio (BD) of 12 the regression analysisdatabase of the original empirical formula is supplementedbased on previous work [8] and a modified empiricalformula for amplitude estimation is proposed throughpolynomial fitting

Y

D minus 00053 middot

B

D1113874 1113875

3+ 00505 middot

B

D1113874 1113875

2minus 0147 middot

B

D+ 015061113888 1113889

middot Ur minus U0( 1113857

(6)

where YD is the dimensionless wind speed Ur U(fD) isthe dimensionless incoming wind speed U0 09St is theonset wind speed point of the ldquounsteady gallopingrdquo based onthe Strouhal number normalization of the section and BDis the width-to-height ratio of the rectangular section Itshould be noted that the empirical formula is applicable onlyin coupled vortex-induced vibration and galloping cases

It should be pointed out that the error of the empiricalformula is a little large due to limited collected data themeasured origin data is scattered which can be observed inFigure 10 and the deviation is supposed to be contracted asdata collected more densely in the future work

5 Conclusions

(1) For the present rectangular suspender whose aspectratio is 12 the vibration response is unrelated to themass and damping parameters under the couplingstate

(2) ere is a Scruton number ldquolocked intervalrdquo whichmakes the ldquounsteady gallopingrdquo response curvesrsquo


slope have no relation to the changes of single massdamping and combined Scruton number eldquolocked intervalrdquo of the present rectangular sus-pender is 124ndash306

(3) e onset wind speed ratio of the vortex-inducedresonance and galloping is determined by theScruton number e magnitude of the ratio de-termines the coupling degree of the ldquounsteady gal-lopingrdquo e results show that there is a Scrutonnumber ldquotransition intervalrdquo which causes themodel vibration to transition from the coupling stateto the uncoupled state For present rectangularsuspender it is 268ndash306

(4) A modified empirical formula for amplitude esti-mation of ldquounsteady gallopingrdquo is proposed and itcan provide a reference for relevant engineeringdesigns

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e authors gratefully acknowledge the support of theNational Natural Science Foundations of China (Project nos51708202 52078504 and 51422806) Scientific ResearchPlan of Hunan Province of China (Project no 2019RS2051)Innovation-Driven Project of Central South University (no2020CX009) and Scientific Research Plan of China StateConstruction Engineering Group (Project no CSCEC-2020-Z-43)

References

[1] T Andrianne R P Aryoputro P Laurent and G ColsonldquoEnergy harvesting from different aeroelastic instabilities of asquare cylinderrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 172 pp 164ndash169 2018

[2] S Cammelli A Bagnara and J G Navarro ldquoVIV-gallopinginteraction of the deck of a footbridge with solid parapetsrdquoProcedia Engineering vol 199 pp 1290ndash1295 2017

[3] G Gao and L Zhu ldquoMeasurement and verification of un-steady galloping force on a rectangular 21 cylinderrdquo Journalof Wind Engineering and Industrial Aerodynamics vol 157pp 76ndash94 2016

[4] F Petrini and K Gkoumas ldquoPiezoelectric energy harvestingfrom vortex shedding and galloping induced vibrations insideHVAC ductsrdquo Energy and Buildings vol 158 pp 371ndash3832018

[5] G Chen Y Yang Y Yang and Li Peng ldquoStudy on gallopingoscillation of iced catenary system under cross windsrdquo Shockand Vibration vol 2017 no 2 pp 1ndash16 Article ID 16342922017

[6] W Sun S Jo and J Seok ldquoDevelopment of the optimal bluffbody for wind energy harvesting using the synergetic effect ofcoupled vortex induced vibration and galloping phenomenardquoInternational Journal of Mechanical Sciences vol 156pp 435ndash445 2019

[7] A Mohamed S A Zaki M Shirakashi M S Mat Ali andM Z Samsudin ldquoExperimental investigation on vortex-in-duced vibration and galloping of rectangular cylinders ofvarying side ratios with a downstream square platerdquo Journal ofWind Engineering and Industrial Aerodynamics vol 211Article ID 104563 2021

[8] H Niu S Zhou Z Chen and X Hua ldquoAn empirical modelfor amplitude prediction on VIV-galloping instability ofrectangular cylindersrdquo Wind and Structures vol 21 no 1pp 85ndash103 2015

[9] EN1991-1-4 Eurocode 1-actions on Structures Parts 1ndash4General Actions-Wind Actions London UK 2010

[10] C Mannini A M Marra and G Bartoli ldquoVIV-gallopinginstability of rectangular cylinders review and new experi-mentsrdquo Journal of Wind Engineering and Industrial Aero-dynamics vol 132 pp 109ndash124 2014

[11] C Mannini A M Marra T Massai and G Bartoli ldquoVIV andgalloping interaction for a 32 rectangular cylinderrdquo in Pro-ceedings of the Sixth European and African Conference onWind Engineering Cambridge UK July 2013

[12] C Mannini A M Marra T Massai and G Bartoli ldquoIn-terference of vortex-induced vibration and transverse gal-loping for a rectangular cylinderrdquo Journal of Fluids andStructures vol 66 pp 403ndash423 2016

[13] C Mannini T Massai and A M Marra ldquoModeling theinterference of vortex-induced vibration and galloping for aslender rectangular prismrdquo Journal of Sound and Vibrationvol 419 pp 493ndash509 2018

[14] Z Chen K T Tse K C S Kwok A Kareem and B KimldquoMeasurement of unsteady aerodynamic force on a gallopingprism in a turbulent flow a hybrid aeroelastic-pressure bal-ancerdquo Journal of Fluids and Structures vol 102 Article ID103232 2021

[15] Z Chen Y Xu H Huang and K T Tse ldquoWind tunnelmeasurement systems for unsteady aerodynamic forces onbluff bodies review and new perspectiverdquo Sensors vol 20no 16 p 4633 2020

[16] T Sarpkaya ldquoHydrodynamic damping flow-induced oscil-lations and biharmonic responserdquo Journal of Offshore Me-chanics and Arctic Engineering vol 117 no 4 pp 232ndash2381995

[17] R N Govardhan and C H K Williamson ldquoDefining theldquomodified Griffin plotrdquo in vortex-induced vibration revealingthe effect of Reynolds number using controlled dampingrdquoJournal of Fluid Mechanics vol 561 pp 147ndash180 2006

[18] S Kumar and S Sen ldquoTransition of VIV-only motion of asquare cylinder to combined VIV and galloping at lowReynolds numbersrdquoOcean Engineering vol 187 no 3 ArticleID 106208 2019

[19] Y Tamura and G Matsui ldquoWake-oscillator model of vortex-induced oscillation of circular cylinderrdquo Journal of Wind En-gineering and Industrial Aerodynamics vol1981 pp13ndash24 1981

[20] Y Tamura and A Amano ldquoMathematical model for vortex-induced oscillations of continuous systems with circular crosssectionrdquo Journal of Wind Engineering and Industrial Aero-dynamics vol 14 no 1-3 pp 431ndash442 1983

[21] C Mannini ldquoIncorporation of turbulence in a nonlinearwake-oscillator model for the prediction of unsteady galloping


responserdquo Journal of Wind Engineering and IndustrialAerodynamics vol 200 Article ID 104141 2020

[22] G Gao L Zhu J Li and W Han ldquoModelling nonlinearaerodynamic damping during transverse aerodynamic in-stabilities for slender rectangular prisms with typical sideratiosrdquo Journal of Wind Engineering and Industrial Aerody-namics vol 197 Article ID 104064 2020

[23] W-L Chen W-H Yang F Xu and C-G Zhang ldquoComplexwake-induced vibration of aligned hangers behind tower oflong-span suspension bridgerdquo Journal of Fluids and Struc-tures vol 92 Article ID 102829 2020


generating from the synergetic effect of coupled vortex-induced vibration and galloping responses WhileMohamed et al [7] tried to suppress the coupled vibrationwith a downstream square plate by varying side ratios

Niu et al [8] established a mathematical model ofldquounsteady gallopingrdquo by combining the vortex excitationmodel of the wake oscillator with two degrees of freedom(DOF) and a quasi-stationary vibration force model ewind speed amplitude response curve was classifiedaccording to the coupling degree e numerical calculationresults were in agreement with the results obtained byidentifying the relevant aerodynamic parameters e windspeed amplitude curve of the ldquounsteady gallopingrdquo responseis determined by two parameters that is the onset windspeed U0 of the coupled vibration and the linear slope of thewind speed amplitude response curve e European windresistance design code [9] suggests that a coupled vibrationnecessitates an onset wind speed ratio between the vortexresonance and the galloping vibration in the range of 07 to15 e studies in [10] showed that the European code isunsafe and the ratio should be greater than the range from45 to 85 so that the coupled vibration can be avoided In[11] the wind-amplitude response curve morphology of theldquounsteady gallopingrdquo under different Scruton numbers wasstudied by adjusting the stiffness support form of themember model in the wind tunnel tests and changing thedamping ratio indicating that a Scruton number exceedingthe range of 50 to 60 is a necessary condition for eliminatingcoupled vibration Mannini et al and Chen et al [12ndash14]proposed a prediction model that was more consistent withthe experimental results measured by modifying the aero-dynamic parameters based on the 2-DOF Tamura wakeoscillator model Mannini et al [11] investigated the in-fluence of the wind attack angle and turbulence on theldquounsteady gallopingrdquo proposing that low turbulence (ap-proximately 3) would enhance the coupled vibration of thevortex-induced resonance and galloping

Niu et al [8] proposed an empirical formula applicable tothe amplitude estimation of the ldquounsteady gallopingrdquo ofrectangular section members Studies have shown that thelinear slope of the response curve under the coupled vibrationstate is a function of the ratio of the width of amember sectionto its height (BD) An empirical formula for amplitude es-timation was proposed by accumulating a large amount ofdata for regression analysis e studies in [15 16] indicatedthat mass and damping are key parameters that influence theamplitude of vortex-induced resonance responses and thisinfluence is not independent e degrees of influence ofsingle parameter variations (mass or damping) on amplitudeare different Hence the mass and damping parameterscannot be used to form the Scruton number to evaluate theamplitude response e use of the Scruton number as anindependent parameter was investigated in this study eauthors in [17 18] showed that differences in the Reynoldsnumber can result in different amplitude responses which isalso the reason behind the conclusions in [15 16] To evaluatethe amplitude the mass and damping parameters are usuallycombined as the Scruton number

With respect to the coupled vibration between thevortex-induced resonance and galloping vibration (ldquoun-steady gallopingrdquo) a comparative study of the parametersensitivity of the independent mass and damping parame-ters as well as the combined mass damping parameterldquoScruton numberrdquo should be conducted considering theReynolds number effect To further study the ldquounsteadygallopingrdquo response mechanism and to consolidate the re-sults of more reasonable amplitude estimation formulas thisstudy considers a typical rectangular section member modelwith a width-to-height ratio of 12 (BD 12) as a researchobject and adjusts the equivalent mass equivalent stiffnessand damping ratio of the model system in order to conduct aseries of wind tunnel tests on the same mass with differentdamping ratios the same damping ratio with different massvalues and the same Scruton number with different massand damping combinations It also measures and draws thecurve of the wind speed amplitude responses to compara-tively study the influence of the mass and dampingparameters

2 Theoretical Model

21 Wake Oscillator Model A diagram of the cylinder flowwake oscillator is shown in Figure 1 e cylinder tail fol-lowing the cylinder flow can form a nearly stable vortex thatoscillates up and down and provides an oscillator that caninteract with the vibration of the cylinder e oscillatormass is formed by the air quality and the fluid-solid coupledshear stiffness of the vortex oscillating up and down is thevibrator stiffness It can be simplified as a vortex-inducedforce mechanics model with a 2-DOF wake oscillator asshown in Figure 2 In the figures U is the incoming flowwind speed D is the diameter of the cylinder L is the halfwidth of the wake vortex H is the height of the wake vortexG is the center of gravity of the wake vortex still air and α isthe angular displacement of the up-and-down oscillation ofthe wake oscillator Furthermore I K and C are the tor-sional mass torsional stiffness and aerodynamic damping ofthe wake oscillator respectively e vibration equation ofthe established Birkhoff model is given by

I middotd2αdt

2 + C middotdαdt

+ K middot α 0 (1)

For a two-dimensional (2D) cylinder Tamura consid-ered the influence of changes in the wake vibrator length inthe vibration period He also proposed a modified Birkhoff2-DOF vortex-induced resonance mathematical model


2

C2L0

1113888 1113889α21113896 1113897 _α + ]α minus mlowast euroY minus ]S

lowast _Y

euroY + 2η +n f + CD( 1113857v

Slowast1113896 1113897 _Y + Y minus

fn]2αSlowast2

CL minus fα + Slowast _Y

v1113888 1113889

(2)





Fy minus12ρU

2D middot A1 middot

_y

U1113888 1113889 + A2 middot

_y

U1113888 1113889

2

+ A3 middot_y

U1113888 1113889

3

+ A4 middot_y

U1113888 1113889

4

+ A5 middot_y

U1113888 1113889

5

+ Δ_y

U1113888 1113889

n

⎡⎣ ⎤⎦ (3)

CFY A1 middot_y

U1113888 1113889 + A2 middot

_y

U1113888 1113889

2

+ A3 middot_y

U1113888 1113889

3

+ A4 middot_y

U1113888 1113889

4

+ A5 middot_y

U1113888 1113889

5

+ Δ_y

U1113888 1113889

n

(4)




2

C2L0

⎛⎝ ⎞⎠α2⎧⎨

⎩

⎫⎬


lowast _Y


Slowast minus

nA2Slowast

] _Yminus nA3

Slowast

v1113888 1113889

3_Y2


⎩

⎫⎬

⎭_Y + Y minus

fn]2αSlowast2


]minus A2

Slowast _Y

v1113888 1113889

2

minus A3Slowast _Y

v1113888 1113889

3


⎩

⎫⎬

⎭

(5)

U

dydt

O

G

y

L L

D

α


M

K

C

IG

y






3 Wind Tunnel Test













Rolling Heaving

Spring

Spring

Fixed Boundary

Fixed Boundary

Pitching

Spring

Spring

Fixed Boundary

Fixed Boundary


X

Y

Zo


1

2

3

5

accelerometer






0 5 10 15 20 25 30 3500

02

04

06

08

10

12Re=16000 Sc=124~241

A4 ξ=0146 m=1266A8 ξ=0146 m=1611

A12 ξ=0146 m=1795A13 ξ=0146 m=2460

YD

UfD

10

0049

(a)

A3 ξ=0200 m=1266A7 ξ=0200 m=1611

A9 ξ=0200 m=1795

00

02

04

06

08

10

12

YD

0 5 10 15 20 25 30 35UfD

Re=16000 Sc=170~241

100048

(b)




A1

827

0284 241

616 293 16000

28A2 0223 189 22A3 0200 170 20A4 0146 124 15A17 0500 425 50A5

1053

0223 241

767 293 16000

28A6 0284 306 36A7 0200 216 25A8 0146 158 19A18 0500 540 64A9

1173

0200 241

852 293 16000

28A10 0284 341 40A11 0223 268 32A12 0146 176 21A19 0500 602 71A13

1608

0146 241

1181 293 16000






0 5 10 15 20 25 30 35 40 45 50 55 60 6500

01

02

03

04

05

06

07

08

09

10

003110

0025

0027

YD

UfD

Re=16000 Sc=268~825


A11 Sc=268A19 Sc=602A18 Sc=540A17 Sc=425

10

10


A1 m=827 ξ=0284A5 m=1053 ξ=0223

A9 m=1173 ξ=0200A13 m=1608 ξ=0146

0 5 10 15 20 25 30 3500

02

04

06

08

10

12

0048

Re=16000 Sc=241

YD

UfD

10






05 10 15 20 25000001002003004005006007008

BD

Slope = -00053(BD)3 + 00505(BD)2 -0147(BD) + 01506

R2=08069

Slop

e


0 5 10 15 20 25 30 3500

02

04

06

08

10

12Re=16000 Sc=124~241

YD

UfD

A1 m=827 ξ=0284A2 m=827 ξ=0223

A3 m=827 ξ=0200A4 m=827 ξ=0146

10

0047

(a)

00

02

04

06

08

10

12

YD

A5 m=1053 ξ=0223A6 m=1053 ξ=0284

A7 m=1053 ξ=0200A8 m=1053 ξ=0146

0 5 10 15 20 25 30 35

Re=16000 Sc=158~306

UfD

10

0051

(b)










Y


B

D1113874 1113875

3+ 00505 middot

B

D1113874 1113875

2minus 0147 middot

B

D+ 015061113888 1113889


(6)



5 Conclusions







Data Availability




Acknowledgments


References






























Fy minus12ρU

2D middot A1 middot

_y

U1113888 1113889 + A2 middot

_y

U1113888 1113889

2

+ A3 middot_y

U1113888 1113889

3

+ A4 middot_y

U1113888 1113889

4

+ A5 middot_y

U1113888 1113889

5

+ Δ_y

U1113888 1113889

n

⎡⎣ ⎤⎦ (3)

CFY A1 middot_y

U1113888 1113889 + A2 middot

_y

U1113888 1113889

2

+ A3 middot_y

U1113888 1113889

3

+ A4 middot_y

U1113888 1113889

4

+ A5 middot_y

U1113888 1113889

5

+ Δ_y

U1113888 1113889

n

(4)




2

C2L0

⎛⎝ ⎞⎠α2⎧⎨

⎩

⎫⎬


lowast _Y


Slowast minus

nA2Slowast

] _Yminus nA3

Slowast

v1113888 1113889

3_Y2


⎩

⎫⎬

⎭_Y + Y minus

fn]2αSlowast2


]minus A2

Slowast _Y

v1113888 1113889

2

minus A3Slowast _Y

v1113888 1113889

3


⎩

⎫⎬

⎭

(5)

U

dydt

O

G

y

L L

D

α


M

K

C

IG

y






3 Wind Tunnel Test













Rolling Heaving

Spring

Spring

Fixed Boundary

Fixed Boundary

Pitching

Spring

Spring

Fixed Boundary

Fixed Boundary


X

Y

Zo


1

2

3

5

accelerometer






0 5 10 15 20 25 30 3500

02

04

06

08

10

12Re=16000 Sc=124~241

A4 ξ=0146 m=1266A8 ξ=0146 m=1611

A12 ξ=0146 m=1795A13 ξ=0146 m=2460

YD

UfD

10

0049

(a)

A3 ξ=0200 m=1266A7 ξ=0200 m=1611

A9 ξ=0200 m=1795

00

02

04

06

08

10

12

YD

0 5 10 15 20 25 30 35UfD

Re=16000 Sc=170~241

100048

(b)




A1

827

0284 241

616 293 16000

28A2 0223 189 22A3 0200 170 20A4 0146 124 15A17 0500 425 50A5

1053

0223 241

767 293 16000

28A6 0284 306 36A7 0200 216 25A8 0146 158 19A18 0500 540 64A9

1173

0200 241

852 293 16000

28A10 0284 341 40A11 0223 268 32A12 0146 176 21A19 0500 602 71A13

1608

0146 241

1181 293 16000






0 5 10 15 20 25 30 35 40 45 50 55 60 6500

01

02

03

04

05

06

07

08

09

10

003110

0025

0027

YD

UfD

Re=16000 Sc=268~825


A11 Sc=268A19 Sc=602A18 Sc=540A17 Sc=425

10

10


A1 m=827 ξ=0284A5 m=1053 ξ=0223

A9 m=1173 ξ=0200A13 m=1608 ξ=0146

0 5 10 15 20 25 30 3500

02

04

06

08

10

12

0048

Re=16000 Sc=241

YD

UfD

10






05 10 15 20 25000001002003004005006007008

BD

Slope = -00053(BD)3 + 00505(BD)2 -0147(BD) + 01506

R2=08069

Slop

e


0 5 10 15 20 25 30 3500

02

04

06

08

10

12Re=16000 Sc=124~241

YD

UfD

A1 m=827 ξ=0284A2 m=827 ξ=0223

A3 m=827 ξ=0200A4 m=827 ξ=0146

10

0047

(a)

00

02

04

06

08

10

12

YD

A5 m=1053 ξ=0223A6 m=1053 ξ=0284

A7 m=1053 ξ=0200A8 m=1053 ξ=0146

0 5 10 15 20 25 30 35

Re=16000 Sc=158~306

UfD

10

0051

(b)










Y


B

D1113874 1113875

3+ 00505 middot

B

D1113874 1113875

2minus 0147 middot

B

D+ 015061113888 1113889


(6)



5 Conclusions







Data Availability




Acknowledgments


References






























3 Wind Tunnel Test













Rolling Heaving

Spring

Spring

Fixed Boundary

Fixed Boundary

Pitching

Spring

Spring

Fixed Boundary

Fixed Boundary


X

Y

Zo


1

2

3

5

accelerometer






0 5 10 15 20 25 30 3500

02

04

06

08

10

12Re=16000 Sc=124~241

A4 ξ=0146 m=1266A8 ξ=0146 m=1611

A12 ξ=0146 m=1795A13 ξ=0146 m=2460

YD

UfD

10

0049

(a)

A3 ξ=0200 m=1266A7 ξ=0200 m=1611

A9 ξ=0200 m=1795

00

02

04

06

08

10

12

YD

0 5 10 15 20 25 30 35UfD

Re=16000 Sc=170~241

100048

(b)




A1

827

0284 241

616 293 16000

28A2 0223 189 22A3 0200 170 20A4 0146 124 15A17 0500 425 50A5

1053

0223 241

767 293 16000

28A6 0284 306 36A7 0200 216 25A8 0146 158 19A18 0500 540 64A9

1173

0200 241

852 293 16000

28A10 0284 341 40A11 0223 268 32A12 0146 176 21A19 0500 602 71A13

1608

0146 241

1181 293 16000






0 5 10 15 20 25 30 35 40 45 50 55 60 6500

01

02

03

04

05

06

07

08

09

10

003110

0025

0027

YD

UfD

Re=16000 Sc=268~825


A11 Sc=268A19 Sc=602A18 Sc=540A17 Sc=425

10

10


A1 m=827 ξ=0284A5 m=1053 ξ=0223

A9 m=1173 ξ=0200A13 m=1608 ξ=0146

0 5 10 15 20 25 30 3500

02

04

06

08

10

12

0048

Re=16000 Sc=241

YD

UfD

10






05 10 15 20 25000001002003004005006007008

BD

Slope = -00053(BD)3 + 00505(BD)2 -0147(BD) + 01506

R2=08069

Slop

e


0 5 10 15 20 25 30 3500

02

04

06

08

10

12Re=16000 Sc=124~241

YD

UfD

A1 m=827 ξ=0284A2 m=827 ξ=0223

A3 m=827 ξ=0200A4 m=827 ξ=0146

10

0047

(a)

00

02

04

06

08

10

12

YD

A5 m=1053 ξ=0223A6 m=1053 ξ=0284

A7 m=1053 ξ=0200A8 m=1053 ξ=0146

0 5 10 15 20 25 30 35

Re=16000 Sc=158~306

UfD

10

0051

(b)










Y


B

D1113874 1113875

3+ 00505 middot

B

D1113874 1113875

2minus 0147 middot

B

D+ 015061113888 1113889


(6)



5 Conclusions







Data Availability




Acknowledgments


References


































Rolling Heaving

Spring

Spring

Fixed Boundary

Fixed Boundary

Pitching

Spring

Spring

Fixed Boundary

Fixed Boundary


X

Y

Zo


1

2

3

5

accelerometer






0 5 10 15 20 25 30 3500

02

04

06

08

10

12Re=16000 Sc=124~241

A4 ξ=0146 m=1266A8 ξ=0146 m=1611

A12 ξ=0146 m=1795A13 ξ=0146 m=2460

YD

UfD

10

0049

(a)

A3 ξ=0200 m=1266A7 ξ=0200 m=1611

A9 ξ=0200 m=1795

00

02

04

06

08

10

12

YD

0 5 10 15 20 25 30 35UfD

Re=16000 Sc=170~241

100048

(b)




A1

827

0284 241

616 293 16000

28A2 0223 189 22A3 0200 170 20A4 0146 124 15A17 0500 425 50A5

1053

0223 241

767 293 16000

28A6 0284 306 36A7 0200 216 25A8 0146 158 19A18 0500 540 64A9

1173

0200 241

852 293 16000

28A10 0284 341 40A11 0223 268 32A12 0146 176 21A19 0500 602 71A13

1608

0146 241

1181 293 16000






0 5 10 15 20 25 30 35 40 45 50 55 60 6500

01

02

03

04

05

06

07

08

09

10

003110

0025

0027

YD

UfD

Re=16000 Sc=268~825


A11 Sc=268A19 Sc=602A18 Sc=540A17 Sc=425

10

10


A1 m=827 ξ=0284A5 m=1053 ξ=0223

A9 m=1173 ξ=0200A13 m=1608 ξ=0146

0 5 10 15 20 25 30 3500

02

04

06

08

10

12

0048

Re=16000 Sc=241

YD

UfD

10






05 10 15 20 25000001002003004005006007008

BD

Slope = -00053(BD)3 + 00505(BD)2 -0147(BD) + 01506

R2=08069

Slop

e


0 5 10 15 20 25 30 3500

02

04

06

08

10

12Re=16000 Sc=124~241

YD

UfD

A1 m=827 ξ=0284A2 m=827 ξ=0223

A3 m=827 ξ=0200A4 m=827 ξ=0146

10

0047

(a)

00

02

04

06

08

10

12

YD

A5 m=1053 ξ=0223A6 m=1053 ξ=0284

A7 m=1053 ξ=0200A8 m=1053 ξ=0146

0 5 10 15 20 25 30 35

Re=16000 Sc=158~306

UfD

10

0051

(b)










Y


B

D1113874 1113875

3+ 00505 middot

B

D1113874 1113875

2minus 0147 middot

B

D+ 015061113888 1113889


(6)



5 Conclusions







Data Availability




Acknowledgments


References






























0 5 10 15 20 25 30 3500

02

04

06

08

10

12Re=16000 Sc=124~241

A4 ξ=0146 m=1266A8 ξ=0146 m=1611

A12 ξ=0146 m=1795A13 ξ=0146 m=2460

YD

UfD

10

0049

(a)

A3 ξ=0200 m=1266A7 ξ=0200 m=1611

A9 ξ=0200 m=1795

00

02

04

06

08

10

12

YD

0 5 10 15 20 25 30 35UfD

Re=16000 Sc=170~241

100048

(b)




A1

827

0284 241

616 293 16000

28A2 0223 189 22A3 0200 170 20A4 0146 124 15A17 0500 425 50A5

1053

0223 241

767 293 16000

28A6 0284 306 36A7 0200 216 25A8 0146 158 19A18 0500 540 64A9

1173

0200 241

852 293 16000

28A10 0284 341 40A11 0223 268 32A12 0146 176 21A19 0500 602 71A13

1608

0146 241

1181 293 16000






0 5 10 15 20 25 30 35 40 45 50 55 60 6500

01

02

03

04

05

06

07

08

09

10

003110

0025

0027

YD

UfD

Re=16000 Sc=268~825


A11 Sc=268A19 Sc=602A18 Sc=540A17 Sc=425

10

10


A1 m=827 ξ=0284A5 m=1053 ξ=0223

A9 m=1173 ξ=0200A13 m=1608 ξ=0146

0 5 10 15 20 25 30 3500

02

04

06

08

10

12

0048

Re=16000 Sc=241

YD

UfD

10






05 10 15 20 25000001002003004005006007008

BD

Slope = -00053(BD)3 + 00505(BD)2 -0147(BD) + 01506

R2=08069

Slop

e


0 5 10 15 20 25 30 3500

02

04

06

08

10

12Re=16000 Sc=124~241

YD

UfD

A1 m=827 ξ=0284A2 m=827 ξ=0223

A3 m=827 ξ=0200A4 m=827 ξ=0146

10

0047

(a)

00

02

04

06

08

10

12

YD

A5 m=1053 ξ=0223A6 m=1053 ξ=0284

A7 m=1053 ξ=0200A8 m=1053 ξ=0146

0 5 10 15 20 25 30 35

Re=16000 Sc=158~306

UfD

10

0051

(b)










Y


B

D1113874 1113875

3+ 00505 middot

B

D1113874 1113875

2minus 0147 middot

B

D+ 015061113888 1113889


(6)



5 Conclusions







Data Availability




Acknowledgments


References






























0 5 10 15 20 25 30 35 40 45 50 55 60 6500

01

02

03

04

05

06

07

08

09

10

003110

0025

0027

YD

UfD

Re=16000 Sc=268~825


A11 Sc=268A19 Sc=602A18 Sc=540A17 Sc=425

10

10


A1 m=827 ξ=0284A5 m=1053 ξ=0223

A9 m=1173 ξ=0200A13 m=1608 ξ=0146

0 5 10 15 20 25 30 3500

02

04

06

08

10

12

0048

Re=16000 Sc=241

YD

UfD

10






05 10 15 20 25000001002003004005006007008

BD

Slope = -00053(BD)3 + 00505(BD)2 -0147(BD) + 01506

R2=08069

Slop

e


0 5 10 15 20 25 30 3500

02

04

06

08

10

12Re=16000 Sc=124~241

YD

UfD

A1 m=827 ξ=0284A2 m=827 ξ=0223

A3 m=827 ξ=0200A4 m=827 ξ=0146

10

0047

(a)

00

02

04

06

08

10

12

YD

A5 m=1053 ξ=0223A6 m=1053 ξ=0284

A7 m=1053 ξ=0200A8 m=1053 ξ=0146

0 5 10 15 20 25 30 35

Re=16000 Sc=158~306

UfD

10

0051

(b)










Y


B

D1113874 1113875

3+ 00505 middot

B

D1113874 1113875

2minus 0147 middot

B

D+ 015061113888 1113889


(6)



5 Conclusions







Data Availability




Acknowledgments


References






























05 10 15 20 25000001002003004005006007008

BD

Slope = -00053(BD)3 + 00505(BD)2 -0147(BD) + 01506

R2=08069

Slop

e


0 5 10 15 20 25 30 3500

02

04

06

08

10

12Re=16000 Sc=124~241

YD

UfD

A1 m=827 ξ=0284A2 m=827 ξ=0223

A3 m=827 ξ=0200A4 m=827 ξ=0146

10

0047

(a)

00

02

04

06

08

10

12

YD

A5 m=1053 ξ=0223A6 m=1053 ξ=0284

A7 m=1053 ξ=0200A8 m=1053 ξ=0146

0 5 10 15 20 25 30 35

Re=16000 Sc=158~306

UfD

10

0051

(b)










Y


B

D1113874 1113875

3+ 00505 middot

B

D1113874 1113875

2minus 0147 middot

B

D+ 015061113888 1113889


(6)



5 Conclusions







Data Availability




Acknowledgments


References


































Y


B

D1113874 1113875

3+ 00505 middot

B

D1113874 1113875

2minus 0147 middot

B

D+ 015061113888 1113889


(6)



5 Conclusions







Data Availability




Acknowledgments


References






























Data Availability




Acknowledgments


References































Documents

Investigation of the Scruton Number Effects of Wind