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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)

293

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 D

D U BC (5)

212 L

L 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 arctany

Uand

yU

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

observed that arctany y

U U when 0.2

yU

. 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) canno