SUBSONIC LIFTING SURFACE ANALYSISWITH STATIC AEROELASTIC EFFECTS

Larry Glen Pearson

I

NAVAL POSTGRADUATE SCHOOL

Monterey, California

TH ESISSUBSONIC LIF IING SURFACE ANALYSIS

WITH STATIC AEROELASTIC EFFECTS

by

Larry Glen Pearson

Thesis Advisor: L. V. Schmidt

June 1972

Appnovnd fax pubtic xeJLejue.; dLU&UJbuXA.on antimiX&d.

Subsonic Lifting Surface Analysis

with Static Aeroelastic Effects

by

Larry Glen PearsonLieutenant, United States Navy

B.S. , Naval Postgraduate School, 1971

Submitted in partial fulfillment of the

requirements for the degree of

MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOLJune 1972

ABSTRACT

A computer program was coded to obtain static aeroelastic effects

on simple planforms through the use of subsonic lifting surface theory.

The program was divided into two major technical areas, aerodynamic and

structural, with matrix notation used to indicate the influence of each in

the final aerodynamic loading distribution. Several typical stability

derivatives were then obtained to make an accurate stability and control

analysis on the desired planform.

TABLE OF CONTENTS

I. INTRODUCTION 9

II. METHODS USED 12

A. GENERAL 12

B. AERODYNAMICS OF THE PROBLEM 13

C. COMPRESSIBILITY CORRECTIONS FORSUBSONIC FLOW 19

D. STRUCTURAL ASPECT OF THE PROBLEM 20

E. TOTAL SOLUTION 21

F. CALCULATION OF STABILITY DERIVATIVES 22

III. RESULTS AND DISCUSSION 26

A. LONGITUDINAL 26

B. LATERAL 29

IV. CONCLUSIONS AND RECOMMENDATIONS 3 5

APPENDIX A. ASSUMPTIONS 36

APPENDIX B. DERIVATION OF THE AERO SUBROUTINE 37

APPENDIX C. DERIVATION OF THE STRUCTURES SUBROUTINE ___43

APPENDIX D. TWO-DIMENSIONAL WING PRESSURE LOADINGCONVERGENCE CHECK 50

COMPUTER PROGRAMS 54

REFERENCES 71

INITIAL DISTRIBUTION LIST 72

FORM DD 1473 • 73

TABLE OF FIGURES

1. Horseshoe vortex lattice geometry 14

2. Finite vortex element, between A and B, of strength/ 14

3. Contour encircling all / 's located at the individual

boxes 17

4. Effects of planform distortion 17

5. Angle of attack distribution for a constant rolling

velocity p 25

6. Ailerons as defined by horseshoe vortices 25

7. Lift-curve slope and moment-curve slope plotted as a

function of leading edge sweep at a Mach numberof 0.5 27

8. Lift- curve slope and moment-curve slope plotted as a

function of Mach number with a leading edge sweepof20.0deg 28

9. Static margin indicator plotted as a function of leading

edge sweep at a Mach number of 0.5 30

10. Rolling moment due to aileron deflection plotted as a

function of free stream dynamic pressure at a Machnumber of 0.5 32

11. Damping in roll derivative plotted as a function of free

stream dynamic pressure at a Mach number of . 5 33

12. Bending and torsion stiffness curves 34

Bl. Simplified horseshoe vortices system for wing lift

distribution 38

B2. Geometry for downwash velocity calculation 38

CI. Elastic axis representation of the wind with the

torques and moments at the station elastic axis points 44

Dl. Geometry for two-dimensional study of chordwisepressure distribution 51

D2. A tabulation of the two-dimensional wing convergencecheck 53

TABLE OF SYMBOLS

[A] aerodynamic influence-coefficient matrix

[AS] symmetric aerodynamic influence- coefficient

matrix

[AA] anti-symmetric aerodynamic influence-coefficient

matrix

An Fourier coefficients for chordwise collocation

technique

n lift curve slope of section of infinite wing

n

/^ wing aspect ratio = b /s

Bn Fourier coefficients for spanwise collocation

technique

b wing span

C planform chord .,

C wing aerodynamic chord = g- j C du

Cave average chord of the planform = s/b

CL

total wing lift coefficient, Lift/qS (+ acts upward)

Cm total wing pitching moment referenced to C/4 (pitching

moment)/qS~c (positive leading edge up)

C L total wing lift curve slope, /<^oC

Cyy, static stability derivative, ° *"/&&.

Cw»c indicator for static margin, m<* /c

C roll damping derivative , • VtA ( f^\

Cg rolling moment due to aileron deflection, *'&§^

C. tip chord

«-oc

Crroot chord

C section pitching moment coefficient about the localc/4 quarter chord

C, section lift coefficient

C» section lift curve slope

h horseshoe vortex semispan

\ section lift

I^Lii incremental aerodynamic loadings over the planform

9 J

M free stream Mach number

NC number of chordwise stations

NS number of spanwise stations

N total number of stations on the planform

p rolling velocity (radians per second) ( + right winggoing down)

J-

—

roll helix angle2V

(vl

1 2free stream dynamic pressure, a scalar constant =—pV

R^ distance from a control point to the left corner of the

horseshoe vortex

S structural influence-coefficient matrix

S wing area

V free stream velocity

Aur incremental downwash velocity, (+ acts downward)

distribution of downwash angles at the three-quarter

icfa chord control points

x axial coordinate

y spanwise coordinate

2 vertical coordinate

at angle of attack (positive is nose up rotation)

0(p final angle* of attack

Q£w rigid input angle of attack

0^s angle of attack due to structural twist

O^ geometric angle of attack«)

Q Prandtl-Glauert planform distortion factor, /4/\-M2

-* u b prototype ,. /0 ,

A scaling parameter = ~rf , = (b/2)b model prototype

1 vortex strength

Vy aerodynamic loading

^) flow velocity potential

(£ the second partial derivative of <j> with respect to x

p air density

2V Laplacian operator

Matrix notation:

L J rectangular matrix

I J row matrix

} j-column vector

L J square diagonal matrix

[ J inverse of a square matrix

ACKNOWLEDGEMENT

My most sincere appreciation to Professor L. V. Schmidt for his

unending patience and professional guidance.

I . INTRODUCTION

In making a stability and control analysis on modern high perform-

ance aircraft, the ability to make span load evaluations on the wing and

tail with static aeroelastic effects taken into account is most important.

It seemed desirable to develop a simple computer program utilizing

current 'state-of-the-art' theories, and proper documentation, so that

design groups might have readily usable methods at hand.

Span load determinations have gone through a long line of

evolution, right up to the lifting surface theories of today. Prandtl's

lifting line theory is considered as the earliest, which was suitable

for simple planforms of high aspect ratio and unswept quarter chords

operating in incompressible flow. Modifications of this method were

made by Glauert, Multhopp, Weis singer, and others until the present

lifting surface theories suitable for low (or high) aspect ratio plan-

forms with swept quarter chords operating in compressible flow were

attained

.

Two lifting surface concepts for subsonic calculations are:

1. Define the surface pressure loading in the chordwise and span-

wise directions by admissible sets of orthogonal functions with

the magnitudes of the unknown coefficients being obtained by

collocation techniques, Ref. 6.

2. Define a grid of boxes over the wing, each of which contains

a horseshoe vortex, Ref. 4. Then determine the strengths

9

of the vortices by imposing the constraint of flow tangency at

an equal number of control points

.

The approach chosen for this study was the second, which is

usually denoted as a vortex lattice method and is thoroughly described

by Belotserkovskii, Ref. 4. The major advantage to this approach

is the convenient use of matrix methods for taking the static aero-

elastic effects into account.

The method of Gray and Schenk, Ref. 1, used initially on the

Boeing B-52 aircraft, was carefully reviewed as background for this

study. Their analysis, which was restricted to aspect ratio wings of

greater than five, employed the Weissinger L-Method to account for

subsonic aerodynamics. The advantage of the lifting surface theory

used for this study is that wings of lower aspect ratio may be ana-

lyzed and the ability to more accurately predict chordwise (as well

as spanwise) distribution of aerodynamic loading is provided. The

method of Ref. 1 did not allow for aerodynamic loadings other than

additional type, whereas the method of this study is not restricted

to loadings with the aerodynamic centers at the local quarter chord.

To account for compressibility in subsonic flow the Prandtl-

Glauert rule was used, Ref. 2, sometimes referred to as planform

Additional type pressure loadings are defined as the air loadings

due to angle of attack only. Generally they have a cotangent type chord-

wise pressure distribution in subsonic flow with a logarithmic type singu-

larity at the leading edge, and vanish at the trailing edge to satisfy the

Kutta condition.

10

distortion. The aeroelastic effects will depend upon the original wing

structure and will vary proportionally with the free stream dynamic

pressure. The span load evaluation at zero dynamic pressure corres-

ponds to the rigid wing results. Symmetric span loads will be appli-

cable to longitudinal stability and control analyses, with anti- symmetric

to lateral-directional.

The program has been restricted to simple planforms with

straight taper, and the influence of fuselage interference was not

taken into account. This was not an oversight but deliberate since

It was felt appropriate at this time to obtain a straight-forward docu-

mentation on a simple wing situation, with applications to more com-

plex planforms and interference effects as a logical extension of this

study.

The approach used in the structural area was restricted to a

basic beam-like wing structure, corresponding to modelling the wing

by an elastic axis with variable stiffness properties along the span

for both bending and torsional moments. An influence coefficient con-

cept for plate-type lifting surface might be a more general approach,

but this was not felt warranted without a specific application in mind.

The purpose of this study was to develop and document a simple

approach for static aeroelastic analysis and airframe stability and

control considerations. This documentation is a necessary prelude to

more serious considerations on actual configurations such as new

current Navy aircraft like the F-14 and S-3.

11

II. METHODS USED

A. GENERAL

In a static aeroelastic analysis there are two distinct technical

areas requiring attention. Both the aerodynamics of the problem and

the structural aspect must be considered.

With the free-stream subsonic Mach number and dynamic pressure

given, the aerodynamic loadings of the rigid wing are given as:

where [ A J= Aerodynamic influence coefficient matrix at the given

Mach number.

m incremental loadings over the wing.

jctf I = rigid wing input angle of attack, specified at

the control stations.

Structural deformations on the wing due to the aerodynamic load-

ings result in a change in the angle of attack.

*M¥W«.} 02)

where q = the free stream dynamic pressure, scalar constant.

[ S ] = the structural influence coefficient matrix.

icx^i = the angular deflections at the specified control

points on the wing due to aerodynamic loadings

.

12

Now the complete solution to the aeroelastic problem is given by:

[M-<JLS1]{^}=(«,] .... (.03a)

or

ff}=[[A]-9tSli'{a,} <-03b)

B. AERODYNAMICS OF THE PROBLEM

The basic approach used in this development is similar to the

lifting surface analysis as described by Belotserkovskii Ref . 4 .

With the concept of a vortex lattice array of horseshoe vortices the

wing is covered with N boxes where:

N = NC x NS (.04)

and NC = Number of chordwise station

NS = Number of spanwise station

In a manner similar to lifting line theory, as developed by Prandtl,

Glauert, Multhopp, and Weis singer (Refs. 2 and 3), the swept bound

horseshoe vortices were placed at the local quarter chord of each box

and the flow tangency requirement imposed at the local three-quarter

chord control point of each box. Figure 1 shows the geometry of the

problem, with A and B denoting the left- and right-hand corners of

the horsehoe vortex element and C the control point.

As a consequence of the Helmholtz vorticity theorems (Ref. 2),

and because inviscid flow was assumed, each horseshoe vortex ele-

ment has constant strength, once established. The Biot-Savart Law

13

Figure 1. Horseshoe vortex lattice geometry

ZSW

Figure 2. Finite vortex element, between A and B,

of strength / .

14

(Ref. 2) then defines the velocity induced at a control station C (see

fig. 2), for a vortex filament of finite length as:

PAU> = —y (cosoC 4 cos^)

( >0 5a)

and the dovvnwash angle as:

&S = -i- (!.)<«» CI 4 ectf ---. (.05b)V 4rrr V

If the vortex element is expanded to plus and minus infinity, the

well known expression for downwash angle in two-dimensional flow

results: to-jl.^ .... (.06)

since cos oc = cos g> =1.0

As previously stated, the flow tangency requirement was applied

at the three-quarter chord control point, and the flow was required to

be parallel to the zero lift line at that point. Hence,

($.)= t* (.07)

Justification for the use of a local three-quarter chord control

point in each vortex lattice box has not been rigorously established.

The concept of the "rearward" aero-center at the wing three-quarter

chord control point, from steady and unsteady two-dimensional wing

aerodynamics, appears to be the genesis of this usage. At present

the modelling is considered satisfactory because the theoretical load-

ings are in accord with experimental results. It is not clear as of

this writing why the preservation of flow tangency at local vortex-

box three-quarter -chord control points preserves the Kutta condition

15

at the wing trailing edge, but the calculated values of local vortex

box loadings do show the preservation of this trait. (Ref. Appendix D)

The value of the downwash angle (tt) 3cvaries at each control

station on the wing, and there are contributions by the bound and

trailing vortices from all of the wing stations. Hence, the flow

tangency requirement may be expressed as a column vector as follows:

where the matrix [A] is the aerodynamic influence-coefficient matrix

as developed in Appendix B.

For the vortex lattice representation with N horseshoe vortices,

the flow tangency requirement was given by:

A

- (%M = im'K] — <-° 9 >

There was a temptation to view the vortex strength at each

control box as the pressure differential of that box. This, however,

is not the case. Again as a result of the Helmholtz vorticity theorems

(Ref. 2), the vortex strengths of the individual boxes may be summed

in the chordwise direction to obtain the total vorticity at each span-

wise station. Hence the vortex strength of a section may be "lumped"

into one vortex equal to the sum of the individual vortices. (See Fig. 3.

The development of the [a1 matrix for spanwise stations

only may be seen in Ref. 1.

16

C=£a/7

Figure 3

.

Contour encircling all / 's located at the

individual boxes .

X Xa) Field X, Y, Z b) Field X ,Y, Z

Figure 4. The effects of planform distortion.

17

From the total vortex strength at a chordal station, the section lift

NC

loadings can be obtained, i.e., rM = %^ .... (.10)

t- \

and by application of the Kutta-Joukowski law

(I) = 2 £ (.11)

The section moment may be obtained by taking the moments about a

reference station of the individual bound vortex elements, inasmuch

as the bound vortex elements represent the local air loading of each

box by the Kutta-Joukowski law.

A two-dimensional study was made to compare the "pressure

distribution" imposed by the vortices and the cotangent type pressure

distribution as prescribed in theory (Ref. 2).

Assuming that the chordwise vortex elements described a chord-

wise pressure distribution, it appeared feasible to use a chordwise

collocation technique for obtaining section lift coefficients and moment

coefficients , C2 $ C^ ,

given by

q= 2-nA> +nlV,, $ -«*= ?<*^

Since

and

4*> - 2Jd * = 1(1- cos ©^c. 2

co2£ = A A c cot I 4 4 g A,* sin n©

it appeared that a solution for the Fourier coefficients could be found

in the following relationship:

COT ©«A S\S4 0, SIM 2©,

mCOT Q

«/2

18

S\m(v»-0©,

I

I

I

I

M

However, the existence of the An 's assumes knowledge of the pressure

distribution over the wing, and correct results were not obtained,

since the vortex strengths did not represent pressure loadings.

A check was made using two-dimensional theory to bear out

these conclusions, and it was found that even for ten chordwise sta-

tions, the pressure distribution was incorrect, and the aerodynamic

center was not located at the one-quarter chord. The leading edge

singularity was not properly represented and the first collocation

point differed from theory significantly. This check can be seen in

Appendix D

.

In light of these results, a straight-forward summation was used

to obtain the section lift coefficients , i.e.,nc

Czc = Z (q-)

L (.12)

C. COMPRESSIBILITY CORRECTIONS FOR SUBSONIC FLOW

The compressibility corrections were obtained by the Prandtl-

Glauert planform distortion method (Ref. 2). The applicable flow field

equation for the velocity potential was given by

0-n£>4> + 4> + a = o (.13)

where the subscript represents second partial derivative. By use of

the Prandtl-Glauert transformation, X- ^oV^^i ^ 14 ^

this reduced equation (.13) to:

<p + <J> 4 $ = O (.15)

19

The effect of the Prandtl-Glauert transformation was to stretch all of

the x-coordinates by the factor *r- ^— . The effects can be

seen in Fig. 4.

The net effects of the planform distortion were as follows:

1) the aspect ratio was decreased; 2) the leading edge sweep was

increased; 3) the taper ratio (^t/C ) was unchanged; and 4) most

importantly, the downwash distribution Jrrf was unchanged, i.e.,

-fi =i"^ <- 16)

Since the downwash angle distribution, \rr\ , is unchanged in

going from the distorted wing to the original wing, the aerodynamic

loading distribution, \ — \ , will be the same on both wings. This is

an important concept in that the aerodynamic loading obtained from

the distorted planform under incompressible flow will be the same as

the aerodynamic loading on the original planform under compressible

flow. Hence the aerodynamic loading is an invariant when going from

the compressible undistorted planform to the incompressible distorted

planform

.

D. STRUCTURAL ASPECTS OF THE PROBLEM

Because wings are flexible or elastic, there are always changes

in angle of attack due to the deflections of the structure. There are

two deflection modes present, torsion and bending, both affecting the

changes in angle of attack distribution on the wing. The effects of

these two modes were called static aero-elastic effects and have been

20

incorporated into the program by the simple elastic-axis beam

representation.

The final angle of attack distribution \°({\ was made up of

two components j OCs [ , and j 0(^.1 , i.e.

[<*fj= M +{«,] (.17)

Now the {o^ s | as shown in Ref. 1, has been determined to be:

M-ltsHf} (.«

See Appendix C for the development of the structural influence

coefficient matrix \_ S J .

E. TOTAL SOLUTION TO THE PROBLEM

By combining equations 12, .17, and .02, the following rela-

tionship was formed:

« [ui-^Hfl = Kl ---- <-i8)

The closed form solution for aerodynamic loading distribution

may be given as:

[&[= [lA]-q l%\] [«t](.03b)

This was the final equation used in the program to obtain a chordwise

and a spanwise loading distribution for a given surface.

21

F. CALCULATION OF STABILITY DERIVATIVES

Five stability derivatives were investigated to check the validity

of the program. These were:

The total wing Ct was obtained by a spanwise collocation

technique, using the section lift coefficients.

+ t

where \j = (^)=cos9 > cJ>2=-sm©

(.19)

d©CO

Let

Then

9 v 4=1 *

CLcAV£

= £ I 2 Bhsmv\Q <s\m© d©

= 5ZB* Jsiu>og s\nG 6q = 5B(

Since by orthogonality

'2^ fTf rOR ^ rl

J sin v\© sin <d© =j

ft V

Therefore

C,= ^ |» (.20,

Since the assumption of linearized wing theory was made,

C L= & C L due to [O^ input of 1.0 rad . uniformly over the

whole wing, i.e. C — 7 S

The C^ 's were obtained by a straight-forward integrationci.

technique. In effect, the section pitching moment at the quarter chord

was found to be: (^ = C^< = 2(^). (X

£- X^l

'

22

Hence, total wing pitching moment is:

(.21)C„_ =§*U¥ l{(x-xv))^e^ 5 c <* l c/4 )

Again Cy^ was evaluated by: Cw ~ AC

where ACy„ is pitching moment change about the wing c/^ point

due tojof V= 1.0 rad. input.

With C^ and Gy^ calculated, the Cm was easily found

by the following relationship:

C = ^^ = ir*v*

dCu C L ,

(.22)

From the anti- symmetric solution of the Aero Matrix (see

Appendix B) , certain rolling moment derivatives were extracted. The

rolling moment coefficients were obtained by a collocation technique

similar to that used to obtain the total wing d L .

RM C r _ 1 fW\ I—* ^ UcCAVE - - yJ ^/ ^ d>

2

= - j-} 2 B

VN<s\v4 r\© cosq specie

8 "=l o"*% roe Y\r2.

O » n*2

Therefore = "TT ~Z~(,23)

* ifo cME

23

The roll damping derivative, C^, was then obtained by placing

the anti-symmetric angle of attack input on the wing, as shown in

Figure 5. At the tip, the angle of attack = XTT = 1 rad., and

Cg was given by:

r - ISl. =AC*

, ,

*'" Hfe^ {<v^( ' 24)

The aileron control effectiveness, Co , was found by defining

the location of two of the boxes to coincide with the location of a

typical set of ailerons, see Fig. 6. By inputing zero angle of attack

everywhere on the wing except at the aileron locations and giving

them 1.0 rad. angle of attack each, positive on one side of the wing

and negative on the other, a rolling moment was produced. Co was

given by:

c _ *Qi _ be*

&*. ^ Sq^ [ofr 0O6. TO SA

24

+1

o<

~ *

Figure 5. Angle of attack distribution for a constant

rolling velocity P.

a =-1.0

Figure 6. Ailerons as defined by horseshoe vortices.

25

III. RESULTS AND DISCUSSION

A. LONGITUDINAL

The results from the symmetric portion of the program were

checked by using longitudinal stability derivatives (i.e. lift-curve

L ,moment-curve slope ^-yy\ , and ). To insure

* * o Cu

that changes in leading edge sweep and Mach number are handled

correctly in the program, checks were made on C L and Cm .

These checks were made for a rigid wing with an aspect ratio of 6 .

and a taper ratio of 0.3.

Figure 7 shows the effects of leading edge sweep on lift-curve

slope C- L , and moment-curve slope C^ . These curves were

obtained by inputing a "do loop" in the program and varying the lead-

quite well with theory, i.e. (C, \ = ==: (.26)

ing edge sweep JV from to 40 degrees. These results agreed

where the Mach number is held constant and the sweep varied.

Figure 8 shows the effects of the compressibility corrections in

the program on C^ and CM . Again a "do loop" was used,cf- o(

stepping through the program for Mach numbers of 0.0 to 0.9. These

results also agreed with theory, i.e. equation (.26), where the sweep

is held constant and the Mach number varied.

26

Cm,

(perRAD)

2.

AR = 6.0"TR=0.3

© C

ACL<x

Mcx

-A_LE (DEG.)

Figure 7. Lift-curve slope plotted as a function of leading edgesweep at a Mach number of 0.5.

27

8.

-2

AR=60TR=0 3

-A^ = 20°

.6

iS—A—&

—

&—&MACH NUMBER

.9

OC

ACLex

m«

Figure 8. Lift-curve slope and moment-curve slope plotted asa function of Mach number with a leading edge sweepof 20.0 deg.

28

It should be noted that the C^ variation, with increasing

leading edge sweep, is in a direction to shift the aerodynamic center

ACaft. Figure 9 shows this more clearly since the value of -~

ci C L

about the wing C/4 corresponds to the distance, in wing m.a.c.

lengths , that the aero center is from the reference axis . A positive

value of — indicates that the aero center is forward of the

c/4 reference point.

B. LATERAL

In order to document the anti- symmetric portion of the program,

two lateral stability derivatives were investigated, roll damping^-J? p

and roll due to aileron deflection Co . The aeroelastic effects wereSou

checked at the same time by choosing an example from Bisplinghoff

,

Ashley, and Halfman, Ref. 3. A typical jet transport with a straight

taper and straight elastic axis defined at 3 5%c is used. The n< for the

wing was 6.15 and the taper ratio was 0.444. The bending and torsional

stiffness properties of the wing were taken from Ref. 3, Fig. 2-29; then

they were scaled to our model by (El)^= —4Ckl) P and reproduced

A

here in Fig. 12.

The checks were made at a constant Mach number while varying

the dynamic pressure (q in p.s.i.) from 0.0 (rigid wing) to 18.0, corres-

ponding to varying the altitude. The effect of aeroelasticity upon roll

damping for this wing, with an unswept quarter chord, is to increase the

damping with an increase-in q. Note that the sign on roll damping is

negative since the moment that is developed is opposite to the direction

29

M,Cl

AR=6.0TR=0.3M = 0.5

40

.A. (DEG.)

Figure 9. Static margin indicator plotted as a function of leading

edge sweep at a Mach number of 0.5.

30

of roll. Figure 11 shows the roll damping derivative plotted as a function

of q. The results of aeroelasticity upon the rolling moment due to aileron

deflection can be seen in Fig. 10. Note that the change in sign corres-

ponds to the control reversal dynamic pressure, qr . The qrwas found

to be 1 1 . 8 p . s . i . , while it wa s shown to be 11.39 p . s . i . in Ref . 3 , by

another method; this is a difference of approximately 4%.

31

-.150-

AR=6.154TR=Q444M=Q50

Figure 10. Rolling moment due to aileron deflection plotted asa function of free stream dynamic pressure at a

Mach number of 0.5.

32

cIp

AR =6.154TR =0444

-.4M =0.50

-.3 -

-.2^^ ,1— i f l I v^

_ ^"v ft} ^-*'

-.1

11114 8 12

q (psi.)

16

Figure 11. Damping in roll derivative plotted as a function of

free stream dynamic pressure at a Mach numberof . 5

.

33

GJ

20

50

40 -\ >w

\ °J^15

30\^

El

X10

20 \

5 10 \

i

.5

?

1.0

Figure 12. Bending and torsion stiffness curves.

34

IV. CONCLUSIONS AND RECOMMENDATIONS

This computer program can provide a 'limited' capability to make

stability and control analyses of modern Navy aircraft. The program, as

it now stands , is limited both to wings structurally compatible to the

elastic-axis concept and to simple planforms . However, slight modifi-

cations could easily handle arbitrarily shaped wings and even a variable

sweep configuration, such as the F-14.

The study is not complete at this point; a logical addition to the

program would be to add the influence of the body radius to the Aero sub-

routine. This would improve the value of the program. If felt warranted,

plate theory could be used for part or all of the structures matrix.

A follow-on study now seems appropriate to make the program more

attractive and useful to design groups.

35

APPENDIX A. ASSUMPTIONS

In writing this program, many theories with basic underlying

assumptions were used. Most of them are associated with thin airfoil

theory, or the structure of the wing. These assumptions, most of them

listed by Gray and Schenk in Ref. 1, are as follows:

1. Potential flow about the wing with no boundary layer effects

,

i.e. inviscid, and without compression shocks, i.e. below

transonic mach numbers .

2. The thickness ratio of the wing is small.

3. The lift-curve-slope, 01 = ^TT i.e., two-dimensionalo *

4. Changes in camber due to bending and torsion of the wing

are neglected

.

5. The changes in angle of attack due to structural deformations

are uniform along each spanwise station.

6. The elastic deformation of the control surfaces is the same

as that of the adjoining wing.

7. The angles of structural deformation are small such that

s\n e= ta,m e = e . cose = \.0

8. Drag-load effects to deformation of the wing were neglected.

36

APPENDIX B. DERIVATION OF THE AERO SUBROUTINE

The basic development of the downwash matrix or aerodynamic

influence matrix was done by Gray and Schenk in Ref. I. An expansion

was made here to include swept-bound vortices and a chordwise, as well

as spanwise, distribution of horseshoe vortices.

The lift or aero loading distribution over an airfoil can be charac-

terized by a system of horseshoe vortices, each of these vortices con-

tributing to the upwash or downwash at points all over the wing. Figure

Bl shows two horseshoe vortices combined to make a wing. Conventional

right hand rule indicates the sense of circulation associated with each

horseshoe vortex. Quite obviously, the more horseshoe vortices used to

approximate a wing, the more accurate the lift distribution obtained.

With the wing divided up into NC X NS stations, the horseshoe

vortices were placed at each quarter chord of the box, and the horseshoe

vortex lattice, as shown in Fig. 1, was arrived at. Now the boundary

conditions or flow tangency requirements , for simplicity of calculations

,

were imposed at the three-quarter chord for each station or box. The

boundary condition implies that the downwash angle ( 77 ) at each con-

trol station is equal to the geometric angle of attack. The downwash vel-

ocity uJ at each station is due to the sum of the downwash velocities in-

duced by each horseshoe vortex over the entire wing.

The chord for each box is equal to the local wing chord divided

by NC.

37

1

e ©

i

Figure Bl. Simplified horseshoe vortices system for

wing lift distribution.

A:(yQjxQ )

y

C:(yc ,Xc)

Figure B2 . Geometry for downwash velocity calculation.

38

Given one horseshoe vortex and one control point on the same

wing, as shown in Fig. B2 , the downwash velocity AuT was calculated

as follows:

S1nQ= (X c-*aVRa SlN<$ = (Xc-XbVRb

cosO= (yc-yAVE* cos<t> = (yc-yBVEB

sin tf= CxB-xAV^B co * *"= (^b-^a^/JJab

oc= O- Y § £ = cj>- JT

S\N C< =r SIN COS tf — COS Q SIN ^

COSC* =r COS0 COS^ + S\M S»N ^

COS g> = COSCJ) COS^T -f S\N <$) S\N ^

The left hand vortex contribution to the downwash velocity was given by:

A , , _ r c i + sit4 q)AW L~ T— 7 ^ B.01

The right hand vortex contribution was given by:

^*=^ c^o" ---- <»<»>

The bound vortex contribution was given by:

^*=^7 i^* im cc"--- (B " 03)

39

If the control point happened to be on a straight line extension of

the bound vortex, both numerator and denominator would go to zero

simultaneously . The contribution from the bound vortex would be zero

and a check was placed into the program and if this situation were to

develop, Ac*^ would be set to zero.

The total contribution to the downwash velocity due to the one

horseshoe vortex on the same wing was:

Au> = kUJ^ 4- Auj u + Auj^

or AuJ - r[(UsinQ) ( H sm<fl

l

(costt-cos^l(B 04)

The downwash velocity contribution at control points all over the

wing, due to a horseshoe vortex system, was calculated in a similar

manner. These can be thought of in matrix format as follows:

ur, ^ (a,uj u 4 au), z + • • • • •+ au\ n

fcU^ 4 AU)U+ .... 4 A(*\MI

Hi+^^2+ " * * ' * ^U NN

where the subscripts indicate the downwash at station i due to a horse-

shoe vortex at station j, etc. Now by factoring a | from each term

in the proper manner:

H=2[M{r}

then the following relationship follows:

40

or

W-W > '• M-^MeI — (B - 07)

The A matrix is the aero-influence matrix as used in the program.

The computations are carried out for contributions from each wing, i.e.

For symmetric loads: t AwS"i= I ^^"^t ^v^l \ (B.03a,b)

For anti-symmetric loads: [ A ^ 1= V A. r.1~ \- ^ 1

1

Then {(Xrl=[AS orAMJ^IE

Also contained in the Aero subroutine is the Prandtl-Glauert plan-

form distortion correction for subsonic compressibility. In essence, all

of the x distances are stretched by a factor of ( l^M^) ', defined as

p . Therefore throughout the subroutine TvBx^) replaces \ V^/j).

(i.e. ^CVV^ instead of Cvc c-yA^ ).

The 'lifting-surface' representation of a wing has been written up

in FORTRAN IV for use with the IBM 360-67 computer. The program was

designed to handle planforms of low to high aspect ratio, swept leading

edges, and a linear taper ratio. The choice of the number of stations on

the wing, chordwise and spanwise, was left arbitrary for easy compati-

bility with most wings and for almost any flap and aileron arrangement.

41

So as to make the program more flexible, subroutines were used

for each phase of the calculations. Subroutine Geom is used for locating

the horseshoe vortices and their control points; subroutine Aero is used

for obtaining the aerodynamic influence (or downwash) matrix; and sub-

routine Struct is used for obtaining the structural influence coefficient

matrix. With the subroutine packages set up as they were, changes to

the program can be easily handled simply by making small changes in the

input or the main program.

Sample calculations were made for arbitrary wing geometries . The

results of these calculations can be seen in the main text.

The desired results were obtained in these calculations. The

corrections for subsonic compressibility, sweepback, and static aero-

elastic effects have been checked and do give the desired accuracy.

42

APPENDIX C. DERIVATION OF STRUCTURES SUBROUTINE

As in Appendix B, the £sl , structures matrix or elasticity matrix,

was developed by Gray and Schenk in Ref. 1. An expansion to fit

'lifting-surface' theory vice 'lifting-line' theory was made for this

program

.

The problem was basically stated as: Given a loading over a wing,

find the deformation imparted to the wing and apparent change in angle

of attack. From the previous subroutine,

!<*<!= tM{|'}% \ (B.07)

The following relationship was obtained from this subroutine:

(c*s )= <j[s]{fc]| (.02)

The assumption has been made that the wing acts like a beam with

the elastic axis representation. (See Ref s . 1 and 4.) The beam bending

and torsional moments have been resolved about the elastic axis at each

spanwise station of the wing. Although the elastic axis may be arbitrary,

it was thought of as a straight line for simplification here (see Fig. CI).

The wing is divided evenly in the spanwise direction with each of

the chordal strips having width 2h. Therefore for the wing with an

assumed unit value of semispan, h = 1/2 (NS) (C.01)

Positive direction for torsion and bending moments are shown in Fig. CI

relative to the elastic axis orientation.

43

AXIS"

Figure CI. Elastic axis representation of the wing with

the torques and moments at the station elastic

axis points .

44

The span loads per unit span length are defined as -—- for the

N vortex lattice boxes in a numbering sequence starting from N = 1 at

the most forward, inboard box proceeding aft to the trailing edge at

N = NC, then stepping to the next outboard set of boxes at the leading .

edge box for N = NC+1. Continue in this manner until the Nth box at

the trailing edge of the tip is reached.

The individual lift force contribution towards the bending of the kth

wing station may be expressed as:

^i = Z tffV(2-S L|)qh (C02)

The presence of the Kroecker delta takes into account that only one-half

of the air loading at the kth station contributes to the bending (and torsion)

moment at the kth station.

Therefore the bending moment at the kth station may be expressed

as:NS

M. = /iUC|H2] Al:(2to(j-to (C.03)

or in matrix notation as:

M-q.lAMl{^ (c.04)

where {.^\ — M**l column vector

{ ^- j = N * * column vector

LA.NM= NS*N rectangular matrix, and

45

[AMI = tf

»/2 8

_ «4l 4

o c

o

> l|/z

a4~

1 £

8

3 _!__?_[ Va_ [_i _|_-

i

i

: i !

The AM matrix was partitioned into NS x NS elements where each ele-

ment is a (1 x NC) row matrix made up of constants as indicated, i.e.

-ji/Ti =[y2 i

/z .... i/

j

_i

—

T

The torsion moment at the kth station may be expressed as:

T%=<3tern (.05)

where

[m- ] = *

O ,AV ...11/ I

lAX

O

T

and

where

AX%,= <**\ (

x

^The zero partition elements to the left of the lead diagonal correspond

to inboard loadings that do not contribute to the moments at stations

outboard

.

46

The torsional and bending moments as developed here were perpen-

dicular and parallel respectively to the streamwise flow at the elastic

axis reference points. In order to reorient them relative to the elastic

axis, they were put through the following rotation matrix:

COsAea, -S\NAe*(C.06)

Now the moment and torque distributions have been resolved to be the

following relationships:

[m] = ^[lWe.] [A \a) - £si*AE^[AT]]{f

1

- (C.07)

The change in angle of attack in the streamwise direction was ob-

tained by the relationship (see Ref . 1):

Td<(C.08)

GJ

where m equals the beam bending moment per unit pitching moment at

the station Y? , and t equals the torsion about the elastic axis per

unit pitching moment. ds => r — (C.09)CosAeA

The above integral equation was changed to an algebraic equa-

tion by summing the contributions at each station.

(•2 t,.T, I•2» «2 r\ZU T, fr,

(GJ),cosA.< ' (Gj^cosA.

47

(CIO)

The influence coefficients my and ty were easily defined by the geom-

etry of the problem and found to be as follows:

-sinA- where the point i is outboard of the point j

where the point i is at or inboard of the point j

cosA^ for station i outboard of station j

for station i at or inboard of station j

If

and t. .=

In matrix form the m^'s and t^'s were written as follows

H-

M-

l"ij ° U11^ 1j

Z- S>\Hkz -SIN A-*, _ _

- S\M Aa2

- S\N A"*,

1 1

1

_ S\U A3Z

1

1

cosA,2

CO-5.A,. cosA 3

O

o1

1

2

Oi

cosA B

COSA321

1

Now

2£W-l-tfflarJW^dr^fecMT

1

}(C.ll)

48

Thus the desired form was achieved and the S matrix was:

+ Wt^x^AllAWl+tcosAllAT]]] (c.12)

The final resultant matrix equation was:

W = g[s]{^}, 02)

The computer program is normalized such that the wing semispan

has unit length. However, an actual wing will have a finite wing span

of say b/2 inches, will operate at a dynamic pressure of q p.s.i. , and

2will have stiffness properties of EI and GJ in units of lb-in .

Recognize that in a scaling analysis the structural twist of the

model will equal the structural twist of the prototype and that they will

operate at the same dynamic pressure. However, the aerodynamic load-

ings will differ by a factor of \

where A = 7u7^P=(^P since (I) =' (C - 13 >

(b/2y

i.e. r^l = 1 (Ml (C.14)

Therefore by equation (.02) we can scale

but analysis of equation (C.12) reveals that \.S ] OC ^/ET " /GJ

sots] p

a[«?4i].

but

49

APPENDIX D. TWO-DIMENSIONAL WING PRESSURE

LOADING CONVERGENCE CHECK

In this appendix, a check was made to ascertain the validity of

viewing the vortex strengths of the control boxes as pressure differentials.

A comparison was made for the C 's obtained by this assumption, using

a collocation technique for obtaining a chordwise pressure distribution,

and the theoretical C 's with a cotangent type pressure distribution.

A two-dimensional section with unit chord was chosen for this

check. The section was divided into N equal chordwise stations with

the vortices placed at the quarter-chord of each station and the control

points at the three-quarter chord of each station (see Fig. Dl).

Since W . H^ ... ,1"^ represent the small vortex strengths concentrated

at the quarter chord for each station, the total vortex- strength was given

by:

Now M c=oVT

t £ AP;=^ =AC C/N

and AC^A^i -£-(B\ (D.01)

The principle of an infinite vortex (Ref . 2) was used for the down-

wash angles, i.e. ^7 — £n £ V "\j J or

{5} = &t«lvP

!(D.02)

where -- =• OC at the three-quarter chord control point, and

(D.03)

50

xccj-j

V IT

e y 10 x o x i oz=x(o x (o :

X=o.oh~H*c* ^ X=l0

Figure Dl. Geometry for two-dimensional study of

chordwise pressure distribution.

51

with y^. — r- • i - iZ? , the local quarter chord point,

and Y^r.— ~ i — QjiP the local three-quarter chord point.

Hence {kC^ = 4 A N t Bl {<*} (D.04)

with OC =1.0 radian for each station.

Now the theoretical & CL for two dimensions was given by

(Ref. 2): &Cp = 4 QjLEf±fl (D.05)* «blM

using the coordinate transformation:

©= cos'[l-*2(|M J

(D ' 06)

Since the check was made at OC = 1.0 radian and a unit chord, the

C\C obtained will be equivalent to C? and should be approximatelyoc

cm27X . The aerodynamic center given by: AC = c€

(D.07)

should be at 0.25c or just 0.25 in this case.

The tabulation in Fig. D2 clearly shows that the correct results

were not obtained by the method as described herein.

52

c^ CDCD CDCM CM

CD OCO 00r<-» 00

O CO00 ento o

O CO CO (MCD CO CM CMnT CD in CD

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53

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REFERENCES

1. Gray, W. L. and Schenk, K.M. , "A method for Calculating the

Subsonic Steady-State Loading on an Airplane with a Wing of

Arbitrary Planform and Stiffness," NACA TN 3030 , December1953.

2. Kuethe, A.M. and Schetzer, J. D. , Foundations of Aerodynamics ,

John Wiley and Sons , 1958.

3. Bisplinghoff , R.L. , Ashley, H. , Halfman, R. L. , Aeroelasticity ,

Addison-Wesley, 19 57.

4. Belotserkovskii, S. M. , The Theory of Thin Wings in SubsonicFlow , Plenum Press, 1967.

5. Etkin, B. , Dynamics of Flight, John Wiley and Sons , 1959.

6. Ashley, H. and Landahl, M., Aerodynamics of Wings and Bodies ,

Addison-Wesley, 1965.

71

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Documentation Center

Cameron Station

Alexandria, Virginia 22314 2

2. Library, Code 0212 2

Naval Postgraduate School

Monterey, California 93940

3. Professor L. V. Schmidt, Code 57Sx 1

Department of Aeronautics

Naval Postgraduate SchoolMonterey, California 93940

4. LT Larry G. Pearson 1

VF-124NAS Miramar, California 92145

5. Chairman, Department of Aeronautics 1

Naval Postgraduate SchoolMonterey, California 93940

6. Mr. H. Andrews 1

NAIR 5301

Department of the NavyWashington, D. C. 20360

72

Security Classification

DOCUMENT CONTROL DATA -R&Dtassitication ot title, body of abstract and indexing annotation must entered when the overall report Is classified)

ORIGINATING ACTIVITY (Corporate author)

Naval Postgraduate School

Monterey, California 9 394

Za. REPORT SECURITY CLASSIFICATIOI-

Unclassified2b. GROUP

IEPORT TITLE

Subsonic Lifting Surface Analysis with Static Aeroelastic Effects

DESCRIPTIVE NOTES (Type ot report and.inclus ive dates)

Master's Thesis; June 1972

iTHORIS) (First name, middle initial, last name)

Larry Glen Pearson

REPORT DATE

Time 1972•a. CONTRACT OR GRANT NO.

b. PROJECT NO.

10. DISTRIBUTION STATEMEN"

7«. TOTAL NO. OF PAGES

74

7b. NO. OF REFS

6

9a. ORIGINATOR'S REPORT NUM6ER(S)

9b. OTHER REPORT NO(S) (Any other numbers that may be assignedthis report)

Approved for public release; distribution limited.

I. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Naval Postgraduate School

Monterey, California 93940

13. ABSTRAC T

A computer program was coded to obtain static aeroelastic effects on simple

planforms through the use of subsonic lifting surface theory. The program was

divided into two major technical areas, aerodynamic and structural, with matrix

notation used to indicate the influence of each in the final aerodynamic loading

distribution. Several typical stability derivatives were then obtained to make an

accurate stability and control analysis on the desired planform.

DD ,

f

n°o

rvM473 (PAGE

S/N 0101 -807-681 1

1 )

73 Security Classification

Security Classification

KEY WORDS

Lifting surface

Aeroelastic

Stability and control

Subsonic compressibility

Low aspect ratio

Swept wings

.T.".,1473 'B'CK.

101 -S07-68?174

Security Classification

t I JUN74

I JUN76

2 4 8 6

2 35 U 8

^2335 52 4 42 6 18*

1

Thes is13471?

P313 Pearsonc.l Subsonic lifting

surface analysis withstatic aeroelastic

HJUN^fcfects. *2p»«*2 3 5 U 8

1 JUN76

£3883355

r

th

1

Thesis 134717P313 Pearsonc.l Subsonic 1 'fting

surface anal /sis withstatic aeroe lastic

effects.

thesP313

Subsonic lifting surface

H"""

analysis with s

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DUDLEY KNOX LIBRARY