Effect of modal damping on bridge aeroelasticity

Journal of Wind Engineeringand Industrial Aerodynamics 77&78 (1998) 421—430

Effect of modal damping on bridge aeroelasticity

Anurag Jain!,*, Nicholas P. Jones", Robert H. Scanlan"

! Hart Consultant Group, 2425 Olympic Blvd., 670E, Santa Monica, CA-90404, USA" Department of Civil Engineering, The Johns Hopkins University, Baltimore, MD 21218-2686, USA

Abstract

Estimation of modal damping is always a critical first step in the analytical estimation of theflutter and buffeting responses of long-span suspended bridges. The impact of under or overestimation of modal damping on the aerodynamic and aeroelastic behavior of suspendedbridges is not clear. This issue surfaced when a seismic retrofit of the Golden Gate bridge usingviscous dampers was proposed. The affect of these viscous dampers on the flutter and buffetingresponse of the Golden Gate bridge needed investigation. This paper considers the affect ofadditional damping imposed by these viscous dampers on the aeroelastic and aerodynamicresponse to wind-induced excitation. This mechanical damping was introduced into theaeroelastic system as additional modal damping. A multi-mode flutter and buffeting analysiswas performed for several levels of modal damping. The results of the analysis revealed thatadditional damping decreased the buffeting response whereas the effect in flutter stability wasmore complex. At different levels of damping not only did the critical flutter velocity change butalso the angle of attack at which it occurred. ( 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction

At the time of this study, the Golden Gate bridge was under a large aerodynamicand seismic retrofit program. Part of the retrofit scheme proposes the installationof viscous dampers to dissipate energy during a seismic event. However, the damperswill respond to wind-induced motion as well and the effect of supplemental mech-anical damping on the buffeting response required investigation. The configurationof the dampers was primarily expected to affect the buffeting response of the verticaldisplacement component only, however, the effect of supplemental damping onboth the vertical and torsional response is evaluated. The impact of supplementalstructural damping is potentially more important and useful from the view point of

*Corresponding author.

0167-6105/98/$ — see front matter ( 1998 Elsevier Science Ltd. All rights reserved.PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 1 6 1 - 5

bridge stability to flutter. For analytical purposes, the supplemental damping wasassumed to manifest itself as additional modal damping. This assumption wasused in assigning several values of modal damping to the aerodynamic system inthe determination of the response to wind-induced excitation. A multi-mode analy-sis procedure [5] was used to estimate the response and is briefly described below.

2. Aerodynamics and aeroelasticity of long span suspended bridges

The state-of-the-art analysis procedure described by Scanlan and Jones [1] isoutlined below. The deflection components of the bridge deck can be repre-sented in terms of the generalized coordinate of the mode m

i(t), the deck width

B and the dimensionless representations of the mode form along the deck hi(x),

ai(x), and p

i(x) as

vertical: h(x,t)"+i

hi(x)Bm

i(t),

twist: a(x,t)"+i

ai(x)m

i(t),

sway: p(x,t)"+i

pi(x)Bm

i(t). (1)

The governing equation of motion is

Ii[m$

i#2f

iu

imQi#u2

imi]"Q

i(t), (2)

where Iiis the generalized inertia of the ith mode, u

ithe ith mode circular natural

frequency, fithe damping ratio-to-critical and Q

iis the generalized force.

Let the lift, drag, and moment per unit span be defined by

lift: ¸"¸!%#¸

"drag: D"D

!%#D

"moment: M"M

!%#M

", (3)

where subscripts ae and b refer to aeroelastic and buffeting, respectively. In general, thebuffeting effects include self-induced buffeting, but these are excluded herein.

If motions are purely sinusoidal then the aeroelastic forces can be representedcompletely as

¸!%"

1

2oº2BAKH*

1

hQº

#KH*2

BaRº

#K2H*3a#K2H*

4

h

B#KH*

5

pRº

#K2H*6

p

BB,(4)

D!%"

1

2oº2BAKP*

1

pRº

#KP*2

BaRº

#K2P*3a#K2P*

4

p

B#KP*

5

hQº

#K2P*6

h

BB, (5)

422 A. Jain et al./J. Wind Eng. Ind. Aerodyn. 77&78 (1998) 421–430

M!%"

1

2oº2B2AKA*

1

hQº

#KA*2

BaRº

#K2A*3a#K2A*

4

h

B#KA*

5

pRº

#K2A*6

p

BB,(6)

where o is the air density, º the mean velocity of the oncoming wind (which isturbulent in general), K"Bu/º is the reduced frequency and H*

i, P*

i, A*

i, i"1,2,6

are the flutter derivatives.The buffeting forces (assumed independent of structural motion, excluding the

steady force component, and with assumed unit aerodynamic admittances) due towind turbulence under assumed slowly varying gust action can be expressed as

¸""

1

2oº2BCCLA2

u

ºB#(C@L#C

D)w

ºD"1

2oº2BL

"(x,t), (7)

D""

1

2oº2BCCDA2

u

ºB#(C@D)w

ºD"1

2oº2BD

"(x,t), (8)

M""

1

2oº2B2CCMA2

u

ºB#(C@M

)w

ºD"1

2oº2B2M

"(x,t) (9)

with CL, C

D, C

M, defined as the coefficients of lift, drag, and moment (referred to deck

width B) of a typical deck section, C@L"dC

L/da, C@

D"dC

D/da, C@

M"dC

M/da and

u"u(t), w"w(t) are the respective horizontal and vertical gust components of thewind.

The generalized force Qi(t) can thus be defined as

Qi(t)"

l

P0

(¸hiB#Dp

iB#Ma

i) dx, (10)

where l is deck span length.

3. Multi-mode formulation

If no variation of the flutter derivatives with the coordinate along the span x, isassumed, and the modal integrals defined as

l

P0

qi(x)r

j(x)

dx

l"G

qirj, (11)

where qi, r

i"h

i, p

i, or a

i, then the new system of equations (in matrix form) in the

Fourier-transform domain is

[E]MmM N"MQM"N, (12)

A. Jain et al./J. Wind Eng. Ind. Aerodyn. 77&78 (1998) 421–430 423

where

Eij"!K2d

ij#iKA

ij(K)#B

ij(K),

Aij(K)"2f

iK

idij!

oB4lK

2Ii

[H*1G

hihj#H*

2G

hiaj#H*5G

hipj#P*

1G

pipj#P*

2G

piaj

#P*5G

pihj#A*

1Gaihj#A*

2Gaiaj#A*

5Gaipj

],

Bij(K)"K2

idij!

oB4lK2

2Ii

[H*3G

hiaj#H*4G

hihj#H*

6G

hipj#P*

3G

piaj#P*4G

pipj

#P*6G

pihj#A*

3Gaiaj#A*

4Gaihj#A*

6Gaipj

],

QMbi"

oB4l

2Ii

l

P0

MLM"(x,K)h

i#DM

"(x,K)p

i#MM

"(x,K)a

iNdx

l.

In the above equations the dij

used is the Kronecker delta function and is defined asdij"1, i"j; d

ij"0, iOj.

The multi-mode flutter problem is solved by considering the homogeneous systemof Eq. (12) and determining the values of K and u for which both the real andimaginary parts of the determinant of matrix [E ] simultaneously become zero. This isthe negative-damping threshold (in the “multi-mode” sense), and the critical flutterspeed is evaluated using these values of K and u.

The power spectral density matrix for the generalized displacements m is developedusing Eq. (12) and is as expressed below

[Smm(K)]"[E(K)]~1[SQ"Q"

(K)]ME*(K)]TN~1, (13)

E*(K) is the complex conjugate of matrix E(K). The equations for displacements(Eq. (1)) can be developed into the power spectral densities of the respective displace-ment components e.g., for h as

Shh(x

A,x

B,K)"+

i

+j

B2hi(x

A)h

j(x

B)Smimj(K), (14)

where i and j are summation indices over the number of modes being used. Themean-square values of the displacements can be evaluated in terms of their respectivepower spectral density functions

p2rr(x

A,x

B)"

=

P0

Srr(x

A,x

B,n) dn, (15)

where r"h, p or a. A covariance matrix for h, p and a is thus obtained.


4. Golden Gate bridge

The main structure of the Golden Gate bridge consists of three suspended spans— a center span of 1280 m (4200 ft) and two side spans of 343 m (1125 ft) each— suspended from two 0.93 m (36.5 in) diameter steel wire cables spaced 27.5 m (90 ft)apart. The cables are each supported on two steel towers, each rising 227 m (746 ft)above water level, and are anchored in massive concrete anchorage blocks at theirends. The suspended structure consists of parallel-chord stiffening trusses, 7.6 m (25 ft)deep, spaced 27.5 m (90 ft) apart directly beneath the cables.

5. Description of analysis

The Golden Gate suspension bridge was analyzed for single- and multi-modeflutter and buffeting. Representative flutter derivatives of the deck cross section, ata #5° and 0° angle of attack, appear in Figs. 1 and 2, respectively. The naturalfrequencies and mode types of the first ten modes of the bridge are listed in Table 1.Experimentally determined flutter derivatives H*

iand A*

i, (i"1,2,4) [2], and an

empirical value of the flutter derivative P*1

("!2CD/K), static force coefficients

(Table 2) along with the finite element analysis of a model of the bridge [3] were usedas basic input in computing the flutter stability and buffeting response of the bridge.The results of the analysis are presented below.

Fig. 1. A*1

flutter derivatives of Golden Gate bridge at 5°.


Fig. 2. A*1

flutter derivatives of Golden Gate bridge at 0°.

Table 1Natural frequencies and mode types of Golden Gate bridge

Mode No. Frequency (Hz) Mode type Modal integrals

Ghihi

Gpipi

Gaiai

1 0.0490 LS 2.63]10~16 3.30]10~1 8.03]105

2 0.0870 VAS 3.20]10~1 7.65]10~15 1.90]10~15

3 0.1124 LAS 1.72]10~14 3.20]10~1 1.24]10~2

4 0.1285 VS 1.90]10~1 8.28]10~14 1.37]10~14

5 0.1340 VAS 3.40]10~1 5.97]10~14 2.66]10~14

6 0.1638 VS 3.40]10~1 4.00]10~13 1.84]10~14

7 0.1916 TAS 6.67]10~12 3.32]10~2 3.20]10~1

8 0.1972 TS 2.49]10~12 2.50]10~1 1.80]10~1

9 0.1988 VAS 1.80]10~1 4.60]10~13 9.61]10~12

10 0.2021 VS 2.60]10~1 1.49]10~15 8.00]10~15

Note: L: Lateral, V: Vertical, T: Torsional; S: Symmetric, AS: Anti-symmetric.

Table 2Static force coefficients of Golden Gate bridge at 0° wind inclination with the horizontal

CD

CL

CM

C@D

C@L

C@M

0.3042 0.2113 0.0044 0.0000 3.2487 !0.1772


6. Buffeting response

The buffeting calculations were carried out using the flutter derivatives for thebridge at 0° angle of attack under a mean wind velocity of 22.2 m/s (50 mph). A terrainroughness length of 0.009 m, a deck elevation of 67 m and a modal damping ratio of0.5%were used in the computations.

Fig. 3. Evolution of Golden Gate bridge phh

and paa with increasing modal damping (f).


The multi-mode buffeting response [4] of the Golden Gate bridge was subsequentlydetermined for modal damping ratios varying between 2% and 26% at a wind speedof 34.0 m/s (76 mph — 100 year return period mean hourly wind speed at deck level).The maximum values of p

hhalong the deck for each value of modal damping are

plotted in Fig. 3. The effect of increased damping on the buffeting response ispredictable, however its detailed behavior is specific to the bridge under considera-tion. A similar analysis was conducted to study the effect of additional modal dampingon the paa response. The results so obtained followed a trend similar to the p

hhre-

sponse.

7. Flutter analysis

Calculations were performed by considering a combination of up to the firsteight modes, since the eighth mode is the first symmetrical torsional mode [5].This multi-mode analysis was carried out for five sets of flutter derivatives datacorresponding to H*

iand A*

i, i"1,2,4 obtained at #5°, #2.5°, 0°, !2.5° and

!5° angles of attack. No flutter was observed for a combination of up to the first sixmodes for any angle of attack. The inclusion of the seventh mode resulted in a fluttercondition for all five angles of attack. It was determined that the least flutter velocityof 22.0 m/s was obtained using the bridge flutter derivatives measured at a wind angleof #5°. The seventh real and sixth imaginary roots of the impedance matrix E fora combination of seven modes are shown in Fig. 4. The combinations of K and u and

Fig. 4. Seven modes flutter analysis for Golden Gate bridge at 0.5% modal damping (7th real and 6thimaginary root).


the corresponding flutter velocity for the various angle of attack and number of modesparticipating in flutter combinations are summarized in Table 3. A single-mode flutteranalysis was also performed for each mode and for each angle of attack. Mode7 yielded the lowest flutter velocity in all cases. The single-mode flutter velocitiesobtained were almost identical to the multi-mode flutter velocities obtained fora combination of seven modes (Table 3). This and the multi-mode flutter mode shapesuggest that the flutter is indeed dominated by mode 7, the first torsional asymmetricmode.

To investigate the impact of supplemental damping, the modal damping wasincreased from 0.5% and the bridge analyzed for multi-mode flutter. For the case of1% modal damping (Fig. 5) the only flutter velocity that could be determined was44.6 m/s (105 mph) compared to a minimum of 22.0 m/s obtained for 0.5% damping.The other flutter velocities identified for the 0.5% modal damping case were 32.7 and37.4 m/s, both significantly lower than the only value obtained for the 1% damping

Table 3Flutter velocities for Golden Gate bridge

Angle !5° !5° !2.5° !2.5° 0° 0° 2.5° 2.5° 5° 5°

No. of modes 7 8 7 8 7 8 7 8 7 8K 1.27 1.27 1.28 1.28 1.34 1.34 1.43 1.43 1.49 1.49w 1.18 1.18 1.17 1.17 1.18 1.18 1.19 1.19 1.19 1.19º

c(m/s) 25.5 25.5 25.2 25.2 24.1 24.1 22.7 22.7 22.0 22.0

Fig. 5. Seven modes flutter analysis for Golden Gate bridge at 1% modal damping (7th real and 6thimaginary root).


case. No flutter velocities could be determined for the cases where modal dampingexceeded 1% of critical value. This analysis was conducted with the #5.0° flutterderivatives (the buffeting computations for the Golden Gate bridge were conductedat 0°). The multi-mode flutter analysis was therefore repeated for the 0° flutterderivatives to study the impact of modal damping on flutter velocity.

For the 0° case the lowest flutter velocity determined at a modal damping of 0.5%was 24.1 m/s (53.9 mph) and 28.8 m/s (64.3 mph) at 1%. From Figs. 1 and 2, it isobserved that the slope of the A*

2flutter derivatives at 0° is much steeper compared to

that at 5.0°.The flat slope of the A*2

at 5.0° will clearly yield higher flutter velocities athigher values of modal damping. Therefore, it is evident that the critical angle forflutter changes from 5.0° for 0.5% damping to 0° at 1% damping.

8. Discussion of results

The effect of assuming additional modal damping on the multi-mode flutter andbuffeting response was studied. It is observed that supplemental damping providedthrough the selection and installation of appropriate external dampers will certainlyincrease the flutter stability and reduce the buffeting response of a long-span bridge.This study also indicates that external dampers can be pursued as a viable option toovercome the problems of low flutter speeds or excessive buffeting deflections ina long-span bridge. A very important issue highlighted by this study is the criticality ofmodal damping estimates. Slight changes in modal damping can produce a signifi-cantly different response and reliable estimation of modal damping is therefore vital.

References

[1] R.H. Scanlan, N.P. Jones, Aeroelastic analysis of cable-stayed bridges, J. Struct. Eng. ASCE 116 (2)(1990) 279—297.

[2] J. Raggett, Golden Gate bridge flutter derivatives, personal communication (1994).[3] T.Y. Lin International, San Francisco, California, USA, Golden Gate bridge finite element analysis,

personal communication (1994).[4] 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.[5] A. Jain, Multi-mode aeroealstic and aerodynamic analysis of long-span bridges, Ph.D. Dissertation,

The Johns Hopkins University (1996).


Effect of modal damping on bridge aeroelasticity

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