A new approach for determining the onset velocity of galloping instability taking into account the nonlinearity of the aerodynamic damping characteristic

JOURNAL OF

windengineering ELSEVIER

Journal of Wind Engineering and Industrial Aerodynamics 69-71 (1997) 303 314

®(Nt~2~alF~i(Ng

A new approach for determining the onset velocity of galloping instability taking into account the nonlinearity of the aerodynamic damping characteristic Claudia Ziller*, Hans Ruscheweyh Institute o]'Steel Construction, R WTH Aachen, Mies-van-der-Rohe-Str. 1, D-52074 Aachen, Germany

Abstract Slim structural designs are known to be induced to vibrate in natural wind. These vibrations are mainly caused by eddies and galloping vibrations. Such oscillations are representing a safety risk and have repeatedly led to damage so that reconstruction has to be undertaken. The aim of this study is the investigation of aerodynamic instabilities in order to predict the response of slender structures in natural wind and to assure their safety. The present study provides aerodynamic coefficients for the calculation of vibrations caused by galloping oscillations either in the vertical mode or torsional mode in natural wind. The investigated section types are slim building components (1/d >~ 10) and they correspond to forms common in practical use. The aerodynamic damping characteristics are measured with section models using free decay tests and a forced oscillation method.The stability parameters such as lift and moment coefficients for various on-flow angles, for variations of the degree of turbulence and for different geometric dimensions of the profiles are measured. Within this study the nonlinear relation of the aerodynamic damping of a system and the wind velocity is demonstrated and discussed. Depending on the characteristic of this nonlinear relation two different methods of determining the onset point of galloping vibrations are introduced. Comparisons between observed torsional flutter responses of existing structures and the results of the section model tests are made. Both methods of determining the onset point for galloping vibrations are used in this case of damage and the results are discussed. Keywords: Galloping vibrations, vertical and torsional mode; Nonlinear damping characteristic; Wind tunnel tests; Section model technique; Stability parameters; Full scale observations

* Corresponding author. E-mail: [email protected]. 0167-6105/97/$17,00 ~.'), 1997 Elsevier Science B.V. All rights reserved. PII S0 1 67-61 0 5 ( 9 7 ) 0 0 1 64-5

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304

I. Introduction Slender structural designs are known to be sensitive to wind-induced vibrations such as vortex shedding, galloping and torsional flutter. Some buildings and their slender structural parts are built more and more frequently with material saving and highly sophisticated methods. This increases the probability of wind-excited vibrations. To predict the response of those structures to aerodynamic instability effects and to assure their safety it is necessary to perform wind-tunnel tests [6--8]. To describe the response of buildings and structures in natural wind the influence of the turbulence intensity, the geometric dimensions of the profiles and of the on-flow angle on these instability effects has to be determined. Galloping vibrations are dependent on turbulence to a considerably greater extent compared to the influence of turbulence effects for vortex-induced vibrations. By determining the onset point where galloping vibrations start off, the nonlinear relation between wind velocity and aerodynamic damping characteristic has to be included. Depending on the characteristic of this relation two different methods of linearization can be used [8]. The stability parameters for galloping vibrations calculated from the measured aerodynamic damping characteristic partly deviate from the known statically determined coefficients. This shows that the aerodynamic force terms which are linked to the vibration movement of the system may not be neglected. The available static coefficients usually lead to conservative results. But there are exceptions in which these static coefficients identify a system as stable although it can become unstable.

2. Theoretical background 2.1. Vertical mode oscillations

For an arbitrarily shaped two-dimensional bluff cylinder with a linear spring and viscous damping restrained to a translational degree of freedom in a smooth uniform flow with the velocity U0o which is subjected to a vertical aerodynamic force F(0, the equation of motion can be written as: m j; + k,,S' + Cyy + F(t) = O, 1

2

F(t) = ~ puoo dcy(~).

(1)

To include the experimental determined steady lift coefficient cv(:0 in an analytical way, the following polynomial form can be used:

(2) "

. = 1

\ U o o /

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305

In the instability point where the aerodynamic vibrations start off and the amplitudes of oscillations are small, it is possible to use the following linearized relation: K1

-

dcy . dc~ - a~,.

(3)

With co~, = 27rf~ s, = x/CSm,b~y = ks,/2mf~y , u~oa = Uoo/fo, d and with the Scruton number S%. = 2 m / p d 2 6~s, the equation of motion becomes

j~+ 2f, y p d 2 (

IAreda~).~ -[- O.)ey 2y

SCsy -}- ~ -

= 0.

(4)

The model will be unstable in translational oscillations if the damping term in Eq. (4) is smaller or equal than zero: 1

Scsy + ~ u~oa a*y ~< 0.

(5)

The second term in the inequality of Eq. (5) is the aerodynamic Scruton number: Ured 2m Sc,,, = ~ - a* = ~ aa,,

(6)

where 6ay is the aerodynamic damping. Thus, the stability condition in Eq. (5) can then be formulated as Scs~, + Sc~r ~ 0.

(7)

If the aerodynamic damping is constant for different velocities, the critical velocity for the onset of translational galloping vibrations can be determined from Eq. (5): 2Scsy Uoy = urea,o,,Lrd - ( 7 ~ y ) L , , d

4m6~y - pd2( _ a , ) f ~ f l .

(8)

Eq. (8) shows that negative values of the lift coefficient a s,, are essential for instability. F o r m a n y profiles the aerodynamic Scruton number Sca is a function of the reduced velocity Urea and is shown in principle in Fig. 1. This relation has to be determined with experimental procedures and shows the nonlinear character of the onset of galloping oscillations. For different Scruton numbers Scs of a vibrating system either the curve or the x-axis have to be shifted as shown in Fig. 1. This nonlinear relation is the reason for the following observation: the increase of the Scruton number which is a function of the structural damping by the same step size from Sc~l to Scs2 and Scs3 results, in fact, in an increase of the onset velocity Ured,O (U~ed,O~ to Ured,02) but with completely different step sizes. In order to determine the onset point of galloping vibrations it is necessary to take the nonlinear relation of Sc~ and urea into account. Depending on the characteristic of this relation two different methods to determine the critical on-flow velocity where galloping vibrations start off can be used.

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306

Sc

" , . _ ~ . Slope n, "A"

stable

g

l Scs3

unstable

Linearization 1 • Linearization 2 Fig. 1. A e r o d y n a m i c Scruton n u m b e r as a function of the reduced velocity.

A very conservative m e t h o d called Linearization 1 for the determination of this instability p a r a m e t e r is to linearize the relation as shown with the dashed line in Fig. 1. Linearization 1 can be expressed as Sc,o, = nlUrecl.

(9)

Eq. (7) then becomes Scsy =

(10)

- - F/1//red,Oy ,

and the critical onset velocity uo is Uoy = Ureo.oyfeyd

Scs~.

f~s,d ( Z nl ).

(11)

The stability p a r a m e t e r with this m e t h o d can be formulated using Eq. (6): a* = 2nl.

(12)

Below the point "A" in the d i a g r a m of Fig. 1 this linearization indicates instability while the real nonlinear curve shows that the profile is stable in this range. Thus, Linearization 1 has to be handled with care. Recent research results are representing a m o r e realistic and therefore m o r e accurate m e t h o d which is called Linearization 2 [1]. The onset point is determined by linearizing the part of the curve where galloping vibrations actually do occur. T h a t means, in the range where the curve is crossing the axis and the a e r o d y n a m i c Scruton n u m b e r becomes negative. This second m e t h o d is indicated with the dotted line in Fig. 1. This linearization can be written as SCay = Scagy q- F/ZUred.

(13)

C. Ziller, H. Ruscheweyh/Z Wind Eng. Ind. Aerodyn. 69 71 (1997) 303-314

307

With Eq. (7) the stability condition can be formulated as Scsv + SCa*,+ n2ured,O~, = 0,

(14)

and the critical onset velocity Uo is Uo~, = Ured,0~,f~vd =f~yd Scsy + SG*, --

(15)

n 2

The stability parameter can be written as * = 2SGy n2 a~, S%, + ScL"

(16)

In the case of SG*, = 0 the advanced method corresponds to the conservative method. Linearization 2 is only valid for velocities u~a above point "A". However, Linearization 2 has the advantage of giving more accurate and realistic results in the range of the aerodynamic characteristic of a profile where galloping vibrations actually do occur. Depending on the damping characteristic of a profile either Linearization 1 or Linearization 2 should be used to determine the onset point of galloping vibrations. 2.2.

Torsional mode oscillations

The same analytical considerations have to be made for the torsional-mode oscillations. The procedure is exactly the same and should therefore not be carried out in detail. The stability condition resulting from the equation of motion for the torsional mode of vibration is Scs~, "Jr- SCat, ~ 0.

(17)

L i n e a r i z a t i o n 1: Scaq 3 = F/I Ured

(18)

leads to the following onset velocity for galloping vibrations in the torsional degree of freedom: Scs~o

U0~o = Ured,0~ofeod = L ~ d ( -- n~)"

(19)

With L i n e a r i z a t i o n 2: Sca~ = Sc** + n2Ured,

(20)

the onset velocity can be determined by the following equation: Sc+~ + Sca~ Uo~o = Ured.o~ f e J

= L¢d

• --

n 2

(21)

308

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3. E x p e r i m e n t a l s e t u p

For the wind tunnel tests a wind tunnel with an open measurement section of the Institute of Steel Construction at the Technical University of Aachen ( R W T H Aachen) in G e r m a n y is used. The turbulence is generated by grids that are installed in the upstream of the working section. The experimental tests are carried out with section models between large endplates (section model technique). Two different measurement procedures to determine the aerodynamic damping characteristics are used: free-decay tests and a forced oscillation method. Using a dynamic experimental setup the effects of vibrations that are known to affect the aerodynamic response of a system are included. Due to the effects of the aerodynamic force terms the lift and moment coefficients found within this investigation differ significantly from the coefficients that are determined from the static results. The wind-tunnel models are rectangular cylinders with different b/d ratios, lsections and T-sections. Within this research project, the geometric dimensions, the on-flow angle and the turbulence intensity are varied.

4. R e s u l t s

Figs. 2 6 show the aerodynamic damping characteristic for vertical and torsional oscillations for some of the investigated profiles. Eqs. (11) and (15) (or Eqs. (19) and (21)) show that a negative slope of either Linearization 1 or Linearization 2 are essential for instability. A negative slope of the Sc~,

200

Rectangular Cylinder b/d = 2 On-Flow Angle ot = 0° Vertical Amplitudes y/d = +_0,3

100

x

x-~ -100

m

~---~

b ---~--- I=2,6%

--4-.--M---

-200 0

x'.\

d m

1=6,4% I=8,4% 1 = 10,8%

""~"~

:-2 - '* "~"X

I

I

I

25

50

75

Ured

100

Fig. 2. Aerodynamic Scruton number, vertical mode, cylinder h/d= 2.

309

C. Ziller, H. Ruscheweyh/J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 303 314

SCa\ 5000

Rectangular Cylinder b/d = 8 Turbulence Intensity 2,6% On-Flow Angle c~ = 0° Vertical Amplitudes y/d = _+1,5

40OO 3000

Moo

~ ,

, Id b

2000 1000

-1000 0

i

i

~

L

I

i

25

50

75

100

125

150

Urea 175

Fig. 3. Aerodynamic Scruton number, vertical mode, cylinder b/d = 8.

Sc.,, 100 [ ] |

I - Section b/d = 2 On-Flow Angle c¢ = 0° Vertical Amplitudes y/d = + 0,3

~ , , ~ "x~ ~'N

-50 ~ ,

-,,

~b _ _o~~. . ,od .~, ~ " b

-100

_

--X--

I=2,6%

-- 4,-.- 1= 6,4% --×---

-150 0

I=8,4% I = 10,8%

X

I

I

25

50

Ured

I

75

100

Fig. 4. Aerodynamic Scruton number, vertical mode, I-section bid = 2.

d a m p i n g c h a r a c t e r i s t i c indicates t h a t the profile m a y be instable. F o r a certain r e d u c e d w i n d velocity, the a e r o d y n a m i c S c r u t o n n u m b e r b e c o m e s negative. T h a t m e a n s there m a y be a p o i n t where this a e r o d y n a m i c S c r u t o n n u m b e r exceeds the s t r u c t u r a l d a m p i n g c h a r a c t e r i s t i c of the system a n d the d a m p i n g t e r m (Eq. (5)) in the

(Z Ziller, H. Ruscheweyh/J. Wind Eng. lnd Aerodyn. 69 71 (1997) 303-314

310

Stay

50

T - Section b/d ~ 1,5 On-Flow Angle c~= 0° Vertical Amplitudes y/d = + 0,3

25

U°°

×';,, "X ~. "-R ~ ~ " ' /

b ~ d

-25

I=2,6% ---~-'--X--

""

....

__

1-6,4% I-8,4% I 10,8% I

I

I

I

0

10

20

30

40

--

-50

---M.._

~ ~ X Ured

50

F'ig. 5. Aerodynamic Scruton number, vertical mode, T-Section bid - 1,5.

equation of m o t i o n (Eq. (4)) will be equal to or even less than zero. Thus, the structure is induced to vibrate. F o r the profiles in Fig. 2 (for the highest turbulence intensity) and Fig. 3 the slope of the curve is positive, the a e r o d y n a m i c Scruton n u m b e r does not change its sign, i.e., the profile is stable for b e n d i n g - m o d e oscillations. These d a m p i n g characteristics show the strong influence of the turbulence intensity. F o r these profiles it can be said that a high turbulence intensity has a stabilizing effect. The slope of the curve becomes smaller and even changes its sign. Fig. 6 presents the d a m p i n g characteristic for the torsional m o d e of oscillation while Fig. 3 shows the nonlinear relation for the bending mode. By c o m p a r i n g both figures it becomes obvious that a rectangular cylinder with these geometric dimensions is sensitive to the torsional galloping and is stable for the b e n d i n g - m o d e oscillation. F o r some of the tested profiles the p a r a m e t e r s Sc* and n2 and the a,*.-values for the models corresponding to the L i n e a r i z a t i o n 2 are given in Fig. 7. They are c o m p a r e d with a*-values given in the Eurocode 1, part 2.4 [2] and with a*-values observed from static tests given in Ref. [3]. In most of the cases, the realistic stability p a r a m e t e r s o* calculated using Linearization 2, differ significantly from the cr*-values given in the EC 1 [2, 3]. F o r example, the a*-values for the T-sections and I-sections from EC 1 [2, 3] m a y lead to very conservative and uneconomical results. But for some profiles (see, e.g., b i d = 2, ~ = 90 ° in Fig. 7) a e r o d y n a m i c coefficients are given which are not safe.

311

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gCa~ 6000 4000

/

/'~

~

Rectangular Cylinder b/d = 8 Turbulence Intensity 2,6% On-Flow Angle ot = 0° TorsionalAmplitudes lp --+ 3°

~ ~

2000 0 -2000 -4000 -6000 -8000

0 ~u'.~ .1, . 0

.b .

j

d

"~.~ X

X

I

t

I

t

i

25

50

75

100

125

Ured 150

Fig. 6. AerodynamicScruton number, torsional mode, cylinder bid = 8.

5. F u l l

scale

observations

Some cases of damage due to wind-induced vibrations in the torsional mode have been reported in the last years. One of them is the failure of a crane. Vibrations of the pendant bars, which are long and slender steel bars with a large bid ratio, have been observed. Investigations have revealed that the pendant bars suffered fatigue failure following a period of wind-induced torsional oscillations. High wind velocities were reportet for the damage situation. The question that arises now is whether the observations made on the cases of damage and the results of the wind-tunnel tests agree. The pendant bars of the crane had a ratio of bid = 8. The wind-tunnel tests showed that a structure with these geometric dimensions is sensitive to torsional galloping vibrations (see Figs. 3 and 6). Using the wind-tunnel test results and by following the calculation procedure described in the theoretical background, the onset velocity u0~ of the collapsed crane bar can be calculated. The structural parameters of the pendant bars (like geometric dimensions, mass, natural frequency and structural damping) are given. Fig. 8 shows the aerodynamic Scruton number as a result of the wind-tunnel tests of profiles with the same b/d-ratio like the pendant bars (see Fig. 6). Both linearization assumptions are made and the structural damping of the damaged pendant bar is indicated. Table l shows the calculated onset velocities using both linearization methods and following the real nonlinear curve.

C. Ziller, H. Ruschewevh,/.L ~Tnd Eng. Ind. Aerodyn. 69 71 (1997) 303 314

312

Rectangular Cylinder b/d On-Flow Angle c~ = 0 °

=

stable

2

0,5 Rectangular Cylinder b/d -- 2 On-Flow Angle ct = 90 °

S

stabEc /

0

!

e

unstabIe -0,5

-10

-1,0

i

i

i

i

i

2

4

6

8

10

][%1

o

+ unstable

-5

-15

,

-1,5

12

0

i

i

i

2

4

6

i

i

8

10

12

[%1

O'*,

5

5

stable

T-Section b/d - 1,5 On-Flow Angle 0t ~ 0 ° R- -

-

-

1'

-

'

2

stable

I-Section b/d = 2 On-Flow Angle c~ = 0 °

0

×

~

~ - - x

unstable -5

O

~5

<>

unstable

-5

.9

O--

C, -10

--

i

toO--

¢,,

~

-Ore

~

__

.O

,'~

-10

-15 0

. 2

.

4

.

. 6

.

8

10

12

I [%]

-15 0

12 [ [%1

X Linearization 2 - - o - - EC 1 [4] static tests [5] Fig. 7. C o m p a r i s i o n [4, 5].

of stability parameters

o-* o f L i n e a r i z a t i o n 2 w i t h v a l u e s g i v e n in t h e E u r o c o d e

1

Observations that were made on site in the time when the crane was damaged showed no torsional flutter responses when the wind velocity was in the range 10-20 m/s. This effect can be explained with the nonlinear damping characteristic discussed in the theoretical background. Fig. 1 explains the reason: using the Linearization 1 the result will be very conservative and leads to low-onset velocities. But Linearization 2 describes the vibrational response of the pendant bar more realistic: the onset velocity is larger and this result agrees with the observations made on site. Wind velocities within this range (25 30 m/s) were measured at the site of the damage. In addition, it was investigated that the wind was blowing from west and the pendant bar profile was attacked in the critical direction. These considerations demonstrate the importance of the enclosure of the nonlinear response of structures in natural wind. For the design of structures or building components the simple conservative theory (Linearization 1) is sufficient and safe. But in cases of damage or even failure it may be necessary to evaluate the exact and realistic character of wind responses.

313

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Sce~

6000 4000

I

I

\

~ ~t" "x.

/

i

"~ \ ~

~-

i

I

Pendant bar of the crane b/d = 8 Turbulence Intensity 2,6% On.FlowAnglee~=O °

2000 0

"-----.I

-2000

--'%

Linearization l - . X

Sc~= 1408

X ' ~

-4000 Uoo

-6000

I, -8000

" \-

~,~ t

0

o,

/ b

25

Linearization2 \ .

X

I

L

r

~,

50

75

100

125

'

Urcd

50

Fig. 8. Aerodynamic Scruton number for the damaged pendant bar.

Table 1 Linearization 1 Linearization 2 Real nonlinear curve

u0~o= 11 (m/s) uoo = 27 (m/s) Uo~,= 28 (m/s)

However, the stability parameters determined within these investigations and the observations and measurements of the collapsed crane bar show a very good agreement.

6.

Conclusions

The aim of this research work was to investigate the vibration behavior of different profiles in natural wind (specially galloping vibrations). Wind-tunnel tests were carried out with freely vibrating section models and with a forced oscillation method [1]. The measured stability parameters ay* (lift coefficient) and a~* (moment coefficient) differ in part considerably from the available statically determined coefficients. This is due to the aerodynamic force terms linked with the vibration movement, which cannot be measured with static procedures. Wind-tunnel tests for establishing these coefficients should therefore only be carried out with a dynamic experimental setup.

314

C. Ziller, H. Ruschewevh/J. Wind Eng. Ind. Aerodyn. 69 71 (1997) 303 314

An i m p o r t a n t p o i n t of t h e i n v e s t i g a t i o n was the i n f l u e n c e of the t u r b u l e n c e i n t e n sity. G a l l o p i n g v i b r a t i o n s are d e p e n d e n t o n t u r b u l e n c e . It is difficult to find a c l e a r c o n c e p t w h e t h e r a p r o f i l e is s t a b i l i z e d o r d e s t a b i l i z e d by t u r b u l e n c e effects. H o w e v e r , it was tried to s h o w a t e n d e n c y . T h e s t u d y d e m o n s t r a t e s the i m p o r t a n c e of the e n c l o s u r e of the n o n l i n e a r d a m p i n g c h a r a c t e r i s t i c of the s t r u c t u r e s in n a t u r a l w i n d in o r d e r to p r e d i c t the r e s p o n s e o f b u i l d i n g c o m p o n e n t s in a m o r e a c c u r a t e a n d realistic way. D e p e n d i n g o n the d a m p i n g c h a r a c t e r i s t i c t w o different m e t h o d s c a n be used. C o m p a r i s o n s b e t w e e n o b s e r v a t i o n s of t o r s i o n a l flutter r e s p o n s e s o f a c r a n e a n d the results of the s e c t i o n m o d e l tests are m a d e . T h e a g r e e m e n t of the results of the full-scale m e a s u r e m e n t s a n d c a l c u l a t i o n s of t h e v i b r a t i o n a l r e s p o n s e w i t h the s t a b i l i t y p a r a m e t e r s d e t e r m i n e d in the w i n d t u n n e l are satisfying.

References Eli M. Hortmanns, Contribution to identification and consideration of nonlinear aeroelastic effects, Doctor Thesis, not yet published, Institute of Steel Construction, RWTH Aachen, Germany 1996 Ein German]. [2] Eurocode 1 (ENV 1991-2-4): Basis of design and actions on structures. Part 2 4: Wind actions, CEN 1994. [3] H. Ruscheweyh, Dynamic Wind Effects on Structures, vol. l/ll, Bauverlag, Germany, 1982 [in German]. [4] K6nig, Bremer, Torsional flutter of tension bars, Final Report No. 530-3/80, Institute for Light Construction, Dresden, Germany, 1980 [in German]. [5] M. Matsumo, H. Shirato, S. Hirai, Torsional flutter mechanism of 2-D H-shaped cylinders and effect of flow turbulence, J. Wind Eng. Ind. Aerodyn, 41 44 (1992) 687 698. [6] H. Ruscheweyh, M. Hortmanns, W. Zillinger, Systematic investigation of wind induced vibrations of slender structures in order to establish specifications for constructions (e.g., DIN 18809 and Eurocode), Final Report of the Research Project No. FEI5.184.R89G, Minister of Transport Germany, vol. 634, 1992 [in German]. [7] C. Ziller, Determination of aerodynamic coefficients of different rectangular profiles for wind induced excitation, Proc. 9th ICWE, New Delhi, India, 1995, pp. 403 418. [8] C. Ziller, M. Hortmanns, H. Ruscheweyh, Determination of aerodynamic coefficients of different profiles for wind induced excitation, Proc. Int. Syrup. on Cable Dynamics, Lfittich, Belgium, 1995, pp. 263 272.