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7AD-A130-476 AERDELASTIC RESPONSE OF
AN AIRCRAFT WING TO RANDOM
ESTABLSHMENT OTAWA ONARD)HIGH SPE.B
HLEE
UNCASSFIED APR83 NAE-LR-613NRC21230 F/G20/4
IhmhhmhINDu
11"1 I.,11,800
1.211 . 11.6
MICROCOPY RESOLUTION TEST CHARTNATIONAL BUREAU OF STANDARDS- I963-A
I * National Research Conseil nationalCouncil Canada de, 1ehrce Canada
AEROELASTIC RESPONSEOF AN AIRCRAFT WING
TO RANDOM LOADS
byB.H.K. Lee
National- Aeronautical EstablishmentC.3, OTIC
OTTAWA ELECTEAPRIL1983 JUL 4198
U83 07 14 064 aAERONAUTICAL REPORT
LR413Ca~nRO .,NRC NO. 21230
docu &m..t h"a komiq,A m PO uue Ji.
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Section des publicationsConseil national de recherches Canadattabliusent asronautique nationalIm. M-16, pl~ce 204Chemin de Montr6alOttawa (Ontario)KIA 0R6
AEROELASTIC RESPONSE OF AN AIRCRAFT WING TO RANDOM LOADS
REPONSE AEROELASTIQUE D'UNE AXLE D'AERONEF ADES CHLARGES ALEATOIRES
by/par
H.H.K. Lee
C.3
H~a Di Amm LeI pnnnC r.Ics md
diss[bistribution/.4 f oijviablt Code
- ~ ~~~ 8L or .- 2*~
SUMMARY
A method for the prediction of the response of an elastic wing torandom loads at flight conditions using rigid model wind tunnel pressurefluctuation measurements is presented. The wing is divided into panels orelements, and the load is computed from measured pressure fluctuations atthe centre of each panel. A series representation with terms of the correlatednoise type is used to curve fit the experimentally determined pressure powerspectra. Two methods are used to calculate the random load spectrum: aconstant correlation approximation, and an exponential spatially decayingcross-power spectrum model for the pressure. The coupling between thestructural dynamics and aerodynamics of a vibrating wing is taken intoconsideration using the doublet-lattice method for computing the unsteadyaerodynamic forces. The acceleration and displacement response spectrahave been computed for the F-4E aircraft for various Mach numbers,dynamic pressures and flight altitudes. The importance of the unsteady aero-dynamic loads induced by the vibration of the wing and input load repre-sentation is illustrated by comparing the theoretical predictions with resultsfrom flight tests.
REkSUMEk
II s'agit d'une m~thode pour pr~dire la rhponse d'une aile 6astiquei des charges al6atoires en vol i partir des mesures de fluctuations depression sur une maquette rigide en soufflerie. L'aile est divie~e en panneauxou 61iments, et la charge est calculhe i partir des mesures de fluctuations depression prises au centre de chaque panneau. Une repr~sentation adrie destermes du type de bruit en corrdlation sert i tracer la courbe des spectresde puissance pression dhterminhs exp6rimentalement. Deux m6thodes sontemployhes pour calculer le spectre des charges alhatoires: l'approximationpar corrdlation des constantes et un module exponentiel du spectre de lapuissance d6croissante dans l'espace pour la pressaon. La relation entre ladynamique structurale et l'ahrodynamique d'une aile en vibration est priseen compte au moyen de la m6thode de rdaeau des doublets pour calculerlea forces a6rodynamiques variables. Lea spectres de rdponse en acchlhrationet en dhplacement ont 6th calculhs pour le chasseur F-4E A divers nombresde Mach, diverses pressions dynamiques et altitudes de vol. L'importancedes charges ahrodynamiques variables induites par la vibration de l'aile et larepr sentation des charges exerches est ddmontrhe par la comparaison desprxdiction th6oriques avec les rhsultats des vols d'euai.
(il)
CONTENTS
Page
SUMMARY ........................................................... (ii)
SYM BOLS ............................................................ (vi)
1.0 INTRODUCTION ...................................................... 1
2.0 ANALYSIS ........................................................... 1
2.1 Dynamic Aeroelastic Equations ......................................... 12.2 Representation of External Load on Wing ................................. 32.3 The Doublet-Lattice Method ........................................... 42.4 Coupling between Structural Dynamics and Unsteady Aerodynamics for a
Vibrating Wing ................................................... 5
3.0 RESULTS AND DISCUSSIONS ............................................ 6
4.0 CONCLUSIONS ........................................................ 9
5.0 REFERENCES ......................................................... 9
APPENDIX A Expressions for K1 and K2 in the Equation for Normalwash ....... 47
APPENDIX B Expressions for Normalwash Factor .......................... 49
ILLUSTRATIONS
Figurepa
1 Schematic of Wing .................................................. 13
2 Panel Representation of Wing ......................................... 14
3 Representation of Wing by Panels and Location of Vortices, Doublets andCollocation Points ................................................ 15
4 Wing and Element Co-Ordinates ....................................... 16
5 Block Diagram for Wing Response to Random Loading ..................... 17
6 Panel Representation of F-4E Wing Planform ............................. 18
7 Approximate Modelling of RMS Pressure Coefficient on Wing ................ 19
8 Pressure Transducer Locations ......................................... 20
(iv)
-1.;,;--
ILLUSTRATIONS (Cont'd)
Figure Page
9 Comparison of Response Acceleration Power Spectral Density from Flight Testswith Theoretical Predictions Using Constant Corrlation Assumption atMO- .78mandQ -22Opud........................................... 21
10 Comparison of Response Acceleration Power Spectral Density from Flight Testswith Theoretical Predictions Using Constant Correlation Assumption atM -0.82 andQ =35Opdf........................................... 22
11 Variation of Acceleration Power Spectral Density with Mach Number andFlight Altitude for Q =220 pd...................................... 23
12 Variation of Displacement Power Spectral Density with Mach Number andFlight Altitude forQ =220 ps...................................... 24
13 Variation of Acceleration Power Spectral Density with Mach Number andFlight Altitude for Q = 350 psf...................................... 25
14 Variation of Displacement Power Spectral Density with Mach Number andFlight Altitude for Q = 350 psf...................................... 26
15 Variation of Acceleration Power Spectral Density with Dynamic Pressure andFlight Altitude for M = 0.51......................................... 27
16 Variation of Displacement Power Spectral Density with Dynamic Pressure andFlight Altitude for M - 0.51......................................... 28
17 Variation of Acceleration Power Spectral Density with Dynamic Pressure andFlight Altitude for M= 0.78......................................... 29
i8 Variation of Displacement Power Spectral Density with Dynamic Pressure andFlight Altitude for M = 0.78......................................... 30
19 Variation of Acceleration Power Spectral Density Normalized with Respect toQ2 with Dynamic Pressure and Flight Altitude for M = 0.51 .................. 31
20 Variation of Displacement Power Spectral Density Normalized with Respect toQ2 with Dynamic Pramre and Flight Altitude for M = 0.51 .................. 32
21 Variation of Accelerationl Power Spectral Density Normalized with Respect toQ2 with Dynamic Pressure and Flight Altitude for M = 0.78 .................. 33
22 Variation of Displacement Power Spectral Density Normalized with Respect toQ2 with Dynamic Pressure and Flight Altitude for M - 0.78 .................. 34
23 Variation of Acceleration Power Spectral Density with Mach Number andDynamic Pressure at 15,000 ft. Altitude................................ 35
24 Variation of Displacement Power Spectral Density with Mach Number andDynamic Pressure at 15,000 ft. Altitude................................ 36
(v)
ILLUSTRATIONS (Cont'd)
Figure Page
25 Variation of Acceleration Power Spectral Density with Structural Dampingfor M-O0.78,Q -220psf........................................... 37
26 Variation of Displacement Power Spectral Density with Structural Dampingfor M -0.78, Q -22Opdf........................................... 38
27 Variation of Acceleration Power Spectral Density with Structural Dampingfor M=O0.82, Q =350psf........................................... 39
28 Variation of Displacement Power Spectral Density with Structural DampingforM =0.82,Q =350 psf........................................... 40
29 Comparison of Response Acceleration Power Spectral Density from FlightTests with Theoretical Predictions Using Exponential Spatially DecayingPressure Cross-Spectrum Assumption for M = 0.78, Q = 220 pf ................ 41
30 Displacement Power Spectral Density Using Exponential Spatially DecayingPressure Cross-Spectrum Assumption for M = 0.78, Q = 220 pf ................ 42
31 Comparison of Response Acceleration Power Spectral Density from FlightTests with Theoretical Predictions Using Exponential Spatially DecayingPressure Cross-Spectrum Assumption for M = 0.82, Q = 350 psf ............... 43
32 Displacement Power Spectral Density Using Exponential Spatially DecayingPressure Cross-Spectrum Assumption for M = 0.82, Q = 350 psf ............... 44
33 Comparison of Response Acceleration Power Spectral Density from FlightTests with Theretical Predictions for M = 0.78, Q = 220 psf .................. 45
34 Comparison of Response Acceleration Power Spectral Density from FlightTests with Theoretical Predictions for M = 0.82, Q = 350 psf ................. 46
B1 Horseshoe Vortex System............................................. 52
SYMBOLS
Symbol Definition
A area of a panel
b wing semi-span
C generalized damping coefficient
average chord
D normalwash factor
NOi
SYMBOLS (Cont'd)
Syrbol Definition
D 0 steady normalwash factor
DI planar normalwash factor
D2 non-planar normalwash factor
E defined in Equation (38)
e semiwidth of panel
f frequency
H frequency response function
I defined in Equation (19)
K defined in Equation (35), also generalized stiffness
Kt ,K2 defined in Equations (Al) and (A2) respectively
k reduced frequency
L Fourier transform of generalized load
Qgeneralized load
M Mach number
P Fourier transform of pressure
p pressure
Q Fourier transform of displacement in generalized co-ordinates, also dynamicpressure
q generalized co-ordinate
r distance between centres of two panels
SL load spectrum
pressure spectrum
SQ spectrum of displacement in generalized co-ordinates
T1 , T2 defined in Equations (23) and (24) respectively
t time
U free stream velocity
(vil)
SYMBOLS (Cont'd)
Symbol Definition
Uc Cconvection velocity
w normalwash
W Fourier transform of normawash
x,y,z Cartesian co-ordinate system
X panel dimension in xi-direction, also frequency parameter defined inEquation (41)
Ax 3 chord of sending panel
Y dimension of panel in y-direction
z displacement in z-direction
adecay coefficient in xi-direction
Pdecay coefficient in y-direction
-y dihedral angle of sending and receiving point
6 boundary layer thickness
damping ratio
'co-ordinate (in y-direction)
X, sweep angle of panel quarter-chord
co-ordinate (in x-direction)
p density
0mode shape function
Wo frequency
Wn natural frequency
Subcripts
L denotes applied load
denotes receiving and sending point
denotes the k-th panel
a, denotes the a and mode of vibration
(vlil).4 ___________
SYMBOLS (Cont'd)
Symbol Defintion
Supedosuipte
denotes the k-th panel
'r' denotes the r-th panel
denotes random component of load due to wing buffeting or turbulence
'A' denotes unsteady aerodynamic component of load due to vibration of wing
denotes non-dimensional quantities
7-1•L r=
-1-
AEROELASTIC RESPONSE OF AN AIRCRAFT WING TO RANDOM LOADS
1.0 INTRODUCTION
In References 1 and 2 the dynamic response of a wing to statistical non-stationary randomloads is investigated using a time-segmentation technique. The time history of the applied load issegmented into a number of time intervals. In each time segment the non-stationary load is repre-sented by the product of a deterministic shaping function and a statistically stationary random func-tion. The total response is obtained by summation of the responses of the wing from each timesegment. This approach is useful when the aircraft engages in rapid manoeuvres. Because of the com-plexities of the analysis only the structural response, without consideration of the unsteady aero-dynamics around the vibrating wing, is analyzed. It is assumed that aerodynamic effects can beapproximately accounted for by using a damping value in the dynamic aerolastic equations that isdetermined either experimentally or from estimation.
In order to make the problem of the coupling between structural dynamics and aero-dynamics of a vibrating wing more amenable to theoretical analysis, the random loading on the wingis assumed to be statistically stationary. This assumption can be justified as long as the flight condi-tions are unchanged within a reasonably long analysis time, when measurements are made of eitherthe load or the wing response. In wind tunnel testing statistical stationarity requires long runningtime compared to the natural period of the mode of vibration under investigation. The random loadsexperienced by a wing of an aircraft arise from turbulence or flow separation at buffet conditions.The use of statistical theory in the investigation of buffeting was pioneered by Liepman (R,.f. 3) whoexamined the problem of the lift force exerted on a two-dimensional thin airfoil moving in turbulentair. Later, he extended the method to wings of finite span (Ref. 4). The analysis was generalized byRibner (Ref. 5) using a model of turbulence represented by the superposition of plane sinusoidalshear waves of all arientations and wavelengths. The correlations between flight and wind tunnel testswith predictions based on statistical approach had been reported by various investigators (Refs. 6-10).Amongst the more recent studies in this subject are those of Mullan and Lemley (Ref. 11), Hwangand Pi (Refs. 12, 13), Jones (Ref. 14) and Butler and Spavins (Ref. 15). All the above studies consideronly the structural response without including the effects of the unsteady aerodynamics around thevibrating wing. In s report, the method of Reference 2 is extended to include the coupling betweenthe structural vibration and aerodynamics of a wing under random loading. The panel method de-scribed in Reference 2, which is a modification of a method proposed by Schweiker and Davis(Ref. 16) for the study of the response of shells to aerodynamic noise, is most suitable for adaptingthe doublet-lattice technique (Refs. 17-19) in computing the unsteady aerodynamics.
2.0 ANALYSIS
2.1 Dynamic Aeroelastic Equations
Consider a Cartesian co-ordinate system x, y and z fixed on the wing as illustrated inFigure 1. The displacement of the wing can be expressed in terms of a set of normal co-ordinatesq, (t) as:
z(xy,t) = 1 ,(x,y)qa(t) (1)a1
where 0. (xy) is the mode shape function of the ath mode, and I is the number of modes required toadequately represent z(xyt) in the form of a series. The dynamic aeroelastic equations governingthe response of q.(t) to a load 9. (t) is given in generalized co-ordinate as:
M. iL + C. q6 + Ka = Q(t) (2)
-2.
for the ath mode. The dots denotes differentiation with respect to time, Ma, C, and K, are. thegeneralized mass, damping coefficient and stiffness of the ath mode respectively. In the above equa-tion the assumption of light damping is made, thus cross damping terms do not appear.
Define the undamped natural frequency wn. as:
KW 2 = 3na M.
and the damping ratio a as:
C
= M.K (4)
Using the above two expressions Equation (2) can be rewritten as follows:
46 + W2 = --I t(5)
The generalized force can be expressed in terms of the fluctuating pressure p(x,y,t) as:
Q (t) = ff 0 (x,y)p(xy,t)dxdy (6)wing
planform
where the integration is taken over the wing surface. Let La(wa), Q0 (wA) and P(xy,w) be the Fouriertransform of R (t), q. (t) and p(xyt) respectively, and introduce a frequency response functionHa (w) defined by:
1Hc (w) = (7)M. [W2a W2+ i2 a nw
Equation (5) can be written as follows:
Qa(w) = Ha(w)L0 (w) (8)
In the term 2, (t) there are two types of forces acting on the surface of the wing. Let the
component of the load due to buffeting or turbulence be represented by RD (t) and that from the
unsteady aerodynamic forces due to vibration of the wing be QA (t), then
Qa(t) = RD (t) + RA (t) (9)
Similarly, the pressure can be expressed as
p(x,,t) = pD (x,y,t) + pA (x~V,t) (10)
and Equation (10) becomes
Q -(w) Ha(w) (L()+)+ LA(W)) (11)S
-3.
where LD (M) and L. (M) are the Fourier transforms of QD (t) and Q. (t) respectively.
2.2 Representation of External Load on Wing
From Equation (6), the generalized force due to the random loading can be expressed in
terms of the pressure pD (xy,t) as follows:
2 D (t) = ff * (xy)pD (xy,t)dxdy (12)wing
plan form
The power spectral density can be written as:
S =(w) ffff Y (xilyi) (x2 'y2 )S PD (x) tyl 9x2' Y2 'W) dX ,dy, ,dx2 ,dy 2 (13)
where S p (x 1 , ,x2 IY2 ,W) is the cross power spectral density of the pressure pD (x,y,t) between
points (x I yI ) and (x2 ,Y2 ) (Fig. 1).
In Figure 2 a panel representation of the wing is shown. From Reference 2, SL D (W) can bewritten as: a
K 2
S~~~ D fCj =S;(4L k = 1 K D ( xlyl ,X 2 ,Y2 ,)dAk dAk (14)
where Ak is the area of the k-th panel and 0k is an average mode shape for the 'a' mode within the
k-th panel defined by1
= - f 0,, (xy)dAk (15)
When the distance rke (Fig. 2) between the centres of the k-th and -th panels is large (Ref. 2), an
approximate form for S D (w) at very low vibration frequencies can be obtained by assuming the
pressure at the centre of a panel to be representative of the pressure field at all points within thatpanel. The pressure spectrum at any point is given by that at the centre of the panel and the pressurecross spectrum between two points in the same panel is taken to be equal to the pressure spectrum atthe panel centre. Equation (14) can then be simplified to the following:
L k () k ) A2 (16)aL k I p
where SkD (Wo) is the pressure spectrum at the centre of the k-th panel. A more accurate expressionp
for S D (O) can be obtained by writing S pD(xl y Ix 2 Y 2 ' W) as follows:La
-Ol*,-X=l -Oly,-Y2IS D (X lyl ,X2 ,y2 ,W) S9kD (w) e e cosp(xo - x 2 ) (17)
p p
-4-
where a and P are spatial decay coefficients in the x and y directions respectively, p - and UC isU'
the convection velocity. Substituting Equation (17) into Equation (16) and carrying out the integra-tion gives the following:
S D(w) ffi SkD (w "-k 2 ik (18)La k= 1 p D a
where
ik = -4 (e- +Yk - +p ) 2 -2 +aXk(C2 +p 2 )+e-0Xk [(02_ p2CoS pXk_ 2asinpXk]}
(19)
and Xk and yk are the dimensions of the k-th panel (Fig. 2).
2.3 The Doublet-Lattice Method
The doublet-lattice method is described in References 17-19, and in this report, only abrief outline of the method and the equations needed in the present analysis will be given.
The wing surface is divided into small elements or panels arranged in strips parallel to thefree stream and all surface edges, fold lines and hinge lines lie on the panel boundaries (Fig. 3). Ineach panel the steady flow is represented by a horseshoe vortex having the bound vortex of the horse-shoe system lying along the quarter-chord line. Superimposed on the bound vortex are accelerationpotential doublets of uniform strength. The control point is located at the three-quarter chord point. Thenormalwash w, at panel 'r' non-dimensionalized with respect to the free stream velocity U is givenby the following:
wr = 2 DrsAC (20)
where Ds is the normalwash factor, and the summation is over all the panels. In a co-ordinate system
having the midpoint of the sending element as origin and the axes aligned with the sending element(Fig. 4), the normalwash factor can be written as (Ref. 20):
Axs f/KT 1 K2 T2 .D x S (K T____ iw' -tanX )/U] d 21rs 8 -e 2 expL-iw(i- (21)
where
r,= [(_)2 + -2] (22)
Ax1 is the chord of the sending panel (Fig. 3), X is the sweep angle of its quarter-chord, and e is itssemi-width. T1 and T2 are the direction cosine functions given as:
T, = coS(ys - Y') (23)
T2 - izcos(y, - -) + ( -)sin(y - -1,)] (24)
where -1 and y are the dihedral angles at the sending and receiving points on the respective panels.K andk2 are given in Appendix A. Equation (21) can be written in the following form:
jI
.5-
Drs= Dor s + D1 rs + D2rs (25)
where D0 s, Dlr s and D2r s are the steady, planar and non-planar normalwash factors respectively,
given in Appendix B.
The normalwash boundary conditions for the lifting surface elements are determined byassuming the surfaces to move normal to themselves. Taking the substantial derivative of Equation (1),the normalwash at panel 'r' is:
W(t) Z Z w'. ) (26)
and w, (t) = at + z (x,y,t) (27)
(L)Cwhere k = - (28)U
i is the average chord and the ' denotes non-dimensional quantities. The Fourier transform of Equa-tion (27) gives:
WO ) + ik & Q'.(W) (29)
2.4 Coupling between Structural Dynamics and Unsteady Aerodynamics for a Vibrating Wing
The external loading on the wing due either to buffeting or turbulence causes vibration ofthe wing which in turn generates unsteady aerodynamic forces on the wing surface. Figure 5 shows ablock diagram illustrating the proceuure in calculating the response of a wing to random loading with
the aerodynamics included in a feedback loop. The aerodynamic load LA (w) can be written as:
LA (co) = ff 0.(xy)P(xy,w)dxdy (30)winplanfotto
where P(x,,co) = I PO(x y,w)QO(W) (31)
P(x,y,w) is the Fourier transform of pA (Xy,t) in Equation (10). Evaluating Equation (30) over each
panel and summing over all panels, LA (co) becomes:
A 1 I k kLa (o) =f pU 2 k a A OCk AkQP(co) (32)
where 'denotes non-dimensional quantities and AC is the pressure coefficient.
The power spectrum S A (co) of the aerodynamic load can be written as:
(iPUY E X ). A (w)AC (W) A (33)
-6-
where S., (w) is the power spectral density of Q,(w) and * denotes the complex conjugate.
In non-dimensional form, Equation (8) can be re-written as:
Q'.H(w)- K.H:(w) L.'A(() + L'D(w) (34)
where I p U2K ° 2f (35)2 M a fna
and H'(w) 1 + i2r -- (36)
The primed quantities for the loads are non-dimensional with respect to p UE2 . Using Equa
tion (32) an expression for SQ, (Mo) can be obtained from the following:
a
where
E K. 1; jk ACk p (w)Ak (38)
To determine ACkp (w) combine Equations (20), (29) and (31) and obtain
akof + ik Z D AC(w) (39)
ax S s kS
where k replaced #k in Equation (27). Hence, for known Mach number and reduced frequency,
Dk, can be obtained from the equations in the previous section for a given wing geometry. The mode
shape factor can be determined from structural calculations. With ACp' known S., (w) can be
obtained from Equation (37), and hence the displacement and acceleration spectra.
3.0 RESULTS AND DISCUSSIONS
The steps involved in computing the response of the wing are fairly straightforward oncethe necessary aeroelastic data are given. As an example, the response of a F-4E wing is studied usingstructural and experimental data supplied by Mullana and Lemley (Ref. 11). Figure 6 is taken fromReference 11 and shows the choice of panel dimensions on the wing. A total of eighteen panels areused in the computations, and the panel centres are marked numerically from 1 to 18.
11 ___ ____ _____
S
.7-
The first ten symmetrical and antisymmetric modes are considered in this example. Thenatural frequencies wn, generalized mass M,, average mode shape ik and damping ratio t, for three
dynamic pressures Q - 220, 350 and 470 pef are tabulated in Mullans' and Lemley's report (Ref. 11).They also obtained buffet data from wind tunnel tests on a 10% scale rigid three-dimensional model.For a clean wing with zero leading edge and trailing edge deflections, the rms pressure coefficientsAC prm on the wing are shown in Figure 7. In this particular example, AC varies practically~rm SPfml
linearly along both constant chord and constant span directions.
The power spectral densities of the pressure fluctuations at various points on the wing forMach number 0.7 and angle of incidence 120 are also given in Reference 11. The transducers are notlocated exactly at the panel centres and there may be more than one transducer per panel or in somecases, none at all. Figure 8 indicates the transducers (numbers correspond to these given in Ref. 11)whose measured outputs are used to represent the fluctuating pressures on the panels where they arelocated.
Reference 11 also gives the mode shape k for the symmetric and antisymmetric modes.
To obtain -- , a cubic spline algorithm is used in the present study. For given Mach number and
reduced frequency, the normalwash factor Dks in Equation (39) can be obtained using the analysisof Section 2.3. Upon substitution in Equation (38), Equation (37) can be solved for SQ, (w) for
aknown power spectrum of the external random load SLD (w). The spectra of the pressure fluctua-
tionsat each of the 18 panels are taken from Mullans and Lemley for q = 470 psf, M = 0.7 and a = 120.For other flow conditions, it is assumed that the spectra remain the same. Following Reference 2 theexperimentally determined power spectral density curves from rigid model wind tunnel tests areexpressed in series form and the coefficients in the series are obtained by a curve fitting procedure.Using proper scaling (Ref. 11) the pressure spectra at flight conditions can be obtained from windtunnel tests.
The spectrum of the generalized force due to the random load, (Eq. (18)) can be calculatedknowing the decay coefficients a and 0 and also the convection velocity Uc. Data for a and ( fromflight tests are extremely scarce. In order to show qualitatively the effect of a decaying pressure fieldand the difference in the response spectrum as compared to that for constant correlation assumptionusing Equation (16), the data from Coe et al. (Ref. 22) are used. It is assumed in this investigationthat (3 =, and a is determined from measurements of surface pressure fluctuations in a turbulentboundary layer. The decay coefficient is expressed in the form
at6 - A (T.) (40)
where 6 is the boundary layer thickness, A and B are constants and
f8X = (41)U
f is the frequency. From Reference 22 A and B are taken to be 0.2 and .3891 respectively forseparated flow, and X0 M 0.001. To calculate the boundary layer thickness, the formula given byBies (Ref. 23) is used, that is,
r0.37 R ' /'[ i111 (42)x XLI
The Reynolds Number R. is calculated based on midspan chord. Taking x to be equal to the
midapan chord. The value of 6 thus calculated when substituted into Equation (40) gives an approxi-mate average value for a which is assumed to be constant at all the panels. It is a function of Machnumber, frequency and flight altitude only.
Figures 9 and 10 show a comparison of the response acceleration power spectral densityfrom flight tests (Ref. 11) with theoretical predictions at M = 0.78, Q =220 psf and M -0.82,Q = 350 psI respectively. The location of the accelerometer is at 84% semnispan and 26% chord. Theconstant correlation assumption using Equation (16) for the input load is applied to all the panels.Two sets of damping values are used: a structural damping ratio of 0.05 for all modes, and dampingratios determined experimentally from flight tests (Ref. 11). The ten symmetric and antisymmetricmodes have been considered in the computations. It is seen from these two figures that at low fre-quencies below 25 Hz the theoretical predictions are fairly good. However, at higher frequencies, thepredicted values are nearly tenfold the experimental flight test results.
The effect of altitude and Mach number on the acceleration and displacement spectra atconstant dynamic pressure Q are shown in Figures 11 to 14 for an assumed value of 0.05 for thestructural damping ratio. At Q = 220 psf, the response at low frequencies does not depend toostrongly on Mach number and altitude, while at the higher frequency range of the spectra the effectof Mach number can be seen. It is of interest to note from these figures that the response decreasesfor increasing Mach number and altitude at low frequencies around 10 liz while at higher frequenciesthe reverse is true.
Figures 15 to 18 show the response at two Mach numbers M - 0.51 and 0.78 for variousvalues of Q and altitude. They all show that the response is strongly dependent on Q. Dividing theresponse by Q2, Figures 19 and 20 show that the curves can roughly be collapsed into a single curveindicating a Q2 dependence. However, at M = 0.78, a similar behaviour is not observed.
At constant altitude of 15,000 ft., the variation of the response with Mach number and Qare shown in Figures 23 and 24. As expected, the response increases with Mach number and Qthroughout the whole frequency spectrum. The effect of varying the structural damping is shown inFigures 25 to 28 for M = 0.78, Q = 220 psf and M =0.82, Q = 350 respectively.
To obtain better agreement between theoretical and experimental results than those shownin Figures 9 and 10 at the higher frequencies, Equation (18) is used to represent the input loadspectrum using values of decay coefficients derived from Equations (40) to (42) and a value of theconvection velocity UC 0.80U. These results are shown in Figures 29 to 32. From the accelerationspectra at M - 0.78 and 0.82 (Figs. 29 and 31), it can be seen that the high frequency range of thespectra has been much improved and agrees well with flight test data. However, at low frequencies,the constant correlation assumption seems to give better agreement. This is probably due to theinaccuracies of the decay coefficients obtained by extrapolation of data obtained at much higherfrequencies.
Figures 33 and 34 also show comparisons between flight test data and theoretical predic-tions. These figures are prepared by combining Fgures 9and0 wth Fgures 29 and 31for M -0.78,Q -270 psI and M -0.82, Q - 350 pef respectively. The predictions from Reference 11 which con-sidered only the structural response are also included to show the effect of unsteady aerodynamics inthe response calculations. At low frequencies below 15 Hz, improvements over predictions whichconsidered only the structural dynamics of the wing are quite noticeab". However, at the higherfrequency range of the spectra, unsteady aerodynamic effects become less important.
.9-
4.0 CONCLUSIONS
A panel method for predicting the acceleration and displacement response of a wing torandom loading has been developed taking into consideration the coupling between the structuraldynamics and aerodynamics of a vibrating wing. The doublet-lattice method is found to be mostsuitable in the present panelling scheme to compute the unsteady aerodynamic forces.
Using available structural data for the F-4E wing together with rigid model wind tunnelfluctuating pressure measurements, the displacement and acceleration response have been computedfor various Mach numbers, dynamic pressures and flight altitudes. The results indicate that theresponse is most affected by changes in dynamic pressure and Mach number.
At low frequencies, improvements in the response predictions over those which consideredonly the structural dynamics but excluded the unsteady aerodynamic forces generated by thevibrating wing are quite significant. However, at the higher frequency range of the spectra, unsteadyaerodynamic effects become less important.
Two methods have been used to represent the spectrum of the input random load. Usingthe constant correlation assumption, comparison of the computed acceleration power spectra withthose from flight tests shows that at low frequencies the agreement is fairly good, but at high fre-quencies, the predicted values are nearly tenfolds the experimental results. Much better agreementat the higher frequencies is obtained when the exponential spatially decaying form of the cross-powerspectral density for the fluctuating pressures is used. However, this results in discrepancies at lowfrequencies which are probably due to the extrapolation of the decay coefficients from data obtainedat much higher frequencies. The results illustrate the importance of a proper representation of theinput load spectrum.
5.0 REFERENCES
1. Lee, B.H.K. Theoretical Analysis of the Transient Response of a Wing to Non-Stationary Buffet Loads.NRC, NAE Aero. Rept. LR-597, National Research Council Canada,April 1979.
2. Lee, B.H.K. A Method for the Prediction of Wing Response to Non-Stationary BuffetLoads.NRC, NAE Aero. Rept. LR-601, National Research Council Canada,July 1980.
3. Liepman, H.W. On the Application of Statistical Concepts to the Buffeting Problems.Journal Aeronautical Sciences, Vol. 19, No. 12, December 1952,pp. 793-800.
4. Liepman, H.W. Extension of the Statistical Approach to Buffeting and Gust Responseof Wings of Finite Span.Journal Aeronautical Sciences, Vol. 22, No. 3, March 1955, pp. 197-200.
5. Ribner, H.S. Spectral Theory of Buffeting and Gust Response: Unification endExtension.Journal Aeronautical Sciences, Vol. 23, No. 12, December 1956,pp. 1075-1077.
6. Huston, W.B. Measurement of Analysis of Wing and TUl Buffeting Loads in a FighterSkopinski, T.H. Airplane.
National Advisory Committee for Aeronautics, Report 1219, 1955.
--- sZ
-10-
7. Huston, W.B. Probability and Frequency Characteristics of Some Flight Buffet Loads.Skopinski, T.H. National Advisory Committee for Aeronautics, Technical Note 3733,
1956.
8. Huston, W.B. A Study of the Correlation Between Flight and Wind Tunnel Buffet Loads.North Atlantic Treaty Organization for Aeronautical Research andDevelopment, Report 111, April-May 1957.
9. Skopinski, T.H. A Semi-Empirical Procedure for Estimating Wing Buffet Loads in theHuston, W.B. Transonic Region.
National Advisory Committee for Aeronautics, RML56EO1, September1956.
10. Davis, D.D. Buffet Tests of an Attack-Airplane Model with Emphasis on Analysis ofWornom, D.E. Data from Wind-Tunnel Tests.
National Advisory Committee for Aeronautics, RML57H13, February1958.
11. Mullans, R.E. Buffet Dynamic Loads during Transonic Maneuvers.Lemley, C.E. Air Force Flight Dynamics Laboratory Technical Report AFFDL-TR-72-
46, September 1972.
12. Hwang, C. Transonic Buffet Behaviour of Northrop F-SA Aircraft.Pi, W.S. American Institute of Aeronautics and Astronautics, Paper 75-70,
January 1975.
13. Hwang, C. Investigation of Steady and Fluctuating Pressure Associated with thePi, W.S. Transonic Buffeting and Wing Rock of a One-Seventh Scale Model of the
F-5A Aircraft.National Aeronautical and Space Administration, CR-3061, November1978.
14. Jones, J.G. A Survey of the Dynamic Analysis of Buffeting and Related Phenomenon.Royal Aircraft Establishment Technical Report 72197, February 1973.
15. Butler, G.F. Preliminary Evaluation of a Technique for Predicting Buffet Loads inSpavins, G.R. Flight from Wind-Tunnel Measurements on Models of Conventional
Construction.North Atlantic Treaty Organisation, Advisory Group for AerospaceResearch and Development, Paper 23 of AGARD CP-204, 1976.
16. Schweiker, J.W. Response of Complex Shell Structures to Aerodynamic Noise.Davis, R.E. National Aeronautics and Space Administration, CR-450, April 1966.
17. Rodden, W.P. Refinement of the Nonplanar Subsonic Doublet-Lattice Lifting SurfaceGiesing, J.P. Method.Kalman, T.P. J. Aircraft, Vol. 9, No. 1, January 1972, pp. 69-73.
18. Kalman, T.P. Application of the Doublet-Lattice Method to Nonplanar ConfigurationsRodden, W.P. in Subsonic Flow.Giesing, J.P. J. Aircraft, Vol. 8, No. 6, June 1971, pp. 406-413.
19. Rodden, W.P. New Developments and Applications of the Subsonic Doublet-LatticeGiesing, J.P. Method for Nonplanar Configurations.Kalman, T.P. Symposium on Unsteady Aerodynamics for Aeroelastic Analyses of
Interfering Surfaces. North Atlantic Treaty Organization, Advisory Groupfor Aerospace Research and Development, AGARD-CP-80-71, 1971.L
-11-
20. Landahl, M.T. Kernel Function for Nonplanar Oscillating Surfaces in Subsonic Flow.AIAA Journal, Vol. 5, No. 5, May 1967, pp. 1045-1046.
21. Hedman, S.G. Vortex Lattice Method for Calculation of Quasi Steady State Loadingson Thin Elastic Wings in Subsonic Flow.Aeronautical Research Institute of Sweden, Stockholm, Report 105,October 1965.
22. Coe, C.F. Pressure Fluctuations Underlying Attached and Separated SupersonicChyu, W.J. Turbulent Boundary Layers and Shock Waves.Dods, J.B., Jr. AIAA Paper 73-996, October 1973.
23. Bies, D.A. Review of Flight and Wind Tunnel Measurements of Boundary LayerPressure Fluctuations and Induced Structural Response.National Aeronautical and Space Administration, NASA CR-626,October 1966.
WO
..... 77 . .
PIG 1: SCH~EMATIC OF WING
y
xx
RkOh PANEL
th PANEL/
FIG. 2: PANEL REPRESENTATION OF WING
-15-
xYC
BOUND VORTEX AND
LINE OF DOUBLETS
DOWNWASHCOLLOCATIONPOINT TALNPOINT VORTI CES
FIG. 3: REPRESENTATION OF WING BY PANELS AND LOCATION OF VORTICES.DOUBLETS AND COLLOCATION POINTS
* 16 -
FIG. : WIN AND ELEMENT RIAE
i 7
~III --
- 17 -
STRUCTURAL RESPONSE
INERTIA
I I
0L ISTIFFNESS
z I WING0lI MOTION
I STRUCTURAL DAMPING0
zw
AERODYNAMIC EFFECTS
AERODYNAMIC
TRANSFER FUNCTION
_RANDOM INPUT
F _I FRWNREPSETRADMLDI
FIG. 5: BLOCK DIAGRAM FOR WING RESPONSE TO RANDOM LOADING
- 18 -
140
180
220
/M
I," /260 w
*1 z
ACTUAL I1I30LEADING EDGE / 1
2 7
'I I~- 380 K
3 420
1;7 1 r 46 5%
41400% 65%
240 200 160 120 80 40 0
WING STATION, INCHES
FIG. 6: PANEL REPRESENTATION OF F-4E WING PLAINFORM
-19.
0 .04 .08 .12 .16
ACPrms
4
FIG 7:APROXMAT MDELIN OFRMSPRSSUE OEFICINTON IN
-20-
NUMBERS ON PANELS CORRESPONDTO TRANSDUCER NUMBERSIN REF. II
05
.4
014
'3
S802
FIG. 8: PRESSURE TRANSDUCER LOCATIONS
0 1.2
-21 -
1o'1
Q 220 PSF, M 0.78
STRUCTURAL DAMPINGRATIO =0.05
FOR ALL MODES
N EXPERIMENTS
I IN
IV
0-
.-l
Cr-)
10 0 0FRQUNYH
FI.9 OPRSNO EPNEACEEAINPWRSETA ESTFRMFIH ET IHTERTCLPEITOSUIGCNTN
CORLTOzSUPINATM07 N 2 S
-22 -
101-
Q 350 PSF, M : 0.82
STRUCTURAL DAMPINGRATIO = 0.05
...................... TOTAL DAMPING RATIOFOR ALL MODES
t00N_.. EXPERIMENTS
-j
i-i
0.
CL
.
zI.-,
_j .w,., IO2't
I0o I01 102
FREQUENCY, HZ
FIG. 10: COMPARISON OF RESPONSE ACCELERATION POWER SPECTRAL DENSITYFROM FLIGHT TESTS WITH THEORETICAL PREDICTIONS USING CONSTANT
CORRELATION ASSUMPTION AT M - 0.82 AND 0 350 PSF
-23 -
101
Q =220 PSF
-- M =0.42, ALT. = 5, 000 FT
................ ............... M =0.51, ALT. =15,000 FTM z 0.78, ALT. x35,000 FT
------------------------ M= 0.89, ALT.= 40,000 FT100-
9N
ct 0~'
0
-0J2
i r-
100 10' 102
FREQUENCY, HZ
FIG. 11: VARIATION OF ACCELERATION POWER SPECTRAL DENSITY WITH
MACH NUMBER AND FLIGHT ALTITUDE FOR 0- 220 PSFam1n
- 24-
V -I
Q 22 P*
M=z2 L. ,00F. ........... M =0 5 , A T 50 0 F
M- .8 L. 500F
409 ------ Mr08,AT 000F
10 11 0
FRQUNYH
FI.1:VRAINOzIPAEEN OE PCRLDNIYWTMAHNME N LGTATTD O 2 S
- 25-
101 -
Q 350 PSF
M= 0.51, ALT. = 2,OOOFT...................... M 0.65, ALT. = 15,000 FT
M 0.82, ALT. = 25,000 FT A0 M0.90, ALT. = 30,000 FT
cml
• ioQ...
100
-
4I- 'a..
a.
u
IO-'
O0 I01 t0'
FREQUENCY, HZ
FIG. 13: VARIATION OF ACCELERATION POWER SPECTRAL DENSITY WITHMACH NUMBER AND FLIGHT ALTITUDE FOR 43= 350 PSF
. . .
* I
ILL
-26
10-5
I
10-8-'
w
2
0 3 :50 PSF. ............. M = 0 65 L . =.50 0 F
- M 0.82, ALT. = 2,000 FT
10-9 --------- M 0.90, ALT. = 30,000 FT
I00 I01 p0FREQUENCY, HZ
FIG. 14: VARIATION OF DISPLACEMENT POWER SPECTRAL DENSITY WITHMACH NUMBER AND FLIGHT ALTITUDE FOR Q - 350 PSF
V S -ad
- 27 -
101-
M_ _ 0. 51 0Q 350 PSF, ALT. = 2, 000 FT. .............. Q 220 PSF, ALT. IS1, 000 FT
Q = 177 PSF, ALT. = 20, 000 FTA------ 0 = 1If4 PSF, ALT. =30, 000 FT
S100
z
a-.
0
03
4 0 0 0
FRQECH
FI.1:VRAIO FACLRTINPWRSETRLDNIYWTDYAI RSUEADFIH LIUEFRM-05
-28-
10-** _ _ __ _ _ _ _ __ _ _ _ __ _ _ _ _
CY 1/-6 -
tI
II'
0- , Y-
350 PSF 0 F
............ 0 = 2 0 P F L . = 1 ,0 0 F
02 7 S1ALT 000F
lor- 0 114PS~ AL. = 0,00 Floo 101 10
FREQUECY, H
FI.1:VRAINOwIPAEEN OE PCRLDNIYWTDYAI4RSUEADFIH ALITD FO M 0.1
-29-
M =0. 78
- -------- Q0 620 PSF, ALT. = 10,000 FT........ Q=414 PSF, ALT. =20,000 FT a:
0Q 220 PSF, ALT. =35,000 FT______Q = 103 PSF, ALT. = 50,000 FT
I I :~ I I f
lo 0 0FREQENCY HZ
FIG.~~~~~~~~~~~I 17 VAITO a:ACLRTONPWRSETALDNIYWTDYNAMIC~~~~~~~ PRSUEADFLGTATTDEFRM-07
- 30 -
.-
IL
3:
z
620, -- ,AT 1,00F
.o........... Q = 1 SF A T 0 0 0 F0 = 20 PF, AT. =35, 00 I0-9- ------ 0=13.SAT 000F
lo 1- 0
FRQECHFI.I:VRAINO IPAEEN OE PCRLDNIYWT
DYAI RSUE N LGTATTUEFRM-07
- 31-
I4 . ...
M 0.51
, 0 350 PSF, ALT. = 2,000 FTN-,-U
S ................ 0 220 PSF, ALT. = 15,000 FTI 0 = 177 PSF, ALT. = 20,000 FT
o " 0 = 114 PSF, ALT. = 30,000 FT
I - 5 -
o.
m 0 - i n.i-..
I .F' ,I l
c-w- I'_ _ __ _ _ _ _ _ _ _ P_ / __ __ _° 11
I I V
10-8
| w , , I
00 I-01 102
FREQUENCY, HZ
FIG. 19: VARIATION OF ACCELERATION POWER SPECTRAL DENSITY NORMALIZEDWITH RESPECT To 02 WITH DYNAMIC PRESSURE AND
FLIGHT ALTITUDE FOR M - 0.51
i
-32-
10-10
io-IO
w
1 -
o i
i-I
a.O-10_2
-biz
wj2
I . i0-13
M= 0.51
Q = 350 PSF, ALT. = 2,000 FT I...................... Q = 220 PSF, ALT. = 15,000 FT
Q = 177 PSf , ALT. = 20,000 FT
Q = 114 PSF, ALT. = 30,000 FTI J -" I I IT- - - -s - I I I ! i I
100 101 102
FREQUENCY, HZ
FIG. 20: VARIATION OF DISPLACEMENT POWER SPECTRAL DENSITY NORMALIZEDWITH RESPECT TO Q2 WITH DYNAMIC PRESSURE AND
FLIGHT ALTITUDE FOR M = 0.51
-33-
M 0.78
_____- - = 620 PSF, ALT. 10,000 FT
. .............. 0 = 414 PSF, ALT. =20, 000 FTN~~~~ =_____ 220 PSF, ALT. =35,000 FTP
--------- 0Q 103 PSF, ALT. = 50,000 FT
U o -*
co
a.'
z0
10-
0-
loo 10110
FRQUNYH
FIG 21 VRIAIO OFACCLEATIN OWE SECTAL ENITYNOMALZEWIT RSPET O 2 WTHDYNMI PESSREAN
FLGTALIUE O -07
- 34 -
10-10
zNi
**
0
0.-
zLiJ
fllM0.7
0 2 Sa-L.=1000F. ........... 4 4 PF-L . 0 0 0 F
z 2 SAT 500F10-4 ------ 0 13 SAT 000F
FLGH 0.78D FR 07
-35.
101-
ALT. =15, 000 FT Ia
M = 0. 42, 0Q 147 PSF a
. ........... m = 0.51, 0 = 220 PSF
M = 0.65, Q = 350OPS F----- M =0.78, Q 509 PSF
N1001
* I
0(L
0
-2-
-3
1 0 0 0 102___ _ _ __ _ _ _FRQECHFI.2:VRAINO CEEAIO OE PCRLDNIYWT
MAHNME N YAI RSUEA 500F.ATTD
- 36-
I.It
. . .. .. .. .. ... .. .. .. .. ..
W, 1r7
CL
D - 8o
I I I I I I I
FREQUENCY, H
FIG.~~~~~~~ 24: VAITO O IPACMN OWRSETRLDNITYITMAHNUBRAN YNMCPRSUE T1500FT ITUDE
- 37.
1o'1
0 220 PSF, M =0.78
- --------- STRUCT. DAMP. =0. 025
STRUCT. DAMP. =0.05.~ ~ ~ ~ ~ ~ ~ ~~~I ............ TUT AP .1
STRUCT. DAMP. =0. 10
ILI_________)
N -SRC.DM.=01
0I(LI~
I, I %
10-
i&iI
-38-
* 0-6-
II
I-
U) AlQ 2 SM 07
- - - - - -- S R C . DA P . 2
4TUT AM.=00
... ...... ...- T U T A P .1STUT-DM. .1
Ir~llo 11 0
wRGECH
0I.2:VRAINO IPAEEN OE PCRLDNIYWTSTUTRLDMPN - .7, 2 S
O IL - - -- - - - - - ' , -- - . - , -P L * . -- . I, --
-39-
101 -
'I
o 350 PSF, M= 0.82
---------- STRUCT. DAMP. = 0.025I,
STRUCT. DAMP. = 0. 05 i
. ................... STRUCT. DAMP = 0. 10STRUCT. DAMP. = 0. 15
S100*I
S' u
100 1,1 102
FREQUECY,"H
a ,:
FIG 27: VAITO.FACLRTO OE PCRLDNIYWT
(S)a. 101 - __ __ _ __ _ __ _ __ _
I -ii2 I I
z I I it
tiC,
1 - - iI I I i li i i
I00 I01 102
FREQUENCY, HZ'
FIG. 27: VARIATION OF ACCELERATION POWER SPECTRAL DENSITY WITHSTRUCTURAL DAMPING FOR M - 0.82,0-,,350 PSF ,
- 40-
lAs
a.~
NL10-
0 35I0F 08
-*-- - - - - S R C . D M . = 0 2
STUT.DMP 00
.- ... ... .. .. .. T U T A P . 1
STRUC. DAP. =0. 1
z09lo 11 0
FRQENY HZ
FI.2:VRAIO FDSLCMETPWR-ETRLDNIYWTSTUTRLDMPN.O) 08, 5 S
- 41 -
1og1
0 220 PSF, M =0.78
STRUCTURAL DAMPINGRATIO = 0.05
........ ...................... EXPERIMENTS
Nocy0
z
(L
0-
-J 1 , 0
LAAt
-42-
10-5 -
N
SI0- 6 _**
z
I-
w
c,
0
zww0.
-J
--
0 220 PSF, M 0.78
STRUCTURAL DAMP.RATIO 0.05
1,, 0- 9
100 10' 102
FREQUENCY, HZ
FIG. 30: DISPLACEMENT POWER SPECTRAL DENSITY USING EXPONENTIALSPATIALLY DECAYING PRESSURE CROSS-SPECTRUM ASSUMPTION FOR
M 0.78Q- 220 PF
I ________.____
-43-
101
0 350 PSF, M 0.82
STRUCTURAL DAMPING
RATIO = 0.05
...................... EXPERIMENTS
N
10OO
0
-
zIl0
I-
M. - -.
mo
IJJ". .,
z °...0 n 2
- J
100 0 ! 102
FREQUENCY, HZ
FIG. 31: COMPARISON OF RESPONSE ACCELERATION POWER SPECTRAL DENSITYFROM FLIGHT TESTS WITH THEORETICAL PREDICTIONS USING EXPONENTIAL
SPATIALLY DECAYING PRESSURE CROSS-SECTRUM ASSUMPTION FORM-O .82, Q-"350 PSF
- 44 -
z
CL
z
w
(L
t0 0 350 PSF, M =0.82
STRUCTURAL DAMP.RATIO 0. 05
100 101 102
FREQUENCY, HZ
FIG. 32: DISPLACEMENT POWER SPECTRAL DENSITY USING EXPONENTIALSPATIALLY DECAYING PRESSURE CROSS-SPECTRUM ASSUMPTION FOR
M-O0.82, Q -350 PSF
- 45 -
101
Q 220 PSF, M =0.78
- -- - REF. 11 (C=0.05)
EQS.(16) AND (37) (C=0.05)
------- EQS.(18) AND (37) (Cz 0.05)
100 FLIGHT TESTS I
CL
0a.
4
-2
0-)
0 0 1021
FRQENY HZ
FI. 3:COPAISN F ESONE CELRAIO PWE SECRA DNSTFRO FLGTTSSWTHTERTCLPRDCIN O
M-.8,Q22 S
Noa
-46-
101 -
0 : 350 PSF, M 0.82
REF. I I (C 0. 05) ,
EQS.(16) AND (37) (C =0.05)
EQS. (18) AND (37)( = .05)
N 0 0 __ F L IG H T T E S T S I I.. ' I jI t*M
I i i ' I.. , t ,-> _-
I-
w IIN
-i:
o I I
-I I I1 I
Ii0- I 1
I- I *9
z I Iil \ / l
0 II 9 I '!
4 0 10 0
< - ,[ --I' I // '
MJ~J - 0.20- 5 S
I / /9 1'
/ I "I
, 9
I10-5" I I m I I I I III
100 101 102
FREQUENCYt HZFIG. 34: COMPARISON OF RESPONSE ACCELERATION POWER SPECTRAL DENSITY
FROM FLIGHT TESTS WITH THEORETICAL PREDICTIONS FORM - 0.82, 0 = 350 PSF
-47-
APPENDIX A
EXPRESSIONS FOR K AND K2 IN THE EQUATION FOR NORMALWASH
In Equation (21), the terms K! and K2 are given by Landahl (Ref. 20) as follows (withsign reversed):
exp(-ik u, )Mr 1K, = -11 - 2\ I (Al)
R(1+ U2
ik 2 r2ik I r, exp(-ik I u,) Mr r,K, =312 + -- +u2) - - + 22 2 2) V2 R I R 2
Mr IU 1 exp(- ik Ul )+ RJ 1 (A2)
where
= (1- M 2 )' (A3)
R= [(R- tanXs)2 + (A2r4]'
(4
U, (MR - i + tanA s)/0l2r, (A5)
kI -(A6)U
For u > 0 (Ref. 19),
I,(Ulkl) (u, k, 2 u)3/2 ikl Io(ul,kI) exp(-ik uI) (A7)
where
IIan(nc-ikl )e x p (- nc u ! )I0(ul ,k j) 2 : (A8)nfl n2 c2 +k2
l m~'' + > .. .. ........ ... .. .. ., . u+ ... ml, ..... + "--+'+ +' .. ." ' t ... ++...... . .....
- 48 -
a, = 0.2418619 a 7 = - 41.1863630
a, = -2.7918027 118 545.98537
113 = 24.991079 a9 = -644.78155(A9)
114 = -111.59196 ao= 328.72755
115 = 271.43549 al = -64.279511
a6 = - 305.75288 C = 0.372
and 312 (Ul fk,) = (2+ikiut)l ) -_____-ik 1 10 (u1 ,ki
+ kOu~lep-i l(AlO)
where
an afexp(ncu 222))21(nk,2c In' ) - k~ + ncu (n~c + k) ik, [2nc +u, (n c + k)J
(All)
For u < 0,
I, (u1 ,kl 2RQ 11 (0,k, RQ 1, (-ul ,k, )+ lHm 11 (-u, ,) (A12)
and
12 (Ul k1 ) 2RQ 12 (0,k,) RQI12 (-u,k, ) + ~im 11 (-u,,k1) (A13)
For steady states, K 1 and K 2 are given as:
Kill -1 - (E- tanX,)/R (A14)
K20 =2+ (i -tanx) (2 + R2/R (A15)
- 49-
APPENDIX B
EXPRESSIONS FOR NORMALWASH FACTOR
In Equation (25), the normalwash factor is given as:
Drs f Don + DIrs + D2rs (25)
From Reference (21), the steady normalwash factor Do can be derived as (Fig. BI):
AxSDors = 2 (V y sinr - Vzcosy r ) (B1)
where
cosob - COSP b 1 - cosw i coso0 + 1VY = (d bxcosAsin s - dbzsinA) 2d+ (R 4 R. - R 2z) '-z -
yr b4v +z R 41r(2+ 47r R' + R2 Ob i Yy 0 0
(B2)
and
COSb - COSN0b 1 - CO840i CoO + 1
Vz = (dby sinA - dbx cosAcos-ys ) - R + + 1 ROY
(B3)
The components of db, Ri and R0 are:
dbx = Rix - RicosO b sinA (B4)
dby = Riy - RicosObcosAcosys (B5)
dbz = Riz - RicosobcosAsinj.s (B6)
and
Rix f X0 + etanA (B7)
Riy = yo + ecowys (B8)
Riz Zo + esiny, (B9)
Ro X etanA (BlO)
Roy M yo - ecoys (Bll)
Roz W zo - esin, (B12)
-50 -
wherexo
X0 (B13)
tanA = 1/ptanX, (B14)
p M2(B15)
Finally, the angle relations are given as:
1cosOb fi (RixsinA + RivcosAcosys + RicosAsiny) (B16)
1cosIPb - (R0xsinA + R0ycosAcosy s + R0 zcosAsiny/) (B17)
RR.Rix
C0 1 - R. (B18)
ROX
cosO0 RO (B19)
From Reference (19), the planar normalwash factor D 1 rs is given as:
Ax S [ ) jDrs fi 2-2)A +B +C] F + ( + A log + 2eA (B20)
(8(r.2 + e)2 + 2
The terms A l , B l and C1 are determined from the following:
A1 f 2{P(-e)-2P(0)+P,(e)} (B21)2e 2
B1 2 e {Pl')- P(-e) (B22)
C, = P1(0) (B23)
where PI (e), Pl (-e) and P(O) are the outboard, inboard and center values of P( ) defined as:
P( = TI KIexp -- (T - tanx)] - K1 1 (B24)
which is obtained by subtracting the steady value K10 from the numerator of the first term of Equa-
tion (21). The term F in Equation (B20) is given by:
-51-
F 1 2 2 (B25)Izi y 2 + z 2 -e 2
where the principal value of the arctangent is taken in the range (Ow). When i 0,
2eF f e 2 ( B 2 6 )
and for << 1, 2 +z 2 > e2 ,
2e i2F = -2+i2 e2 e 2 (B27)
and2n-4
4e4 7 (-1)n 2ez )
01 1 - (B28)( 2 +i 2 -e 2 )2 n-2 2n-1 + 2-e 2
The non-planar normalwash factor D2 is:
Axs
D 2 r= 9 1(92 + )A2 +B 2 +C2 ]FDrs 167r12
+ (3 + e) 2 + ([ ( 2 + 2 2 )y + ( 2 z2)e] A 2 + ( 2 + i 2 + e )B 2
+ ( + e)C 2) e)2 1 ([( 2 + 2) ( 2 - i2 )e A 2
+ ( 2 +i2 - e)B 2 + ( - e)C 2) (B29)
A2 , B2 and C? are determined from Equations (B21) to (B24) with the subscripts in those equationschanged to '2 For small values of 1, the following equation for D2., is used:
AxSe 2(g2 + i 2 + e2 ) (e2 A2 + C2 ) + 4 e2 B2 0 +
-8W(y2 + i - e2) [(y +e)2 + 12] [(g - e)2 +~2 V e2~y+i) 2 +B 2 +~
(B30)
where a is given by Equation (B28). This equation is generally used except when ( 2 + V - e2 )/2e7z < 0.1 in which case Equation (B29) is used instead.
- 52-
r
x0b
F-Yo R(x0, y01z0)
RYr
ZO
FIG. B1: HORSESHOE VORTEX SYSTEM
REPORT DOCUMENTATION PAGE / PAGE DE DOCUMENTATION DE RAPPORT
REPORT/RAPPORT REPORT/RAPPORT
LR-613 NRC No. 2123018 lb
REPORT SECURITY CLASSIFICATION DISTRIBUTION (LIMITATIONS)CLASSIFICATION DE SECURITE DE RAPPORT
Unclassified Unlimited2 3
TITLE/SUBTITLEITITREISOUS-TITR E
Aeroelastic Response of an Aircraft Wing to Random Loads4
AUTHOR(S)/AUTEUR(S)
B.H.K. Lee5
SERIES/SERIE
Aeronautical Report6
CORPORATE AUTHOR/PERFORMING AGENCY/AUTEUR D'ENTREPRISE/AGENCE D'EXgCUTION
National Research Council Canada7 National Aeronautical Establishment High Speed Aerodynamics LaboratorySPONSORING AGENCY/AGENCE DE SUBVENTION
8
DATE FILE/DOSSIER LAB. ORDER PAGES FIGS/DIAGRAMMESCOMMANDE DU LAB.
83-04 60 349 10 11 12a 12b
NOTES
13
DESCRIPTORS (KEY WORDS)/MOTS-CLES
1. Wings - load distribution2. Wings - test results
14SUMMARYISOMMAIRE
A method for the prediction of the response of an elastic wing to random loads at flightconditions twing rigid model wind tunnel pressure fluctuation measurements is presented. The wing isdivided into panels or elements, and the load is computed from measured pressure fluctuations atthe centre of each panel. A series representation with terms of the correlated noise type is used tocurve fit the experimentally determined pressure power spectra. Two methods are used to calculatethe random load spectrum: a constant correlation approximation, and an exponential spatiallydecaying cros-power spectrum model for the pressure. The coupling between the structural dynamicsand aerodynamics of a vibrating wing is taken into consideration using the doublet-lattice method forcomputing the unsteady aerodynamic forces. The acceleration and displacement response spectrahave been computed for the F-4E aircraft for various Mach numbers, dynamic pressures and flightaltitudes. The Importance of the unsteady aerodynamic loads induced by the vibration of the wing andinput load representation is illustrated by comparing the theoretical predictions with results fromflight tests.
1
~I
T
