An experimental study on vortex-induced vibration of a circular cylinder tower at a high wind speed

JOURNAL OF

windengineePing ELSEVIER

Journal of Wind Engineering and Industrial Aerodynamics 69-71 (1997) 731 744

~ l ~ i ~

An experimental study on vortex-induced vibration of a circular cylinder tower at a high wind speed Tetsuya Kitagawa "'*, Toshihiro Wakahara b, Yozo Fujino u, Kichiro Kimura b "Department of Civil Engineering, Nihon University, 1-8 Kanda-Surugadai, Chiyoda-ku, Tokyo 101, Japan bDepartment of Civil Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyou-ku, Tokyo 113, Japan

Abstract A wind-tunnel experiment using a circular cylinder tower rocking model is conducted to study the characteristics of the across-wind response. Special attention is paid to the vortexinduced vibration which occurs at a wind speed higher than the occurrence wind speed of the ordinary vortex-induced vibration. Included in this study are the discussions on the effects of the turbulence of wind and the damping of the model. The wavelet analysis clarifies that the tip-associated vortices which are formed rather intermittently around the model top are concerned with the cause of the vibration. Keywords: Circular cylinder tower; Vortex-induced vibration; Wavelet; Wind velocity fluctuation measurement

1. Introduction The v o r t e x - i n d u c e d v i b r a t i o n of a circular cylinder occurs a r o u n d the w i n d speed where the p e r i o d i c v o r t e x - s h e d d i n g frequency coincides with one of the n a t u r a l frequencies of the structure. M a n y studies have been m a d e on the v o r t e x - i n d u c e d v i b r a t i o n , a n d m o s t of t h e m c o n s i d e r the p h e n o m e n a only a r o u n d the a b o v e m e n t i o n e d wind speed. O n the o t h e r hand, a l t h o u g h s o m e e x p e r i m e n t a l studies using circular cylinder m o d e l s i n d i c a t e the possibility of o c c u r r e n c e of v o r t e x - i n d u c e d v i b r a t i o n at a w i n d speed higher t h a n t h a t of the o c c u r r e d wind speed of the o r d i n a r y v o r t e x - i n d u c e d v i b r a t i o n (Vvs), the m e c h a n i s m of that v i b r a t i o n has n o t been clarified. F o r a t w o - d i m e n s i o n a l c i r c u l a r cylinder, Durgin et al. [1] m a d e a w i n d - t u n n e l e x p e r i m e n t to investigate the a c r o s s - w i n d r e s p o n s e of a circular cylinder. T h e y

* Corresponding author. E-mail: [email protected]. 0167-6105/97/'$17.00 @, 1997 Elsevier Science B.V. All rights reserved. PII S01 67-6 1 05 ( 9 7 ) 0 0 2 0 1 - 8

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T. Kitagawa et al./J. Wind Eng. Ind. Aerodvn. 69 71 (1997) 731 744

obtained the response of the ordinary vortex-induced vibration as well as that of the vortex-induced vibration at a wind speed three times as high as V,~. According to their consideration, the mechanism of that vibration was explained as subharmonical vibration due to the K a r m a n vortex-shedding. On the contrary, Matsumoto et al. [2] indicated the three-dimensional vortex-shedding in a two-dimensional model as a cause of the vibration at a wind speed higher than V,s. They used a uniform circular cylinder cable as the experimental model and measured the wind velocity fluctuation behind the model at various positions in span-wise direction. As the result of spectral analysis of the data, the K a r m a n vortex-shedding frequency varied in span-wise direction, which had been similarly observed in the experiment by Bearman et al. [3]. M a t s u m o t o et al. mentioned that long periodic vortex-shedding was produced due to the span-wise difference of the K a r m a n vortex-shedding frequency and the long periodic vortex could cause a kind of vortex-induced vibration at a wind speed higher than Vv~. For tower-like structures, Wootton [4] carried out a wind-tunnel experiment using circular stacks and obtained a peak of the across-wind response at a wind speed 2 times as high as Vvs. He described that the vibration was induced due to the flow influenced by the tip of the model but no evidence was shown. Similarly, Kawai [5] made a wind-tunnel experiment using circular cylinder rocking models and obtained not only the response of the ordinary vortex-induced vibration but also the response peak at a wind speed 2.5 times as high as V,~. In the power spectra of the across-wind response, the peak due to an aerodynamic force different from that of the K a r m a n vortex shedding appeared. He inferred that this force could lead to the response peak at the wind speed 2.5 times as high as V,~. In the present study, this phenomenon observed in tower-like structures will be called as the "vortex-induced vibration at a high wind speed" (VHW), and a windtunnel experiment is conducted to investigate its characteristics. A cantilever circular cylinder that vibrates in across-wind direction is used as the experimental model. The model response and the wind velocity fluctuation behind the model are measured and their frequency components are studied. Furthermore, the wavelet transform is applied to the data of the wind velocity fluctuation behind the model to investigate the detailed characteristics of the fluctuation due to the vortices.

2. Experimental model and measurement setup The wind-tunnel experiment was carried out at the circulation-type wind tunnel in the Institute of Technology, Shimizu Corporation. The test section of the wind-tunnel was 2.1-2.4 m high, 2.6 m wide and 18.9 m long. The experiment was conducted under two types of approaching wind: uniform flow whose profiles are shown in Fig. la and Fig. lb, and turbulent flow produced by roughness blocks whose profiles are shown in Fig. 2a and Fig. 2b. These profiles were based on the data measured at the mean wind speed, V, of 10m/s. In Fig. la and Fig. lb, the flow was almost uniform but the boundary layer generated by the friction of the floor surface is seen for H = 0-200 mm (H: height from the floor of the test section). In Fig. 2a, the curve calculated by the

Z Kitagawa et al./,~ Wind Eng. Ind. Aerodyn. 69-71 (1997) 731 744

733

1000

800

EE600

400 I

0 5 10 1 5 0 5 10 15 20 Mean windspeed (m/s) Turbulence (%)

(a)

(b)

Fig. 1. Profiles of the uniform flow: (a) mean wind speed, (b) turbulence.

I

V=V0

I

(HIHo

I ((x=~

(

'

510 20 30 40 50 Mean wind speed(m/s) Turbulence(%) (a)

(b)

Fig. 2. Profiles of the turbulent flow: (a) mean wind speed, (b) turbulence.

power law, V = Vo(H/Ho) ~, is also illustrated with a dotted line, where Vo = 10 (m/s), Ho = 500 (ram) a n d ~ = 0.22. As s h o w n in Fig. 3, a circular cylinder tower model was used, with a height, Hc, of 500 m m a n d a diameter, D, of 20 mm. The cylinder was s u p p o r t e d by a plate spring a n d allowed the v i b r a t i o n in across-wind direction. The n a t u r a l frequency, f~, was

734

77 Kitagawa et al./J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 731-744

Acce!ero; _ Wind D

•

10 mm•:~" I-type probe

II ¢.n O -O

3 n

3

D =20 mm--~ .°,,.,.°,...." .....

,.:.::/. Floor

Platespr;ng'~ Fig. 3. Experimental model.

17.5 Hz and the critical d a m p i n g ratio, ~, was 0.28%. An accelerometer was installed on the model top to measure the across-wind response and an I-type hot-wire a n e m o m e t e r was used to measure the wind velocity fluctuation behind the model. The position of the a n e m o m e t e r was 5D d o w n s t r e a m and 1D aside from the model center. Also the height was varied in the range of H - 200 500 m m with a 10 m m pitch to investigate the span-wise characteristics of vortex shedding.

3. Across-wind response

Fig. 4 shows the relationship between the reduced wind speed, V/f,D, and the reduced across-wind response (RMS) of the model, y/D, which was obtained under the uniform flow shown in Fig. la and Fig. lb. In addition, the result for the case of a larger d a m p i n g is shown with black dots. F o r ~ = 0.28%, the ordinary vortexinduced vibration (VIV) was observed at V/flD = 5.7. The response peak at V/f,D = 17 is V H W mainly discussed in the present study. The time histories of the across-wind response at V/f,D = 5.7, where VIV occurred, and VffnD = 17, where V H W occurred are, respectively, shown in Fig. 5a and Fig. 5b. While the amplitude of VIV was nearly constant, that of V H W changed slowly. Comparatively, in the case of ~ = 0.66 0.81% in Fig. 4, the amplitude of VIV at V/fnD = 5.7 decreased significantly whereas the response amplitude of V H W a r o u n d

T. Kitagawa et al./J. Wind Eng. Ind. Aerodyn. 69-71 (1997) 731 744

0.12

735

I I I I I I I I

9__

o ~=0.28%

0"1- x~=0.66~0.81%#~ it 00 rr 0.08 II ',Vortex-induced v

.,I-,,,*

0.06 -

tt

vibrati°n Vortex-induced vibrationata highwindspeed

0.04

/

-

"13 0 n"

oo2t 0~

0

10

20

30

Reduced velocity

40

V/f,D

Fig. 4. y/D versus V/f,D (under the uniform flow).

~" 5

°o <

-5 0.0

1.0

2.0

3.0 4.0 Time (sec)

5.0

6.0

3.0 4.0 Time (sec) (b)

5.0

6.0

(a)

g0 -~ -2 o 0.0

1.0

2.0

Fig. 5. Time history of the across-wind response: (a) V/f.D = 5.7, (b) V/f,D = 17.

V/f~D = 18 did not decrease. This result is consistent with Kawai's result [5]. The reason of the phenomenon is thought to be the strong non-linearity of the aerodynamic force, but further study is necessary to clarify the mechanism. Fig. 6 shows the relationship of y/D versus V/fnD in the case of ~ = 0.28%, which was obtained under the turbulent flow shown in Fig. 2a and Fig. 2b. The response of

736

Z Kitagawa et al.,/J. Wind Eng. Ind. Aerodyn. 69-71 (1997) 731 744 /'~ 1 "-1 u.,c

~

~

i

I

I

I

i

i

~,=0.28% ~ 0.1- l~--..~.Vortex_induced vibration :; 0.08-

~"

---"E'0.06:'0.04"0 o

-~ O.02rr"

0 Fig. 6.

{

i

L

i

L

I

0 20 30 40 Reduced velocity V/fD

y/D versus V/f,D( u n d e r

the t u r b u l e n t flow).

VIV was observed at V/JnD = 5.7. However, the response peak of V H W was not identified and the discussion on this phenomenon will be made in Section 5.

4. Power spectra of across-wind response and wind velocity fluctuation behind model For the case of ~ = 0.28% under the uniform flow, the power spectra of the response and the wind velocity fluctuation behind the model are studied to investigate the cause of VHW. The power spectra of the across-wind response of the model at V/f,D = 5.7, where VIV occurred as shown in Fig. 4, is shown in Fig. 7a. The peak at f/f, = 1 (.~Ji~: non-dimensional frequency) was remarkable and it is thought that the Karman vortex shedding frequency coincided with f,. In the following, the Karman vortex shedding will be called as "2-D vortex shedding" (two-dimensional vortex shedding) to distinguish it from other vortices produced due to three-dimensional effects. In Fig. 7b, the power spectra of the wind velocity fluctuation behind the model is illustrated as a function off/f~ and H/Hc (non-dimensional height). The peak at J/in = 1 was found over the whole height (H/H¢ = 0.4 1.0) where the measurement of the wind velocity fluctuation behind the model was done, although the peak around the model top was small. It means that the 2-D vortex shedding whose frequency is equal to f , exists along the whole height. On the other hand, the small peak around fir, = 2 is due to high harmonics of the fluctuation by the 2-D vortex shedding around the model top and is not a substantial phenomenon.

T. Kitagawa et al./J. Wind Eng. Ind. Aerodyn. 69-71 (1997) 731-744

103

.i

........

~" 10 ~ E 10 I

o.4 o

10° ~o 101

~

10 "2

737

_

10.3

HIHc O . 8

n° 10-4 0.1

1.0

10.0

f/f .

2.0

1.,.,

(a)

(b)

6.0

4.0

f/f.

Fig. 7. Power spectra: (a) across-wind response, (b) wind velocity fluctuation behind the model at various heights (V/f,D = 5.7).

"~ 1°°'.,I Tip-associated ,o ~vortex

~,o"L

.,~t,03

,~'-f. /12-D v o r l e x

s/s=0.7

II

-../\

/ shedding

s/s =2

/

J

j

0.4

10.5

0.1

1.0

s/s

(a)

0.0

' '10.0

1.0 ~ (b)

6.0

2.0 4.0

f/f,

Fig. 8. Power spectra: (a) across-wind response, (b) wind velocity fluctuation behind the model at various heights ( V/f,D = 11 ).

Fig. 8a shows the power spectra of the across-wind response at V/f,D = 11, where the response amplitude was small as shown in Fig. 4. Comparing with Fig. 7a, with the wind speed increasing the peak due to the 2-D vortex-shedding frequency moved tof/J~ = 2. On the contrary, another peak was found atf/f. = 0.7, which is thought to be due to a different vortex from the 2-D vortex. The power spectra of the wind velocity fluctuation behind the model at V/JnD = 11, illustrated in Fig. 8b, provides the information about the height where that vortex appears. In Fig. 8b, the peak due

T. Kitagawa et al. ,/J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 731-744

738

~-

10 ~

i

f.

10 I o

10°

2ovo,,ex7;,ng

v

0.4

10.2 ~. 10 ,3 m

0

10.4 0 O_ 10 .5 O.

1.0

0

10.0

f/f

(a)

(b)

. . v

6.0

4.0

f/f,

Fig. 9. P o w e r spectra: (a) across-wind response, (b) wind velocity fluctuation behind the model at various heights V,'y~D = 17).

to the vortex was observed at f/f, = 0.7 and H/Hc = 0.78 1.0. In the following, this vortex will be called "tip-associated vortex" since it appears only around the model top. In addition, the 2-D vortex shedding frequency atJXfll = 2 was not observed in the region higher than H/Hc = 0.78. In Fig. 9a, the power spectra of the response at V/J~D = 17, where V H W occurred as shown in Fig. 4, is shown. Both peaks due to the 2-D vortex shedding and the tip-associated vortices moved to the higher-frequency region with the wind speed increasing. At this wind speed, it is seen that the frequency of the tip-associated vortices coincides withfn. This coincidence is also confirmed in the power spectra of the wind velocity fluctuation behind the model. In Fig. 9b, the peak due to the tip-associated vortices was located at f / f l - 1 and H/Hc - 0.78 1.0. Therefore, it is considered that V H W is caused by the tip-associated vortices and this consideration is consistent with Wootton's conjecture [4] described in Section 1. Also, the tiny peak at f/Jn = 1 was barely observed in the range of H/Hc = 0.4 0.78, which is thought to be due to the wind velocity fluctuation deriving from the across-wind vibration of the model. The relationship of VS[I,D versus y/D and vortex-shedding frequency is summarized in Fig. 10. The frequency of the tip-associated vortices as well as the 2-D vortexshedding frequency were proportional to the wind speed. The response peak due to V H W was observed at a wind speed where the frequency of the tip-associated vortices coincided with f,. It is considered that the tip-associated vortices are generated by three-dimensional flow around the model tip. Although some experimental studies about the flow around the tip of a cantilever circular cylinder have been reported [6,7], more detailed measurements are needed.

T. Kitagawa et al./J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 731 744

739

5.0 4.0

2-D Vortex

3.0 "~

2.0 1.0 0.0 0.12' ~ =0.28%

_

o.1

0.08

Vortex-induced vibration -

0.06

Vortex-induced vibration-

"(1.) o

2 [D.

E 0,04 "o {,3 "1 "13

0.02

122

00

10 20 30 Reduced velocity

40

V/fD

Fig. 10. Relationship between vortex-shedding frequency and across-wind response.

5. Case under turbulent flow As shown in Fig. 6, the response peak due to V H W was not seen under the turbulent flow whose profiles were shown in Fig. 2. In order to investigate the reason, the power spectra of the across-wind response is studied although the wind velocity fluctuation behind the model was not measured in this case. Fig. 1 la shows the power spectra of the across-wind response at V/fnD = 5.7 under the turbulent flow, where VIV occurred as shown in Fig. 6. The peak of fn was remarkable and it is considered that the 2-D vortex-shedding frequency coincides with fn- At V/fnD = 11 (Fig. llb), the peak due to the aerodynamic force by the 2-D vortex-shedding moved to f/f, = 2 in accordance with the wind speed increasing. Comparatively, in the case under the uniform flow at this wind speed, the peak due to the tip-associated vortices was found at f/fn = 0.7 as shown in Fig. 8a. However, in this case, that peak was not observed. At a higher wind speed of V/fnD = 17 (Fig. 1 lc), the peak due to the tip-associated vortices was not found. According to those results, there are two possibilities which explain the reason why V H W did not occur in the case under the turbulent flow. Firstly, the generation of the tip-associated vortices can be prevented by the turbulence of the approaching flow.

71 Kitagawa et al./J. ~7nd Eng. Ind. Aerodyn. 69 71 (1997) 731-744

740

~-

10 3 f~

10 ~ oE o

10 ~ 10 ° 10 1

10 3 n°

10 4 0.1

i

i

i

i I,IIF

I

,

,

, ,~

1.0

0.0

(a) -~

10 ~

T

10 o

f2-D vorle× s h o d d i n 9

g

10 ~ 10 ; l O `3 10 4

n

10 .5 0.1

i

=

i

i

,

i iiii

,

,

,

,,,,

1.0

10.0

YIo

(b) S v~

10 2 10 ~

A2

-D vorlex

10 ° o

f/f

shedding

=3.1

10 1 10 ~ mQ. l O 3 10 4 o

0-

10 5 0.1

,

,

,

. . . . . . . .

i J,,,I

1,0

I0.0

//f

(c) Fig.

(a)

1 I.

Power

spectra

of

across-wind

response

under

the

turbulent

flow: (a)

V/JI, D

= 5.7, (b)

V.,J,,D = 11,

V/JaD = 17.

Then, V H W is not led because of no aerodynamic force to make resonance. Secondly, although the tip-associated vortices is generated, the response of V H W is not able to be distinguished from the response of buffeting. In fact, the response curve at V/f~D > 10 in Fig. 6 was larger than that in Fig. 4. Also, in Fig. 11 b and Fig. 11 c, the frequency component except for fn was larger than that in Fig. 8a and Fig. 9a.

7~ Kitagawa et al./J. Wind Eng. hid Aerodyn. 69 7l (1997) 731 744

741

However, the measurement of the wind velocity fluctuation behind the model is necessary to investigate the phenomenon in detail.

6. Wavelet analysis of fluctuating wind velocity behind model In order to investigate the characteristics of the vortex shedding, wavelet transform [8] is applied to the time history of the wind velocity fluctuation behind the model (d = 0.28%), which was obtained in the case under the uniform flow. The wavelet transform is a mathematical technique for non-stationary signal analysis. Bases localized in both of time domain and frequency domain are used for integration, which allows us to unfold the time-history signals into time and frequency. The continuous wavelet transform of the function, f (t), is the inner product betweenf(t) and the base, ~9(t), called wavelet, which gives the wavelet coefficients, W:

W(I, t') = (f(t)lO(t)) = ~

Or*t,(t)f(t) dt, y

where ~b* is the complex conjugate of ~b, ~bt,t,(t)is the wavelet family of translated and dilated wavelets: 1 /t - t'\ I/s,.,,(t)=~l/st~ )

(l,t'~R,l ¢0),

where I is the scale dilation parameter corresponding to the width and the frequency band of the wavelet, and t' is the translation parameter corresponding to the position of the wavelet. In addition, ~9 should satisfy the admissibility condition

co =

i

I<'1

< oo

(~((o) =

i

exp( - kot)~s(t) dt).

Since any function can be a wavelet if it satisfies the admissibility condition, the choice of the wavelet implies arbitrariness. Then, m a n y kinds of wavelets were developed [-8 12]. In the continuous wavelet transform, the scale dilation and translation parameter can vary continuously, whereas in the discrete wavelet transform, those parameters are discrete. Additionally, while in the continuous wavelet transform the base is non-orthogonal, the discrete wavelet transform allows an orthonormal projection. In the present study, Morlet's continuous wavelet [12] is used (Fig. 12a).

qs(t)=exp(ikpt)exp(-ltl~)

(kp: wave vector).

As shown in Fig. 12b, the frequency band of the wavelet varies depending on the scale dilation parameter, I. Fig. 13a shows the time history of the wind velocity fluctuation behind the model, which was obtained at V/fD = 5.7 and H/He = 0.7. V/fD = 5.7 is the wind speed

742

i~ Kitagawa et al./J. Wind Eng. Ind. Aeroclyn. 69 71 (1997) 731-744 --

Realpart

Imaginary pad

Time domain

Frequency domain

(a)

(b)

Fig. 12. Morlet's wavelet: (a) time domain, (b) frequency domain.

ga "O

-~ 0

(a)

r-

I

(b)

g 5 ".~,

8-5
0.0

0.5

1.0

1.5

2.0

Time (sec) (c)

Fig. 13. (a) Wind velocity fluctuation behind the model a! H/Hc = 0.7, (b) wavelet coefficient (real part), (c) across-wind model response (V/f,D = 5.7).

where VIV o c c u r r e d as s h o w n in Fig. 4, a n d H / H c = 0.7 is a height where the 2-D vortex s h e d d i n g was f o r m e d d o m i n a n t l y as s h o w n in Fig. 7b. T h e result of the wavelet t r a n s f o r m of the time history d a t a s h o w n in Fig. 13a is illustrated in Fig. 13b as c o n t o u r . In Fig. 13b, the real p a r t of W is shown. T h e h o r i z o n t a l axis is time a n d the vertical axis is n o n - d i m e n s i o n a l frequency, f / f , , which was used in place of the scale d i l a t i o n p a r a m e t e r in o r d e r to facilitate u n d e r s t a n d i n g . The brightness of the c o n t o u r m e a n s t h a t W is positive a n d the d a r k n e s s m e a n s negative. P e r i o d i c a l fluctuations of W were seen a r o u n d f / l ~ = 1 a n d t h r o u g h o u t the time (0 2 s), which m e a n s t h a t the 2-D vortex is g e n e r a t e d periodically. The a c r o s s - w i n d response was a h a r m o n i c oscillation with c o n s t a n t a m p l i t u d e (Fig. 13c).

T. Kitagawa et al./J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 731 744

743

~8 O) c-I o)

-o e-

0

(a)

1

(b) v

,- 2 .9 ,~ 0

o0_2 0.0

0.5

1.0

1.5

2.0

T i m e (sec) (c)

Fig. 14. (a) W i n d velocity fluctuation behind the model at H/H~ = 0.92, (b) wavelet coefficient (real part), (c) across-wind model response (V/f,D = 17).

Fig. 14a shows the wind velocity fluctuation behind the model at V/f~D = 17 and H/Hc = 0.92. V/f,D = 17 is the wind speed where V H W occurred as shown in Fig. 4, and H/Hc = 0.92 is the height where the tip-associated vortices were formed as shown in Fig. 9b. At this wind speed, the frequency of the tip-associated vortices coincided with fn (Fig. 9a and Fig. 9b). That coincidence is also observed in the contour of W (Fig. 14b), which is the wavelet transform of the time history in Fig. 14a. Fluctuations of the brightness and the darkness are seen to occur aroundf/fn = 1. C o m p a r ing with Fig. 13b, the fluctuations atf/fn = 1 are to be non-stationary. This suggests that the tip-associated vortices are generated rather intermittently. The response amplitude of V H W (Fig. 14c) changed slowly, and the cause is considered to be the intermittent generation of the tip-associated vortices.

7. Concluding remarks A wind-tunnel experiment using a cantilever circular cylinder model was carried out to investigate the characteristics of the vortex-induced vibration at a high wind speed. The main findings are summarized as follows: 1. Both the vortex-induced vibration at a high wind speed and the ordinary vortexinduced vibration were observed under uniform flow, while under turbulent flow the vortex-induced vibration at a high wind speed did not occur.

744

I2 Kitagawa et al./J. Wind Eng. Ind. Aerodvn. 69-71 (1997) 731-744

2. The amplitude of the model response due to the vortex-induced vibration at a high wind speed was not influenced by increasing the critical damping ratio of the model whereas that of the ordinary vortex-induced vibration decreased significantly. 3. The vortex-induced vibration at a high wind speed appears to be caused by tip-associated vortices, which are generated around the tip of the model and whose frequency is proportional to the wind speed. 4. The result of the wavelet analysis applied to the wind velocity fluctuations behind the model indicated that the tip-associated vortices were generated intermittently. 5. The amplitude of the vortex-induced vibration at a high wind speed changes slowly due to the intermittent generation of the tip-associated vortices.

References [1] W.W. Durgin, P.A. March, P.J. Lefebvre, Lower mode response of circular cylinder in cross-flow, Trans. ASME J. Fluids Eng. 102 (1980) 183 190. [2] M. Matsumoto, T. Nishizaki, J. Aoki, M. Kitazawa, H. Shirato, Rain-wind induced vibration of inclined cables as velocity restricted vibration, J. Struct. Eng. A 40 (1994) 1059 1064 (in Japanese). [3] P.W. Bearman, N. Tombazis, The effects of three-dimensional imposed disturbances on bluff body near wake flows, Preprint 2nd BBAA, 1992. [4] L.R. Wootton, The oscillations of large circular stacks in wind, Proc. Inst. Civil Eng. (1969) 573 598. [5] H. Kawai, Vortex-induced vibration of tapered cylinder, J. Wind Eng. 59 (1994) 49 52 (in Japanese). [6] T. Okamoto, M. Yagita, The experimental investigation on the flow past a circular cylinder of finite length placed normal to the plane surface in a uniform stream, Bull. JSME 16 (1973) 805 814. [7] T. Fox, G.S. West, Fluid-induced loading of cantilevered circular cylinders in a low-turbulence uniform flow, part 2: Fluctuating loads on a cantilever of aspect ratio 30, J. Fluids Struct. 7 (1993) 15 28. [8] M. Farge, Wavelet transform and their application to turbulence, Ann. Rev. Fluid Mech. (1992) 395 457. [9] Y. Meyer, Orthonormal wavelets, In Wavelets, Springer, Berlin, 1989, pp. 21 37. [10] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (7) (1989) 909 996. [11] D.E. Newland, Harmonic wavelet analysis, Proc. Roy. Soc. London A 443 (19931 203 225. [12] A. Grossman, J. Morlet, SIAM J. Math. Ann. 15 (1984) 723 736.