Mechanism of Wind & Rain/Wind Induced Cable Vibrationscontribute.modot.mo.gov/csb/documents/07-TheGenerationMechanism… · Mechanism of Wind & Rain/Wind Induced Cable Vibrations

Wind Induced Vibration of Cable Stay Bridges Workshop, April 25-27, 2006

Mechanism of Wind & Rain/Wind Induced Cable Vibrations

The Generation Mechanism of Inclined Cable Aerodynamics-Rain-Wind Induced Vibration and

Dry-State GallopingMasaru MATSUMOTO

Kyoto UniversityKyoto, JAPAN

Inclined Cable aerodynamicsHigh Speed Vortex Induced Vibration

Velocity-Restricted ResponseGalloping Instability

Divergent ResponseVelocity Restricted Response

?

Cable Attitude to Wind

wind

wind

β*=arcsin(cosα�sinβ)β :yaw angleβ* :effective yaw angle

α :vertical angle

β

90°-β*

γ =90°-arccos(sinα�sinβ/(sin2α+cos2α�cos2β)1/2

vibrating plane

cable in-planecable vertical axis�(cable in-plane)

vibration plane

horizontal axis

front stagnation point

normal to cable in-plane

γ

γ

γ

γα

90°-β

Inclined Cable Inclined Cable AerodynamicsAerodynamics

The substantial factors

The upper water rivulet formation on a cable surfaceThe axial flow in a near wakeCritical Reynolds number

Cable Aerodynamics in related to KarmanVortex

Karman Vortex

The Karman Vortex Street (KV)

Theodore von Karman,

1881~19631)/(sinh =ahπ

Stable Condition of Vortex Street in Potential Flow, [Theodore von Karman, 1911.]

Karman Vortex�(KV)

Karman Vortex�(KV)

Karman vortex (KV)

Karman vortex can be sensitively influenced by

various factors.

Lock-in Phenomena? (KV)

At near resonance velocity to Karmanvortex, the large amplitude response might be excited by motion-induced vortex, but not by Karman vortex. (Lanivier, Zaso etc.)

Splitter Plate with suitable OR (KV)

Cable model

Splitter plate

Wind

D=50mm L=700mm

Opening ratio(OR) of Splitter plate was changed to suitably control the Karman vortex

Change of Opening Ratio(OR) of Splitter Plate (KV)

OR:90%OR:40%OR:10%

OR is changed from 0% to 100%

Fluctuating Lift Force v.s. OR (KV)

0 10 20 30 40 50 60 70 80 90 1000

0.05

0.1

0.15

0.2

Open ratio of splitter plate [%]

Fluc

tuat

ing

win

d fo

rce

coef

ficie

nt C

L′

CL’

β=0˚, U=6.0m/s

The intensity of Karman Vortex becomes significantly weak if OR decreases.

0 10 20 30 40 50 60 70 80 90 1000

510

1520

2530

3540

45

0

0.005

0.01

Open ratio of splitter plate [%]Frequency [Hz]

P.S.

D. [

(N/m

)2 /Hz]

PSD of fluctuating lift force

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/D 2A [mm]

U [m/s]U/fD

f=3.561Hzm=1.477kg/mδ(2A=10mm)=0.00242Sc(2A=10mm)=2.4337

0 10 20 30 40 50 60 70

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/D 2A [mm]

U [m/s]U/fD


0 10 20 30 40 50 60 70

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/D 2A [mm]

U [m/s]U/fD


0 10 20 30 40 50 60 70

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/D 2A [mm]

U [m/s]U/fD


0 10 20 30 40 50 60 70

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/D 2A [mm]

U [m/s]U/fD


0 10 20 30 40 50 60 70

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/D 2A [mm]

U [m/s]U/fD


0 10 20 30 40 50 60 70

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/D 2A [mm]

U [m/s]U/fD


0 10 20 30 40 50 60 70

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/D 2A [mm]

U [m/s]U/fD


0 10 20 30 40 50 60 70

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/D 2A [mm]

U [m/s]U/fD


0 10 20 30 40 50 60 70

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/D 2A [mm]

U [m/s]U/fD


0 10 20 30 40 50 60 70

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/D 2A [mm]

U [m/s]U/fD


0 10 20 30 40 50 60 70

0

0.5

1

1.5

Free Vibration Test (KV)

0%10%20%30%40%50%60%70%80%90%100%OR

Enlarged maximum amplitude

Vortex induced vib. terminates at near U/fD=15and another steady response appears

β=0˚

External Stimulation v.s. KarmanVortex (KV)

Longitudinally Fluctuating Stimulation(LFS) (KV)

In-phase stimulation at up and down sides separation points

0 5 10 15 20 25 30 35 40 45 500 5 10 15 20 25 30 35

40 45 50

0

0.5

1

Pulsator Frequency [Hz]Frequency [Hz]

P.S.

D. [

(m/s

)2 /Hz]

Due to Karman vortex

0 5 10 15 20 25 30 35 40 45 500 5 10 15 20

25 30 35 40 45 50

0

0.01

0.02

0.03


P.S.

D. [

(m/s

)2 /Hz]

0 5 10 15 20 25 30 35 40 45 500 5 10 15 20

25 30 35 40 45 50

0

0.02

0.04


P.S.

D. [

(m/s

)2 /Hz]

PSD of Velocity in a Wake (by LFS) (KV)

OR10% OR70% OR100%

β=0˚, U=6.0m/s, measured point�1.5D downr stream, 1.0D beneath

weak intensiveIntensity of Karman vortex

No appearance of stimulating frequency peak

Vertically Sinusoidal Stimulation(VSS) (KV)

Out-phase stimulation at up and down separation points

123456789100%

Wavelet Analysis of Fluctuating Lift Force(by VSS) (KV)

Wing frequency�3HzU=6.0m/s

OR

Wing frequency

Karman vortex frequency

β=0˚

WindWind

RainRain

Water rivuletWater rivulet

Axial flowAxial flow

Water rivulet and axial flowWater rivulet and axial flow

Inclined Cable Inclined Cable AerodynamicsAerodynamics

The substantial factors

The upper water rivulet formation on a cable surfaceThe axial flow in a near wakeCritical Reynolds number

Axial Flow

Visualized axial flow by light strings for a proto-type cable

(AX)

wind

Axial flow velocity -approaching wind velocity diagram of a proto-type stay-cable

(AX)

((ββ*=40*=40°° --5050°°, where , where ββ*: equivalent yawing angle)*: equivalent yawing angle)

0 2 4 6 80

2

4

6

8

Mean wind velocity [m/s]

Va

[m/s

]

Visualized Flow Field around an Visualized Flow Field around an Inclined Cable Inclined Cable (AX)(AX)

Visualized axial flow by flags in a wake

of yawed cable (β=45°, V=1m/s)

Visualized intermittently Karman vortices

and axial vortices by fluid paraffinof yawed cable (β=45°, V=1m/s)

Wind tunnel tests (AX)

Top view of wind tunnel

Wind

Wind tunnel walls

β

α =0º

wind tunnel testswind tunnel tests((αα=0=0°°, , ββ=45=45°°) )

Typical Response (β=45°, in smooth flow, without water rivulet) (AX)

0 1 2 3 4 5 6 7 8 9 1011121314 0

20

40

60

80

1002A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1

1.5f=2.096Hzm=0.509kg/mδ (2A=10mm)=0.00373Sc(2A=10mm)=1.138

- 0.45

- 0.4

- 0.35

- 0.3

- 0.25

- 0.2

- 0.15

- 0.1

- 0.05

0

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

V/ fD

2A/D

V-A diagram V-A-δ diagram

Galloping Latent vortex

Experimental set up (three end conditions)Experimental set up (three end conditions)(without rivulet) (yawing angle=45deg) (without rivulet) (yawing angle=45deg)

(AX)(AX)

three end conditions1.without wall2.without wall and with end plates3.with walls installed windows (φ=170mm)

wind tunnel(without wall)

wind

walls and windows

Karman vortex v.s. low frequency component of lift and fluctuating velocity

in wake (AX)

If Karman vortex would be mitigated, low frequency lift/velocity appears.

P.S.D. and wavelet analysis of P.S.D. and wavelet analysis of the the unsteady lift forceunsteady lift force on stationary cable on stationary cable

model (without rivulet) model (without rivulet) (AX)(AX)

0 5 10 15 20 250

0.5

1

1.5

2 x 10- 3

Frequency [Hz]

P.S.

D. [

N2 /Hz]

40 20 10 8 7 6 5 4V/ fD

β=0°, in smooth flow, V=5.0m/s

0 4 8 12 16 200

0.5

1 x 10- 4

Frequency [Hz]

P.S.

D. [

N2 /Hz]

40 20 10 8 7 5 4V/ fD


Wavelet analysis ofWavelet analysis of fluctuating wind velocityfluctuating wind velocityin the wake of stationary yawed cable model in the wake of stationary yawed cable model

(AX)(AX)

(without rivulet, β=45°, V=4m/s, in smooth flow)

Axial Flow v.s. Intensity of Karman Vortex

Axial flow velocityAxial flow velocity (Va) in a wake (Va) in a wake (AX)(AX)

0 4 8 12 16 20 240

0.2

0.4

0.6

0.8

1

1.2

X/ D

Va/V Wind

X

Y

Hot wire

(1) without wall (2) without wall and with end plates

(3) with walls installedwindows (φ=170mm)

((VV=8m/s, =8m/s, ββ=45=45°°))

Definition of coordinates, X, Y, Z Definition of coordinates, X, Y, Z (AX)(AX)

Cable model

Wind

Y

Hot wire elementX

Cable model

Z

Wind

Wavelet analysis of Wavelet analysis of fluctuating wind velocityfluctuating wind velocitymeasured at various points in a wake measured at various points in a wake (AX)(AX)

X/D=7.7

X/D=12.7

X/D=17.6

(1) without wall (2) without wall and with end plates

(3) with walls installedwindows (φ=170mm)

((ββ=45=45°, VV=8.0m/s, =8.0m/s, Y/DY/D=1.8, =1.8, Z/DZ/D=0.5)=0.5)


(AX)(AX)



wind

walls and windows

Time history and the wavelet analysis of the unsteady Time history and the wavelet analysis of the unsteady crosscross--flow displacement (flow displacement (unsteady gallopingunsteady galloping))

0 10 20 30 40 50-0.01

-0.005

0

0.005

0.01

Time [sec]

Dis

plac

emen

t [m

]

18 18.2 18.4 18.6 18.8 19-4

-2

0

2

4 x 10-3

Time [sec]

Dis

plac

emen

t [m

]

40 40.2 40.4 40.6 40.8 41-4

-2

0

2

4 x 10-3

Time [sec]

Dis

plac

emen

t [m

]

free cablefree cable--end condition (without end condition (without wall, without endwall, without end--plates)plates)

((VV=5.0m/s, =5.0m/s, ff00=2.87Hz, =2.87Hz, ffkk=8.6Hz)=8.6Hz)0 10 20 30 40 50-0.01

-0.005

0

0.005

0.01

Time [s]

Dis

plac

emen

t [m

]

Intensity of Karman vortex shedding (AX)

� The most intensive: with end plates and without walls (no galloping appears)

� Weak: without end plates without walls (unstable galloping appears)

� The weakest : with walls installed windows (galloping appears)

Upper water rivulet formation

Rain Effect (WR)Water rivulet formation

The effect of rivulet on response and stationary force (its location)

Pa+(σ /R)=P0

Pa:the wind pressure on rivulet surface

σ :the surface tensile force of water rivulet

R :the local curvature radius of water rivulet

P0 :the hydro-pressure of water rivulet

The formation of the upper water rivulet and its location on an inclined surface can be determined by the following formula

The Relation of the Position of Upper Water Rivulet and its Response (WR)

θl,θw

θlθwθs

Windward EdgeHorizontal Line

Stagnation point

Leeward Edge

Rivulet

Artificial rivulet (WR)

wind

Cable diam.:50mm

Artificial rivulet

cable length:1100mm

Width:3.6mm��thickness:1.6mm

The detail stationary lift and dragThe detail stationary lift and dragforce coefficient subject to rivulet position force coefficient subject to rivulet position

(WR)(WR)

-0.3-0.2-0.1

00.10.20.30.40.5

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180upper rivulet position [deg.] (θ )

C L

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

upper rivulet position [deg.] (θ )

CD

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180


C L0

0.10.20.30.40.50.60.70.8

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

upper rivulet position [deg.] (θ )C

D

for nonfor non--yawed yawed circular cylindercircular cylinder((ββ=0=0°°))

for yawed for yawed circular cylindercircular cylinder((ββ=45=45°°))

Rivulet position effect on unsteady lift force and Strouhal number induced by Karman vortex

shedding (WR)

0

0.050.1

0.150.2

0.25

0.30.35

0.4

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180


L f (L

ift fo

rce

ampl

itude

) [N

]

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180


L f (L

ift fo

rce

ampl

itude

) [N

]



0.1

0.12

0.14

0.16

0.18

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180upper rivulet position [deg.]

S t0.15

0.17

0.19

0.21

0.23

0.25

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

upper rivulet position [deg.]

S t

Aerostatic/fluctuating force v.s. rivulet position �β=0˚�(WR)

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

Position of water rivulet (deg.)

Stea

dy w

ind

forc

e co

effic

ient

CL ′

CL CD CL’

0 20 40 60 80 100 120 140 160 180-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8


Stea

dy w

ind

forc

e co

effic

ient

CL

0 20 40 60 80 100 120 140 160 1800

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Stea

dy w

ind

forc

e co

effic

ient

CD

At near 50deg., Karman vortex becomes weak,then drag decreases and stationary lift appears.

Scanlan’s Flutter Derivatives(R. H. Scanlan,

1974)( )

⎭⎬⎫

⎩⎨⎧

+++=b

HkHkVbkH

VkHVbL ηφφηρ *

42*

32*

2*

122

21 &&

⎭⎬⎫

⎩⎨⎧

+++=b

AkAkVbkA

VkAVbM ηφφηρ *

42*

32*

2*

122 )2(

21 &&

2DOF

L/M : the unsteady lift force/pitching momentper unit span length

η/φ : the heaving/torsional displacementsρ : the air densityb : a half chord lengthHi*, Ai* (i=1-4) : the flutter derivativesk : the reduced frequency (k=bω/V)ω : flutter circular frequency

1DOF η, φ motion

2DOF η, φ coupled motion

superposition

1914~2001Scanlan

-150

-100

-50

0

50

100

0 20 40 60 80 100

V/fD

y

ƒ Æ=54° ƒ Æ=56° ƒ Æ=58°

ƒ Æ=60° ƒ Æ=70° ƒ Æ=90°

Aerodynamic derivative H1*�β=0˚�

at sub-cr. Re number (WR)

-150

-100

-50

0

50

100

0 20 40 60 80 100

V/fD

y

without wivulet ƒ Æ=40° ƒ Æ=46°ƒ Æ=48° ƒ Æ=50° ƒ Æ=52°

unstableunstable

Aerody. unstable rivulet position

HH11**

U/fD U/fD

P.S.D. of the P.S.D. of the fluctuating lift forcefluctuating lift force on on stationary cable model with artificial stationary cable model with artificial

upper rivulet upper rivulet (WR)(WR)

0 20 40 60 80 100 120 140 160 180

010

2030

4050

0

0.02

0.04

without rivulet

Rivulet position θ [deg.]Frequency [Hz]

P.S.

D. [

(m/s

)2 /Hz]

0 20 40 60 80 100 120 140 160 180

0

10

20

30

40

0

1

2x 10-3

Rivulet position θ [deg.]

without rivulet

Frequency [Hz]

P.S.

D. [

(m/s

)2 /Hz]



Rivulet positionRivulet position effect on velocityeffect on velocity--amplitude amplitude diagrams of nondiagrams of non--yawed cable model (yawed cable model (ββ=0=0°°) )

(WR)(WR)

0 1 2 3 4 5 6 7 8 9 10 11 12 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1

1.5f=1.734Hzm=0.823kg/mδ(2A=10mm)=0.003745Sc(2A=10mm)=2.047

0 1 2 3 4 5 6 7 8 9 10 11 12 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1

1.5

f=1.734Hzm=0.823kg/mδ 2A=10mm)=0.003745Sc(2A=10mm)=2.047

0 1 2 3 4 5 6 7 8 9 10 11 12 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1

1.5f=1.734Hzm=0.823kg/mδ 2A=10mm)=0.003745Sc(2A=10mm)=2.047

θ=47°

θ=54°

θ=58°

0 20 40 60 80 100 120 140 160 180

010

2030

4050

0

0.02

0.04

without rivulet


P.S.

D. [

(m/s

)2 /Hz]

P.S.D. of the fluctuating lift force on stationary cable model

θ=56°

θ=66°

θ=94°

0 1 2 3 4 5 6 7 8 9 1011121314 0

20

40

60

80

1002A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1

1.5

f=2.089Hzm=0.588kg/mδ (2A=10mm)=0.00336Sc(2A=10mm)=1.178

0 1 2 3 4 5 6 7 8 9 1011121314 0

20

40

60

80

1002A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1

1.5


0 1 2 3 4 5 6 7 8 9 1011121314 0

20

40

60

80

1002A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1



(WR)(WR)

0 20 40 60 80 100 120 140 160 180

0

10

20

30

40

0

1

2x 10-3

Rivulet position θ [deg.]

without rivulet

Frequency [Hz]

P.S.

D. [

(m/s

)2 /Hz]


CFD analysis (3D LES) (WR)

Flow field and stationary and unsteady forces of non-yawed cable with upper rivulet

2000.0021000020D×20D×0.2D189×131×3140°

2000.0021000020D×20D×0.2D189×131×3110°

2000.0021000020D×20D×0.2D189×131×390°

2000.0021000020D×20D×0.2D189×131×380°

2000.0021000020D×20D×0.2D189×131×370°

2000.0021000020D×20D×0.2D189×131×366°

2000.0021000020D×20D×0.2D189×131×360°

2000.0021000020D×20D×0.2D189×131×356°

2000.0021000020D×20D×0.2D189×131×352°

2000.0021000020D×20D×0.2D189×131×350°

2000.0021000020D×20D×0.2D189×131×346°

2000.0021000020D×20D×0.2D189×131×340°

2000.0021000020D×20D×0.2D189×131×3no rivulet

Total timeTime stepReDomain Grid Rivulet position

Computational cases of large eddy simulation of flow around a cable model (WR)

�

Entire domain��Grid around cable modelComputational grid (WR)

Strouhal number

0

0.05

0.1

0.15

0.2

0.25

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

Rivulet position

St

Drag force coefficient

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

Rivulet position

Cd

Lift force coefficient

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

Rivulet position

Cl

Amplitude of lift force

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

Rivulet position

Amp. of L

Strouhal number (by fluctuating lift force,■: no rivulet) (WR)

Drag force and lift force coefficient(■: no rivulet) (WR)

Amplitude of fluctuating lift force (■: no rivulet) (WR)

0T/4 T/4 2T/4 3T/4 4T/4

L

t Fluctuating lift force about 0T/4

�

about T/4 about 2T/4

about 3T/4 about 4T/4Pressure contour around cable during one period of fluctuating lift force (no rivulet) (WR)

0T/4 T/4 2T/4 3T/4 4T/4

L

t

Fluctuating lift force about 0T/4

�


about 3T/4 about 4T/4Velocity vector around cable during one period of fluctuating lift force (no rivulet) (WR)

Rivulet position 60 degree

0T/4 T/4 2T/4 3T/4 4T/4

L

t


�


about 3T/4 about 4T/4Velocity vector around cable during one period of fluctuating lift force (θ=60°) (WR)

0T/4 T/4 2T/4 3T/4 4T/4

L

t


�


about 3T/4 about 4T/4Pressure contour around cable during one period of fluctuating lift force (θ=66°) (WR)

0T/4 T/4 2T/4 3T/4 4T/4

L

t Fluctuating lift force about 0T/4

�


about 3T/4 about 4T/4 Velocity vector around cable during one period of fluctuating lift force (θ=66°) (WR)

High Speed Vortex Excitation

3D Karman Vortex Shedding along Cable Axis (WR)

Instantaneous water rivulet Instantaneous water rivulet distribution distribution (WR)(WR)

V=10m/s

Wind Tunnel Tests (WR)

rivuletrivulet

aa aabb

Position of artificial rivuletPosition of artificial rivulet

rigid cable modelrigid cable model((DD=0.05m)=0.05m)

windwind

rivuletrivulet0.0320.032DD0.0720.072DD

θ

Position Position and length of artificial rivuletand length of artificial rivulet

Wind tunnelWind tunnel

P.S.D. of fluctuating velocity in a wake (a:θ=65°(a/l=0.35), b:θ=55°(b/l=0.24), in

smooth flow, V=8.0m/s) (WR)

0 10 20 30 40 500

0.1

0.2

0.3

0.4

Frequency [Hz]

P.S.

D. [

(m/s

)2 /Hz]

40 20 10 8 7 6 5 4V/ fD

0 10 20 30 40 500

0.1

0.2

0.3

0.4

Frequency [Hz]

P.S.

D. [

(m/s

)2 /Hz]

40 20 10 8 7 6 5 4V/ fD

0 10 20 30 40 500

0.1

0.2

0.3

0.4

Frequency [Hz]

P.S.

D. [

(m/s

)2 /Hz]

40 20 10 8 7 6 5 4V/ fD

2)2) θ θ =55=55°°/65/65°°

1)1) θ θ =65=65°° (St:0.167)(St:0.167)

3)3) θθ =55=55°° (St:0.189)(St:0.189)

11 22 33

windwindaa aabb

windwind1.51.5DD

0.40.4DD

hot wirehot wire

P.S.D. & Wavelet analysis of fluctuating lift force

(a:θ=65°(a/l=0.35), b:θ=55°(b/l=0.24), in smooth flow, V=8.0m/s) (WR)

0 10 20 30 40 50

10- 5

100

Frequency [Hz]

P.S.

D. [

N2 /Hz]

40 20 10 8 7 5 4V/ fD

P.S.D.P.S.D. Wavelet analysisWavelet analysis

not so clearnot so clear

intensive unsteady power appears intensive unsteady power appears at high reduced velocity rangeat high reduced velocity range

V-A diagrams (uniform rivulet θ=47°, 54°, 58°, in smooth flow)

(WR)

0 1 2 3 4 5 6 7 8 9 10 11 12 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1

1.5

θ θ =54=54°°

Galloping (G)Galloping (G)


0 1 2 3 4 5 6 7 8 9 10 11 12 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1


θ θ =47=47°°

0 1 2 3 4 5 6 7 8 9 10 11 12 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1

1.5 f=1.734Hzm=0.823kg/mδ (2A=10mm)=0.003745Sc(2A=10mm)=2.047

θ θ =58=58°°

Non Non -- Galloping Galloping (NG)(NG)

Non Non -- Galloping Galloping (NG)(NG)

V-A diagrams (in smooth flow) (WR)

a:a:θθ=58=58°°(a/l=0.31), (a/l=0.31), b:b:θθ=47=47°°(b/l=0.33)(b/l=0.33)

0 1 2 3 4 5 6 7 8 9 10 11 12 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1


a:a:θθ=58=58°°(a/l=0.31),(a/l=0.31), b:b:θθ=54=54°°(b/l=0.33)(b/l=0.33)

0 1 2 3 4 5 6 7 8 9 10 11 12 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1


(NG)(NG) (NG)(NG) (NG)(NG) (G)(G)

V-A diagrams (in smooth flow) (WR)

a:a:θθ=65=65°°(a/l=0.41), (a/l=0.41), b:b:θθ=55=55°°(a/l=0.12)(a/l=0.12)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/ D2A [mm]

V [m/ s]

V/ fD0 20 40 60 80

0

0.5

1


a:a:θθ=50=50°°(a/l=0.31), (a/l=0.31), b:b:θθ=55=55°°(b/l=0.33)(b/l=0.33)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80

0

0.5

1


a:a:θθ=65=65°°(a/l=0.36), (a/l=0.36), b:b:θθ=55=55°°(b/l=0. 24)(b/l=0. 24) a:a:θθ=65=65°°(a/l=0.31), (a/l=0.31), b:b:θθ=55=55°°(b/l=0.33)(b/l=0.33)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80

0

0.5

1


0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80

0

0.5

1

1.5 f=2.874Hzm=1.784/mδ (2A=10mm)=0.00169Sc(2A=10mm)=2.00

V-A & V-A-δ diagrams (in smooth flow) (WR)

- 0.03

- 0.02

- 0.01

0

0.01

0.02

0.03

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

V/ fD

2A/D

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80

0

0.5

1

1.5 f=2.874Hzm=1.784/mδ (2A=10mm)=0.00169Sc(2A=10mm)=2.00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80

0

0.5

1


- 0.03

- 0.02

- 0.01

0

0.01

0.02

0.03

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

V/ fD

2A/D

a:a:θθ=50=50°°(a/l=0.31), (a/l=0.31), b:b:θθ=55=55°°(b/l=0.33)(b/l=0.33) a:a:θθ=65=65°°(a/l=0.36), (a/l=0.36), b:b:θθ=55=55°°(b/l=0.24)(b/l=0.24)

Movement Effect of Upper Water Rivulet

Observed water rivulet movement on prototype scale cable model during rain-wind induced vibration (MR)

The time history of cable vibration and rivulet movement (MR)

0 5 10 15 20Time [s]

Cab

le m

otio

n

0 5 10 15 20Time [s]

Riv

ulet

mot

ion

0 5 10 15 20Time [s]

Cab

le m

otio

n

0 5 10 15 20Time [s]

Riv

ulet

mot

ion

(large cable amplitude)(large cable amplitude) (small cable amplitude)

Rivulet movement (MR)

Rivulet movement behaves non-uniformly and non-stationary along

cable axis even stationary cross-flow vibration.

Rivulet movement andcable vibration (MR)

When Karman vortex intensively sheds, cable vibration is mitigated.

Rivulet tends to move in synchronization to cable motion, when Karman vortex shedding becomes weak.

Rivulet movement is controlled by Karman vortex shedding when it becomes intensive.

Simulation of rivulet movement during cross flow vibration (MR)

θ (t) rivulet movement

(actual state)

wind

η cross flow vibration

=

θ (t)=θ0+φ

φ rotational motion

+

(assumed state)

θ0

η cross flow motion

wind wind

The unsteady lift forces can be expressed by use of aerodynamic derivatives (MR)

( ) ( ){ }

( ) ( ){ } 0cossin24

sincos24

*4

*3

*2

0

022

20

*3

*2

0

0*1

2

=⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

++−−

+⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

++−+

ηλληφωρω

ηλληφωρη

HHHDm

D

HHDHm

D&&&

η=η0sinωtφ=φ0sin(ωt+λ)=(φ0sinλ/(η0ω))dη/dt+(φ0cosλ/η0)ηR=(D/2)(φ0/η0)

where

Characteristics of vibration and rivulet motion (Cosenteno et al) (MR)

Time history of rivulet position and cable motion

rivulet positioncable motion

Rivulet position effect on velocity – damping diagrams of yawed cable (MR)

:moving water rivulet:fixed water rivulet

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0 20 40 60 80 100

V/fD

δ

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0 20 40 60 80 100

V/fD

δ-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0 20 40 60 80 100

V/fD

δ

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0 20 40 60 80 100

V/fD

δ

d) θ=72°°c) θ=66°°

a) θ=50°° b) θ=56°°

(β=45°, λ=180°, R=0.0698)

Rivulet Movement Effect (MR)

Wind tunnel test results suggest that the rivulet movement synchronized

to cable motion affects the cable vibration more or less, however, the

fundamental response characteristics of cable vibration can be framed by

the neutral rivulet position.

Reynolds Number Effect

Critical Reynolds Number

Re=UD/ν

ν�1.5x10-5

D=0.15m

if Recr=(3-5)x105, then U=30m/s-50m/s.

Reynolds number effectsReynolds number effects on circular on circular cylinder aerodynamics (cylinder aerodynamics (ScheweSchewe) ) (RE)(RE)

(a) r.m.s. of the lift fluctuations

(b) Strouhal number of the lift fluctuatioSr=fD/u∞

(c) drag coefficient

Reynolds number effectsReynolds number effects on inclined cable on inclined cable aerodynamics (NRCC) aerodynamics (NRCC) (RE)(RE)

lift force coefficientdrag force coefficient

Reynolds number effects (RE)

Surface roughness

High wind velocity

Larger diameter

Styrene foam Max.17m/sD=158mm

Change of Reynolds number

Cable Model with surface roughness (non-yawed state) (RE)

Cable Model (β=0°) with Surface Roughness (Styrene Foam)

Cable Model (β=0°)

Cable Model (β=45°)

Static forces�with surface roughness�(RE)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

Reynolds Number (*105)

Stea

dy w

ind

forc

e co

effic

ient

CD

Wind Velocity (m/s)0 3 6 9 12 15 18

0 0.5 1 1.5 2-0.6

-0.4

-0.2

0

0.2

0.4

0.6


Stea

dy w

ind

forc

e co

effic

ient

CL


0 0.5 1 1.5 20

0.2

0.4

0.6

0.8


Fluc

tuat

ing

win

d fo

rce

coef

ficie

nt CL

′


0 0.5 1 1.5 20

0.2

0.4

0.6

0.8


Fluc

tuat

ing

win

d fo

rce

coef

ficie

nt CL

′


0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4


Stea

dy w

ind

forc

e co

effic

ient

CD


0 0.5 1 1.5 2-0.6

-0.4

-0.2

0

0.2

0.4

0.6


Stea

dy w

ind

forc

e co

effic

ient

CL


β=0˚

CL CD CL’Decrease of drag

stationary liftDecay of Karman vortex intensity

Critical Re. number range

Increase of dragNon-app. of stationary lift

β=45˚

Super critical Re. number range

Flow field�with surface roughness�β=0 �̊(RE)

0 0.5 1 1.5 2-50-40-30-20-10

01020304050


Diff

eren

ce o

f win

d ve

loci

ty (%

)


0 0.5 1 1.5 2-50-40-30-20-10

01020304050


Diff

eren

ce o

f win

d ve

loci

ty (%

)


0 0.5 1 1.5 2-50-40-30-20-10

01020304050


Diff

eren

ce o

f win

d ve

loci

ty (%

)


wind

(a)(�) (�)

0 0.5 1 1.5 2-0.6

-0.4

-0.2

0

0.2

0.4

0.6


Stea

dy w

ind

forc

e co

effic

ient

CL


(a)

(c)

(b�

Velocity difference at critical Re number range where Karman vortex is suppressed

Non-symmetry flow field

Appearance of Stationary lift

0T/4 T/4 2T/4 3T/4 4T/4

L

t


�



At critical Re. number range (RE) Decrease of drag

Velocity-difference btw up and down sides

Stationary lift

wind

Relative angle of attack due to vib.

Amplified velocity at down side

exciting force

Aerodynamic derivative v.s. Re number (RE)

Frequency of forced heaving vibration was changed to change

the reduced velocity under the fixed wind velocity to change

Reynolds number.

Wo. Surface Roughness

0 2 4 6 8 10 12 14 16 18 0

20

40

60

80

2A/D 2A [mm]

U [m/s]U/fD


0 5 10 15 20 25 30 35

0

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6 8 10 12 14 16 18 0

20

40

60

80

2A/D 2A [mm]

U [m/s]U/fD


0 5 10 15 20 25 30

0

0.1

0.2

0.3

0.4

0.5

0.6

Free vib. test�β=0˚�(RE)

w. Surface roughness

Response with unsteady amp. appears at cr. Re.

number range

Aerodynamic derivative H1* �w. SR�β=0˚,

without rivulet �(RE)

-70

-60

-50

-40

-30

-20

-10

0

0 20 40 60 80 100 120

U/fD

H1*

Re=47000 Re=62000 Re=93000

-700-600-500-400-300-200-100

0100200300

0 50 100 150 200

U/fD

H1*

Re=109000 Re=120000 Re=130000

Re=135000 Re=140000 Re=166000

Sub-crtical Re. number range Cr. Re number range

Super cr. Re number range

unstable

Summary

The Role of KarmanVortex on Inclined Cable Aerodynamics

Karman Vortex Mitigation by

Axial FlowRivulet formationCritical Re numberTurbulenceThe others

The Role of Karman Vortex

Axial Flow(AF) Mitigation of

Water Rivulet (WR) ►► KarmanVortex

Reynolds Number (RE) ▼▼

GallopingHigh Spewed Vortex Exc.

If Karman Vortex is controlled,

Low frequency fluctuation component of flow field and lift appears. Which would excite the galloping or high speed vortex exc..

Non-symmetrical flow field appears, then stationary lift force can be generated. Which would excite the galloping.

Future Subjects

How to Control the Cable Vib.?

Mechanical Countermeasure ?Aerodynamic Countermeasure ?

How to encourage the 2D KarmanVortex?

Thank you for your

kind attention.

P.S.D. and wavelet analysis of P.S.D. and wavelet analysis of the the unsteady lift forceunsteady lift force on stationary cable on stationary cable

model (without rivulet) model (without rivulet) (AX)(AX)

0 5 10 15 20 250

0.5

1

1.5

2 x 10- 3

Frequency [Hz]

P.S.

D. [

N2 /Hz]

40 20 10 8 7 6 5 4V/ fD


0 4 8 12 16 200

0.5

1 x 10- 4

Frequency [Hz]

P.S.

D. [

N2 /Hz]

40 20 10 8 7 5 4V/ fD


Wavelet analysis ofWavelet analysis of fluctuating wind velocityfluctuating wind velocityin the wake of stationary yawed cable model in the wake of stationary yawed cable model

(AX)(AX)

(without rivulet, β=45°, V=4m/s, in smooth flow)

Aerostatic/fluctuating force v.s. rivulet position �β=0˚�(WR)

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5


Stea

dy w

ind

forc

e co

effic

ient

CL ′

CL CD CL’

0 20 40 60 80 100 120 140 160 180-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8


Stea

dy w

ind

forc

e co

effic

ient

CL

0 20 40 60 80 100 120 140 160 1800

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Stea

dy w

ind

forc

e co

effic

ient

CD

At near 50deg., Karman vortex becomes weak,then drag decreases and stationary lift appears.


(WR)(WR)

0 1 2 3 4 5 6 7 8 9 10 11 12 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1


0 1 2 3 4 5 6 7 8 9 10 11 12 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1

1.5

f=1.734Hzm=0.823kg/mδ 2A=10mm)=0.003745Sc(2A=10mm)=2.047

0 1 2 3 4 5 6 7 8 9 10 11 12 0

20

40

60

802A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1

1.5f=1.734Hzm=0.823kg/mδ 2A=10mm)=0.003745Sc(2A=10mm)=2.047

θ=47°

θ=54°

θ=58°

0 20 40 60 80 100 120 140 160 180

010

2030

4050

0

0.02

0.04

without rivulet


P.S.

D. [

(m/s

)2 /Hz]


0T/4 T/4 2T/4 3T/4 4T/4

L

t


�




(AX)(AX)



wind

walls and windows

Time history and the wavelet analysis of the unsteady Time history and the wavelet analysis of the unsteady crosscross--flow displacement (flow displacement (unsteady gallopingunsteady galloping))

0 10 20 30 40 50-0.01

-0.005

0

0.005

0.01

Time [sec]

Dis

plac

emen

t [m

]

18 18.2 18.4 18.6 18.8 19-4

-2

0

2

4 x 10-3

Time [sec]

Dis

plac

emen

t [m

]

40 40.2 40.4 40.6 40.8 41-4

-2

0

2

4 x 10-3

Time [sec]

Dis

plac

emen

t [m

]

free cablefree cable--end condition (without end condition (without wall, without endwall, without end--plates)plates)

((VV=5.0m/s, =5.0m/s, ff00=2.87Hz, =2.87Hz, ffkk=8.6Hz)=8.6Hz)0 10 20 30 40 50-0.01

-0.005

0

0.005

0.01

Time [s]

Dis

plac

emen

t [m

]

Observed water rivulet movement on prototype scale cable model during

rain-wind induced vibration

The time history of cable vibration and The time history of cable vibration and rivulet movementrivulet movement

0 5 10 15 20Time [s]

Cab

le m

otio

n

0 5 10 15 20Time [s]

Riv

ulet

mot

ion

0 5 10 15 20Time [s]

Cab

le m

otio

n

0 5 10 15 20Time [s]

Riv

ulet

mot

ion

(intensive cable response)(intensive cable response) (weak cable response)

Static forces�with surface roughness�(RE)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4


Stea

dy w

ind

forc

e co

effic

ient

CD


0 0.5 1 1.5 2-0.6

-0.4

-0.2

0

0.2

0.4

0.6


Stea

dy w

ind

forc

e co

effic

ient

CL


0 0.5 1 1.5 20

0.2

0.4

0.6

0.8


Fluc

tuat

ing

win

d fo

rce

coef

ficie

nt CL

′


0 0.5 1 1.5 20

0.2

0.4

0.6

0.8


Fluc

tuat

ing

win

d fo

rce

coef

ficie

nt CL

′


0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4


Stea

dy w

ind

forc

e co

effic

ient

CD


0 0.5 1 1.5 2-0.6

-0.4

-0.2

0

0.2

0.4

0.6


Stea

dy w

ind

forc

e co

effic

ient

CL


β=0˚

CL CD CL’Decrease of drag

stationary liftDecay of Karman vortex intensity

Critical Re. number range

Increase of dragNon-app. of stationary lift

β=45˚

Super critical Re. number range

Flow field�with surface roughness�β=0 �̊(RE)

0 0.5 1 1.5 2-50-40-30-20-10

01020304050


Diff

eren

ce o

f win

d ve

loci

ty (%

)


0 0.5 1 1.5 2-50-40-30-20-10

01020304050


Diff

eren

ce o

f win

d ve

loci

ty (%

)


0 0.5 1 1.5 2-50-40-30-20-10

01020304050


Diff

eren

ce o

f win

d ve

loci

ty (%

)


wind

(a)(�) (�)

0 0.5 1 1.5 2-0.6

-0.4

-0.2

0

0.2

0.4

0.6


Stea

dy w

ind

forc

e co

effic

ient

CL


(a)

(c)

(b�

Velocity difference at critical Re number range where Karman vortex is suppressed

Non-symmetry flow field

Appearance of Stationary lift

At critical Re. number range (RE) Decrease of drag

Velocity-difference btw up and down sides

Stationary lift

wind

Relative angle of attack due to vib.

Amplified velocity at down side

exciting force

Future Subjects

How to Control the Cable Vib.?

Mechanical Countermeasure ?Aerodynamic Countermeasure ?

How to encourage the 2D KarmanVortex?

Thank you for your

kind attention.

Quasi-Steady Theory for Inclined Cable with Fixed rivulet ?

Wind

Cable model(D=0.05m)

Wind tunnel wall

30% Perforated splitter plate (PSP)

Axial Flow Simulation by a Perforated Splitter Plate(30%) (QS)

Cross-flow Vibration affected by PSP (QS)

0 1 2 3 4 5 6 7 8 9 1011121314 0

20

40

60

80

1002A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80 100 120

0

0.5

1


0 1 2 3 4 5 6 7 8 9 1011121314 0

10

20

30

40

502A/ D 2A [mm]

V [m/ s]V/ fD0 10 20 30 40 50 60

0

0.2

0.4

0.6

0.8

1


β=0°, with a 30% perforated splitter plateV-A diagram wavelet analysis of the lift force

β=45°, without a splitter plate V-A diagram wavelet analysis of the lift force

Probably PSP for non-yawed cable would be simulated to yawed cable aerodynamics from the point of axial flow effect. (QS)

Observed water rivulet movement on prototype Observed water rivulet movement on prototype scale cable model during rainscale cable model during rain--wind induced wind induced

vibrationvibration

V=10m/s

Observed water rivulet movement on prototype scale cable model during

rain-wind induced vibration

The Problems of the Stationary Lift and Drag Forces in Applying to

Quasi-steady Theory (QS)

Real inclined stay cables

Vθ

axial flowwater rivulet

y.


Vrα

θwind

Aerostatic force coefficients test in wind tunnel

Vθ


Vrα

y.

θ


wind

Wind

Cable model(D=0.05m)

Wind tunnel wall

30% Perforated splitter plate (PSP)

Axial Flow Simulation by a Perforated Splitter Plate(30%) (QS)

Various situations ( Position of water rivulet, axial flow and PSP ) (QS)

PSP

water rivulet

Vθ

y.

wind θ PSP

water rivulet

Vr

α

(a) Wind tunnel tests situation, β=0°, with PSP

PSP

water rivulet

Vθ

y.

wind θPSP

water rivulet

Vr

α

(b) Wind tunnel tests situation, β=0°, with fixed PSP (ε=180°)

y.

water rivulet

Vθwind θ

water rivulet

Vr

α

(c) Wind tunnel tests situation, β=0°, without PSP

PSP

water rivulet

Vθ

y.

windθ-α PSP

water rivulet

Vr

α

(d) Wind tunnel tests situation, β=0°, with fixed water rivulet, with PSP


Vθ

y.

wind θ

water rivulet

Vr

α

axial flow

(e) Wind tunnel tests situation, β=45°, without PSP


Vθ

y.

wind θ

water rivulet

Vr

α

axial flow

(f) Actual situation

The Stationary Lift and Drag Coefficients (β=0°, in smooth flow, V=8.0m/s ) (QS)

0 20 40 60 80 100 120 140 160 180- 0.5- 0.4- 0.3- 0.2- 0.1

00.10.20.30.40.50.60.7

position of water rivulet θ [deg.]

Stea

dy li

ft fo

rce

coef

ficie

nt C

L

0 20 40 60 80 100 120 140 160 1800.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

position of water rivulet θ [deg.]St

eady

dra

g fo

rce

coef

ficie

nt C

D

::ε ε =176=176°° x:x:εε =178=178°° oo::ε ε =180=180°° ::εε =182=182°° +:+:εε =184=184°°**Position of PSP

Angle of Attack, Drag, Lift and Lateral Forces (QS)

D:drag force

L:lift forceF:lateral force

waterrivulet

V

Vr

wind θ

α=tan-1 (y/V).

dCF/dα diagrams under various situation (QS)

0 10 20 30 40 50 60 70 80 90 100 110120 130140 150160 170180- 4

- 2

0

2

4


dCF/

dα

(a) Wind tunnel tests situation, (a) Wind tunnel tests situation, ββ=0=0°°, with PSP, with PSP(b) Wind tunnel tests situation, (b) Wind tunnel tests situation, ββ=0=0°°, with fixed PSP (, with fixed PSP (εε=180=180°°))(c) Wind tunnel tests situation, (c) Wind tunnel tests situation, ββ=0=0°°, without PSP, without PSP(d) Wind tunnel tests situation, (d) Wind tunnel tests situation, ββ=0=0°°, with fixed water rivulet, with PSP, with fixed water rivulet, with PSP(e) Wind tunnel tests situation, (e) Wind tunnel tests situation, ββ=45=45°°, without PSP, without PSP

0 20 40 60 80 100 120 140 160 180- 0.6- 0.4- 0.2

00.20.40.60.8

11.21.41.6


Stea

dy w

ind

forc

e co

effic

ient

sCD

CL

Steady Wind Force Coefficients (QS)

o:o:ββ=0=0°°, without windows, with PSP, without windows, with PSP::ββ=0=0°°, without windows , without windows

x:x:ββ=45=45°°, with 110mm windows, with 110mm windows

dCF/dα Diagram (β=0°, with PSP) (QS)

0 10 20 30 40 50 60 70 80 90 100 110120 130140 150160 170180- 4

- 2

0

2

4

position of water rivulet θ [ �]

dCF/

dα

Galloping Galloping dCdCFF/d/dαα < < 00Quasi-steady theory

Quasi-steady Approach

Wind tunnel tests(ex. β=45° situation)

Using yawed cable model Using yawed cable model (QS)(QS)

Wind

Actual situation

RelativeWind

Wind

Quasi-steady Approach (QS)

Wind tunnel tests(Correct situation)Actual situation

RelativeWind

The non-linear analysis based on the quasi-steady theory (QS)

CF=A1+A2α+A3α2+A4α3+A5α4+A6α5+A7α6+A8α7

α= tan-1 (y/V)

=(y/V)-(1/3)(y/V)3+(1/5)(y/V)5-(1/7)(y/V)7

α : angle of attack

y : the cross-flow response velocity

Comparison of V-A diagrams ; wind tunnel tests (β=45°) and the quasi-steady analysis

(CF:β=0°,with PSP) (QS)

0 1 2 3 4 5 6 7 8 9 10 11 0

20

40

60

80

1002A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80

0

0.5

1

1.5

(a) without water rivulet


0 1 2 3 4 5 6 7 8 9 10 11 0

20

40

60

80

1002A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80

0

0.5

1

1.5

(b) θ=0°


0 1 2 3 4 5 6 7 8 9 10 11 0

20

40

60

80

1002A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80

0

0.5

1

1.5

(d) θ=66°


0 1 2 3 4 5 6 7 8 9 10 11 0

20

40

60

80

1002A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80

0

0.5

1

1.5

(f) θ=180°


0 1 2 3 4 5 6 7 8 9 10 11 0

20

40

60

80

1002A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80

0

0.5

1

1.5

(c) θ=56°


0 1 2 3 4 5 6 7 8 9 10 11 0

20

40

60

80

1002A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80

0

0.5

1


(e) θ=72°

OO : wind tunnel : wind tunnel test (test (ββ=45=45°°))

OO : quasi : quasi –– steady steady analysis analysis ((CCF:F:ββ=0=0°°,,withPSPwithPSP))

X : limit cycleX : limit cycle

ΔΔ: : VrVrcrcr

(Parkinson)(Parkinson)

: : VrVrcrcr

(Den Hartag)(Den Hartag)**

0 2 4 6 8 10 12 14 16 18 20 22 24 0

20

40

60

80

100

1202A/ D2A [mm]

V [m/ s]

V/ fD0 100 200 300 400 500

0

0.5

1

1.5

2

e) Sc=248

f=0.833Hzm=1.894kg/mδ(2A=10mm)=0.231Sc(2A=10mm)=248

0 1 2 3 4 5 6 7 8 9 10 0

20

40

60

80

100

1202A/ D 2A [mm]

V [m/ s]

V/ fD0 50 100 150 200

0

0.5

1

1.5

2

b) Sc=46.2


0 1 2 3 4 5 6 7 8 9 1011121314 0

20

40

60

80

100

1202A/ D 2A [mm]

V [m/ s]

V/ fD0 50 100 150 200 250 300

0

0.5

1

1.5

2

c) Sc=125

f=0.828Hzm=1.867kg/mδ (2A=10mm)=0.117Sc(2A=10mm)=125

0 2 4 6 8 10 12 14 16 0

20

40

60

80

100

1202A/ D2A [mm]

V [m/ s]

V/ fD0 100 200 300

0

0.5

1

1.5

2

d) Sc=169

f=0.825Hzm=1.903kg/mδ(2Α=10μμ)=0.155Sc(2A=10mm)=169

0 1 2 3 4 5 6 7 8 9 10 0

20

40

60

80

100

1202A/ D 2A [mm]

V [m/ s]

V/ fD0 50 100 150 200

0

0.5

1

1.5

2

a) Sc=12.3

f=0.826Hzm=1.922kg/mδ(2Α=10μμ)=0.0113Sc(2A=10mm)=12.3

Comparison of V-A diagrams ; wind tunnel tests (β=45°,without rivulet) and the quasi-

steady analysis (CF:β=0°,with PSP) (QS)

OO : wind tunnel test : wind tunnel test

((ββ=45=45°°))

OO : quasi: quasi--steady analysissteady analysis

((CCFF::ββ=0=0°°,,with PSPwith PSP))

Comparison of V-A diagrams ; wind tunnel tests (β=45°) and the quasi-steady

analysis (CF:β=45°,without PSP) (QS)

0 1 2 3 4 5 6 7 8 9 10 11 0

20

40

60

80

1002A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80

0

0.5

1

1.5


(b) θ=66°

0 1 2 3 4 5 6 7 8 9 10 11 0

20

40

60

80

1002A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80

0

0.5

1


(a) θ=56°

0 1 2 3 4 5 6 7 8 9 10 11 0

20

40

60

80

1002A/ D 2A [mm]

V [m/ s]

V/ fD0 20 40 60 80

0

0.5

1

1.5


(c) θ=72°

OO : wind tunnel test: wind tunnel test ((ββ=45=45°°))

OO : quasi: quasi--steady analysis (steady analysis (CCFF::ββ=45=45°°,,without PSPwithout PSP))

ΔΔ : : VrVrcr cr (Parkinson)(Parkinson) , : : VrVrcr cr (Den (Den HartagHartag))**

Quasi-Steady Theory (QS)

Still we have to discuss more in application of Quasi-Steady Theory.

Documents

Mechanism of Wind & Rain/Wind Induced Cable Vibrationscontribute.modot.mo.gov/csb/documents/07-TheGenerationMechanism… · Mechanism of Wind & Rain/Wind Induced Cable Vibrations