simple model of rain-wind-induced vibrations of stayed cables_2003_wilde

Journal of Wind Engineering

and Industrial Aerodynamics 91 (2003) 873–891

Simple model of rain-wind-induced vibrations ofstayed cables

Krzysztof Wilde*, Wojciech Witkowski

Department of Structural Mechanics, Faculty of Civil Engineering, Gda !nsk University of Technology,

G. Narutowicza 11/12, 80-952 Gda !nsk, Poland

Received 21 March 2002; received in revised form 10 December 2002; accepted 24 February 2003

Abstract

This paper proposes a single-degree-of-freedom model of rain-wind-induced vibrations in

stayed cables. It is assumed that the frequency of the circumferential motion of the upper

rivulet is equal to that of cable and the rivulet amplitude is set constant for a given wind speed.

The obtained results are verified with the existing experimental data showing that these

assumptions capture the qualitative properties of the phenomenon. The explicit, analytical

expressions are derived for the aerodynamic damping and exciting force. Finally, a linear

SDOF model is derived for simple estimation of the amplitude of cable vibrations induced by

wind and rain.

r 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Stayed cables; Vibrations; Rain-wind-induced vibrations

1. Introduction

Cable vibrations of large amplitude, induced by wind and rain, were at firstobserved on the Meikonishi Bridge in Nagoya, Japan [1]. It was found that thecables, which were stable under wind action, were oscillating under a combinedinfluence of wind and rain. The observed oscillations attained amplitudes of theorder of 55 cm under wind of velocity 14m/s. The subsequent study revealed that thisphenomenon could not be accounted for by either vortex-induced oscillations or awake galloping. The frequency of the observed vibrations was much lower than the

*Corresponding author. Tel.: +48-58-347-2051; fax: +48-58-347-1670.

E-mail address: [email protected] (K. Wilde).

0167-6105/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0167-6105(03)00020-5

critical one of the vortex-induced vibrations. Further field observation revealed thatthe cable oscillations took place in the vertical plane and were mostly of single mode.With the increase of the cable length, the higher modes, up to the 4th one, appeared.The frequencies of these modes were confined to the range of 1–3Hz. It was alsoobserved that, during the oscillations, a water rivulet appeared on the lower surfaceof the cable. This rivulet, characterized by a leeward shift, oscillated incircumferential direction.A later wind tunnel investigation [1], carried out for three different cable

frequencies, i.e.: 1, 2 and 3Hz, showed that this phenomenon appeared for windvelocity from 7 to 14m/s regardless of the tested frequency. A particular care wasexercised towards the rivulet formation. It was observed that there were, in fact, tworivulets: one on the upper cable surface and the other one on the lower surface. Theserivulets oscillated in circumferential direction at the same frequency as that of thecable. Their formation point depended on the wind velocity, which has also beennoted by Bosdogianni and Olivari [2].The measurements of the aerodynamic force with the rivulets formed separately

[1–3] showed the negligible role of the lower rivulet, since it is formed in the wakebehind the cable. It has been concluded that the aerodynamic interaction betweenthe oscillating upper rivulet and cable is the primary cause of wind-rain-inducedoscillations.Further studies by Matsumoto et al. [4,5] reported that there might be

another factor triggering the rain-wind-induced oscillations, namely an axial flowgenerated at the near wake of the inclined cable and associated with the 3-D flowcharacteristic.The foundations for the modelling of the rain-wind-induced vibrations have

been laid down by Yamaguchi [6]. His study reveals that the Den Hartogmechanism (indicated in [1]) cannot explain the rain-wind oscillations pheno-menon. The proposed two-degree-of-freedom model couples the plunge motionof the cable with the circumferential motion of the upper rivulet. Thenumerical simulations showed that when wind speed is close to 10m/s, thecircular rivulet frequency coincides with that of the cable yielding a veryrapid growth of the cable amplitude. In this model, however, the frequencyof the rivulet is a function of wind velocity, which has not been confirmedexperimentally.Gu and Lu [7] also proposed two-degree-of-freedom model. In this model, the

equilibrium of forces, including inertia forces associated with the rivulet and cablemotion, yielded a set of two coupled ordinary differential equations. The numericalstudy led to the concept of dangerous zones describing the stability of the cable dueto the instantaneous rivulet position.In this paper simplification of the two-degree-of-freedom models is studied. The

SDOF model is based on the analysis of one mode that describes theaerodynamically coupled oscillations of the rivulet and the cable. Linearization ofthe proposed model enables the explicit assessment of the aerodynamic damping andexciting forces, and provides very simple formula for estimation of the cableamplitude of wind-rain-induced vibrations of stay cables.

K. Wilde, W. Witkowski / J. Wind Eng. Ind. Aerodyn. 91 (2003) 873–891874

2. Single degree-of-freedom model

The SDOF model of wind-rain-induced oscillations, derived hereafter, is based onthe following assumptions:

(1) The in-plane, small amplitude vibrations of a cable with a small sag areconsidered,

(2) The rivulet frequency equals that of the cable [1,2],(3) Amplitude ratio of the rivulet and cable is constant for given wind speed [6], and

can be modelled by a function describing the dependence of the rivuletamplitude on wind speed.

(4) Initial position of the upper rivulet is a function of the wind speed [1],(5) Mass of the rivulet is negligible compared with that of the cable,(6) The considered mode of oscillations, its frequency, the properties of the cable

are taken from [6] and the steady wind force coefficients are after [6,7].

A cable under action of the incoming flow of velocity U0 has an inclination angle aand yaw angle b (Fig. 1a). The effective wind speed and the angle of attack in theplane normal to the cable axis are given by

U ¼ U0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2 bþ sin2 a sin2 b

qð1Þ

and

g ¼ arcsinsin a sin bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cos2 bþ sin2 a sin2 bq

0B@

1CA: ð2Þ

Based on the assumptions given above, the equation of the in-plane motion for thecable takes the following form

.y þ o2y þ 2xso ’y ¼ �F

m; ð3Þ

α

α

γ

β

β

directionof motion

U0

U

U0

U0 sinβ

U0 sinβ

U 0 co

sβ

U0 cosβ

U0 sin sinβ α

β α U0 sin sin

yy

(a) (b) (c)

motion plane

Fig. 1. (a–c) Cable orientation.

K. Wilde, W. Witkowski / J. Wind Eng. Ind. Aerodyn. 91 (2003) 873–891 875

where y is the vertical displacement of the cable in the motion plane (Fig. 1c), xs isthe structural damping ratio, o is the cable circular frequency, m denotes the mass ofthe cable per unit length. The term F in Eq. (3) is the in-plane aerodynamic force perunit length of the cable and the upper rivulet. The aerodynamic force is computedusing the steady force coefficients taken for the instantaneous relative wind velocityUrel and the instantaneous relative angle of attack f� (Fig. 2a) defined by thefollowing formulas:

Urel ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðR’y sinðyþ yiÞ þ U sin gþ ’yÞ2 þ ðR’y cosðyþ yiÞ þ U cos gÞ2

q; ð4Þ

f� ¼ arctanR’y sinðyþ yiÞ þ U sin gþ ’y

R’y cosðyþ yiÞ þ U cos g

; ð5Þ

where the cable radius is denoted by R ¼ D=2; yi is the initial position of the upperrivulet, measured counterclockwise from the vertical axis. The oscillations of therivulet, y; are assumed to be harmonic, i.e.

y ¼ am sinðotÞ; ð6Þ

where am denotes the amplitude and o is the rivulet frequency equal to that of thecable.Yamaguchi [6] showed that the rivulet-cable amplitude ratio of the considered

mode depends on wind speed. The function describing the amplitude ratio has apeak at the wind speed coinciding with the largest amplitude cable vibrationsand rapidly decreases for smaller and larger wind speeds. In this study the amplitudeof the rivulet is considered to be a function of wind speed U0 in the followingform:

amðU0Þ ¼ a1 exp �ðU0 � UmaxÞ

2

a2

; ð7Þ

where a1; a2 and Umax are constants to be determined for a given cable. Note that forU0 ¼ Umax the nondimensional rivulet amplitude am equals a1 and for other valuesof U0 it gradually vanishes. Function (7) models small decrease of the rivulet

φ

θ i

θi

θi

θ

θ

θ + θ i

U

Urel Urel

FL

FD

F

y

y

Rθ

γ

Rθ

∗

(a) (b)

Fig. 2. Relative flow (a) and action of quasi-steady wind force (b).


amplitudes in the neighbourhood of Umax: Experimental study [1] showed thedecrease of the amplitude of order of 12%. Function (7) also models no rivuletcondition by rapid reduction of rivulet amplitude for small and high wind speeds.Flamand [3] showed that if wind speed is smaller than 7m/s the upper part of thecable remains dry whereas if wind speed exceeds 12m/s the upper rivulet is pulledaway by the flow.Projection of the components of the aerodynamic force FD and FL (Fig. 2b) on the

y axis becomes

F ¼U2

relDr2

ðCLðfeÞcos f� þ CDðfeÞsin f

�Þ: ð8Þ

In Eq. (8) r is the fluid density, CD; CL denote the drag and the lift coefficient,respectively. Angle fe; used in the experimental studies [6,7] is computed by thefollowing formula:

fe ¼ f� � y� yi: ð9Þ

3. Numerical simulations

The cable under consideration has the following properties [6]: mass per unitlength m ¼ 10:2 kg; diameter D ¼ 0:154m, frequency f ¼ 2Hz and structuraldamping ratio xs ¼ 0:002: The coefficients CD; CL taken from [6] (for the case d=D ¼0:1; where d and D are rivulet and cable diameters, respectively) and [7] are depictedin Fig. 3. Their values interpolated in the range of interest, are given by

CD ¼ 0:0831f3e � 0:885f2

e � 0:5382fe þ 1:5555; ð10Þ

CL ¼ 1:0081f3e þ 1:7625f2

e þ 0:2507fe � 0:3909; ð11Þ

-100 -80 -60 -40 -20 0 20 40angle of attack �e (deg)

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

stea

dy w

ind

forc

e co

effi

cien

ts C

D, C

L

CD

CL

GuYamaguchi

Fig. 3. Steady wind force coefficients for cable with rivulet.


for Yamaguchi’s data [6] and

CD ¼ 5:1350f5e � 0:8484f4

e � 2:1984f3e þ 0:6219f2

e � 0:0931fe þ 1:0204; ð12Þ

CL ¼ � 12:6840f5e � 11:6705f4

e � 1:6217f3e þ 1:4189f2

e

þ 0:5279fe � 0:1758; ð13Þ

for Gu and Lu [7]. Angle fe is expressed in radians. Following Hikami and Shiraishi,the inclination and the yaw angles are assumed to be 45�.In order to corroborate the assumption made with regard to Eq. (7), Fig. 4 depicts

the cable amplitudes versus wind speed for CD; CL taken from [6]. The amplitudesare taken from the steady-state response at time above 40 s and are computed withrespect to the new equilibrium positions determined by each wind speed. The rivuletamplitudes, am; varies from 0.05 to 0.45. The numerical simulations are carried outusing Runge–Kutta scheme of the fourth order for the initial conditionsy0 ¼ 0:001m, ’y0 ¼ 0 or y0 ¼ 0:03m, ’y0 ¼ 0: As it can be observed for all consideredrivulet amplitudes, the cable amplitudes grow steadily to attain maximum value atwind speed of about 9.5m/s. This and the assumption of the zero rivulet amplitudesat wind speed smaller than 7m/s and higher than 12m/s [3] yield the following valuesfor the coefficients in Eq. (7): Umax ¼ 9:5 m=s; a1 ¼ 0:448 and a2 ¼ 1:5842: Theassumed, in the following simulations, variations of the rivulet amplitude, y; togetherwith the initial position of the rivulet, yi; [1] are shown in Fig. 5.Firstly, the cable response for three different cable frequencies, i.e.: 1, 2 and 3Hz

are studied. In Fig. 6 the calculated cable amplitudes are compared with theexperimental ones [1]. Note that, despite the quantitative difference between thenumerical and the experimental results, which descends from different cablecharacteristics, the qualitative character is preserved. That is, with the growth ofthe cable stiffness, the amplitudes decline. This fact was also observed in [7]. Thelargest responses are independent of both the wind velocity and frequency, in a sense

5 6 7 8 9 10 11 12 13wind velocity U0 (m/s)

0

1.25

2.5

3.75

5

am=0.05

am=0.1

am=0.15

am=0.2

am=0.25

am=0.45

cabl

e am

plitu

e (c

m)

Fig. 4. Cable response due to different cable amplitudes.


that they occur for the U0 ¼ 9:5m/s regardless of the cable stiffness. Note that thereare no significant differences between results obtained with Yamaguchi’s forcecoefficients and those obtained by Gu and Lu.Variations of the phase shift, denoted by c; between cable displacements and

aerodynamic force are studied following the concept presented in [1]. At selectedamplitude of cable oscillations, called ‘set-amplitude’, the phase shift between peaksof the cable displacements and the aerodynamic force are measured. In [1], the setamplitude is 5 cm while in this paper it is chosen as 2 cm. The tests are carried outassuming the cable frequency to be equal to 2Hz. For the reference the steady-statecable amplitudes from the numerical study and the experiment are plotted in Figs. 7aand b. The phases, c; for the cable without rivulet are shown in Figs. 7c and d.The numerically determined phases are constant functions of U0; while those

5 6.5 8 9.5 11 12.5 14

wind velocity U0 (m/s)

20

40

60

80

θ iθ

(deg

)

initial position of rivulet

Hikami&Shiraishiinterpolation

Fig. 5. Variation of the rivulet amplitudes vs. wind speed.


0

2

4

6

8

10

cabl

e am

plit

ude

(cm

)

f requency1 Hz2 Hz3 Hz

Gu&Lu

Yamaguchi

5 6 7 8 9 10 11 12 13 14 15wind velocity U 0 (m/s)

0

5

10

15

20

frequency1 Hz2 Hz3 Hz

(a) (b)

cabl

e am

plit

ude

(cm

)

Fig. 6. Cable responses for different frequencies: (a) numerical simulation, (b) Hikami and Shiraishi [1].


from the experiment vary. However, they also lie in the neighbourhood of �90�.Negative sign of the phases indicates the damping characteristics of the aerodynamicforce F :The computed phases for the cable with rivulet (Fig. 7c) are similar to those of the

no rivulet case for wind speed below 7.5m/s and above 11m/s. For wind speed rangefrom 7.5 to 11m/s the phases are positive and about 90�. The positive sign of thephase shifts indicate the exciting characteristics of aerodynamic force F : The changein the sign of the phase shift was also observed in the experiment [1] (Fig. 7d).The nature of the aerodynamic force can be described by the following formula [1]

Ft ¼ Fj jsinðcÞ; ð14Þ

5 6 7 8 9 10 11 12 13wind velocity U 0 (m/s)

0

1

2

3

4

5

cabl

e am

plitu

dey

(cm

)

set amplitude 2 cm

Gu&Lu

Yamaguchi

5 6 7 8 9 10 11 12 13 14 15wind velocity U0 (m/s)

0123456789

cabl

e am

plitu

dey

(cm

)

set amplitude 5 cm

5 7 9 11 13

wind velocity U 0 (m/s)-180

-90

0

90

180

phas

e la

g �

(deg

)

with rainwithout rain

Gu&Lu

Yamaguchi

6 7.5 9 10.5 12 13.5 15

wind velocity U0 (m/s)-180

-90

0

90

180

phas

e la

g ψ

(de

g)


5 7 9 11 13


-5

0

5

Ft (

N/m

)


Gu&Lu

Yamaguchi

6 7.5 9 10.5 12 13.5 15


-10

-5

0

5

10

Ft (

N/m

)


(a) (b)

(c) (d)

(e) (f)

Fig. 7. Comparison between numerical and experimental results: (a) cable response (numerical),

(b) Hikami and Shiraishi [1], (c) phase lag (numerical), (d) Hikami and Shiraishi [1], (e) Ft numerical

and (f) Hikami and Shiraishi [1].


where F is the magnitude of the exciting force and c is the phase shift. When force Ft

is positive, it indicates an exciting character of force F and when Ft is negative, F isregarded as a damping force. The comparison between the numerical and theexperimental results is illustrated in Figs. 7e and f, respectively. Note that the forcechanges from the damping one to the exciting, and then again to the damping one.The range in which the positive sign of Ft occurs (Fig. 7e) corresponds to that wherethe phase shift is positive (Fig. 7c) which, in turn, corresponds to the steady-stateamplitudes larger than 2 cm (Fig. 7a). There are no significant differences betweenresults based on different steady force coefficients.Fig. 8 shows an example of time history of the aerodynamic force for wind speed

U0 ¼ 9:5m/s. In this case the computations were performed using the steady-windforce coefficients from [6]. The force is nonlinear and periodic. At the beginning ofthe motion (Fig. 8b) the force precedes the response of the cable, exhibiting thus theexciting characteristic. Note that, as the time unfolds, another component appears.In the steady-state response, seen in Fig. 8c, this component has a significantamplitude and lags behind the response of the cable. It indicates that the

0 25 50 75 100time (s)

3.5

4

4.5

5

5.5

forc

e am

plitu

de F

(N

/m)

A B

2 3 4 5time (s)

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

cabl

e am

plitu

de y

(m

)

3.5

4

4.5

5

5.5

6

forc

e am

plitu

de F

(N

/m)

Fy

97 98 99 100time (s)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

cabl

e am

plitu

de y

(m

)

3.5

4

4.5

5

5.5

forc

e am

plitu

de F

[N

/m]Fy

(a)

(c)(b)

Fig. 8. Time histories of cable displacement cable and force for CD; CL from Yamaguchi [6]:

(a) time history of aerodynamic force F for U0 ¼ 9:5m/s and (b) transient motion—A, (c) steady-state

response—B.


aerodynamic force acting on the cable with moving rivulet can be expressed by twocomponents.

4. Simplified models of the aerodynamic force

In the model presented in the previous section it is not possible to assess neitherthe exciting component of the aerodynamic force nor the aerodynamic damping.Therefore, the formula for the aerodynamic force is expanded and expressed in termsof the cable velocity ’y: Three simplified models are considered. Model 1 assumeslinearization of all trigonometric functions, models 2 and 3 additionally assumeslinearization of steady-state force coefficients by tangent and least square fit,respectively.

4.1. Model 1

It is assumed that the drag and lift coefficients change in an arbitrary way. Theexpansion is based on the following assumptions descending from preliminarynumerical simulations:

1. The aerodynamic force F can be expressed by terms containing cable velocity ’y upto the first power.

2. The second (and higher) powers of the rivulet amplitude am are small incomparison with am and therefore are neglected.

3. The term R’y cosðyþ yiÞ in Eqs. (4) and (5) is much smaller than U cos g and isconsidered negligible.

4. The function arctanðyÞ in Eq. (5) is Taylor-expanded retaining only the linearterm. The equilibrium position for expansion descends from inclination and yawangles and is assumed as g:

5. The sine and cosine functions in Eq. (8) are also expanded about g retaining onlylinear terms.

As a result, the relative velocity, the relative angle of attack and the aerodynamicforce read

UrelDffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðR’y sinðyþ yiÞ þ U sin gþ ’yÞ2 þ ðU cos gÞ2

q; ð15Þ

f�DarctanðgÞ þðR’y sinðyþ yiÞ þ U sin gþ ’yÞ=U cos g � g

1þ g2; ð16Þ

fe ¼ f� � y� yi; ð17Þ

F ¼U2

relDr2

ðCLðfeÞðcos g� sin gðf� � gÞÞ þ CDðfeÞðsin gþ cos gðf� � gÞÞÞ:

ð18Þ


The substitution of formulas (15) and (16) into (18) yield the polynomial of thethird order with respect to ’y: The terms involving the same powers of ’y had beencollected together followed by the removal of the higher powers. Consequently, theterms involving the same powers of am had been collected retaining only the zero andthe first powers. The partially linearized equation becomes

.y þ o2y þ 2xso ’y ¼ �1

mðFdampðtÞ þ FexcðtÞÞ; ð19Þ

where

FdampðtÞ ¼ ’yðZ1 þ amZ2ðtÞÞ ð20Þ

is the aerodynamic damping force and

FexcðtÞ ¼ F1 þ amF2ðtÞ ð21Þ

is the exciting force. The coefficients in (20) and (21) are given in Appendix A.Dividing the right-hand side of Eq. (20) by 2 ’ymo yields the formula for aerodynamicdamping ratio

xa ¼Z1 þ amZ2ðtÞ

2mo: ð22Þ

Examination of the coefficients Z1 and Z2 indicates that the damping ratiodepends on time. In contrast to the cable without rivulet, for which the dampingratio is solely dependent on CD [8]. Here, due to the presence and the oscillations ofthe upper rivulet, the aerodynamic damping is a function of CD; CL and time. Thismodel is applicable to problems with steady-force coefficients rapidly varying withthe instantaneous angle of attack fe:

4.2. Model 2

This model, apart from all the assumptions of the previous section, additionallyassumes that the functions for CD and CL can be represented as linear functions ofthe angle fe i.e.:

CD ¼ D1fe þ D2; ð23Þ

CL ¼ L1fe þ L2; ð24Þ

where the coefficients are defined as follows

D1 ¼dCD

dfe

��fe¼feq

e

; ð25Þ

D2 ¼ CDðfeqe Þ; ð26Þ

L1 ¼dCL

dfe

��fe¼feq

e

; ð27Þ

L2 ¼ CLðfeqe Þ ð28Þ


and

feqe Dg� yi: ð29Þ

In Eq. (29) use is made of the fact that the equilibrium point for f�; given by (16),may be assumed as g; and since g is the function of the inclination and the yaw angle,feqe depends only on wind velocity U0: The coefficients D1; D2; L1; L2 and feq

e are tobe determined for each value of U0:The sine term of ’y sinðyi þ am sinðotÞÞ in Eq. (16), upon the Taylor expansion, is

replaced with the term am o cosðotÞsin yi: Thus the expression for the aerodynamicforce become

F ¼U2

relDr2

ððL1fe þ L2Þðcos g� sin gðf� � gÞÞ

þ ðD1fe þ D2Þðsin gþ cos gðf� � gÞÞÞ: ð30Þ

Grouping the terms in the above equation yields the formula for the damping andthe exciting force, i.e.

FdampðtÞ ¼ ’yðZ3 þ AD sinðot þ yDÞÞ; ð31Þ

FexcðtÞ ¼ F3 þ AE sinðot þ yEÞ; ð32Þ

where all the coefficients in (31) and (32) are the functions of g;D1;D2;L1;L2;feqe ; am

and are defined in Appendix A. Numerical simulations, based on Eqs. (31) and (32),reveal that the oscillating part of the damping force and the constant term F3 inexpression for the exciting force have a negligible effect on cable response. Thus,

FdampðtÞ ¼ Z3 ’y; ð33Þ

FexcðtÞ ¼ AE sinðot þ yEÞ: ð34Þ

The formula for the aerodynamic damping ratio is then

xa ¼Z3

2mo: ð35Þ

Finally, the equation of motion reads

.y þ o2y þ ’y 2xsoþZ3

m

¼ �

1

mAE sinðot þ yEÞ: ð36Þ

Formulas for amplitude AE and phase shift yE are given in Eqs. (A.14) and (A.15).Note that since Z3 is time-invariant, Eq. (36) represents a harmonic oscillator drivenin harmonic fashion. In this model the exciting component of the aerodynamic forceissue from the presence and oscillation of the rivulet.The phase shift between motion of the cable, y; and rivulet, y; is found to be

yyy ¼ 90� þ yE: ð37Þ

The phase shift between cable and rivulet is a function of wind speed, initial rivuletposition, cable orientation and its radius.


4.3. Model 3

In this model the functions of the stead-wind force coefficients are also linearized.In this model the coefficients in Eqs. (23) and (24) are found using first-orderpolynomial fitting with the constrains expressed by (26) and (28), satisfied on anglefeqe (29). The fitting is conducted for angles, fe; from the range feq

e � 15�pfepfeqe þ 15�: The transformations proceed in the same fashion as in model 2

yielding Eqs. (33)–(36).

4.4. Numerical results from simplified models

The numerical results obtained from the simplified models for data both from [6,7]are compared with the solution of the full model (3) (Fig. 9). The frequency of thecable was set to 2Hz. It may be observed that the greatest discrepancies arepronounced for wind speed around 9.5m/s. Model 1 underestimates the amplitudesregardless of the used aerodynamic coefficients. Model 2, with CD and CL linearizedby tangent, gives larger amplitudes of the cable. This is due to the fact that the valuesof steady force coefficients vary rapidly with the changes of angle fe: Better resultsare obtained from model 3, where the curves CD and CL are linearized by fitting onthe selected range of fe: Generally speaking, the nonlinear curves of steady forcecoefficients can not be represented by a linear function. Model 3 describes theprocedure of the optimal linearization of the aerodynamic force for estimation of themaximum cable amplitude.Fig. 10 presents the time histories of the damping and exciting force components.

The steady-state responses are computed for models 1, 2 and 3 for U0 ¼ 9:5m/s. Formodel 1 both aerodynamic force components are periodic and nonlinear. Note thatthe exciting force computed for Gu and Lu’s force coefficients has the additional


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

cabl

e am

plitu

dey

(m)

solutionexact

model 1model 2model 3


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

cabl

e am

plitu

dey

(m)

solutionexact

model 1model 2model 3

(a) (b)

Fig. 9. Cable amplitudes from different models: (a) CD; CL from Yamaguchi [6] and (b) CD; CL from Gu

and Lu [7].


peaks coinciding with the peaks of the damping component. Models 2 and 3represent the force components by sine function and thus neglect the additionalpeaks in the time histories. The amplitudes of forces from models 2 and 3 are largerthan those obtained from model 3. Note that there are no differences in modellingthe forces by models 2 and 3 for the Yamaguchi’s force coefficients. The cableamplitudes at wind speed U0 ¼ 9:5m/s (Fig. 9) are similar for those models. Thederivative of Yamaguchi’s CD curve (Fig. 3) has small values and do not vary in therange of interest (fe about 20

�). Thus, there are no significant differences betweenlinearization by tangent (model 2) and constrained fitting (model 3). The derivativeof CD curve, given by Gu and Lu, has large values and changes considerable in therange of interest, yielding large error in values of CD used in simulations by model 2.Therefore, there are differences in modelling the damping (Fig. 10b) and excitingforce component (Fig. 10d) by models 2 and 3. Those differences are reflected on the

98 99 100time (s)

-1.5

-1

-0.5

0

0.5

1

Fda

mp (

N/m

)

98 99 100time (s)

-1.5

-1

-0.5

0

0.5

1

Fda

mp (

N/m

)(a) (b)

(c) (d) 98 99 100

time (s)

-1.5

-1

-0.5

0

0.5

1

1.5

Fex

c (N

/m)

model 1 model 2 model 3

98 99 100time (s)

-1.5

-1

-0.5

0

0.5

1

1.5

Fec

x (N

/m)

Fig. 10. Time histories of the damping and exciting component of the aerodynamic force: (a) damping

force (CD; CL from Yamaguchi), (b) damping force (CD; CL from Gu and Lu), (c) exciting force (CD; CL

Yamaguchi) and (d) exciting force (CD; CL from Gu and Lu).


computed cable amplitudes (Fig. 9b). Model 2 overestimates the exact solution forwind speed U0 ¼ 9:5m/s by 0.036m, while model 3 by 0.0184m.The maximum amplitude of the oscillating cable is determined by the added

aerodynamic damping and the amplitude of the exciting force. The contribution ofthe aerodynamic damping, resulting form the presence of the rivulet, can be assess bythe ratio

G ¼xaxs

¼Z3

2moxsð38Þ

Fig. 11 shows the cable aerodynamic damping, computed from model 3, for thecable with and without the rivulet. The no rivulet case is determined through theformula derived by Macdonald [8] for the in-plane cable motion. The cable damping,for Yamaguchi’s data, is larger than for Gu and Lu’s force coefficients. This isattributed to the differences in values of the CD and CL curves and their firstderivatives (Fig. 3). In the range of interest, both CL curves are similar, while themean value of CD for Gu and Lu is around 1.2 and for Yamaguchi’s data is 1.55.This difference is a primary factor for the differences in computing coefficients D1;D2; L1; L2 which determine the aerodynamic damping (Eq. (A.13)). Note that theaerodynamic damping computed for Gu and Lu’s data is lower than damping of thecable without rivulet for all considered wind speeds.The amplitudes of the exciting forces, computed for the Yamaguchi and Gu and

Lu steady force coefficients, versus wind speed are shown in Fig. 12. The forceenvelopes are computed from model 3. The largest amplitudes for both aerodynamicdata are obtained for U0 ¼ 9:5m/s. The maximum amplitude for Yamaguchi’s datais larger than the force amplitude for Gu and Lu force coefficients. The Yamaguchi’sCD and CL overestimate the effect of rivulet since they are determined for the rivuletof a relatively large size. However, the computed amplitudes of the cable oscillations

5 7 9 11 13wind velocity U 0 (m/s)

1

2

3

4

Γ(no

ndim

.) Yamaguchi

MacDonald(without rivulet)

Gu&Lu

Fig. 11. Contribution of the aerodynamic damping vs. wind speed.


are similar to those obtained from Gu and Lu’s coefficients (Figs. 6 and 9), sinceboth components of the aerodynamic force are overestimated.

5. Conclusions

The phenomenon of rain-wind-induced vibrations of the stayed cables has beenstudied. The derived SDOF models assume the circumferential oscillation of theupper rivulet with the same frequency as the cable and constant rivulet—cableamplitude ratio for given wind speed. The aerodynamic force has been described byquasi-steady formulation. These assumptions oversimplify the problem, since theydo not address the problem of adhesive forces acting between the rain water and thesurface, the flow of the water on the cable and most of all the effects of the threedimensional air flow around the oscillating cable and rivulet. Nevertheless, theproposed models describe the phenomenon by simple formula that can be easilyadopted for estimation of the maximum amplitudes of cable oscillations induced bysimultaneous action of wind and rain.The study on the linearized models have revealed that the aerodynamic force

acting on the cable may be considered as a superposition of the damping force andthe exciting one. These forces depend explicitly on the oscillation of the rivulet aswell as on the cable orientation and the steady-wind coefficients. The factor thatplays the major role in determining the maximum amplitude of the oscillating cableis the amplitude of the exciting force.The proposed models have been based on the available aerodynamic data for cable

with rivulet. The applied force coefficients have been determined for the horizontalcables, and thus, the effects of axial flow could not be incorporated. In addition there

Fig. 12. Amplitudes of the exciting force from model 3 vs. wind speed.


are no systematic experimental studies, that describe all variables of the wind rain-induced vibrations of cables required for, not only qualitative but also quantitative,verification of the proposed models.

Acknowledgements

The contribution of Prof. Yamaguchi, Saitama University, Japan to the presentedwork is greatly appreciated. The authors are thankful to Prof. Gu, Tongji University,China, for provided research results.

Appendix A

A.1. Coefficients for model 1

The aerodynamic damping force:

Z1 ¼DUr2c5

ðCDc7 þ CLc8Þ; ðA:1Þ

Z2 ¼DRor cosðotÞsinðyi þ am sinðotÞÞ

c5½CDðc2ð4þ g2Þ þ c1ðc4c5 � gc6ÞÞ

þ CLðc1c5 � c2ð3c3 þ c4 � 2gþ c4g2 � g3ÞÞ: ðA:2Þ

The exciting force:

F1 ¼DU2r2c5

½CDððc21 þ c22Þðc2c6 þ c1ðc4c5 � gc6ÞÞÞ

þ CLðc1c22g2 þ c31c5 � c32ðc3 þ c4 � 2gþ c4g2 � g3Þ

þ c21c2ð�c4c5 þ gc6ÞÞ; ðA:3Þ

F2 ¼DURro cosðotÞsinðyi þ am sinðotÞÞ

2c5ðCDc7 þ CLc8Þ; ðA:4Þ

where

c1 ¼ cosðgÞ; ðA:5Þ

c2 ¼ sinðgÞ; ðA:6Þ

c3 ¼ tanðgÞ; ðA:7Þ

c4 ¼ arctanðgÞ; ðA:8Þ

c5 ¼ 1þ g2; ðA:9Þ

c6 ¼ 2þ g2; ðA:10Þ


c7 ¼ c21 þ c22ð5þ 2g2Þ þ 2c1c2ðc4c5 � gc6Þ; ðA:11Þ

c8 ¼ c2ðc1 þ 2c1g2 þ c2ð�3c4 � 2c4 þ 4g� 2c4g2 þ 2g3ÞÞ: ðA:12Þ

A.2. Coefficients for models 2 and 3

The aerodynamic damping force:

Z3 ¼1

Ro sinðyiÞðD1c11 þ D2c12 þ L1c13 þ L2c14Þ: ðA:13Þ

The exciting force:

AE ¼ am

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðD1c9 þ L1c10Þ

2 þ ðD1c11 þ D2c12 þ L1c13 þ L2c14Þ2

qðA:14Þ

yE ¼ arctanD1c11 þ D2c12 þ L1c12 þ L2c14

D1c9 þ L1c10

; ðA:15Þ

where

c9 ¼ �RU2rðc21 þ c22Þðc2ð1þ c5Þ þ c1ðc4c5 � ð1þ c5ÞgÞÞ

c5ðA:16Þ

c10 ¼ �RU2rðc1c22ð�1þ c5Þ þ c31c5 þ c21c2ð�c4c5 þ gþ c5gÞ þ c32ð�c3 � c4c5 þ gþ c5gÞÞ

c5

ðA:17Þ

c11 ¼1

c25Uor sinðyiÞðR2ðc21ð2c4c5 � 2g� c5g� c5yiÞ

þ c22ð6c4c5 þ 2c4c25 þ c3ð4þ 3c5Þ � 6g� 5c5g� 3c5yi � 2c25yiÞ

þ c1c2 2ð1þ g2Þ þ 2c25ðc4 � gÞðc4 � yiÞ�

þ c5ð1� 4c4gþ 2g2 þ 2gyiÞÞÞ; ðA:18Þ

c12 ¼R2Uor sinðyiÞðc21 þ c22ð3þ 2c5Þ þ 2c1c2ðc4c5 � ð1þ c5ÞgÞÞ

c5; ðA:19Þ

c13 ¼R2Uor sinðyiÞ

c25ððc21c5 þ c1c2ð2c4ð�1þ c5Þc5 þ 2g� c5gþ c5yi � 2c25yiÞ

þ c22ð�2� 4c23 þ 3c5 � 2c24c25 þ 4c4c5gþ 2c4c

25g� 2g2 � 2c5g2

� 2c4c25yi � 2c5gyi � 2c25gyi þ c3ð�6c4c5 þ 6gþ 3c5gþ 3c5yiÞÞÞ ðA:20Þ

c14 ¼R2Uor sinðyiÞc2ðc1ð�1þ 2c5Þ þ c2ð�3c3 þ 2ð�c4c5 þ gþ c5gÞÞÞ

c5:

ðA:21Þ


References

[1] Y. Hikami, N. Shiraishi, Rain-wind-induced vibrations of cables in cable stayed bridges, J. Wind. Eng.

Ind. Aerodyn. 29 (1988) 409–418.

[2] A. Bosdogianni, D. Olivari, Wind and rain-induced oscillations of cables of stayed bridges, J. Wind

Eng. Ind. Aerodyn. 64 (1996) 171–185.

[3] O. Flamand, Rain-wind-induced vibration of cables, J. Wind Eng. Ind. Aerodyn. 57 (1995) 353–362.

[4] M. Matsumoto, N. Shiraishi, H. Shirato, Rain-wind-induced vibration of cables in cable-stayed

bridges, J. Wind Eng. Ind. Aerodyn. 41–44 (1988) 2011–2022.

[5] M. Matsumoto, T. Saitoh, M. Kitazawa, H. Schirato, T. Nishizaki, Response characteristics of rain-

wind-induced vibration of stay cables of cable-stayed bridges, J. Wind Eng. Ind. Aerodyn. 57 (1995)

323–333.

[6] H. Yamaguchi, Analytical study on growth mechanism of rain vibration of cables, J. Wind Eng. Ind.

Aerodyn. 33 (1990) 73–80.

[7] M. Gu, Q. Lu, Theoretical analysis of wind-rain-induced vibration of cables of cable-stayed bridges,

APCWE-5, J. Wind Eng. 89 (2001) 125–128.

[8] J.H.G. Macdonald, Separation of the contributions of aerodynamic and structural damping in

vibrations of inclined cables, J. Wind Eng. Ind. Aerodyn. 90 (2001) 19–39.


simple model of rain-wind-induced vibrations of stayed cables_2003_wilde

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