Measurement of model aeroelastic deformations in the wind tunnel at transonic speeds using stereophotogrammetry

APPENDIX BASIC EQUATIONS FOR DETERMINING MODEL DEFLECTIONS The direct linear transformation method of solving the colinearity condi- tion of photogrammetry was developed by Abdel-Aziz and Dr. H. M. Karara of the University of Illinois in 1971 (ref. 6). It establishes a direct linear rela- tionship between the comparator coordinates of points and the corresponding object space coordinates. As such, it does not require fiducial marks in the photographs, the computation of partial derivatives, nor linear approximations of unknowns in the solution, as would be required in the conventional solution.

Although originally conceived for use with nonmetric cameras (such as 35-mm or 70-mm cameras) in close range photogrammetry, it can also be applied to metric cameras.

Given a set of fixed reference points, this procedure will compute model space coordinates and deflections. The procedure can then transform the model space coordinates from the axis system of the fixed reference points to the model axis system showing model deflections. To avoid introducing errors by printing a photograph or diapositive from the negative, the coordinates of a point are read directly from the negative.

The basic theoretical concept used in photogrammetry is that the photograph or image, being a perfect plane, is a central projection of the object as shown in figure 13 and described in reference 7. Implicit in this concept is the con- dition of colinearity of the image point on the photograph, the projection center of the camera, and the object point on the model.

Determining the Transformation Coefficients The method first solves for the transformation coefficients relating two- dimensional film measurements with three-dimensional object space measurements.

'Equations (I) are the basic formulas derived by Abdel-Aziz and Karara (ref. 6) for the direct linear transformation method of solving the colinearity condition -% LIX + L2Y + L3z + L 4 _:+ :0 L9x + L10Y + L11z + I (1) L5x + L6Y + L7z + L8 _+ :0 L9x + L10Y + L11z + I where x, y, and z are object space coordinates (model), X and Y are film measurement coordinates, and LI, ., L11 are the transformation coefficients.

APPENDIX

Since there are 11 transformation coefficients to be determined, a minimum

of 6 fixed reference points is required in each photograph. For these 6 known

points, equations (I) can be used to provide the following 12 equations relating

film readings and object space points:

(I) L1xI + L2YI + L3zI + L4 = -XI(L9Xl + L10Y I + L11z1 + I)

(2) L1x2 + L2Y 2 + L3z2 + L4 = -X2(L9x2 + L10Y 2 + L11z2 + I)

(3) L1x3 + L2Y 3 + L3z3 + L4 : -X3(L9x3 + LI0Y3 + L11z3 + I)

(4) L1x4 + L2Y 4 + L3z4 + L4 : _X4(L9x4 + L10Y 4 + L11z4 + I)

(5) L1x5 + L2Y 5 + L3z5 + L4 = "X5(L9x5 + L10x5 + L11z5 + I)

(6) L1x6 + L2Y6 + L3z6 + L4 = -X6(L9x6 + L10Y 6 + L11z6 + I)

> (2)

(7)

L5x I + L6Y I + L7z I + L 8 = -YI(L9Xl + L10Y I + L11z1 + i) (8) L5x 2 + L6Y 2 + L7z 2 + L 8 = -Y2(L9x2 + LlOY 2 + L11z 2 + i) (g) L5x 3 + L6Y 3 + L7z 3 + L 8 = -Y3(L9x3 + L10Y 3 + L11z 3 + i) (10) L5x4 + L6Y 4 + L7z4 + L8 = -Y4(L9x4 + L10Y 4 + L11z4 + I) (11) L5x 5 + L6Y 5 + L7z 5 + L 8 = -Y5(L9x5 + L10Y 5 + L11z 5 + I) (12) L5x6 + L6Y 6 + L7z 6 + L 8 = -Y6(L9x6 + L10Y 6 + L11z6 + I) where the subscripts I, 2, 3, • ., 6 in x, y, z, X, and Y denote point numbers for the six fixed reference points. The coordinates x, y, and z for the fixed reference points are known by measuring their location.

Rewriting equations (2) to solve for the transformation coefficient LI, . ., L11 we have

APPENDIX

(I) xiL I + YiL2 + ziL 3 + XIXIL9 + XIYILI0 + X1zIL11 + L4 = -XI

(2) x2LI + Y2L2 + z2L3 + X2x2L 9 + X2Y2LIO+ X2z2L11+ L4 = -X2

w w m u (3) x3LI + Y3L2 + z3L 3 + X3x3L 9 + X3Y3LIO + X3z3L11 + L4 = -X3 (4) x4L I + Y4L2 + z4n 3 + X4x4L 9 + X4Y4LIo + X4z4L11 + L4 = -X4 (5) x5L I + Y5L2 + z5L 3 + X5x5L9 + X5Y5LI0 + X5z5L11 + L4 = -X5 (6) x6L I + Y6L2 + z6L 3 + X6x6L 9 + X6Y6LIO + X6z6L11 + L4 _ -X6 (3) (7) xIL 5 + YIL6 + zIL 7 + YIXIL 9 + YIYILI0 + Y1z1L11 + L8 : -YI i m m w (8) x2L 5 + Y2L6 + z2L 7 + Y2x2L9 + Y2Y2LIo + Y2z2L11 + L8 = -Y2 (9) x3L 5 + Y3L6 + z3L 7 + Y3x3L9 + Y3Y3LIO + Y3z3L11 + L8 : -Y3 (10) x4L 5 + Y4L6 + z4L 7 + Y4x4L 9 + Y4Y4LI0 + Y4z4L11 + L8 = -Y4 (11) x5L 5 + Y5L6 + z5L 7 + Y5x5L 9 + Y5Y5LI0 + Y5z5L11_+ L 8 = -3 5 (12) x6L 5 + Y6L6 + z6L 7 + Y6x6L 9 + Y6Y6LI0 + Y6z6L11 + L8 = -Y6 By using equations (3) and the values of x, y, z, X, and Y for the six fixed reference points, the values of the transformation coefficients LI, . ., L11 are determined for both cameras.

Thus, there are 12 equations for determination of the 11 transformation coefficients LI, ., L11 using the minimum number of fixed reference points for calibration purposes. By incorporating additional reference points, additional equations are available and result in more multiples of these equation sets. Because of physical accuracy limitations on measurements and film read- ings, the multiple solutions of 11 equation sets will not provide a unique set of coefficients for the transformation. With 6 fixed reference points, there will be 12 values for each transformation coefficient. Therefore, a least- squares procedure is utilized to provide for the best transformation coefficients

APPENDIX for a given set of experimental data. Generally, 10 to 20 fixed reference points are used since additional fixed reference points improve the accuracy of the transformation coefficients. The accuracy also would be improved if the refer- ence points surround the model and one must avoid having the fixed reference points in the same plane.

Determining Model Space Coordinates Having determined the transformation coefficients for both cameras, equa- tions (I) can be rewritten to relate the stereo camera arrangement to any other common target points.

Camera I: (L I + XiL9)xi + (mR + xim10)Yi + (L3 + XiL11)zi = -(L4 + Xi)_ (4a)

l

(L5 + YiL9)xi + (L6 + YiLI0)Yi + (m7 + YiL11)zi -(L8 + Yi)J Camera 2: (4b) (L I + XiL9)xi + (L2 + XiLIO)Yi + (L3 + XiL11)zi = -(m4 + Xi)_ (L5 + Yim9)xi + (L6 + YiLIO)Yi + (L7 + YiL11)zi -(L8 + Yi)J where i denotes the target point.

Since the coefficients L I, ., L11 and X i and Yi are known for all points common to both cameras, the values x i, Yi, and zi can be computed for these points. Again, since there are four equations to solve for three unknowns, a least-squares method is employed to get the best solution. Employ- ing additional cameras will produce additional equations and improve the accuracy of the solution.

The accuracy of the method can be further improved by using an image refine- ment process by correcting the film coordinates X and Y for lens distortion, film deformation, and comparator errors. The technique is described in refer- ence 8 and the program is described in reference 9.

REFERENCES I. Igoe, William B.; and Baals, Donald D.: Reynolds Number Requirements for Valid Testing at Transonic Speeds. Facilities and Techniques for Aero- dynamic Testing at Transonic Speeds and High Reynolds Number, AGARD CP No. 83, Aug. 1971, pp. 5-I - 5-4.

2. Haines, A. B.: Further Evidence and Thoughts on Scale Effects at High Sub- sonic Speeds. Windtunnel Design and Testing Techniques, AGARD-CP-174, Oct. 1975, pp. 43-I - 43-12.

3. Bartlett, Dennis W.; and Harris, Charles D.: Effects of Wing Trailing-Edge Truncation on Aerodynamic Characteristics of an NASA Supercritical-Wing Research Airplane Model. NASA TM X-3024, 1974.

4. Bartlett, Dennis W.; and Harris, Charles D.: Aerodynamic Characteristics of an NASA Supercritical-Wing Research Airplane Model With and Without Fuselage Area-Rule Additions at Math 0.25 to 1.00. NASA TM X-2633, 1972.

5. Braslow, Albert L.; and Knox, Eugene C.: Simplified Method for Determina- tion of Critical Height of Distributed Roughness Particles for Boundary- Layer Transition at Mach Numbers From 0 to 5. NACA TN 4363, 1958.

6. Abdel-Aziz, Y. I.; and Karara, H. M.: Direct Linear Transformation From Comparator Coordinates Into Object Space Coordinates in Close-Range Photogrammetry. Paper presented at Close Range Photography Symposium (Urbana, Illinois), Jan. 1971.

7. Moffitt, Francis H.: Photogrammetry. Second ed. Int. Textbook Co., c.1967.

8. Karara, H. M.; and Abdel-Aziz, Y. I.: Accuracy Aspects of Non-Metric Imageries. Photogrammetric Engineering, vol. XL, no. 7, July 1974, pp. 1107-1117.

9. Marzan, G. T.; and Karara, H. M.: A Computer Program for the Direct Linear Transformation Solution of the Colinearity Condition and Some Applications.

Paper presented at Symposium on Close Range Photogrammetric Systems (Champaign, Illinois), 1975.

TABLE I.- LOCATION OF OPTICAL TARGETS Chord location Span Chord location on wing, x/c on body center location line, x'/c Y 0.15 0.25 0.35 0.45 0.65 0.85 0.95 b12 -0.046 0.247 X X X • 309 X X X X X -. 247 X X X -. 309 X X X X X X X -.371 X X X X -. 402 X X X X 1 096 -.433 X X X X 1 241 1.384 -.464 X X X X X X X 1.526 -.495 X X X X 1.669 ,.526 X X X 1.812 -.557 X X X X 1.955 -.588 X X X 2.O97 -.618 X X X X -.649 X X X X X X -. 680 X X X -.742 X X X X -.804 X X X X -. 866 X X X X -. 897 X X X X X -.928 X X X X -. 959 X X X X

TABLEII.- TUNNEL TESTCONDITIONS

Temperature Reynolds number Dynamic pressure

Mach

number

K °F per meter per foot Pa lb/ft 2 2.0 x 106 6.6 x 102

I .20 425

322 120 20 349

I .20 4.1 85O

322 120 13.5 40 698 .95 322 120 7.5 2.3 20 349 425 4.6 85O .95 322 120 15.1 40 698 .80 2.6 425 322 120 8.5 20 349 .80 85O 322 120 17.1 5.2 40 698 I 4) -,.-t v C C R O_ d / q 1 I_ O_ _ d /I II .,-t I N) .o G) Q) L c_ o .,.4 C C .o I ao ,-4 C ,-4 N) I -'4 Inboard section assumed rigid \ / \ I

/

\ /

\ \ / \ / i / \\I / / t -y < -_ +y Figure 3.- Top view of wing and fuselage showing pattern of optical targets on lower surface of model.

2O

l

I / s'l\ \ \

!i / / "-a \ _

tl/ I \/\

Figure 4.- Wind-tunnel installation for stereophotographs of model.

All dimensions are in cm (in.).

L-76-1772

Forward camera

Aft camera Figure 5.- Stereophotographs from two cameras.

f/

J

/

.M 0-,'_ ,_-I .P It _ L-- C_ 0 [] .0 C_ 0 _. _. o o.

w f- • ,_ d a/x' aJnssaJd _to Ja_,ua 0 U •,.4 _ g., NI I_ ,---t 0C) ,-.4 _ e'_ o © [] @ El _i .,-I r._ I I o, O_ ° ,r,I

F- _o

I °r'l r-4 _.,

o

ii (i) ill 0"l '_ ,,-,t co :l

II

A 0 • u o r'_ !

t'..- II

o'"

I.l"% II J c_ a_ o,1 I I I I I OJ o o t !

I I I r",,,,i o I

_Ul '0

.9

t

_l _.

\

\

i, \

i

,_1 \, 1

, _:_ _r_

k

t!

\ , : , I _ -,. H e ,/ II 4

.d

I I I o o o _r, O0 ._1 L L 1 i _J L L | _ i 1 _1 cq. ,_.

"u! ' 'u! ' 9 II L) q J _l \!, ".-" L, *,4 i: \!

I I I

:i I

j • • i

!il !l!

¢y II J o u-_ o c_ ,_; • L .... L.... I t ...................................

_. _ "U! ' "u! • 9 q • _ w4 • -- • m ...... I I(:0 _I m

\-

A _ _m L c)

T-

IR \ ...... 4 -- ti o- o • , ° II c_ c_ c_ O_ o --t'_l J__l " L _._---- 1_-- L _----J • L__,.__ JE-- u_ _ _ ca r_ u._ o u,_ o _ • W:_J9 I I L L ] i ¢ L I i L 1 I "u! " g "u! ' g q i !

i o d)

I 0

-_ r_

g

.,-4 \ \ \ ....... :_ \ m I

m II II '

o.

_ .

li ..... dl ..... • ___ _- II _o --" t J 1 L J I ,J Ul_'g I I l I ! I I I I "ul ' 9 26 L _ __q -\ _ -\ ....

_r .......

,q.

, \

. ----_______ L__L _ __ -i. I [ l 04 r--I wo'9 L__ • [ / _ i I c_ "ul ' g II r..)

I o b-- "0 ® -_ --] __ r..,

_1 .... JJ

r_

,li

I ,__ I f.L_- 1 \ _ 1_÷__ 1 .........

I l " I -" -_, i I [ I L ___ L L L__ J_ __I______]___ J oo "u! " g o.

a_ o O0 o a_ o i ] t: II' -,--t

!i-

II {D" ] o'- oo II 4_ o_ .,-4 o

q

ii .0 (II CO

-i

LO iI OJ o" ,L/ 4-_ II II I I I I I I I I !

0 u'_ 0 q u_ e.i _ • '-' ILl:) ' 9 WD'9 "_ [ ] I 1 I I I I ,,_r oq i_, ¢,.I 0 _ '_.

"u! ' (2 .,-4 "U! '9

2B

o.

L- ............

>,{

I

o' II 4.)

ii _) o I I oo !

C (I) r,..

-,-4 m __n_ i <> i -- CO (.1 i '_ Q t I.

t t II II !

{7 o II II I I I I I o c,,I L W:) ' 9 I , , I I L l I e, t _ .o _ c., c3 oR.

"u! ' 9 "uJ ' g

q

__m m.

I

[!1

1 /

II • _ L o" II r_ o

oDO._ 4

o

o. II _.)

I o v co

g

-,-I A II II o'" II J I I I I I w_ ' 9 I I I L I I oo.

"u! ° 9 3o © q = 20 _19 Pa (425Ib/sq if) [] q = 40 698 Pa {850Ib/sq ft) O , -.4 8 ,deg -.8 -- -I.2 ....

-1.6 "0.

--_ 3--- ----_ _ -- ,8 .............

8,deg -I.2 \ -I.6 - 2.0 ..... ] ....

- 2.4 o .2 .3 .4 .5 .6 .7 .8 .9 1.0 Y b/2 (a) M = 0.80.

Figure 9.- Variation of spanwise wing twist in tunnel determined from stereophotographs.

0 q = 20 349 Pa {425 Ib/sq if) [] q = 40 698 Pa (850 Ib/sq ft) Ol w ,,4 -- -.8 8,deg

\

-I.2 -1.6 a=4 = -2.0 L .w-- u ,4 --

,,q

-.8

<

\

\

8,deg - 1.2 ----

\

- 1.6 -2.0 ....

] a=8 °

t

-3.4 1.0 .6 .7 .8 " ,9 .2 .5 0 .I .3 .4 Y

_2

(b) M = 0.95.

Figure 9.- Continued.

0 q = 20 349 Pa (425 Ib/sq if) [] q = 40 698 Pa (850 Ib/sq if) !

i !

-.4 8,deg --,8

-____[

-I.2 -.4 -.8 -I.2 e,deg - 1.6

\

-2_.0

\

-2,4 -2.8 1:1=8 °

I I I

.I .2 .5 .4 .5 .6 .7 .8 .9 1.0 Y

b/2

(c) M = 1.20.

Figure 9.- Concluded.

-I .2 surface surface 0 E) q = 20 349 Pa M25 Ib/sq if) Upper Lower --I. 0 ED [] q = 40 608 Pa (850 Ib/sq ft) i \ Cp -,2 - _ - t .............

.2 ,4 .__L,=

_=,°: _ =°= [

o=,o_ =oi= ___i__

,6 -I.2 -I.0 --.8 _ - --.6 _ ,_- --_ _=-41 ___ -- -J i ........

cp-4 ___ ._!-

--,2 ..................

0=8°; _ =0.80 °=4°; b_2 =0.80 .4 0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 x/c X/C (a) M = 0.80.

Figure 10.- Effect of dynamic pressure on wing pressure distribution Y at --- of 0.48 and 0.80.

b/2 --I.2 I ......

Upper Lower -- surfacesurface -- 1.0 _ q = 40 698 Pa(850Ib/sq ft) _1__ _ _ q = ZO 349Pa(425Ib/sq if) --.6 C:' : _ Cp --.2 -_ \,_ .- a=4°; _ =0.48 .4 --I.2 --I.0 --.8 --.6 Cp -. 4 -.2 0 _ .......

N

11;4°; _ =0.80 °=8°; b-_ =0.80 .40 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 I.O X/C x/c (b) M = 0.95.

Figure I0.- Continued.

tower Upper surface surbce q = 20 349 Pa (425 Ib/sq ft) q = 40 69B Pa L850 Ib/sq ft) [] [] -.8 -.6 ....... _ ----_ J. ¢== i \ _-_ I -,4 i I b ! ; Cp -.2 !

t I I J, I .2

g-

.4 -.8 t ........

Cp J_ I "-F_ _J O- !

o2 ---- = 8°; _ = 0.80 I (_=4°; _ =0.80 [.

"40 ' .2 • 4 ,6 .8 1.0 0 .2 .4 .6 .8 1.0 x/c X_ (C) M : 1.20.

Figure I0.- Concluded.

.6

............

.4

.2 ¸

a=4 °

1.2

© q = 20 349 Pa (425 Ib/sq ft) [] q = 40 698 Pa (850 Ib/sq ft)

• 8 L_

\

.6

\\

.2

(]=8 °

O. 2 .3 .4 .5 .6 .7

.8 .9

I0

Y

b/Z

(a) M = 0.80.

Figure 11.- Effect of dynamic pressure on variation-of wing semispan load distribution.

.8

I

.6 .4 c c i i ' i

n<Cal

"-_ .....

.... t !

.2

\

a=4° t .... I I I 1.2 o q = 20 349 Pa (425Ib/sq if) [] q = 40 698 Pa (850Ibl sq fl) 1.0 .8 .6

\

.4

\

\

.2 (3=8 ° .9 I.O .5 .7 .8 __ .3 .4 .6 Y

b/2

(b) M = 0.95.

Figure 11 .- Continued.

.6

.4

.2

13{=-4 °

1.0

0 q = 20 349 Pa (425 Ib/sq ft)

\

[] q = 40 698 Pa (425 Ib/sq ft)

\

.8

\

\

.6

C C

n( av)

.4

.2

0=8 °

.7 .8 .9 1.0

.5 .5 .6

.4

Y

_z

(c) M = 1.20.

Figure 11.- Concluded.

0 a =4 ° [] u = 8° -- -- -- 2 (a = 4°) l __ -.4 8,deg -.8 -I.2 ....

-1.6 IJJ IB 2 _-- leO ....

::%,.,\

\

n \ .8 \ ,,,,, L.J.

c c '-,....

.4

"x)-.. \

\

.2

&

,q

i 0 1.0 .2 • 3 .4 .5 .6 .7 .8 .9 Y

b/2

Figure 12.- Effect of an increase in angle of attack on spanwise variation of wing twist and semispan load at M = 1.20 and q = 20 349 Pa (425 lb/ft2).

The dashed line indicates double the value of either wing twist or semispan load at _ = 4 ° .

4O Negative Rear nodal point ection cenler or fwd nodal poin!

Principal

\

\

\

Objed Figure 13.- Sketch showing basic theoretical concept used in photogrammetry.

3, Recipient'sCatalog No.

1. Rel_t No, 2. Government AccessionNo.

NASA TP-1010 5. Report Date 4. Title and Subtitle October 1977 MEASUREMENT OF MODEL AEROELASTIC DEFORMATIONS IN THE WIND TUNNEL AT TRANSONIC SPEEDS USING 6. Performing Organization Code STEREOPHOTOGRAMMETRY 8. Performing Organization Report No.

7 Author(s) L-II092 Joseph D. Brooks and Jerry K. Beamish 10 Work Unit No 505-11-16-08 9. P_f_ming Or_nization Name and Addre= NASA Langley Research Center 11 Contract or Grant No.

Hampton, VA 23665 13 Type of Report and Peciod Covered 12. S_nsoring Agency Name and Addr_s Technical Paper National Aeronautics and Space Administration 14. Sponsoring Agency Code Washington, DC 205h6 15. _pplementary Not_ Fort Worth Division, General Dynamics Corporation, Jerry K. Beamish: Fort Worth, Texas.

16. Abstract This investigation was conducted to evaluate a stereophotographic method of determining the aeroelastic deformations of an airplane model under aerody- namic load in the wind tunnel, This is a Joint NASA and General Dynamics pro- gram. Wind-tunnel tests were conducted in the Langley 8-foot transonic pressure tunnel on the wing of a O.0625-scale model of the TF-8A supercritical-wing research airplane to obtain simultaneously the aerodynamic forces and moments, pressure distributions, and stereophotographs; these tests were conducted at Mach numbers of 0.80 A 0.95, and 1.20, and at free-stream dynamic pressures of 20 349 Pa (h25 lb/ft z) and 40 698 Pa (850 lb/ftf).

The accuracy of the stereophotographic technique in determining wing deflec- tions was within 0.013 cm (0.005 in.) under static conditions. This value trans- lates to an error in wing twist of O.10 ° inboard and increases to 0.20 ° outboard.

When the model is under aerodynamic load in the wind tunnel, the accuracy of the stereophotographic technique of determining wing deflections increased to 0.052 cm (0.020 in.) when compared with static wing loadings because of the dynamic motion of the model in the tunnel.

At transonic speeds, the wing deflections and wing twist generally do not increase linearly with an increase in either angle of attack or dynamic pressure and Reynolds number.

18 Distribution Statement t7 Key Words(Suggestedby Author(s)) Aeroelastic deformations Unclassified - Unlimited Wing twist Stereophotography Photogrammetry Subject Category 05 19 Security Oa_if (ofthisre_rt) 20 SecurityCla_if (ofthis _ga) 21 No of Pages 22 Dice" hl $_.00 Unclassified Unclassified

,I

"For sale by the National Technical Information Service, Springfield, Virginia 22161 NASA-Langley, 1977