Vortex Shedding Induced Vibrations of a Light Mast

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BBAA VI International Colloquium on: Bluff Bodies Aerodynamics & Applications

Milano, Italy, July, 20-24 2008

VORTEX SHEDDING INDUCED VIBRATIONS OF A LIGHT MAST

Bjarni Bessason, and Jnas Thr Snabjrnsson Engineering Research Institute

University of Iceland, Hjardarhaga 2-6, 107 Reykjavik , Iceland e-mails: [email protected], [email protected]

Keywords: Wind, vortex shedding, vibrations, acceleration, full-scale data, countermeasures.

Abstract. The paper reports a case study of wind induced vortex shedding vibrations of a steel light mast. The vibrations are examined, full scale observations reported and the coun-termeasures devised to reduce the motions are described. The across wind motion of the mast was observed in windy weather shortly after its installation. The amplitude of deflection at the tip of the mast was estimated to be few centimetres and considered unsatisfactory by the owner. A measurement program was initiated to map the vibrations of the mast and to corre-late them with wind data. The recorded data showed that the initiation of the intense vibra-tions as well as the amplitude of the vibrations was in fair agreement with predictions based on theory of bluff body aerodynamics. To reduce the vibrations cylindrical wind flaps were added to the light mast to counteract the formation of vortexes. Comparison of recorded vi-bration data before and after the installation of the wind flaps show that the flaps signifi-cantly reduced the vibrations

Bjarni Bessason and Jnas Thr Snbjrnsson

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1 INTRODUCTION In the summer of 2005 a frame type mast supporting flight control lamps for the approach

control of incoming flights to the N-S runway of the airport in Reykjavik, Iceland, was in-stalled. The Reykjavik airport is located in the middle of the city and surrounded by inhabited areas and dens road system. This affected the construction of the light mast in two ways. Firstly, it was considered preferable to design an elegant figuration of the mast that could fit nicely into the surroundings. Secondly, the optimal location of the new light frame interfered with a road intersection and therefore it was necessary to design a cantilever beam stretching out over part of the intersection. The original structure is shown on Fig. (1).

Soon after the installation of the light frame, significant across wind motion of the cantile-ver beam was observed in windy weather by people passing and stopping at the intersection. This was reported to the owner of the mast, the Icelandic Civil Aviation Administration. Field observations confirmed incidental vertical swings of the cantilever beam in modarate winds. The amplitude of deflection at the tip of the beam was estimated to be few centimetres. The Aviation Administration considered these vibrations unacceptable and decided to investigate the problem in order to seek a solution to reduce the swings.

The main objective of this paper is to report the investigation of the observed vibrations and the countermeasures carried out in order to reduce the motions. The analysis includes both full scale measurements as well as analytical evaluation of the dynamic behaviour of the light mast.

Figure 1: The flight mast with the cantilever beam stretching out over the intersection to the left. The yellow bubbles on the top of the beam are the flight control lamps.

2 WIND INDUCED VIBRATIONS

2.1 Vortex shedding The flow induced vibrations of the cantilever were traced to the much studied vortex shed-

ding process [1], which can create problems in a variety of contexts in wind engineering. The vortices shed from a bluff body as the flow region is separated induce a fluctuating force on


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the structure that can lead to vibrations. The intensity of this force acting on the structure con-trols the amplitude of vibration of the structure that primarily depends on the cross-sectional shape of the structure and the mean wind velocity [2]. The phenomenon is well described in the literature [3] as well as in design codes (see for instance Ref. [4] EN 1991-1.4, Annex E ) and design guidelines, such as Ref. [5] by ESDU.

The vortex shedding for rectangular sections is characterized by the Strouhal number, St, of the flow:

vt

n DSV

= (1)

where V is the wind velocity, D is the dimension of the cross-section perpendicular to the wind direction, and nv is the vortex shedding frequency. The Strouhal number depends on the cross section type, the Reynolds number of the flow and the amplitude of the resulting vibra-tion. The customary value for circular cylinders is 0.2, but it can be found in the literature, in handbooks and is supplied in design guidelines and codes for various cross section types.

If the vortex shedding frequency coincides with a natural frequency of vibration of a mem-ber (or an assembly of members), ns, and the damping (structural and fluid) is sufficiently small for the cross flow forces to 'lock on' to the structural natural frequency then large ampli-tude oscillations result. The wind velocity generating this condition is termed the critical wind velocity, i.e. V ~ Vcrit, and is evaluated as:

scrit

t

n DVS

= (2)

The resulting oscillations self limit when the amplitude of the vibration becomes comparable to the size of the vortex. This limit is dependant on the member mode shape, but is often found to be of the order of 1 cross section diameter which is rarely an acceptable limit.

The actual vibration amplitude of a structure under the influence of vortex shedding are governed both by the structures inherent damping characteristics and by the mass ratio be-tween the structure and the fluid it displaces. These two effects are often combined in the Scruton number sometimes referred to as a stability parameter, which is defined as:

24 e T

c

mSD

pi

= (3)

Here me is average mass per unit length, T is the total critical damping ratio, is the density of air and D is the across wind dimension of the cross section. Although the damping parame-ter in Eq.(3) represents the total damping, the damping is almost totally due to structural damping since the aerodynamic fluid damping is usually negligible. As Sc increases then the response amplitude proportionally decreases when locking occurs. For a narrow band excita-tion a sinusoidal excitation model gives, in principle, an adequate representation of the vibra-tion amplitude [2], this can be simplified to:

max

24 c ty kCD S Spi

=l

(4)

Here ymax is the maximum vibration amplitude, Cl is a mode and amplitude dependent lift co-efficient. k is a parameter depending on the structural mode shape, given as:


4

h

j0h 2

j0

(z) dz(z) dz

k

=

(5)

For a cantilever the mode shape j(x) can be taken as (x/H), where H is the cantilever length. For uniform or near-uniform cantilevers, can be taken as 1.5 and then k = 1.6.

At wind speeds away from V ~ Vcrit, the locked-in narrow banded behaviour becomes more episodic, and the response decreases. The range of V/Vcrit for which the oscillations remain significant is considerably wider for lightly damped structures than for heavily damped struc-tures which points to the dominant role of structural damping in determining the amplitude of wind vortex induced oscillations and the range of excitation wind speeds.

There are several guidelines available to an engineer which can be used in assessing the susceptibility of structures to wind induced vortex shedding. These include for instance Ref. [5] (ESDU 90036) and Ref. [4] (EN 1991-1.4 Annex E). Both approaches are based on the simple description of the process involved provided her above but with some additions and accompanying datasets.

The methodology of Ref. [5] takes into account many parametric effects, such as: Rough-ness of the vortex generation surface, the effect of atmospheric turbulence, span-wise correla-tion due to end effects, effects of tapered or stepped diameter members, effect of mode shape on the excitation force, influence of the response amplitude on the excitation force, reduction of response as the wind velocity deviates from the critical wind velocity and includes predic-tion of low amplitude broad banded responses. The basic technique is to derive the response amplitude at the critical velocity as a function of damping using the standard equations for the member properties. The response is then calculated by interpolating the derived function for the correct value of structural damping. In the case of non-critical wind velocities the response can be corrected based on the damping level and response amplitude at the critical velocity and for low amplitude broad banded responses a modified set of equations are presented and used.

The methodology presented in Ref. [4] is similar to the ESDU procedure, but is put for-ward in somewhat simplified form that does not attempt to account for all the parameters that might influence the response. It is to some extent based on the work of Ruscheweyh et al [6], which developed a mathematical model for predicting the vortex-excited vibration taking into account the increasing correlation length of the exciting force with increasing vibration ampli-tude, called the correlation length model.

2.2 Modal analysis and estimation of critical wind velocity for vortex shedding The light mast is a slender steel frame structure. The horizontal beam at the top of the mast

is made of welded U-profile (400x400mm) while the columns have a closed rectangular sec-tion of same size. All corners are very sharp and therefore susceptible to vortex shedding, but the U-profile section of the cantilever should cause a breakdown of flow regularity to some degree by the irregular nature of the flow trapped within the section. The structure has low internal damping and therefore when excited the duration of vibration is considerable.

A 3D beam finite element (FE) model was made of the light mast in SAP2000 [9]. All cross-sections and dimensions were based on the engineering drawings of the light mast. To take into account the weight of the flight lamps and other fittings the dead weight of the steel structure was increased by 10%. The main results of the modal analysis are shown in Table 1. As seen from the table some of the vibration modes were out of plane while the others were in


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the plane of the structure. Field observations showed that it was the vertical motion of the can-tilever beam that was most apparent. Therefore modes no. 2, 4 and 6 are of concern, while the out-of-plane modes can be ignored. In Fig. (2) and Fig. (3) modes 2 and 4 are shown. Mode 2, which is the most likely candidate for vortex induced vibrations has a natural frequency of 2.3 Hz. This value corresponds closely to the natural frequency for an equivalent cantilever beam fixed at the column connection (2.4 Hz). For this mode the modal analysis showed that the deflection at the middle of the span BC (see Fig. (2)) was only 6% of the deflection at point A, thus explaining why the swings were only observed at the tip of the cantilever but not else-where along the mast.

Mode no.

Natural period

(s)

Natural frequency

(Hz)

Main description of mode

1 0.52 1.94 Out-of-plane. Lateral movement of cantilever beam. 2 0.43 2.33 In-plane. Vertical movement of cantilever beam. 3 0.32 3.10 Out-of-plane. Both columns moving in phase 4 0.25 4.06 In-plane motion. Whole structure is moving. 5 0.22 4.45 Out-of-plane motion. Torsion. 6 0.13 7.87 In-plan motion. Whole beam is moving.

Table 1 - Results of the modal analysis of the flight mast.

Figure 2: Mode 2, f2=2.33 Hz. The cantilever beam moving up and down.

Figure 3: Mode 4, f4=4.06 Hz. The whole structure is moving in plane.

Using Eq. (1) the critical velocity for vortex shedding can be deduced from the Strouhal number, based on the natural frequency of the excited mode of the structure and the character-

A

B C D


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istic width of the structure. Based on Ref [4] (EN 1991-1.4 Annex E) the Strouhal number is ~0.12 for a square rectangular section with sharp corners. Ref. [5] (ESDU 90036) gives a Strouhal number in the range of 0.11 to 0.125.

Assuming mode 2, with a natural frequency of 2.3 Hz to be the critical mode of vibration with regard to across wind induced vibrations and since the width of the section is 0.4 m, the critical mean wind velocity was found to lie between 7 and 8 m/s, which is a common wind velocity level in Reykjavik. Higher wind speeds, or 12-15 m/s, are required to excite mode 4, and to excite mode 6 wind speed in the range 24-28 m/s are necessary. Mode 4 and 6 involve movement of the whole frame structure. Those modes are not likely to be involved in vortex or wind induced motion, due to lack of correlation in the wind action over the full frame structure. Open non circular cross sections such as U sections are also prone to galloping os-cillations [4], but the onset wind velocity is higher than the wind speed necessary to excite mode 2 by vortex shedding, or about 11 m/s.

2.3 Estimation of vortex shedding Vibration amplitude based on Eurocode The vibration amplitude of a structure under the influence of vortex shedding can be esti-

mated by aerodynamics methods. Both the ESDU method in Ref. [5] as well as the methodo-logy given in Annex E in Eurocode 1 (see Ref. [4]), were applied. In the response estimation the cantilever beam was basically assumed to be fixed at the first column connection (i.e. at point B in Fig. (2)) and vibrating in mode 2. As an example, the following equation from Ref. [4] can then be used to estimate the maximum vertical deflection at the tip of the beam (point A in Fig. (2)):

,max

1 1A W lat

t c

K K c DS S

= (6)

Here St ~0.12 is the Strouhal number, Sc is the Scruton number, KW is the effective correlation factor, clat is a lateral force coefficient and D is the vertical dimension of the cross-section as before. Knowing the critical damping ratio, the mode shape and the equivalent mass per unit length of the beam for the mode of vibration makes it possible to evaluate the Scruton number given by Eq.(3) and estimate the maximum tip displacement of the beam due to vortex in-duced effects. The two methods tested gave similar results. Assuming a critical damping ratio of 1%, the ESDU [5] estimate was 10 cm but the Eurocode [4] estimate was 6 cm. These re-sults, especially the Eurocode result, corresponded well to the measured value of roughly 6 cm as presented in Section 3.2.

It should be noted that this type of estimation is quite sensitive to the choice of critical damping ratio. Assuming a lower damping ratio of 0.5%, which is quite reasonable for this type of structure, would result in a considerably higher estimate for the tip displacement of the cantilever, i.e. 12 cm using Eurocode [4] and 15 cm using ESDU [5].

3 RECORDINGS OF STRUCTUREAL RESPONSE AND WIND DATA

3.1 Instrumentation To confirm the source of the problem and to map the situation before taking any actions it was decided to carry out a measurement program at the site. The frame was instrumented with a uni-axial vertical accelerometer located at the tip of the beam, and wind velocity and wind direction meters above the other supporting column, see Fig. (4). Monitoring of the vibrations and the


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wind also gave the opportunity to compare the behaviour of the light mast before and after installation of countermeasures and thereby to see the effectiveness of such measures. The accelerometer continuously recorded data with a sampling rate of 32 Hz, i.e. 2.76 million val-ues in one 24-hour day. Referring to the natural frequency of mode 2 from the FE-analysis, this corresponded to approximately 14 samples for one oscillation, which was considered suf-ficient. The wind velocity meter recorded the average wind speed and wind direction for each minute. The wind direction is given as clockwise angle, , from the north, i.e. 0 is north, 90 is east, 180 is south, etc. The plane of the light mast faces north and south, respectively. The data were recorded by a data acquisition station located in a box beneath column BE (see Fig. (4)). The data were transmitted via mobile telephone communication to the laboratory for processing.

Figure 4. Location of accelerometer, and wind speed and wind direction meters.

3.2 Recorded data Data were recorded over a period of six days. When there was little wind the measurements were stopped by a phone call and then started again with another phone call. Otherwise the measurements and the data transfer were completely automatic. During these six days 86 hours of data were collected. Fig. (5) displays an example of the recorded time histories along with wind data for 13 hours of continuous recording. The recorded acceleration and the evalu-ated displacement at the tip of the beam are displayed in the top two figures, respectively, whereas the wind speed and the wind direction are displayed in the two bottom figures, re-spectively. The time axis is the same for all four graphs. The wind is blowing from south dur-ing these 13 hours, as demonstrated by the bottom graph of Fig. (5). From the graphs displaying the response on one hand and the wind velocity on the other, it is clearly seen how the oscillations of the cantilever beam are strongly correlated to the wind speed. Whenever the wind speed reaches the 6 to 8 m/s range the beam starts vibrating intensively. This wind ve-locity range is in fair agreement with the critical wind velocity predicted by Eq. (1) in Section (2.2). The maximum displacement during these 13 hours was 6.43 cm, which corresponds to a peak-to-peak motion of almost 13 cm. This was in fact the maximum recorded deflection in the 86 hours of available data. Another way to present the data is to plot the maximum dis-placement within every minute versus mean wind speed for the corresponding minute, see Fig. (6). This figure shows, that the peak displacements are within 2 cm, except when the wind speed is above 5.5 m/s. Furthermore, the largest peak displacements (the top 5 values) are ob-served when the wind speed is in the range 7 to 8.5 m/s which again is in fair agreement with the vortex shedding behaviour discussed in Section (2).

7.4 m

Wind speed and wind direction meter

A B C D

E F 30.4 m

Vertical accelerometer


8

12 14 16 18 20 22 24-20

-10

0

10

20Ac

cele

ratio

n - (m

/s2)

12 14 16 18 20 22 24-8-6-4-202468

Disp

lace

men

t - (cm

)

Time - (h)

12 14 16 18 20 22 240

2

4

6

8

10

Win

d sp

eed

- (m

/s)

12 14 16 18 20 22 240

90

180

270

360

Win

d di

rect

ion - ()

Time - (h)North

East

South

West

North

Figure 5. Example of recorded data for 13 hours (20 August 2005).


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2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

Peak

di

spla

cmen

t - (cm

)

Wind speed - (m/s)

Figure 6: Maximum displacment versus mean wind speed within 1 minute window for 13 hours of recorded data from 20 August 2005 (see also fig. (6)).

3.3 Data processing Fig. (7a) shows an example of an acceleration time series recorded during 3 hours of log-

ging and Fig. (7b) shows a 10 second long segment from that same time series. The segment displayed is typical of the behaviour observed within the many bands of increased motion of the cantilever beam during the 3 hour observation period. These bursts of motion are induced by vortex shedding as the wind velocity reaches the critical level. From the time series seg-ment in Fig. (7b) it can be seen that the signal is a narrow-band process and nearly a regular sine wave. Therefore, even without frequency analysis, it can be deduced from the time series alone that the natural frequency of the motion was ~2.3 Hz. This result was in good agree-ment with the results from the modal analysis for mode 2 (Table 1). The nature of these bands of acceleration bursts induced by the vortex shedding process is further displayed in Fig. (8), which shows an example of the shape and duration of a complete band of increased motion. As can be seen from the figure, the duration of the gust induced motion is about 2 minutes, from initiation to termination. Power spectral density of the time series in Fig. (8) is displayed in Fig. (9). As the power spectral density function demonstrates, the response is a narrow-banded process dominated by the second mode, or the first vertical mode of vibration. Further system identification gave a natural frequency of 2.31. The damping estimation gave values between 0.5% and 1% of critical.

The narrow-band nature of the signal simplifies the data processing, since instead of inte-grating the acceleration time history numerically, which usually calls for data filtering and corrections for linear trend, etc., the deflection at the tip of the beam can be estimated by as-suming that the acceleration is a sine wave, which can be analytical integrated as:


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Figure 7: a) Data for 3 hrs. (12:00-15:00, 17 August 2005) from the accelerometer b) 10-second-long segment enlarged from the upper graph.

0 20 40 60 80 100 120 140-15

-10

-5

0

5

10

15

TIME (s)

ACCE

LER

ATIO

N (m

/s2 )

Figure 8: An example of a vortex shedding induced acceleration burst.

a)

b)

Acce

lera

tion

- (m

/s2 )

Acce

lera

tion

- (m

/s2 )

Time (s)

Time (h)


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-80

-60

-40

-20

0

20

40

Normalized frequency ( pi rad/sample)

One

-sid

ed PS

D (d

B/ra

d/sa

mple

)

Periodogram Power Spectral Density Estimate

PSD estimate of the recorded signalPSD of a model output

Figure 9: Power spectral density estimate of the gust induced acceleration time series displayed in Fig. (8).

2

222

( ) sin(2 )1( ) sin(2 )(2 )

a t A n t

d t A n tn

pi

pipi

=

=

(7)

where a stands for acceleration as function of the time, t, A is the maximum amplitude of the acceleration when described as sine-wave, d is the deflection as function of time, and n2 is the natural frequency of mode 2. This means that multiplying the recorded acceleration values with the constant 1/(2pin2)2 gives a good estimate of the deflection as a function of time. No other data processing was therefore necessary for the acceleration data in order to estimate the deflections.

4 AERODYNAMIC COUNTERMEASURES As mentioned before, the unpropitious rectangular shape of the cross-section of the beam,

sharp corners as well as low internal damping combine to promote vortex shedding excitation. Several measures to reduce the effects of vortex shedding have been suggested in the litera-ture and many have been successfully applied (see for instance Ref. [7] and Ref. [8]).

Suppression of vortex induced vibration can be achieved using mechanical means by mak-ing changes in member properties and structural detailing. Increase in the natural frequency, stiffness or damping, either singly or in combination, will generally reduce the oscillation amplitude. Another approach is to apply aerodynamic means to reduce the amplitude of re-sponse by disrupting the fluid processes involved in the vortex formation. The interaction af-fects the shedding mechanism primarily by interfering with the wake and reducing spanwise coherence. A mechanical solution would generally be used at the design stage whereas a fluid dynamic solution would be considered if in-situ problems were encountered, such as in the case discussed herein.


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Several possible solutions were considered, such as: To stiffen the beam; To add damping to the system, for instance by using a tuned damping device; To smoothen the corners of the cross-sections; To reduce the dimensions of the cross-section gradually from the column top to the

end of the cantilever beam; To make holes in the vertical webs of the cross-section of the cantilever beam; To design wind flaps on the beam.

In evaluating the possible solutions, issues like cost and change of appearance of the struc-ture were of concern, in addition to the expected efficiency of the method chosen. Eventually it was decided to install cylindrical wind flaps on the top and bottom of the cantilever beam. This rounds of the cross section and may introduce a positive Reynolds number effect. In ad-dition a 25 mm air gap was intentionally designed between the original beam and the cylin-ders in order to allow the wind to bleed through the section and thereby counteract the formation of vortices. This implementation does not require redesign of the original cross-section and provided an opportunity to try another solution if the vibration amplitude were insufficiently reduced. The chosen solution is shown by drawing on Fig. (10) and by photos on Fig. (11).

Figure 10: The original U-profile of the horizontal beam in the light mast, and the new cylindrical wind flaps with air gap for reducing vortex excitation.

5 RECORDINGS AFTER INSTALLATION OF COUNTERMEASURES In order to see the effects of the circular wind flaps data was recorded after their installa-

tion. The flaps added some weight to the structure and the response frequency of the recorded data was slightly reduced to 2.2 Hz.

After the installation of the wind flaps data were recorded for a total of 145 hours during 7 days compared to the 86 hours on 6 days before the installation. In order to see the effects of the countermeasures all the data, before and after installing the flaps, were divided into 10 minute intervals. Within each interval the peak deflection of the beam end and the mean wind speed and the mean wind direction were evaluated. The peak deflection versus the 10 minute mean wind speed for all data recorded before the installation is shown in Fig. (12a) and the same for all data recorded after the installation is shown in Fig. (12b). Comparison of the two

100 mm

25 mm

100 mm

400 mm

25 mm

Originally cross-section

New wind-flap

New wind flap

Air gap


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graphs demonstrates that the flaps are effective and reduce the vortex induced motion by about 70%. The displacements recorded after installation of the flaps are all less than 1.5 cm. Wind direction may of course have played some role in how the beam was excited. Fig. (12) displays data from all recorded wind directions for both cases, i.e. before and after the instal-lation of the flaps.

(a)

(b)

Figure 11: The light mast after installation of wind flaps with air cap on the cantilever beam. (a) The cantilever as seen from Northwest. (b) The intersection and the light mast seen as seen from Northwest.


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0 2 4 6 8 10 120

2

4

6Pe

ak di

spla

cmen

t - (cm

)a)

0 2 4 6 8 10 120

2

4

6

Peak

di

spla

cmen

t - (cm

)

Wind speed - (m/s)

b)

Figure 12: Peak deflection versus mean wind speed for 10 minute intervals, based on recorded data. a) Before installation of flaps (all available data) and (b) After installaion of flaps (all available data).

6 CONCLUSIONS A steel frame structure supporting approach lights for air traffic control was installed at

Reykjavik airport. The frames substructure, a cantilever beam, showed significant across wind motion in windy weather.

Analysis and measurements revealed that this behaviour was due to vortex shedding ef-fects. The phenomena are well documented and the formulation given in codes and the litera-ture was found to give an accurate prediction of the observed behaviour. However, it should be noted that the response estimates depend strongly on the critical damping ratio which is generally not a well known parameter.

Countermeasures in the form of cylindrical wind flaps on the top and bottom of the canti-lever beam to reduce the induced vibration amplitude were proposed and implemented. The recorded motions before and after installation of the flaps clearly indicate significant reduc-tion in the vibration amplitude of the cantilever light mast. Recorded peak deflections were roughly 6 cm before the installation of flaps but were reduced to less than 2 cm after the im-plementation.


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REFERENCES

[1] M. Matsumoto. Vortex shedding of bluff bodies: A Review, Journal of Fluids and Structures, 13 (7-8), 791-811, 1999.

[2] J. D. Holmes, Wind loading of Structures, Spon Press, 2001. [3] R. Hffer. Processes of buffeting and vortex forces in turbulent wind. Journal of Wind

Engineering and Industrial Aerodynamics, 64 203-220, 1996. [4] Eurocode 1: Actions on Structures - Part 1-4. CEN TC 250, prEN 1991-1-4(E), 2004. [5] Engineering Sciences Data Unit (ESDU) Data Item Numbers: Structures of non-

circular cross section: dynamic Response to Vortex Shedding, 90036. [6] H. Ruscheweyh, M. Hortmanns and C. Schnakenberg Vortex-excited vibrations and

galloping of slender elements Journal of Wind Engineering and Industrial Aerodynam-ics 65 (1996) 347-352.

[7] M. M. Zdravkovich. Review and classification of various aerodynamic and hydro-dynamic means for suppressing vortex shedding. Journal of Wind Engineering and In-dustrial, 7 (2), 145-189, 1981.

[8] R. Dutton and N. Isyumov. Reduction of tall building motion by aerodynamic treat-ments. Journal of Wind Eng. and Industrial Aerodynamics, 36 (1-3) 739-744, 1990.

[9] SAP2000, Version 10, Integrated Software for Structural and Analysis and Design. Computer & Structures Inc. Berkeley, California, USA.

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Vortex Shedding Induced Vibrations of a Light Mast