unsteady aerodynamics for advanced configurations · unsteady aerodynamics for advanced...
TRANSCRIPT
FDL TDR-64-152PART I
UNSTEADY AERODYNAMICS FOR ADVANCEDCONFIGURATIONS
PART I-APPLICATION OF THE SUBSONIC KERNEL FUNCTIONTO NONPLANAR LIFTING SURFACES
TECHNICAL DOCUMENTARY REPORT No. FDL-TDR-64-152, PART I
COPY ___ 7
MAY 1965 "l' " ' ' '/' d?
AIR FORCE FLIGHT DYNAMICS LABORATORYRESEARCH AND TECHNOLOGY DIVISION
AIR FORCE SYSTEMS COMMANDWRIGHT-PATTERSON AIR FORCE BASE, OHIO
Project No. 1370, Task No. 137003 r4 I I
(Prepared under Contract No. AF 33(657)-10399 byThe Space and Information Systems Division
North American Aviation, Incorporated, Downey, California;H. T. Vivian and L. V. Andrews, Authors)
NOTICES -
W 75,hen Government drawings, specifications, or other data are used for anypurpose other than in connection with a definitely related Government procure-ment operation, the United States Government thereby incurs no responsibilitynor any obligation whatsoever; and the fact that the Government may haveformulated, furnished, or in any way supplied the said drawings, specifications,or other data, is not to be regarded by implication or otherwise as in anymanner licensing the holder or any other person or corporation, or conveyingany rights or permission to manufacture, use, or sell any patented inventionthat may in any way be related thereto.
Copies of this report should not be returned to the Research and Tech-nology Division unless return is required by security considerations,contractual obligations, or notice on a specific document.
400 - )1gy 1965 - 448-48-1023
C ,
'0
This Document ContainsMissing Page/s That AreUnavailable In TheOriginal Document
BestAvailable
Copy
FOREWORD
This report covers the research conducted by theSpace and Information Systems Division of NorthAmerican Aviation, Inc., Downey, California, forthe Aerospace Dynamics Branch, Vehicle DynamicsDivision, Air Force Flight Dynamics Laboratory,Wright-Patterson Air Force Base, Ohio, underContract No. AF 33(657)-10399.
The work was performed to advance thestate of the art of flutter prevention for flightvehicles as part of the Research and TechnologyDivision, Air Force Systems Command's exploratorydevelopment program. This research was conductedunder Project No. 1370 "Dynamic Problems in FlightVehicles," and Task No. 137003, "Prediction andPrevention of Dynamic Aerothermoelastic Instabilities."Mr. James Olsen of the Vehicle Dynamics Division,Air Force Flight Dynamics Laboratory was the ProjectEngineer.
Mr. L. V. Andrew was the Program Manager forNorth American Aviation. Mr. H. T. Vivian, underthe guidance of Dr. H. Ashley, laid out the formof the solution and wrote the computer program.Dr. E. R. Rodemich made many significant contri-butions to the numerical analysis.
The contractor's designation of this report isSID 64-1512-1.
This technical documentary report has been reviewedand is approved.
Walter J. MykytowAsst. for Research and TechnologyVehicle Dynamics Division
ABSTRACT
In this part, equations for pressure distributions and generalizedaerodynamic forces are derived for a thin nonplanar lifting surface in simpleharmonic motion at subsonic speeds. A digital computer program, written inFortran IV, is also presented herein. The computer program will generateup to a ten by ten matrix of generalized aerodynamic forces when given datafor the geometry of a planar lifting surface with a folded planar tip, the flightMach number, the reduced frequency of motion, and some control constants.Control surface deflections are not accounted for in this study.
The kernel function method given by Watkins, Runyan, and Woolston(Reference 1), which relates the pressure distribution to the downwash on aplanar lifting surface, has been extended and applied to a nonplanar liftingsurface. Hsu's technique (Reference 4) of employing Gaussian quadratureformulas ;.s used when integrating the product of the kernal function and thelift function over the planform area.
Recommendations are made to extend the method to account for bluntedleading edges and the accompanying airfoil thickness and to account forcontrol surface deflections.
4 iii
I
CONTENTS
Section Page
I INTRODUCTION .I.. . . . . . . 1
II FUNDAMENTAL EQUATIONS OF FLUID MOTION .... 5
III LINEARIZED EQUATIONS OF MOTION ...... 9
IV THE ACCELERATION POTENTIAL ....... 15PLANAR WINGS ......... 15NONPLANAR WINGS . ....... 15
V THE BOUNDARY CONDITIONS ... .... 21
VI APPLICATION TO A WING WITH A FOLDED TIP ... 31DESCRIPTION OF THE COMPUTER PROGRAM . • 39USE OF THE COMPUTER PROGRAM .... 41
VII RESULTS . . . . . . . . . .. 49
Viii CONCLUSIONS AND RECOMMENDATIONS ... 53
REFERENCES AND BIBLIOGRAPHY ...... 55
APPENDIX. COMPUTER PROGRAM LISTINGS. 57
iv
I
ILLUSTRATIONS
Figure Page
1 Generalized Curvilinear Planform . . . . . . 102 Planar Wing With Planar Symmetrically Folded Tips . 323 An Optimum Set of Collocation and Integration Points . • 324 Functional Flow Diagram-Main Program . . . . . 405 Sample Data Sheets . . . . . . . .. . 446 Lift and Moment Coefficients Vs Mach Number for
Aspect Ratio 2. 0 Rectangular Wing at Various FoldAngles . . . . . . . . . . . . 50
7 Lift and Moment Coefficients Vs Fold Angle for65 Triangular Wing at M.,* 0.8 . . . . . 50
8 Coefficients of Lift and Moment Due to Pitch VsReduced Frequency for 65-Degree Triangular Wingat M = 0.8 and Various Fold Angles . . . . . 51
9 Lift Curve Slopes Vs Mach Number for 65-DegreeTriangular Wing at Various Fold Angles . . . . . 52
Tables
Table Page
1 Spanwise Parameter in the Kernel Function . . . . 34
V
II
SYMBOLS
a Local speed of sound (constart irx linear theory)
bo Ro')t semichord
b (s) Local semichord
Cp Pr,.ssure coefficient
Cp Time independent factor of os cillatory, part of Cp
i,j,k Unit vectors parallel to the coordinate axes
k Reduced frequency, wbo/Uw
kl kr I
K Kernel function
K 0 , K1 , KZ Modified Bessel functions of the second kind
M Local Mach number (same as free stream Mach number,Mco, in linear theory)
n Coordinate measured normal to vv-ing surface
n(x, s, t) Contribution to n of elastic deforrxiation of the wing
nt (x, s) Contribution to n of wing thicknes s
n (x,s) Time independent factor of n(x, s, t)
p Pressure
q Fluid velocitr
R Gas constant
LUst of Symbols continued on next page.
Vi
r2I yrq)2 + (z- )
s Curvilinear coordinate on the wing (page 10)
T Absolute temperature
t Time
U, I Subscripts indicating upper and lower wing surfaces
u, v, w Components of perturbation velocity
U, W x- and z-components of V
U00 IVi (speed at infinity)
V Uniform fluid velocity at infiity
W Time independent factor of normal velocity
x, y, z Cartesian coordinates
X0 X- :
Yo Y- TI
Y Ratio of specific heats'
Y(s) Local angle between wing surface and xy-plane
Time independent factor of pressure difference betweenwing surfaces, P j- u
i 1, Cartesian coordinates
Coordinate on the wing (page 22)
Velocity potential
Perturbation velocity potential
Time independent factor of cp
Acceleration potential
p Fluid density
W Angular frequency (radiana per unit time)
vii
I. INTRODUCTION
The first published numerical method for solving the subsonic pressuredistribution problem for planar lifting surfaces undergoing simple harmonicmotion was developed at NASA's Langley Research Center by Watkins,Runyan, and Woolston (Reference 1). Watkins, et aL, presented twomethods of handling the numerical integration of the kernel function in theregion where high-order singularities exist. Both methods involved a denseconcentration of integration points in the neighborhood of the singularity.Using these methods, it is possible to obtain downwash integrals, in termsof the pressure-loading coefficients, at any arbitiary set of points on thesurface (e. g., at all the kinematic downwash points known from previouslydetermined vibration mode data). However, in order to reduce the runningtime on the computer, the downwash integrals were obtained at a selectedset of collocation points, suchas those at intersections of quarter, half, andthree-quarter chord stations and like half-span stations. When downwasheswere matched exactly (and thus, boundary conditions) at these collocationpoints, responsibility was placed upon the user to evaluate the kinematicdownwashes there. A least-square error surface fitted to the mode data wascommonly used to evaluate them. Furthermore, if the user desired that theboundary colditions be satisfied at a greater number of points, it was neces-sary that he use a correspondingly greater number of loading functions.
Procedures were then described by Rodaen and Revell (Reference Z)and the correct form of the equations were presented by Fromme (Reference3) for calculating pressure-loading coefficients which match a greaternumber of kinematic downwashes than coefficients, in the sense that the sumof squares of amplitudes of differences of complex numbers are minimized.Since it was still the responsibility of the user to evaluate the kinematicdownwashes at the collocation point, least-square error procedures wereused twice: once implicitly and once explicitly.
Hsu (Reference 4) significantly advanced the logical development of thekernel function approach when he established an optimum set of collocationand integration points. He started with the previously established chordwisepressure functions based on steady-state, two-dimensional, incompressibleaerodynamics, and with spanwise loading functions, based on steady-statelifting-line theory. He concluded that there is sufficient reason to believethat these functions display the proper characteristics near the edges oflifting surfaces oscillating in a compressible fluid.
Manuscript released by authors February 1965 for publication as an RTD Technical Documentary Report.
!I
Returning to the two-dimensional case, Hsu established that if thechordwise distribution of modal deflections (and thus downwashes) is accu-rately represented by a polynomial of degree ZN-I and is approximated bya polynomial of degree N-i, then the integral for the sectional load is eval-uated with zero error by a N-point Gaussian quadrature if the differencebetween the accurate and the approximate representation of the downwashesis made equal to zero at each of the N points (i. e., the chordwise collocationstations). Conversely, still for the two-dimensional case, Hsu establishedthat if the product of the pressure function and the kernel, divided by theJacobi-Gauss weight factor (which produces the square root singularity atthe leading edge), is accurately represented by a polynomial of degreeZN-1, then the integral for the downwash at any one of the collocationstations is evaluated with zero error by a N-point Gaussian quadrature.These N points are then made the chordwise integration stations.
For the spanwise direction, using lifting-line theory, Hsu similarlyestablished M-spanwise collocation stations and M + 1 interdigitated spanwiseintegration stations plus the conditions under which the Gaussian quadraturecan be used with zero error.
It is important to note that the kernel of the integral equation for thedownwashes in unsteady, three-dimensional, compressible flow cannot beaccurately represented by a polynomial of finite degree. It is equallyimportant to note, however, that, because of the edge characteristics of thepressure and loading functions, the Gaussian quadratures employed at Hsu'soptimum point set evaluate the integrals with the least squared error fora given number of integration points. We have yet to match the boundaryconditions using Hsu's method.
The downwash matching problem in Hsu's approach is basically thesame as in Watkin's approach; we merely have a more logical choice ofpoints at which to match them. In the examples Hsu used to demonstrate hisapproach, he chose to use the same number of pressure-loading functions ascollocation points. However, the approach is not dependent upon that choice.If a smaller number of pressure-loading functions are used, then the pro-cedures described by Rodden, Revell, and Fromme may be used to computepressure-loading coefficients which yield a minimum sum of squares ofamplitudes of differences in down-washes.
a
A need has arisen for application of the kernel function method to
rnon-pLanar lifting surfaces on future aerospace vehicles. Application
is also required to more conventional non-planar surfaces such as T-tail,
-tall. and wing-vertical tail combinations.
Professor H. Ashley outlined the application to the folded tip con-
figuration. A computer program based on Ashley's work was developed for
steady-state flow by L. Johnson, et al., of the Los Angeles Division of
North American Aviation, Inc.
The work reported herein is based on Professor Ashley's outline.
lowreve-, the expression for the kernel has been greatly simplified by
D]r. E. R. Rodemich of North American Aviation, Inc., Space and Information
Systems Division.
3
il. FUNDAMENTAL EQUATIONS OF FLUID MOTION
Consider a body immersed in a compressible, nonviscous, perfectfluid and assurfte the fluid flow to be isentropic and irrotational. Under theseconditions, a velocity potential , exists:
where j is the velocity vector of a fluid element and V is the gradientoperator (See Reference 4. ) Also under these conditions, the isentropic(constant entropy) pressure-density relationship is valid. Thus,
a2 =Iap\ YRa : :Y RT (Z)MOB
Other equations which govern the flow are the continuity equation forconservation of mass
at5-+ "(p =0 (3)
and Euler's equations for conservation of momentum
Dq _-15F = Vp (4)
Dwhere, 5t the substantial derivative, is
D 8Dt = 5t (5)
These equations may be combined, as described in Reference 5, to yieldthe nonlinear, unsteady flow equation
2 F2g 23 c2lZ L + 6 (6)
a 2- [ at2 z(q Vat ~-
5
Consider, then, that the fluid motion consists of a perturbation super-
imposed on a uniform stream velocity V= Ui + Wk parallel to the xz-plane
of a rectangular Cartesian coordinate system. Then the velocity potential
may be expressed as the sum of a uniform part and a perturbation part
$- Ux + Wz + (7)
and, similarly, the velocity vector becomes
= V + VT= Vq? (8)
The pressure coefficient at any point in an isentropic flow field is
P - p.Cp p= U= (9)
where p - pm.is the difference between local pressure and free-stream pres-sure, U. = IVI, and 1/2 n- 173 is the frpp-at-rejs .. " . xin
TC l.e-', equation (.... rec 5) for isentropic flow[Yz~ - q + 2 av ]Y-1
SI1 + i- MI ( - (10)
p 2I 2 00(l -lYMC UJ
A complete statement of the fundamental problem requires specifica-tion of the boundary conditions. The boundary conditions at infinity depend
upon the free-stream velocity. When it is less than the speed of sound inthe fluid, the disturbances to the flow die out and are not felt at infinity.When it is greater than the sonic speed, then in the region where disturbancesare felt, even at infinity, the component of flow due to the disturbance is
directed away from the source of disturbance and otherwise the free-streamflow is undisturbed. The boundary conditions at the surface of the bodyrequire that the flow be tangent to the surface everywhere on the body. This
condition is satisfied by the equation
F- D (X, Y, Z t) 0
where
B (x, y, z, t) = 0 (1Z)
is the equation for the position of the surface at any time t, and the substan-
tial derivative D/Dt is defined by Equation 5.
7
111. LINEARIZED EQUATIONS OF MOTION
Linearization of the equations of motion is not dependent upon an
explicit form of the body equation, Equation 12, so long as the normal
derivatives of the equation are everywhere nearly perpendicular to the free-
stream direction. Thin lifting surfaces at small angle of attack satisfy this
condition and are treated herein and in Parts 2 and 4 of this report. The
special considerations required for thick bodies and high angles of attack are
treated in Parts 3 and 5. The following development is, therefore, restrictedto thin airfoils.
We first obtain the specialized form of Equation 7 when the uniform
stream velocity lies along the x-axis; i. e., W = 0 and, therefore, V = Ui,U,0 = U. The velocity potential is
- Ux + q (13)
ard the velocity vector of a fluid element becomes
q (U + u) i + vj + wk (14)
where
v , and=
The perturbation velocities u, v, and w are assumed to be much smallerthan the free-stream velocity; i. e., u, v, w << U.
The linearization procedure when applied to Equation 10 yields the fullylinearized pressure coefficient
C u + U . (15)U 2 ao
9
and when applied to Equation 6 yields the fully linearized unsteady flowequation
a89' 89' 89'- 1 28B9' 89' 89 (16)2! Z + 2 2 2 + 2U + = O 1 1 6 )
ax By Oz aco ax at
Next, we write the body equation for a thin, nonplanar lifting surface(Figure 1), in terms of a curvilinear coordinate
s = s (y)
s represents the integral of distance along the line of the mean position ofthe airfoil from the centerline to y,
s (y) =f y ds (17)0
A Section A-A
L<I S
-. A
Figure 1. Generalized Curvilinear Planforrn
10
The position of the surface in terms of s and n (the normal to S), is
separated into two parts; one part for the upper (or iner) surface, and the Aother for the lower (or outer) surface
B (x, n, s, t) = n - n (x, ) - (x, s, t) (18a)
I -- ., + P-V -V - * U J)
where nT(x, s) represents the thickness of the airfoil and n(x, s, t) repre-
sents the elastic deflection of the airfoil. In accordance with Equation 11,we use the operator
i a + + k' (19)
ax as On
on Equation 18a to get
VB (x, n, s, ti n (x, s)+-n(x, s, t)
n (x , s) + nx a, t)+ k
OB T
Substitution into
8B,,Su (x, s, n, t) -- = + (Ui +v) V B = 0 (Z0)at U.(0
of the equation for the upper surface, after higher order terms have beendiscarded, gives
+U+ n(x, s, tI+ Ix, s) (Z1)8n 'at ax ax T
11
The same procedure, for the lower surface, gives
8' =1- in (x, s. t)- n (X , -) 1an 'at . .x s
1* Finally, we restrict the analysis to that class of problems in which theeffects of thickness on the time dependent forces can be neglected. By letting
nT(x, s) = 0 (23)
we get a single expression for the boundary condition
-( + U-x ) n (x, s, t) (24)
It is evident from Equation 24 that, when the motion of the surface issimple harmonic motion,
n(x, s, t) = W(x, s) et (25
then,
jo (x, S, n, t) =(x, s, n) elWt (26)
Substitution of Equations 25 and 26 into Equations 15, 16, and 24gives
= - (27)
2- 1 D2 VV (P - 2 (28)
a 2 Dt 2
89' Dii
Bn Dt (29)
t12
where
- + 1i (30)
and
-- + - + - (31)ax2 as2 n2
To this degree of approximation, M M, and a ac.
13
IV. THE ACCELERATION POTENTIAL
PLANAR WINGS
Equation 27 shows that the. calculation of pressure requires takingderivatives of the velocity putential, and Equation 29 states the boundarycondition which must be satisfied. Watkins, Runyan and Woolston(Reference 1) solved this problem for planar surfaces in terms of a seriesof pressure functions, or acceleration potential functions (4), where,
(x",y, Z) T (x,y, z) (32)
Integration of Equation 32 gives the general expression for the velocitypotential in terms of the acceleration potential:
.x x. X- ~- f " "[ i U(x, y, z) = e J_ e 0 (, Z) dX (33)
if the velocity at infinity is V = Ui. The velocity potential due to a pulsatingdoublet satisfies Equation 28 (in the planar case s = y and n = z); and, becausethe order of operators is interchangeable, the acceleraion potential alsosatisfies Equation 28.
A complete discussion of the application of boundary conditions in theplanar case is given in Section 6-4 of Reference 5. The application in thenonplanar case is discussed in less detail in the following text.
NONPLANAR WINGS
In the nonplanar case, the acceleration potential at a point (x, *, N)due to a pulsating doublet located at the point (t, r, n), or (. rj,1) in thedirection of n is
x, i, N) &.A 6 (34)
where
= cosy (T) siny (I) (35)
R = VX - ) 2+ 8 2 1y - TO + (z ~2I (36)
and Y(*1) is angle between the wing and the xy-plane at the point (, 1, ).The perturbation velocity potential may be built up from a distribution of
doublets of acceleration potential over the wing. If A is the infinitesimal
doublet strength at (g,a , i ), the contribution toq from the doublet at this
point, from Equations 33 and 34, is
x- x - a13 a 8 j
-AxsN) a - Wn C
(37)where
RI = x- Z+ 03 1(y-1)+ (z- Z]
and the velocity component normal to the surface at (x, sN) is
lim 8 Aw(x,sO) = N-0 O Aq(xs,N) (38)
where
- cosy (y) - - - siny (y)j- (39)
Note that when the operator 8/8n is applied in Equation 37, the partials
8/83 and 8/8n maybe replaced by - 8/8z and - 8/8y, respectively.Substitution of Equation 37 into Equation 38 gives
.i x__ x- iwo(X - MR')/Uj3 2
A U lima p eAw(x,s,0) = A e N--0O R' dX
-CO R
(40)
where the operator P is
16
P - cosy (y) siny (y) cosy (n) "sin (,j) (41)I 8z Oy 8i z 9y,
and finally
A = 67pd d a- (42)41r p
and Ap is the complex amplitude of the difference in pressure on the upperand lower sides of the surface at ( o),
(goa-) = Pu (9,o') - T (9,-) (43)
and dtd0- is the incremental area of the doublet sheet.
The normal wash at (x, s, 0) given by Equation 40 is that due to apoint pressure doublet at ( , a, n = 0). The total normal wash is the integralover the surface of all the pressure doublets,
W(x, s,0) = - 1 a )K(x s, ao,M) dgd" (44)
U 41r pU 2
where f denotes the Mangler formula for evaluating infinite integrals
(Reierence 7 ), and the kernel of the integral equation is (omitting thearguments w, M for brevity)
r x 02Kira)-io i= - MR'/UB
o n 0 e P RN-0
where
R r +z
r 1 2 + Z 0
and
x 0 x-, Yo y y-1,andz 0 zo10
17
Now, by putting k wr/U and v = X/Br1; tlhen, by puttingv -M 1 l+ V
I - M the integral in Equation 45 may be written
S0 i w(X - MR')/U62 d o -iklU
o R1 d% f Vf +-Co u
where
x - MR0u I - -~
1 21
By breaking up the interval of integration into three subintervals andin the first two integrals letting u = w/i
1 -k w o -k1w u1 -ik 1 u
-=if 1f e 1 dw -jdo 0 V, v, 1 7 0 +u
or,
1 -k w Ul -ik u
I = (k )-i f dw- du (47)0 0 ol - w 0
where Ko(k 1 ) is the modified Bessel function of the second kind and zerothorder of the argument k 1 .
To obtain the analytic form of the kernel, we write the operator P(Equation 41) in a more convenient form, taking advantage of the fact that1o is a function of only r1 when x0 is held constant
PIo Cos [v(Y) - r1 8r0
+ Izocosv(Y) - Yosinv(y)J 0zo~ 0 ysiny(-q)JI (r.L~' (0. r r 8
18
The resulting kernel is
K(x Isfcr) 5 TPK(x fs~a)
0i T 1K I(x o , r) + T K (x,s, -))eU 2 22~ (48)
where
T= Cos [v(Z)
10 -0 -0YT2 = - Cosy(s) - r iv2 O cosv(r) - sinV(r)I
(48a)
S, ( k (k xEe -k 1u 11we -k 1 wK (x ~) k1 (k) e1 + ik1 f 2 ew
U 1 -ik Iu
+ ik, f ue dd +u
+(x 2 0 0rx (Mr 1 + Ru -k2 'x0 RSLT kR 2ki 3+i
1 -k w 1 2-k w- i we 1 dw -ik 2 f w e d
0o VF7 I - ,I TU-w
- ik 1 ue -ikl~ U1 u u+k-ik 1u d
19
2r1 5 TIP o o'~
R = Xo+ 13rI
X -MR
r
13
kt - U(48c)
1 U
and Kl(k1 ) and K,(kI) are modified Bessel lunctior-s of the second kind andfirst and second orders. The sub-bar indicates division by sTiP, esg.,r= rl/STIP.
20
V. THE BOUNDARY CONDITIONS
The remainder of the problem is to match the boundary conditions; i. e.,to find a pressure-loading function (, r) which, when inserted into
Equation 45 and integrated over the surface, yields the kinematic down-washes at selected points on the surface w(x, s, 0).
In subsonic flow, the behavior of the pressure distribution is known in
the area of the wing edges from a few of the exact solutions in lifting surfacetheory. In the neighborhood of the leading edge, the pressure should behaveas
lim
6- 0
In the neighborhood of the trailing edge and all edges parallel to the
free-stream direction, the pressure should behave as
lim
6-- 0
where 6 is the distance to the wing edge. Both Hsu and Watkins employ a
linear superposition of functions that satisfy these conditions. Hsu's
function differs from Watkins only in that for any given number of terms in
the series, Hsu's terms are linear combinations of Watkins terms. We use
a normalized form of the function given by Watkins:
U /2 - N M
U6a a a' ( ) (49)A ( )-b(a) I anm - n()(9
n=O m=O
where
fo( ) /1
21
U 29) = 1.0
U(~ -(2g U 1 +U 2 ) 35 nn Un- Un-2); ( +3 <
and
bTE - tLEb() = 2
The anmis are unknown pressure coefficients to be determined by matching
the kinematic downwashes at the selected points (xj, sr) on the surface.Substitution of Equation 49 into Equation 44 leads to the matrix equationgiven by Rodden and Revell (Reference 2) Equation 39, for the point set
Xj, 8r
-1 [rDiml am (50)
where, in this case,
1 1
DJ 1 - o- ff-x - g,_)ded (51)-ll f n r
We now reexamine the fundamentals of the problem before proceedingto evaluate the integrals in Equation 51 and thence to solve Equation 50.
One of the basic reasons for development of the kernel function methodis that pressure distributions over a continuous lifting surface are smoothcontinuous functions that can be represented with reasonable eccuracy by aseries of analytic functions. We point out that fn(g) can be written
f (9) =0 )
1l -
-2
Iand, therefore, the inner integral in Equation 51 may be written
I f (52)
where
P(o) = al- g)R(x.-, ,r
pn(g) = (1 - 'Z )Un(')(x - ,- ,-); 1 _n
Now, we assume for the moment that the kernel K(X- - 9, sr, (r) can be
represented with reasonable accuracy by a polynomial in V. Then, Pn(t)
is also a polynomial in g, and the Chebyshev-Gauss quadrature formula may
be used to obtain the exact value of the integral expression (52); i. e.,
1 P( d -) K M -(5afd= K Pn(gk)+E (2a)
-1 - kul
where
iZr P (ZK)()2 K(Z)!
n
and,
lXI <1.0
The error term E is zero if P is a polynomial of degree -< 2K - 1.n
A more accurate formula which utilizes the fact that Pn(1 .0) = 0 is
used by Hsu (and by us)
f n (i) = ---- (+)Un()
23
Then, the inner integral in Equation 51 may be written1
I,= fn ()d (53)-1
where
Fo - r - )
Fn(t) = (1 +t) Un (t) K (x-, s ,.) 1 -n
This is evaluated bv the L-point Jacobi-Gauss quadrature with the weightfunction V(1. - 6)/(1 + j,) (see Reference 8, Chapter 8). The resultingformula
LW k F n (k) (54)
k=l
is exact if Fn(i) is a polynomial of degree :5 ZL-1, which corresponds todegree ZL for Pn(t ). Putting = -cos 0, the polynomials
* m~ =Cos (m +.L)ecos 1 0
2
are orthogonal with respect to the weight function 11 - )/(i + )
-I +g 0, m An.f ' m(Mtn(&) d&
This is easily verified by expressing the integral in terms of 0. Referringto formulas in Reference 8, Chapter 8, Section 8.4, it can be shown, usingthese polynomials, that Equation 54 takes the form
i L
f I -g F(id =H F (55)n 2L + I- +2 k=l
Z4
i!
where
H = (i- k)Hk
and
k= -COS 1
In two-dimensional, steady, incompressible flow, there is an optimum
set of chordwise collocation stations (xi) for the determination of sectionallift, depending upon the order of the polynomial required to adequately repre-sent the downwash distribution
- / 2jx =- j = 1, 2, .... N. (56)
Since the behavior of the integrand for the chordwise loading is apt toexhibit similar characteristics near the surface edges, it is inferred that thisset should also yield the best approximation in three-dimensional, unsteady,compressible flow. Note that the number of collocation points is not requiredto be the same as the number of integration points. As will be seen later, itis only necessary that the total number of downwash collocation points beequal to or greater than the number of pressure coefficients anm. When Nchordwise integration stations are used, the quadrature used to evaluate theinner integral of Equation 51 is exact for integrands represented by apolynomial of degree -< 2 N - 1.
Hsu shows that an optimum set of interdigitated spanwise collocationstations and integration stations exists for evaluation of the outer integralin Equation 51. By reasoning similar to that used to establish the chordwisecollocation stations, it was established that the optimum spanwise collocationstations are
r= -cos Tr, r = 1, 2, . M. (57)
It was observed that the quadrature for the integral of difference between theactual and polynomial approximation of the spanwise loading is zero when theactual loading is precisely represented by a polynomial of degree _< 2M - 1,and the polynomial approximaiion is of degree =N - 1.
25
.. .. .. . - - - -_ . .. . .. .. . . . . .. . . . . . .. .
Then, by substitution of Equation 55 into Equation 51,
11 1l~ VD 1 - G (x., s, or d a (58)
nrn ri f nm- --1 (s r
where
Lz m _ 2 x r:m 2 (1 - k) (-ar E) " k'±rGor ZL+ 1 k= 1( ~ ( Xj~~r
k11
L
nm ZL+ 1 _k ( k n k -r (xj*-)k' -2rJ ) ; I n
Hsu established the form of the Gaussian quadrature and the spanwiseintegration stations. The difficulties of the singularity'of the kernel at• E= -n the #4'..,.-.- t-~.~
. and ... y of UifLerentiation with respect to (r (he uses thesteady-state lifting line formula to derive the form of the quadrature) areavoided by removal of the singularity at (r = sr and then by an integration byparts.
We first integrate by parts to get
i 1 1 _ 1 - G ( _ -, f , M)D nm j -r -
nm 8 - 1 (s r -E)
which corresponds to Equation 58 in Reference 9. The Gaussian quadratureformula is developed and shows that when the number of integration btations
is one greater than the number of collocation stations, and if they are inter-digitated in the prescribed way
26
M+ 1 (1 -__p) Gnm_ ,__p
nm 8r p M + 1 2= (sr - g p
- w (M + 1) Gnm (ij, r , 2r) (60)
where
rirS=-COS --r
-r M+I
and
2p - 12(M+l) r
r1= 1, Z, . . . M
Evaluation of the second term in the brackets requires the observationthat the multiplier of K? in Equation 48 goes to zero whenever the collocationpoint is in the plane of the doublet sheet located at the integration point.Therefore, the finite part of the integral of the K2 term is zero, and the entirecontribution comes from the K1 term. Inthis case, y(s) = y(a-) and
U_.z = (_r - )2 = 0.
It can be shown that
-2, x >K (x-. Sr, Sr ) =
Thus, the chordwise integral which defines Gnm (xj, Ar, sr) is
nm (xj r -r -2 f f n ed7
If the range of integration is extended to I = 1 by making the integrandzero for E > Xj, the integral cannot be well appruximated by a polynomialbecause it has a jump discontinuity at 9 = xj. To overcome this difficulty,we write
27
xjri fn( X ( ) " I
Gn(X!, st, r = I ) 1 e 3 1I dtnm f ( - n-1
(61a)
-2 rfX - f ( ) d .
The second term here depends on the integrals
x.
h (x .) f (g)d9n -1 n
which may be evaluated exactly. We have
hxx1h'j (x jd+ si + Fi777oCj) f fJ Fl- d x
f - + sin x + +-I
This list may be extended as far as it is needed.
The first integral in Equation 61a is considered as an integral over-1 < < 1 with zero integrand when g > xj, and Equation 55 is applied. Theresult is
G s,) = - 2r m ( k)om -r-r 2.L + I l
k 1 ei X (j1k< Xj
m-2 ( g 0(x)
(61b)
28
G (" o'm(1 - k 2 ) Ur() 2 e knm2j r - ZL+ I k-- 1 U k
k< Xi
- 2T gn (R), n> 1.
Equations 59 and 61 are used to evaluate the Di Is given byEquation 60. Equation 50 is then used to determine the preseure coefficientsanm to match the downwashes (i. e. the Wi/Uls) at the collocation points(xja sr). Once the pressure coefficients are determined, the generalizedforces are computed. A polynomial expression for the ith modal deflectionsnormal to the surface,
N Mn(i) (a, Z b (i) V T
0 0 VLLv=0 i± =0
and Equation 49 are substituted into the equation
TIP XTESf f (i) (j)
TIP XLE
to getN M N M
Q T2 (1/2 PUl I I a (j ) b (i)n = 0 r = 0 v= 0 0 nm v AQnmvi±i 62 .
where
1 1
fnvf 7om+ u Fn(*) dEd,-1 - E+- 1
The quadrature formula given lby Equation 55 with L set equal to N may beused to evaluate the inner integral. For evaluation of the outer integral, aquadrature formula with the weight function /I- , analogous to Equation60 for a nonsingular integral, may be used. This is not practical for a wing
29
with folded tip, for which a different representation should be used on eachplane part of the surface. Here, the spanwise integral over one part of thewing may have a square root factor at one end of the interval of integration,or at neither end. Such integrals may best be evaluated by a suitableapplication of ordinary Gaussian quadrature with a weight function of 1. 0.The points and weights for this quadrature method are not given by simpleformulas such as Equation 55. They are listed in many places, (e. g.Reference 5).
30
VI. APPLICATION TO A WING WITH A FOLDED TIP
The planform may be any continuous surface. However, the computerprogram was developed to treat either a planar or nonplanar planform of thetype show:* in Figure 2. (Only one-half the planform is shown.) To facilitatemodifications of the program, comment cards are placed throughout theprogram to indicate where changes may be made to handle other nonplanarsurfaces like that of the Paraglider.
In the application of the kernel function method to the planform shownin Figure 2, the computer program calculates from the equations of theleading and trailing edges the collocation and integration points for whichthe integrands of Equation 51 must be evaluated. For demonstrationpurposes, we calculate the collocation and integration points for values ofL = 4, N = 6, and M = 10 and show the results in Figure 3.
XLE s tan% LE, XLETIP S FLan X LE + (s - s )tanXLFL LTP
xTE = 2b 0 +atan% TE, XTETIP 2bo+ sF tan ATE+(s SF tanXTETI
L FL
b(s) = 1/2 (xTE -XLE)
x = /2 (xLE +xTE)
x = x + b(s) x
= 9 + b(u)
x. = - cos N-- - r, j = , ,. . .x Cos F _+ll,2,... N.
2k - 19k = - cos L + r, k 1,,... L.
s s STIP
31
2b. ALETIP
A' ATEA'TETIP
TP 5TIP ~
Figure 2. Planar Wing With Planar Synmmetrically Folded Tips
*INTEGRATION STATIONS (L-4)
0 CHORDWISE COLLATION STATIONS (N *6)AND SPAM'ISE INTEGRATION STATIONS
+ DOWNWASH COLLOCATION POINTS (M - 10)
Figure 3. An Optimum Set of Collocation and Integration Points
32
2 = -COS-M-l m = 1,2,.. 0M.
i 2p- 1
0p = - Cos 2(M+ 1) r p ,,.. .,(M+ 1).
We have also constructed a table of equations (Table 1), for thefunctions yo, zo, rl, T1 , and T 2 for use in the expression for the kernel,Equations 48 through 48c. In Table 1, the subscript FL indicates the foldline.
A difficulty .is encountered in the spanwi.se integration because thekernel function has a finite discontinuity at the fold line. For example, notein Table 1 the change in T1 and T 2 for receiving or collocation points on thewing as the sending or integration points shift from the port tip to the wingand from the wing to the starboard tip.
If we consider the kernel as a function of and.o for fixed values ofxands,
q( ,_.) = K (x
this function may be broken up into a simple discontinuous part g** and apart g* which is continuous across the fold lines:
= 9 + g**(Z',o) (L3)
To do this, define
g( LEFL+) g( ,EFL-), E > gFL
= o <o <g (64)-rL CrL
g9 a -EFL-)- g(9F aF+) < <rFL
and then define g* by Equation 63.
More explicitly, for _r > (rL, define
b b(g: + ~ +FL =(F L +-LE(FL) + TE FL J
33
[ I- 0. - - - -p~ 0 0
0 0 a 0- a a - -
0. b bp .4 + a.
0 * N.4 N N
II It + a * .~ .~
- * ,~ .4I-. -~ b * b
S a *.4
.0 - - - - -I. N*6 ~
N -~
-~ I..I.. 3..
0. Np ~ N _ +
? + .~.4o N Y. N
0. N * b ~+ b +
b 'g * * I
U * .~ * Nb - I
I - +b
* *I-.. 0.
Q) - -.- - -1. .g N* .4
3.4 4 b + a .~.- S ~ N.4 Nb a . b~ *I' b
0 I aa ____ 0.
G) p0
~. .~ 4)
- aI' 0 0
~ - ~ 0 a . 0* 0 - -
a b3.4 0~ - a.
Vt~ a. ~ d
0.-~~ -- av 0 ________
~ .~b0 Ni.. a ~ a
U ~. - 0. .4
4, p *g b bN N b
b a N(ri - I *~ * I .4
b b a~I.4 - ~ .4
? b a a0
0) - ___ - -
N.~
N - 141
.3 3..3..
C~i 0. -~ N b.44
.4
Cl) F.. )- .. bN ~ -~ a
aa .3
-- 4 -. 0. b .44 + N.4 .~ ~. 3..
- * a0) b b * 4.* * * N a aa .4 a
.0 - b a~ *
--
0 .4 N .g0. ~ .4 *
3.. 0 b h 4.44 .4
a 4 a aN N N
v .~ a *2- ~ h b
a a .4* S C
-- .3
0. 3..p +
I-.a 0 ~ o
- .40 0 .4
b
a0
~.1
~ 0 0 - - N
34
Then
=ST =Wg (,_ = K (x- F SFA )E - K (x- FLS,_ OF
L L L L
in which
=STY(2:) = YTIP in K (ST - starboard tip)
=W (W - wing)AT-) = 0 in K
A similar formula applies when a- < - E FL g ** may be written in a form
which indicates that it is independent of a- in each tip region:
g ST( ) > EFL
g*( 0" , < < o"(65)s -FL - L
* PT(a), E < - z- gFL
With the use of Equations 63 and 65, Equation 51 may be rewritten as
1 1i i f 1 mf fD- / T n () g*,o) d do"mn 87rn(g )
-i -l
1**
+8-r n( g ST (
1 -ZFL
+1 f-f0' **'-) f 4-7 c d-gPr1 n PT -1
The first of these three double integrals is calculated according to Equation60. In the others, the inner integral may be evaluated exactly. Theconstants
35
1
u = f _i 7 o- d o"
FL
are given by formulas
0 = -4 siFL L .F ]
1 2 3/2
etc.
In terms of these constants
1 1 -
rnn 8r . ..
f1=f 7~ 09) m* Z f f(-) g* * .d dE-1 -1
1
+8"rn_ n(G) gT ) + ( l m gPT1 ( )j dE
The last integral in this formula is evaluated by Equation 55.
In the evaluation of 0. , , given by Equation 62 for the plane case, it isassumed that modes numbers i and j are either both symmetric in a-, or bothantisymmetric. Then the contribution to the integral for o- < 0 is the sameas the contribution for a- > 0. Let the deflection in the i t l mode be given by
N M
V = 0 1 = 0 V l FL
n M N M
v= 0 0 V1 O:F L
36
Then
N M N M
0..=z 2 1Z P 2 )SC I I Iij yTIP ( 2 P a= 0 m = 0 V 0 0
a 1( :(Ii) '(1) + P ~) ( - 1nm vg nm v j. v k. nmvkL
in which
L 1i(l) 1 - rn +. f 1 F()ded_nmvii -0 1 n
1 1
.i T F 2 m - F ()d d
Note that the inner integrals are the integrals of polynomials in multiplied
by (1 - E)/(l + ), by virtue of the relation
b(ff) + - VLE (7r ) + T (2 )]
Hence, the inner integral may be evaluated exactly by either Equation 52a orEquation 55 if enough points are used in the formulas. For the limits v _< 5,n -< 4 used in the computer program, six points are sufficient for Equation52a, five points for Equation 55. Equation 52a was used with K = 6. (.Thischoice was made arbitrarily; it would be just as good to use Equation 55. )
In the integrations over a., six-point Gaussian integration with weightfunction 1. 0 was used. The basic formula is
1 6
J f(v)dv = hif(vf) (66)0 A=
exact for f(v) a polynomial of degree at most 11. The constants occurringin this formula are given in the subroutine FORCE. -They may be derivedfrom a table given by Scarborough (Reference 6, p. 148).
37
In I ,Equation 66 is applied by puttingnmvgl
nT 1. v0" = 0. V
_- FL
The resulting expression is
6 6(1) = T m + k
nFL 2= 1 k= 1
in which
_EJ 9 FL V,L
k()= b() k + -2 gE(fff ) + TE(ZP )
In I (n ) , the transformation
Tr = 1 - (1 - o )v_ -FL
was used. This makes the v-integrand behave like a polynomial at the ends
of the interval. The resulting formula is
b 6m~ 3 F +
L k=1
k (1 - k )Fn(k)
in which,
2o-0= -l ( I - _2 F )vf
k = b(g) k + - 1 LE + TE
38
DESCRIPTION OF THE COMPUTER PROGRAM
A functional diagram of the computer program is given in Figure 4.With the exception of two subroutines named MSIMEC and MSIMER, all ofthe programs are written in Fortran IV. These two subroutines, written inmachine language, are used for complex and real matrix inversion,respectively. There are certain limitations related to the various othersubprograms, which are listed below.
Subprogram Limitations
Subroutine Data
NCC The number of chordwise collocation stations mustbe _ 10.
NCS The number of spanwise collocation stations mustbe _ 9.
NDATA The number of sets of data must be - 10.
N The number of chordwise pressure modes _ 5.
M The number of spanwise pressure modes 5 5.
Subroutine Zen
MODES This is the number of modes used in the calculationof generalized forces, and must be - 10.
NPTS For a planar wing, this is the number of points atwhich the deflection is given in the horizontalsurface and must be 5 66. For a nonplanar wing,there must be -5 66 points for the deflections in thehorizontal surface and _5 66 points for the deflectionsin the vertical surface.
39
START
The first set of data Is read. This Includes items 1-14 In theDATRD data array, the Mach numbers and frequencies, Item 41, and
perhaps some of the mode data that will be needed later.
F Subroutine DATAl performs most of the calculations that areDATA] Independent of Mach number and frequency, such as setting up
the arrays of collocation and integration stations.
This is the beginning of a loop that is gone through for each com-
3 DATA2bination of Mach number and frequency specified. SubroutineDATA2 sets up the matrix (Di.) of Equation 50.
ZE4AR In subroutine ZEN, any niecessary data for the oscillation modesis read in. The left side of Equation 50 is set up for each mode.
Equation 50 is solved for the pressure coefficients for each modeMSIMEC by MSIMEC.
Subroutine FORCE computes the generalized force matrix. TheFORCE program returns to step 3 until the Mach numbers and frequencies
are exhausted, then to step 1.
(.) RETURN TO BEGINNING FOR NEXT CASE
(cc) LOOP ON NUMBER OF SETS OF MACH NUMBERSAND FREQUENCIES
Figure 4. Functional Flow Diagram-Main Program
40
USE OF THE COMPUTER PROGRAM
The following rules apply to optimum use of the computer program:
1. Before attempting to use the computer program compute thecoefficients of polynomials of minimum order in x and y thatwill adequately represent the modal deflection distributions.Least-square fitted surfaces are very useful for this purpose,since weighting factors may be used to obtain a better fit inspecial regions.
2. Set the number of chordwise collocation points M equal to oneplus the highest of the orders in x; and, unless there is specialreason to reduce the number of chordwise integration points, setL equal to M.
3. Set the number of spanwise collocation points R equal to oneplus the highest of the orders in y. This establishes the numberof pressure coefficients anm at M x R. Their values arecomputed by matching exactly the downwashes at the M x R span-wise collocation points.
The iapul.data is read by the subroutine DATRD. Use of this sub-routine requires that, on each data card, the first 72 columns are sixfields of width 12, as indicated on the sample date sheets (Figure 5).The first field contains an integer giving the location in the data array inwhich the number in the second field is to be stored. The numbers in theremaining fields are stored in consecutive locations. If a field is blank, thecorresponding location in the data array is unchanged. DATRD reads anynumber of cards. A minus sign in column 1 indicates the last card to beread; if this minus sign is not present, DATRD continues with the next card.The storage locations of the data on a card are not affected by the sign incolumn 1. All floating point numbers must be written with decimal points.All integers must be at the right of their fields.
The data array is set up as follows:
1. N The number of cbordwilse preaure mo&s
2. M The number of spanwise pressure modes
3. NCC The number of chordwih collocation points
4. NCS The number of spanwise collocation points
5. NDATA The number of sets of values of Mach number andfrequency vo be used.
6. NSYM Indicaoc fx symmetric (NSYM = +1) ocantisymmetric (NSYM = -I) modes of oscillation.
41
7. SFOLD Distance spanwise from wing center line to fold line
8. STIP Semispan
9. DO One-half of the root chord
10. ALFA1 The fold angle (in degrees)
11. x LE The swetp angle of the leading edge (in degrees)
12. XTE
13. XLETp
14. XTETIp
21-30. Values of Mach number. NDATA of these must be entered.
31-40. Values of reduced frequenc NDATA of these must be entered.W. DO/U.
41. NMOD The number of modes
42. JD Indicator for the type of input data for a mode. IfJD=I, the deflections are given at a set of points on
the wing (and tip). If JD=2, the coefficients of
polynomials for the deflection of the wing and tip
are given. If JD=0, the current mode and subsequentmodes are not given by data. They are the same asthe corresponding modes which were used for the
previous frequency and Mach number.
43. NPTSW Number of points on the wing at which deflections
are given (used if JD=I1).
44. NPTST Number of points on tip at which deflections aregiven.
51-71. Deflection coefficients on the wing. The coefficients are stored as follows:
51 52 53 54 55 56 57 58 59
00 +a x+a x2 +a x3 +a x4 +a x5 + y(a 01+a x+a21x10 +a0 x 20 a30 a40 a50 012
60 61 62 63 64 65+ a31 x3+ a 41 A + y2(a 02+a12 x+a 22x2+a x3 )
66 67 68 69 10 71
+ y3(a 03+a 13x+a 2 3x2) + y4(a 0 4 +a14 X) + a0 5Y5
42
76-96. Deflection coefficients on the tip. Same storage rule as above, except all locationsare increased by 25.
98. Indicator that no more modes areto be read after the present one.(For current and subsequentfrequencies and Mach numbers.)
101-299. Deflection data at points on thewing, in the order x1, s1, n1, x2 ,s2, n2, etc.
301-499. Deflection data at points on the tip.
Items 1-41 must be read in the first set of data. There may be anadditional set of data for each mode, for each Mach number and frequencycase. After the indicator DAT(98) has been given a non-zero value, no moredeflection data will be read. After JD has been given the value zero, nodata will be read for the higher numbered modes.
The data in Figure 5 is for a 600 triangular wing folded at 75 percentsemispan, at an angle of 30*. The root chord is 5. 0 feet, making BO = 2. 5.Three modes: plunge, pitch, and a third nonrigid mode are considered. TheMach number is 0. 7, and six frequencies: 10, 20, 30, 40, 50, and 60 cpsare used. The speed of sound is taken to be 1000 ft. /sec. This gives reduced
frequencies of 0. 157, 0.314, 0. 471, 0.628, 0. 785, and 0. 942.
Three spanwise and three chordwise pressure modes, six chordwiseand eight spanwise collocation stations are specified.
In the set of cards numbered 15 - 30, which give deflection data, someof the cards have been omitted. Otherwise, this is a complete set of data for
a computer run.
43
-------------------------------------
00
00
o ZI 14.... .......
oy~ %g
z w
w b
F- I (/2) c co 1 I
H- 0-It
N -
o
44
z 4M
w 44
w 0
0
o zLU 42
oX 1.2U
CD 0 4J
4 c rlc
o. .. . . .. . i n\ < . .. . ' . . v . . ) Q
w 0qat . r w t, Of c4L0E
La.~ ~~~ ~ ~ :::$ c:' o c c.*** . . . . . . . . . . . . . . . L
0001: J M-U' -- I 0 -__ U--
05
-- CV)
0
0 0) 0
H- 43
%o z
a. 0-0
w 0 00
z 0o0 0
.4-. r443 n2
0 0,
00 V
2
ccd
o ov1 0
*)
oo
I L I
HNO
'C N
w(n-
DZI
UI 1
4 '1 L
U-i U - -- LN ::i :-L- IM 0 t- - IU - 0 L- z
46
Iin
00
0 0
4 43 0
4.
...... .' . ~ 4
0- a 4 H4.) cN
-~ 0 4
wl ci
x (In
- 47
434
UW 0
V U)4-
09
0 4 43 43 4)
* CL ID -J U* 0
CL) 4-b00 1.0
- -1- 111 ~
w 0 "
C,,
0- %0 % nt 0Ut -
LL. r4 0440
48
1Vl1. RESULTS
The computer program was applied to the rigid modes of two wings.
ASPECT RATIO 2. 0 RECTANGULAR WING FOLDED AT 80 PERCENTSEMISPAN
CLais plotted as a function of Mach number in Figure 6. Thedashed curve is a plot of the approximation
27rA. R.
Z +V4 + (A.R.)2 (1_MZ)
where A.R. is the aspect ratio. This is formula (6-31) of Reference 5, forthe case of a rectangular wing.
TRIANGULAR WING WITH FOLDED TIPS
The configuration used was a triangular wing with a sweep angle of65 degrees, folded at 60 percent semispan.
Figures 7 and 8 show CLaand Cmc as functions of fold angle(at M = 0. 8), and as functions of Mach number. Figure 9 is a plot ofunsteady generalized forces for rigid oscillations of the wing in the pitchingmode.
49
-3.0
0 =10*
-2.0 0 =30-
CLa 0 =900°
0.00.0 0.5 1.0
SM-0-
Figure 6. Lift and Moment Coefficients Vs Mach Number forAspect Ratio 2. 0 Rectangular Wing at Various Fold Angles
-3.0
-2.5
- 2 .0 C . -
CL-
-1.5
-1.0
-0.5 I0 30 60 90
FOLD ANGLE (DEGREES)
Figure 7. Lift and Moment Coefficients Vs Fold Angle for650 Triangular Wing at M, = 0.8
50
-3.0 LIFT
CL.
0.00
0.0 0.5 1.0
-3.0 MOMENT. .
G;=0
01.0
0.5 1.0
Figtw* 8. ft md Yoinst owffoaients Tit Haah Ymbor ftr 650Tiripar Wing at~ Yarlow Fold Iftlo
-8.0 LIFToe 0=
-REAL PART
--- IMAGINARY PART
-6.0
.00 y e1 0 300
00, 0= 600
-4.0 /0'o .# 00 .10 01
-2.0
0.0 .000.00.
- k
-6.0 MOMENT 11 =
- REAL PART-
-4.0 - IMAGINARY PART e
-2.00-0
0 .0e 1 .0.0 , 1 00 . - 0 0
0.0 0.500
0.- k -
Coeffioleati Of Lift and !Minit D to Pitalh Vf Udta" ftsquAWmq fora650 Trisngular Wiug at K& - 0.8 mmd Various Fold Angles
52
Vill. CONCLUSIONS AND RECOMMENDATIONS
The results obtained by the nonplanar kernal function method showthe expected trend with increasing fold angle, which agrees with theobserved e-perimental trend. For zero fold angle, the method reduces tothat already used by Hsu (Reference 4).
Possible extensions of the method include the treatment of moregeneral configurations, or of other specific configurations, such as the T-tail. Also, any generalizations proposed for the planar case should beconsidered here, such as the problem of a nonplanar wing with a controlsurface.
The formila that was used for the kernel function (page 19) should beuseful in any future developments using the three-dimensional kernel function.The previously available formula was much longer. A special case of thatformula is given in Reference 12. The use of the simplified kernel function,together with Hsu's method of integration, results in greatly reduced com-puter running times. This makes it practical to use the kernel functionmethod as a tool in the preliminary analysis of new wing configurations.
53
REFERENCES AND BIBLIOGRAPHY
1. Watkins, C. E., H. L. Runyan, and D. S. Woolston. On the KernelFunction of the Integral Equation Relating the Lift and DownwashDistributions of Oscillating Finite Wings in Subsonic Flow. NACAReport 1235 (1955).
2. Rodden, W. D. and J. D. Revell. The Status of Unsteady AerodynamicInfluence Coefficients. SMF Fund Paper No. FF-33 (January 1962).
3. Fromme, J.A. Least Squares Approach to Unsteady Kernel FunctionAerodynam ics, AIAA Journal, Vol. 2, Number 7 (July 1964).
4. Hsu, P. T. Calculation of Pressure Distributions for Oscillating Wingsof Arbitrary Planform in Subsonic Flow by the Kernel Function Method,Part 1. Aeroelastic and Structures Research Laboratory, MassachusettsInstitute of Technology Technical Report 64-1 (October 1957).
5. Bisplinghoff, R. L., H. Ashley, and R. L. Halfman. Aeroelasticity.Massachusetts: Addison-Wesley Publishing Co., Inc. (1957).
6. Scarborough, J. B. Numerical Mathematical Analysis. Baltimore:The John Hopkins Press (1958).
7. Mangler, K. W. Improper Integrals in Theoretical Aerodynamics.Report No. Aero. 2424, British R. A. E. (1951).
8. Churchill, R. V. Operational Mathematics. New York: McGraw-HillBook Company, Inc. (1958).
9. Watson, G. N. A Treatise on the Theory of Bessel Functions. NewYork: The Macmillan Company (1948).
10. Titchmarsh, E. C. The Theory of Functions. Oxford, England: OxfordUniversity Press (1939).
11. Watkins, C. E., D. S. Woolston, and H. J. Cunningham. A SystematicKernel Function Procedure for Determining Aerodynamic Forces onOscillating or Steady Finite Wings at Subsonic Speeds. NASA TechnicalReport R-48 (1959).
1Z. Ashley, H., S. Windall and M. T. Landahl. New Directions in LiftingSurface Theory, AIAA Journal, Vol. 3, Number 1 (January 1965),pp. 3-15.
55
APPENDIX. COMPUTER PROGRAM LISTINGS
57
0000000,000000,00.00000000000000.00.000000
000 0 00 0.0 0 0 0 0 0,0 0:0 0*0 00 0 0 00 0 a0 00 00 00 00 0000000:000000000000000000000000000000
0-0 -
-. 0 0- t
4A .0 Z
0 a-ZL)
NNO.
co > - -0X 0Z
N -4W n -
N' 1--0
-~ W ti
In~0 *40 t-- - U
N~e - )L o C"- b-4 -.0M0n0. 0.-
N-Z 00.% 0
NI,4- U.-I-n z :-_-z -. 4 ft 00 * n 14 3W% X 0 tI- N3 Mg
a0t/ 0 0 0W r 0. 04 D LM .- 4 0. X0. 00 1++ 0 -< -0-4 -I + %0 -4r
X. Z' Ui 0.0 0. Li.- I,-ZZZ -ZHI 1 n
0 A I - - - ' - iL) z .-4014 "_- j.~ N 33 jmM m -0- 0~ W- - 7-1 -I : i_
N~ IN 0L +4. 0 N *--- In t.- -4
0.-.40.~~~~.. %0 I NC ~ I I-
- .40.0. 0. L 58
m0 0 0 0000 t4%t.tt~ 0 NILAnImmmm0 0000 00 0 0 00 0 0 0 OQOQ 000r ,* - -I
00000000000 00-0000000000000000000000000000000000000000000*00000000000000000000000000000 00000000000000000000000000
0 0x
z H
W 0
0 U
0X*
CC- 0 0
r- Wm ~L a
S.- -
ob S. 0L4 N- N4 C U..o
LL~~ 0.I.J.b. 0 -b Z o
P-44 Qt cc 0 0 .U.n - W L3 I.
S. 10 0 - N)WV~ "4 "4 .1-0 &- u C
cc 0. %0 0t NN "4_1-
"4 co 00-4 0 NNC -. 44 x(.P412- g. tIW (0 * 0 I.-C of ' 4
C) 0e oC C N to*1 4 W CDv %s.w W- 0. 0.%oo0. - 1-0 ic) .Z v
W W2:X 4 0x0 . W W. OCL m 0 G. . UW N'P-4 0CD X: 0x ow
W, cc CN4 It 1.. - (D LIXW DM4 0. M..- U Z 0.' 0-j Z-5
- Z "4N( Z 6-' - N0 cc N 14*-0 0 - f + _i 0 + t U. -Z &A >
0 *. ftC n %0 N +WW C 0+ Wa 0'. -0 -X<N~ 0-4 "4W 0 0 IL 14 1 0 J- al JO 0-in 4 0 0 &AI NH .4- -"+ L *I-" .W .- t-04O..j40aWx c CieCC cc lb 1 X 4/) e. (AL' - -M -X0 8*14"41 C 0..WC 00% . -4- 4 - W.0 4 mWW- W 0t4 0 4 L %o,- e W'.I4 -4Z <0-4O "4 JZO- DZ-J ZN -Oo 5N
ft~ N u N -1-+ 1- I-H"Z b-IN -. 14 1-"4LL ZX) WCf7 ~CW OWW4<"4Lf..JW4V0.JLIJW0W<W< tW< ft 6-40
0. oe - <1 W- 4-ccC HZ -cc Z Z Z W 'WW-._
0 t-. tn10~ a- CI 0 mfl0 4 tA %0 o0 "4 00-4 0 ItNO 4. 4 0
59
00000
00000,0o 00,00)000.00
z
qj
.4
I-
0
z
w
I-
z4u
w"~4
U.
U
w
0: 0.
S. N
w 4I--
c. '-ON 0
o-4
60
000000000000000000000 0000000000000000www ww W() 0'0 (), 0' 0% 0' o'0%0 000 000000 ."F~4.4.4,4.4
000 000 000 0000a 0 000 0".g'4~~~e~.,e....00 000-00000 0000 0000 000,000000 000 00000000000000000000000000o00000000000000000000 00000000 0000000 00000 00000 000 000000000000000000000 000000000000000000QO0000
M 0
wz
40.
B- ob 4'.
W4 ZU0'-4'W."' o 0(,0.
>.NU (A(1 -V4AN %a JL Z s'q.Ix'. N .0 ONi 0
VCv-0W a 4 U4 i'.g.4z Vf) ON
CID ) >_> bN tu W' (fl.(A
-. NZ 0.4j.- s- 3 - 3 14 xw xtn* a z u - 3C0 X01 Nof-N-O ob o 0 z ccz c: 4 L. 4o.00 .) 4 -4 140C4 M0 0. X .
in 3 0-Z - CL 30.3: z x oN-ZO.Z- W LU)L u-
op <0. -4 a.f CA(N .0. 0. 0 01.0x
-.- WIWu0 94 0 (A 4 A A t 0sk. '.-. ) _#o <1 4-$- 0 1--
%-Ora MO Ci 0..O 0.CL0. Z< .0
- oz Jtn. 000 '.-0 *1n ~4 9. tD .J -Y .- z Z w
Old 4A0. .W .- t!-9-I ft0.tn o.ocic o ~ o4 0uZ X 3
.J.J W 0. a--L t .M n00X4--Z Z- _jc .4 w ZZ- 0 0 X. eI
.-<fl) OI $.<~. 00-D 4 M Z) ZZ-0 -4N tn >_ X- LL. .t N.L L o00
0~~L '..4'N ..- f U. IL U. 4A - .. 0 N - InX CL jOOX r4 XN <Z f-OO00oz + r- - tn 0 )0 - tn cA 0
Wtmfl40mxoo - -- *. %0- 0 (m *r..4 L.
X X * .. '..---%-BOO-- O 04. "' B--0 0-O0-4U09- 4*. .... .4N,-,-0<ZZZZ zZZ-O040w NI--- 'N.- MZZZZZZZZ--44 no N I-NO 00--0A B-ON
4N w00 0 0 9I ~ MN 4N 4 N W4 CN tA N-'-N w -t0 "B-
0 ~00 00000N N
-4 04U%-4 0
61
0 0000 mOO 0000 00 00 0000 0000 00 00 0000000
" NINNN *NNNNNN N mm mmmmm4 t% I t t.* % n n %I.-4 " 4 " ". ,-4 " "W4 4 P4 4 4 4
4 4 4 P.4 e-4 e-4 -4 " .4 4 m 9414 " . -4 q-146 -4 .4 -4 W4 Vol "4 9-4
cO 00000000000 0000 0000 00 00 jOOOO000 000
0 0000000000000 00 000.00900 00 00 000000900
z
10 z
DOD N a D
N m N -4
IZ - U 64
-0000 0011 S. J 0-.14 . UL
0 11 mI .L=0 - ZZII M U- x 1
-- - *-- O ci N * L) P4 a !) ol> *-U
,-4.-l.-4 -W1 L I . - + L*1 L------- A Ar4 -0% C MW MM 0LLW NULtL N 11- - - 1Z -
H 11 fi t<-ZO#I-+ ~ IIZ 04 * - 3 i 0-4-. 4 4 - .04 N( ti )C f1 tN Ctn 4 i (1) LINO f- n --. N C %H-0 0 <OO < X- 4- - -J .J..% WU It 110(j- AN0 ) LN 0 + 4 -
L4O0I4V1 LL W'~ W'.L Z'L Z<. W Z- W NWL-ZZMW0 C jZ04U fII .<L
- 0000-- 1 - It- 0 .- It- t' U-IMC 11 OLU I.e --4-= V) 11 L - II- ifH
0----1-1-I ~ 10.0*- 1- 0
H Hlil I4~4 4C~LOLII4 . * -+ 0~.tf tOL-%s-
4 ~ L L)1.uD II tIL--~L~1-%-- '~t* ~-Z
I- 44<.JL.-4*--4-J-WII N-ON.U+0.ON-+-621
00 00 000 U 0000000 00000000000 0000 00
00 0,0000 000 0,00000 000 0000 0000 00000000 00 00000 000 000 0000.0000 00000 000000000 0000 00000000 0000 00 000000 000 000000000000 00000000000000000000000
z0-
P4
'.4 ZW of
4 9. +M In
6.. - - -'-dl (14 -4 4w
I.- W <NW r4 _ L U.U.UL
4.. s- -. - -3 -1 - (AC
L) C1 0 b4 I.J."J %
_j +A U. z 0 0Z ADUr
<LI 6. ' 4 %%%%.N 0 -4 0L) . * -f CA.-.. 1: CX X-- I- W (/N m 4 NN - I
f.-M 00-, '40--w <.44 ' < I- C
*.- U'-e 0 l 0zU. LU. LI)L ( U...)z x r- -'"4-- JNN..t-itU 0 . ft
>XtA - l-,..i4N s444ZZZZ * 0 U. xI'-.-4V) I- - N n%N *-...tOIAU.. U.- .L/)tI)LlLUUU.Ii 0 * 0 fn (flX Z %0 - NO %0 :.J _j4 1- t0 0 0+ I + 1 0O. 0 -4W CD 0
'4 0 S.fn Z-U .**Z ZZZ 0 4AU *4 I%. = x'-4 . WVZ AAA---l+Z NW.-40 4-4
*U .. UULJOO nn* 1 0.-4 qu4~I- 4~ l-00000O 4 It N If 0 OUZ -Z_7OZI_l-
<N .N NW.WW=)N N N ZN N N N O.XC.X-4-4L*N--4N #-4W<<4
.4 W "- b--< z zzzzmmtnm~u w. N H zomC.I-wo0 0 X _WO W>0s-4UZLLL)'-4LUWOO Z
-4
U
63
00008000000000O0000000000~ NN
0000O 00 000 001%00000 00 00 00 00 00 000 00 000000 0000000 00 00 000 000 000 000000000000000 0000 00 000000 0000 000 000 00 000000
S. In
M S.
W U,.04.
,.cz * - Il U.0. -4 & # I4-- f
- .9 r4 ) -
14NU N U0..Z . 0
- CL X.~coO) 0 x N D..4 W m.dI U.
in S. Z V) u
N N 4h 6- . In
IN> eN cm 0 (n
N rZ 1-' - a. N Wj
- - ZU) -
ag-0 C 4 V
.40 .- 4 2: " Z - ( N' -J*..4M - -WU-<. i-
-- ZO.Z' N -U.. N W
a' .- )C 1-- 00.4C3i
0 w<4 W-4fA .4- -NIn-O 0 C U N Z to 0.
0Z -i t L S. . I.) -uu.0.. N .. (14a4>
X .U U.0. CC XLL N014 - Z W _.j >0' -M u44 U) 0. 0. C,4 N
30. CC 0 '- ~-') 0.J - 4
z Co 0. ft 0 (1 Go - *00 6-4.
N C-4 *-LL - CDI-. CL - 0 o) .Z~ wI $ ON Z m~< - Z OY*' O-Z 0 0 N U I- CL 0-0.4 1 *1 4 s-1
l"0> U. 0-.4 0 00 Z .-4)()( Ua-XWMLZ-U ,- ><in ,% tLU.' c -Lu% o I.-a + N - + * * s-s CLb4
CNS~'N'~U40 X #4-NN *X W-0 m'- H WZ0o.-4XN4CZwx -IN LL N> I- - M. I Ne-
WtA,-0mxoo Z-a -1 a-0.)(- 0.0* s-0OXNZ<-Q0UOUUZZO-I--40 " XZ,-CIO 4h. 9- W " *%. 0- Z X
3CX ,,U -,U .",0 11 4a-0 0 - a+ 4 c)C4 OU. " ID'4% %. " -- - ) - C * o 0 J0Qa. D a-N *~ x t N 0 -Wt
DZZZZZZZZL4U). -N O0s-0.-N) 0N N 40N 00000000oZZnw-1aN 0 0'-- N CC- 0--oI U %0 i-ai-X XX X W-M X 11 -J.-4X- N 0C~f-4N.4U.5- 4 W -J 4
0
64
0o0000000 r0o000 000 00 0000 00 0000000000 000
Nm N N NY N tomm of in (n M I m % 4 -t NIt- 4 It , -t% in In in in In In W% In tA V% o ~o
00 00000000 000000000000 00000000000000 00000 00 000000000000 00000000000000000 0
00000000000000000000000000000000000000000000000000000000000000000000000000
x
-~ 40
2: - 0
04 0.Ll0" W4 N
ft tu 40CY Oe O
N . a-% j N N
w W
04 ab N . -1-1 -1 -
,- X 44 <jJ~
W- 0 W 00 NnD 4 Y-4
. 0- 0 1- - -XXz x 0 ~~U.U.. -x
z - 0 Ce400 .0n 0 0.J CCa t0 w- -W-
00-O0 000
I- 0 a -44 - --- 0 -->- 1-u -0U. MMO -J XX 0 0$.-ND4>
0 - - W(O W N 40Z*:-L LLA ZNaLt LtOzz=) '- 1Z) 0- W -0 tm c U NN NV-N 0N NN
-In)-w -- Z -0 X 0 e 4 * aa eu0- -Z ->-N ) 7 o cn .4 I- >-0> I - - - 0 ---- '
*n01 0 aZ .40 N)L..0,- 6IU-6 ,*"mN_ -S- -X_ -4N -4 N . 4
00 .N0w -& 4W X X u *C .6-40 0 n X U (---X-X- 0 xx0NW.-.'--
* o 4itO00J , -- 3~ 00 *7 0% 0
~~-W WN~ N4 N- ',4 0 O-U0 .A ZN4IL.u..zz0> .4u.~.*~n ~ N ~ O
~ I IL)
N N~~~n Ii-~--O65
000000 00 0000 00 0000 0 0 00000 00 000000 0000
N ofb % IA '0 r- 0 0% 0 .-4 N m~ 4J 1A 10 r 0 O0 0 0-1 N m 4 in 0 1- w0 0 .4o N m' ~4 in '0 P'- mo
N N N1 N N- N N N4 N N N N N N N N m N4 N N N N N N1 N N N N N N N1 N N N1 NY N NS
0000000 00 0000 00000 0000000 0000 00000000
00000000000000)OC OO)O000000 0000000 000000000000 000000000000
0* z
9 :3 -4~ _.
w * g
40 :c
>: '-I lid U.
o - U. -. - .- e
'-4- 1 - -
< I (V V"
z 1 ~ ''~1.e J- Z* .* .. X CD L)WW 0. -
- *CDCD 4 00
.4 I/AI 41' 0 4 4 6- + + 0. (n)
W. '-3 Z A :cf tAf _j >- 3Ez .4
(a N V) In4 () < 0.-L) 0 z- If, 0
1.- -0 f lc1 3 :3 1(U JA C 0. qo$e4 -4 0 N
Z N * -4 11 1 - oil- .4 CC 4 0 O.4' - -oN I
0- *WH)C <-<.4-< Hf 1*0 11 * it 3cX WL
* Zj W 2: 6-4 - .4<'- D 041 C n0 -I U. 0 z O -. 12: .4
W W ."40--4 -' 4 L 6W-4 0. _.j X-4N _.j - 0- 0 * . - 46
W 0 >-X~ Z + U.)ZU.)W w -P%4W d til 1111W1 CD X S- 0.
D 0.4 1 W In+ 0 )- *=)- : r ~* It N :) W 0 0 n Wo
ZEOO00ON JN.4-JNZb4ZZZ W-Z 4 4--OVI~
N1 0 az% e-d 10Z'04-- 6 ~* 'o U..0 o-n .O 'o n n -4 - Z %o %0
- W N .-- 4~ If .4 U Z Z I W -4I J.J-H -.. '4-'- I-- N-.-4- HN
4 ZO Z ILI ) (n )InZ >- Z 0--Z z-- --- 1HI
0 3::c X X 0.044.40 3cu0<wWuwa IU
'0 .0.00 .0 %0
r4 -444 -4 -4
66
0000 000000000 0
VO 000 000000 "-4-4-
00000000000000oo0o0oo00000000000000000000000000000000000C
*e 0
CL U.N
w
Sd cic
0. 4 U.
-n b4 LL
LA LL7 WL
<4 X O-
-44 * - 4
.4 * l
fn N Z% L-S3f .In t( -4
t~W %0 .0 W -4 (n P4M - u - -.D.
X -I.--I- N04 U N Z-
0 <m0wo0wCZowo OO~ZM>.U3033UL. UUWW
o-4
00()%00 U. 0
%0%000 0 UN rfUNP4 r4 r4 v4 r.4 L f
67
000n00000 L000000000000O0
NNNNN NNN mmmmmmm000000000000000000000000000 ~l~ lb(l' (b WbEt I I I0000 000000000000000 0000 00000000 00000000000000000000000o0000 000 00000000000000000000
z -- W
S. (A 1-P-4 C )0
N u~ *
0o 0 :z s .. w 6-
WU N(n (D0 ftI--z (Au
:I COJ ->- *N (
-Z Z-Lf
.0 0. -4
o 0 0 * 0
ft 0 - <N L)U.O . U
4;o0 l'- z
4 u-w wLL - UO W Ncf
1:n~ Z C3-0x 4c0 NXLL I-i- N (A l(X
C -4 N ) '4% OLJ * -t .) tfl
W0Z0ZZ~+ * * U 010LLL)04. +4- ++ , - -+LLL
MZZZZ<6-.Li-00 000 L")ZtD'AIL+ ..-. >-Z
> Z -4)0 -4 :- -4 N 0fU U~ 0 0 g> -4N 4>
(i)~~~ ~~ 000 X X-L. 0 Ui -N) 0 :)0- N' X X o- )- c0
0%
68
0000000000000 000 00000000O0000000000O000
00000000 00000 0000000000000000000000000000 0000 00000 0000000 00000000 000. ~0 0 00 00000 000 000000 000 0000 000 00000 00000000000000000 00000 000000000000000000000000
UNe
%0 fn
Z 04
W O0
N r400 0 (n
N .0
co *Otfl
0. 0dCO(
N 00
~0 *0
.~Nf' N0 00-0 N40
NZ.0 *t!
Z~ Z N+I-(
l'- - %"%1 0 6 0 -yU n M n -4N - %U
N NWnNv=) m M .
N H -4 If N t IUI tION'.. -Z44 -tO . 0NN t - m*A 0 %-1
Z0rt eccUC 3 0 1 )0MML LL 10 ( L 0.--> *00(5XL.4~N N -- ' U ) UU.* U
W~i 04N c -nD - . -
ZZ *Nt- 1- *HN*tfZ'%+ + I*+69U
00000 000000 00 0000 0000000
0 00 000000 00 000 0000 0000000 0000 0000 00000 00 0000000 000000000000000000000000000000 0000 00000 0000000000
0.U. U. N
CNJ+1
U~Ji
++ 0tn I
ob 0 00
N* 0 ~ ** O
N le 0 *0
Q 0 IN% * 0--
0 -WU. N N -40 vr4 NN040 4x 0 j M * s-LL U * 0
Il W* AIt *u -- 4v** *
W11 *I-wLj4 I U~f)LL. U. 1
o ~ *Z*Nv* - 'Nr4"4N 11 ItZ 000 4.U4 CO U 4 4 X 0Z 0 0 4.4
'-o 0 W
.; NOH liIAN 1111 0N zz wZ N(. 0 II tu- It ii HO It 0 cr
N N
70
0 0 0 000 P4 -4V4 i-4 V4 V-4 r4 04 V4,4 N N NY N N'N NI N NY N enl M (n* Mf M ("~ cn m M -tIt -t4.44444.....4.....44444 -tt-t4.t% I t" tI "tI r' *% t* t% t t' t' tI tI l tI 4.4t4.4
00000000,000 00 0-00000000000000000'000000 00 0000:00 0000 00 00 0000 00 00 0000 0000000 00000000000000000 00-000000 000000 000000000 0000 0000000 00000 0'000 000000 000000
0
-0 0Z0 V4 Ob N
'fl-u -
.. 0 N 0- %U,4 V- -
U 0. o0N 0- 0o
0. . --
.N N 4;
co. >- >-0X ( 0 0
N-0 x 4 0
N 0 0 W 4
4.94Z 0 0.4
0.
tu, 0 '-4 ID
U%
CM N- U .
NI- x0 4.%0 - 0 0 w0-
I '0.o- - M- U 0.9 N 0 0,
Ifl.-44.4 40< I.N 0.U -- - r4 0, m aC.*ZU 4. 0 0 N , z
Sfl> 00 XOO X- o -- 0 1.44U N 4. '41. -- ,04 0 4 t $4 V4.0N
N 0 U -0 v-4 .4 . XQ- X *0 0 II.I--
*.tf -4 0 ~ t x u % 0c.D -uiI -0 %tU.,
3:~ ~ ~ ~ X %% Z 4.0 0-W % -4 0w(
ZIA> Z0.ON >O- OW-- 4n. 0 V - - - c MtUO".
wNx x XX = O 11 0Z0 4. -j U, _j
N u UU- U0 aw '% Z 0 un 0 _ X .0 N _-- Z-- u -1 Z a a
OOZ.4(3 W . Z W- N Lcoa..... 01- 4.
WU~-4Z0 00 4. 4.tZ.-.W4 Z--' 4 4
Z-- 00 ZZZ -. -40 * N- 1 .I--I DC.71.
000OO0O00000 000000 000000oooo0000004.000000000000000000000000000000000000000oO00oooooooooooooooooo000o
0000OOOOQOOOQ00oo0c000000ooooooo0 0 0 ooo
z
W0
0 X-zz
'-4 z6. 01
P14
6..
-4 -
N to
6.4
oo u NS. N ZL-U
M. L 0 -. -6NO M i It LL CLN0N f, L M *- N N -"NI: U 0 tnIl ' 1-No C(A- 0 n- NN 11 N= =4 0
+. + Z -% t u 6.-I 4 7 04 S. b a '> N4r4 N X 64 - k4 00 > - Z - 4 b X( 6-iN
CCCQ Z -Oou) -t. **1 + It 4 Ill-tN + 0 ).4 aU.14I-- DW,- >-.-~ " NI4I Hu-4 04 W) - Z - .t:r = * N N Ne-iOX444 (A) 0 1N-J If -- 3 N 11 c N %0 W's VA -%0s G%4%t.4NN--.+c00 (A 0 .-0I I J-, .44a6- N .N N N .JZ 0. 6.6 CL.. 00 -H N - I-:-WUI-- P-4 .4U
wJ ---.. J>.J > 4 -N-ZO Z 3:Z IX4&Z Z --u.6-.CC.-4 WJ tN 0N =N~LomN Do NP4 N 0 !- 0 4 f N cqi0-4 V4 4- .4 N N CNI 4 m
72
0000000 00 000000-00 00000000000
0000000 0000000000000 00 00000000000000000000000000 00000000000000000000000000 0000000 000
31 3
.40 VcN.
to Na 0
0b *>. o* o
n Xo C)N - .
NM *M 0-X_ 0++6" W 3C N- * -O0. -1 0 X Z - *
-. 0 S. 6-d >. N
U4. 1^~ b0 a + - a _j-* U00 w 4NmfZ XC.- -. 4 0 4 2tfn NL u PCi Z 0*Z-
0"4 * I- + % t0' a~4) U.OL 00 0.co 0 94 0~ 4C A 9 Z 0-30-- N NN -N Nr4 NN"N N 00-JWO " %O.-4No N C4- N H~ - W
s-N N + Z-.J - - - 0 - . -.. sa'0 6-0 N N I-- N -4 - -+ a .
P-IV.j~N _JwooUw4w +~ *Z 4. ~f
0% rn 0
73
00 000o 00 0000 00 00 0000000 00000 0000000
0 0 000 0 000 000000V00000- 1 4V- NNNNN NN Oof mm mm m omoMoo&n n 0 m00minwin00m0ninuN n0n00mm0i i00 inn N n n0i i i0i m N t i00 00000 000 00000 000 0000 0000O000000000000000000000000000000000000'000000000000
N %0 I
IU U - .o U.
-- OD) 0. "O W%>,-L) U4* t0% 0-4(i)N MIC 0%. -
'P.5 ofU- t V4 r.- b-d>-J 0 L 9--00 0 - 3
ONJU NUL 0 0%' 1-0 X0 X 00 et:0% -
IaOX - N 00 a "-o -.4
4 0 0, 1- 1, D -N4Z (A) 0%0 C7 Ln I-N . 0-- NO "0 * N0. X: NJCL Ln 0' Z q>- .N - (P.Ll 40 ZN)L. Z I~ 0n 0 0 0%-t"J 4.S. ZU -4 Ill 10l0
N-0 * .Q 9-04 a% 0 C4.IA) tD u r4P0 P4 CID .N4-Z Z " -d.- N~ 0 N <4. 0 .( N 0 N - 4. C) I- ae xNYI-4 0- - 9-41 0% 0. ZO 0 clW1-- 0 -4 0.4 t'- 0 0-i 1 4-<I)a W 1t < 00 t 0. I
OV.. 0 .tD r . x - Y4InQ.nto u -.-. . (n0 0 1--- I. z 0 Q- -. . - M0 ce D* (n L e.4 0.0. W 0/ :r c 1- <
ON% 0~. 0 a b '0.0N 0 - N UL 00 N. Q)J 3:U. *-~. -Z~U' co CL 0 .. *. CA 1--6. --400 ftN0- LL0 4- o a 1r .
U. cet~ O. o in % D ~ I oI o eI in ft N- HU. X Z ULL)D0% C 9 1 "44O 0O0-3o., M %0 - L L0Ocu>cW m UUUMX 0 X 0 uJ 000O3 O O IX- 4 tI )0 .t
l..- Ia '0_ Id%,N % " 1 j > + + c
UU.
74
00 00000000 0000000000 00 000 00 0000000000
000*0000000000000 00O000000 00 00000 000000000000000000000000000000 000 000000 0000000000000000o0O00000 0000 000000000 0000000000000 00000000000 0000 000000000000
LIL)
r)4 +M 0E 0 -
* U. U.c.tI.- < + x+
NU 1 % N o*
* * 4 o - 4.-4- *4- VV4 r4 " 4.4 DI 0--.- 4 z 1 : .4
0 ( -40 .4 =44 N < NHN W*4 O . -++0 "wIzn* _ 04-_ d_ 4 J4, _0wN wOO OZZ if if aJN-W W*C d0 * 0-A ft~ *.. . * 14 - , 0 * . 4 >.
x o - N + -~NC W II,4 N0 0N.-4 Ns4 0 N4 No o- 0 NN- XCie CD I )IN Ok'.t 4 4 N r4 o41N N NJ.11*- N H 1P4- 0 t- NN-
0~~( 0M rN >( 0 -4Z N0 N (DI ~
N r4V N C N-4
75
tMY 00 0 0 0 0 0 000 MMC, 000%
O00000000000.0000OOOo! 000000000000000000000o000000000000000000000o
~000000000000000000000o
'-44
10
I-
4 0
-L N 0
* b ZoL <~(L
r4 NN < o
0 l = '-4N.. ON
I.J it N IIII
* +. -t zLm
N N s
U- 4 - L 4
0 0. x WZ N 4 N c c* -i
O x o U.. * * 0 --NZ; +: U .. - X ~ f
If 6- N4 U- H + 4 % .
0 -1) < -4 + l- uccI-Z I-O WWNC% D N X( C C 0 'W<0l
C: * -Z 4 N0 r4 - J- 1 LLL00 4
O 0 P4 It L II 0u I N 1 I 1 t * ~- -
0 J-40 1.1. I.S - U- -ULL- X W WW ~ ~ 1 Go 0%.- '-4 0 rr4N
0-4 4 44 + onie -t-
76c
0 C, II UU- O NJ lli~l .- Q~-
0 000 0000 00000 0000 00 000 00000 00 00000 00 0NM r m *a In'1- 0% Q .6NM 44 W% %Or0 V- w 0' 0 -4 NM 4U %a I,, ' w 0 00-4 N m4 4 mn '0t- w0 0 0 00 00 0 9-4 '.4 p.4 94 e-4 p4 V.4 P4 p4 W- 4 N NY NY NY N N N N n fn~ m4 m4 m~ f% mf fn'0 N0O %0 .0 .o 0*%0 %0 %0 .0 %0 %0 N% .0 '0 %0 k 0 . 0 '0 VO .0 4 0 %00000000 0 0 0000000000000000000,00000000000000000000O0000 000000000000 000 0Q0000 000000 000 000000 000000000000 000 000 000 0000000 00000
w
co
0
"4
to 0
P4 x
z; 4 - S
Sw. U. cow
4.Z W- U) o
lb. D LLZ6-
4 0
ON %0
IL '4*WZ U
o'4N- W bou - xX-x - -. N t- J. D ' 0 -4 0. . "0L'4U 4. HC x 4n *-XZ ZO
Zf.4N- Z Z-' ZX>0L i - t4 ." . f b- aZ Z- .
"0000 ZOO + .J-4W N -4 -J,-4 0-4 0 4 .-4 CL.- *.4b-4.-4LC. I- .- e. "Z W aU. O~-e 4 N W H e N X ~N > Id HW 1 Nwc notno>-Xww Z " N'.- '-+-On-!IO 6 L0 -~X - z- c'D mZZZ..jl-WWOON.jx - N 6-4 )- N X .4- N d*A cWWWOn WON N Xvd-NX04Ztfl-tA -Wr4 t- N no -HN 4)I- M ao 6. NZ~ .O . ZO ..-. '-' H 4 1-) N .-4 0.H -1 0--10
77
0000000000000000 000 000000000000000000
0000 000000 0 000 000 0000 000000 00 0000000000000000000000000 0000 000000000000000000000 000000000 0000000 00000000000000
00
LL CD L1- 0.: 4Z~ (-%I- + r
LL ~ ~ ~ - 6- II- 1-. jr
Z I- Z - -- t-CL : 0 L b- "I L -
.4 - - - - - .1 0: - 4- Lu_
N / 4 < 1UW. f - -xi i 0 Zu1-1CL 0 1:n WW - X D" i C -0- c ME 04 o. I-7 =(D
cz ~ ~ -0. < 1:4 -4- 1 7 -0 -Z
D b j-7 0- 1 e4 1 - -10N nr - 1oV)~~ DZD- +: -. *-Z 7nI- -+ -
1. Zz- -.1- a1 I. " It 4- .- *.%. 11 11 -1 i
HzI t0IIlW.- -j LD 1 i s jx0- II 1 11H.0 WIf-HL t0. LL oi LL 0 i f WW.lZ' U Ce- 0-'c P--*.. 0 Zj .d- 1a< J-
LI) i-i i- - I Z J 0 -4 II.4 ii H l .J4
78
000000
0
8 z
-wz
79
000 0MA 00000 00 00000 00000 0000 0000
000000 0 www000w)%0%0 0%0%000 0 0 00 0 0 00 0 000 0000000000000000000000000000000000000 00 000000 0000000 00 000000000000000000 0000000000000000000000
4
00
0 z L
. -4
0 Z 0 L0 q0 I .
- I- 0. N U
0 00 ~iN CL
N0 .0 4kI00. 0 a r- -
. 00 NU64
o0.- 4z l. - 0 -Z(t 0 n4 0 N >- N)0)o .. < z e. ft 0.
fn 00 * t X. t
9- )N 4 + : CLx() N 0 -, ~4
" %k. LL +0 -a CL >( X "N -.- 'OODz >. - Z >- xZ )<
N---V).4 0 -Z*(- 0 - 4-4 1A.-I04 'W 0 1 0 .-4m-. N" .- 4 0 0. 0 0
Wu 0DN 4D - -Z-0Z - .40 11 -4ZZZ e-) 0 -4 I NN.-"4 +Z ftm "M-000<z 10 It 0- - 6I C)00- s-W 0 0-4-..4
o AU-.4,4 W l'Z1- -Zazz in0-Z 0111o-Ig*ZNtom -'w<Ow4w
N WW 40utnZ-N iNt- -4 Z *- 64 11 It 1-- 1- 2: 4 -
000 0 00Ln -4N In -4 #-4 -4
L.)
80
OOAO 00
p4 .40t000e
000000000000000000000000
W%F..
N0
0
4"> %0
%.0
wI-- (A NZ I-N
0A -4.
81
P-1 V-4-4 NNN NNNNNNNMfM
0 000000 00000 00000000000 0000000000000000 0000 000 00
0000000 000000000
-
0
3
0
0D-4 NC U
co co r- ,'0cc 4Uj f 0. X -* I
x 1-- 0. 0. * 0. &SU.. N~rc M'O4 :Do
(A) * - -
b.. J.4 U- If H- 11 1-- 11
U. D> HLXULLXO)<U.)(o)wzu.w n - W. .- U.. (0 w. .- w 0 LL. &~ w
N rcn %t In 1O r- co
82
00000000 00 000000000000 00000000 0000 000
00000000 00 000 000 00 0000 00 0000000 000 000
V 9-
0 z0D
x9 49 __CID4 Ci L WZ
.. m~ N u 0 w M
0. ae* to of coz co4 U.WU - w~
- mz I.- cox
W 0n3 009- 4zx"Z
K ZZ40ZOn-ZZO ) i
0 9 C -e-' D U M . n
q% W-4 CD _ _ j Z .
5dW = - 0( -O AZ.
ZW Wo U.J 0cAn9K 0 tu DW0*CD (A. 0
K- ( -ZWdUe - :) (J ,e t-4 0 In to
tWW ZWO C 0 omZ 4 r4 ft t-4 1 1 %O%.-4<-NNMoxoZ."-4,-x" 099'4 MOWW044N4N<N<.NN%
tn- aU6, 0 . 0- W 0
W Wo W -W 0u<. 4 a-4DWOnWE OULb- X IfN
ce ~ ~ 0 .U) > W 0 *L( 4 :- ux J _ 0 ma-JfN-Zo" Q > .-J i ~ 0 00 0 0
14 0 K 11 LL I-- <. 0 -1 < xx u9I--I9 ------
W 0.W
WmC
83
W~ 00-4 N M -stlt %0 P- 0 0% 0 -4N M Mt %f 0 f- OD 0' 0 '-4 Ncq --t in %0 t- a 0 0 P4 NV M~ *
0 000 0000 00 00 0(cO0 00000 00 000 000 00000n )t nLn n t mn i n I % oinw I1nI inA in IAALnII m AAIL mA IA in u) inmminu n n
00000000 00000 00000 00000000 000 00 000 0000 000 000000 00000 000 00 0000 00 000 000 0000 00 000000 0 0000 0 00 0 000000000000000C)000
co
co
Cc
-J
be
"-r J . 4 l -t 4 " 4 4 - dr+ t f . f - " r 4 - 4 0 bv - f - o a
< 4V 4- t0W
CiN 30C 3<x<<4< x-
N- 1'- I- - - - ' - -44x x j<x x x x x x xtA 94 t^000oo-4 1-- 0 Mtn ot 01)MS00.-0 04 00
S. S S. . 0* ,-4 . S
N wN -4~4-44.4~ -?~ I -r- . .4..-84,-
0000 0 0 0 00 00 0 00 0 0 0 000 0 0 0 000 0 0000O, 4v4P 4 41 4v 4o4v
mm mm mna nani nin in mm m m in mn mn m in mn mn un mn m m in mn mn m m min mm I in LA n mm mn0000000000 00000000000000000 000 0000000
z
w
9-94 3++ 10
itI z z
9-4 P49 t t b0
S. + 1 . Z N 1O 0 N t-P.4 P4 -44iM1AA.4 v4 0 NM%0t-0O N a& N t- f.- 6. 0.l' tn otni4 M + 0- .+ 4 - 0. " ft t t l.-+ 4 -4 0. 1- ft I 0 t"
s( -Do~t-0* I-- .J0000-40* 4w- N -J<4<I- < <0 0 -
V-
mn Mo~~ 0n 10.
94<
85
0000000000000000000000000000000000000m E 4" -.0 0 0 V4 N (f 4 MIA r0- M 0% 0 -4 N tf 4 MA -4 t- M 0% 0 '-4 N Mt~4 MA NO t- W
N NNNNNN m MM4 -t 44 t 4
0000000000000 000 0000 00 000 00000000000000000000000900000000000000000000000
z zat D
LU -j
zN
>
X -4
-. -0 .-4-4 N.- 4 LD0. 0.4bLa. N 4 0.- 0 ft Z
.0.0 %0%0 -. 4 X N 4 (n 0 0 D fn 0 4N4 JN ..- 4 t-- < n%4lID. < < t- +N ft"f t"f + .- + CD o+ cn* +- 4 x t . t
0COcO N 4 *-4~ GOW 00000<40* *4 *4-w* *. L 000to
Li, wI z
Xf U.Lfl4 - X ".-= W ~--tl- 0 Q0O
%0 %00<4 .4 co- 0% -N <4S 4 4.
86
000000000000000000000000000 0000000 000
0000000000000000000,000000000O000000000000000000000000000000000 000000 00000000000000000000000000 0000 00 0000000000
CI
'a
LU
x.
0.-4
zz z :o- IZ
t .-4.J r: t!)N
.4 -4N o - - O 4 M .0 M 4 r- 40d4r tOI n n V N %t 0
0 *0 -it4 c* 4 490 0 0J* c o o *~~4< 0 N <<<000a*
z0
bd~~ LL0~U 0 - Z >n )-- mm m <
-4 .4 .4 "4
87
0000000 00000 w0% ~00O4N000&A000 w0O0 4N tOOa -0 0 0 "4
W WC 0 00 j %C.0000000 0 0 00 - 4 4-4- - 44P4P4NN44 4 . 4 .- 4, 4 4NN NN NN NN NN NN NN NN NN NN NN N
00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000 000000000000000
-o
z
w
z z
-4 z + -
ob0-4 z 0. +,0 t.4of
<. -4 At 00 jGO<co*IS. . S~4 S +. S.5.S. . S S
'. O.4U~O.'O 'ON Cia~c'O-O'O~'.~OD- m
0wu
a x -- xx x 4 x0 3:1-- X4
.4 4 N* * -4 '
88
a 000000000000000 000 00 00000000 0000 0000
N N N N N N N m mn mn mn m m '4.4 4 %r % %t4 -r N44 n mn mn mn W% mn an n, m'
in i Wm 4nIn kl U% n I 4nm 0 m n k tA n 0m i 0 & InIn UN LA0 W in 0 m00000Q000000000 00000 0000 000000000000oooooo00000000000000 00 00000000 00000 0000~00000000000000 0000 00000000 0000 00000
bi
zz
1 1 4 1 1-4
o- ft 4-4t
Nt 5n df 4 m 1 4 ( 1 r- -4 n
.0 .ott% -. ' - ft*brn 4% 0, 4. 4 .
*4***A..I* * to co co n *V4 *.<00*4 000000 at 4
%0 *0 .0LA r- %0 N N N co 0-4. N MN NYN 4 4 4 N NY mn (A C
4 t 44 W4N en 444 4
89
0 f-4 N m 4 tA %0 t- w 0% 0 u-4 N m 4 A %01%- 0 0% 0 04 N m 4t U% 0 f- 0 0% 0 u-4 N mf 4 U% '0
000000000000000000000000000000000000000000000000000000000000000000000000000 )00 0 00000000 0000 000000000 00000000000
co
OD
ccL.
-t
z z
Izz ZI III -I I -44 -
4I 0. -4 II I@- o. ?. 0. " 0. ft
S.. 0. 0. 0. t- t. 0.4, 0. (n rS. ob--* a- enI N000' so - c ~(n.4 P-4Ui tn .4 r4 o~f I'-.t--I.,"4lM .-4 0 tN 14 I4 4- 1
~9O< OFM 0 3 1-- Cy L 0CDX I--
-iL 0U 00 -jL Uw01- .4
m -4 (n v4 co G .
90
000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000 00000
b
2
00
3 occ o
+ + + 0oc I- a: c 0w oi L W W 0x 4 LU t
E~~ b. v4 s-4 N ~ ~ "4 W Or4 U 4 AM0 '
ID
Lu
co LiJ
cow Lu * *L '(4nJ .
91
000000000000,0000000000C000000O00000,000
NNN NNMMM t - %4+ 4 t4 t tmMO nO
00 0 00 00 0000 m0 0i 00 00 00 00 0 min0 m0omn60n6mnin0 no m o0000000 000000000000000000000000,000 0000000000 0000000 0000 0000 000000000000000
b
w z 00
0D z
z W
t!:) 0 00tD- of z 4.- z
_j oC 03Z n X - ZW ftxO DO
* Z <OUt- ZZ cc~
b* ua WXDU>XZ -i- I-
X OqQ.- 0 O - 0 -JC U)Z c
N - i'4 LL :3 xD M t Z n- 0 W 0 %0 00.1CZ -4 0Z0f
- - ZI-<0w0-
L -Wd 02~ >2. " -I - o 00r2: - D40.o .. 0I-..J0f.J( r4 .4Z
tn 0 "ti -- n o
LU0 W:0 m~." Q'--W4 L<
X X-J -3*.-JC.DW)-ZO 00 > _j -6.4 in OZ OIt LV) <r -4 -4 -4 <+
0 L) UI-)W3eZ L m-01- u. < 0. -t- 0 .- +(.-nL + n ) u w V2: ~ ~ ~ ~ ~ L CL-Z UL ~ 0 . WN..~4**4II'. N N w
-W 0 W .0ZVo~Z *-4-~ *' 4- 4-u4U)IU Z-I-2:t- 01-- I-40W 044N 4N4(.JN
I- ~ -.. o-cl0
~( *W V)Zc~>LflW e9
0000000 00000000-0000 000000000 000000000
000000000 000000 000000000000000000 00000000000 000000 00 000000 0000 000 0000 00 000
4 -+z 4 4 .4 44- 0' ,P-+ ++ ++----+ 4V 4 4S
I- NN 1v4 t~ *M
9-4 * b 93
0000 o00000000000000 000000000000000000
IA~ 0 00 0 0 0 0 00 0"4r 4 -4@44 @-'404'4 '4.-. NC N NN N N N N N N in4' (AM M MW Mm .4t .t .4 .-t %t .44 4.4 -t.4't 4. .. -t. . t 4.4 44 4.t .4r 4t .4 4r %t' -t .4. 4.. 4.4.4.4.4.4. -t
inBA mBA oA in in Ln kn w% mA u% "A o Ln tA mA in in BA mA BA tn BA in wA BA &n in in kn-tA BA BA in in0000O0000-0000 0000 0000000 000 0000000000 0000-000'0000-00000000900.00000000000000000000000000000000000'00 0,000000000000000
w
-4 NN N LL30. 0.ft 0 0N
4 N N U. Zn0.
*. f Z N N %0 0 NtA 4'4 44.(m4.I'4 4 0 -M .01P- 0 . N '4 N 0 * -rrfflBAl + *.+ + .-4 ft"0.0 + +. + _j ftf.
4444 *N '010 * 0 I- N 040 0 * 4W- I-I-N 4 <
N
U 94
000 0000o00OQ0000000000 0000 000000 000000
lBk in WN enn W% in n g M% in n i S in IA Wn W% WVnW in n W U% UN W% n U% W% n n Wn N n w% n n0 00 00 000000 00.00 00'000000 00'000000 000 000.0 0000-0,00000 00 00 0000,00.0000 0000 0000 000000000-000oo o00 00 000000000 0000 0,000 00 0000 '
z
S>
zi x
N r NC-
0LL
zx
CY 0CYX -44 0.IiI *0 tD .. 0.C D Yx0 x0.4 Cie 4 N.<-44CAfl0<"t. ( A - 0f
0 - .%xx+ - x~b +4 x~r+ t S 0.0.0 + a.~4 4.- ux0
tA 0 0.0* .. o
NO AC 0
95.
f-0~ 00 - f -P 0M WM 0 00000 0%~ 0 000% 00 7 00:0 00 0 00 0 00
u.n in niln &A sran m~ mn I m n in n W N In n n M an n % N UN ia n n M~ an A aNan In tn an In00000*00 000 000000000 00 0000000000 0000000000000000 0000000000000000000000000000000000000 000000000000000000000000000
b
-iz
~N 4N b N N NN0. w ft ft ft ".
N.-4N (N 0.44 t0. + + 00 tO + 5 4 t. . 4+4C
wz
4 yU 41-aX<XX0- 4X 4C fa yt)c -
0 r4 N 4nr4 V4*r .4 P4
96
0000000 000 000000 0000 00 000 000 00 0000600
O000O00000O00000O0000000000.0oo 00000oo~00000000000000000000000000000'000000 00 0000000000000 0000 00 000000000 000 000000000
zUw
Z 0
CL N
z z co z* .4N N NY
.& & 0. 4bM r4 4 04 V N tq%04
+ + S.. . + +.4 +.4 U. *-4S.. + + X K+ Z N - -- -+
z
040
97
000000000000000000 000000000000 0000000f- 00 0 -4 N Mf 4 M %0 I- 0 0' 0 9-4 NM 4~ 0J %t0 t- 0 0' 0".4 NMt Wi 0 in* W (0 0 P4 IN Mi4.4 4 LAt LIM m im n LA n mn In W% %0 %00 0 0 04 4 %.00 1'- f'- r- f- r-- e'-P I- - - -OD 0 0 0CW% In 9% UN LA in Wn iN in in UN n W Wt Wn W% W% in it In In U% W% W% aN UNi In UN A an aN W% W% aN n i%000000000)O,0000000000-00000000000000000c00o0O000000000000 O00000 000 00000000000 0o-00000090 000 00 00000000 000 000000 000000(:
to
z
u
zz
NN
10 10%0 %t It% 0 '0.0 ft. 0. ob 0. -t ft ft- o4 Z
%0 r. v-4 0N .0 40tn %0 -'o "4 w4 4 10 0 C14N 0.Z + x + +z ft 2 0. + N CO Z + x + + z ..
OX "t00- OI C - DC < C O*.
uiu cc
NN
98
9 0000000000000000000000000000000000000
m V InW% % W 0In % U inm UN9% n 0NO%0 O % %00 0 % %00 00 0 0 0 0 ' 0%0
#A0000000000000n00000000000000000000000I i i U W i W i 9 9 9 W V W I U I I0000000 00 0000 00 0000 00000000 0000 000000OOOOOOOOO0QOO00000 00 000000000000 000000
0000000000000000000
0u
r!U
V4 N , N N0mMts
+ *~t a 0 " 0. 4kf t t f
N 4 4 C"0r 4 N 0- t "40 0tl- .0? f-- 04 n( 0 4t 4 -
4 00 c c n . - 41-4 '0- 1.4?-
N cn It V%
N N NN
99
0000000000 000000000000. 0 0000.000, 004 000
N NN NN N NN( NM mmm mmmm MM4 %r 4f rl 4 %r 444r4 tn &A IAk I In mmIAI%~ 0 '0 ~0 '00 '0 % '0 0 0 0 0 0 0 0 0 0 0 0 %a S00.0 0 0,*.% .0 N0 %000' %00 0 0 .00O
S0 00000000000000000000000000000000-0000000 000 000 00 0000000 0000 0000 0000 000-000
z
N
zz -4 zN -3
N N obN~N N
'n IA '0 tt.AS rn 10 NS. 0. . ft S. -s S.D~ f-0S.C~ r- oo M o
e. 6 . ft 4NC-4OIA r4 -4 tr-Of- 4 t-4 in -4.4N.4N o.' S.N44MV f-CWN.4N + , .~ + + o. f . + + o. + + < + < NtI .S.+N*4-s.
'0 <0 '0t iw0- .U 1w0-1U L01-1- < C
N0C1 N N OD 0%~N N At < N N
1.00
UNIIn 4444NO o o4 4 444 f- f- I- I- f- P- f- r~- r-'r- CO000 OC 00D00 O O %0 ON 0
UN U% U% W% Wnw W% n W% Wn W% V% .n Wn n V an n mn W% U% an wt W% W% i% WN In W% W% i% W% in In i% in %n
0000Q000000000000C~00000000000 0 0 0 000
oooqoooaooo0000'000000
0VOOOOOOOOO00000000
.j
0+
Z 4z iz
NINA NNNN I
zt 0- : tn 0*
t- 0q%
- - %- fl-t
.: , f + 0,ft . . a S 4. S.' t - 0 -
0 P- 0 - 4 0 0 %. t- P4O4 M- %0 0v- .4b 4 4- #4 tn 04- - *
a. 41 + + n 41 o+ + a0. + + (n+e *
0000* * * 4 4 4 ~ 4 4 4 a 4 4 4 4i
N )("Ct4in O O%0 t-
M i
101
0000000000000000000000000000000000000
000000000 00 00 00 00000 00 0000000 0 0 P4 44 4- P"NNNNNNNNNNMM
00000000000000000000000000000000000000000000000000 000000000000000000000000
-J
zz
N L
%o f in- .4c - o .< 0 c 30 * e - C) .0fl MW 4
2: aYC aDYC ao aCC 0 aFO 00.U DO x a-b. CL~a.J
210
0000000 0000 00000 00000 00 000000000
Nm ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ U Zt in w iqvm ti t %ov n% -wo v Nm
0000 000 0000 0000 0000.00 000000 0000 00000 00000 0000 000000 000000000000 00000 00000000000 00000000 00000000 0
00-J9-MI
+zu
wn( u t.4+ .
ODin-.- r
-4oa+ -:4W L rN 94 c 4 . 1 -t z NXNxP 0 0
*. CC
(I) W W
0n 0: U (A.JI-- U AI.- cc s- u I-Mauiw wv
0 cow
U10
"4 N (7'00 0 V4 N MC 4 t n0 t W 0' 0 P4 N MW 4U N 4 U) P- U) M 0 0 V4 N M 4t M~'0 1'- W
0000000 000 0000000000000 00C'N NNNNCJN'N NN
0I 0I 0i L)'I) 0i 0d Li 0i 0I 0i 0I LI LI 0i En) Li) L) WI 0i (A) L) 0i 0I 0I Li) CWOI L) 0i 0i 0d LI) 0i CA) 0) IAwwwwwwwwwwwww~wwwwwLwwwwwwwwwwwwwwwww
z
.4.4
0xW
LU Z 0..
00 o- II
o LZc Uw 0. 0 c
w Ix f)Z CO U)x U. L
t-4 LU 00w
z1~4 )C Wz
Z L) xJ
" .W 0 2:I-= U. -Z Wu cy w 0 xz wC~C m( D ~-Z CA=) 0w + I- t-U. oCI Ln wii L c
-i zU)04 4 U. 4J LL %. 't- ODLUW :D N - o0 -4 x x w mC 0. N -4 -4Li)J0ICY N + It(DIt+ +ZZ+ ow'4WI- +++ 0 +
0 Cd)A.- w' cLUCLi 4LW0 '4 *0J (b4WC.I l CXL) 0 ( wWujwo 4-LU0 W' 4 - CWJ e<- -4.J 0 M *? 2C 4 i: LLCV)WW CC CCOCeW0 -C
z
w we .., Da0d > < -0 0. Or < w < -0 < 0 -Ja 0 -i I-- 33 .4 ILIto J(0tL1. '4J00 x x -)-0I-- '44 w _jj_ j 0 M 1--
44 CC u
Li) .4 WU LU
104
00000000000000000000000000000
uwwwwwwwwwwwwwwwwwwwwwwwwwwuwwwwwwww
wz
naco-
m,0F
z
G0or- I-c 0 In tfl1.0 in4V4r4 ' " 4 N* V-4 N -4 V4 -4-4 #4.-4 4N N V4 .4
+ +.4+ + + + -+ . . . . .. + + +. .4-o u* wwww wwow j UWOWtu u iu w wwJ Mwwwwwow w 0 0
-Jw
('I 4
105
0 0000 000000 00 0000 00 000000 0000 00000001040 fl- w Qh 0 '-4 N mn t In %0 ?1- w at 0 '.4 N% M 4* WN *0 1'- 00 Ell 0 .4 C) (q %tW 'o r.- Go Ql% 0'i N0 000,4NNP4-.4NwN4- 9N N N NN N M MM MMM MMM4 4 %
wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwW W W w~ W A W 4W J Wn 0~ w W W W W 0 W 0 0 VJ W (A IW W~ VW Wi 00~L 0 w W V) 0J $A LU 4A
'4c-
OD
U)wzz
4 0 U..
z z
z w
L)r
N w
0 Lu<
O '.4 -4 )l o .- 4. -J co ~ U co0 .-4 -4 co -4.-4.4 CVN M. CD N LN~ D - 4 -4- N N en ol #-4+ + Y+ + X D +-+L + + - ++.-4++ CO++++
0 w- -twn W WW>w J< &w W0.w.w w w waci oo w ow w w ow Iw w jW W LLU W W<LL LL N LL L U.LL u. LL mLL LL ( U. U. .L L L LLLL LL L..LU..
tu )X1 ~0 I- c 0 <Z .- 4-i 1-4 1--J- -J 0 1-UCI--400- - 0X -- 0-Z Lco x U. IL W0u)I-- OUwA-L'fou UU.~u)U.U0.U.L0)U)..-JLLU)U)JULX
43 4
106
wwwwwwwwwwwwwwwwwwwwwwwuwwwwwwwwwwuwwww
4
z
4
6-
%0% -- 00 % C
1- 4MW.z4- 4f nN 4 c 4 i+ nI
-'-
4UU
IZ 0 LiLLnL U u t^L U 0 I . L uU -a LLt
107 <
wwtu~wwwwwuwwiwwwwwwwwwwwwwwuwwwwwwwwwwww
wz
U)U
Ii <Z r-4Z Z
+ s-4 -
z0) U.
-0 LU 04
4'- Nt+ 1-+ % W 4+ W ZZ J 4 -4 N -. 4ON -%t %0%c 0c L0 %0
+ + + e- m.. .. ++i ++ - 4 . ..
o000 00000 000 00 00000000 000000 0000 00 000
*~~~~~~~~~~4t 4.....444. 4.4.4. 444444 in 4....44444,.,.444W WWW WWW WW W WW WWW W wWWWWWW WWWWW WWW u uWw W wWIA-I IA IA WAI WA 0f1 W l 0I0 4nA 4n IAIAI IA 0A.14, IA IA %n IA 0I IA IA IA Ia v I 0, inV)0IA VAwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww
w M00M0M0 cO M0 0 00co0o 000000000coco o o d00000;a 00 do co( o0oc oc
44.a-J
zD Ito.
c-4coA
n
z M
IA Z
-4 1- U.L) 0-1
~~z
o I- - 0 0 1-- -4.V-4 ZZW-"i 0C 0 o O 0 CD-4 en 0% M~ -40~ f 470%M in e+ W 0 >+ +.t + + + + + +N++ ~+ +v 4.
o W4.er4WW WWW W 4W W 0WWL4X W 4 WOWWWWWWO+W0 CC < N C I - C C C 4 &. CD~ CI fJCC* X - C: JM A :- 4f- LL X J tL W 0 U. W UMO IN LW U. U.U. 4 0 .4 U.U .wU. w w m wUw
-jw(A 0.4 049x to <)a h n f aM C0 1 owx 3C. . Q4 a (CcWE)
W ..JI IAxUj I-AUcO <<1-0M IA L.I e- -i- UIU.AI- j U .I-t0fl
109
OO)OOOOOOOOOOOOOOOOOOOOOQOOOOOOOOOQOOO
in tA in U) M In %A 0 . 0 0 0000%0 %0 %0 t- t-- Pk, f- f~- f~- t- - COCOCDCOCOCD00CD00 0'O
wwwwwwwwwwwwWWWWWWWWWLUWWWW WW WWw w ww wWW WW W WW W W
cc zqa <
U-Wu D
ccLU
ooLUWn
CC U. J)
o z*L I--
4 Z 4JII- cl iD C;rWV c
-1U
4A -4<<
0'-10
0000000000 000000000000000000000A000000
0t00400404000 m n n n0 OOmt Anmmi 0% 0
wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww
.4
w-j
zz
00
D4 0
-,i
0uj
(flu 0
z fn % 'CD - 1010 10%0 <4 't r t'-
(l--0% O0'O'% a 1- N N% N 03 vo rj a:) 4 4 -4 -4 't 4 4 0. .+.N. .+ .+..4 .++. ..+ +N + + 4A ) It %r.
o iL UUj0 jL w~~w wwwwoU L WW4W UWWW 0W W Z f--'.-a *No e. fj w ce. . : O CC:C J 40 CQ C - 0 * * * . 0 * Wtl- L..LL LLL LL L L LL LL LU LLU..U. It LLLtILIL W U.LLLL L UM n 4 tC O0
-jwU#A 0 L ~0 <00DC -1 0C0. 040.O.040 C C 0-C < U LJ) U L) U U --W C : i< :L -- - C 0 x< <0 -- 1 : tJ -. .. J U U WWWWWWWWLco t n 1 LW 0 u -jLL0 L ILL LL )L U 000000000
07 -4 04lL -t O CC<000 0 0 OD-LUIILL
N 0 -J J - -J J 3
0 0 N L LL L tLL111
000ot 10 ta%0% 0r%
w iw
zD.
0
0e
co0D
.4
0 jt
+0c
0u0 -t0iL %
0 i0 %LI
Security Classification ,,_,_. .. ....... . .
14. LINK A LINK B LINK CKEY WORDS ROLE WT ROLE WT ROLE W1
INSTRUCTIONS
1. ORIGINATING ACTIVITY: Enter the name and address imposed by security classification, using standard statementsof the contractor, subcontractor, grantee, Department of De- such as:fense activity or other organization (corporate author) isauing (1) "Qualifled requesters may obtmin copies of thisthe report. report from DDC."
2a. REPORT SECURITY CLASSIFICATION: Enter the ov,- (2) "Foreign announcement and dissemination of thisall security classification of the report. Indicate whether report by DDC Is not authorized.""Restricted Data" Is included. Marking is to be in accord-ance with appropriate security regulations. (3) "U. S. Government agenzles may obtain copies of
this report directly from DDC. Other qualified DDC2b. GROUP: Automatic downgrading is specified in DoD Di- users shall request throughrective 5200. 10 and Armed Forces Industrial Manual. Enterthe group number. Also, when applicable, show that optionalmarkings have been used for Group 3 and Group 4 as author- (4) "U. S. military agencies may obtain copies of thisIzed. report directly from DDC. Other qualified users
3. RkPORT TITLE: Enter the complete report title in all shall request throughcapital letters. Titles In all ceses should be unclassified. .-If a meaningful title cannot be selected without classifica-tion, show title classification in all capitals in parenthesis (S) "All distribution of this report is controlled. Qual-immediately following the title. ified DDC users shall request through
4. DESCRIPTIVE NOTES: If appropriate, enter the type of .___report, e.g., interim, progress, summary, annual, or final. If the report has been furnished to the Office of TechnicalGive the inclusive dates when a specific reporting period is Services, Department of Commerce. for sale to the public, indi-covered. cate this fact and enter the pr:ce, if knowr.
5. AUTIIOR(S): Enter the nadie(s) of author(s) as shown on 11. SUPPLEMENTARY NOTES: Use for additional explann-or in the report. Entet last name, first name, rAiddle Initial, tory notes.If military, show rank and branch of service. The name ofthe principal .uthor is an absolute minimum requirement. 12. SPONSORING MILITARY ACTIVITY: Enter the name of
the departmental project office or laboratory sponsoring (pay-6. REPORT DATE: Enter the date of the report as day. Ing for) the research and development. Include address.month. year. or month, year. If more than one date appearson the report, use date of publication. 13. ABSTRACT: Enter an abstract Giving a brief and fuctual
summary of the document inrjIcative of the report, even though7a. TOTAL NUMBER OF PAGS: The total page count it may also appear elsewhere in the body of the technical re-should follow normal pagination procedures. Le.. enter the port. If additional space is required, a continuation sheet shall'number of pages containing information, be attached.
7b. NUMJER OF REFERENCF.S: Enter the total number of It is highly desirable that the abstract of classified reportsreferencescited in the report, be unclassified. Each paragraph of the abstract shall cni with
8o. CONTRACT OR GRANT NUMBER: If appropriate, enter an indication of the military security classification of the In-the applicable number of the contract or grant under which formation In the paragraph, represented as (rS). (s). (C). or (U)the report was written. There Is no limitation on the length oJ the abstract. Ilow-
8b. 8&-, & 8d. PROJECT NUM1BER Enter the appropriate ever. the suggested length is from 150 to 22S words.military department Identification, surh as project number. 14. KEY WORLS: Key words are technically meaningful termssubproject number. system numbers, task number, etc. or short phrases that characterize a report and may be use'd as9a. ORIGINATOR'S REPORT iUMDBER(S): Enter the offl- index entries for cataloging the report. Key words must becial report number by which the document will be ident'f4ed selected so that no security classification is required. Identiand controlled by the originating activity. This number must fiers, such as equipment model designation, trade name. militarybe unique to this report, project code name. geographic location, may be used as key
9b. OTHER REPORT NUMBER(S): If the report has been words but will be followed by an Indication of technical con-
assigned any other report numbers (either by the originator text. The assignment of links, rules, and weights is optional.or by the sponsor), also enter this number(s).
10. AVAILABILITY/LIMITATION NOTICES: Enter any ltis-itatlons on further dissemination of the report, other than those
SSecurity Classificatior,
UNCLASSMl IMSecurity Classification
DOCUMENT CONTROL DATA - R&D(Security cleallcatlon of title, body of abstract and indexing annotation must be entered when the overall report Is classified)
IORIGINATING ACTIVITY (Corporate author) 28. REPORT SECURITY CLASSIFICATIONNorth American Aviation, Inc.I UNCLASSIFIED
Space and Information Systems Division 2b. GROUPDowney, California N/A
3. REPORT TITLE
Unsteady Aerodynamics for Advanced Configurations, Part I - Application of theSubsonic Kernel Function to Nonplartar Lifting Surfaces
I. DESCRIPTIVE NOTES (Type of report and Inclusive dates)
Final Report5. AUTHOR(S) (Last name. first name. Initial)
Vivian, H. T.Andrew, L. V.
6. REPORT DATE 7a. TOTAL NO. OF PAGES _ 7NO. OF REp3
May 1965 110 13a. CONTRACT OR GRANT NO. AF33 (657 )-10399 9a. ORIGINATOR'S REPORT NUMBER(S)
b. PROJECT No. 1370 FDL-TDR-64-152, Part I
c. Task 137003 9b. OTER IlPORT NO(S) (Any othernumb.re thattmaybe assglned
d. this STD 64-151
10. A VA IL ABILITY/LIMITATION NOTICES
None
11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY
Air Force Flight Dynamics Laboratory (FDDS)Wright-Patterson AFB, Ohio h5h33
13. ABSTRACT
Equations for pressure distributions and generalized aerodynamic forces are de-rived for a thin nonplanar lifting ourface in simple harmonic motion at subsonicspeeds. A digital computer program, written in Fortran IV, is also presented.The computer program will generate up to a ten by ten matrix of generalized aero-dynamic forces when given data for the geometry of a planar lifting surface witha folded planar tip, the flight Hach number, the reduced frequency of motion, andsome control constants. Control surface deflections are not accounted for in thisstudy. The kernel function method given by Watkins, rhnyan, and Woolston(Reference 1), which relates the pressure distribution to the downwash on a planarlifting surface, has been extended and applied to a nonplanar lifting surface.Hsuls technique (Reference h) of employing Gaussian quadrature formulas is usedwhen integrating the product of the kernel function and the lift function over theplanform area. Recommendations are made to extend the method to account forblunted leading edges and the accompanying airfoil thickness and to account forcontrol surface deflections.
DD IJAN6S 1473 UNCASSIFClIE caSecurity Classification