A study on aeroelastic forces due to vortex-shedding ...

Wind and Structures, Vol. 12, No. 1 (2009) 63-76

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A study on aeroelastic forces due to vortex-shedding by reduced frequency response function Xin Zhang* School of Civil Engineering, Zhengzhou University, Zhengzhou, China

Zhanying Qian, Zhen Chen and Fanna Zeng Henan Provincial Agency of Quality Supervision for Construction Industry, Zhengzhou, China (Received March 12, 2008, Accepted December 16, 2008)

Abstract. The vortex-induced vibration of an -shaped bridge deck sectional model is studied in this paper via the wind tunnel experiment. The vibratory behavior of the model shows that there is a transition of the predominant vibration mode from the vertical to the rotational degree of freedom as the wind speed increases gradually or vice versa as the wind speed decreases gradually. The vertical vibration is, however, much weaker in the latter case than in the former. This is a phenomenon which is difficult to model by existing parametric models for vortex-induced vibrations. In order to characterize the aeroelastic property of the -shaped sectional model, a time domain force identification scheme is proposed to identify the time history of the aeroelastic forces. After the application of the proposed method, the resultant fluid forces are re-sampled in dimensionless time domain so that reduced frequency response function (RFRF) can be obtained to explore the properties of the vortex-induced wind forces in reduced frequency domain. The RFRF model is proven effective to characterize the correlation between the wind forces and bridge deck motions, thus can explain the aeroelastic behavior of the -shaped sectional model. Keywords: force identification; bridge deck sectional model and vortex-shedding.

1. Introduction Vortex-shedding is an interesting phenomenon that happens to bluff bodies. It becomes the major concern when the structure, such as the deck of cable supported bridges, is flexible and damped very lightly. The general problem of vortex-induced vibration has been subjected to intensive studies. Early researchers, such as Iwan and Blevins (1974) and Billah (1989) developed wake-body-coupled models in which the body oscillation and wake oscillation are coupled through interaction terms. Other researchers such as Chen, et al. (1995), Ehsan and Scanlan (1990) and Gupta, et al. (1996), utilized single dynamic equation with aeroelastic terms, which can be identified by experiments. Experimental studies on fluctuating wind forces on prisms with various sectional properties are * Associate Professor, Corresponding Author, E-mail: [email protected]

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Xin Zhang, Zhanying Qian, Zhen Chen and Fanna Zeng

plenty. Kim and Sakamoto (2006) used forcing devices to “reproduce” the free vibration of a circular cylinder in the cross-flow direction and measure the fluctuating lift forces to investigate, among other issues, the work done by the fluctuating lift force and the phase between the body motion and fluid force. It was reported that the phase between the fluctuating lift force and the cylinder displacement changes abruptly as the reduced velocity increases. Takai and Sakamoto (2006) studied torsional vibrations of rectangular prisms and classified the vibration into six patterns depending on the width-todepth ratio. For more complex cross sections, Choi and Kwon (2003) studied the effect of corner cuts on the Strouhal number of rectangular cylinders with various dimensional ratios and various angles of attack. The test results show that the Strouhal number of the model with various corner cuts has a fluctuating trend as the angle of attack changes. For each cutting ratio as the angle of attack increases at cutting ratio above 15o, the Strouhal number decreases gradually, and these trends are more evident for larger corner cut sizes. Experimental studies are also reported by Cheng and Tanaka (2005), Tamura and Dias (2003) Itoh and Tamura (2002), Leonard and Roshko (2001), Mills, et al. (2003) and others. It can be inferred from the literatures reviewed above that many of the current studies on vortexinduced force on prism structures are based on indirect measurements of the fluid force. It would be useful if the fluid force can be identified directly from the response of the oscillating body, and the properties of the aeroelastic system can be investigated by direct relating the fluid force time history with the body motion time history. It is particularly the case when complex patterned vortexshedding exists as what has happened to the Sectional model studied in this paper.

2. The sectional model The bridge deck model (Fig. 1) studied in this paper is excited by the motion-induced vortices, which are generated at the leading edge of the girder. At low wind speed, the flow separates from the leading edge and reattaches to the sectional model. Because the lower surface of the model is open, exposing the transversal and axial girders to the reattached wind flow, complex flow pattern is generated. As wind speed increases from 2.5 to 4.0 m/s, the reattachment point moves downstream, finally, the flow does not re-attach. During this process, the model experiences a transition of the predominant vibration mode from the vertical to the rotational degree of freedom. The transition of the predominant vibration mode indicates the bifurcation of the bridge behavior in a time-varying wind speed environment: the vibration could switch between vertical and rotational degree of freedom due to small change of the mean wind speed. The frequency ratio of the vertical and rotational vibration (3.3 Hz /5.3 Hz) was designed to match the frequency ratio of the prototype. Therefore the transition of the predominant vibration mode is considered as the inherent characteristic of the problem that can not be bypassed by setting the frequency ratio to a different value. A

Fig. 1 The Sectional Model (Dimension in mm)

A study on aeroelastic forces due to vortex-shedding by reduced frequency response function

65

theoretical explanation of this phenomenon might be complex and it is beyond the scope of this research, which would rather focus on “curve fitting” instead of explaining the phenomenon. In this case, it would be easier to identify the dynamic wind forces directly to obtain the frequency domain properties of the wind forces, i.e. to find a linearized mapping between the bridge deck motion and the aerodynamic forces: f(K) = H(K)x(K), where f(K) is the dynamic wind forces in reduced frequency, K, domain; H(K) is the reduced frequency response function (RFRF) and x(K) is the displacement of the structure in reduced frequency domain. Therefore, the objective of this paper is to propose a method for the direct identification of the aerodynamic forces so that the above mentioned reduced frequency response function can be computed. For the sake of completeness, the RFRF is identified for both wind speed increasing and decreasing cases.

3. Method for the identification of dynamic wind forces For a two-dimensional sectional model with degrees of freedom in vertical and rotational direction, the vibration equation can be written as: X· ( t ) = A X ( t ) + B f ( t ) c

c

Y ( t ) = CX ( t ) + Df ( t )

(1)

T T T where X ( t ) = { x ( t ), x· ( t ) } is the state variable;

Ac = is the state matrix; B c =

0

I

–1

–1

–M Km

(2)

–M Cm

0

is the input matrix; –1 M –1 –1 C = [ C d − C a M K C v −C a M C m ] is the output matrix; D = CaM −1 is the feed through due to the

⎧ Fh ⎫ measurement of acceleration and f(t) = ⎨ ⎬ , the wind force vector. ⎩ Fa ⎭ In the expressions above, x(t) = {h(t), α(t)}T is the displacement time history vector comprising of vertical displacement, h(t), and rotational displacement, α(t); M is the mass matrix of the model, Km and Cm are the stiffness and damping matrix respectively; Cd Cv Ca are the influence factor for displacement, velocity and acceleration measurement respectively; Fh(t) is the lifting force and Fα (t) is the rotational moment. In this study, only displacement is measured, therefore, C = 1 0 0 0 and D is a null matrix. 0100 In the equation, all the wind forces, including the vortex-shedding forces, have been put into the force vector f(t), so that the state matrix is purely structural. In discrete time format, the same system above is: X(i + 1) = AX(i) + Bf(i) Y(i) = CX(i),

(3)

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Where A=e B=

A c ∆t

∆t

∫0

e

and

Ac τ

B c dτ .

(4)

in the equation, i = 0, 1, 2, ..., n indicates the sequence of the sampled data. To perform the computation using displacement time history only, a similarity transform is performed, i.e. use X * = TX as a new state vector, where T = C is the transform matrix. CA After the transform, the discrete state space equation is *

*

*

*

X (i + 1) = A X (i) + B f(i) *

*

Y( i) = C X ( i )

(5)

⎧x(i) ⎫ * –1 * * –1 * where, A = TAT , B = TB and C = CT . The state vector is X ( i ) = TX = ⎨ ⎬ , where x(i) ⎩ z(i) ⎭ is the discrete measurement of displacement and z(i) is a variable containing a force term. The structure of matrix A* can be certified to be in the form of (Jakobsen and Hansen; 1995) 0 I –[ a0 ] –[ a1 ]

*

A =

(6)

and the output matrix C* = 1 0 0 0 . 0100 Substituting A* B* and C* into Eq. (5) leads to ⎧x(k + 2) ⎫ ⎪ ⎪ ⎧f(k + 1) ⎫ [ I [ a1 ] [ a0 ] ] ⎨ x ( k + 1 ) ⎬ = [ b1 b0 ] ⎨ ⎬ ⎪ ⎪ ⎩ f(k ) ⎭ ⎩ x(k) ⎭ *

(7)

*

where [ b 1 ] = [ I 0 ]B and [ b 0 ] = [ [ a 1 ] I ]B . This equation describes an autoregressive model with external input (ARX), whose parameters, [ b 0 ], [ b 1 ] , are predefined by the autoregressive parameters [ a 0 ], [ a 1 ] . The autoregressive coefficient matrix can be identified directly from the free vibration of the sectional model under no-wind condition with an AR model. If k + 2 measurements are available, Eq. (7) becomes ⎧ x(3) x(4) … x(k + 2) ⎪ [ I [ a1 ] [ a0 ] ] ⎨ x ( 2 ) x ( 3 ) … x ( k + 1 ) ⎪ … x(k) ⎩ x(1) x(2)

⎫ ⎪ ⎧ f ( 2 )f ( 3 ) … f ( k + 1 ) ⎫ ⎬ = [ b1 b0 ] ⎨ ⎬. ⎪ ⎩ f ( 1 )f ( 2 ) … f ( k ) ⎭ ⎭

⎧ h(i) ⎫ where x ( i ) = ⎨ ⎬ , (i = 1, …, k + 2) are the displacement time history measurements. ⎩α(i) ⎭

(8)


Multiplying Eq. (8) by a pseudo-inverse of matrix [ b 1 b 0 ] , the wind least squares sense: ⎧ x(3) x(4) ⎧ f(2) f(3) … f(k + 1) ⎫ ⎪ + ⎨ ⎬ = [ b1 b0 ] [ I [ a1 ] [ a0 ] ] ⎨ x ( 2 ) x ( 3 ) … f(k) ⎭ ⎩ f(1) f(2) ⎪ ⎩ x(1) x(2) +

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force is estimated in the … x(k + 2) … x(k + 1) … x(k)

⎫ ⎪ ⎬, ⎪ ⎭

(9)

+

where [ b 1 b 0 ] is the pseudo inverse of [ b 1 b 0 ] so that [ b 1 b 0 ] [ b 1 b 0 ] = I. Eq. (9) offers one set of the possible force time histories that can drive the model to vibrate in the recorded way.

4. Identification of reduced frequency response function (RFRF) for the wind forces The identified wind forces are random processes with predominant frequencies. The effect of such forces is affected by factors such as the instantaneous phase angle between the wind force and the model displacement. The distribution of the random phase angle is thus important in understanding the response of the model. This information can be obtained by introducing the wind force reduced frequency response function (RFRF). f(K) H ( K ) = ----------- , x(K)

(10)

where K is reduced frequency; the complex valued H(K) is the response function in reduced frequency domain; x(K) and f(K) are Fourier transformed displacement and wind forces in reduced frequency domain, respectively. Noted is that the fluid is the system to be identified with model displacement as the input. The absolute value of H(K) reflects the magnitude of the wind forces activated by unit displacement at reduced frequency K. The phase angle of H(K) indicates the wind forces are leading ahead or lagging behind the model motion. In extracting the reduced frequency response function, a chirp signal of the input term, i.e. the body displacement time history with continuously changing frequencies, is preferred so that a fine resolution of the RFRF is possible. While the frequency of the free vibration is not changeable, a chirp signal can be obtained in dimensionless time domain by controlling the mean wind speed to vary at a very low changing rate. When there is continuously changing wind speed, the interval of the dimensionless time is defined as ( t )- ∆t , ---------∆S = U B

(11)

where ∆S is the interval of dimensionless time S, U ( t ) is the positive instantaneous average value of wind speed over time step ∆t centered around time t and B is the deck width. Thus, the dimensionless time may be defined by S=

t

∫t

0

U ( t )- dt ---------B

(12)

The estimated wind forces, together with the displacement time history, can then be transferred to dimensionless time domain, i.e. f(S) and x(S), and re-sampled at even sampling rate in dimensionless time domain. By adjusting the wind speed gradually for U1 at time t0 to U2 at time t, we change the

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nB nB reduced frequency from K 1 = ------- to K 2 = ------- , where n is the body oscillation frequency. U1 U2 If the lock-in reduced frequencies fall between K1 and K2, the frequency response function of the vortex-shedding forces can be expressed in reduced frequency domain as shown in Eq. (10).

5. The experiments and results As shown in Fig. 2, the model was suspended by eight springs to allow for vertical and rotation free vibration. The lateral motion was restrained by thin wire. The spring constant and the spring anchorage point were so chosen such that they have a combined stiffness producing the vertical frequency of 3.3 Hz and the rotation frequency 5.3 Hz. Two laser displacement sensors were installed below the front and rear edge of the model to record the vertical and rotational displacement. The sampling frequency was 50 Hz. One hotwire anemometer was installed in front of the model to record the horizontal wind speed. No turbulence generating device was used. The typical turbulence intensity is less than 2 percent. Therefore, the oncoming wind was considered smooth. The testing program was performed in two steps: the structural parameters under no-wind condition were identified first by triggered free decay vibration and [ a 0 ], [ a 1 ] and [ b 0 ], [ b 1 ] were obtained; the response of the sectional model when the smooth wind flow was applied was then recorded. The wind changed from higher wind speed to lower wind speed monotonously and then ramp up to a higher wind speed, covering the wind speed range of interest. The changing rate of the wind speed was controlled to be as slow as 1m/s/500s, to allow full development of the vortexshedding response. As is shown in Fig. 3, due to the special cross section shape, the model is active both in the vertical and rotation direction within the wind speed region between 2 to 4 m/s due to the vortexshedding excitation. Measurements of the dynamic wind forces on the fixed model by force transducers show Strouhal Numbers of 0.85 for vertical motion dominant period and 0.79 for the rotational motion dominant period. (The Strouhal number is computed as S t = N s D/U where Ns is the vortex-shedding frequency D is the characteristic dimension of the model and U wind speed) It can be observed that in the wind speed decreasing case, the vortex-shedding locks onto the

Fig. 2 The Suspension System


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Fig. 3 Displacements and Wind Speed vs. Time

rotational motion first and sticks to the rotation motion, showing a roughly symmetric amplitude envelope. The lock-in in the vertical motion is suppressed, with only weak signs of existence after the rotational lock-in region has passed. In the wind speed increasing case, however, the vortexshedding locks onto the vertical motion first and then comes to a sudden stop, giving way to the lock-in in the rotational direction. The rotational lock-in then becomes predominant. Both the vertical and rotational amplitude envelopes are not symmetric in this case. In order to have a better understanding of the data, the proposed force identification method is applied. In the analysis, the wind forces are identified for wind speed decreasing case and increasing case separately to avoid confusion of the data in reduced frequency domain. In the analysis, the data was de-noised before the force identification is performed by using the wavelet decomposition which has good performance in maintaining the original phase relations. To test the effectiveness of the proposed method, a comparison study was performed. A segment of measured displacement time history in Fig. 3 between 1200s-1500s was subjected to the proposed procedure to identify the force vector, which was subsequently used as the driven force on the same numerical model to produce the reconstructed response signal. A segment of results are shown in Fig. 4. It can be seen that the discrepancies between the two signals are acceptably small. Therefore the force identification method proposed in the paper works reasonably well.

6. The wind speed increasing case The identified wind forces for the wind speed increasing case are shown in Fig. 5. As expected, in an operational environment, the estimated input forces of a dynamic system should contain noise. However, the prediction errors are usually wide banded in frequency domain. Therefore, if the result

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Fig. 4 The Measured and Reconstructed Signal Driven by the Identified Force

Fig. 5 Identified Wind Forces (Wind Speed Increasing Case)

is used in frequency domain instead of time domain, it would be more reliable as meaningful part of the signal may be concentrated in a narrower frequency band while the errors spread over a wider frequency band, resulting in a low intensity of energy. The fast Fourier transform (FFT) result of the identified forces in reduced frequency domain are shown in Fig. 6. It can be seen that the lifting force has two major reduced frequency bands. One is around 0.6 (dimensionless) and the other falls between 0.83 to 0.93. The vertical response of the model mainly concentrates between 0.58 to 0.7, matching the lower frequency center of the lifting force. The rotational moment also has two peaks covering from 0.58 to 0.7 and from 0.83 to 0.93. The rotational displacement shows predominant frequency between 0.83 to 0.93. Very weak responses exist around 0.6. These observations might suggest that the rotational moment at the lower frequency region is


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Fig. 6 FFT of Displacements and Wind Forces in Reduced Frequency Domain (Wind Speed Increasing Case)

generated by the vertical movement and the lifting force at the higher frequency is related to the rotational motion. The phase angle of the RFRF is shown in Fig. 7. In the figure, “A” represents rotation direction; “H” denotes vertical direction; “AH” denotes the rotational moment due to displacement in the vertical direction. The phase angle of AH is the angle by which the rotation moment leads the vertical displacement. “AA, HH, HA” are defined in the similar way. Because the wind forces are random processes with predominant frequencies, the phase angles show dispersals. However, these figures do show concentrations of phase angle within upper and lower lock-in region. From reduced frequency 0.58 to 0.7, the rotational moment slightly lags behind the rotational

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Fig. 7 Phase Angle between the Wind Force and the Model Displacement (Wind Speed Increasing Case)

displacement (Angle AA), whose effect is positive damping and negative stiffness (be noted the minus signs in matrix Ac of Eq. (2)). Therefore, even the rotational moment is large within this region, the model response remains small. In the plot “HH”, there is a migration of the phase angle from 1800 to 00, generating negative aeroelastic damping. The vertical motion predominates due to smaller effective vertical damping. It is also observed in plot “AH”, the correlation between the rotational moment and vertical displacement is clear, suggesting the aeroelastic coupling between rotational and vertical direction. From reduced frequency 0.8 to 0.93, the rotational moment is always leading the rotational displacement by an angle slightly less than 1800 (Angle AA) reducing the effective rotational damping. Therefore, rotational response becomes the major response within this region. On the other hand,

Fig. 8 Absolute Value of the FRFs of Wind Forces (Wind Speed Increasing Case)


73

the phase angles between lifting force and vertical motion (Angle HH) are symmetrically distributed around 1800. The wind force contributes to positive stiffness. The vertical motion is, therefore, small. The aeroelastic coupling at this reduced frequency is still clear (Angle HA), indicating the lifting force is generated by the rotational motion. Between 0.7 and 0.8, there is a transition region in the subplot AA and HH. The phase angles are distributed symmetrically (or almost so) around zero line causing a cancellation of the aeroelastic effects. This corresponds to the transition period between 1500s and 1600s in Fig. 3. In Fig. 8, the absolute values of the wind force RFRF are shown at the reduced frequencies where the denominator in Eq. (10) is large. The conventions of this figure are the same as Fig. 7.

7. The wind speed decreasing case The identified wind forces are shown in Fig. 9. The reduced frequency representations are shown in Fig. 10. It is noticed that the magnitude of the FFT of vertical displacement is considerably lower than the same parameter in the wind speed increasing case. The figures show there is lifting force at the reduced frequency of the rotational movement due to the aeroelastic coupling effect. The lifting force, however, generates weak vertical response at this frequency. The phase angle between the lifting force and vertical motion is shown in Fig. 11, subplot Angle HH. Around the reduced frequency of 1, where the lifting force is active, the vertical displacement leads the force by an angle less than 1800, hence the model is experiencing positive aeroelastic damping and aeroelastic stiffness. The vertical displacement is damped out. The aeroelastic coupling is indicated in the subplot Angle HA. From 0.8 to 1, the phase angle between the lifting force and rotational displacement continuously change from 00 to an angle slightly less than 1800, and stay constant until 1.2 is reached, indicating the coupling term is changing from displacement related to velocity related and then changing to displacement related again. In the

Fig. 9 Identified Wind Forces (Wind Speed Decreasing Case)

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Fig. 10 FFT of the Wind Forces in Reduced Frequency Domain (Wind Speed Decreasing Case)

rotational direction, the situation is similar to the previous case, except for the distribution of the phase angle is more scattered. In Fig. 12, the absolute value of RFRF AA and HA are shown at the rotational lock-in region. The value of RFRF AA, to certain accuracy, is in consistence with the former case despite the large noise component. So is the value of RFRF HA.

8. Conclusions The aeroelastic effects of the complex patterned vortex shedding from a sectional model of

-


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Fig. 11 Phase Angle between the Wind Forces and the Model Displacement (Wind Speed Decreasing Case)

Fig. 12 Absolute Value of FRFs of Wind Forces (Wind Speed Decreasing Case)

shape are successfully studied by the proposed reduced frequency response function. This RFRF model is realized through a time domain identification scheme for aerodynamic forces due to wind with slow monotonously increasing or decreasing speed and may serve as an alternative to the existing methods when the vortex shedding pattern is too complex to be described by a parametric model. The beauty of the proposed method is that it can clearly picture the linearized mapping relation between the displacements and aerodynamic forces. Through the proposed method, strong aerodynamic coupling effect between the vertical and rotational motion of the bridge is discovered; aeroelastic features of the bridge are explained by the phase of the RFRF giving insights into whether the aeroelastic effect is damping related or stiffness related.

Nomenclature [a0], [a1] A, A*, Ac [b0], [b1] B, B*, Bc C, C*

: Coefficient matrices : State matrices : Coefficient matrices : Input Matrices : Output matrices

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Cd, Cv, Ca Cm D f(t), f(i), f(K) h(t), h(i) H(K) i, k K Km S T U(t) x(t), x(i), x(K) X(t), X(i), X*(i) Y(t), Y(i) α(t), α(i)


: Measurement influence factor : Structural damping matrix of the model : Feed through matrices : Wind force : Vertical displacement : Frequency response function of wind force : Indices : Reduced Frequency : Structural stiffness matrix of the model : Dimensionless time : Transform matrix : Instantaneous averaging wind speed : Displacement vectors : State vectors : Output measurements : Rotational displacement

References Billah, K.Y.R. (1989), “A study of vortex-induced vibration”, Ph.D. dissertation, Princeton University, Princeton, N.J. Chen, S.S., Zhu, S. and Cal, Y. (1995), “An unsteady flow theory for vortex-induced vibration”, J Sound Vib., 184(1), 73-92. Cheng, S. and Tanaka, H. (2005), “Correlation of aerodynamic forces on an inclined circular cylinder”, Wind Struct., 8(2), 135-146. Choi, C.K. and Kwon, D.K. (2003), “Effects of corner cuts and angles of attack on the Strouhal number of rectangular cylinders”, Wind Struct., 6(2), 127. Ehsan, F. and Scanlan, R.H. (1990), “Vortex-induced vibration of flexible bridges”, J. Eng. Mech., ASCE, 116(6), 1392-1411. Gupta, H., Sarkar, P.P. and Mehta, K.C. (1996), “Identification of vortex-induced response parameters in time domain”, J. Eng. Mech., ASCE, 122(11), 1031-1037. Itoh, Y. and Tamura, T. (2002), “The role of separated shear layers in unstable oscillations of a rectangular cylinder around a resonant velocity”, J. Wind Eng. Ind. Aerod., 90, 377-394. Iwan, W.D. and Blevins, R.D. (1974), “A model for vortex-induced oscillations of structures”, J. Appl. Mech., 41(3), 581-586. Jakobsen, J.B. and Hansen, E. (1995), “Determination of Aerodynamic Derivatives by a System Identification Method”, J. Wind Eng. Ind. Aerod., 57, 295-305. Kim, S. and Sakamoto, H. (2006), “Characteristics of fluctuating lift forces of a circular cylinder during generation of vortex excitation”, Wind Struct., 9(2), 109-124. Leonard, A. and Roshko, A. (2001), “Aspects of flow-induced vibration”, J. Fluids Struct., 15, 415-425. Mills, R., Sheridan, J. and Hourigan, K. (2003), “Particle image velocimetry and visualization of natural and forced flow around rectangular cylinders”, J. Fluid Mech., 478, 299-323. Takai, K. and Sakamoto, H. (2006), “Response characteristics and suppression of torsional vibration of rectangular prisms with various width-to-depth ratios”, Wind Struct., 9(1), 1. Tamura, T. and Dias, P.P.N.L. (2003), “Unstable aerodynamic phenomena around the resonant velocity of a rectangular cylinder with small side ratio”, J. Wind Eng. Ind. Aerod., 91, 127-138. CC

ARTICLE IN PRESS

Journal of Wind Engineering and Industrial Aerodynamics 92 (2004) 493–508

Effect of relative amplitude on bridge deck flutter Xin Zhanga, James Mark William Brownjohnb,* a

School of CEE, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore b School of Engineering, University of Plymouth, Drake Circus, Plymouth PL4 8AA, UK Received 26 March 2002; accepted 23 February 2004

Abstract Self-excited wind forces on a bridge deck can be non-linear even when the vibration amplitude of the body is small. This phenomenon is evaluated in this paper. Experiments detecting the nonlinearity are performed first, with the concept of ‘‘relative amplitude’’, i.e. the amplitude of the externally triggered free vibration relative to the envelope of the ambient response of an elastically supported rigid sectional model. Two types of sectional model, a twin-deck bluff model (model A) and a partially streamlined box girder model (model B) are tested with two extreme cases of relative amplitude. Based on the flutter derivatives of model B, a flutter boundary prediction is subsequently carried out on a cable-supported bridge to manifest the changes of critical flutter wind velocity due to different relative amplitudes. The effect of relative amplitude on flutter derivatives and on the flutter boundary reveals, from the structural point of view, a complex relationship between the self-excited forces and the ‘‘structural vibration noise’’ due to turbulence that is inherent in the interaction of the ambient wind with the structure. Although the aeroelastic forces are linear when the body motion due to an external trigger is not affected significantly by this turbulence, they are postulated to be nonlinear when this ‘‘vibration noise’’ cannot be neglected. r 2004 Elsevier Ltd. All rights reserved. Keywords: Flutter; Relative amplitude; Sectional model testing

1. Introduction In the formulation of self-excited wind forces on cable-supported bridge decks, a linearized flutter derivative model is often used [1]. The amplitude dependency of the *Corresponding author. School of Engineering, University of Plymouth, Drake Circus, Plymouth PL4 8AA, UK. Tel.: +44 (0) 1752 233673; fax: +44 (0) 1752 232638. E-mail addresses: [email protected] (X. Zhang), [email protected] (J.M.W. Brownjohn). 0167-6105/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2004.02.002

ARTICLE IN PRESS 494

X. Zhang, J.M.W. Brownjohn / J. Wind Eng. Ind. Aerodyn. 92 (2004) 493–508

Nomenclature A state matrix of discrete state space model A0 mechanically triggered vibration amplitude Aij ðKÞ; Bij ðKÞ variables in Eij B input matrix of discrete state space model Bd width of the bridge deck C output matrix Ci ðkÞ output covariance D feed through matrix E½ expectation operator E impedance matrix Eij element in impedance matrix G input matrix for covariance dynamics Gri sj modal integral h; p; a displacements of the rigid body in vertical, lateral and rotational direction, respectively hi ; pi ; ai ith vertical, lateral and rotational modes Hi ; Ai ; Pi ði ¼ 1; y; 6Þ flutter derivatives Ii generalized inertia K reduced frequency l bridge deck length pðtÞ buffeting force Ra relative amplitude U wind speed X state vector Y displacement ZðtÞ structural function Ssiþ1 variable vðtÞ measurement noise t time o; oi circular frequency x% participation factor vector of structural modes at flutter dij Kronecker delta function D characteristic measurement representing the ambient vibration magnitude

flutter derivatives has been indicated by previous researches [2,3]. From their points of view, nonlinearity does not exist for the small amplitude case. As a matter of fact, however, the self-excited wind forces due to a pulse response of the deck can be nonlinear even if the response amplitude is small. Historical study [1] showed that the flutter derivative presentation of the interactive forces holds strictly for sinusoidal oscillation or exponentially modified sinusoidal motion of decay rates less than 20%. Therefore, for a flexible bridge in the

ARTICLE IN PRESS X. Zhang, J.M.W. Brownjohn / J. Wind Eng. Ind. Aerodyn. 92 (2004) 493–508

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wind, the flutter analysis predicts the critical wind speed for a smooth sustained sinusoidal motion of the deck, which indicates the bridge is going to lose its stability due to negative damping. A ‘‘smooth’’ sinusoidal motion is rarely the real case when there are disturbances in the aerodynamic forces. An initially small amplitude sinusoidal motion can only be considered as a ‘‘noisy’’ sinusoidal one. It needs to be answered in the first place whether or not the noisy sinusoidal motion will grow beyond the ‘‘environmental noise’’ and become large enough in amplitude making itself smoother (the noise part then becomes relatively unimportant) before a flutter prediction is applied. It is necessary, in this case, to investigate how the self-excited forces change due to the effect of the ‘‘noise’’ in the structural motion resulting from turbulence in the oncoming flow and signature turbulence. The changes in the interactive forces will be indicated by a change in the flutter derivatives.

2. The relative amplitude effect The discussion in this paper is limited to an elastically supported rigid sectional bridge model subjected to a pulse input to trigger the model to vibrate. The interactive forces under investigation are the self-excited wind load generated by the response due to pulse excitation of the rigid body. A pulse function is a Dirac delta function with a finite small duration. The pulse input is a practically achievable approximation of the delta function for the input force. In all the cases discussed below, the absolute values of vibration amplitude are small. The relative amplitude is then defined as the triggered vibration amplitude of the model relative to ‘‘structural noise’’ in the vibration due to the turbulence of the wind excitation. To quantify it, the relative amplitude is defined by Ra ¼ A0 =D in the simulation of Fig. 1, where A0 is the mechanically triggered vibration amplitude and D is a characteristic measurement representing the ambient vibration magnitude. Relative amplitude is thus defined in one dimension. This definition can be easily extended to a multi-dimensional case. 1.5

A0

1

∆

0.5

0

-0.5

-1

0

5

10

15

20

25

30

35

40

45

Fig. 1. Definition of relative amplitude.

50

s



This study is to manifest that the relative amplitude may affect flutter derivatives and the flutter boundary. This effect reveals, from the structural point of view, a relationship between the self-excited forces and the ‘‘structural vibration noise’’ due to random component of wind forces. If Ra -N; the triggered vibration is totally smooth. If Ra -0; the triggered vibration is severely affected by the turbulence. This turbulence related part of the response exists for either smooth or turbulent flow due to the fact that small fluctuations in the oncoming wind are inevitable and the signature turbulence is an inherent part of bluff body dynamics. This turbulence response forms the environment for the pulse response to exist. It could be a different story if the pulse response of the sectional model is strong and affects the surrounding flow, or the pulse response is small with all the turbulence properties of the aeroelastic system unchanged. The difference will be illustrated by the various flutter derivatives identified with different relative amplitudes.

3. Using output covariance as Markov parameters Experiments with the large relative amplitude, i.e. transient vibration testing, are common. It is difficult, if not impossible, to identify directly the flutter derivatives when the pulse response amplitude is smaller than the magnitude of the ambient vibration. However, a nominal pulse response for the ambient vibration can be retrieved mathematically by using output covariance [4,5]. This corresponds to identifying the extreme case where the pulse response amplitude tends to zero. Therefore, in the study of relative amplitude effect, it is possible to compare two extreme cases, i.e. large Ra transient vibration case and zero Ra ambient response case. In current experiments, except for the relative amplitudes, all other experimental conditions remain the same. It is reasonable to state that if the output covariance produces a state matrix considerably different from the physically triggered case, there is a relative amplitude effect on the flutter derivatives. It is a classical result in stochastic identification that the output covariance can be used as the discrete version of pulse response function, i.e. Markov parameters, of a deterministic linear time invariant system, when the measurement noise in the equipment and process noise (random component of the wind excitations) is white (wide-banded in the practical sense) and zero mean. A proof is presented in Appendix A. It is assumed, in the experiments, the displacement measurement is zero mean. If the deck has a static displacement due to the static wind load, it can be treated as a centered process, i.e. Y ðtÞ ¼ Y ðtÞ E½Y ðtÞ;

ð1Þ

where E½ denotes expectation operation. Therefore, it is always assumed that the identification algorithm is dealing with a zero mean process.


497

The input term, buffeting forces on the bridge deck model, is random and unknown and is assumed to be wide banded noise because a real-world process will not be white noise. The approximation is a widely adopted one in the application of system identification techniques as long as the noise frequency band is considerably larger than the system frequencies band. The errors so generated should be tolerable. A typical turbulence power spectral density is shown in Fig. 2 at wind speed of 17:5 m=s: It can be observed that the turbulence is wide banded and therefore meets the requirement for the application of the output covariance method. One consideration is that within the time period of measurement of the ambient vibration, the system is actually non-stationary due to uncontrollable experimental conditions. Therefore, in calculating the output covariance, some numerical consideration has been taken to overcome the problem. The structure function was introduced by Kolmogorov [6] in his study of locally isotropic and homogeneous stochastic turbulence. It is defined by ZðtÞ ¼ lim

T-N

1 T

Z

T

½Y ðtÞ Y ðt þ tÞ2 dt ¼ 2ðCt ð0Þ Ct ðtÞÞ;

ð2Þ

0

where Ct ðtÞ is the auto-covariance function of the measurement after time t þ t: For ambient vibration, the covariance function for zero lag time is constant, and the structure function is the sum of the negative covariance function and a constant. If the measurement shows slow fluctuation or has a time varying trend, i.e. shows marked derivation from a stationary behavior, the method of structure function may be advantageous because the structure function can tolerate more low-frequency noise than the correlation function [7]. Fig. 3 shows a typical calculated output covariance Ci ðkÞ (k ¼ 1; 2yN 1 is the number of lags) from the ambient response by structure function. In the figure, H

Fig. 2. Power spectral density of lateral turbulence U ¼ 17:5 m=s:



Fig. 3. A typical output covariance.

and A denote vertical and rotation direction, respectively. It can be observed that the signal is quite smooth and suitable to be used in system identification. 4. Experiments and results Two sectional models (Figs. 4 and 5) were tested. The experimental environment was kept the same for the same model in either ambient or transient vibration testing. The setup is shown in Fig. 6. The model was restrained in the lateral direction to allow two-dimensional tests. Eight vertical springs were hung from two frames to hold the sectional model in position. These springs were pre-tensioned to avoid non-linearity. The stiffness of the springs was chosen to make the model vibrate in a suitable frequency, so that a suitable range of reduced wind speeds can be covered by the experiment. The separations between the springs could be adjusted according to the required rotational frequency. Laser displacement sensors were used to measure the displacement history of the model, with one below the front edge and the other below the rear edge to record the vertical and rotational displacement. A single hot-wire anemometer was mounted in front of the sectional model at mid-span location to record the wind speed. The typical turbulence intensity of the oncoming wind is less than 2%, e.g. 1.7% at wind speed of 17:5 m=s: Therefore, the wind is considered smooth. The following discussion will be limited to smooth flow case. For turbulent flow cases, more researches are needed. The experimental procedure for the tests is as follows: (1) One of the bridge deck models, model A or model B, was installed. (2) Lateral DOF was constrained with thin wire for 2DOF experiment. Measurement of the system properties was made with free decaying vibration of the model by giving it an initial displacement when there was no wind.


499

Fig. 4. Model A: twin deck bluff model (dimension in mm).

355 227

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perspex coat 92

91

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Fig. 5. Model B: streamlined box girder model (dimension mm).

Spring Model String

String Spring

Fig. 6. Setup for free vibration test (one end).

Light of the Displacement Sensor



(3) Five-minute ambient response records of the model under the action of wind were recorded. (4) Three to five free-decay tests were conducted under the same condition as in step (3) except for the initial conditions. (5) Wind speed was increased. Experiments in steps 3 and 4 were repeated. The wind velocities varied from minimum to maximum wind speed at a reasonable interval. (6) After the wind speed reached the upper limit, the other model replaced the first one. Experiments from (2) to (5) were repeated. Flutter derivatives were then identified from the transient vibration and the output covariance of the ambient vibration. The identification method used was ERA (eigensystem realization algorithm) [8]. Figs. 7, 8 show the flutter derivatives vs. reduced wind velocity (RU) of model A and model B identified with transient and ambient vibration. For model B, the quality of A2 identified from ambient case is poor; the dotted line in Fig. 8b is used as the fitting curve for it. The features of some of the flutter derivatives are very different for different relative amplitudes, suggesting nonlinearity in the self-excited wind load. In the flutter derivatives for the twin deck bluff model (model A), differences are observed in H2 ; H3 and H4 : However, there is no big difference in Ai ; ði ¼ 1; y; 4Þ: For Model B, H1 ; H2 and H3 are among the most sensitive parameters to the relative amplitude. In the transient vibration, where the value of relative amplitude is high, the interaction between the vertical and rotational DOF is strong within lower range of the reduced velocity both in terms of aeroelastic stiffness and damping; while in the ambient response ðRa ¼ 0Þ; this effect is reduced greatly. A4 is also sensitive to the relative amplitude.

5. Flutter analysis considering relative amplitude effect Differences between the flutter derivatives identified from transient and ambient vibration testing suggest further study is needed to investigate the effect of relative amplitude on the flutter boundary of a full bridge. Fig. 9 gives the three-dimensional view and finite element mesh of a bridge with partially streamlined box girder cross-section (sectional model B). Table 1 summarizes the material properties and other dimensions required in calculation. The main span is 1410 m; with side span of 530 and 280 m: The steel box sections are 22 m wide and 4:5 m deep. The towers are box section, 6 m by 6 m at the bases and 4.5 by 4.5 at the tower tops. To facilitate the inclusion of an aeroelastic load model, 3-D beam elements were used to model the deck structure. Spar elements (having no flexural stiffness) were used to present the main cable and hanger. They have the facility to accommodate the initial strain value. The tower was analyzed using beam elements with tension, compression, torsion and bending capabilities. The modal


501

Fig. 7. (a) 2DOF H (Model A), (b) 2DOF A (Model A).

analysis was conducted by using commercial software, ANSYS. Table 2 summarizes the modal analysis results. The frequency domain method developed by Jain et al. [9] is used to perform the flutter instability analysis, which requires solving an aeroelastically influenced eigenproblem. The flutter condition is obtained at the reduced frequency satisfying equation jEðKÞj ¼ 0:

ð3Þ



Fig. 8. (a) 2DOF H (Model B), (b) 2DOF A (Model B).

The general term of the impedance matrix EðKÞ is given by Eij ¼ K 2 dij þ iKAij ðKÞ þ Bij ðKÞ; pffiffiffiffiffiffiffi where i ¼ 1 and dij is the Kronecker delta function defined through ( 1 i¼j : dij ¼ 0 iaj

ð4Þ

ð5Þ


503

Fig. 9. Plot of the bridge.

Table 1 Material properties of the bridge Cable Young’s modulus of cables Young’s modulus of hangers Area per hanger Area of each cable

Main span Side span

Box girder deck Young’s modulus Axial area of steal Second moment of area for vertical bending Second moment of area for lateral bending Torsional rigidity Towers Young’s modulus Average axial material area of each leg Average second moment of area of each tower leg for longitudinal bending Average second moment of area of each tower leg for lateral bending Average torsional rigidity of each tower leg

193 kN=mm2 140 kN=mm2 0:0021 m2 0:29 m2 0:31 m2 200 kN=mm2 0:73 m2 1:940 m4 37:07 m4 4:5 m4 20 kN=mm2 20:37 m2 66:8 m4 68:24 m4 113:1 m4


504

Table 2 Dynamic properties of the bridge Mode no.

Mode type

Frequency ni (Hz)

1 2 3 4 5 13 14 16 26 28

L, 1st S V, 1st S L, 1st AS V, 1st AS V, 2nd S V, 2nd AS LT, L, 2nd S; T, 1st S V, 3rd S V, 3rd AS T, 1st AS

0.0688 0.1277 0.1591 0.1646 0.1897 0.2498 0.2816 0.3246 0.4022 0.45853

Note: S ¼ symmetrical; AS ¼ anti-symmetrical; L ¼ lateral; V ¼ vertical; LT ¼ lateral-torsion; T ¼ torsion and Structural modes used in flutter boundary computation are normalized with respect to mass matrix.

Fig. 10. E-matrix (transient vibration case).

Aij and Bij are shown in Appendix B. The non-trivial solution x% of the aeroelastically influenced eigenvalue problem E x% ¼ 0

ð6Þ

indicates the relative participation of each structural mode under the flutter condition. Fig. 10 is the analysis result using flutter derivatives from transient vibration testing. This is equivalent to defining two surfaces, one for the real part and the other for the imaginary part of jEj: These two surfaces are functions of reduced velocity


505

RU and vibration frequency f : The intersection of these surfaces with RUBf plane is obtained by linear interpolation. Then the zero contour curves of the real surface (solid line) and imaginary surface (dashed line) are obtained with piecewise linear approximation and their intersections can be determined either numerically or graphically. These intersection points define the flutter conditions. There are two flutter frequencies identified: 0.27 and 0:39 Hz; corresponding to flutter wind speed 52.7 and 75:9 m=s; respectively. The flutter modes corresponding to them are shown in Fig. 11. The result shows, for flutter derivatives identified from transient vibration, that flutter may happen at two different wind speeds with different flutter frequencies and modes. More flutter conditions may exist beyond the

Fig. 11. (a) First flutter mode, (b) second flutter mode.



U

Uamb

Utran

t Fig. 12. Non-stationary flutter boundary.

reduced velocity range covered by the experiments. However, only low reduced velocity solutions are important in the practical sense and are calculated. For flutter derivatives corresponding to the pulse response of ambient vibration ðRa ¼ 0Þ; there is no flutter condition found below wind reduced velocity value of 8. Based on the concept of relative amplitude, there should be two thresholds UAmb and UTran of the wind speed. The former is for a signal to diverge in the context of ambient response, the latter is for a larger amplitude signal to diverge after going outside ‘‘ambient vibration envelope’’. If a signal cannot diverge in the former case, i.e. UoUAmb ; it would not go to the stage of latter case. Theoretically, the structure is stable to wind excitation. If the signal diverges from ambient response to larger amplitude but subsequently decays, i.e. UAmb oUoUTran ; the response is bounded. If the signal diverges in both cases, i.e. UAmb oU&UTran oU; the system is not stable. It is interesting to consider the case where UTran oUAmb : Under this condition, the judgment of the onset of flutter based on UAmb is not safe, since there could be other sources of excitation contributing to the dynamic response. A strong gust, for example, could push the bridge deck outside the normal ambient vibration envelope. After the gust, the bridge deck may oscillate back from outside into the ambient vibration envelope. Before the gust, the critical wind speed for flutter is Uamb ; after the gust, however, the critical wind speed may change to Utran ; which may be lower than Uamb : This is shown in Fig. 12. If flutter happens in this way, a transient rather than steady critical flutter wind speed exists. It will be conservative to use UTran as the design flutter wind speed in this case.

6. Summary In this study, non-linearity in self-excited wind forces is detected through the concept of relative amplitude. By comparing the flutter derivatives identified from triggered free decay and output covariance of ambient vibration of elastically supported rigid sectional bridge models, the relative amplitude effect on the interactive wind forces is manifested. This effect, from the structural point of view, reveals a complex relationship between the self-excited forces and the ‘‘structural


507

vibration noise’’ due to the turbulence excitations. Although the aeroelastic forces are linear when the body motion due to an external trigger is not affected significantly by the turbulence, they are non-linear when the noise component in the vibration due to the turbulence is not negligible. The experiments were conducted in smooth flow; more experiments are needed to have a discussion on the relative amplitude effect under turbulent condition. Conclusions apply to the specific sectional type tested in the study. Flutter derivatives of the partially streamlined box girder section identified from transient and ambient vibration testing were used to manifest the relative amplitude effect on flutter boundary. The flutter derivatives from ambient vibration were found aeroelastically stable, while the flutter derivatives from transient vibration were not. Therefore, as far as this section type is concerned, a smaller relative amplitude effect appears to have a stabilizing consequence.

Appendix A. Output Covariance When the measurement noise v in the equipment and process noise p (random component of wind forces) are white and zero mean [10], E½X qT ¼ 0 and

E½X wT ¼ 0;

ðA:1Þ

where qðiÞ ¼ BpðiÞ and wðiÞ ¼ DvðiÞ: B and D are input and feed through matrix of a state space model, respectively. The Lyapunov equation for the state covariance matrix is Ssiþ1 ¼ E½X ði þ 1Þ X T ði þ 1Þ ¼ ASsi AT þ Q;

ðA:2Þ

where Q ¼ E½qðiÞ qT ðiÞ and A is the state matrix. If output covariance is defined by Ci ðkÞ ¼ E½Y ði þ kÞ Y T ðiÞ;

ðA:3Þ

Ci ð0Þ ¼ E½Y ðiÞ Y T ðiÞ ¼ CSsi C T þ R;

ðA:4Þ

then

where R ¼ E½wðiÞ wT ðiÞ: If we define G ¼ E½X ði þ 1Þ Y T ðiÞ ¼ ASsi C T þ S;

ðA:5Þ

where S ¼ E½qðiÞ wðiÞ; Eq. (A.3) gives, for k ¼ 1; 2; y Ci ðkÞ ¼ CAk1 G: This produces a new state-space model of A; G; C; Ci ð0Þ:

ðA:6Þ



Appendix B. Coefficients of the Impedance Matrix Elements The coefficients of the impedance matrix element Eij are [9] rB4d lK ½H1 Ghi hj þ H2 Ghi aj þ H5 Ghi pj þ P1 Gpi pj 2Ii þ P2 Gpi aj þ P5 Gpi hj þ A1 Gai hj þ A2 Gai aj þ A5 Gai pj

ðB:1Þ

rB4d lK 2 ½H3 Ghi aj þ H4 Ghi hj þ H6 Ghi pj þ P3 Gpi aj 2Ii þ P4 Gpi pj þ P6 Gpi hj þ A3 Gai aj þ A4 Gai hj þ A6 Gai pj ;

ðB:2Þ

Aij ðkÞ ¼ 2zi Ki dij

Bij ðkÞ ¼ Ki2 dij

where K ¼ Bd o=U is the reduced frequency and Ki ¼ Bd oi =U is the reduced ðm ¼ 1; y; 6Þ are flutter derivatives and the modal frequency of mode i; Hm ; Am ; Pm integrals Gri sj are obtained by integration over the deck, which is the primarily aerodynamic load source: Z l dx ðB:3Þ Gri sj ¼ ri ðxÞ sj ðxÞ ; l 0 where ri ¼ hi ; pi or ai ; sj ¼ hj ; pj or aj :

References [1] R.H. Scanlan, J.J. Tomko, Airfoil and bridge deck flutter derivatives, J. Eng. Mech. ASCE 97 (6) (1971) 1717–1737. [2] R.H. Scanlan, Amplitude and turbulence effects on bridge flutter derivatives, J. Struct. Eng. ACSE 123 (2) (1997) 232–236. [3] M. Falco, A. Curami, A. Zasso, Nonlinear effects in sectional model aeroelastic parameters identification, J. Wind Eng. Ind. Aerodyn. 41–44 (1992) 1321–1332. [4] C. Hoen, N. Moan, S. Remseth, System Identification of Structures Exposed to Environmental Loads, Structural Dynamics-EURODYN’93, T. Moan et al. (Eds.), Balkema, Rotterdam, 1993, pp. 835–843. [5] J. Bogunovic Jakobsen, E. Hjorth-Hansen, Determination of aerodynamic derivatives by a system identification method, J. Wind Eng. Ind. Aerodyn. 57 (1995) 295–305. [6] A.N. Kolmogorov, Local structure of turbulence in non-compressible flow with high Reynolds numbers, Dokl. Acad. Sci., USSR 30 (1941) 301–305. [7] J. Solnes, Stochastic Processes and Random Vibration Theory and Practice, Wiley, New York, 1997. [8] J.N. Juang, Applied System Identification, Prentice-Hall, Englewood Cliffs, New Jersey, 1994. [9] A. Jain, N.P. Jones, R.H. Scanlan, Coupled flutter and buffeting analysis of long-span bridges, J. Struct. Eng. ASCE 122 (7) (1996) 716–725. [10] P.V. Overschee, B.D. Moor, Subspace Identification for Linear Systems, Theory–Implementation– Application, Kluwer Academic Publishers, Dordrecht, 1996.

ARTICLE IN PRESS

JOURNAL OF SOUND AND VIBRATION Journal of Sound and Vibration 283 (2005) 957–969 www.elsevier.com/locate/jsvi

Some considerations on the effects of the P-derivatives on bridge deck flutter Xin Zhanga,, James Mark William Brownjohnb a

School of CEE, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore b School of Engineering, University of Plymouth, Drake Circus, Plymouth PL4 8AA, UK Received 16 June 2003; received in revised form 18 May 2004; accepted 25 May 2004 Available online 11 November 2004

Abstract Using two degrees of freedom (dof) experimental flutter derivatives to perform three-dimensional flutter analysis for a cable-supported bridge is a widely practiced method. It is important to consider the Pderivatives effect to have more accurate analysis for a long-span bridge. Through a case example, this paper studied some of the issues relating to the P-derivatives effects on flutter. The operational condition in twodof experiments was discussed. It was suggested that due to the strong aeroelastic coupling effect of the sectional model studied in this research, there was an inherent weakness of two-dof experiments. The effect of the P-derivatives was studied for an example bridge by comparing the flutter analysis results using twodof and three-dof experimental flutter derivatives. r 2004 Elsevier Ltd. All rights reserved.

1. Introduction Techniques predicting the flutter boundary of cable-supported bridges developed either in frequency domain [1,2], or in time domain [3] are to solve negative damping driven flutter problems. The structural system by means of its deflection and time derivatives taps off energy from the wind flow. If the system is given an initial disturbance, its motion will either decay or diverge Corresponding author.

E-mail addresses: [email protected] (X. Zhang), [email protected] (J.M.W. Brownjohn). 0022-460X/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2004.05.031


X. Zhang, J.M.W. Brownjohn / Journal of Sound and Vibration 283 (2005) 957–969

Nomenclature Ai ; ði ¼ 1; :::; 6Þ flutter derivatives Aij ðKÞ variables in E ij B width of the bridge deck Bij ðKÞ variables in E ij dm infinite small mass E impedance matrix E ij element in impedance matrix G ri sj modal integral hi ith vertical mode H i ; ði ¼ 1; :::; 6Þ flutter derivatives

generalized inertia Ii K reduced frequency l bridge deck length pi ith lateral mode Pi ; ði ¼ 1; :::; 6Þ flutter derivatives U,V,S matrices ai ith rotational mode oi circular frequency x¯ participation factor vector of structural modes at flutter Zi the ith full bridge mode shape

according to whether the energy of motion extracted from the flow is less than or exceeds the energy dissipated by the system through mechanical damping. The dividing line between the decay and divergent case, namely, sustained sinusoidal oscillation, is recognized as the critical flutter condition, the threshold of negative damping. The energy flow between the structure and the surrounding flow is characterized by two sets of parameters: flutter derivatives [4] and structural mode shapes. The flutter derivatives are measured via experiments with rigid sectional models, which do not show instability within the reduced velocity range covered by the experiments. By incorporating the effect of mode shapes of a flexible structure, these flutter derivatives may change some particular aeroelastic modal damping of the interactive system from positive to negative. At the critical wind speed, it is reasonable to postulate that a single aeroelastic mode will approximate the total response. This assumption is justifiable from observation of the fact that typically just one predominant mode will become unstable and dominate the flutter response of a three-dimensional bridge model in the wind tunnel. It has been common to use the combination of a set of mechanical modes, namely the modes of the bridge deck under non-wind condition, as the flutter mode to perform the flutter analysis. While the flutter mode is described in three dimensions, the flutter derivatives are usually identified experimentally from a two-dimensional sectional model, which represents a strip of the full bridge deck. Totally 18 flutter derivatives describe the aeroelastic property of a sectional model with three degrees of freedoms (dofs). Due to the difficulties in the identification of these 18 parameters, two-dof results are usually obtained in the experiments. The use of two-dof flutter derivatives in three-dimensional flutter analysis is supported by the assumption that the P-related derivatives have a stabilizing effect on flutter. If they were omitted, the analytical result would be conservative. In the two-dof experiments, however, confinement must be applied to the sectional model to prevent the model from oscillating in the lateral direction. If the sectional model experiences aeroelastic coupling, the confinement in the lateral direction may affect the model motion in other directions through the aeroelastic coupling effect. In this case, the two-dimensional experimental result could be affected.

ARTICLE IN PRESS X. Zhang, J.M.W. Brownjohn / Journal of Sound and Vibration 283 (2005) 957–969

959

Previous researches pertaining to the identification of two- and three-dof flutter derivatives performed by Singh et al. [5], Chen et al. [6] and others mainly concentrate on the identification algorithm. The flutter derivatives are considered as constitutive quantities, independent of measurement methods. This is true when the experimental condition is controlled ideally. Under the operational condition, however, the two-dof identification result may be affected by external factors, such as the physical lateral restraint. Furthermore, the lateral confinement on the sectional model is equivalent to applying additional restraints on the prototype. These restraints confine the structure in the aeroelastic sense rather than allowing it to vibrate freely without causing any aeroelastic forces in the lateral direction. For a structure that is more confined, a higher flutter wind speed is usually expected. This might not give rise to a conservative design. A previous study by Katsuchi et al. [7] on Akashi-Kaikyo Bridge suggests that the lateral derivatives have a significant effect on the flutter wind speed. When P-derivatives are considered, in their particular case, there is a notable decrement of the flutter wind speed. In order to discuss these issues, it is needed to distinguish the following three cases where (A) the sectional model is tested in two-dof experiments with the third degree of freedom confined; (B) the sectional model is tested in three-dof experiments for 18 flutter derivatives among which P-related flutter derivatives are assigned zero and (C) the sectional model is tested in three-dof experiments for 18 flutter derivatives all of which are used in the flutter analysis. Based on the classification above, it should be Case B instead of Case A that reflects the assumption of omitting the P-related derivatives in the flutter analysis of a full bridge. These three cases are studied in this research for one particular bridge section type.

2. Experimental flutter derivatives A partially streamlined box girder sectional model with extended wings on each side (Fig. 1) was tested [8]. The suspension and measurement system is shown in Fig. 2. The system identification method used is eigensystem realization algorithm (ERA) [9]. The identified flutter derivatives of two- and three-dof are shown in Fig. 3a–c. In Fig. 3c, the aeroelastic coupling between the lateral and the rotational dof is indicated by large value of P3 : Due to the aeroelastic coupling effect, the orthogonality of the modal coordinate for the sectional model is affected. Under this situation, the restraints in the lateral direction, which are considered orthogonal to the rotational dof under no wind condition, now can be felt by the rotation motion. The behavior of the rotational vibration is expected to be affected by the lateral restraining force. Therefore, the two-dof experiment may not produce accurate results. This could be part of the reason for the differences in the flutter derivative (H 1 H 4 ; A1 A4 ) between two- and three-dof experiments. It should be mentioned that the differences might also result from the inaccuracy of the identification.


X. Zhang, J.M.W. Brownjohn / Journal of Sound and Vibration 283 (2005) 957–969 355 227

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Fig. 1. Streamlined box girder model (dimension mm).

Spring Model

Spring Light of the Displacement Sensor Spring

Light of the Displacement Sensor

Fig. 2. Setup for free vibration test (one end).

It seems that the other coupling term between rotational and lateral dof, A6 ; will not affect the two-dof experiments as much as P3 does because, in the two-dof experiment, the lateral motion is very weak and thus can only generate a small coupling force through A6 to affect the rotational motion. The same discussion applies to P6 and H 6 for the aeroelastic coupling in vertical and lateral dof. In this case example, the existence of the aeroelastic coupling between the lateral and the other two dofs suggests that all the three dofs are coupled together inherently in wind. Constraints on the lateral dof may block the energy flow among the three dofs. An analysis which omits the Pderivatives might not be accurate. To evaluate the P-derivatives effect on flutter, it is needed to distinguish between the three different cases mentioned in the section of introduction. A study on a full bridge is performed in the following part.

3. The suspension bridge and aeroelastic modeling The main span of the example bridge is 1410 m, with side spans of 530 and 280 m. The steel box sections are 22 m wide and 4.5 m deep and the shape is the same as the box girder sectional model. The towers are box section, 6 6 m at the base and 4.5 4.5 at the tower tops.


961

Fig. 3. Flutter derivatives: (a) (H i ; i ¼ 1; . . . ; 6), (b) (Ai ; i ¼ 1; . . . ; 6), and (c) (Pi ; i ¼ 1; . . . ; 6).

To facilitate the inclusion of an aeroelastic load, 3-D beam deck formulation was used to model the deck structure. Spar elements (no flexural stiffness) were used to represent the main cable and hanger. They have the facility to accommodate the initial strain value. The tower was analyzed using beam elements with tension, compression, torsion and bending capabilities. Modal analysis was first conducted. The resultant first 10 deck modes are listed in Table 1. After these modal parameters were obtained, they were used for the flutter instability prediction. The frequency domain flutter prediction method developed by Jain et al. [2] was used. The nature of this method is to solve an aeroelastically influenced eigen-problem: jE j ¼ 0:

(1)



Table 1 Dynamic properties of the bridge Mode no.

Mode type

Frequency ni (Hz)

1 2 3 4 5 13 14 16 26 28

L, 1st S 1st V, S L, 1st AS V 1st AS V 2nd S V 2nd AS L 2nd S; T 1st S V 3rd S V 3rd AS T 1st AS

0.0688 0.1277 0.1591 0.1646 0.1897 0.2498 0.2816 0.3246 0.4022 0.45853

Note: S=symmetrical; AS=anti-symmetrical; V=vertical; LT=lateral-torsion; and T=torsion.

The general term of the impedance matrix E is pffiffiffiffiffiffiffi where i ¼ 1;

E ij ¼ K 2 dij þ iKAij ðKÞ þ Bij ðKÞ;

Aij ðkÞ ¼ 2zi K i dij þ P2 G pi aj

rB4 lK ½H 1 Ghi hj þ H 2 G hi aj þ H 5 G hi pj þ P1 G pi pj 2I i þ P5 G pi hj þ A1 G ai hj þ A2 G ai aj þ A5 Gai pj

(2)

ð3Þ

and rB4 lK 2 ½H 3 G hi aj þ H 4 G hi hj þ H 6 Ghi pj þ P3 G pi aj 2I i þ P4 G pi pj þ P6 G pi hj þ A3 Gai aj þ A4 Gai hj þ A6 G ai pj :

Bij ðkÞ ¼ K 2i dij

ð4Þ

In Eqs. (2)–(4), r is air density, B is the deck width, l is the deck length, K ¼ Bo=U is the reduced frequency, K i ¼ Boi =U is the reduced frequency of mode i, H m ; Am ; Pm ; ðm ¼ 1; . . . ; 6Þ are flutter derivatives and dij is the Kronecker delta function defined as dij ¼

1; 0;

i ¼ j; iaj;

(5)

The modal integrals G ri sj are obtained by integration over the length of the deck, which is the primarily aerodynamic load source Z

l

ri ðxÞ sj ðxÞ

G ri sj ¼ 0

dx ; l

(6)


963

where ri ¼ hi ; pi or ai and sj ¼ hj ; pj or aj are the ith and jth mode shapes in the vertical, lateral and rotational direction, respectively.

4. The flutter analysis result The analysis is carried out in three steps. In the first step, the analysis is performed with two-dof flutter derivatives (Case A), and the critical wind speeds for flutter are obtained. In the second step, two-dof flutter derivatives are obtained from three-dof flutter derivatives by setting the P-related derivatives to zero for all the reduced velocity range, i.e., P1 P6 ¼ 0; A5 ¼ A6 ¼ 0; H 5 ¼ H 6 ¼ 0 (Case B). In the third step, analysis is carried out with three-dof flutter derivatives (Case C). The first step indicates that the self-excited load in the lateral direction is constrained; the second step assumes that there is a self-excited load relating to lateral vibration but it is neglected; and the third step fully considers self-excited forced in three dofs. Eq. (1) is doubled since both the real and imaginary parts of the determinant have to be zero. Corresponding unknowns are reduced frequency K or wind velocity U and vibration frequency o: These equations are highly nonlinear in both unknowns not only through the dependence that appears in the expression of the elements in impedance matrix, but also through the flutter derivatives that are implicit in these expressions. A graphical method was proposed by Astiz [10]. In this method, it is necessary to first compute jE j for an array of K o values: this is equivalent to defining two surfaces, one for the real part and the other for the imaginary part of jE j: The intersection of these two surfaces with K o plane is obtained by linear interpolation. Then the zero contour curves of the real surface and imaginary surface are obtained with piecewise linear approximation and their intersections can be determined either numerically or graphically. The intersection points define the flutter condition. Figs. 4a–c show the plot of contour lines for impedance matrix of zero determinant value from the three study cases mentioned above. The intersection points of the solid line (zero value contour line of real part of the determinant of the impedance matrix) and the dash line (zero value contour line of imaginary part) define the flutter condition. Table 2 summarizes the analysis cases and the corresponding flutter wind speeds and frequencies. It can be observed that there are two flutter conditions found in the analysis of Case A and B, but only one flutter condition found for Case C. The lowest wind speed for flutter occurs in Case B, i.e. the case which uses the three-dof flutter derivatives with all P-related flutter derivatives being assigned zero. The lowest flutter wind velocity in this case is 38.5 m/s. The flutter frequency is 0.252 Hz. Two-dof flutter derivatives give rise to the highest flutter wind speed: 53 m/s and flutter frequency 0.267 Hz. Case C produces the flutter wind velocity 48m/s and flutter frequency 0.332 Hz. The observation in Figs. 4a–c suggests that the flutter frequency in Case C is very close to the second flutter frequency in Case B. This might suggest that the fundamental flutter mode in Case C corresponds to the second flutter mode in Case B. If this is true, omitting the P-related derivatives may change the shape of the fundamental flutter mode. This phenomenon is quite reasonable because in Eqs. (3) and (4), flutter derivatives are coupled with the modal integrals. Any changes in the flutter derivatives will be equivalent to changes in the participation of the



Fig. 4. E Matrix in (a) Case A, (b) Case B, and (c) Case C.

Table 2 Flutter speeds and frequencies Case

Deck mode combination

52.7 38.5 Case C

1, 2, 3, 4, 5, 13, 14, 16, 26, 28

Flutter speed, U flutter (m/s)

48.2

75.9 51.1 NA

Flutter frequency, f flutter (Hz) 0.267 0.252 0.332

0.392 0.341 NA

NA: Solution not found within the reduced velocity range covered by experiment.

modal integrals and finally equivalent to changes in the analytical modal flutter result. This statement, however, needs to be verified by solving not only the analytical flutter frequencies but also flutter mode shapes.


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The solution of E flutter x¯ ¼ 0

(7)

at flutter determines the participation magnitude of each mechanical mode. Because the flutter frequency is solved numerically, the determinant of impedance matrix obtained at flutter is not strictly zero, i.e. jE jflutter 0:

(8)

Solving Eq. (7) directly will not always give a reasonable result. Some care should be exercised to solve the equation, due to the numerical sensitivity of the system at flutter. It was argued [11] that typically for suspension bridges, a single torsional mode is the most likely mode to dominate flutter while the participation of other modes may not significantly alter the outcome of the analysis. Therefore, for practical reasons, the preset value should be assigned to the entry corresponding to such flutter dominating modes. Otherwise, misleading results may be obtained. However, some analytical results of multimode flutter analysis of long-span bridges indicate that because of the closely spaced natural frequencies and three-dimensional mode shapes, the aerodynamic coupling mechanism among modes becomes complex. The flutter is not always initiated by the fundamental symmetric torsional mode [12,13]. These results seem to be sensitive to the structural and aerodynamic characteristics of the system. A systematic method to solve the flutter mode is equivalent to finding an exact singular matrix ~ so that the numerically obtained impedance matrix can be approximated with its main E; structure maintained and the solution of equation E~ x¯ ¼ 0 producing approximately the real eigenvector. In this study, the solution is obtained by singular value decomposition (SVD) of the impedance matrix, i.e. E ¼ USV T ; where U and V are orthogonal singular vectors matrices and S is the diagonal singular value matrix. By assigning the last singular value to zero, Eq. (7) is approximated and solved as follows: S1 0 T ¯ (9) V T x¯ ¼ 0: U Ex 0 0 Figs. 5–7 show the calculated flutter mode shape. All the figures indicate that the vertical motion predominates in the flutter mode. It can be seen that the fundamental flutter mode shape corresponding to Case C is similar to the second flutter mode in the other two cases. The P-derivatives push the lower one of the two unstable modes to higher reduced wind velocity, which is outside the experimental range. It seems the energy in the vertical and rotational modes flows into the lateral mode and is dissipated by the P-derivatives in Case C. In Case B, however, by setting the P-related derivatives to zero, the modal integrals in the impedance matrix containing lateral mode component are deactivated. The energy exchanging process is stopped. The energy stays inside the system resulting in instability of the system. On the other hand, the higher frequency flutter mode, which occurs in all three case studies, seems to show a different behavior. For this flutter mode, the P-derivatives must be contributing



Fig. 5. (a) First flutter mode in Case A, and (b) second flutter mode in Case A.

energy to the response via the modal integrals resulting in a lower flutter wind speed in Case C than in Case B. The mechanism for flutter derivatives to affect the full bridge flutter boundary seems complex. All the discussions above should be considered as a specific case study of a more general problem.


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Fig. 6. (a) First flutter mode in Case B, and (b) second flutter mode in Case B.

5. Conclusion In comparison with two-dof flutter derivative case (Case A) and three-dof flutter derivative case (Case C), the analysis in Case B, where three-dof flutter derivatives with P-related flutter derivatives being set to zero gives rise to the lowest flutter wind speed. This might confirm the



Fig. 7. First flutter mode in Case C.

assumption that the P-related flutter derivatives have a stabilizing effect on flutter. The two-dof flutter derivatives produce the highest wind speed for instability, indicating that the current practice of using two-dof experimental results is not necessarily conservative. In the absence of P-derivatives, not only the predicted flutter wind speeds but also the flutter frequencies and mode shapes may be different from what are predicted by using the full set of 18 aeroelastic parameters, indicating the complexity of the mechanism for the aeroelastic parameters to affect the flutter of a full bridge. Conclusions apply to the specific case example in this study.

Reference [1] R.H. Scanlan, The action of flexible bridges under wind. Part I: Flutter theory, Journal of Sound and Vibration 60 (2) (1978) 187–199. [2] A. Jain, N.P. Jones, R.H. Scanlan, Coupled flutter and buffeting analysis of long-span bridges, Journal of Structural Engineering 122 (7) (1996) 716–725. [3] X. Chen, M. Matsumoto, A. Kareem, Time domain flutter and buffeting response analysis of bridges, Journal of Engineering Mechanics 126 (1) (2000) 7–16. [4] R.H. Scanlan, J.J. Tomko, Airfoil and bridge deck flutter derivatives, Journal of Engineering Mechanics 97 (6) (1971) 1717–1737. [5] L. Singh, N.P. Jones, R.H. Scanlan, O. Lorendeaux, Simultaneous identification of 3-dof aeroelastic parameters, Proceedings of the Ninth International Conference on Wind Engineering, Wiley Eastern, New Delhi, India, 1995, pp. 972–981. [6] A. Chen, X. He, H. Xiang, Identification of 18 flutter derivatives of bridge decks, Journal of Wind Engineering and Industrial Aerodynamics 90 (2002) 2007–2022. [7] H. Katsuchi, N.P. Jones, R.H. Scanlan, Multi-mode coupled flutter and buffeting analysis of the Akashi-Kaikyo Bridge, Journal of Structural Engineering 125 (1999) 60–70.


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[8] X. Zhang, Aeroelastic Analysis of Super Long Cable-supported Bridges, PhD Thesis, Nanyang Technological University, Singapore, 2003. [9] J.N. Juang, An eigensystem realization algorithm for modal parameter identification and modal reduction, Journal of Guidance 8 (5) (1984) 620–627. [10] M.A. Astiz, Flutter stability of very long suspension bridges, Journal of Bridge Engineering 3 (3) (1998) 132–139. [11] A. Jain, Multi-mode Aeroelastic and Aerodynamic Analysis of Long-span Bridges, PhD Thesis, Johns Hopkins University, Baltimore, USA, 1997. [12] T.T.A. Agar, The analysis of aerodynamic flutter of suspension bridges, Computers and Structures 30 (1988) 593–600. [13] T. Miyata, H. Yamada, Coupled flutter estimate of a suspension bridge, Journal of Wind Engineering and Industrial Aerodynamics 37 (1988) 485–492.