An Analysis of Binary Flutter of Bridge Deck Section

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Journal of Sound and Vibrat i on (1978) 57(4), 471-482

AN ANALYSIS OF BINARY FLUTTER OF BRIDGE DECK

SECTIONS

Y. NAKAMURA

Research I nsti t ute or Appl i ed Mechani cs, Ky ushu Uni versi ty , Fukuoka, Japan

(Recei ved 20 Sept ember 1977)

This paper is concerned with an analytical and experimen tal study of binary flutter of

bridge de ck sections. A set of analytical formulas giving the frequency and rate of grow th o f

oscillation, the position of the equivalent center of rotation and the phase differencebetween bending and torsion near the critical flutter point is presented . The formulas

provide an analytical basis for the previously prop osed me thod of classification of binary

flutter of bluff structures. The results of wind tunnel experim ents on mod els with simple

geom etrical shap es confirm that the present formulas are applicable to a variety of struc-

tures ranging from a flat plate to much mo re bluff bridge dec k sections.

1. INTRODUCTION

Flutter is a flow-induced, self-excited oscillatory instability of elastic structures. Since the

spectacular collapse of the Tacoma Narrow s Bridge in 1940, flutter of long-span suspension

bridges in wind has been a subject of serious engineering concern, and prediction of thecritical flutter spee d is currently one of the mos t important design proced ures for modern

long-span suspension bridges.

Most bridge deck sections are not streamlined so that the flow around them is necessarily

separ ated. Unfortunately, howe ver, there is no satisfactory aerodynam ic theory for any

separ ated flow, particularly an unsteady one, and hence no com plete theory of flutter of

bluff structures is now available. It is also true, on the other hand, that m ost bridge deck

sections are not very bluff so that they still behave as efficient lifting surface s. Accordingly,

there is some resemblance between aerofoil and bridge deck flutter, and a number of authors

have dealt with the two phenome na as similar, if not actually identical. In the measur ement

of oscillatory aerodynam ic forces and mom ents of bridge deck sections [l], Scanlan and

Tom ko, while benefiting from the aerofoil study as much as possible, yet show ed its distinct

limitations as regar ds applicability to bridge deck sections.

In a previous pape r [2], it was shown that flutter of bluff structures in many coupled degre es

of freedom can be classified into three types depending upon the mode of energy transfer

from fluid to structure; the three types of flutter m entioned are herein referr ed to as the

classical type flutter, the single degr ee of freedo m type, and the intermediate type, respec-

tively. The classification could shed new light on the underlying physical mechan isms of

flutter of bluff structures in many d egree s of freedo m that had often remained unclarified in

the past investigations. In the case of binary flutter, in particular, a mo re practical meth od of

classification was also prop osed and the experim ents could confirm its usefulness. This

meth od of classification was based on the observation of the rate of grow th of oscillationrather than that of the energy transfer from fluid to structure, on an intuitively corre ct

assumption that the former would be proportional to the latter.

47 100222460X/78/0422-0471 $02.00/O 0 1978 Academ ic P ress Inc. (London) Limited



47 2 Y. NAKAMURA

In the present paper, the analytical counterpart of the above-mentioned method of classi-

fication of binary flutter-that is, to predict the rate of growth of oscillation with use of

given aerodynam ic coefficients-is considered . The aerodynam ic coefficients required for the

analytical approach may be obtained experimen tally either by a free oscillation technique

[l, 31 or with a more sophisticated forcing apparatus [4, 51. First, a discussion of the aero-

elastic characteristics of a bridge deck binary system is presented, and following from this aset of useful analytical formulas is derived for the solution to the flutter equation : name ly,

for the frequency and rate of grow th of oscillation, the position of the equivalen t center of

rotation and the phase difference between bending and torsion. The formulas obtained are

remarkably simple so that they are capab le of clear physical interpretation. In particula r,

the formu las can be rewritten in forms su itable for the previously proposed classification of

binary flutter. The results of some wind tunnel experiments are also presented to confirm the

applicability of the formulas to all the three types of flutter m entioned above; in short, they

are applicab le to flutter of a variety of structures rangin g from a flat plate to much more

bluff bridge deck sections. It is thus shown tha t the present formu las are very useful not only

for such practical applications as trend studies in the design of long-span suspension bridges,

but also for academic purposes of obtaining clear insight into the complicated physical

nature of flutter of bluff structures.

It is well known that most bluff structures are susceptible to vortex excitation at wind

speeds where the frequency of the von Karman vortex street approaches one of the natura l

frequencies of the structure. However, structures of the type herein considered are not, in

general, very bluff, as wa s mentioned earlier. Their aeroelastic responses are then character-

ized by violent flutter in stabilities occurring at relatively high wind speeds, which are some-

times preceded by very weak vortex excitation at much lower wind speeds. In this paper,

attention is focused on the flutter instab ilities at higher w ind speeds with vortex excitation

being left outside of consideration.

2. EQUATIONS OF MOTION

Consider a two-dimensional section of a bridge deck which is endowed with two degrees

of freedom, y and 0, and balanced mechanically about its midchord rotation axis. The

model is suggested in Figure 1. For this model, the linearized equations of motion are

mji + m(2nfy)2y = L[y] + L[B], ze + z(2nf)2 8 = M[y] + M [e], (L a

where m and I are the mass and mass moment of inertia per unit span, respectively,& and

f ere the still-air frequencies in the y and 0 degrees of freedom, respectively, L[ ] and M[ ]

are the aerodynamic lifts and moments about the midchord, respectively, in which [ ] means

that the lifts and mom ents are functionals of the argumen ts, and the dot denotes the differen-

tiation with respect to the time 1. For simplicity, structural damp ing is neglected in the present

analysis.

IY

V 8-9%

L

Figure 1. Bridge deck section in a uniform flow.



BINARY FLUTTER OF BRIDGE DECK SECTIONS 473

The aerodynamic lifts and mom ents are respectively expressed in terms of the aerodynamic

coefficients as

UYI + Jwl = +PW2b) {CJYPI + CJm (3)

A4bl + M PI = 4P?2bY {chf[Y/b1 CMW , (4)

where p is the air density, V is the horizontal wind speed, 26 is the deck width, and C,[ ] andC,[ ] are the aerodynam ic lift and moment coefficients, respectively.

The equations of motion are then non-dimensionalized for convenience:

II” + R2 ~9 q = ( r2/47r2 P) {C,[V] + CJOI], (5)

6” + 02 0 = (P/27? v){C&] + C,[Q]},

where T = 2T f t is the reduced time, in which f is the frequency of oscillation in wind, q(T) =

y(t)/b is the reduced vertical displacement, p = m/(pb2) is the reduced mass, v = Z / (p b4 ) s

the reduced mass moment of inertia, R = f v / f e is the uncoupled frequency ratio, 0 = fe/f

is the frequency ratio, P= V/ ( f b ) is the reduced wind velocity, and the dash denotes thedifferentiation with respect to T .

The solution of equations (5) and (6) is assumed in a form of exponentially modified

oscillations :

q(T) = rloexp O(T)=Ooexp[(&+i)T+$], (7,8)

with

‘lo/e0 = X (9)

where q. and O. are the amplitudes, b is the logarithmic rate of growth of oscillation, &J s thephase angle between q (T ) an d O(T ) , i = (-1)1’2, and, a s was pointed out in reference [2], X

represents the position of the equivalen t center of rotation on an assum ption of 141< 1. It

is further assumed that the aerodynamic lifts and moments for exponentially modified

oscillations may be replaced by those which correspond to purely sinusoidal oscillations.

That this assumption is correct at least for ]/II @ 1 was shown by Scanlan and Tom ko [I]

and this evidence is supplemented by the results of the experiment reported in what follows.

For example, CJ?] is given by

c,[v] = (CL,, + i&i) V(T), (10)

where CL,, and CL,, are the real and imaginary parts of the frequency response function,

both of which are functions of the reduced wind velocity l?

Introduction of equations (7) to (9) into equations (5) and (6) yields the following four

equations, when the exponential functions are eliminated from both sides of the equations

and then the real and imaginary parts are separated :

[@‘/4x2) - 1 + R2 g2] X = ( P2/47r2p) (CL,, X + CL,, cos t$ - CL,, sin b), (11)

(b2/47r2) 1 + c2 = ( V2/27r2v) {(CM,, cos 4 + CM ,, sin 4) X + CM eR}, (12)

j?X = ( Y2/4np) (CL,, X + CUR sin 4 + CL,, cos 4), (13)

B = ( P2/271v)K-C,,, sin 4 + CM ,, cos 4) X + CMB I}. (14)

Equations (11) to (14) can be solved to give rr, A’,#Iand 4, but such a solution would no doubt

be complicated as too many parameters would then be involved. The derivation of simple



474 Y. NAKAMURA

formulas for c, X , /I and $J in which only the most essential p arameters are retained was the

aim of the work reported here. This is effected by observing some of the characteristics of the

aerodynamic coefficients on the one hand and imposing some conditions on the solution on

the other.

3. ANALYSIS OF THE AERODY NAM IC COEFFICIENTS AND THE SOLUTION

First of all, the range of the reduced wind velocity of interest is well above that of vortex

excitation; say, P> 10, approximately, is assumed . For this range of the reduced w ind

velocity, C,,, is much sm aller in magnitude than CLBRor typical bridge deck sections as well

as for a flat plate [4, 51. Thus

IC,,,/ 6 ICLsRI. (15)

Second , the analysis of torsional flutter of bluff structures [6] indicates that m ajor portions

of C,[8] and C& I] are the contributions of the angle-of-attack motion, those of the angular

velocity motion being much less. That is, the following relations are obtained to the first

order of approximation :

(CL,, + GJ tl = (CL,, + iCLOd (24 PI 4, (16)

(CM,, + iCM,d II = (Go, + iCd(W PI v’. (17)

Or, alternatively,

C‘lI = (27rV) CLtW Chf,, = (270) C&ft?R, C,,, = - WY ) C&V ,

ChfVR - (27t/I? ChftW (18)

Accord ing to the Theodorsen unsteady aerofoil theory, the four relations in equation (18)

hold for a flat plate at high reduced wind velocities. It should be mentioned, however, that

for bluff structura l sections, all these relations may be valid over a much wider range of thereduced wind velocity, although the degree of the approximation and the lowest reduced

wind velocity for which they are valid depend on the bluffness of a section.

Regarding the solution, the stability of the torsional branch is considered in the subse-

quent analysis, whereas the bending branch is assumed to be stable : that is,

IX/< 1, (19)

which states that the equivalent center of rotation should not be very far away from the

midchord. In addition, the phase angle between bend ing and torsion is assumed to be small

in magnitude: that is,

I91< 1. (20)Experience has shown (for example, see reference [2]) that these two conditions are satisfied

for most bridge deck sections for which R @ 1. A further assumption to be imposed on the

solution is

IBI 4 1. (21)

Thus, oscillations near the critical flutter point, either decaying or growing, are herein

considered.

4. THE USEFUL APPROXIMATE FORMULAS

A considerable simplification of equations (11) to (14) is now feasible in the light of theanalysis of the preceding section. Equation (12) is first taken up. It turns out that the first

term on the left-hand side and those associated with the brace of the right-hand side are



BINARY FLUTTER OF BRIDGE DECK SECTIONS 41 5

relatively small and can be discarded. Equation (11) is then considered, in which the first

term in the brace of the left-hand side is discarded whereas on the right-hand side only the

second term in the brace is retained with cos 4 replaced by unity; similar analysis is appl.ied to

equations (14) and (13). The sim plified expressions for (T, X, /I and 4 are thus obtairred in

the following form :

d = 1 + (P/2$ v) C&& 9& ( 22 )

X = { az/47r’ ~(-1 + Rz a')}LoR, ( 3)

P = (P2/2nv) (CM,, + CM ,, X), (24)

/IX = ( P2/47rp) (C,,, X + CLe, + C,,, sin 4). (2.5)

It is important to mention that equations (22) to (25) are presented in the order of their

successive determination. That is, when the value of reduced wind velocity is prescribed,

the frequency ratio ~7 s first determined by specifying the value of CM eR,which is a function

of l? The evaluation of the position of the equivalen t center of rotation X follows by speci-

fying the value of C,,, together with r~ ust determined: similar procedure is applied for /3and 4.

Substitution of the first two relations of equation (18) into equations (23) and (24) leads

to the following alternative expressions for X and /!:

X = { P3/8n3 ~(-1 + R2 ')}L,,,, (26)

J = (P2/27rv){C&@, (27c/P)C,,, X}. (27)

For practical applications, specification of the phase angle is not always required. Then, use

of equations (22), (26) and (27) is more advan tageous with the aerodynamic coefficients

involved being reduced to only three in number.Because of the simplicity of the present formulas, several useful conclusion s may be ob-

tained. First, consider equation (22) for the frequency ratio (T. Since the value of C,W ,, for

most bridge deck sections is positive, 0 > 1 holds for P > 0; in other words, as the wind

speed is increased from zero, the frequency of oscillation decreases m onotonically from,& .

Second, equation (26) indicates that under the condition of R < 1

-1 +R202<0, (28)

as the wind speed is increased from zero. This, combined with C,,, < 0, leads to X > 0: that

is, the equivalent center of rotation of a bridge deck section moves toward the leading edge

with an increase in l? Third and lastly, by multiplying equation (25) by X and adding it to

equation (24), one obtains

B = P2(C‘&, + CL,,, sin 4 (X/2)}/27r(v + pX2), (29)

where

C’MB1 Chf,, + CM ,, X + (C,,, + C,,, X) X/2. (30)

Equation (29) is important because a classification of binary flutter is possible on this basis.

Let the aerodynam ic mom ent coefficient for the sinuso idal torsional oscillation co rrespond ing

to X be C,&,, + iC,& and consider that ChB, is expressed in terms of the aerodynamic

coefficients corresponding to the midch ord. After a little man ipulation , one finds that

C,&,, is identical with equation (30). One notes also that v + pX2 in equation (29) represents

the reduced mass moment of inertia with respect to X. Then, w riting



47 6

where

Y. NAKAMURA

/3i = P2 C~Bi/Zrc(V /LP), fi2 = P2 CL,, sin 4(X/2)/27r(v + pXz), (32933)

one easily finds that the rate of grow th of oscillation, /?, consists of two physically different

terms, one representing the rate of growth of oscillation for the system in a single degree of

freedosn in torsion with respect to X, and the other being that proportional to sin 4, where

C#Js the phase angle between ben ding and torsion. The classification of binary flutter is nowfeasible as

‘(1) the class ical type flutter, if p2 9 fir ;

(2) the single degree of freedom type flutter, if /?i B pz;

(3) the intermediate type flutter, if fil 2: Bz.

Thi:s method of classification of binary flutter is identical with what was proposed earlier

[2], but analytical expressions have now been formulated, as equations (31) to (33), for the

first time. O ne finds, by going back to equation (27), that the stability of a bridge deck section

is directly controlled by the pitch dam ping (-CM ,,) regard less of the type of flutter. T hat th is

is characteristic of bridge deck flutter will be men tioned later.The assumptions made about the aerodynamic coefficients and the solution in deriving

the present fo rmulas are by no means seriously restrictive, and a s will be exemplified by the

experiment described in section 5 , they may be satisfied for typical bridge deck sections a s

well as for a flat plate provided these have m ass and m ass moment of inertia of sufficiently

large values. In short, the present formu las are applicab le to flutter of a variety of structures

ranging from a flat plate to much more bluff bridge deck sections. It may be added that equa-

tion (24), when it is reduced to p = 0, is basically similar to the formula of reference [7]

which had been obtained on the basis of the Theodorsen unsteady aerofoil theory. Further,

consider application of the Theodorsen unsteady aerofoil theory to equation (27) for the

case where the high frequency approximation is made. Here, the high frequency approxi-

mation implies that in the circulation function, which is C(k) = F(k) + iG(k), where k =

2~17, F(k) = 3 an d G(k) = 0 are assum ed. T he aerodynamic coefficients for a flat plate in

this approximation are, with the virtual mass terms neglected,

CL@R - n, Chf,, = nr/4 and C,,,,, = - 7r2/41? (34)

Inserting equation (34) into equations (22), (23) and (27), and letting j = 0 in equation (27),

one obtains

VF = d27r(l - R2)l{(l/P) + (1/4v)], (35)

for the critical flutter speed. By replacing the arithmetic mean, [,u-’ + (4~)~‘l/2, in equation

(35) by the geometrical mean, (~,LLv)-‘/~,quation (35) is reduced to

FF = d27r(l - R2) 6. (36)

It is interesting that equation (36), if (27~)“~= 2.506 . . ., is replaced by a slightly different

constant of 2.623, is identical with Selberg’s empirical formula [8] which is known to be

exceptionally accurate for a flat plate.

5. EXPERI MENTAL SET UP AND TEC HNIQUE

The experiments were conducted in a low-speed wind tunnel with a 3 m high by 0.7 m

wide working section. The basic model which w as used consisted of a thin flat plate, 6 mm in

thickness, 0.66 m in span, w ith a uniform chord of O-4m, and with stiffening trusses. It had alarge thin plate m ounted at each end to ensure two-dimensional air flow near the ends. The

model is illustrated in Figure 2.



BINARY FLUTTER OF BRIDGE DECK SECTIONS 411

Alum inium brackets attached to the end plates were elastically hinged with cruciform

supporting frames by the use of flexure spring s. The entire system w as suspended by four

equal coil springs which were clamped at one end on the horizontal rod of each supporting

frame. Horizontal restraining wires of sufficient length were also attached to the top and

bottom of the vertical rod of each supporting frame. Thus, the model w as allowed to give

both vertical and torsional ‘motions with downstream and spanwise displacements prevented.The vertical displacement was detected with a contactless optical displacement follower,

whereas the torsional one was measured by the use of strain-gauges cemented on the surfaces

of the flexure springs.

In addition to the basic model, three simple deck section mod els were chosen in the present

experiment: that is, a flat plate with six equal-spacing stringers, and that with stringers and a

center barrier, both of a height to deck width ratio of U.075, and an H-section with a girder

to deck width ratio of 0.09. Any of these three m odels w as easily obtained by attaching seg-

ments of light polystyrene plate with negligible weights to the basic flat plate m odel.

The free oscillation experiment wa s divided into tw o parts. The first part of the experiment

was concerned with the model experiment : i.e., (1) both the vertical and torsional responses

1/ - -End p lo te

Figure 2. Flat plate m odel with end plates. All dimensions are in millimeters.

occurring simultaneously; (2) the torsional response occurring with the vertical response

prohibited at X. This resulted in curves of b and fil together with X versus J? The second part

of the experiment was concerned with the following responses : (1) the torsional response

with the vertical response prohibited at the midchord; (2) the vertical response with the

torsional response prohibited. The measurem ents of the frequency and growth rate of

oscillation yielded curves of CWBR, M ,, and CL4, versus v which are required for equations

(22), (26) and (27).

In the first experiment, the still-air frequencies were f, and f e 1.60 Hz and 3.20 Hz,

respectively, from which R = 0.50 was obtained. In addition, p and v = 150 and 100, respec-

tively, whereas the mech anical dam pings, either vertical or torsional, were of an order of

0.03. The experimental wind speed ranged from 5 m/s to 1 8 m/s, approximately. The system

used in the second part of the experiment was identical with that used in the first part of the

experiment except for the case of the H-section model where the lighter system was necessary

for the precise measurements of small frequency variations at low wind speeds. In most

cases, oscillations at small amp litudes were found to be of exp [;.t] type where 1 is a constant,

so that the measuremen ts of b were done at 8, = 3 degrees, approximately.

All the experiments were made w ith the model at zero mean incidence : i.e., with the deck

surface parallel to the incident flow. Change in the mean incidence during the tunnel runs

occurred due to the asymmetry of the flow caused by the presence of the stiffening trusses ,but it was small and neglected; the maximum change encountered was about 0.5 degree for

the A at plate model.



47 8 Y. NAKAMURA

6. EXPERIMENTAL RESULTS AN D DISCUSSIONS

Figures 3(a) to 3(c) show the results of the binary flutter experime nt for the flat plate, th e

flat plate with stringers and the H-section whe re the logarithmic rates of grow th of oscillation

/3 and /I1 togethe r with the position of the equivalent center of rotation X are plotted as

functions of the reduc ed wind velocity I?

0.5

I I I /

Cc)

0

I iP

..

AP,

nX

2.0

1. 0

4

-0.25

I 1 I I

I5 20 25 30

1 I 1 I

IO 15 20 25

Figure 3 .I ‘osition of equivalent center of rotation X and logarithm ic rates of growth B and jll us. reducedwind veloc itx . X is measu red from midchord (positive towards leading edg e); n, experimental; heavy line,theoretical: b represents the logarithmic rate of growth for binary system; 0, experimental; heavy line,theoretical: /31 represents the logarithmic rate of growth for torsional system with respect to X; 0, experi-mental. (a) Flat p late; (b) flat plate with stringers; (c) H-section.

0.5

0.25

a

F

: 0

-0.25

(0) ’I 1 1

Xa

.--_::

2.0

n

P

I.0

0%

0

25 30 35 40

vI I I I

(b)

According to Figure 3(a), the instability of the flat plate obviously belongs to the classical

type flutter. The instability of the flat plate w ith stringers in Figure 3(b) still belongs to the

classical type flutter, but the presenc e of the stringers results in an increase in PI and a decr ease

in the critical flutter speed . The H-section results in Figure 3(c) show that the critical flutter

speed is further decre ased, and becaus e /? N j?r over the entire velocity range tested, flutter isnow of the single degre e of freedo m type. How ever, there is also an indication that flutter a t

still higher wind sp eeds would tend to be of the classical type.




The results of measurement of the aerodynamic coefficients for all the four models are

presented in Figures 4(a) to 4(c). Figure 4(a) indicates that all the data for CL ,, roughly

collapse on a single curve exc ept for a sharp drop exhibited by the H-section results at low

wind speeds. The sharp drop suggests the proximity to the critical wind speed for vortex

excitation which is P= 7.0, approximately. Figures 4(b) and 4(c) show rather spectacular

variations of both C M,, and CM,, with Pfor the H-section, which appear to be characteristicof structures with sufficient bluffness [6] ; again, the effects of vortex shedding at around

P = 7.0 are evident.

Iv r

Figure 4. Experimental aerodynamic coefficients vs. reduced wind velocity. ??Flat plate; c , flat platewith stringers; X, flat plate with stringers and a center barrier; a, H-section.

7. EXPERIM ENTAL CHECK OF THE VALIDITY OF THE APPROXIMA TE

FORMULAS

The theoretical estimates for j? and A’, which wer e obtained by applying equations (22),

(26) and (27), are presented as heavy lines in Figures 3(a) to 3(c); the theoretical estimates for

/? include allowance for the experimen tal mechanical damping of O-03 . The agreem ent

between theory and experiment is reasonably good through all the three figures for both /I

and X except for the case of /3 in Figure 3(b) where there is a slight discrepancy due to someunknown reasons. It is wor th pointing out that the present formu las are valid not only for

the classical type flutter (Figure 3(a)) , but also for the single degree of freed om type (Figure



48 0 Y. NAKAMURA

3(c)). In other w ords, the validity of the formulas does not depend on the type of flutter. Note

also that they are valid for any of the three oscillations (decaying, steady state and grow ing);

in particular, this supports the validity of the aforementioned assumption that the steady

state oscillation aerodynam ic coefficients may be replaced by those corresponding to decaying

or growing oscillation, and vice versa.

It was shown [9] that a center barrier is sometimes effective in augmen ting the aeroelasticstability of a bridge deck section. The present inv estigation provides a good example for this

favourable effect of a center barrier. F or example, compare the aerodynam ic coefficients

I I / I

20 25 30 35

2.0

I.0

4

0

Figure 5. Position of equivalent center of rotation and logarithmic rates of growth vs. reduced wind velocityfor a flat plate w ith stringers and a center barrier. See Figure 4 for key to symbols.

between the flat plate with stringers and that with stringers an d a center barrier, which are

shown in Figures 4(a) to 4(c). It is obvious that the attachment of a center barrier yields a

remarkable increase in pitch damping, whereas the rest of the aerodynamic coefficients

indicate little change. It follows from equation (27) that this should result in a considerable

increase in the critical flutter speed. That this prediction is correct is show n in Figure 5,

where, again, the agreement between theory a nd experiment is reasonably good.

8. SOME REMARKS ON THE CHARAC TERISTICS OF BRIDGE DECK FLUTTER

It may be worth mentioning the following po ints that characterize bridge deck flutter in

comparison with the aircraft counterpart. First, the pitch damp ing can directly control the

stability of a bridge deck section even for the case of the classical type flutter. In sharp contrast

to this, the in-phase aerodynamic coefficients have dominant contributions in most aircraft

flutter [lo, 111 . In other w ords, the critical flutter speed for an aircraft wing is often de ter-

mined, though approx imately, by applying the so-called frequency coalescence criterion

which states that the frequencies of the bending and torsional branches become coalescedas the critical flutter speed is approached. Because of the presence of CM,, in equation (27),

however, the phenomen on of frequency coalescence is not found in bridge deck flutter.




According to Zimm erman [lo], the frequency coalescence criterion can be applied to a

short-time explosive flutter w here the variation of the phase angle becomes steep near the

critical flutter point. The classical type flutter of bridge deck sections, however, appears to

be rather mild, as exemplified by Figure 3(a) and also because the phase variation can be

assumed small. Another point of interest is the weakness of the couplings of equations (22)

to (25) in bridge deck f lutter.,Again , this is not the case for aircraft flutter; one then finds thatthe equations for r~and X, and also those for b and 4 , are both strongly coupled.

A close examination of the equations of motion reveals that all the above-mentioned

characteristics of bridge deck flutter may be attributed to the absence of the inertial coupling.

A numerical experiment for a flat plate with an uncoupled frequency ratio of 0.5 was designed

I.0

0

$=0.75

0.5

1X6/b = 0. I

I p.2

T;;;-1

0

040.2 0.3

0.3

0

/ I 1

0.5 0.75 I.0 1.25

Figure 6. Effect of the position of center of gravity X,/b on the natural mode frequencies for a flat plate.X,/b is measu red from midchord; nume rical exp eriment is based on Theodorsen unsteady aerofoil theory.

to demonstrate the effect of the inertial coupling on the variations of the natu ral mode

frequencies with wind speed, where the position of the center of gravity, X ,, m easured from

the midchord, is varied for 0 < X,/t, < 0.3. The analysis w as based on the Theodorsen un-

steady airfoil theory and followed the method of Goland and Luke [12], where the aero-

elastic m odes of exp [At] type, in which 1 is a complex number, were assumed, and the Wagner

“lift deficiency” function was utilized to compute the unsteady aerodynam ic forces corres-

ponding to the exp [At] type motion. The results are shown in Figure 6, which indicates thatas X, /b increases from zero the frequencies of the two branches become closer at around the

critical flutter speed V ,, so that the applicability of the frequency coalescence criterion be-

comes restored.

9. CONCLUDING REMARKS

A set of analytical formu las applied to binary flutter of bridge deck sections has been

presented in this paper . The frequency and rate of growth of oscillation together with the

position of the equivalen t center of rotation near the critical flutter point are obtained by

solving three simp le equation s where the aerodynam ic coefficients involved are only three in

number. The rate of growth of oscillation consists of two terms of physically different origins;one represents the rate of growth of oscillation corresponding to a system in a single degree

of freedom in torsion abou t the equivalen t center of rotation, and the other is that propor-



482 Y. NAKAMURA

tional to sin 4, whe re 4 is the phas e ang le between bending and torsion. A comp arison of

which of these two is predominant provides an analytical basis for the previously prop osed

meth od of classification of binary flutter o f bluff structures. The simplicity of the formu las

may be attributed to the absence of the inertial coupling in bridge deck flutter. The assum p-

tions on which the present formulas are based are not seriously restrictive, and the formu las

are expe cted to be applicable to a variety of structures ranging from a flat plate to muchmore bluff bridge deck sections.

ACKNOWLEDGMENT

The author is indebted to Messrs K. Watanabe and T. Yoshimura for their help in con-

ducting the experimen tal wor k.

REFERENCES

1. R. H. SCANL AN and J . J . TOMKO 1971 Journal of t he Engi neeri ng M echanics Di vi sion, Proceedi ngs

of the Ameri can Societ y of Civ i l Engineers 97, EM6, 1717-1737. Airfoil and bridge deck flutterderivatives.

2. Y. NAKAMURA and T. YOSHI MU RA 1976 Journal of the Engineeri ng M echanics Di vi sion, Pro-

ceedi ngs of the Ameri can Society of Civ i l Engineers 102, EM4 ,685-700. Binary flutter of suspen-

sion bridge deck sections.

3. N. SHIRAISHIand K. OGAWA 1975 Proceedi ngs of the Japan Society of Ci vi l Engineers 244,

23-35. An investigation on aeroelastic respon ses of suspension bridges d ue to the non-linear

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characte ristics of the aerodynam ic forces in self-excited o scillations of bluff structures.

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Society of Civ i l Engineers 97, ST1 1, 2653-2678. Newport bridge superstructure.10. N. H. ZIMMERMANand J . T. WEISSENB UR GE R 964 Journal of Ai rcraft, Ameri can Insti tut e of

Aeronautics and Astronautics 1, 190-2 02. Prediction of flutter onset spee d based on flighttesting at subcritical speed s.

11. G. T. S. DONE 1969 Aeronaut i cal Research Council , Report s and M emoranda No. 3554. A studyof binary flutter roo ts using a meth od of system syn thesis.

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An Analysis of Binary Flutter of Bridge Deck Section

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