The 2012 World Congress on

Advances in Civil, Environmental, and Materials Research (ACEM’ 12)

Seoul, Korea, August 26-30, 2012

ON GALLOPING INSTABILITY OF STAY CABLES OF

CABLE-STAYED BRIDGES

Masaru Matsumoto

Professor Emeritus of Kyoto University, Kyoto, Japan

ABSTRACT

Aerodynamic response, including rain and wind-induced vibration(RWIV)

and dry state galloping(DG), of stay cables of cable-stayed bridges are studied

from the points of generation mechanism. It has been verified that water rivulet

formation(WR), axial flow (AF)in a wake of inclined cable and critical Reynolds

number(Recr) can excite individually and cooperatively can excite

aerodynamically inclined cable, that Karman vortex(KV) has essentially affected

these excitation mechanism caused by these factors, WR, AF and Recr and,

as a summary, that aerodynamic response of stay cables is “unsteady

galloping”(UG).

1. INTRODUCTION

Recent cable-stayed bridges have lengthened their main span length

with more than 1000m , including Sutong Bridge(1088m China)m Stone cutters

Bridge (1018m China) and Russky Island Bridge(under construction 1104m

Russia). Their lengths of stay cables are more than 600m. Stay cables are

extremely low frequency and low damped structures, in consequence, they are

significantly and sensitively excited by wind effect. Since Hikami[1] had reported

firstly in the world this particular wind-induced vibration aided by precipitation in

1986, rain-wind induced vibration(RWIV) and dry-state galloping(dry galloping)

of stay cables of cable stayed bridges have been observed at a lot of long

spanned cable stayed bridges in the world. Their amplitudes are too large and

frequencies of vibration events are too much to produce serious damages, in

particular, at connecting joint parts to girders. Wind-induced vibration of stay

cables truly become crucial issues in design of bridges. In order to establish

effective and reasonable countermeasure to suppress their vibration, ones

Keynote Paper

should verify the precise generation mechanism of these stay cable wind-induced vibration. However, their generation mechanisms are not clarified, even though huge number of studies on cable aerodynamics (Matsumoto et al.[2,to11], Verviewe[12], MacDonald[13], Larose[14], Chen[15], , Katsuchi et al[16], Kimura et al.[17]) because of complicated phenomena between fluid and structures. Matsumoto [2, 3] has pointed out three major factors, those are formation of upper water rivulet, axial flow in near wake and critical Reynolds number, play definitely substantial role for generation mechanism of wind-induced vibration of stay cables. Each of these factors can excite aerodynamically inclined cable individually and/ or cooperatively(Matsumoto et, al[11]). Observed violent vibration of stay cables in the fields must be galloping. Galloping has been widely known to be cross-flow divergent type fluid-induced vibration, that is bending flutter, and its generation mechanism is thought to be caused by negative slope of lift coefficient associated to pitching angle, that is known as den Hartog Criterion[18]. Parkinson[19] and Novak[20] successfully explained the non-linear response characteristics of galloping of square cylinder by use of Krylov- Bogolivoff method, basing on quasi-steady theory. On the other hand, Nakamura and Hirata[21] reported that the other type galloping exists, that is so called “Low-Speed Galloping(LSG), which cannot be explained by den Hartog Criterion. LSG can be observed for bluffer rectangular cylinders with B/D (B:along-wind length, D;cross flow length) less than 0.75 and at lower reduced velocity range than Vr<1/St. (Vr=V/f0D and St:Strouhal number). Its generation mechanism is explained by Nakamura and Hirata[21] as deformation of flow by body motion not affected by past flow field, it means without fluid memory. On negative slope of lift coefficient, dCF/dα, the particular flow fields around bluff body, those are “inner circulatory flow”(Bearman and Trueman[22]) and “ the flow which generates reattachment type pressure”(simply flow with reattachment type pressure”)(Nakamura and Tomonari[23]) should play definitely important role. On the other hand, negative slope of lift coefficient, dCL/dα<0, can be observed at the stalling pitching angle of thin airfoil. The reason of local negative slope of lift force is flow separation from airfoil, and formation of separation bubble on its surface. Rinoue[24] studied on the separation bubble characteristics, those are short bubble and long bubble, corresponding on properties of lift coefficient – pitching angle diagram. Furthermore, research group of Central Research Institute of Electric Power Industry, Japan, has studied on the aerodynamic behavior of transmission line with snow, and they verified that the negative slope

of lift is caused by flow reattachment, then galloping can be excited in relation to den Hartog Criterion.(Shimizu et.al[25], Matsumiya et. al[26]). Nakamura and Hirata[21] studied on the galloping instability of rectangular cylinders with various side ratios, B/D without and with splitter plate(SP) in a wake. They explained the role of SP is interruption of two shear layers in near wake, mitigation of Karman vortex(KV) shedding and sequential flow undulation generated cylinder motion. Matsumoto[27] studied on the effect of SP in awake on aerostatic forces of rectangular cylinders with various side ratios. Furthermore, aerodynamic forces associated to aerodynamic damping, that is Scanlan derivative(Scanlan and Tomko[28]) H1* has measured for these rectangular cylinders. Though bluff rectangular cylinders, with B/D =0.5, 0.4. and 0.3, with SP, their lift coefficients slope indicate positive (dCF/dα>0), all of their H1* is positive (H1*>0) at wide reduced velocity range. Taking into account that H1*>0 means cross flow vibration, that is galloping, can be excited, This galloping cannot be apparently explained basing on den Hartog Criterion. This kind of galloping is, in consequence, not QG. Nakamura and Hirata[21] also reported the appearance of galloping for these rectangular cylinders with SP. Therefore, another type of galloping should be generated under the particular flow field around bluff body. According Nakamura’s explanation, this galloping is definitely caused by the flow undulation affected by body motion in the past time, that is fluid-memory. Since this galloping is absolutely generated by unsteady body motion, in consequence, it might be called as unsteady galloping(UG), in contrast to QG. In this note, complicated aerodynamic behavior of inclined stay cables of cable stayed bridges is explained in relation to QG and UG taking account into flow fields generated by upper water rivulets, axial flow in near wake and critical Reynolds number, individually or cooperatively. 2. QUASI-STEADY GALLOPING

First of all, the generation mechanism of negative slope of lift coefficient, dCF/dα, has been explained by Bearman and Trueman[22] as follows. When Flow comes to rectangular cylinder with certain positive pitching angle, and separated shear layer from the leading edge of lower surface of cylinder approach without reattaching on the sharp trailing edge, then inner flow particle of closed space between cylinder surface and separated shear flow cannot be go out from this space into a wake, because of difficulty of fluid supply from wake to this space due to enough small inlet space at the trailing edge. Then the fluid

particle should return up-stream side without going out to maintain fluid volume in this closed space. Thus intensive” inner circulatory flow“ is generated on lower surface. This “inner circulatory flow “can produce intensive negative pressure on lower face of cylinder, it means generation of negative slope of lift force. On the contrary, Nakamura explained the generation mechanism of negative slope of lift force by appearance of reattachment-type pressure distribution on lower surface, which indicates significantly low pressure zone near leading edge of body and significantly pressure recovery at near trailing edge, through testing by use of D-shaped cylinder which does not have sharp trailing edge but round trailing edge. Therefore this particular flow which can generate reattachment type pressure distribution should be mechanism of negative slope of lift force. However, both explanations by Bearman and Nakamura can be identical from the point of that separated flow from the leading edge approaches to the trailing edge without reattachment on side face. Nakamura called the conventional galloping “High Speed Galloping(HSPG)” which appears for rectangular cylinders with B/D between 0.75 and 2.8 at higher reduced velocity than inverse value of Strouhal number, that is Vr=V/f0D>(1/St), related to negative slope of lift force coefficient, that is dCF/dα. On the other hand, Galloping which appears for bluffer cylinder with B/D less than 0.75 at lower reduced velocity than 1/St, is called as “Low Speed Galloping(LSPG)”. They explained the generation mechanism of LSPG by non-fluid memory, which is instantaneously deformed flow field under less affected by Karman vortex by body motion because of KV sufficiently lower frequency than the one of body motion. They verified that the instantaneous pressure distribution on side face indicates ” reattachment pressure distribution” which is generation of galloping, during cross flow motion as unsteady effect of motion, which completely differs from stationary( quasi-steady) situation. This LSPG should be Unsteady Galloping (UG) discussed below in details.

On the other hand, the aerostatic properties of transmission line with snow have been studied by research group of Central Research Institute of Electrical Power Industry Japan verified that the negative slope of lift force was generated by the flow reattachment and formation of separation bubble. (Shimizu et.al.[25] and Matsumiya et.al.[26]) Shimizu and et.al[25] clarified that flow reattached on body surface and formation of separation bubble corresponding to appearance of negative slope of lift force by CFD analysis. Matsumiya and et.al29] reported flow fields corresponding to its aerostatic

forces. On behavior of separation bubble near stalling angle of airfoil NACA0012,

Rinoue[24] explained that short bubble, which produced more intensive negative pressure, and long bubble characterized the intensity and distance of negative pressure on airfoil surface depending on change of pitching angle, and burst event of separation bubble changed short to long bubble. These unsteady characteristics of separation bubble can also generate negative slope of lift force. In summary on negative slope of lift force of bluff bodies, there are two different kinds of flow fields, those are “ inner circulatory flow” or flow field which produce the reattachment type pressure distribution, both are non-reattachment flow, and unsteady separation bubble by flow reattachment. Basing on quasi-steady theory, taking into account generation of relative pitching angle, αre=(dy/dt)/V, due to cross-flow motion, when structural section moves downward, that is generation of positive pitching angle, intensive negative pressure on down-side surface of section generated by “inner-circulatory flow” or pressure recovery by “flow reattachment” on the upper-side surface. Both of these flows can generate downward lift force, then galloping should be excited because of coincidence of directions of motion and force. 3. UNSTEADY GALLOPING

As briefly explained above, Nakamura and Hirata [] studied on galloping of rectangular cylinder with B/D<0.8 and at low reduced velocity, that is Vr=V/f0D<1/St (St: Strouhal number) , named by Low Speed Galloping(LSPG). They pointed out that its generation mechanism is particular unsteady flow which appears during cross flow oscillation of cylinder. This flow is different quasi-steady flow generated by relative angle of attack, αre=arctan(dy/dy/V) and produce unsteadily “reattachment pressure distribution type” flow. It should be noted that LSPG is completely free from den Hartog Criterion, it means these bluff cylinders (B/D<0.8) show positive lift slope, dCF/dα>0, (Nakamura, Tomonari, xxx). Furthermore, they pointed out that similar unsteady flow can be observed in the case of rectangular cylinder with splitter plate (SP) in an wake as shown in Fig.4 (Nakamura and Hirata[21]).

redu(Nak

B/D=15Dwith rectadecrdCFshouquascylincylinenou

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(b)B/D=0.5 Fig. 4 CD-α diagram , CF-α diagram , CF’-α diagram and St(D)-α diagram of

rectangular cylinders with B/D=0.3 and 0.5 at the states of without SP and with SP In these figures, it should be noted that CL is lift coefficient defined as

structural axis. On the other hand, flutter derivatives of these rectangular cylinders at the state of without SP and with SP. in terms of aerodynamic damping, H1* defined by Scanlan(Scanlan and Tomko[28]), has been measured by forced vibration method. (a) B/D=0.3 (b) B/D=0.5 Fig.5 Flutter derivatives in terms of aerodynamic damping, H1* of rectangular cylinders with B/D=0.3 and 0.5 at the state of without SP and with SP (red line: with SP, black line: without SP, blue line : calculated from Theodorsen function)

H1* is defined as follows[28]: (dy2/d2t)+2ζ0ω0(dy/dt)+ω0

2y=(ρb2ωF/m)H1*(dy/dt) +(ρb2ωF2/m)H4*y (1)

where m:mass per unit length, ρ: air density, b: half chord length (=B/2), ζ0: heaving damping ratio, ω0:heaving circular frequency , ωF: flutter(galloping) frequency (=:ω0), H4*:flutter derivative in term of aerodynamic stiffness

Therefore, H1*>0 means aerodynamic unstable effect, and if (ρb2ωF/m)H1* is larger than 2ζ0ω0 , galloping appears. Quasi-steady state being satisfied, following formula should be satisfied: H1*=(1/π)(-dCF/dα)(D/B)Vr (2)

0

0.5

1

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-10 -8 -6 -4 -2 0 2 4 6 8 10α [deg]

C D

without S.P.with S.P.

-1-0.8

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0.40.6

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C L

without S.P.with S.P.

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C L ' BPF

without S.P.with S.P.

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St

0 40 80 120 160 200 240 280

-200

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H1 *

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U/fD

0 20 40 60 80 100 120 140 160

-100

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0 20 40 60 80

H1*

U/fB

U/fD

As shown in Fig.5, all rectangular cylinders with B/D=0.3, 0.5, 0.6, 0.9 at the state of with SP, galloping should be excited, however their slopes of lift coefficient, dCF/dα, are positive. This kind of galloping cannot be explained by the conventional quasi-steady theory. This galloping should be excited by substantial unsteady interaction between fluid and motion of body. This galloping can be called as “unsteady galloping(UG)” in contrast to “quasi-steady galloping(QG)”. What kind of flow field can excite UG? When SP is installed in a wake, Karman vortex(KV) should be mitigated/weakened in near wake, but flow undulation generated by body cross-flow motion still survive even though installation of SP. (Hirata[30]) This flow undulation around oscillating body must be generated by body cross-flow motion in the past time. In another expression, this undulation flow can be said “fluid memory”. Through a series of wind tunnel tests, it can be evaluated that mitigation of both of KV and upstream influence of flow interaction between two separated shear layers in a near wake plays definitely important in enhancement of “flow undulation” or “fluid memory“, then, unsteady lift force with phase lag to excite galloping, that is H1*>0. On the other hand, rectangular cylinders with more than B/D=5 at the state of with SP, do not indicate UG anymore, because H*<0.

At the state of without SP, it has been known that galloping of rectangular cylinders with B/D more than 2.8 cannot be observed, This aerodynamic property can be observed from H1* diagrams. At the state of with SP, critical B/D is 5, it means rectangular cylinders with B/D=3.0 and 4.0 still indicate galloping instability, because of time average flows of these cylinders do not reattach on side face because of curvature of separated shear layers should be mitigated by mitigation of KV, on the other hand, rectangular cylinders with B/D more than 5, time averaged flow reattach on side face of cylinder, in consequence galloping of these rectangular cylinders is stabilized. On the other hand, Assi, Bearman and Kitney[31] reported that circular cylinders with fixed SP in a wake with length from 0.2D up to 2D show violent galloping. This galloping seems to be UG similarly with bluff rectangular cylinders with from B/D=0.3 to B/D=0.9 at the state of with SP in a wake. Kawai[32] verified by CFD analysis(discrete point vortex method) out the SP installed in a wake of circular cylinder can produce the undulating flow locked in cylinder cross-flow motion because of interruption of two separated shear layers by SP . Furthermore, another case of appearance of UG, in addition to the case of with SP, is discussed as follows.

UG could be excited at the particular situation of when flow reattaches on the surface or edge of bluff body. One can identify the flow reattachment by CL-α diagram, CD-α diagram, CL’-α diagram and St-α diagram. When flow attachment arises, the local peak (summit or canyon) of CL, accompanying with local CD canyon and the local minimum value of CL’ (fluctuating lift coefficient) caused by KV, and rapid change of St are observed. The complicated response

characteristics of circular cylinder with symmetric protuberances in relation of KV mitigation is shown in Fig.9(by Matsumoto and et.al[33]).

(a) Circular cylinder with symmetrical protuberance (each protuberance size:0.032D thickness, 0.072D width, D:diameter of cylinder(50mm)) at θ measured from horizontal line)

(b) CL-θ diagram (c) CD-θ diagram (c) CL’-θ diagram (e) St-θ diagram Fig.9 Aerostatic properties of circular cylinder with symmetrical protuberances (Matsumoto[33])

Appearance of stationary “bias flow” for geometrical symmetrical section has been observed at circular cylinder at critical Reynolds number( Schewe[34], Larose[14], Matsumoto[11], Sato[35], Liu[36]), side-by-side arranged rectangular cylinders (Alam and Zhou[37], Okajima[38], Matsumoto[39] and circular cylinder with movable (rotatable) splitter plate (Assi, Bearman and kitney[31], Cimbala and Garg[40]). On bias flow would be not discussed in this paper.

-0.3

-0.2

-0.1

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0 30 60 90 120 150 180θ [deg.]

CL

U=6.0[m/sec]U=8.0[m/sec]

0

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1.5

2

0 30 60 90 120 150 180θ [deg.]

CD

U=6.0[m/sec]U=8.0[m/sec]

0

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0 30 60 90 120 150 180θ [deg.]

CL'

U=6.0[m/sec]U=8.0[m/sec]

0

0.1

0.2

0.3

0.4

0.5

0 30 60 90 120 150 180θ [deg.]

St

U=6.0[m/sec]U=8.0[m/sec]

On the other hand, cross-flow responses and CL-α diagram (L is defined as structural axis, in these cases) of circular cylinder with symmetrical protuberances at the position of θ=48° (a), θ=50°(b), θ=55°(c), θ=58°(d) and θ=60°(e) are shown in Fig.11, respectively.

(a) θ=48° (SC=2mδ/ρD2=3.145)

(b) θ=50°(SC=3.144)

(c) θ=55° (SC=4.254) Fig.11 CL-α diagrams and cross-flow responses of circular cylinder with symmetrical protuberances at θ=48°, θ=50° and θ=55°.

The lift slope and cross-flow response drastically change bounded at

-1

-0.5

0

0.5

1

-10 -8 -6 -4 -2 0 2 4 6 8 10α [deg.]

CL

U=6.0[m/sec]U=8.0[m/sec]

-1

-0.5

0

0.5

1

-10 -8 -6 -4 -2 0 2 4 6 8 10α [deg.]

CL

U=6.0[m/sec]U=8.0[m/sec]

-1

-0.5

0

0.5

1

-10 -8 -6 -4 -2 0 2 4 6 8 10α [deg.]

CL

U=6.0[m/sec]U=8.0[m/sec]

0

0.5

1

1.5

2

0 10 20 30 40 50 60U/fD

2η /D

0

25

50

75

1002η [mm]

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2η /D

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75

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2η /D

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50

75

1002η [mm]

0 2 4 6 8 10U [m/sec]

around at θ=50° where flow attaches, as shown in Fig.11 As far as slope of lift coefficient is observed as negative at θ=48° and θ=50°, almost zero at θ=55°, θ=58° and θ=60°, respectively. Therefore, cross-flow responses at higher reduced velocity than motion induced vibration(MIV), including unstable response at θ=60°, should be UG or at least hybrid type of QG and UG at θ=48° and θ=50°.

As another example of UG, cross-flow response of circular cylinder at critical Reynolds number. Recently Liu and et.al[36] carried out significantly tough wind tunnel test at high wind velocity up to 80m/s in order to realize subcritical, critical and transient critical Reynolds numbers. At critical Reynolds number, 3.3- 4.4 ×105, steady lift, up to CL=1.6, suddenly appears in variation of Reynolds number caused by bias flow. Drag crisis appears like step.. It is interested in the appearance of cross-flow response at two narrow Reynolds number zones, corresponding to two particular Reynolds number zones where drastic change of CL appears. A similar cross-flow response of circular cylinder at critical Reynolds number has been reported by Kimura and et.al[17]. This cross-flow seems to be UG, because cross-flow vibration cannot produce exciting force basing on quasi-steady explanation. At critical Reynolds number, It has been verified that flow reattaches and separation bubble can be generated by CFD analysis by Basu[41] and high Reynolds number wind tunnel test by Sato et.al[37].In particular, Sato reported that the separation bubble was formed at one side face, it means un-symmetrically.

Thus, UG of bluff body can be excited under the “undulation of separated flow” synchronized to body motion less affected by KV, that is significantly “mitigated KV” by such as installation of SP in a wake, or at the condition near “flow reattachment”. However, un-cleared subjects still exist in the generation mechanism of UG, more precise research should be needed.

4. RWIV (RAIN AND WIND –INDUCED VIBRATION) AND DRG

(DRY-ATATE GALLOPING) OF INCLINED STAY CABLES As described at 1.Introduction, the fundamental factors are “axial flow” in

near wake of yawed/inclined cable, formation of “upper water rivulet” on cable surface and the state of critical Reynolds number. 4.1 The Role of Axial Flow(AF)

First of all, the effect of “axial flow” is explained. Fig.15(a) and (b) show

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(b) PSD (Left figure) and its wavelet value (right figure) of yawed(β=45°) cylinder

(c) PSD (Left figure) and its wavelet value (right figure) of non-yawed cylinder with 30% PSP

Fig.18 Analogy of unsteady lift forces of yawed cable and non-yawed cable with PSP(Matsumoto [9]) (a) Non-yawed (β=0°) with PSP (30%) (b) yawed cylinder with

β=45° Fig.17. Comparison of PSD of fluctuating lift force and its Wavelet value and their cross-flow response of non-yawed (β=0°) cylinder with PSP (30%) and yawed (β=45°) cylinder. (at the case of (a), f0=4.538Hz, m=4.75Kg/m, δ0=0.00216(at 2y0=0.2D), Sc=7.168, and the one of (b), f0=2.096Hz, m=0.509Kg/m,δ-=0.00373 (at 2y-=0.2D), Sc=1.138) (Matsumoto[9])

Furthermore, on unsteady cross-flow response of yawed (β=45°) at

0 4 8 12 16 200

0.5

1x 10

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Frequency [Hz]

P.S

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40 20 10 8 7 5 4V/fD

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0.4

0.6

0.8

1

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

subcvelocKV beco

(a)

(b)

(c)

Fig.1corre

Beca

-

S.T.

D. o

f flu

ctua

ting

velo

city

[

m/s

ec.]

critical Reycity, Vrcr,, is mitigateomes smal

Test view yawed(β=

cross flowblue data)KV frequz/D=1.25)

STD subttime-averaamplitudesec: blue

18 unsteaesponding

Unsteadyause of am

0 20-4

-2

0

2

4 x 10-3

Dis

plac

emen

t [m

]

0 20-1

-0.5

0

0.5

1

Win

d V

eloc

ity [m

/sec

.]

0-0.1

0.05

0

0.05

0.1

0

ynolds numof diverge

ed, on thell, as show

of Yawed

=45°) cylin

w response) and fluctuuency (fK) (bottom re

racted by aged by 5

e B.S. filterbroken lin

ady respounsteady

y intensitymplified cro

40 60 80Time [sec.]

Displacement (Natural

40 60 80Time [sec.]

Fluctuating wind velocity (K.V.

20 4020 40

mber, atent-type UGe contrary, w in Fig.18.

(β=45°) cy

nder (righ

e at U=4muating veloK=8.0-10.5ed data)

its mean v.0sec (fK

red by f0 (fne nse of yaKV intensi

y of axial oss-flow re

100

l frequency B.P.F.)

100

frequency B.P.F.)

60Time [sec.]

60

lower redG significa

when KV

ylinder(leftht figure)

m/s Band-pocity in a w5)(measure

value of STB.P.F) :redf0B.P.F.), s

awed (β=ity

flow migesponse ap

0 2 0

10

20

30

40

50

60

2A/D 2A [mm]

0 2

0

0.2

0.4

0.6

0.8

1

1.2

8080

duced veloantly large V is intens

t photo) an

pass filterewake (U=4med positio

TD in 100sd-solid linsubtracted

45°) circu

ght cause ppears in r

4 6 8 10

f = 2.57Hzm = 0.660kgδ (2A=10mmSc (2A=10m

20 40 60 80

θ

100100-4

-2

0

2

4x 10-3

Am

plitu

de [m

]

ocity than amplitude

sive, respo

nd cross flo

ed by f0 (2m/s) Band pon x/L=0

sec of fluctne, and ST

by its mea

ular cylind

unsteadyrelation to

12 14 U [m/s]U/fD

g/mm) = 0.003

mm) = 1.22

0 100

θ

onset redappears w

onse ampl

ow respon

2.4-2.8 Hzpass filtere

0.25, y/D

ctuating veTD of respan value in

der at V=

y KV intesuppressio

uced when itude

se of

z)(top ed by =1.0,

locity ponse n 100

4m/s

nsity. on of

KV, this unsteady response at V=4m/s is thought to be o sort of UG. 4.2 The Role of Upper Water Rivulet

Artificial single rivulet is used to clarify the role of rivulet on non-yawed (β=0°) circular cylinder. Rivulet is this rectangular shape with 0.062D in length and 0.032D in width and its position was change from θ=0°( at front stagnation point) to θ=180° (at center point of rear face of cylinder). CL(θ), CD(θ), CL’ (θ)and St(θ) were measured at various rivulet position. As shown in Fig.19, protuberance position sensitively characterizes them. From these characteristics, “flow reattachment” occurs at θ=50°, as explained before, those are showing local peal of CL, local minimum CD, significant mitigation of KV and drastic change of St at θ=50°. Furthermore, it should be noted that position of “flow reattachment” is identical in both cases of single protuberance and symmetrical ones. Fig. 20 shows PSD of fluctuating lift force. It is clarified that complicated and drastic changes of KV characteristics at near “flow reattachment” at θ=50°. Galloping appears at high reduced velocity at θ=54°, corresponding negative slope of CF, therefore this galloping might be almost QG with less effect of UG.

(a) CL-θ diagram and CD-θ diagram

(b) L’(fluctuating lift force caused by KV)-θ diagram at V=8m/s

-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

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

upper rivulet position [deg.] (θ )

L f (L

ift fo

rce

ampl

itude

) [N

]

(c) SFig.1(β=0

Fig.2sing

varioθ=56aerofromprotuAs dgallo

(a) w

0.15

0.17

0.19

0.21

0.23

0.25

S t

00

0.02

0.04

Fre

P.S.

D. [

(m/s

)2 /Hz]

St-θ diagra19 CL, CD0°) circular

20 PSD ofle protuber

On the oous protub6°, θ=58°, odynamicalm these Huberance adiscussed oping at θ=

without pro

0 10 20 30 40 50 60

upper riv

1020

3040

5equency [Hz]

m D, L’ and cylinder w

f fluctuatinrance

other handberance p

θ=60°, θ=lly unstable

H1* diagraat the posibefore, g

=52° would

otuberance

70 80 90 100 110 120 130

vulet position [d

0 20 40 60 80 1

50Rivulet p

St dependwith single p

ng lift force

, aerodynaositions, θ=70° and e, that is ams, non-ytion of θ=5alloping a be hybrid

(rivulet), θ=

0 140 150 160 170 180

eg.]

100 120 140 160 180

without rivulet

position θ [deg.]

ding on prprotuberan

e of non-y

amic derivθ=40°, θ=4θ=90°. As

appearanceyawed (β=50°, θ=52°at θ=50° a type of QG

=40°, θ=46

rotuberancnce (Matsu

yawed (β=

ative, H1*46°, θ=48s explainee of cross-=0°) circul, and θ=54

at least shG and UG.

6°, V, θ=48

e position umoto[11])

0°) circula

, are show°, θ=50°,

ed before, -flow vibratar cylinde

4° should should be U.

8°, θ=50° a

of non-ya

ar cylinder

wn in Fig. 2θ=52°, θ=H1*>0 m

tion. Thereers with sshow galloUG. The o

and θ=52°

awed

with

21 at =54°, eans

efore, single oping. other

(b) θFig.2vario 4.3T

of fobefoaerothouflow 5. C

follow1. G

a2. D3. W

ue

4. Sm“ain

5. TpQ

θ=54°, θ=521 Aerodyous protub

The RoleUnsteady

ormation oore, but fodynamicalught to be w

on aerody

CONCLUConclusi

ws: Galloping oand “UnsteDen HartogWhen wakunsteady fessential roStay cablesmajor factoaxial flow “n particulaThese factposition caQG can be

6°, θ=58°, namic dererance pos

e of Reyny galloping

of separatefor yawedlly unstablwhy coopeynamic inst

USIONS ons of thi

of bluff bodeady Gallopg criterion ike undulaflow geneole for UG s of cable sors, those a“ in a near r at near tors can en generate, in conseq

θ=60°, θ=ivative, H1sitions. ( 2

nolds Nug of non-yaed bubble d/inclined le as show

erative effetability.

s study o

dy can be ping (UG)”is not alwation gene

erated by excitation.stayed bridare “formatwake of incritical Re

excite mae significanquence, ex

70°, θ=90°

1*, of non-y0=10mm

umber awed/inclinon cable cable unswn in Fig.

ect of forma

on inclined

classified .

ays necesserated by

body-mot. dges can btion of rivunclined/yaweynolds nuinly UG, nt negativexcited.

° -yawed (β=(0.2D), fy=

ned cable msurface atsteady ga.13(Kimuraation of se

stay cab

into “Quas

sary to excKarman

tion aroun

be aerodynulet” on cabwed cable mber. but rivulet

e lift-forces

=0°) circula=2.0Hz)

might be ext neat Recalloping ba and et. paration bu

ble aerodyn

si-steady G

ite UG. vortex(KV)

nd body p

namically eble surfaceand “Reyn

t formations slope, tha

ar cylinder

xcited beccr as explabecomes al[17]). Thubble and

namics ar

Galloping (

) is mitigplays defin

excited by te by rain enolds numb

n at partiat is dCF/d

r with

cause ained more his is axial

re as

QG)”

ated, nitely

three effect, ber “,

cular dα<0,

6. Unsteady cross-flow response of yawed/inclined cable at lower reduced velocity than critical reduced velocity of divergent type galloping, should be a sort of UG, generated by unsteady of KV intensity caused by unsteady axial flow.

As Further study, effective and practical countermeasure to stabilize

aerodynamic response of inclined cables is definitely needed. ACKNOWLEDGEMENT

In this study, the authors acknowledge many former graduated students of bridge engineering laboratory of Kyoto University for their wind tunnel tests, data analysis .

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