Determination of flutter derivatives for the great belt bridge

Journal of Wind Engineering and Industrial Aerodynamics, 41-44 (1992) 153-164

153

Elsevier

Determination of Flutter Derivatives for the Great Belt Bridge N.K. Poulsena, A. Damsgaardb and T.A. Reinholdb aiMSOR, The Technical University of Denmark, DK-2800 Lyngby, Denmark bDivision of Ocean, Wind and Information Technology, Danish Maritime Institute, 99, Hjortekaersvej, DK-2800 Lyngby, Denmark

Abstract A new method which combines control theory and system identification techniques has been used to extract flutter derivatives from section model tests for the Great Belt East Bridge. Test~ were conducted by exciting the section modal simultaneously in vertical and torsional modes of vibration. Tests were primarily conducted in smooth flow for various ratios between vertical and torsional frequencies of vibration. Limited testing was also conducted in turbulent flow and for different angles of attack. The analysis technique described allowed extraction of seven aerodynamic derivatives including coupled motion derivatives from a simple set of tests. This paper describes the control theory and system identification approach used and discussesthe limitations encountered. Results are compared with flutter derivatives obtained by other researchers. The method offers a reasonably i obust technique for automating the process of extracting aerodynamic derivatives from recorded time histories of coupled vertical and torsional motion.

1. INTRODUCTION Tests were carried out to determine aerodynamic derivatives for the boxg!rder deck section selected for the Tender Design of the Great Belt East Bridge. The Great Belt East Bridge is being designed as a suspension bridge with a length of 2696 metres between anchor blocks ancl a clear span of 1624 metres between the main towers as shown in Figure 1. The Great Belt East Bridge will provide a vehicle crossing over the main international shipping channel between the islands of Zealand and Funen in Denmark. The entire 2696 metre deck between anchor blocks will be cable-supported and will have no fixed vertical supports of the main towers. Thus, the bridge will contain the longest cable-supported deck in the world, even after completion of the Akashi bridge in J.~pan. The cross-sectional shape of the box-girder selected for the tender design is shown in Figure 2. This streamlined deck section is the product of extensive section model tests to investigate the stability of various proposed deck crosssections which have been carried out by a Joint Venture o~ the Danish Maritime institute (DMI) and SINTEFStructural Engineering. 0167-6105/92/$05.00 © 1992 Elsevier Science Publishers B.V. Ag rights reserved.

154

/ ~62~1,

_ 5~5

........

I I

L

/ .........................

1824

. . . . . . . . . . . .

.J,

4r.,l~, 8

qT

.

53%

,~2v

r, 1 44'~44

~I

Figure 1. Schelilatic Drawing of Great Belt East Bridge.

I 2s

I

1 21.2s

H = sso (4.4o)

F

roo,,o, 37.5

S.C. ',,IF

H9.1 I I

L 7s0

i

h_. ~ . 4

0

237s 8:

I

7s0

387s (31.0)

I I

Model dimensions are given in ram. Figure 2. Cross Section of Box-Girder Deck for Great Belt East Bridge. The objective of the test programme was to provide data on aerodynamic derivatives for use in assessing the stability and buffeting response of the bridge in its completed condition as well as during the construction stages when the vertical and torsional frequencies of vibration may be closer together than in the completed bridge. • Testing was performed in DMI's Boundary-Layer Wind Tunnel and data analysIs was pertormed using techniaues and computer programmes developed by the first Author at IMSOR, the Technical University of Denmark. The studies were sponsored by A/S Storebaeltsforbindelsen, the developer of the Great Belt crossing. The structural designer is CBR Engineering, a Joint Venture of COWlconsult, B. Hojlund Rasmussen and Rambell & Hannemann A/S.

155

To the Authors knowledge, this represents the first application of system identification and control theory techniques to the problem of extracting aerodynamic derivatives from bridge-section model tests. Consequently, the paper focuses both on the analysis techniques and on the practical problems encountered. Results are also presented and compared with aerodynamic derivatives extracted by other researchers for streamlined shapes.

2. STRUCTURE OF THE MATHEMATICAL MODEL

Following the arguments of Scanlan [1 - 2], the mathematical model is given by the following coupled differential equations: I~ + pol~ + ~oh = H l h + H2~ + H3a + H4h

(1)

+ Ooa + ¥oa = A1 h + A2~ + A3a + A4a

(2)

These equation represent a general linearized form for self-excited forces under the assumption of small sinusoidal vertical (h) and torsional (a) motions with negligible horizontal motion effects. Frequently, the structural model parameters (~o, po, Yo and Oo) are presented in the following form: Do

=

(2n fh) 2

go

=

2 ~,h (2n fh)

Yo

=

(2n fa) 2

Oo

=

2 ~:~ (2n fo)

(3)

where fh and fa are the free resonance frequencies for the vertical and torsional motions and the parameters ~,h and ~,n are related to the structural damping effect. The goal of the testing and analysis is to determine the dependence of the aerodynamic derivatives H 1 through H4 and A1 through A4 on wind velocity. It is a basic assumption that the aerodynamic parameters are zero under zero wind conditions so that ~o, IJo, Yo, Oo describe the test rig. In the common formulation of equations 1 and2 [3 - 4] the derivatives H4 and A4 are omitted as it is expected that the vertical position of the deck (h) has no effect on the torsional frequency, torsional damping or vertical stiffness. However, equation (1) already contains the term ~oh on the left-hand side, so maintaining the H4h term on the right-hand side does not change the solution technique. Consequently, only the A4 parameter was omitted in this study. For wind velocities greater than zero, the motion of the bridge is then given by' I~ + (IJo- H1)'h + (~o- H4) h- H20- H3o = 0 + (oo-A2) a + (yo-A3) ct-Ali~ = 0 i.e. two second order, coupled differential equations.

(4)

(s)

156

The wind effect is a shift in frequencies and damping terms - and a coupling between the two directions of motions. The analysis is consequently divided into two sub-problems, namely system identification and parameter determination. The system identification problem is to estimate from the data, the parameters p, 13,p, ~, o, y, and 8 in the following equations: I~ +pla +.Bh + p ~ + ~ a = O

(6)

~+oo+yo+6fi=O

(7)

The estimation procedure is performed for each value of the wind velocity for which tests are conducted including zero wind conditions. The parameter determination problem involves two parts. The first is simple to compute the aerodynamic derivatives for a specific test by comparing the estimates obtained in the system identification analysis with the estimates obtained with zero wind conditions: A1

=

- 6,

A2

=

Oo - o,

A3

=

¥o "

H 1

=

•o

H2

=

- p,

H3

=

-~,and

H4

=

13o- P

¥,

" P,

(8)

The second paf~ of the parameter determination problem is to ensure that the parameters are extracted for similar amplitudes of vibration since some of the parameters may be amplitude dependent and to use statistical procedures to extract reliable estimates for the parameters. Tests were repeated ten times at each velocity in order to produce a large enough data base for statistical analysis of the parameters.

3. SYSTEM IDENTIFICATION The two most basic elements in system identification are the data and the model structure. The model structure is assumed to be given by equations 6 and 7 which represent an external model structure. In order to obtain a more suitable structure the mathematical description is rearranged into an internal model structu~'e.

157

The state vector, xt, is introduced as:

(9)

Xt --" (ht, ht, clt, ~zt)T

Then the bridge section can be described by the internal or the state space equations:

Ii 1

xt =

-p

0

0

-]K

-p

0

0

-8

Xt ;Xto = XO

1

-¥

(10)

-o

or in short: )tt

-- A xt ; Xto = Xo

Here Xo is the initial state (operator excitation). The system matrix, A, is dependent on the parameters in the 0-vector. =

(11)

(ht)

which is to be estimated. The output or the measurements are given by: Yt =

=

tit

1

0

0

0

01

00]

xt

(12)

or in short: Yt = CT xt

(13)

Since the data is sampled data, it is convenient to formulate the model in discrete time. The discrete time equivalent to (10) and (13) is: Xn + 1 = (])Xn

Xno = Xo

Yn = CTxn

(14)

where the system matrix: (~ = exp(ATs)

(15)

depends on the parameter vector, 0. Ts is the sampling period. In the experiments where the wind is smooth, the signal to noise ratio is rather high and it is possible to use a procedure based on a deterministic method. The method, the SS-method (State Space), which is to be described in this section is also applicable in stochastic situations.

158

The disturbances result in noise in the output equation, i.e. the model structure is: Xn+l

= ~Xn

Xn o -

Xo

Yn = CTxn + en

(16)

where en is a zero mean stochastic variable. The solution to (10) or (14) depends on the parameters in O. The method consists in adjusting the elements in 0 until the solution to (10) matches the measurements, i.e. to minimize the loss function:

1

N

ix

(17)

i=l

Here Yi is a vector containing the actual measurements of heave and pitch and yi is the solution to (10) or (14). The method has several excellent properties. First, it is simple. Second, it does not raquire that the noise in the output equation is uncorrelated. Thirdly, the method is designed to estimate long range parameters, such as the damping factors o and g. This is specially significant when the factors are close to zero. The result from the method is of course the estimate of the parameters. Another result is the estimate of the parameter uncertainty, which is given by:

The hessian matrix, •0 2 JN, is evaluated in the method, since is uses NewtonRaphson iterations in the search for the parameters which minimize the loss function (15). The validity of the results of the method is naturally dependent on the assumption that the model structure is correct. If the system is poorly excited (e.g. primarily excited by the wind) then the loss function becomes less sensitive to change in specific (combinations of) parameters (e.g. the frequency describing the vertical motion). This result is a hessian matrix which has eigenvalues close to zero and consequently, according to (16) a large variance estimate for the specific (combination of) parameters. Under some specific situations, i.e. when only one mode is excited, it is not possible to estimate all parameters from one experiment.

4. EXTRACTION OF THE AERODYNAMIC DERIVATIVES The testing and analysis programme carried out to determine aerodynamic derivatives from the section model turned out to be an interactive process with several measurement phases followed by analysis and evaluation phases. Several lessons were learned which allowed the final iteraction to be conducted quickly and efficiently.

159

The first and foremost lesson learned was that the bridge model had to be excited initially in both modes to a sufficient amplitude to allow the initial cot,ditions and initial response to the excitation to be well defined. Failure to excite both modes sufficiently resulted in significant variations in the estimated parameters or resulted in a failure to converge to a solution despite selection of various initial guesses or starting points for the iteration. In some cases where only one mode was excited, an attempt was made to fix parameters for the other mode based on extrapolation or interpolation of data from plots of the paramters. This proved only marginally successful since some of the cross-coupling terms were quite sensitive to the values of frequency of motion or damping specified. The most consistent and reliable results were those produced from tests where both modes of vibration were excited by the operator and the initial response to that excitation was well defined. Thus it was more difficult to estimate the parameters when the response became unstable (i.e. when the torsional motion began to increment or become larger with every cycle). The estimation of parameters in unstable conditions was further complicated by an extremely strong vertical damping of the section for high wind speeds which essentially produced vertical motion typical of a supercritically damped system. Thus at high wind speeds, the parameters had to be estimated from very short segments of the records. Nevertheless, when the section mode! was properly excited it was possible to produce reliable estimates of all seven parameters from a relatively simple set of tests. Each test was repeated ten times in order to allow a statistical analysis of the parameter estimates. Thus each data point for aerodynamic derivative results shown in Figures 3 through 9 represents the statistical analysis of parameters from ten tests. Another implication of the fact that the method required a clear initial excitation and response in both modes of vibration was that estimation of parameters was more difficult in turbulent flow and became nearly impossible at high wind speeds for this quasi deterministic type of analysis. The aerodynamic derivatives for the Tender Design deck section which were produced from the tests are shown in Figures 3 through9. Scaling of the aerodynamic derivatives and plotting of the results are carried out using the torsional or vertical frequency of oscillation at the actual wind speed used in that test. This normalization approach was adopted in order to provide results which are consistent with and directly comparable with those of Scanlan [3 - 4]. The results may be related to still air frequencies by determining the frequency shifts in torsion and vertical motion from the A3* and H4* curves, respectively. Figures 3 through 9 include sketched curves representing aerodynamic derivat0ve results for an a0rfod and a streamlnned deck sectnon denoted by aurfo01 and deck section "1" from Scanlan [3 - 4]. The trends agree well between these tests and those reported by Scanlan. •

,

.

•

•

•

,

I I A I I

5. SUMMARY AND CONCLUSIONS

Tests were carried out to determine aerodynamic derivatives for the boxgirder deck section selected for the Tender Design of the Great Belt Brid~.~ in Denmark. Tests were conducted using three different model suspension configurations with three significantly different ratios of torsional to vertical frequencies of vibration.

160 .

I Stiff- Nt/Nv= 1.45

÷ Flexi - NtJNv=2.80 41 Flexi- Nt/Nv=1.32

~C

~

2";

Flexi - Nt/Nv=3.44 X

.

÷;I

Q

1 STREAMLINED OECl~'i

O

A

ml=

Stiff- Inclined + 3 IB Stiff- Inclined -3

L ~

.••

~÷

.p

A

-4-

ill=

÷

C

!

5

0

10

15

20

25

30

REDUCED VELOCITY- U/(Nv*B)

Figure 3. Aerodynamic Derivative A l* Compared with Results Reported by Scan lan [4]. 0,1 OJ

.... STREAMLINED DECK 1

~ -0,1 !

~ -0,2 ,4-

.0,3 .~

,

,,,

,

,,~

÷ Stiff- NUNv= 1.45

-0,4"

-0,5-

÷

A m

Flexi • Nt/Nv=2.80 Flexi • N t / N v = l . 3 2 Flext • Nt/Nv=3.44

X

~ -0.6"

. Stiff- Inclined + 3

a

Stiff- Inclined -3

-0,7-

-0.8

,.

5

10

15

REDUCED VELOCITY - U/(Nt*B)

Figure 4.

Aerodynamic Derivative A2* Compared with Results Reported by Scanlan [4].

IOl 3.0-

~ff'N~Nv=1, I 45

2,5!

m~ ~.. 2.0-,

F~xi"NUNv=2.80r F~xi"N~Nv=l'32/ F~xi"Nt/Nv=3-44L SI~"Inclined+3 /

A A

AIRFOIl_ 1.5-

8

1.0-

0.5-

0.0

5

0

10

REDUCED VELOCITY - U/(Nt*B)

Figure5. Aerodynamic Derivative A3* Compared with Results Reported by $canlan [4]. 0J

L

!

-5-

NUNv=I.45 St•iff.

-10'

.o-~

.

-!5"

-200

5

10

15

20

25

30

REDUCED VELOCITY- UI(Nv*B)

Figure 6. Aerodynamic Derivative H1* Compared with Results Reported by Scanlan [4].

162

/

.

[ sT. ,ML,.Eoo c , ]

Stiff- NtJNv= 1.45 -4Flexi - NUNv=2.80

tt

-4,~-

Flexi - NUNv= 1.32

A

Flexi- NUNv=3.44 Stiff - ~nclined + 3

lm Stiff- Incliw~,~d-3

o=

i

-1

5

10

15

REDUCED VELOCITY- U/(Nt*B)

Figure 7. Aerodynamic Derivative H2* Compared with ResultsReported by Scan. lan [4]. |

-2'

I". . . .

~

"-

°n

~A

!

~STREAMLINED DECK 1 l

3.

"4'

Stiff- Nt/Nv= 1.45 -4-

i Flexl - Nt/Nv=2B0

4-

Flexl • NI/Nv= 1,32

A

AA~ "

Flexi • Nt/Nv=3,44

-8"

-10

X Stiff. Inclined + 3 II

_

A

A

Stiff- Inclined-3

5

10

15

REDUCED VELOCITY- U/(Nt*B)

Figure 8. Aerodynamic Derivative H3* Compared with ResultsReported by Scan lan [4].

163 OJ I 1

(iF~ x ~f- !~- 1

"t]

_(_

~: -2 I

E~

Stiff- N t / N v = 1 . 4 5

-4'

1

-I-

-4Flexi - N t / N v = 2 . B O

-t-

Flexi - N t / N v = 1.32

•4-

4"

-4-

Flexi - N I / N v = 3 . 4 4 X

8

Stiff - I n c l i n e d + 3 I

Stiff - I n c l i n e d -3

-8

0

5

10

15

20

25

30

REDUCED VELOCITY - U I ( N v * B )

Figure 9. Aerodynamic Derivative H4* Compared with Results Reported by Scanlan [4]. The bridge motion became unstable at lower wind velocities as the ratio of torsional to vertical frequency of vibration was reduced. Since tests were conducted at three significantly different frequency rations ranging from 1.4 to 3.4 it was possible to determine and compare aerodynamic derivatives for stable and unstable conditions at the same reduced velocity. The results indicate that the aerodynamic derivatives are independent of frequency ratio although it was more difficult to determine values for some of the parameters under unstable conditions (incrementing motion) because the records suitable for analysis were quite short. Extraction of aerodynamic derivatives from the analysis of a number of tests at the same wind speed is considered important to producing reliable estimates. The aerodynamic derivatives have been calcuf,~ted in the form suggested by Scanlan [4]. In addition to the basic 6 aerodynamic derivatives [4], a fourth aerodynamic derivative for heave has been computed which corresponds to an increased vertical stiffness [2], caused by increasing wind speed. The aerodynamic derivative data produced from these tests are compared with curves reported by Scanlan [4] and general trends were found to agree. An attempt was made to determine the aerodynamic derivatives in turbulent wind using the method outlined in this paper. At low wind speeds, this determination was successful, but considerable scatter was found in the derived parameters. At higher wind velocities, the measured response contained too much excitation due to the turbulent wind and the operator imposed excitation and response was buried to such an extent that a meaningful analysis was not possible. The results that were derived did not indicate any significantly different parameter values in turbulent wind than those measured in smooth wind conditions.

164

6. ACKNOWLEDGEMENTS

This project was made possible by funding from A/S Storebaeltsforbindelsen The authors would like to express their appreciation to Mr. C. Tolstrup and Mrl Eilif Svensson of A/S Storebaeltsforbindelsen for their support. The authors would also like to acknowledge and express their appreciation for the support and critical review provided by Mr. E. Hjorth-Hansen of SINTEFStructural Engineering.

=

REFERENCES

1 Kumarasena, T., Wind Response Prediction of Lon.q-Span Bridges, Ph.D. dissertation, Johns Hopkins University, Baltimore, Maryland, October 1989. 2 Scanlan, R.H., "Wind-Excited Dynamics of Bridges: A Critical Review", Proceedings of 2nd Asia-Pacific Symposium on Wind Engineering, Beijing, China, June 1989. Scanlan, R.H. and J.J. Tomko, "Airfoil and Bridge Deck Flutter Derivatives", Journal of the Enqineerin.q Mechanics Division, ASCE,Vol. 97, No. EM6, December, 1971, pp. 1717 - 1737. Scanlan, R.H., "State-of-the-Art Methods for Calculating Flutter, Vortex-Induced, and Buffeting Response of Bridge Structures", Federal Highway Administration Report, FHWA / RD-80 / 050, Washington, D.C., April, 1981.