The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 2012

Coupling investigation on Vortex-induced vibration and Galloping of rectangular cylinders

Claudio Borri a, Shuai Zhou *,a, b, Zhengqing Chen b

a CRIACIV/Department of Civil and Environmental Engineering, University of Florence, Via S. Marta, 3, Firenze, Italy

b Wind Engineering Research Center of Hunan University, Changsha, Hunan, China * First and corresponding author

ABSTRACT: Vortex induced vibration (VIV) is caused by regular alternatively shedding vortices from structure surface; the critical wind velocity where vibration onset can be approximately calculated by Strouhal relation. While, according to quasi-steady galloping theory, the galloping occurrence is originated from aerodynamic instability of specific structure cross sections. The negative slope of lift force coefficient of cross sections is an essential condition to predict the onset velocity of galloping. For those structures, where critical wind velocity by VIV and galloping are very close, the actual structural vibration type is unknown. With the aim of knowing more about the mechanism of this new type vibration, several comparable experiments of rectangular cylinders were conducted; the main results are reported in the present paper. The different experimental cylinders varied in aspect ratios and Scruton numbers. When the predicted critical wind velocity of VIV and galloping were very close to each other, the test results indicated that the cylinders tended to undergo a very different vibration type. There was neither VIV lock-in phenomenon nor galloping divergent type vibration, but consistently increasing vibration amplitude with increased wind velocity. While, for the cases of predicted VIV and galloping critical velocities were deviated largely, the VIV lock-in and divergent type galloping were observed separately. And it is seems that the interaction of VIV and galloping brought relatively large VIV amplitude during lock-in range.

KEYWORDS: Interaction; VIV; Galloping; Amplitude; Rectangular cylinders; Onset velocity

1 INTRODUCTION Slender structures such as the hangers of large span arch bridges, lateral beams of electricity transmission towers .etc, are inclining to wind induced vibrations. In particular, for the rectangular cross section hangers, the VIV and galloping are the two most frequently occurred vibrations. (Chapin and Bearman, 2005a; Ge, 2008). To predict the critical wind velocity at which VIV onset in engineering practices, Strouhal relation was used as the most efficient method. The Strouhal number was determined by the structural cross section, Reynolds number, surface roughness, turbulence intensity .etc. Over the

ing VIV, mathematical models were developed to describe the vortex excited force and predict the VIV lock-in range; well known reviews about these topics were given by Bearman (1984), Sarpkaya (2004) and Williamson and Govardhan (2004). These models can be basically classified into single Degree of Freedom (SDOF) and two DOF two types in total. Moreover, the SDOF model subdivided into force coefficient based model and negative aerodynamic damping based model. With the beforehand identified aerodynamic parameters, the prevailing Van der Pol-type nonlinear model was verified to reflect the VIV lock-in very well.

292

Similarly, in order to determine the galloping instability critical point, a relation derived from quasi-steady galloping theory is also available. It takes the consistently changed static force which was caused by relative motions between structure and flow as the original force for galloping instability. The negative slope of lift force coefficient of cross section is an essential requirement for the galloping occurrence, and the onset velocity is positively proportional to the Scruton number and natural frequency. The vibration is considered to divergent after the critical wind velocity threshold which calculated from quasi-steady galloping theory. Nevertheless, Ziller and Ruscheweyh (1997) suggested a new approach to determine the onset velocity of galloping instability, taking into account the nonlinearity of the aerodynamic damping characteristic. Hortmanns and Ruscheweyh (1997) even developed a method to calculate galloping amplitudes considering nonlinear aerodynamic coefficients measured with the forced oscillation method.

s of VIV and galloping are very different, so that the critical wind velocities of these two kinds of vibration can be obtained separately. Therefore, for certain structures, if the critical wind velocity of VIV and galloping are very close, the vibration mechanism is hard to be predicted. More importantly, the better understanding of the mechanism is pretty helpful for the following design of vibration suppressions measures, like the design of tuned mass damper.

2 THEORETICAL ANALYSES

2.1 VIV The nonlinear vortex induced excitation force model suggested by Scanlan and Ehsan (1990), which was characterized by Van der Pol-type oscillator, the vortex induced excitation force was expressed as follow:

22

1 2

2

1 ( , ) ( , )( ) [ ( ) (1 )2

( , ) 1( ) ( ) sin( )]2 L

v x t v x tP t U D Y KD U

v x tY K C K tD

(1)

Where, ( , )v x t = the structural displacement response; 1( )Y K , 2 ( )Y K , , ( )LC K , =the aerodynamic coefficients; = the air density; U =mean wind velocity; D =cross flow dimension;

= vortex shedding frequency. For structures undergoing VIV by wind action, the vibration frequency was almost identical to the natural frequency, hence, the aerodynamic stiffness term was negligible (Zhu, 2005); then Eq.(1) became:

22

1 2

1 ( , ) ( , ) 1( ) [ ( ) (1 ) ( ) sin( )]2 2 L

v x t v x tP t U D Y K C K tD U

(2)

Moreover, during the large amplitude vibration period of lock-in range, the vortex shedding force can also be neglected compared to the motion induced one, then the final vortex induced excitation force formula was simplified as:

22

1 2

1 ( , ) ( , )( ) [ ( ) (1 ) ]2

v x t v x tP t U D Y KD U

(3)

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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 2012 Marra et al. (2011) had proved its accuracy to describe the VIV by the beforehand identified aerodynamic coefficients 1( )Y K , . The critical wind velocity at which vibration starts off is set by Strouhal relation, defined as following:

f DStU

(4)

Where, f = natural frequency; St = Strouhal number which is associated with cross section, Reynolds number, surface roughness, turbulence intensity .etc; D = cross flow dimension; U = mean wind velocity.

2.2 Galloping The aerodynamic force of galloping was originated from continuously changed static force, which was caused by continuously changed relative wind attack angle between structure and approaching flow. (Chen, 2005; Borri et al, 2002).Ignoring the unsteady flow around the structure, the quasi-steady galloping theory was employed. Uniform flow passed through the bluff body, the 2D flow condition was shown in Fig.1.

U

Fig.1. Plane of 2D flow condition

Under the coordinate system of wind direction, the static drag force and lift force assume following form:

212 DD U BC (5)

212 LL U BC (6)

Where, DC = drag force coefficient; LC = lift force coefficient; = relative wind attack angle; B = cross flow dimension; U = mean wind velocity. The combined vertical force yF obtained:

294

2

22

2

1 sin cos21 1sin cos2 cos1 tan sec2

y D L

D L

D L

F U B C C

U B C C

U B C C

(7)

Where:

2 2

arctan

U U yy

U

(8)

While, dyydt

= structural vibration velocity. Spreading the Eq. (7 0 ,

then Eq. (7) became:

22 2

2

1 1[( ) ( 2 ) ( )]2 2

nL L Dy D L

dC d C dCF U B C Cd d d

(9)

Before the large amplitude galloping occurs, the tiny vibration velocity y is negligible compared to the approaching wind velocityU , and one can assume that:

arctan y yU U

(10)

The relations between arctan yU

and y

Uwas plotted in Fig.2, from which can be clearly

observed that arctan y yU U

when 0.2yU

. Actually, at the very tiny vibration period, the

high order terms of formula (9) are negligible with respect to the fundamental ones, hence, the formula can be simplified as following:

2

2

1 ( )21 ( )2

Ly D

LD

dCF U B CddC yU B Cd U

(11)

The galloping instability onset velocity criU obtained by the simplified aerodynamic force of formula (11):

4cri

LD

mUdCB Cd

(12)

295

The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 2012 Where, m = equivalent mass; = damping ratio; = circular natural frequency; the meaning of other symbols just as mention above.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(dy/dt)/U

Fig.2. Comparison of arctan yU

and y

U

The quasi-steady galloping theory can interpret the galloping onset threshold very clear, and it can be considered that a divergent type vibration after the critical point will appear which is due to the negative damping. However, for the vibration regime after the starting point, the theory cannot help in predicting it. Because, the high order terms of formula (9) cannot be ignored anymore, on the contrary, they played a very important role to the aerodynamic force.

3 EXPERIMENTAL DETAILS Taking into account the practical engineering background of arch bridge hangers, four groups of rectangular cylinders were selected as section model wind tunnel tests objects. The four groups of cylinders had different cross section aspect ratios, Scruton number and natural frequencies. Static forces tests and flexibly mounted section model tests were performed for each group separately. The section model wind tunnel tests installation set up are shown in Fig.3. Regarding the flexibly mounted section model tests, as shown in the second column of Fig.3, were the flexibly mounted rigid section models. Eight springs supported the model as stiffness system; the physical mass of section model, participating mass of springs and the supporting system mass formed the total mass system. Four force transducers were equipped at the bottom of the model system, connecting the lower four springs with fixed boundary. Vibration signals of heaving, pitching, rolling mode can be easily extracted from the recorded data of the four force transducers.

296

With respect to the static force tests, just as shown in the third column of Fig.3, rigid section models with end plates were vertically installed in the wind tunnel, the static force measurement scale equipped at the bottom of model.

Fig.3 Overview of the four test groups. The consecutively first row to the fourth row indicates the details of Group 1to Group 4, separately. And the first column is the 2D cross section dimensions of each group; flexibly mounted section model rests setup photos are shown in the second column; while the last column is the static force tests photos. Table.1~Table.4 shows the details of each group Damping ratio identification tests using the best decay estimation method, data analysis was conducted by the recorded decaying signals after the imposed initial displacement. The natural frequencies could also be obtained by FFT of the recorded decaying time series. Strouhal number identification tests were conducted with the same section model; the final values were determined by the Strouhal relation. The galloping instability coefficient was calculated from the static force coefficient curves according to the quasi-steady theory. It was definitely noted that the identical model is characterized by different galloping instability coefficients ( ) and Strouhal numbers (St) when the width and height of the cross section weren t identical. Finally, with the previously obtained Strouhal number and the natural frequencies of section model, the critical wind velocity of VIV can be predicted by Strouhal relation. Meanwhile, the galloping instability onset velocity can also be calculated based on quasi-steady galloping theory.

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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 2012

Table.1 Main parameters of Group 1

v

vf

Table.2 Main parameters of Group2

v

vf

298

Table.3 Main parameters of Group3

v

vf

Table.4 Main parameters of Group4

v

vf

The static force coefficient and galloping instability result curves are shown in Fig.4. The wind attack angles were defined as 0 when the section model anti-wind with their short edges, therefore, the wind attack angles must be 90 when the model anti-wind with their long edges. The static force coefficient results were shown in the first column of Fig.4. And the galloping

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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 2012 instability coefficients were calculated exclusively through the static force coefficient curves according to the quasi-steady galloping theory.

0 10 20 30 40 50 60 70 80 90-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

Stat

ic fo

rce

coef

ficie

nts

CD CL CM

0 10 20 30 40 50 60 70 80 90-5

0

5

10

CD+d

CL/d

0 10 20 30 40 50 60 70 80 90-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

Stat

ic fo

rce

coef

ficie

nts

CL

CD

CM

0 10 20 30 40 50 60 70 80 90-5

0

5

10

CD+d

CL/d

0 10 20 30 40 50 60 70 80 90-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

Stat

ic fo

rce

coef

ficie

nts

CL

CD

CM

0 10 20 30 40 50 60 70 80 90-5

0

5

10

CD+d

CL/d

0 5 10 15 20 25 30 35 40 45 50-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

Stat

ic fo

rce

coef

ficie

nts

CD CL CM

0 5 10 15 20 25 30 35 40 45-5

0

5

10

C D+d

C L/d

Wind attack angle ( ) Wind attack angle ( )

Fig.4. Static force coefficients and galloping instability coefficients curves vs. wind attack angles for the four experimental groups. The 1st to the 4th rows are representing the results of the Group 1to Group 4, separately. While, the first column are the static force coefficient curves and the second column are the galloping instability coefficient curves.

300

4 RESULTS AND DISCUSSIONS In this section, the aerodynamic responses were reported by comparing the time series and vibration frequencies of heaving, pitching and rolling modes. The final response curves of each group were also shown together with the corresponding estimated critical normalized wind velocity of VIV and galloping. In particular, for each run of every case, enough time was allowed for the vibration to stabilize before collecting the test data.

0 10 20 30 40 50-50

-40

-30

-20

-10

0

10

20

30

40

50

Hea

ving

(mm

)

0 2 4 6 8 100

5

10

15

20

25

0 10 20 30 40 50-50

-40

-30

-20

-10

0

10

20

30

40

50

Rol

ling

(mm

)

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50-5

-4

-3

-2

-1

0

1

2

3

4

5

Pitc

hing

(°)

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Time (s) f (HZ)

Fig.5. Tests example of Group 3: the section model anti-wind with short edge, at U/fD=23.97, which is related to the measured maximum point of galloping instability .The first row is the time series and corresponding frequency spectrum by FFT of heaving mode; the second row is for rolling mode and the third row is for pitching mode.

Let s take the group 3 as an example to illustrate the data analysis. Just as shown in Fig.5, these are vibration details of group 3 (short edge facing the flow), at the normalized wind velocity of U/fD=23.97. It can be clearly observed from the comparison of three different modes that the

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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 2012 pitching and rolling modes are about 10 times smaller (and therefore negligible) with respect to the heaving mode. Despite the vibration amplitude up to A/D=0.5, the heaving mode still vibrates at the identical frequency to the natural frequency f=1.51HZ and maintains the continuously steady amplitude. While, for the cases of model anti-wind with long edge of group 3, normalized wind velocity at U/fD=8.89, the vibration details are shown in Fig.6. The peak amplitude of VIV lock-in range was selected. Very similar to Fig.5, in this case, the mode vibrates dominantly at the heaving mode and at the single natural frequency; more importantly, time series also show the steady amplitude vibration.

0 10 20 30 40 50-50

-40

-30

-20

-10

0

10

20

30

40

50

Hea

ving

(mm

)

0 2 4 6 8 100

5

10

15

20

25

0 10 20 30 40 50-50

-40

-30

-20

-10

0

10

20

30

40

50

Rol

ling

(mm

)

0 2 4 6 8 100.0

0.4

0.8

1.2

1.6

2.0

0 10 20 30 40 50-5

-4

-3

-2

-1

0

1

2

3

4

5

Pitc

hing

(°)

0 2 4 6 8 100.0

0.4

0.8

1.2

1.6

2.0

Time (s) f (HZ)

Fig.6. Tests example of Group 3: the section model anti-wind with long edge, at U/fD=8.89, which is related to the maximum point of VIV lock-in range .The first row is the time series and corresponding frequency spectrum of heaving mode; the second row is for rolling mode and the third row is for pitching mode.

302

The conclusive experimental measured vibration curves of group 1~group 4 are shown in Fig.7, together with the predicted critical wind velocity of VIV and galloping. Unfortunately, the experimental data of group 1and group 4 were not enough when the vibration curves were straightly increasing with the increased wind velocity. Nevertheless, it can be clearly seen from other curves that the measured maximum normalized amplitudes are very large although the Scruton number of each model are over 10. For the cases of group 2 and group 3, which the estimated critical wind velocity of VIV and galloping are within the range of 1.5 times, an interesting vibration phenomenon occurred. There is neither self-limited VIV lock-in nor divergent type galloping appearing, whereas the vibration amplitude consistently increases with the increasing wind velocity. It can be obviously observed that there with still no indications of trend of the response amplitude to stop increasing even the normalized amplitude A/D rising up to 0.7. Moreover, the vibration time series of each wind velocity component, especially for the large amplitude ones, were very similar to the form of Fig.5. Recorded vibration time series always maintain the astonishing steady amplitude, and vibrated at the natural frequencies. In particular, the factor of 1.5 times between predicted critical wind velocity of VIV and galloping is also recommended by the Eurocode at which the specific research should be undertaken. However, for the cases of group 1 and group 3, the section models facing wind with their long edges, the estimated critical wind velocity of VIV and galloping were deviated significantly. The separated VIV lock-in range and divergent type galloping phenomenon were observed separately. Meanwhile, it is worth noting that the measured peak amplitudes of VIV lock-in are extraordinary large. The Scruton numbers of these two groups are 12.83, 19.83 respectively, and the experimental measured non dimensional amplitude reach 0.15, 0.29 respectively. In contrast to wind induced vibrations of circular section structures, it is very uncommon for these rectangular section models to oscillate up to such large amplitudes with the relatively large Scruton number. Therefore, one can reasonably assume that the negative aerodynamic damping of galloping instability contributes to the large amplitude VIV to some degree.

0 5 10 15 20 250.00

0.05

0.10

0.15

0.20

0.25

0.30

A/D

response V

cri of VIV

Vcri

of Galloping

0 5 10 15 20 250.00

0.05

0.10

0.15

0.20

0.25

0.30

A/D

response Vcri of VIV Vcri of Galloping

0 5 10 15 20 250.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

A/D

response V

cri of VIV

Vcri

of Galloping

0 5 10 15 20 250.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

A/D

response V

cri of VIV

Vcri

of Galloping

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The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 2012

0 5 10 15 20 250.0

0.1

0.2

0.3

0.4

0.5A

/D response V

cri of VIV

Vcri

of Galloping

0 5 10 15 20 25 30 35 400.0

0.1

0.2

0.3

0.4

0.5

A/D

response V

cri of VIV

Vcri

of Galloping

0 5 10 15 20 250.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

A/D

response V

cri of VIV

Vcri

of Galloping

0 5 10 15 20 250.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

A/D

response V

cri of VIV

Vcri

of Galloping

U/fD U/fD

Fig.7. Conclusive test results: non dimensional vibration displacement A/D vs. normalized wind velocity. The first row to the fourth row represent the wind induced vibration curves of group 1~ group 4 respectively. And the entire first column is with respect to the condition of section model of each group anti-wind with short edge, while the whole second column is for the conditions of section models anti-wind with long edge.

5 CONCLUDING REMARKS Four groups of tests on rectangular section model were performed in wind tunnel, each group of models are varied in aspect ratio and Scruton number. The critical wind velocity of VIV and galloping were also predicted with the beforehand identified aerodynamic coefficients. Main characteristics of result curves allow following remarks. With respect to the circumstance that estimated critical wind velocity of VIV and galloping are very close, e.g. within the range of 1.5 times, an interesting experimental phenomenon occurs. The interaction of VIV and galloping aerodynamic forces creates a new oscillation type: it shows neither lock-in VIV nor divergent type galloping characteristics. The cross flow vibration amplitudes consistently increases with the increased wind velocity, moreover, the vibration time series of each steady wind velocity always maintains constant amplitude. The VIV aerodynamic force seemed to suppress the occurrence of divergent type galloping. When the predicted critical wind velocity of VIV and galloping diverge significantly to each other, the fully measured VIV lock-in range demonstrates the relatively large non dimensional amplitude which the circular section structures cannot achieve at the same Scruton number. According to the quasi-steady galloping theory, the circular section will never attain the galloping instability negative damping; whereas the rectangular models investigated in the present work always being the models differentiated in aspect ratio and Scruton number (B/D, Scr, Table 1~4). Hence, it is reasonable to assume that the interaction of VIV and galloping contribute to the distinctively large VIV amplitude.

304

6 ACKNOWLEDGEMENTS The Authors are grateful to the Natural Science Foundation of China under the Grant 50738002, 50908085 and graduate students innovation fund of Hunan province; furthermore, the partial support of CRIACIV for his research stay in Germany is also acknowledged.

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[2] Ge, Y.J., Xiang H.F., 2008. Recent development of bridge aerodynamics in China. Journal of Wind Engineering and Industrial Aerodynamics 96,736-768.

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[4] Sarpkaya, T., 2004. A critical review of the intrinsic nature of vortex-induced vibrations. Journal of Fluids and Structures 19, 389 447.

[5] Williamson, C. and Govardhan, R. (2004). Vortex-induced vibrations. Annual Review of Fluid Mechanics, 36(1), 413-455.

[6] Ziller, C., Ruscheweyh, H., 1997. A new approach for determining the onset velocity of galloping instability taking into account the nonlinearity of the aerodynamic damping characteristic. Journal of Wind Engineering and Industrial Aerodynamics 69-71, 303-314.

[7] Hortmanns, M., Ruscheweyh, H., 1997. Development of a method for calculating galloping amplitudes considering nonlinear aerodynamic coefficients measured with the forced oscillation method. Journal of Wind Engineering and Industrial Aerodynamics 69-71, 251-261.

[8] Simiu, E., Scanlan R.H., 1996.Wind effects on structures:fundamentals and applications to design, 3rd Edition. John Wiley& Sons, New York.

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[10] Marra, A.M., Mannini, C., Bartoli, G., 2011. Van der Pol-type equation for modeling vortex-induced oscillations of bridge decks. Journal of Wind Engineering and Industrial Aerodynamics 99,776-785.

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