Krzysztof Wilde：Simple model of rain-wind-induced vibrations of stayed cables

Journal of Wind Engineering

and Industrial Aerodynamics91(2003)873–891

Simple model of rain-wind-induced vibrations of

stayed cables

Krzysztof Wilde*,Wojciech Witkowski

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

G.Narutowicza11/12,80-952Gda!nsk,Poland

Received21March2002;received in revised form10December2002;accepted24February2003 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 veri?ed 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.

r2003Elsevier 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?rst observed on the Meikonishi Bridge in Nagoya,Japan[1].It was found that the cables,which were stable under wind action,were oscillating under a combined in?uence of wind and rain.The observed oscillations attained amplitudes of the order of55cm under wind of velocity14m/s.The subsequent study revealed that this phenomenon could not be accounted for by either vortex-induced oscillations or a wake 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:wild@pg.gda.pl(K.Wilde).

0167-6105/03/$-see front matter r2003Elsevier Science Ltd.All rights reserved.

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

critical one of the vortex-induced vibrations.Further ?eld observation revealed that the 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 con?ned to the range of 1–3Hz.It was also observed that,during the oscillations,a water rivulet appeared on the lower surface of the cable.This rivulet,characterized by a leeward shift,oscillated in circumferential direction.

A later wind tunnel investigation [1],carried out for three different cable frequencies,i.e.:1,2and 3Hz,showed that this phenomenon appeared for wind velocity from 7to 14m/s regardless of the tested frequency.A particular care was exercised towards the rivulet formation.It was observed that there were,in fact,two rivulets:one on the upper cable surface and the other one on the lower surface.These rivulets oscillated in circumferential direction at the same frequency as that of the cable.Their formation point depended on the wind velocity,which has also been noted 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 wake behind the cable.It has been concluded that the aerodynamic interaction between the oscillating upper rivulet and cable is the primary cause of wind-rain-induced oscillations.

Further studies by Matsumoto et al.[4,5]reported that there might be another factor triggering the rain-wind-induced oscillations,namely an axial ?ow generated at the near wake of the inclined cable and associated with the 3-D ?ow characteristic.

The foundations for the modelling of the rain-wind-induced vibrations have been laid down by Yamaguchi [6].His study reveals that the Den Hartog mechanism (indicated in [1])cannot explain the rain-wind oscillations pheno-menon.The proposed two-degree-of-freedom model couples the plunge motion of the cable with the circumferential motion of the upper rivulet.The numerical simulations showed that when wind speed is close to 10m/s,the circular rivulet frequency coincides with that of the cable yielding a very rapid growth of the cable amplitude.In this model,however,the frequency of the rivulet is a function of wind velocity,which has not been con?rmed experimentally.

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 cable motion,yielded a set of two coupled ordinary differential equations.The numerical study led to the concept of dangerous zones describing the stability of the cable due to the instantaneous rivulet position.

In this paper simpli?cation of the two-degree-of-freedom models is studied.The SDOF model is based on the analysis of one mode that describes the aerodynamically coupled oscillations of the rivulet and the cable.Linearization of the proposed model enables the explicit assessment of the aerodynamic damping and exciting forces,and provides very simple formula for estimation of the cable amplitude of wind-rain-induced vibrations of stay cables.

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2.Single degree-of-freedom model

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

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

considered,

(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 rivulet amplitude 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 coef?cients are after [6,7].A cable under action of the incoming ?ow of velocity U 0has an inclination angle a and yaw angle b (Fig.1a ).The effective wind speed and the angle of attack in the plane normal to the cable axis are given by

U ?U 0??????????????????????????????????????????

cos 2b tsin 2a sin 2b q e1Tand

g ?arcsin sin a sin b ??????????????????????????????????????????cos 2b tsin 2a sin 2b

q 0

B @1

C A :

e2T

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

.y to 2y t2x s o ’y ?àF

m

;

e3T

Fig.1.(a–c)Cable orientation.

K.Wilde,W.Witkowski /J.Wind Eng.Ind.Aerodyn.91(2003)873–891875

where y is the vertical displacement of the cable in the motion plane (Fig.1c ),x s is

the structural damping ratio,o is the cable circular frequency,m denotes the mass of the cable per unit length.The term F in Eq.(3)is the in-plane aerodynamic force per unit length of the cable and the upper rivulet.The aerodynamic force is computed using the steady force coef?cients taken for the instantaneous relative wind velocity U rel and the instantaneous relative angle of attack f ?(Fig.2a )de?ned by the following formulas:

U rel ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

eR ’y sin ey ty i TtU sin g t’y T2teR ’y

cos ey ty i TtU cos g T2q ;e4Tf ??arctan R ’y sin ey ty i

TtU sin g t’y R ’y

cos ey ty i TtU cos g ;

e5T

where the cable radius is denoted by R ?D =2;y i is the initial position of the upper

rivulet,measured counterclockwise from the vertical axis.The oscillations of the rivulet,y ;are assumed to be harmonic,i.e.

y ?a m sin eo t T;

e6T

where a m denotes the amplitude and o is the rivulet frequency equal to that of the cable.

Yamaguchi [6]showed that the rivulet-cable amplitude ratio of the considered mode depends on wind speed.The function describing the amplitude ratio has a peak at the wind speed coinciding with the largest amplitude cable vibrations and rapidly decreases for smaller and larger wind speeds.In this study the amplitude of the rivulet is considered to be a function of wind speed U 0in the following form:

a m eU 0T?a 1exp

àeU 0àU max T2

2 ;e7T

where a 1;a 2and U max are constants to be determined for a given cable.Note that for U 0?U max the nondimensional rivulet amplitude a m equals a 1and for other values of U 0it gradually vanishes.Function (7)models small decrease of the rivulet

(a)

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

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

876

amplitudes in the neighbourhood of U max:Experimental study[1]showed the decrease of the amplitude of order of12%.Function(7)also models no rivulet condition by rapid reduction of rivulet amplitude for small and high wind speeds. Flamand[3]showed that if wind speed is smaller than7m/s the upper part of the cable remains dry whereas if wind speed exceeds12m/s the upper rivulet is pulled away by the?ow.

Projection of the components of the aerodynamic force F D and F L(Fig.2b)on the y axis becomes

F?U2

rel

D r

2

eC Lef eTcos f?tC Def eTsin f?T:e8T

In Eq.(8)r is the?uid density,C D;C L denote the drag and the lift coef?cient, respectively.Angle f e;used in the experimental studies[6,7]is computed by the following formula:

f e?f?àyày i:e9T

3.Numerical simulations

The cable under consideration has the following properties[6]:mass per unit length m?10:2kg;diameter D?0:154m,frequency f?2Hz and structural damping ratio x s?0:002:The coef?cients C D;C L taken from[6](for the case d=D?0:1;where d and D are rivulet and cable diameters,respectively)and[7]are depicted in Fig.3.Their values interpolated in the range of interest,are given by

C D?0:0831f3

e à0:885f2

e

à0:5382f et1:5555;e10T

C L?1:0081f3

e t1:7625f2

e

t0:2507f eà0:3909;e11T

angle of attack

e (deg)

s

t

e

a

d

y

w

i

n

d

f

o

r

c

e

c

o

e

f

f

i

c

i

e

n

t

s

C

D

,

C

L

Fig.3.Steady wind force coef?cients for cable with rivulet.

K.Wilde,W.Witkowski/J.Wind Eng.Ind.Aerodyn.91(2003)873–891877

for Yamaguchi’s data [6]and

C D ?5:1350f 5e à0:8484f 4e à2:1984f 3e t0:6219f 2

e à0:0931

f e t1:0204;e12TC L ?à12:6840f 5e à11:6705f 4e à1:6217f 3e t1:4189f 2e

t0:5279f e à0:1758;

e13T

for Gu and Lu [7].Angle f e 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.4depicts the cable amplitudes versus wind speed for C D ;C L taken from [6].The amplitudes are taken from the steady-state response at time above 40s and are computed with respect to the new equilibrium positions determined by each wind speed.The rivulet amplitudes,a m ;varies from 0.05to 0.45.The numerical simulations are carried out using Runge–Kutta scheme of the fourth order for the initial conditions y 0?0:001m,’y 0?0or y 0?0:03m,’y 0?0:As it can be observed for all considered rivulet amplitudes,the cable amplitudes grow steadily to attain maximum value at wind speed of about 9.5m/s.This and the assumption of the zero rivulet amplitudes at wind speed smaller than 7m/s and higher than 12m/s [3]yield the following values for the coef?cients in Eq.(7):U max ?9:5m =s ;a 1?0:448and a 2?1:5842:The assumed,in the following simulations,variations of the rivulet amplitude,y ;together with the initial position of the rivulet,y i ;[1]are shown in Fig.5.

Firstly,the cable response for three different cable frequencies,i.e.:1,2and 3Hz are studied.In Fig.6the calculated cable amplitudes are compared with the experimental ones [1].Note that,despite the quantitative difference between the numerical and the experimental results,which descends from different cable characteristics,the qualitative character is preserved.That is,with the growth of the cable stiffness,the amplitudes decline.This fact was also observed in [7].The largest responses are independent of both the wind velocity and frequency,in a sense

5

6

789101112

13

wind velocity U 0 (m/s)

01.25

2.5

3.75

5

c a b l e a m p l i t u e (c m )

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

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878

that they occur for the U 0?9:5m/s regardless of the cable stiffness.Note that there are no signi?cant differences between results obtained with Yamaguchi’s force coef?cients 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 selected amplitude of cable oscillations,called ‘set-amplitude’,the phase shift between peaks of the cable displacements and the aerodynamic force are measured.In [1],the set amplitude is 5cm while in this paper it is chosen as 2cm.The tests are carried out assuming the cable frequency to be equal to 2Hz.For the reference the steady-state cable amplitudes from the numerical study and the experiment are plotted in Figs.7a and b .The phases,c ;for the cable without rivulet are shown in Figs.7c and d .The numerically determined phases are constant functions of U 0;while those

wind velocity U 0 (m/s)

20

40

6080

θi θ (d e g )

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

wind velocity U 0 (m/s)

246810c a b l e a m p l i t u d e (c m

)

wind velocity U 0 (m/s)

5

10

15

20

(a)

(b)

c a b l e a m p l i t u

d

e (c m )

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

K.Wilde,W.Witkowski /J.Wind Eng.Ind.Aerodyn.91(2003)873–891879

from the experiment vary.However,they also lie in the neighbourhood of à90 .Negative sign of the phases indicates the damping characteristics of the aerodynamic force 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 range from 7.5to 11m/s the phases are positive and about 90 .The positive sign of the phase shifts indicate the exciting characteristics of aerodynamic force F :The change in 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]

F t ?F j j sin ec T;

e14T

wind velocity U 0 (m/s)

c a b l e a m p l i t u

d

e y (c m )

wind velocity U 0 (m/s)

wind velocity U 0 (m/s)

p h a s e l a g (d e g )

0p h a s e l a g ψ (d e g )

wind velocity U 0 (m/s)

-10

-505F t (N /m )

wind velocity U 0 (m/s)

F t (N /m )

(a)

(b) (c)

(d)

(e)

(f)

https://www.360docs.net/doc/dc15587082.html,parison between numerical and experimental results:(a)cable response (numerical),(b)Hikami and Shiraishi [1],(c)phase lag (numerical),(d)Hikami and Shiraishi [1],(e)F t numerical and (f)Hikami and Shiraishi [1].

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

880

where F is the magnitude of the exciting force and c is the phase shift.When force F t is positive,it indicates an exciting character of force F and when F t is negative,F is regarded as a damping force.The comparison between the numerical and the experimental results is illustrated in Figs.7e and f ,respectively.Note that the force changes from the damping one to the exciting,and then again to the damping one.The range in which the positive sign of F t occurs (Fig.7e )corresponds to that where the phase shift is positive (Fig.7c )which,in turn,corresponds to the steady-state amplitudes larger than 2cm (Fig.7a ).There are no signi?cant differences between results based on different steady force coef?cients.

Fig.8shows an example of time history of the aerodynamic force for wind speed U 0?9:5m/s.In this case the computations were performed using the steady-wind force coef?cients from [6].The force is nonlinear and periodic.At the beginning of the motion (Fig.8b )the force precedes the response of the cable,exhibiting thus the exciting characteristic.Note that,as the time unfolds,another component appears.In the steady-state response,seen in Fig.8c ,this component has a signi?cant amplitude and lags behind the response of the cable.It indicates that the

time (s)

c a b l e a m p l i t u

d

e y (m )

time (s)

3.54

4.55

5.5

f o r c e a m p l i t u d e F [N /m ]

(c)

(b)

Fig.8.Time histories of cable displacement cable and force for C D ;C L from Yamaguchi [6]:(a)time history of aerodynamic force F for U 0?9:5m/s and (b)transient motion—A,(c)steady-state response—B.

K.Wilde,W.Witkowski /J.Wind Eng.Ind.Aerodyn.91(2003)873–891881

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

4.Simpli?ed models of the aerodynamic force

In the model presented in the previous section it is not possible to assess neither the exciting component of the aerodynamic force nor the aerodynamic damping. Therefore,the formula for the aerodynamic force is expanded and expressed in terms of the cable velocity’y:Three simpli?ed models are considered.Model1assumes linearization of all trigonometric functions,models2and3additionally assumes linearization of steady-state force coef?cients by tangent and least square?t, respectively.

4.1.Model1

It is assumed that the drag and lift coef?cients change in an arbitrary way.The expansion is based on the following assumptions descending from preliminary numerical simulations:

1.The aerodynamic force F can be expressed by terms containing cable velocity’y up

to the?rst power.

2.The second(and higher)powers of the rivulet amplitude a m are small in

comparison with a m and therefore are neglected.

3.The term R’y coseyty iTin Eqs.(4)and(5)is much smaller than U cos g and is

considered negligible.

4.The function arctaneyTin Eq.(5)is Taylor-expanded retaining only the linear

term.The equilibrium position for expansion descends from inclination and yaw angles and is assumed as g:

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

linear terms.

As a result,the relative velocity,the relative angle of attack and the aerodynamic

force read

U rel D

???????????????????????????????????????????????????????????????????????????????????????

eR’y sineyty iTtU sin gt’yT2teU cos gT2

q

;e15T

f?D arctanegTteR’y sineyty iTtU sin gt’yT=U cos gàg

1tg2

;e16T

f e?f?àyày i;e17T

F?U2

rel

D r

2

eC Lef eTecos gàsin gef?àgTTtC Def eTesin gtcos gef?àgTTT:

e18TK.Wilde,W.Witkowski/J.Wind Eng.Ind.Aerodyn.91(2003)873–891

882

The substitution of formulas(15)and(16)into(18)yield the polynomial of the third order with respect to’y:The terms involving the same powers of’y had been collected together followed by the removal of the higher powers.Consequently,the terms involving the same powers of a m had been collected retaining only the zero and the?rst powers.The partially linearized equation becomes

.yto2yt2x s o’y?à1

m

eF dampetTtF excetTT;e19Twhere

F dampetT?’yeZ1ta m Z2etTTe20Tis the aerodynamic damping force and

F excetT?F1ta m F2etTe21Tis the exciting force.The coef?cients in(20)and(21)are given in Appendix A. Dividing the right-hand side of Eq.(20)by2’ym o yields the formula for aerodynamic damping ratio

x a?Z1ta m Z2etT

2m o

:e22T

Examination of the coef?cients Z1and Z2indicates that the damping ratio depends on time.In contrast to the cable without rivulet,for which the damping ratio is solely dependent on C D[8].Here,due to the presence and the oscillations of the upper rivulet,the aerodynamic damping is a function of C D;C L and time.This model is applicable to problems with steady-force coef?cients rapidly varying with the instantaneous angle of attack f e:

4.2.Model2

This model,apart from all the assumptions of the previous section,additionally assumes that the functions for C D and C L can be represented as linear functions of the angle f e i.e.:

C D?D1f etD2;e23T

C L?L1f etL2;e24Twhere the coef?cients are de?ned as follows

D1?d C D

d f e

f e?f eq e

;e25T

D2?C Def eq

e

T;e26T

L1?d C L

d f e

f e?f eq e

;e27T

L2?C Lef eq

e Te28T

K.Wilde,W.Witkowski/J.Wind Eng.Ind.Aerodyn.91(2003)873–891883

and

f eq e D

g ày i :

e29T

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,

f eq e depends only on wind velocity U 0:The coef?cients D 1;D 2;L 1;L 2and f eq

e are to be determined for each value o

f U 0:

The sine term of ’y

sin ey i ta m sin eo t TTin Eq.(16),upon the Taylor expansion,is replaced with the term a m o cos eo t Tsin y i :Thus the expression for the aerodynamic force become

F ?U 2rel D r 2

eeL 1f e tL 2Tecos g àsin g ef ?àg TT

teD 1f e tD 2Tesin g tcos g ef ?àg TTT:

e30T

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

F damp et T?’y eZ 3tA D sin eo t ty D TT;e31TF exc et T?F 3tA E sin eo t ty E T;

e32T

where all the coef?cients in (31)and (32)are the functions of g ;D 1;D 2;L 1;L 2;f eq e ;a m

and are de?ned in Appendix A.Numerical simulations,based on Eqs.(31)and (32),reveal that the oscillating part of the damping force and the constant term F 3in expression for the exciting force have a negligible effect on cable response.Thus,

F damp et T?Z 3’y ;

e33TF exc et T?A E sin eo t ty E T:

e34T

The formula for the aerodynamic damping ratio is then x a ?

Z 3

2m o

:e35TFinally,the equation of motion reads

.y to 2

y t’y 2x s o t

Z 3m

?à1m A E sin eo t ty E T:e36T

Formulas for amplitude A E and phase shift y E are given in Eqs.(A.14)and (A.15).

Note that since Z 3is time-invariant,Eq.(36)represents a harmonic oscillator driven in harmonic fashion.In this model the exciting component of the aerodynamic force issue from the presence and oscillation of the rivulet.

The phase shift between motion of the cable,y ;and rivulet,y ;is found to be

y y y ?90 ty E :

e37T

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

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

884

4.3.Model 3

In this model the functions of the stead-wind force coef?cients are also linearized.In this model the coef?cients in Eqs.(23)and (24)are found using ?rst-order polynomial ?tting with the constrains expressed by (26)and (28),satis?ed on angle

f eq e (29).The ?ttin

g is conducted for angles,f e ;from the range f eq e à15

p f e p

f eq e t15

:The transformations proceed in the same fashion as in model 2yielding Eqs.(33)–(36).

4.4.Numerical results from simpli?ed models

The numerical results obtained from the simpli?ed models for data both from [6,7]are compared with the solution of the full model (3)(Fig.9).The frequency of the cable was set to 2Hz.It may be observed that the greatest discrepancies are pronounced for wind speed around 9.5m/s.Model 1underestimates the amplitudes regardless of the used aerodynamic coef?cients.Model 2,with C D and C L linearized by tangent,gives larger amplitudes of the cable.This is due to the fact that the values of steady force coef?cients vary rapidly with the changes of angle f e :Better results are obtained from model 3,where the curves C D and C L are linearized by ?tting on the selected range of f e :Generally speaking,the nonlinear curves of steady force coef?cients can not be represented by a linear function.Model 3describes the procedure of the optimal linearization of the aerodynamic force for estimation of the maximum cable amplitude.

Fig.10presents the time histories of the damping and exciting force components.The steady-state responses are computed for models 1,2and 3for U 0?9:5m/s.For model 1both aerodynamic force components are periodic and nonlinear.Note that the exciting force computed for Gu and Lu’s force coef?cients has the additional

wind velocity U 0 (m/s)

c a b l e a m p l i t u

d

e y (m )

wind velocity U 0 (m/s)

c a b l e a m p l i t u

d

e y (m )

(a)

(b)

Fig.9.Cable amplitudes from different models:(a)C D ;C L from Yamaguchi [6]and (b)C D ;C L from Gu and Lu [7].

K.Wilde,W.Witkowski /J.Wind Eng.Ind.Aerodyn.91(2003)873–891885

peaks coinciding with the peaks of the damping component.Models 2and 3represent the force components by sine function and thus neglect the additional peaks in the time histories.The amplitudes of forces from models 2and 3are larger than those obtained from model 3.Note that there are no differences in modelling the forces by models 2and 3for the Yamaguchi’s force coef?cients.The cable amplitudes at wind speed U 0?9:5m/s (Fig.9)are similar for those models.The derivative of Yamaguchi’s C D curve (Fig.3)has small values and do not vary in the range of interest (f e about 20 ).Thus,there are no signi?cant differences between linearization by tangent (model 2)and constrained ?tting (model 3).The derivative of C D curve,given by Gu and Lu,has large values and changes considerable in the range of interest,yielding large error in values of C D used in simulations by model 2.Therefore,there are differences in modelling the damping (Fig.10b )and exciting force component (Fig.10d )by models 2and 3.Those differences are re?ected on the

9899100

time (s)

-1.5-1-0.500.51F d a m p (N /m )

9899100

time (s)

-1.5

-1-0.50

0.51F d a m p (

N /m )

(a)

(b)

(c) (d)

98

99100

time (s)

-1.5

-1-0.500.51

1.5F e

x c (N /m )

98

99100

time (s)

-1.5

-1-0.500.511.5F e c x (N /m )

Fig.10.Time histories of the damping and exciting component of the aerodynamic force:(a)damping force (C D ;C L from Yamaguchi),(b)damping force (C D ;C L from Gu and Lu),(c)exciting force (C D ;C L Yamaguchi)and (d)exciting force (C D ;C L from Gu and Lu).

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886

computed cable amplitudes (Fig.9b ).Model 2overestimates the exact solution for wind speed U 0?9:5m/s by 0.036m,while model 3by 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 of the aerodynamic damping,resulting form the presence of the rivulet,can be assess by the ratio

G ?x a x s ?Z 32m ox s e38T

Fig.11shows the cable aerodynamic damping,computed from model 3,for the cable with and without the rivulet.The no rivulet case is determined through the formula 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 coef?cients.This is attributed to the differences in values of the C D and C L curves and their ?rst derivatives (Fig.3).In the range of interest,both C L curves are similar,while the mean value of C D for Gu and Lu is around 1.2and for Yamaguchi’s data is 1.55.This difference is a primary factor for the differences in computing coef?cients D 1;D 2;L 1;L 2which determine the aerodynamic damping (Eq.(A.13)).Note that the aerodynamic damping computed for Gu and Lu’s data is lower than damping of the cable without rivulet for all considered wind speeds.

The amplitudes of the exciting forces,computed for the Yamaguchi and Gu and Lu steady force coef?cients,versus wind speed are shown in Fig.12.The force envelopes are computed from model 3.The largest amplitudes for both aerodynamic data are obtained for U 0?9:5m/s.The maximum amplitude for Yamaguchi’s data is larger than the force amplitude for Gu and Lu force coef?cients.The Yamaguchi’s C D and C L overestimate the effect of rivulet since they are determined for the rivulet of a relatively large size.However,the computed amplitudes of the cable oscillations

wind velocity U 0(m/s)

Γ(n o n d i m .)

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

K.Wilde,W.Witkowski /J.Wind Eng.Ind.Aerodyn.91(2003)873–891887

are similar to those obtained from Gu and Lu’s coef?cients (Figs.6and 9),since both components of the aerodynamic force are overestimated.

5.Conclusions

The phenomenon of rain-wind-induced vibrations of the stayed cables has been studied.The derived SDOF models assume the circumferential oscillation of the upper rivulet with the same frequency as the cable and constant rivulet—cable amplitude ratio for given wind speed.The aerodynamic force has been described by quasi-steady formulation.These assumptions oversimplify the problem,since they do not address the problem of adhesive forces acting between the rain water and the surface,the ?ow of the water on the cable and most of all the effects of the three dimensional air ?ow around the oscillating cable and rivulet.Nevertheless,the proposed models describe the phenomenon by simple formula that can be easily adopted for estimation of the maximum amplitudes of cable oscillations induced by simultaneous 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 and the exciting one.These forces depend explicitly on the oscillation of the rivulet as well as on the cable orientation and the steady-wind coef?cients.The factor that plays the major role in determining the maximum amplitude of the oscillating cable is the amplitude of the exciting force.

The proposed models have been based on the available aerodynamic data for cable with rivulet.The applied force coef?cients have been determined for the horizontal cables,and thus,the effects of axial ?ow could not be incorporated.In addition

there

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

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

888

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, veri?cation of the proposed models.

Acknowledgements

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

Appendix A

A.1.Coef?cients for model1

The aerodynamic damping force:

Z1?DU r

2c5

eC D c7tC L c8T;eA:1T

Z2?DR or coseo tTsiney ita m sineo tTT

c5

?C Dec2e4tg2Ttc1ec4c5àg c6TT

tC Lec1c5àc2e3c3tc4à2gtc4g2àg3TT :eA:2T

The exciting force:

F1?DU2r

2c5

?C Deec2

1

tc2

2

Tec2c6tc1ec4c5àg c6TTT

tC Lec1c2

2

g2tc3

1

c5àc3

2

ec3tc4à2gtc4g2àg3T

tc2

1

c2eàc4c5tg c6TT ;eA:3T

F2?DUR ro coseo tTsiney ita m sineo tTT

2c5

eC D c7tC L c8T;eA:4T

where

c1?cosegT;eA:5Tc2?sinegT;eA:6Tc3?tanegT;eA:7Tc4?arctanegT;eA:8Tc5?1tg2;eA:9Tc6?2tg2;eA:10TK.Wilde,W.Witkowski/J.Wind Eng.Ind.Aerodyn.91(2003)873–891889

c 7?c 21tc 22e5t2g 2

Tt2c 1c 2ec 4c 5àg c 6T;

eA :11Tc 8?c 2ec 1t2c 1g 2tc 2eà3c 4à2c 4t4g à2c 4g 2t2g 3TT:eA :12T

A.2.Coef?cients for models 2and 3

The aerodynamic damping force:

Z 3?1

R o sin ey i T

eD 1c 11tD 2c 12tL 1c 13tL 2c 14T:

eA :13T

The exciting force:

A E ?a m ?????????????????????????????????????????????????????????????????????????????????????????????????????????

eD 1c 9tL 1c 10T2teD 1c 11tD 2c 12tL 1c 13tL 2c 14T2

q eA :14Ty E ?arctan D 1c 11tD 2c 12tL 1c 12tL 2c 14

D 1c 9tL 1c 10

;

eA :15T

where

c 9?à

RU 2r ec 21tc 2

2Tec 2e1tc 5Ttc 1ec 4c 5àe1tc 5Tg TTc 5

eA :16T

c 10

?à

RU 2r ec 1c 22eà1tc 5Ttc 31c 5tc 21c 2eàc 4c 5tg tc 5g Ttc 3

2eàc 3àc 4c 5tg tc 5g TT5

eA :17T

c 11?

1

c 5

U or sin ey i TeR 2ec 21e2c 4c 5à2g àc 5g àc 5y i Ttc 22e6c 4c 5t2c 4c 25tc 3e4t3c 5Tà6g à5c 5g à3c 5y i à2c 25y i T

tc 1c 22e1tg 2Tt2c 25ec 4àg Tec 4ày i T

à

tc 5e1à4c 4g t2g 2t2gy i TTT;eA :18Tc 12?R 2U or sin ey i Tec 21tc 22e3t2c 5Tt2c 1c 2ec 4c 5àe1tc 5Tg TTc 5

;

eA :19T

c 13

?R 2U or sin ey i Tc 25

eec 21c 5tc 1c 2e2c 4eà1tc 5Tc 5t2g àc 5g tc 5y i à2c 25y i Ttc 22eà2à4c 23t3c 5à2c 24c 25t4c 4c 5g t2c 4c 25g à2g 2à2c 5g 2

à2c 4c 25y i à2c 5gy i à2c 25gy i tc 3eà6c 4c 5t6g t3c 5g t3c 5y i TTT

eA :20T

c 14

?R 2U or sin ey i Tc 2ec 1eà1t2c 5Ttc 2eà3c 3t2eàc 4c 5tg tc 5g TTTc 5

:

eA :21T

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

890

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