techniques for measurement of dynamic stability derivatives in ground test facilities

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l

AGARDograph 121

n

a.

(m

V .

Q

<

<

Techniques for Measurement

of Dynamic Stability Derivatives

in Ground Test Facilities

by C. J . Schueler, L. K. Ward and

A.

E. Hodapp, J r

•

OCTOBER 1967

ROYAL AIRCRAFT

ESTA3L?SHM

r

Nr

1

8

APR

m s

LIBRARY



AGARDograph

121

NORTH ATLANTIC TREATY ORGANIZATION

ADVISORY GROUP

FOR

AEROSPACE RESEARCH

AND

DEVELOPMENT

( O R G A N I S A T I O N

DU

T R A I T E

DE L'

ATLANTIQU E NORD)

TECHNIQUES FOR MEASUREMENT

OP DYNAMIC STABILITY DERIVATIVES

IN GROUND TEST FACILITIES

by

C.J.Schueler, L.K.Ward and A.E.Hodapp, Jr

October 1967

This is o ne of a series of publications by the AGARD-NA TO Fluid Dynamics Panel.

Professor Wilbur C.Nelson

of

The University

of

Michigan

is

the editor.



PREFACE

T he au t h o r s w is h t o ex p r e s s t h e i r ap p r ec i a t i o n t o t h e n u mero u s

o r g an i za t i o n s an d i n d i v i d u a l s who s u p p o r t ed t h e p r e p a r a t i o n o f t h i s

AGARDograph by making infor ma t ion and m at er ia l av a i l ab le .

I n p a r t i cu l a r , t h e au t h o r s a r e i n d eb t ed t o t h e A r no ld E n g i n ee r i n g

Development Ce nte r , Arnold Air Force S ta t i on , Tenn essee, USA, fo r

approva l o f the use of AEDC m at er i a l and as s i s t a nc e in pr ep ar a t io n

o f t h e f i g u r e s .

Per sona l thank s a re ex tended to James C .Usel ton fo r c omp i l ing

in fo rm ati on on Magnus fo rc e and moment t e s t i n g , Dr W.S.Norman f or

h i s co n s u l t a t i o n o n s ev e r a l o f t h e s e c t i o n s , A r t h u r R . W all ace f o r

checking p or t io ns of th e m anu scr ip t s , Mrs L .P . Dean, Mrs J.A .Y oung,

and Mrs W.C.Scott fo r typ ing t he manu scr ip t and r e f er en ce r es ea rch

an d t o Mrs T . R . H e st e r f o r c o o r d i n a t i n g t h e p r e p a r a t i o n o f i l l u s t r a t i o n

work.

i l l



C O N T E N T S

P a g e

SUMMARY II

R E S U M E i i

P R E F A C E i i i

LIST OF FIGURE S V

NOTATION AND ABBREVIATIONS xi

1. INTROD UCTION 1

2. S T A T E M E N T O F T H E P R OBL E M 3

2.1 DEVELOP MEN T TRE NDS 3

2.2 EQUA TION S OF MOTION 4

2.2.1 Axes Sys tem s 4

2.2.2 Inerti al Mo me nts 5

2.2.3 Aero dynamic Mom ents 7

2.2.4 Single Degree of Freedom 9

2.2.5 Stabil ity Deri vati ve Coe ffic ients 11

3. CAP TIVE MODE L TESTING TEC HNIQUE 13

3.1 GENER AL DESCR IPTION OF TEC HNIQUE 13

3.1.1 Model Piv ots 13

3.1.2 Sus pensi on Syst em 18

3.1.3 Balance Tar e Dam ping 21

3.2 DYNAM IC STA BILITY DE RIVATIVES - PITCH OR YAW 24

3.2.1 Th eo ry 25

3.2.2 Rep rese ntative Syst ems for Test s of Comp lete Mode ls 55

3.2.3 Rep rese ntativ e Syst ems for Test s of Control and

Lifting Sur fac es 69

3.3 DYNAM IC ROL L DE RIVATIVES 71

3.3.1 The ory 71

3.3.2 Repr ese ntative Syst ems 74

3.4 MAG NUS EFFE CT 76

3.4.1 Type s of Magnus For ces 76

3.4.2 Test Pro ced ure 77

3.4.3 Repr esentat ive Syst ems and Drive Units 77

3.5 PRES SURE MEA SUR EME NTS 81

3.5.1 Tra nsduc er Insta lla tio n 82

3.5.2 Pres sure System Resp onse 82

3.5.3 Cal ib rat ion Te chnique 83

R E F E R E N C E S 84

BIBLIOGRAPHY 92

FIGURES 120

DISTRIBUTION

iv



LI S T O F FI G U R E S

Page

Fig.1 Meth ods of measuring dynamic stability derivati ves 120

Fig .2 Axes syst ems 121

Fig.3 Orientation of coordinate systems 122

Fig.4 Multipiec e crossed flexures 123

Fig.5 Cross-flexure, cruciform and torque tube pivot s 124

Pig .6 Small scale instrument flexure pivo t 125

Pig .7 Knife edge pivot 126

Fig.8 Cone pivot - three degre es of freedom 126

Fig.9 Ball bear ing pivo t 127

Fig.10 High amplitude forced oscillat ion balance with ball bearing pivot 128

Fig .11 Gas bearing pivot 129

Pig. 12 Gas bearing load charac terist ics 130

Fig.13 Damping charact eristics of AEDC-VKF gas bearing 131

Pig .14 Gas beari ng for roll- damping system 132

Pig.15 Spherical gas bearing 132

Pig.16 Two-dimensional oscillator y apparatus 133

Fig.17 Half-m odel oscillating rig 134

Fig.18 Half-m odels and reflection planes 135

Pig.19 Support interference effects

(a) Ik = 2.5 . d

s

/d = 0.4 , a = ±2 deg 136

(b) Ik = 2.5 . d

s

/d = 0.8 , a = ±1.5 deg 137

Pig.20 Support interference effect s

(a) Ho = 3 , d

s

/d = 0. 4 , a = ±2 deg 138

(b) H,, = 3 . d

s

/d = 0.8 . a = ±1.5 deg 139



Page

Pig. 21 Support interference effect s

(a) Ik = 4 , d

s

/ d = 0 . 4 . a = ± 2 d eg 140

(b) Ik = 4 , d

s

/d = 0.8 . a = ±1.5 deg 141

Pig. 22 Variation of static and dynamic stability coefficient and base

pressure ratio with sting diameter for Z

s

/d = 3 142

Fig. 23 Model balance and support assembly for hotshot (AEDC-VKP Tunnel P)

tunnel 143

Fig.

24 Free oscilla tion balance system 143

Pig.

25 Transverse rod and yoke system installed in test section tank 144

Pig.

26 Dynamic stability derivatives versus amplitude of oscillation,

Mac h 10 145

Fig.27 Damping derivatives for the flared and straight afterbody

configurations measured in conjunction with a series of simulated

transverse rod configurations, forced-oscillation tests, amplitude

1.5 deg , Mac h 5 146

Pig. 28 Mag netic suspe nsion system, end view 147

Fig. 29 Basic "V" configu rati on magnet ic suspe nsion system 147

Pig. 30 Contribution of aerodynamic, still-air and structural damping to

tota l dampi ng 148

Fig. 31 Sketch of the vacuum system used to evaluate still-air damping 148

Pig.

32 Variation of structural damping with static pressu re 149

Fig.

33 Variation of the ratio of model still-air damping to model

aerodynamic damping with Mach number 149

Fig. 34 Variation of the angular viscous- damping-moment paramete r with

angular restoring-moment para meter 150

Fig.

35 Tric ycli c pitc hing and yawing 150

Fig.

36 Typical free-flight epicyclic pitching and yawing moti ons of a

nonrolling statically stable symmetrical missi le 151

Pig.

37 Typical free-flight epicyclic pitching and yawing moti ons of a

rolling statically stable symmetrical missi le 151

Pig.

38 Typical free-flight pitching and yawing motion of rapidly rolling

statically unstable symmetrica l missile 152

vi



Page

Fig.39 Moti on transformatio n

152

Pig.40 Comp lex angular motio n

153

Pig .41 Exponentia l decay of the modal vectors 153

Pig.42 Phase diagram 154

Pig.43 High frequency free-os cillat ion balance for Hotshot tunnel tests 155

Fig.44 Model displacement system

156

Pig.45 Free oscillati on balance and displac ement m echanism 157

Pig.46 Large amplitude

(±15

d e g ) ,

free-oscillation balance

158

Fig.47 Free-os cillat ion transvers e

rod

balance

159

Pig.48 Photographs of the free-oscillation (±35 deg) transverse rod

balance

160

F i g . 4 9 F r e e - o s c i l l a t i o n b a l a n c e 16 1

Pig .50 Tra nsv er se rod ba lan ce for low de ns i ty hyp er so nic f low tun nel 162

Fig .51 Photograph of the as sembled ba la nce 162

F i g . 5 2 S m a ll s c a l e f r e e - o s c i l l a t i o n b a l a n c e 163

Fig .53 Bas ic mass in je c t io n ba lan ce and porou s model 163

P i g . 5 4 D ynamic s t a b i l i t y b a l an ce s u i t a b l e f o r g a s i n j ec t i o n 164

Fi g. 55 O ltr on ix dampometer 165

F i g . 5 6 T y p i ca l c a l i b r a t e d s l o t t e d d i s c s and p r i n c i p l e o f d ampo mete r

opera t ion 165

F i g . 5 7 F r e e - o s c i l l a t i o n d a t a r e d u c t i o n 166

F i g . 5 8 S t i n g b a l an ce a s semb ly o f t h e f o r c ed - o s c i l l a t i o n b a l an ce s y st em 166

F i g . 5 9 S ma ll amp l i t u d e ( ±3 d eg ) , f o r c ed - o s c i l l a t i o n b a l an ce 167

Fig .6 0 Dynamic s t a b i l i t y co nt ro l and r ead out sys tem 168

F i g . 6 1 O n - l i n e d a t a s ys te m f o r f o r c e d - o s c i l l a t i o n d yn am ic s t a b i l i t y

t e s t i n g 1 6 9

v i i



Page

P i g . 6 2 C o n t r o l p a n e l f o r f o r c e d - o s c i l l a t i o n s y s te m 170

F i g . 6 3 F o r c e d - o s c i l l a t i o n b a l a n c e 1 71

P i g . 6 4 S ys te m b l o c k d i ag r am f o r f o r c e d - o s c i l l a t i o n b a l a n c e 172

F i g . 6 5 P h o t o g ra p h o f t h e f o rw a r d p o r t i o n o f t h e f o r c e d - o s c i l l a t i o n

ba la nc e mechanism 173

Pi g.6 6 Schem at ic v iew of model and dr iv in g- sy st em components 174

P i g . 6 7 D e t a i l s o f o s c i l l a t i n g m e ch an is m 175

F i g . 6 8 B l oc k d i ag r a m o f e l e c t r o n i c c i r c u i t s u s e d t o m e a su r e m od el

dis pla ce m en t of ap pl i ed moment 176

F i g . 6 9 F o r c e d - o s c i l l a t i o n d yn am ic s t a b i l i t y b a l a n c e 177

F i g . 7 0 Y a w - r o l l b a l a n c e a nd a c c e l e r o m e t e r s 178

F i g . 7 1 F o r c e d - o s c i l l a t i o n b a l a n c e w i t h e l e c t r o m a g n e t i c e x c i t a t i o n 179

P i g . 7 2 C uta wa y v ie w o f t o r s i o n a l f l e x u r e p i v o t w i t h s t r a i n g a g e s f o r

f o r c e d - o s c i l l a t i o n b a l a n c e w i th e l e c t r o m a g n e t i c e x c i t a t i o n 179

Fi g . 73 Mag net ic model sus pen sion system 180

Pi g.7 4 Cone model for ma gnet ic sus pe nsio n 181

P i g . 7 5 B a s i c L c o n f i g u r a t i o n m a g n e t i c s u s p e n s i o n s y s te m 1 81

P ig .76 Magne t i c susp ens ion sys t em 182

P ig .77 Magne t i c suspe ns ion work ing se c t io n . 7 i n . x 7 i n . hyp er son ic

wind tu nn el 183

F ig .78 Schemat i c d i ag ram o f t he susp ens ion magnet a r r ay o f a s i x -

component mag net ic wind tu nn el ba lan ce and sus pen sion system 184

F i g . 7 9 M odel s u b j e c t e d t o a f o r c e d o s c i l l a t i o n i n r o l l w h i l e s u sp e n de d

in th e s ix - componen t ba l a nce 185

F ig .80 View o f co n t ro l su r f ac e d r iv e 186

F i g . 8 1 A p p a r a tu s f o r e l e v a t o r o s c i l l a t i o n 187

F i g . 8 2 A p p a r at u s f o r e l e v a t o r o s c i l l a t i o n 188

F ig .83 Ske tch o f co n t ro l su r f ace p luc k ing mechan ism 189

v m



Page

P i g . 8 4 E l a s t i c mod el s u s p en s i o n and e l ec t r o m ag n e t i c o s c i l l a t o r f o r

o s c i l l a t i o n i n p i t c h 190

P i g . 8 5 F r e e o s c i l l a t i o n w i th a u t o m a t i c a l l y r e c y c l e d f ee d ba ck e x c i t a t i o n 191

P i g . 8 6 H a l f- mo d e l o s c i l l a t i n g r i g 192

Fig .87 Rol l -dam ping sys tem 193

Fig .88 Ai r be ar in g ro l l - dam pin g mechanism 193

F i g . 8 9 A i r b ea r i n g r o l l - d am p i n g b a l an ce 194

Fig .90 Dynamic s t ea dy - r o l l ba la nce 195

P i g . 9 1 F o r c e d - o s c i l l a t i o n b a l a n c e 196

P i g . 9 2 P i c t o r i a l d ra w in g , f o r c e d - o s c i l l a t i o n b a l a n c e 197

F i g . 9 3 I l l u s t r a t i o n s o f t h e v a r i o u s M agnus e f f e c t s

(a) Magnus for ce at low an gl es of at ta ck 198

(b) Magnus for ce at la rg e an gl es of at ta ck 198

(c) Magnus ef fe c ts on ca nte d f i n s 198

(d) Magnus ef fe c ts on dr ive n f i n s 198

F i g . 9 4 E f f ec t o f s p i n p a r ame t e r o n t h e M agnus c h a r a c t e r i s t i c s 199

Fi g. 95 Eff ect of boundary la ye r on th e Magnus for ce 200

Fig .9 6 E le c t r i c motor dr i ve Magnus force sys tem 201

Pig .97 Magnus force sys tem, tu rb in e dr ive w i th in th e model 201

Fig .98 Magnus force sys tem, tu rb in e dr ive w i th in th e model 202

Pi g.9 9 Magnus for ce ba lan ce 203

Fig .10 0 Magnus ro l l -da mp ing ba lan ce w i th a i r be ar i ng s 203

F i g . 1 0 1 V a r i a t i o n o f b ea r i n g t emp e r a t u r e w i t h s p i n r a t e f o r b y p as s

a i r co ol in g 204

P i g . 1 0 2 O i l m i s t l u b r i c a t i n g s y st em and o t h e r i n s t r u m en t a t i o n 2 05

Pig .10 3 Two typ es of Magnus ba lan ce gage se c t io ns

(a) Sandia s t r a in gage ba lan ce 206

(b) AEDC-VKP s t r a i n gage ba la nc e 206

i x



Page

Pig. 104 Compar ison of semiconductor and conventional gage outpu ts 207

Pig.105 Experimental response magnitude and phase- angle calibratio n for

1.38 in. of

0.030

in. nominal-diam eter tubing connected to a gage

3

vol ume of 0.02 in. for air at sea-lev el condit ions 208

Pig.106 Pump for sinusoidal oscilla ting pressur e 209

Fig.107 Cam-ty pe pulsat or calibra tor 209



NOTATION AND ABBREVIATIONS

N o t a t i o n

A r e f e r e n ce a r ea o r g en e r a l co n s t an t

C t o t a l d amp in g co e f f i c i en t o r g en e r a l co n s t an t

C

x

t a r e d ampin g co e f f i c i en t

C

A

(- C

X

) t o t a l a x i a l - f o r c e c o e f f i c i e n t , - 2F

x

/yq

D

V

(

2

;

A

C

c

c r o s s - w i n d co e f f i c i e n t , 2P

Y

/Po>VpA , (C

y

= C

c

)

C

D

t o t a l d rag co e f f i c i en t , 2D/Po,VgA

C

L

l i f t c o e ff ic ie n t , I h / p ^ K

C j r o l li n g - mo men t co e f f i c i en t , 2M

x

/Pa)VpAd

C

l p

BCj/B(Pd/2V

c

)

C

/ r

B C j / 3 ( r d / 2 V

c

)

C

l / 3

dC,/dyS

C

t/

g BCj /3(5d/2V

c

)

C j g * 3 C j / 3 S *

C„ pitching moment coefficient o f a mode l which

h a s

trlagonal o r greater

symmetry, 2M

y

-/

/

o

a

,VgAd

C

« p a

C

n p a

=

C

n r

C

m a

C

m a

—

=

s

=

C

mp/3

C

mq

"

C

n /3

"

C

n 4

C

M a

C

B

p i t c h i n g moment c o e f f i c i e n t , 2M

y

//Oa>V*Ad

C

mpr = 3 V

5

(

P d / 2 V

c '

3

'

r d / 2 V

c >

C

mp /

3 = 3

2

C

m

/ 3 ( P d / 2 V

c

)B ^

C

m p

4 = 3

2

C

m

/ 3 ( P d / 2 V

c

) 3 C f i d / 2 V

c

)

C .

a

= dC

ff l

/3(qd/2V

c

)



J

m/3

= 3C

m

/3/3

ma

ma

c

N

(-c

z

)

C

N ,

C

Na

C

N a

C

n

C

np

C

npq

C

n p a

C

npa

C

n r

C

n/3

C

n/3

= 3c

m

/Ba

= d C

m

/ d ( ad /2V

c

)

n o rm a l f o r c e c o e f f i c i e n

= BC

N

/B(qd/2V

(J

)

= dC

N

/Boc

= 3 c

N

/ 3 ( a d / 2 V

c

)

yawing moment c o ef f i c i e

= 3 C

n

/ d ( P d / 2 V

c

)

= d

2

C

n

/ d ( P d / 2 V

c

) d ( q d / 2 V

c

)

= 3

2

C

n

/ d ( P d / 2 V

c

) B a

= d

2

C

n

/d(Pd/2V

c

)d (6cd /2V

c

)

= 3 c

n

/ d ( r d / 2 V

c

)

= 3c

n

/3/?

= 3C

n

/BC3d/2V

c

)

'yR

D

d

d.

P

X

(-

P

A>

F

Y

.

V - ° »

V

c '

'z/AoV^

side force coeffi cient, 2P

y

/

/

c

a

,V

2

A

cycles

t o

damp

to a

given amplitude ratio

R

velocity o f sound in a cylindrical tube

drag force

o r

general constant

model reference length, base diameter

sting diameter

voltage

forces along

t h e XYZ axes,

respectively

frequency o f oscillation

xii



f ( 0 ) v i s c o u s d a m p i n g

a s a

f u n c t i o n

o f

0

f ( 0

o

) v i s c o u s d a m p i n g a s a f u n c t i o n o f 0

Q

H

x

, H

y

, H

z

m o m e n t s o f m o m e n t u m a b o u t t h e X , Y a n d Z

a x e s ,

r e s p e c t i v e l y

H a n g u l a r m o m e n t u m v e c t o r

I

x

, I

y

, I

z

m a s s m o m e n t s o f i n e r t i a a b o u t t h e X , Y a n d Z

a x e s ,

r e s p e c t i v e l y

i = / ( - I )

T ,

J , k

u n i t v e c t o r s

i n t h e X , Y a n d Z

d i r e c t i o n s , r e s p e c t i v e l y

i . j * , k

u n i t v e c t o r s

i n t h e X , Y a n d Z

d i r e c t i o n s , r e s p e c t i v e l y

J

z x

. J

x y

, J

y z

p r o d u c t s

o f

i n e r t i a

K t o t a l s t i f f n e s s

K j f l e x u r e s t i f f n e s s

L l i f t f o r c e

I m o d e l l e n g t h

o r

t u b i n g l e n g t h

l

3

s t i n g l e n g t h

o f

c o n s t a n t d i a m e t e r

H v e c t o r r e s u l t a n t

o f t h e

e x t e r n a l m o m e n t s a c t i n g

o n t h e

m o d e l ,

( M

X

T

+ M

y

J +

M

z

k )

o r ( M

x

t + M

y

5 + M

z

£ )

M m o m e n t a b o u t

Y

a x i s

( M

y

)

M j o u t - o f - p h a s e m o m e n t

M

r

i n - p h a s e m o m e n t

M

x

, M

y

, M

z

m o m e n t s a b o u t

t h e X , Y a n d Z

a x e s , r e s p e c t i v e l y

M

X p

=

d M

x

/ d P

M

x S

*

= B M

X

/ B S *

M

&

=

B M

y

/ B < 5 c

U

e

= BM

y

/B0

U g = BM

y

/B(9

I L ^ , f r e e - s t r e a m M a c h n u m b e r

x l i i



n number

P.Q.R scalar components

of cd in

the

X

,

Y and Z

directions

(P = P

0

+ P, Q = q

0

+

q.

R

= r

0

+ r)

P =

P d / 2 V

C

P

s

s t e a d y - s t a t e r o l l i n g v e l o c i t y

p , q , r p e r t u r b a t i o n s of P.Q.R

P

t

i n p u t p r e s s u r e

p,,,

m e a su re d p r e s s u r e at gage

P

b

mode l base p r es su re

Pa,

free-stream static pressure

p m e a n p r e s s u r e

O o ,

f r e e - s t r e a m d y n a m i c p r e s s u r e

R r a t i o

o f

f i n a l a m p l i t u d e

t o

I n i t i a l a m p l i t u d e

R e f r e e - s t r e a m u n i t R e y n o l d s n u m b e r

r t u b i n g r a d i u s

o r

r a d i a l d i s t a n c e

t t i m e

U . V . W s c a l a r c o m p o n e n t s

o f

V

c

in th e

X

,

Y

a n d

Z

d i r e c t i o n s

( U

= u

Q

+

u ,

V = v

0

+

v ,

W - w

0

+ w)

u , v , w p e r t u r b a t i o n s of U,V,W

V

c

or V

v e l o c i t y

o f t h e

m o d e l c e n t e r - o f - g r a v i t y w i t h r e s p e c t

t o t h e

m e d i a

(Ul + v j + wk, v

0

=

w

0

=

0)

V volume

of

pressure gage

V

t

volume of tube

XYZ body-fixed axes system (Pig.3)

or

components

of

external forces

X

e

Y

e

Z

e

earth-fixed axes system (Fig.2)

X Y Z nonrolllng axes system (Fig.3)

X.Y

Z„

stability axes system (Fig.2)

o S S

xiv



X

T

Y

T

Z

T

tunnel-fixed axes system (Inertial refe rence. Fig ure 3)

X

W

Y

W

Z

W

wind axes system (Fig.2)

x,z transfer distance

x

c g

distance from

the

model nose

to the

model piv ot axis (rearward po sit ive)

a angle

o f

attack

/3 a n g l e o f s i d e s l i p

y r e f e r e n c e p h a s e a n g l e

A

log

d e c r e m e n t

B *

c o n t r o l s u r f a c e d e f l e c t i o n a n g l e

0 I n s t a n t a n e o u s a n g u l a r d i s p l a c e m e n t f r o m t h e r e f e r e n c e f l i g h t c o n d i t i o n

0± o u t - o f - p h a s e p o s i t i o n

0

Q

o s c i l l a t i o n p e a k a m p l i t u d e

o r

i n i t i a l a m p l i t u d e

0

T

i n - p h a s e p o s i t i o n

A. d a m p i n g r a t e

p

v i s c o s i t y

p

w

f r ee - s t r e am o r amb i en t d en s i t y

£ complex ang le of at ta ck (v + iw)/V

c

o r , f o r s ma l l an g u l a r d i s p l a ce

m en ts, /3 + ioc

< p a r b i t r a r y p h a se a n g l e o r r o l l p e r t u r b a t i o n

X

H,0 ,$ Eule r ang les ; ang les of yaw, p i t c h and ro l l . F ig ure 2(b)

(? = 0

O

+ 0 . 0 = 0

O

+ 0. $ = 0

O

+ cp)

ip,0.4> p e r t u rb at io n s of * , 0 , $

H v ec t o r an g u l a r v e l o c i t y o f t h e n o n - r o l l i n g co o r d i n a t e s y st em w i t h

r e s p ec t t o i n e r t i a l s p ace (O i + q j + r k )

c o an g u l a r c i r c u l a r f r eq u en cy

co. n a t u r a l damped c i r c u l a r f r eq u en cy

lo v ec t o r an g u l a r v e l o c i t y o f t h e b od y co o r d i n a t e s y st em w i t h r e s p e c t t o

i n e r t i a l s p ace (P i + q j + r k , q

0

= r

0

= 0)

xv



<ad/2V„

( ' ). C )

(")

n

Subscripts

a

t

E

i

i o r 2

Abbreviations

AEDC

PWT

VKF

AGARD

BRL

DTMB

HSD

LRC

LTV

reduced frequency parameter

first

a nd

second derivati ves with respect

t o

time

t

vector notation

nonrolling coordinate system notation

aerodynamic

tare

effective values

initial conditions

local values

Initial pitch axis

o r

refer ence flight conditi ons

new pitch a x e s , times,

o r

nutatio n

and

precession mode s

wind

on

vacuum

free-stream

Arnold Engineering Development Center

( U S A i r F o r c e ) ,

Arnold Center,

Tennessee.

Propulsion Wind Tunnel Facility, AEDC.

von Karman

G a s

Dynamics Facility, AEDC.

Advisory Group

f o r

Aerospace Research

and

Development,

a

Division

o f

the North Atlantic Treaty Organization, NATO.

Ballistic Research Laboratory, Aberdeen Proving Ground, Maryland.

David Taylor Model Basin (US N a v y ) , Carderock, Maryland.

Hawker Siddeley Dynamics Ltd, Coventry, England.

Langley Research Center ( N A S A ) , Langley Field. Virginia.

Ling-Temco-Vought A erospace Corporati on, Dallas, Texas.

xv 1



MIT Massachus etts Institute of Technology, Cambridge , Massachuse tts.

NACA National Advisory Commit tee on Aeronautic s (now the National Aero

nautics and Space Agency,

N A S A ) .

NAE National Aeronautical Establi shment, Ottawa, Canada.

NASA National Aeronautical and Space Agency, Washington, DC.

NOL Naval Ordnance Laboratory, White Oaks , Maryland.

OAL Ordnance Aerophy sics Laboratory, Dangerfield, Texas.

ONERA Off ice National d'Etudes et de Rec her che s Ae'rospatiales (National

Bureau of the Aerospace

Research),

Paris, France.

PU Princeton University, Princeton, New Jersey.

RAE Royal Aeronautical Establish ment, Farnborough, Hants, England.

SU Southampton University, Southampt on, England.

UC University of California, Berkeley, California.

UM University of Michigan, Ann Arbor, Michigan.

WADD Wright Air Development Division, Wright-Patters on Air Force Base,

Ohio.

xvli



S E C T I O N 1

INTRODUCTION

Analysis o f aircraft and missile aerodynamic performance and stability requires

knowledge o f t h e forces and mome nts which act under conditions o f steady and unsteady

flight. One of the first studies o f aircraft stability wa s that of Lanchester

1

around

1900, followed later by more complete and rigorous analysis by Bryan

2

, which is still

the basis for much of the dynamic stability work today.

Requirements for supplying a erodynamic charact eristic s for perfor mance analysis

have been met, to some extent, by relying on theories supported by experimental m easure

ments t o establish t he validity o f t h e theories. In some cases, th e theory h a s only

limited application especially when viscous flow h a s a pro nounced influence on the

flow field, as in the case o f high speed flow (hypersonic speeds and above) and low

Reynolds numbers. There are some situations for which the phenom ena cannot be described

adequately by theory. In other cases theory is not applicable because th e configuration

and flow field are so complex that t h e many approximations required invalidate a wholly

theoretical approach.

Thus,

in prac tice , strong reliance is placed on experimental

results for use in analyzing th e modes of motion o f t h e aircraft o r missile.

Early experimental techniques making u s e o f oscillating models to dynamically simulate

rigid modes o f motion for determining stability derivati ves are well summari zed by

Jones

3

.

It is of interest t o note that no fundamental changes in the basic experimental

methods ha ve taken place since that ti me, 1934, although much work h a s been done toward

refinement o f t h e metho ds, equipment and instrumentation. Little emphasis wa s placed

on dynamic stability experiments until about 1940, a nd from that time to 1945 experi

mentation wa s primar ily concerned with subsonic flow and tests were made for small

perturbat ions. Wind tunnel experiments and flight tests within t h e early forties were

directed more toward static stability and control problem s. With t h e advent of high

speed aircraft, missiles, rockets, and re-entry vehicles , increased emp hasis h a s been

placed

on

performance

and

dynamic stabil ity pro blem s since missions demand precis e

control

no t

only

in

small disturba nces from level flight,

but

also

in

large scale

maneuvers. A s a result, dynamic stabil ity measu rements have become increasingly important

at supersonic, hypersonic and hypervelocity speeds and at large perturbations. T h e

experimental methods suggested by earlier experiments have been used in different forms,

and many improvements have been made in the experimental techniques since about 1950.

These improvements have been in the areas o f balance system, model and support system

design, instrumentation, data acquisition, and data reduction.

In 1954 Valensi

4

presented a review o f t h e techniques primari ly in use in Europe for

measuring oscillatory aerodynamic forces and moments o n models oscillating in wind

tunnels.

Th e following year AGARDograp h 11

(Ref. 5)

by Arnold wa s published on the

subject of dynamic me asurements in wind tunnels. T he most recent summary o f techniques



for measuring oscillatory derivatives in wind tunnels was published by Bratt

6

in 1963

as a part of the AGARD Manual on Aeroelasticity. The paper discus ses the basic principles

employed in measuring derivat ives and gives some account of the associated instrumentation.

Much of the report deals with the methods and instrumentation employed in England.

Since these earlier works, the emphasis on dynamic stability testing has increased,

and many reports have been prepared on the subject, so there exists a need for a review

and summary of the current dynamic stability testing techniques in use at supersonic

speeds and above.

There are many experimental techniques which are suitable for measuring dynamic

stability d erivati ves; however, only those techniques that are in most common use in

wind tunnels will be described. In addition, many different systems exist for measuring

the same quantities and, since they differ primarily in the specific application, no

attempt will be made to describe all systems; only systems considered to be representa

tive and those having unique features will be discussed. The report is organized

primarily around the aspects of dynamic stability testing in wind tunnels at supersonic

speeds and abov e, although m uch of the report is pertinent to aspec ts of testing at

transonic and subsonic speeds. Although free -flight testing in ground test facilities

encompasses the use of the wind tunnel and aeroballistlc range for measurements of

dynamic stability derivative s, these t echniques are not included as a part of th is

AGARDograph because other publications on these subjects have been published and are

in preparat ion

7, 8

.



S E C T I O N 2

S T A T E M E N T O F T H E P R O B L E M

2. 1 D E V E L O P M E N T T R E N D S

Aircraft or missiles in high speed flight are free to respond to disturbances with

motio ns involving many degrees of freedom. During vari ous stages of aerodynamic design

of a vehicle, as the configuration is developed, dynamic chara cteristics must be known

and progressiv ely refined to a high degre e for the final configura tion performa nce

specification. The designer must necessarily be concerned with the interpretation of

aerodynamic stability data and with the evaluation of the influence of their accuracy

on an integrated system design.

There are three methods of obtaining stability data:

(i) Theory and empirical data,

(li) Wind tunnel and aer oballistic range model testi ng,

(iii) Sub- or full-s cale flight testi ng.

During the preli minary design stage, it is most expedient to rely primar ily on theory

and empirical data for performance evaluation whereas, in the later stages, design

concepts inevitably require evaluation through the use of some scale model tests. As

a final phase in the design development, sub- or full-scale flight tests are required

to verify performance and obtain in-flight measurements for verification of theory and

prediction methods.

There are certain speed regimes where theoretical results may be inaccurate due to

insufficient or inaccurate basic data. For example, in the transonic speed regime

(0.9 < Mo, < 1. 5) theoretical meth ods for dete rmining dynamic stab ility deriv ative s are

very limited, and, therefore, reliance is place d on methods involving the use of

empirical data on similar configurations and correlation plots showing the manner in

which the stability derivatives vary with Mach number for similar configurations.

Within the superso nic and hypers onic regio n (1.5 < M^, < 10), theoreti cal met hods

are available for estimating der ivatives for bodies of revolution and relatively simple

lifting configuratio ns. At Mach numbers above 10 in the hypervelocit y speed regime,

theories generally fail to predict the derivatives because visc ous effects have a

strong influence on the vehicle aerodynamic characteristics.

In order to Improve the accuracy of aerodynamic de rivat ives that are used in per

formance pre dic tions and to validate theoret ical estima tes, model tes ts in ground test

facilities are recognized as being essential to the development of aircraft, missiles

and re -entry vehic les.



Study of vehicle motions in flight involves consideration of many degrees of freedom.

When the vehicle is considered as a rigid body, six degrees of freedom are required.

Since the vehi cle is not a rigid stru cture, additional relati ve motio ns of components

such as aeroelastic deflection of lifting surfaces and movable controls constitute

additional degrees of freedom.

The vehicle motions may be separated into high frequency and low frequency motions,

where high frequency motio ns studies such as flutter are primarily concerned with

structural elastici ty and unsteady aerodynamics. At low frequencies the moti ons involve

rigid body motions and are studied as dynamic stability using quasi-static aerodynamics.

The mode of model testing discusse d in the present A GARDog raph is concerned with this

latter type motion and the determination of rotary damping derivatives.

In scaling down a vehic le for model t ests the paramet er tha t must be considered in

addition to the Mach number and Reynolds number is the angular velocity or frequency

of model oscillat ion in relation to full-scal e conditions. The scaling param eter,

called the reduced frequency, in the form cdd/2Va, contains the frequency of oscilla tion

and represents the ratio of some characteristic dimension of the vehicle to the wave

length of the oscillation. This type of scaling parameter also applies to dynamic roll

tests and has the form Pd/2Vo, .

Experimental techniques for measuring dynamic stability derivatives in ground test

facili ties may be classified in many different ways. The classifica tion that will be

followed here is presented in Figur e 1. The two major classi ficati ons are derived on

the basis of whether the model is free to move in all basic degrees of freedom or

whether it is restricted in several degre es of freedom. The sub topics shown in Figure 1

for discussion were selected as those methods currently being employed for most high

speed testing; however, it should be emphasized that the outline does not show the case

of the model fixed in the test section with pert urba tions generated in the flow. As

noted in the Introduction, only those methods which apply to captive model testing

techniques will be discussed in this AGARDograph since free-flight tests in the wind

tunnel and range are the subject of AGARDographs published and now in preparation

7, 8

.

Additional general information on meth ods not described herein may be found in Reference 9.

2. 2 E Q U A T I O N S O F M O T I O N

In order to place the role of dynamic stability testing in perspective, it is helpful

to examine the equations of motion. Although the mathe matica l treatment of vehicle

dynamics is well known and found in many te s t s

10

'

12

, a brief derivation of the equations

of motion is included to illustrate the basis for the experimental techniques used in

the study of dynamic stability.

2.2.1 Axes Sys tem s

The equations of motion will be developed using the axes systems shown in Figures 2

and 3. Model orientation is determined relati ve to the tunnel fixed axes X

T

Y

T

Z

T

(inertial reference). The XYZ axes are fixed in the model with the X and Z axes

in the model's plane of symmetry. When the origin of this system lies at the center

of gravity, these axes are then defined as the body axes. The angular orientation of

the XYZ system with respect to the X

T

Y

T

Z

T

system and inertial space is given by the



Euler angles ¥ , 0 , and $ . Somet imes it is convenient to use the nonrolli ng axes

system (XYZ). Thes e are special body axes whose angular orientation in space relative

to the X

T

Y

T

Z

T

system is determined by the angles ? and 0 . This axis system can

be used even though the model is rotating about the X

axis.

The origins of all

systems are assumed to occupy the same point in space

(Pig.3).

2.2.2 Inertial Mom ents

The angular momentum of the model may be expressed as

H = /(r x V

p

) dm ,

(1)

where r is the radius vector from the origin of the rotating XYZ axes to the parti cle

p of mass dm and V is the velocity of the particle p relative to inertial space.

The radius vector r is measured relative to the body fixed XYZ

axes;

therefore,

in order to evaluate the velocity of p relative to inertial space, it is necessary

to use the well-known transformation for the rate of change of any vecto r from fixed

to rotating axes as follows:

dr

V

P

= dT

dF

= — + co x r ,

dt

where

Pi + Qj + Rk

Assuming the model is a rigid body, the equation reduces to

V = co x r ,

since r is then not a function of time.

Substituting t his equation into Equation (1) yie lds

H = J F x (co x F) dm .

This expression is then integrated over the vehicle to obtain

fi = L j ] [ i ] W .

where

[j]

=

LiJkJ

[I] -

{co} =

-J

-J

P"

Q

R

YX

ZX

-J

XY

-J

Y

-J

- J

xz

yz



The moment ac t i ng on th e ve hi cl e i s eq ual to th e r a te of change of angu lar momentum

2M .

H

d t

dH _ _

= — + w x H

d t

Th i s ve c to r equa t io n can be r educed to t he fo l low ing s ca l a r f o rms:

I

X

P + ( I

z

- I

y

)RQ - J

X Z

(R + QP) + J

X Y

(RP - 0) + J

Y Z

(R

2

- Q

2

) = £ M

X

)

I

Y

Q + ( I

x

- I

Z

)PR - J

X Y

(P + QfO + J

Y Z

(PQ - R) + J

X Z

( P

2

- R

2

) = £M

Y

)

I

Z

R + ( I

Y

- I

X

)QP - J

V 7

(Q + PR) + J

Y 7

(QR - P) + J

Y V

(Q

2

- P

2

) (2M

7

) .

'YZ

V

'xz

N

'XY'

(2)

Roll pitch

and ya w

rates

for the XYZ

system

may be

obtained from Figure

3 as

follows:

P

= $ - * sin 0

Q = 0 cos $ + ¥ cos 0 s in $

R

=

y cos

0 cos $ - 0 sin $ .

In orde r

to

simplify

t h e

nonlinear differential equations given above

to a

form

which facilitates analysis o f t he motio n, vehicle symmetry will be assumed, and the

motion will be restricted to small dist urbances from a reference condition. L e t

P = P

0

+ P ,

¥ =

0 + 0 .

e

t c. , where

th e

zero subscripted quantities

a r e t h e

reference

conditions and the unsubscripted lower case quantities are perturbations from this

reference c ondition.

For t h e following discuss ion, the reference condition is selected such that

q

0

= r

0

=

0

Q

=

0

O

= 0 and the

reference quantities

p

0

and

cp

Q

are not

necessarily

small. Considering vehic les having symmetry about t h e X Z plane ( J

w

„ = J „ „ = 0) and

xy y z

n e g l e c t i n g p r o d u c t s o f p e r t u r b a t i o n q u a n t i t i e s , t h e e q u a t i o n s o f m o t io n beco me

V "

J

x z< r + QP)

= £M),

I

y

q + ( l

x

- I

z

) P r + J

X Z

P

2

= ( 2 M )

Y

I

z

r + ( I

Y

- I

x

)Pq - J

X Z

P = (2M)

Z

T he r o l l , p i t c h a nd yaw r a t e s a t t h i s c o n d i t i o n a r e g i v e n a s

Po = ^o

p = cp - \p s in 0

q = 0 co s $ + 0 cos 0 s i n $

r = yi co s 0 co s $ - 0 si n $ .

(3)

(4)



For models having triagonal or greater symmetry, the nonrolling coordinate system

XYZ may be conveniently used when these axes correspond to the mode ls' principal

inertia axes. Since the body is moving with respect to the Y and Z axes (rotating

about the X axis), the restriction of symmetry is necessary, otherwise the moments of

inertia with respect to the XYZ system will be time varying. For the above case,

the angular momentum expression is given as

H

I

Y

Pi + Iqj + Irk ,

where

I = I

Y

= I

z

q - q cos $ - r sin $

r = r cos $ + q sin $

$ ~ f P(t) dt , 1*1 » \-\b sin 0\

Jo

Note that, even though $ = 0 for the nonrolling coordi nates, its XY plane is

rotating with respect to the X

T

Y

T

plane at the angular rate -xft sin 0 .

The equations of motion referenced to the nonrolling coordinates are given as

dH .

— + n x

H

=

S M

,

dt

where

M

O i + qj + rk .

These equations of motion reduce to

13 +

l i * -

I * *

I

X

PF

I

x

P q

=

=

=

(2M)

X

( 2 M )

?

( 2 M )

2

(5)

2.2.3 Aerod ynamic Mome nts

The general pr ocedure for expanding th e aerodynamic moment relat ions is that deve

loped by Bryan

2

. It is assumed that the moments are functions of the steady-state

reference conditions and the instantaneous value s of the distur bance veloci ties, con

trol angles, and their time derivatives. A Taylor series expansion of an aerodynamic

reaction yields this form. For example, the aerodynamic reaction A = A(U.V, ... ) is

expressed as



A(U,V,...)

=

A (u

0

, v

0

. . . . )

+ — (U - u

0

) +

i

Buy

o

+

( B T )

0

( V

-

V

°

) +

( S )

0

( U

-

U

O >

2 +

B

2

A \

(U

- u

Q

)(V -

v

Q

)

+

\ B u B v /

0

2

^

2 I

v

' ' 0 '

B

2

A \

where

\Bv

U

= u

0

+ u , etc.

This expansion is linearized by neglecting products of the perturb ation quantities;

therefore,

for the

above example

the

linearized aerodynamic reaction be comes

A(U,V,...) A(u

n

. v

n

, . . . ) f (—

1

u + I — I v + ... .

Aerodynamic reactions

can be

expanded with respect

t o t h e

variables

U

,

V , W

,

P ,

Q , R , the control d eflecti on angles, and time derivatives o f these variables. For

a given configuration, conditions

of

symmetry

and

past experience concerning relative

effects o f these derivatives can be used to reduce the number of derivatives needed

to describe

the

aerodynamic moment system.

F o r a

spinning vehicle having t riagonal

or

greater symmetry, the moment equations can be reduced to

M

x =

M

xo + l - ~ \ P + (r r r ) 8*

VWo w

)M

Y

\ /B M

Y

\ / 3 M

Y

\ /BM

Y

The term (BM

Y

/Bv)

0

in th e p i t c h equ at io n and th e term (BMg/Bw),, in th e yaw

e q u a t i o n a r e u s u a l l y n e g l i g i b l e e x c e p t f o r t h e c a s e w hen t h e v e h i c l e i s s p i n n i n g r a p i d l y .

These term s are caused by th e Magnus ef fe ct . They ar e g iven in a more fa m i l ia r form

a s f o l l o w s .

Let

< * ) ,

'

YV

° ' W o "

Z

'°



Taking th e first term as an example and assuming that it is primar ily a function

of P , its Taylor series expansion about the condition P = 0 is given as

'BM

M

Yv 0

( P....) = M

Yv 0

( 0. . . . ) +( - ^ ) (P - 0) +

v O l

Bp

The first term on the right hand side is usually negligible; therefore, this expression

may be reduced to

BM

Y

T

P

M

Yvo<

p

) -rr^J

Similarly,

The pitching- and yawing-moment equatio ns are then given as

/ B M

Y

\

/ BM

Y

\ / B M

V

\ /B

2

M

V

"•

= M

K I 7 ) /

+

( l r )

0

i +

^ '

r + l

^' "•

m

The terms M

x o

, M

Y 0

, and M

z o

in Equations ( 6), (7), and (8), respectively, are

the moments about the X , Y and Z axes while the model is at the reference flight

condition. The reference condition is often chosen such that these terms are zero.

Equations (3) describe the motion of a model which has three degrees of freedom in

rotation about a point fixed in space . Unique test equipment is required in orde r to

obtain these moti ons using a captive model test technique. Two balances which have

this capability are described later in Section 3.1.1. In most capti ve model dynamic

stability tests, the model is restrained to a single degree of freedom, i.e., rotation

about a single

axis.

An example of a typical dynamic stability b alance would be a

single-degree-of-freedom system using a flexure pivot. The equation of motion for a

system of this t ype will be developed in the following discussion.

2.2.4 Single Degree of Freedom

The velocity components in the body axis for the case of a model mounted on a single-

degree-of- freedom flexure pivot at an angle of attack a and oscilla ting symm etrically

at a small angle 0 are

u - \

w

(a + d ) = Va , cos a - d \

m

sin a

V = 0

i



10

W

s

Vo,

sin (a + d) - d v

w

cos a + V^ sin a

i

(9)

P = 0

Q = a = £ . ii = 0

R = 0 .

The reference conditions used here

are

those that exist when

the

model

is at the

angle

of attack oc . Under these conditions. Equation (2) reduces to

I

Y

q =

( 2 M )

Y

.

Only the derivation of the equation for a single degree of freedom in pitch is presented,

since the derivations for osci llations in yaw and roll are similar and are treated in

detail

in

many

of the

refere nces included (see,

e.g.,

Reference 4) .

An elastic restraint

is

provided

in the

balance

to aid in

restori ng

the

model

to the

reference attitude following a displac ement. When the aerodynamic moment is expanded

to include all significant terms and the resto ring moment Kj((X +

0 )

and the damping

moment C.6 of the restra int are added to this to obtain the total moment, the equation

of motion in pitch is written as

I

Y

q = M

Y0

+ M

u

u + M

q

q +

\ v +

M^w + Kj(0 + a ) +

C

x

0

.

where

M

u

=

(BM

Y

/ d u )

0

, M

q

=

(BM

Y

/3q)

0

M

w

= (BM

Y

/Bw)

0

, M* = (BM

Y

/Bw)

0

.

After including the relationships for the linear and angular compo nents of velocity

(Equation (9)), the equation above reduc es to the form

I

Y

0 + (-M

q

- Va,M£ cos a - C . ) 6 +

+ (MjjVa,

sin a -

i^Va,

cos a - i . ) 6 - M

yo

+

KjOC

= 0 . (10)

In

the

undisturbed case, conditions

of

equilibrium give

M

y o

= KjOC ,

and, for small va lu es of a , Equation (10) becomes

I

Y

0 - (-M

q

- V ^ - C,)0 + (-Kj - M

w

V

eo

)t9 = 0 . (11)



M

0a

Met

M

0a

M

(9t

=

=

=

(Mq + V„,M*) ,

Cj .

VL/fc.

K j .

11

Defining th e damping-moment pa ram ete r and th e res tor in g-m om ent para m ete r by Mg and

Wg , r es pe c t iv e l y , where

aerodynamic

damping-moment parameter of

e l a s t i c r e s t r a i n t c o ns id e r e d

as a t a r e

aerodynamic

s ta t i c -moment parameter o f

e l a s t i c r e s t r a i n t c o n s id e r e d

as a t a r e .

With the se d e f i n i t io n s , Equat ion (11) becomes

id - m

h

+ m\

e&

)d - ( M

M

+ m\

ea

)d = 0 .

Following

the

procedure presented

in

Reference

4,

similar expressions

can be

written

for t he equations o f motion in the body axis for ya w and roll.

2.2.5 Stability Derivat ive Coeff icients

The stability derivative coefficients

are

obtained

in

this section using the refere nce

condition

?

0

= d

Q

= w

Q

= v

0

= q

0

= r

Q

= 0

. This

is a

typical condition

for a

sym

metrical model oscillating about

its

zero-angle-of -attack trim.

T h e

NACA system

11

will

be used here with

t he

time constant defined

as t* = i/ u

0

. The

aerodynamic moments

are defined

in

this system

as

, 2

M

x

= pV

2

A*C

z

(12)

M

y

= p\

2

c

A lC

m

(13)

M

z

= p \

2

c

A lC

n

. (14)

where V

c

= / ( U

2

+ v

2

+ w

2

)

and it is necessary that the refere nce length ( I ) be defined in half lengths; e.g.,

I = d/2

,

I = b/2 , etc.

The coefficient form

of the

derivative

( BM

y

/Bw)

0

is

obtained, using Equation (13),

as follows:

Since (Bv

c

/Bw)

= w/V

c

, the

first term

on the

right-hand side

o f

Equation (15)

i s

negligible.



12

T he a n g l e o f a t t a c k i s d e f i n e d t o f i r s t o r d e r ( s m a l l a n g l e s o f a t t a c k ) a s

ot

where

u ^ V

o — "c

Substitution of this definition into Equation (15) yie lds

^BM'

^

:

'VIC. ,

where

J S .

Ba

The coefficient form of the derivative (BM

Y

/Bq) is obtained, using E quation (13),

as fo l lows:

©..-*-ft

(16)

T he c o e f f i c i e n t C i s d e f i n e d , u s i n g t h e t i m e c o n s t a n t a nd t h e d e r i v a t i v e f rom

t h e r i g h t - h a n d s i d e o f t h i s e q u a t i o n , a s

3

"«

=

t*(^)

=

l 9 V 3 ( q Z A

c

) ]

0

.

Using t h i s d e f in i t i o n , Equa t ion (16 ) becomes

^ M

v

,

= P ^

A l t

^

T he r e m a i n i n g c o e f f i c i e n t s a r e o b t a i n e d u s i n g t e c h n i q u e s s i m i l a r t o t h o s e o u t l i n e d .

Us ing Eq ua t io ns (6 ) , ( 7 ) , and (8 ) and th e above methods fo r ob ta in i ng the co e f f i c i en t s ,

Eq ua t ion s (12 ) , ( 13 ) , and (14 ) can be w r i t t en in t he fo l lowing c oe f f i c i en t f o rms , where

I = d /2 :

M

x

= |

P

V

2

A d C ,

0

+

C / P

S )

+ C

8

*

S

*

['

A d |c

m0

+ C.JX + c

mo

v

ina

ma

v2V„

+ c

mq

^2V„

+ C,

M

z

=

j p V

2

A d

C

n 0

+

C

n p

p

+ C

n /

g ( - - 1 + B L - ( -

9

\

2 V

9

\2Vm

>2V,

n p a

v2V„

a



13

S E C T I O N 3

C A P T I VE M O D E L T E S T I N G T E C H N I Q U E

3.1 G E N E R A L D E S C RI P T I O N

O F

T E C H N I Q U E

Methods for obtaining rotary deriv ative s in pitch, yaw or roll involve restraint of

the model

in

several degrees

o f

freedom

and

measurement

of

mome nts required

to

sustain

an oscillation

for

specified conditions

o f

frequency

and

amplitude (forced o scillation

technique)

or

measurement

of

model motion after

it has

been disturbed (free oscilla tion

technique).

The basic units

in the

captive model testing system

are the

pivots,

the

support

system and,

in the

forced oscilla tion t echnique, mecha nisms

for

forcing

the

model

oscillation. T h e pivot is a very important part o f a dynamic stability test system,

and

the

selection

of the

type

o f

pivot

and

desi gn will depend

on

several fac tors, some

of which have

a

direct bearing

on the

quality

of

data that

may be

obtained with

t h e

system. Also o f importa nce is the suspe nsion system and the influence it may have on

the aerodynamic measurements. Detailed descriptions

o f

model pivots, suspension systems,

and balance tare damping characteristics

are

presented

in

Sec tions 3.1.

1 to

3.1.3.

3. 1. 1 Model Piv ots

Pivots

of

various types which

are

employed

in

dynamic stability balances include

ball bearings,

g a s

bearings, knife edges, cones,

and

crossed flexures.

The

selection

of a particul ar type pivot will dep end on the system operating requirements and the

environment

in

which

it

will

be

used.

One of the

most important factors that must

be

considered

in the

design

of a

dynamic s tability balance

for

aerodynamic damping measur e

ments

is the

damping that

the

system contribute s

to the

meas ured wind

on

damping.

Aerodynamic damping decreases with increased Mach number

and

becomes

a

small per centage

of the measured damping at the hypers onic Mach numbers. O ne exception that will be

discussed

in

Section

3,1.3 are g as

beari ngs which generally have negligibl e tare

damping. Because

o f i ts

importance, ta re damping will us ually

be one of the

principal

considerations

for

selection

of the

pivot type

o r

bearing

for

roll damping balance

systems.

3.1.1.1 F l ex u r e

High pivot friction

may be

avoided

by

employing crossed flexures

o f th e

general

type shown

in

Figure

4.

These bend

in

such

a way

that

the

center

of

rotation remai ns

at essentially a fixed position. T h e crossed flexure provides stiffness, which is

necessary

to

overcome static moments

o n

models tested

at

angles

o f

attack

by the

free-

oscillation technique. Flexure piv ots

are

used

in

many applica tions,

in

particular

scientific instruments, because they have many favorable characteristics. They

a r e

simple

in

desig n,

yet

rugged

in

constructio n

for

their flexibility; they

do not



14

introduce hysteresi s

o r

backlash

in the

system; they

do not

require lubrication;

and

they

are not

susceptible

to

wear. Their main disad vantage

is

that they

are not

suitable

for large angles o f rotation because of stres s considerati ons, especially under com

bined loadings.

The

design

of

flexure piv ots

for a

particular application requires

consideration

of the

model loads, mom ents,

and

oscill ation frequency

for

dynamical

similitude. Experience

at the

AEDC-VKF

h a s

shown that crossed flexures have linear

stiffness characteristics and structural damping that is repeatable even though the

flexure

is of

multipiece construction. This method

of

construction

is

desirable because

the flexures can be machines as a single unit and then cut into sect ions for the in

dividual flexures.

The

theory

for

design

of

symmetrical crossed-flexure piv ots

is

given in References 13 through 16.

Examples o f flexure piv ots, including cruciform and torsion tube pivot s, that have

been used

in

various applications

are

shown

in

Figure

5. For

each

o f

these,

t he

pivot

tare damping varies inversely with the frequency o f oscillat ion. Strain gage s mounted

to

o ne

member

of the

crossed flexure

are

employed

to

obtain

a

signal v oltage which

is

proportional

to the

angular dis placeme nt.

For some applications, such as tests in very small tunnels or low density tunnels

which have

a

small test core,

it may be

impractical

to

machine flexure pivots

of the

type shown in Figure 5. Th e AEDC-VKF h a s developed a small dynamic stability balance

using

a

commerc ially available instrument fl exure pivot

of the

type shown

in

Figure

6.

The characteristics o f this pi vot were found to be repeatable and linear. Angular

motion time histories

are

measure d with

an

angular transducer described

in

Section

3.2.2. 1.

3 .1 .1 .2 Kn ife Edge and Cone

Knife edge and cone p iv o ts of th e typ e shown in F ig ur es 7 and 8 have rec en t l y been

u s e d i n d yn am ic s t a b i l i t y b a l a n c e s y s t e m s b e c a u s e t h e y a r e n e a r l y f r i c t i o n l e s s .

T he p r i n c i p a l d i s a d v a n t a g e o f e i t h e r t h e k n i f e e d ge o r c on e p i v o t i s t h a t som e

method o f r e s t r a in t must be employed to ho ld the p iv o t s i n con t ac t and f ine ad jus tme n t s

a r e n e c e s s a r y t o e l i m i n a t e b i n d i n g .

The s ln g le - de gre e -o f - f r e edo m kn i f e edge p i vo t shown in F igu re 7 was used in a s i n g l e -

degree-of-

f reedom f r e e -o s c i l l a t i o n ba l a nce sys t em deve loped a t DTMB. Here kn i f e edges

a r e used to cons t r a in t he mode l and a l so fo rm the ax i s o f r o t a t i on .

A s i m i l a r t ec hn iq ue ha s a l so been used by DTMB fo r a t h r e e -d egr ee - o f - f r eed om ba la nce

system; however , i n th i s ca se , th e model mount ing system i s a cone on the end of t he

s t i n g i n s e r t e d i n a c o n i c a l d e p r e s s i o n c o n t a c t w i t h i n t h e m od el ( F i g . 8 ) . T he m od el

i s f r e e t o r o l l , p i t c h , and yaw abou t t he cone and i s he ld on th e p iv o t by mode l d r ag .

In each o f t hese cases , mode l a t t i t ude i s measu red f rom pho tograph ic r eco rds o f mode l

mot ion.

3 . 1 . 1 . 3 B a l l B e a r i n g

Ball bearing pivots may be used in a dynamic stabil ity balance system for pitch and

yaw derivative measurements if high resultant model loads are to be supported and if

aerodynamic damping

is

large

so

that bearing dampi ng constitut es

a

small part

of the



15

measured damping. Experience h a s shown that a ball bearing pi vot is likely to have

about two orders o f magnitude higher damping than that of a flexure pivot designed to

carry comparable loads. Although th e ball bearing pivot h a s t h e advantage o f being

suitable for large angles of rotation, bearing dam ping i s a function of the amplitude

of oscillation

and is

often nonrepeatable.

A

ball bearing pivot used

in an

AEDC-VKF

free-oscillation balance is shown in Figure 9.

Ball bearing pivots are quite fre quently used in forced-os cillation balances when

high amplitudes o f oscilla tion cannot b e obtained with flexure pivots. Although t h e

bearing damping is high, its effects may be eliminated in the measurements by measuring

the forcing torque on the model side of the bearings, a s shown in Fig ure 10. For such

a balance system, angular contact bearings are used with preloa d t o eliminate free play

in t he bearings.

Ball be arings are used in Magnus force and moment ba lance s ystems where mod els are

spun to high rotational speeds. In this application, b earing friction is not important

since

the

measurements

are

static forces

and

moments

in the yaw

plane; however, bearing

heating may be a problem at extremely high rotational speeds.

Until recently, roll-damping balance systems have been designed with ball bearings;

however, ne w systems in development are designed with g a s bearings to overc ome bearing

damping, wear, requirements for lubrication, and sometime s cooling. Angular motion

time histories may be measured with t he type o f angular transducer described in Section

3.2.2. 1.

3.1.1.4 Gas Be ar ing

Although g a s film lubrication is an old concept, it is only since about 1950 that

the study

o f g a s

bearings

h a s

noticeably accelerated.

T h e

reason

for

this

is

that

special bearing requirements can often be best satisfied with a g a s bearing. G a s

bearings were used in special wind tunnel drag-fo rce balances a s early as 1956; howe ver ,

they were no t applied to balances for dynamic stability in pitch and yaw measurements

until about 1961 (Ref.

1 8 ) ,

and roll dampi ng until 1964 (Ref.

1 9 ) .

Applications in wind

tunnel testing techniques have been extensive since these early developments. This is

understandable since all of the desirable features o f a pivo t, which include high load-

carrying capability, low damping, and unlimited rotation, a re combined in a gas bearing

pivot; whereas flexures and ball be aring pivot s have only some o f these characteristics.

Gas bearing pivots are used extensively for dynamic stability testing in the AEDC-VKF

at hypersonic speeds where th e aerodynamic damping i s very low. It is difficult to

measure damping accurately with a flexure pivot ba lance under this condition becaus e

of

the

high percentage

o f

tare damping.

An example o f a journal-typ e g a s bearing pivot used in an AEDC-VKF one-degree-of-

freedom dynamic stability balance for measuring pitch o r y a w derivatives is shown in

Figure 11. Th e inner o r core section o f t h e bearing is supported on each s ide by a

sting, and the model is mounted to a pad on the outer movable ring. Supply g a s ,

generally dry nitrogen, is supplied through th e core to a radial set of orifices in

the center o f t he bearing. Filter s are provided in th e core to remove foreign particles

that will cause bearing fouling. A closed g a s supply and retur n system is not used

at th e AEDC-VKF, since test s have shown that t h e small flow rates have negligible

influence on base pressur es and wakes. Bearings have been developed at the AEDC-VKP



16

with two radial rows o f orifices spaced to improve t h e yawing-moment carrying

capability. Bearings

are

usually designed

for

specific applications; however, com

mercial gas bearings now exist that may be used in one-degree-of-freedom balance

systems.

This

is an

important fact since design, development

and

fabrication

of gas

bearings

can

become v ery time-c onsuming

and

costly.

The gas bearing shown in Figure 11 was designed with inherent orifice compensation

according

to the

procedure outlined

in

Refere nce 20. Analysi s

o f

this type

o f

bearing

is a problem of compres sible flow with friction in three-dimensional passages that

change area with loading. Treatm ent

of

this problem consisted

of

reducing

the

complex

analysis o f t h e bearing to the analysi s o f a simple, flat plate model. A s will be

shown later, these simplifying assumptions lead

to a

model realistic enough

to

yield

acceptable results for design.

The

ga s

bearing shown

in

Figure

11

differed from

the

design model

o f

Reference

20

in t he follo wing ways: ( i) the gas plenum wa s located in the fixed center port ion of

the bearing

to

supply

gas

through radial orific es

to

float

t he

outer moving ring

and

(ii) plat es were added t o t h e ends o f t h e bearing. These provided a self-centering

feature required

for

proper operation

and

some degree

of

lateral restraint.

In

addition,

the end plat es were designed with shields t o direct the exhaust gas toward the center

of

the

bearing; other wise,

gas

exhausting ra dially would impinge

o n the

inside

of the

model

and

possib ly influence

the

damping measur ements.

The

beari ng shown

in

Figure

11

is capable o f carrying high radial lo ads; however, the lateral load cap ability i s

small, y et

adequate

for

most purposes. Where side loads

are

large, additional

gas

supply orifices and pads should be provided on the

ends.

The performance characteristics necessary for evaluating th e journal g as bearing

for application

to

dynamic sta bility te sting include

the

radial load capa city, bearing

pressure,

and the

tare damping. Cur ves illustrating

the

relationship

of

radial load

to critical pressure

are

presented

in

Fig ure 12.

T h e

critical pressure

is

defined

a s

the pressure loading for an outer shell eccentricity r atio o f approximately 0.5 of the

bearing clearance. Perm issi ble loading

is

nearly

a

linear function

of the

supply

pressure

and

agrees with

the

performa nce predicted

by

theory.

As noted previ ously, the one most desirable characteristic of the gas bearing for

use

in

dynamic stability balance sys tems

is the

extremely

low

tare damping. Dat a

on

the damping of the gas bearing in Figure 11 ar e presented in Figure 13 as a function

of frequency

of

oscillation

and

radial load.

In

true viscous damping,

the

damping

moment, M

0

d , is defined as a moment pr oportio nal to Mg which i s invariant with

oscillation amplitude

and

frequency. This

was not

found

to be the

case

for the

bearing

when evaluated over

a

range

o f

frequencies

and

loads

at 11.5 de g

oscillat ion amplitu de.

The results

of the

evaluation

o f t h e

bearing damping pre sented

in

Figure

13

shows

an

inverse relationship with frequency as is the case for flexures and a dependency on

radial load.

T he

damping

is in the

form

Mg =

Cc o

b

,

w h ere C a nd b d e pe n d o n r a d i a l l o a d . I n a c t u a l p r a c t i c e , t h e s e s m a l l v a l u e s o f

ta r e damping ar e us ua l ly ne gl ec te d . For example , th e damping shown in F ig ure 13 i s

on ly 0 . 8% of t h e ae rodynamic damping fo r a t y p i ca l b lun t cone r e - e n t ry co nf i gu ra t ion

at Mach number 10.



17

Gas beari ngs have been used extensively in industry for appl ication to machinery

operating at high spee ds where bearing friction is very important. Satisfa ctory opera

tions at high rotational speeds with very low friction makes the gas bearing very

attractive for use in roll-damp ing balance systems. Until recently, ball bear ings

were used exclusively; however, they have many disadvantages, such as significant tare

damping, wear, lubrication requirements, and heating, that can be overcome with the use

of gas bearings. A typical example of gas bearings applied to a roll-da mping balance

is shown in Figu re 14.

Bearing pivots described up to this point only allow rotation about one axis. Many

problems encountered in flight can only be investigated experimentally using a balance

which has rotational freedom about t hree mutually perpendicular axes. A prime example

of a flight phenomenon of this type would be the coupling of the rigid body pitch and

roll modes. A three-degree-of-freedom balance provide s this capability and in addition

simultaneously perfo rms the functions of the several one-degree-of-freedom balances

as it incorporates into one balance the ability to measure pitch, yaw, and roll damping.

The single-degree-of-freedom balances still have an application as they can be used

to restrain the motions to pure pitch, yaw, or roll in situations where this becomes

necessary.

An inherently compensated, spheri cal, gas journal bearing shown in Figur e 15 has

been developed at the AEDC-VKF for use as the pivot in a three-degree-of-freedom

dynamic stabilit y balance. The inner core of this bearing consists of four spherically

surfaced pad s arranged in a manner which allows the bearing to resi st loads in any

radial direction. Gas is supplied through an orifice located at the center of each

pad. This configuration is capab le of supporting maximum radial l oads of 100 lb using

nitrogen or 160 lb using helium. Changes in the bearing core design will allow the

bearing to support radial loads of approximately 350 lb. The damping inherent in the

spherical bearing, like that of the cylindrical bearings, is extremely low.

Continuous motion histories provide the da ta required to determine the aerodynamic

derivatives. For very small flexures or bearing pivots, it is desirable to measure

the motion without physically contacting the moving parts of the system. A device

referred to as an angular transducer was originally developed by the AEDC-VKP for use

in dynamic stability balance s yst ems

2

. The variable reluctance angular transducer,

shown mounted on a cylindrical g as beari ng in Figure 11, consists of two E-c ores

mounted on the outer movabl e ring of the bearing. When the coils contained in the

E-cores are excited, magnetic paths are established through the E-cores, their adjacent

air gaps, and the eccentric ring. As the bearing rotat es, the reluc tance of the

magnetic path s through the E-core s and eccentric ring remain constant, while the re

luctance of the air gap changes. Thi s change in relucta nce pro duc es an analog signal

proportional to the angular displacement.

Instrumentation for the spherical ga s bearing consists of three mutuall y perpe ndicular

angular transducers

(Pig.15).

The pit ch and ya w transducer s located on the forward

portion of the instrument ring operate from an eccentric which is concentric in roll

and ecce ntric in pitc h and yaw, whereas the roll transducer located in the aft portio n

of the ring oper ates from an eccentric which is concentric in pitch and yaw and e ccentric

in roll.



18

Gas bearing technology is growing very rapidly, and many texts have been prepared

on

the

methods

for

design

and

analysis

(see the

Bibliography). Refer ence 22, Vol.1,

contains detailed summaries o f design methods, material selection, and measurement

techniques,

and

Vol.II

is

devoted

to

applications.

3.1.2 Sus pensio n Syste m

The de sig n of th e susp ens ion sys tem wi l l depend for the most p a r t on th e type of

mode l t o be t e s t e d and the measuremen t s r eq u i r ed . Mode ls may be ge ne ra l ly ca t eg or i ze d

as (1 ) two-d imens iona l mode l s wh ich span the t e s t sec t ion and a r e suppor t ed f rom the

s i d e w a l l s , ( i i ) t h r e e - d i m e n s i o n a l , s e m is p an m o de ls w hic h a r e m o un te d o n a r e f l e c t i o n

p l a n e s u p p o rt e d by t h e t e s t s e c t i o n s i d e w a l l , a nd ( i i i ) t h r e e - d i m e n s i o n a l , f u l l - s p a n

mode l s wh ich a r e supp or t ed by a s t i n g , a s id e s t r u t , o r a magn e t i c susp ens io n sys t em.

Wi th th e exc ep t ion o f magne t i c susp ens ion , each mode l moun t ing syst em p r es en t s a

p o t e n t i a l p r ob l em when b e i n g u s ed f o r t e s t s o f s om e t y p e s of c o n f i g u r a t i o n s , t h a t b e i n g

th e in f lu en ce o f t he sup por t on th e ae rodynamic measuremen t s . When th i s p rob lem canno t

b e d e a l t w i t h i n c a p t i v e mo de l t e s t i n g b y p r o p e r s u p p o r t d e s i g n o r e v a l u a t i o n o f t h e

i n f l u e n c e o f t h e s u s p e n s i o n s y st e m , f r e e - f l i g h t t e s t i n g i n t h e win d t u n n e l an d r a n g e ,

and t e s t i n g wi th magne t i c mode l susp ens io n i s u sed .

Suppor t sys t em s fo r t he t h r e e t yp es o f mode l s commonly in ve s t ig a t ed in wind tun ne l s

may be ge ne ra l ly g rouped in to two ca t e go r i e s : ( i ) t he wa l l suppo r t method fo r t he two-

d i m e n s i o n a l a nd t h r e e - d i m e n s i o n a l s e m is p a n m o d e ls a nd ( i i ) t h e s t i n g , s t r u t , a nd

m a g n e t i c s u s p e n s i o n s y s te m s u p p o r t f o r t h e t h r e e - d i m e n s i o n a l f u l l - s p a n a nd

body-of-

r e v o l u t i o n m o d e l s .

3 . 1 . 2 . 1 W all S u p p o r t

The wa l l sup por t me thod fo r t e s t i n g ha l f mode l s has t h e advan tage tha t , f o r a g iven

t e s t s ec t io n s i z e , a l a rg e r model can be t e s t ed and thu s a l a rg e r Reyno lds number can

be ob ta in ed than wi th a comple t e mode l. The l a rg e ha l f model a l s o s im p l i f i e s t h e

m od el c o n s t r u c t i o n , i n s t r u m e n t a t i o n i n s t a l l a t i o n , a nd r o u t i n g i n s t r u m e n t a t i o n w i r e s and

p r e s s u r e t u b e s o u t of t h e m od el t h r o u g h t h e l a r g e c o n t a c t a r e a s a t t h e t u n n e l w a l l s .

L a r g e a n g l e s of a t t a c k c a n b e o b t a i n e d w i th w a l l - s u p p o r t e d m o d e ls w i t h o u t i n t r o d u c i n g

i n t e r f e r e n c e s from s t i n g s u p p o r t s . I n a d d i t i o n , m e a s u r in g s y s t e m s a nd i n s t r u m e n t a t i o n

f o r d yn am ic s t a b i l i t y t e s t s o f t h e c o m p l e t e m od el o r c o n t r o l s u r f a c e s c a n b e l o c a t e d

o u t s i d e t h e t u n n e l , s o t h a t e q ui pm e n t s i z e i s n o t a n i m p o r t a n t f a c t o r . I n m any t e s t s

o f wings o r con t ro l s u r f ac es , t h i s i s t he on ly method o f suppo r t t h a t can be used to

a c q u i r e dy na m ic s t a b i l i t y d a t a w i t h o u t a l t e r i n g t h e m od el c o n t o u r f o r an i n t e r n a l

ba l a nce . An example o f a two-d imen s iona l and th r ee -d im en s io na l mode l mounted on th e

tun ne l w a l l s , w i th equ ipmen t f o r measu r ing dynamic s t a b i l i t y , i s shown in F ig ure s 16

and 17 t o i l l u s t r a t e t h e p r i n c i p a l a d v a n t a g e s o f t h e w a l l - s u p p o r t me th od d e s c r i b e d

2 3

.

When mode l s a r e moun ted d i r e c t l y on the t un ne l wa l l s , t h e e f f e c t o f t h e t u nn e l

bou nda ry - l a ye r f l ow has t o be con s ide r ed , s in ce i t may change the f l ow f i e ld o ver t h e

model . As note d by van der Bl ie k

2 4

, t he wa l l boundary l ay e r i n f lue nce d the l i f t and

drag co e f f i c i en t on d e l t a wings and wing-body com bina t ion s mounted on the s ide wa l l s

and had a t endenc y to p romote se pa ra t io n and s t a l l nea r t h e wing ro o t . Al though s im i l a r

c o n c l u s i o n s h av e n o t be en f o r m u l a t e d f o r dy na m ic s t a b i l i t y t e s t s , t h e i n f l u e n c e on

s t a t i c t e s t s r e s u l t s sh o u ld b e s u f f i c i e n t t o i n i t i a t e c o n ce r n a b o ut t h e i n f l u e n c e o f

t h e t e s t s e c t i o n w a l l b o u nd a ry l a y e r .



19

In order t o alleviate the influence o f the'wall boundary layer, several m etho ds

have been used with success

at

subsonic speeds

by

vario us investigators. These include

boundary-layer removal through porous walls o r slots ahead o f t h e model and th e in

stallation o f V-shaped vortex generators on the tunnel wall ahead o f t h e attachment.

Formation

o f

large disturbed areas

o n the

model close

t o t h e

wall

may be

prevented

by

using fences attached to the model o r displacing t h e model from th e tunnel wall by

inserting shims between the model and walls. Each of these meth ods h a s t h e disadvantage

of altering t h e shape o f t h e model configuration and not completely eliminating t h e

influence o f th e wall bo undary layer.

Tests have shown that wall boundary-layer effects are largely eliminated by using

reflection planes o f th e types illustrated in Figure 18. With this method, a reflection

plane is introduced a s t h e longitudinal plane o f symmetry and is displaced from t h e

wall to allow passage of the wall bo undary layer flow. Fo r some mounting method s,

contouring in the flow pa ssage may be necessar y to eliminate choking o f t h e flow and

the resultant ineffectiveness o f t h e reflection plane. T h e reflection plane can be

fixed

to the

wall

and the

model rotated,

o r t h e

model

can be

fixed

to the

plate

and

rotated as a unit. When t he model i s rotated on the reflection plate , as is generally

done in dynamic stability tests, a gap must be allowed, which ma y introduce leakage

around the model reflection-plane junction and change the local flow field. Tests o f

very blunt models on reflection planes will produce significant boundary-layer separa

tions and thus alter th e mode l flow field. Informa tion on reflection pla ne design i s

given in Reference 26.

3 . 1 . 2 . 2 S t i n g S u p p o r t

Sting supports are principally used in tests o f complete ve hicle configur ations and

are inserted into t h e model base through t he center o f t h e wake to minimize their in

fluence

o n the

model flow field. Careful attention must

be

given

to the

design

of

the sting to insure that it provides a stiff support; however, modifica tion o f t h e

model contour may be necessary to accommodate the sting. Such a modification is a

compromise between a small diameter sting o f constant length for a distance behind

the model and one which will ca rry th e aerodynamic loads without significant movement

of the pivot axis. Often t he consequences o f t h e compromise are not clear, and there

fore tests are required to evaluate t h e influence.

Static tests

27

have shown that both th e sting diameter and sting length hav e an

influence on force and base pressure data for most configurations. Some forced-

oscillation systems have shaker motors in enlarged sections o f t he sting support, to

simplify t he overall design and reduce the complexity o f t h e driving linkages. In

these cases it is important that t he enlarged section be located an ample distance

downstream o f t h e model base to permit proper formation o f t h e wake. Very few experi

ments have been conducted to investigate specifically t h e influence o f th e sting

diameter and length o n dynamic stabilit y measureme nts.

A series o f tests have been conducted at the AEDC-VKP (Ref.28), over a range o f

Reynolds numbers, to investigate the influence o f t he sting diameter and length o n

the dynamic stability o f a 10-deg half-angle cone with a flat

base.

To determine t h e

relative effects o f sting length, distu rbances downstream o f t h e model base were

introduced by placing a 20-deg conical windshield at vario us stations along the sting

from

0.75d

to 3d. Relati ve interference caused by the sting diameter size wa s created



20

by repeating

t he

tests with diameters

o f

0.4,

0.6, and

0.8d.

The

test result s i ncluded

dynamic stability, static stability,

and

base pressure data, since base pressure s

are

very sensitive to the degree of Interference t hat t h e support presents to the base

flow.

Representative results

are

shown

in

Fig ure s 19,

20, 21, and 22 for

Mach numbers

2.5,

3, and 4

over

a

range

o f

Reynolds numbers

and

sting l engths using thr ee sting

diameters.

Evidence

of

interference caused

by the

support length

may be

judged

by the

deviation of the base pressure ratio from the reference curve corresponding to the

longest sting length

(i

s

/d =

3).

Boundary-layer transition

is

generally

at the

model base,

at a

Reynolds number which

is slightly less than the value where t he minimum base pressur e occurs. An analysis

of

the

results based

on the

base pressure data

and

schlieren photo graphs

f or L / d = 3

shows that the Reynold s number range wa s sufficient for a transitional and fully turbulent

wake. There

i s

evidence

in the

data presented

in

Figure

19 to

confirm that some support

geometry configurations can have an influence on the dynamic stability derivatives.

However, it

appears that

if the

support

is

designed usi ng crit eria

f or

static force

and base pressure

t e s t s

27

,

it is

unlikely that

the

dynamic stability data will

be in

fluenced. This observation does

no t

apply

to

configur atio ns which have contoured bases,

since these

are

more critical with regard

to the

Influence

of the

support because

t h e

wake condition h a s a direct influence on the pres sure distribu tion around the base. A

further complication ar ises

in

this case because

the

base must

be

partially removed

to

allow movement of the model over the sting. In cases where the relative influence o f

the support system

on

damping resu lts cannot

be

assessed

by

capti ve model testing, wind-

tunnel, free-flight, and aeroballistic range tests are necessary to obtain compara tive

results.

Some examples

o f a

very rigid sting

and

support

for a

high frequency, free-os cillat ion,

dynamic stability balance system are shown in Figure 23 and a more conventional support

for

a l o w

frequency system

is

shown

in

Figure

24.

3 . 1 . 2 . 3 S t r u t S u p p o rt

Where limitations

o n

model amplitudes

are

imposed

by

mechanical interference between

the model base

and

sting, these

may be

overcome

by

mou nting

the

model

on a

transverse

rod support system

so

that

it is

free

to

rotate

on the rod

with

no

restraints.

An

example of one such system us ed in the AED C-VKF 50-in. M ach 10 tunnel is shown in Figure

25.

The balance, consisting o f a gas-bearing pivot, model lock, and position

trans

ducer, is

contained within

the

model; nitrogen

gas for the

bearing

and the

model posi

tion lock and instrumentation leads pas s through the transverse rods.

Extension

of the

transverse

rod

into

the

model will affect

the

flow over

the

model

primarily

in the

region

aft of the rod and

along

the

sides

of the

model.

A

region

of

flow separation will exist on the model ahead o f t h e rod. Since th e interference flow

field caused

by the rod

will depe nd

on the

boundary-laye r flow

and the

position

of the

interference on the model surface, it is to be expected that meas urements obtained with

a transverse

rod

support system will

be

dependent upon

t he


and

the Rey nolds number

of the

flow.

An

example

o f

such

a

dependence

is

shown

in

Figure

26,

where data obtained with

a

sting support

are

included

for

comparison.

It is

apparent

from these results, for a 10-deg half-angle cone at Mach number 10, that both Reynolds

number

and


are

important parameters.

It

should

be

noted t hat

when

the

Reynolds number

is

large

t he

influence

of

Reynolds number

and

amplitude

dis

appear

and the

resul ts obt ained with

a

sting support

and

transverse

rod

support

are



21

in agreement. Howeve r, at low Reynolds numbers, very large differences can exist in

results obtained by the two metho ds, t hus indicating significant adverse effect s from

the transverse rod.

The Influence

o f t h e

size

o f t h e

transverse

rod

supports

at

Mach

5 is

shown

in

Figure 27 for a blunt c one cylinder m odel with and without a base flare. A cylindrical

ro d and double-wedge rod were tested with roughness on the model nose to produce

turbulent flow and thus to eliminate separation ahead o f t h e base flare. The results

show no significant effects caused by rod size for diameters up to 30 % of the model

diameter and no effect s caused by rod configuration, i.e., a cylindrical rod as com

pared to a double- wedge rod. A significant reduction of the damping occurr ed when

roughness and the transverse rods were eliminated for the configuration with th e flared

base, because o f t h e r o d wake influence on the region o f separated flow ahead o f t h e

flare.

Although very fe w systematic investigations h ave been conducted to investigate t h e

effects

o f

transverse

rod

geometry

on

dynamic da mping ove r

a

range

of

model configura

tions,

Mach numbers, Reynolds numbers, and amplitudes, t h e data available tend to in

dicate that (i) rod influence becom es negligib le at the high Reynolds numbers when t h e

flow is turbulent and (ii) rod size effects are small pr ovided the rod is small com

pared to the model

size.

Tests with transverse rod support s should be supplemented

with te sts employing a sting support balance system to provide comparative data at

low amplitudes, where the influence o f t h e transverse rod is most pronounced.

3. 1 . 2 . 4 Magnet i c Model Suppor t

Magnetic suspension provides a means for suspending a model in a test unit without

any physical support, and thereby eliminates any questions about the experim ental

results being influenced

by the

method

o f

physical support (sting

or

strut).

T h e

magnetic suspension system consists o f a series o f electr omagnets which prod uce magnetic

fields and gradients o f magnetic fields to provide forces and moments o n a

model.

T h e

model h a s a mag netic moment which may be a part of the model if a permanent magnet i s

used in the model construction. Other metho ds for providing a magnetic moment are

described in Ref ere nce 29. Included in the system for control is an optical system

which monitors t h e position o f t h e model by light bea ms passing through t he test

section and focused onto sensors. When t h e system is in equilibrium, forces and

moments caused by gravity, aerodynamics, and inertial effects, for the case o f a moving

model, balance o u t t h e applied mag netic forces and moments. Changes in model position

activate th e control system, and curre nts flow through t h e magnetic coils to produce

forces and moments o n the model t o hold it in equilibrium.

A magnetic susp ension system with V-magnet o rientation d eveloped at the AEDC-VKF

(Ref. 30)

for static force, mom ent, and wake me asurem ents is shown in Figures 28 and

29.

A similar system, but with L orientation o f t h e magnets h a s been develo ped by

MIT (Ref. 31) to obtain static force, moment, and dynamic stability measurements.

3.1.3 Balance Tar e Damp ing

3 . 1 . 3 . 1 D e s c r i p t i o n o f D ampin g

Wind tunnel m odel s have to be mechanically suspended on a bearing o r spring to

pro

vide oscillatory motion. Exceptions to this are recent develo pments o f magnetic model



22

suspension systems which have been refined to the degree that damping data have been

obtained

29

. Mechanical susp ension introduces damping which must be accounted for in

the analysis.

Aerodynamic damping as a rule cannot be measured directly with most balance systems,

since it is a part of the measure d total damping; theref ore it must be obtained in

directly as the difference between the total and tare damping. With gas bearing pivot s

this is generally not necessary , since they usually hav e a negligible amount of tare

damping. It is customa ry in writing the linear differe ntial equations expressing an

oscillatory motion to assume that the total damping, which is the aerodynamic plus tare

damping, is propo rtional to

0 .

Therefore the motion in pitch for a free oscillation

may be expressed by

I

¥

0 - (M

5a

+

U

h

) 0

- ( M

5a

+

W

e t

) 6

= 0 .

where the subscript a identifies the aerodynamic term and t identifies the tare

term.

The term tare damping is applied here to denote the operation of non-aerodynamic

influences which, by r esisting t he motion and absorbing energy from the system,, reduce

the amplitude of motion. A constant-amplitu de oscilla tion would continue to exist if

damping forces of vari ous kinds did not exist. Damping forc es arise from the mechanical

system damping, damping capacity of the materials used in the system, and air damping.

Although it is convenient to classify the contributing factors in this manner in dis

cussing tare damping as applied to dynamic stability testing, the mechanical system

damping and damping capacity of the materials are inseparable in the measurements and

constitute the term M^

t

. Air damping is important only insofar as determining the

magnitude of M^

t

from air-off tests.

Mechanical system damping involves distinguishable part s of the system and arises

from slip and other boundary shear eff ects at mating surfa ces, interfaces, or joints.

Energy dissipation here may occur as the result of dry sliding (Coulomb friction)

lubricated sliding (viscous forces ) or both.

The damping capacity of materials, or material damping, refers to the energy dis

sipation that occur s within the material when the structure is undergoing cyclic stress

or strain. These internal energy los ses may be considered as being due to imper

fections in the elastic properti es of the material. In a truly elastic member, not

only is strain proportional to stress but also, on removal of the stress, the material

retraces the stress-strain relationship that existed during the application of the

stress. Even if mate rial s are stressed within the elastic limit, some energy is lost.

Energy dissip ation in a material during a stress cycle leads to a hyste resis loop in

the stress-strain curve and the a rea of the loop denotes the energy lost per cycle.

Air damping resu lts from a loss of energy caused by restrictive effects as a mass

moves through a medium such as air. The damping may result from dynamic pressure,

vis cou s effects, and vort ex formation and shedding; however, the relative ef fects of

each are not known. Theore tical studies of the visc ous damping force on a sphere and

cylinder oscillating in a fluid

3 2 , 3 3

show that each will experience forces proportional

to the velocity and square root of the fluid density. Experimental air-damping data

for circular and rectangular plates, cylinders, and spheres are presented in Reference

34 and show the effect of pres sure , vibrator y a mplitude , frequency, shape, and surface



23

area on the air damping for models that a re mounted on the end of a cantilever beam.

The results show that air damping associated with plat es is directly proporti onal t o

the ambient pressure and amplitude i s indicative o f t h e influence o f dynamic press ure.

Low amplitude results for cylinders oriented p erpendicular to the plane o f oscillatory

motion

are in

agree ment with

a

theory

by

S t o k e s

33

and are

proportional

to the

square

root of the density but show no vari ati on with frequency.

It is preferable to design dynamic-stabi lity balance system s in such a way that t h e

tare damping is small compared t o t h e aerodynamic damping. F o r some classes o f models

at hypersonic and hyperveloci ty speeds, where t he aerodynamic damping is low, the tare

damping can approach, or even exceed, t h e aerodynamic damping. An example o f t h e

contribution o f t h e mechanical system and air damping relative to aerodynamic damping

is shown in Figu re 30. It is apparent from this example that, a s t h e Mach number Is

increased, extreme care must be taken to obtain me asure ments o f tare damping with a

high degree o f accuracy, so that t he difference constitute s an accurate measurement

of t he aerodynamic damping.

3 . 1 . 3 . 2 E v a l u a t i o n of S t i l l - A i r Dam ping

Mechanical and material damping can be determined by tests in still air at

atmos

pheric pressure; however, this can introduce large errors in some cases, since t h e

damping th us obtained includes air damping. To determine t h e magnitude of air damping,

tests are made in a chamber o f t h e type shown in Figure 31 to permit variation in the

ambient pr essure. Th e wind tunnel can also be used for this test; however, i t s

environment must be free o f even sma ll amou nts o f airflow and turbulence. A typical

set of tare dam ping meas urements for a blunt cone model are shown in Figure 32 for a

range of pressure s from atmosphere to vacuum. T he level o f damping at a vacuum re

presents the mechanical system and material damping, since the air damping is essen

tially zero.

An

example

o f th e

magnitude

of the air

damping

a s a

percentage

o f t h e

aerodynamic damping is shown in Figure 33 for a series o f model shapes. At supersonic

speeds the air damping is on the order of 2 to 4%, but increases significantly with

Mach number, even approaching 100% o f t h e aerodynamic damping at Mach 10.

The necessity for accurate tare da mping resu lts h a s been mentioned and specific

reference made t o some o f t h e sources o f damping in a mechanical system, o ne being

energy dissipation in joints o f multiple piece systems. To ensure that t are damping

is repeatable and co nsistent, a number o f tare damping re cordings are usually made

following removal and replacement of the model and flexure o r pivot system until tare

damping is shown to be repeatable.

In cases where

t h e

axial and/or normal forces

are

large, measurements

are

made with

the pivot loaded to determine i t s influence on the mechanical damping and pivot

stiffness. In most cases the pivot static loading will have little effect on the tare

damping characteristics.

Since th e model undergoing test will oscillate at a frequency different from that

of th e tare damping tests at a wind-off condition, it is necessary to establish how

the energy loss o r tare, M g

t

, at a vacuum , will va ry with frequency. Fo r flexure-

type pivots Smi th

35

notes that the tare damping p e r cycle is a constant fraction of

the energy stored in the spring at the be ginning o f t h e cyc le, independent o f t h e

frequency. This is equivalent t o M ^

t

=

C/co ,

which wa s found earlier in experiments

by Welsh and Ward

36

.



24

3 . 1 . 3 . 3 D am pin g C h a r a c t e r i s t i c s o f D i f f e r e n t M e t a l s

Damping characteristics

of

mater ials exposed

to

cyclic stress

are

affected

by

several

factors, which must

be

considere d

in th e

selection

o f

materials

for

flexure pivo ts.

Some

of the

factors

are

(1) condition

of the

material

-

composition

and

effect

o f

heat treatment,

(ii) state

o f

internal s tress

-

effect

o f

machi ning

and

changes

in

stress

due to

cyclic motion

and

temperature h istories,

(iii) stress Imposed

by

service conditions

-

type

o f

stress, magnitude, stress

variations, and environmental conditions.

Factors which include

the

magnitude

of the

stress, stre ss history,

and

frequency

may

be si gnificant at one operati ng condition and unimportant at another. Final selection

of

a

material should

be

based

on

tests

o f a

specimen

in th e

configurat ion

in

which

the

material will

b e

used

and

under si mulated operat ing conditions.

It should

be

recognized that,

in

most systems where oscillations

o r

vibrations

occur, the damping introduced by the properties of the materials employed is, in general,

only one factor contributing to the total damping. Nevertheless, a knowledge of the

characteristics

of

structural material s

is

important

and the

choice

o f

material should

be guided by its damping characteristics.

Materials having

a

Young's modu lus which ranged from

6.2 x 10

s

for

magnesium

to

28.5 x 10

6

lb/in.

2

f or Armco 17.4 stainless steel have been tested in the VKF to provide

material damping data

for use as a

guide

in

selecting materials

for

flexure-pivot

designs.

An

example

o f t h e

material damping results

for a

range

o f

frequencies

and

beam stiffnesses

is

shown

in

Figure 34. Additional information

on

material damping

may

be found

in

Reference

37.

3. 2 D Y N A M I C S T A BI L I T Y D E R IV A T IV E S - P I T C H O R Y A W

The basic methods

for

obtaining dynamic stability meas urements

are

well known, dat ing

back to early 1920 when Briti sh r ep ort s o n the subject began to appear. It was found

that aerodynamic pitch damping derivatives consisted of two parts, M and M

a

, and

experiments were devised

to

account

for

these terms separately. These experiments were

successful; however,

it was

concluded that

the

technique

was not

satisfactory

for de

ducing M

a

at high speeds (transonic) or high frequencies. Some mode s o f oscillation

such

as

short-period damping

can be

evaluated satisfactorily

if the sum of the

damping

derivatives,

for

example

(c +

C

m ( 3 t

) ,

is

known, even t houg h

the

separate val ues

of

C a nd C ^ a re unknown. Insta nces m ay exist at supersonic speeds in which the

application

of

test d ata

t o

calculations

m ay

require

the

damping derivatives

in

some

way other than

the

combined form; however,

the

general practice

h a s

been

to

measure

the derivatives in combi ned form. Wind tunnel techniques that are in most general u s e

at supersonic speeds and above can be summarized as the free-oscillation and forced-

oscillation methods.

The free-oscillation method is one of the earliest used and is considered to be th e

simplest from

the

vie wpoint

o f

equipment

and

instrumentation.

The

model, restrained

by

a

pivot,

is

deflected

to an

initial a mplitude

and

released,

and the

decaying motion

is observed.



25

W ith t h e f o r c ed - o s c i l l a t i o n t e ch n i q u e , a r e s t r a i n e d mo de l i s f o r ced t o p e rf o rm

s i mp l e h a rmo n i c mo t io n by mean s o f an o s c i l l a t o r , t r a n s m i t t i n g f o r c e s t h r o u g h a l i n k ag e

to a to rqu e member which i s r i g id ly c onnected to the model. Normal ly th e s t ru c tu ra l

damping of ba la nce sy s tems used a t sup er so nic and hyp er so nic speeds i s very low and

the to rq ue member s t i f fn es s i s a l so low . Thi s r e qu i r es th e sys tem to be ope ra ted a t

or near i t s r eso na nt f r equency . In ord er to ob ta i n da ta over a w ide f r equency rang e ,

t h e mod el moment o f i n e r t i a o r t h e s t i f f n e s s o f t h e s t r u c t u r a l r e s t r a i n t may b e v a r i ed .

An a l t e r n a t i v e me th od t h a t may b e u s ed f o r t e s t s o v e r a w id e r an g e o f o s c i l l a t i o n

f req ue ncie s i s r e fe r re d to as ine xo rab le forc in g ; and as th e name im pl ie s , t he model

i s d r i v en i n a co n t r o l l e d man ne r by r i g i d l i n k ag e .

M ost f o r c ed - o s c i l l a t i o n s y s tems emp lo y s p e c i a l i n s t r u m en t a t i o n t e ch n i q u es wh ich

u t i l i z e a f e ed back co n t r o l s y st em f o r co n t r o l l e d o s c i l l a t i o n o f t h e mod el when n e g a t i v e

d amp in g i s p r e s en t . I n ad d i t i o n , t h e s e co n t r o l s y st ems p r o v i d ed amp l i t u d e co n t r o l f o r

measur ing p o s i t i v e damping .

B oth f r e e - o s c i l l a t i o n and f o r c e d - o s c i l l a t i o n b a l a n c e s may b e f u r t h e r c a t e g o r i z e d a s

e i t h e r s ma l l - am p l i t u d e o r h i g h - a mp l i t u d e s y st ems . A s ma l l - amp l i t u d e b a l an ce meas u r e s

lo ca l va lu es of damping and p i t c h in g moment s lop e , s in ce the model o s c i l l a t e s a t

am pl i tudes of about one or two de gre es , and i s o f t en used to ob ta in da ta a t a ng les of

a t t a ck o t h e r t h an ze r o . The b a l an ce s t i f f n e s s o r r e s t o r i n g moment f o r l a r g e - am p l i t u d e

b a l an c es i s n o r ma l l y ze r o o r n ea r z e r o . F o r ce d - o s c i l l a t i o n s y s tems may b e o p e r a t ed a t

about ±15 de gre es , as compared to am pl i tu des which are on ly l i m i te d by the ba lan ce

s u p p o rt f o r f r e e - o s c i l l a t i o n b a l a n c e s .

Th e ma t h ema t i ca l t r e a t m en t o f s y s t ems w i t h f r ee - o r f o r c ed - o s c i l l a t i o n mo t io n may

be found in many t e x t s ; however , th e methods a re t r e a t e d in th e next s ec t io n for th e

sake of comple teness .

3 .2 .1 Theory

T he t h eo r y f o r l i n ea r an d n o n - l i n e a r o n e - d eg r ee - o f - f r eed o m and t h r e e - d eg r ee s - o f -

fre ed om f r e e - o s c i l l a t i o n s y st e m s w i l l be t r e a t e d . The f o r c e d - o s c i l l a t i o n t h e o r y w i l l

be developed , fo l lowed by a br i e f t r e a tm en t o f lo ca l and ef fe c t iv e v a lu es of damping

an d f i n a l l y t h e ae r o d y n ami c t r an s f e r eq u a t i o n s w i l l b e d ev e l o p ed .

3 . 2 . 1 . 1 F r ee O s c i l l a t i o n ( O n e- D eg ree -O f -F r eed om)

The eq ua t ion of motion for a body in pi tc h was de r iv ed in Se ct io n 2. 2 and has th e

same ge ne ral form as th e equ at i on s of motion for a body in yaw and r o l l . For f re e-

o s c i l l a t i o n s y s tems h av i n g s t r u c t u r a l s t i f f n e s s , t h e eq u a t i o n o f mo t io n h a s t h e fo rm

1,0 - (M

0a

+ \A

e t

)0 - (M

ea

+ H

B t

)9 = 0 . (17)

After introducing ne w nomenclature for simplification o f t h e derivation o f t h e solution

to Equation (17), it becomes

l

y

d + c d + K d = 0 . (18)



26

T he g e n e r a l s o l u t i o n o f t h i s e q u a t i o n d e s c r i b i n g f r e e - o s c i l l a t i o n m o ti on c an b e

w r i t t e n a s

6

r t r t

Aj6 + A

2

e

(1 9 )

where Aj and A

2

a r e a r b i t r a r y c o n s t a n t s and v

1

and r

2

a r e t he ro o t s o f t he

c h a r a c t e r i s t i c s e q u a t i o n . T h es e a r e o b t a i n e d a s

21,

' C

2r[

(20)

bu t , s i n ce the undamped na tu r a l c i r c u l a r f r equency i s co = i / (K / I

y

) , a n d l e t t i n g

C / 2 I

y

= a , Equ atio n (20) becomes

r

1 > 2

= - a ± / ( a

2

- co

2

) .

The case o f Interest is a <

co ,

where the roots are complex, representing the condition

where th e system will oscillate. The general solution t o Equation (19) for this case

is

where

i.

2

d = A j e ^ " ) * + A ^ * " "

1

^ .

a ± bi

a - c/2I

v

(21)

b

=

f

C

c o ^ (undamped natural frequency)

Equation (21) ca n also b e written as

d

- ( C / 2 I

y

) t

&

0

e c o s (aj

d

± -cp)

(22)

d

Q

and cp

being arbitrary constants. Equation (22) represents

a

harmonic oscillation

with damping. When cos

(co^t

-

cp)

= 1 , the envelope encompassing the points of tangency

with the displacement

i s

described

by

d

=

d

0

e

( C / 2 I

y

) t

From thi s relationship, the logarithmic decrement

l o g

e

( 9 / 0

l

) i s

used

t o

obtain

t h e

damping term C . An often used parame ter for measuring the damping coefficient is

the time required

to

damp

t o a

particular amplitude ratio. Letting

d \ and 0

2

re

present a mplitudes occurring at times tj and t

2

, respectively, and letting

6*

2

/(9j = R , we obtain the following equation for the damping moment:

s i n c e

C = - 2 I

y

f l o g

e

R / C

y r

* 2

_

*l

c

yR

f

•

(23)



27

Prom Equat ion ( 2 1 ) f o r t h e na tu ra l damped f r eque ncy , t h e s t a t i c moment i s

K = I

Y

OJ2 + i

Y

( c / 2 I

y

)

2

. (24 )

The term I

y

( C / 2 I

y

)

2

i s

u s u a l l y v e r y s m a l l

an d can b e

n e g l e c t e d .

As shown i n E q u a t i o n s ( 1 7 ) a n d ( 1 8 ) , t h e dynamic damping and s t a t i c m om en ts c o n s i s t

of moments d u e t o t h e a e r o d y n a m i c s a n d moments produced by t a r e s i n t h e ba l anc e s ys tem .

T h e t a r e s a r e ev a l ua ted f rom w ind-o f f m eas u remen ts , a n d t h e f i na l ae rodynam ic dynamic

damping and s t a t i c m om en ts a r e g i v e n by

C = -

M

0w

=

~ ^

M

0a

+

M

0t^w

M

0a

=

<

M

(?a +

M

£t>w " ^

U

h \

K = _ M

0w

=

- <

M

ea

+

M

6»t^w

M

0a

=

(

M

0a

+ M

0t^w ~ (

K

e t \ '

where th e su bs cr ip t s w and v den ote w ind-on and vacuum co nd i t io ns , r e sp ec t iv e l y .

S ince th e s t r u c t u ra l damping moment param eter va r i es in ve r se ly as th e f r equency of

o s c i l l a t i o n , i n c o r p o r a t i o n o f E q u a t i o n s ( 2 3) an d ( 24 ) a l o n g w i th t h e eq u a t i o n s abo ve

g i v es

M

ffa

= 2 I

y

l o g

e

R

(f/C

yR

)

w

-

(f/C

yR

)

v

£

"(9a

= i i W * ) : - « * > j ] •

I n c a s e s w he re t h e b a l an ce s y s tem h as n o s t r u c t u r a l s t i f f n e s s ( s y st ems u s i n g a g a s

b ea r i n g o r b a l l b ea r i n g p i v o t ) , t h e t a r e d amp in g may b e ev a l u a t ed i n d ep en d en t l y o f t h e

model . Se m i-em pir ic al means may be needed to f ind t he ta re damping at th e f requency

and the mot ion ax ia l load th a t w i l l be expe r ience d in the w ind tu nn el . Equat ion (23)

i s s imply used to y ie ld

M

5w

= 2 I

y

f l o g

e

R /C

yR

.

The ta re damping is noted as M^

t

and

Mg

a

= 21

Y

f log

e

R /C

yR

- Mg

t

.

The aerodynamic s t i f fn e s s i s ob ta in ed f rom Equat ion (24) as

M

0a = h

w

l •

The de r i va t i ve s , when reduced to a d im ens ion les s form for p i t ch in g mot ion , become

C

mq

+ C

ma = U^MjoJid

2

)



28

It should be noted that the above equations were developed for a linear model. It

will be shown later, in Section 3.2.1.2, that erroneous values of damping may result

if the linear equations are applied to model s which have a nonlinear restoring moment.

It will also be shown in Section 3.2.1.2 that Equation (22) may be generalized as

0 =

* o < ° >

co(0)

co(t)

-(C/2I

Y

)t

r v

..

e

Y

cos [T(t) -

cp]

where T(t) = jco(t) dt

and E quation (23) now becom es

C = -2I

y

f l o g

e

[R(OJ

2

/O;J)*]/C

.VR

3 . 2 . 1 . 2 E f f e c t s o f N o n l i n e a r i t i e s

Many aerodynamic configurations of interest have damping rates and oscillation

frequencies that vary with time. Quite often these paramete rs are functions of the

dependent variable and its time derivatives; therefore the differential equation which

descr ibes the angular motion is nonlinear. In many cases, these nonlinearities are

slight,

and the amplitude-time history may be broken into intervals over which the

motion is approximately exponentially damped. It has often been assumed that, by

applying the linear theory over these a pproximately exponentially damped segments, an

accurate estimate of the aerodynamic damping may be obtained. This is true only when

the aerodynamic stiffness is constant. Ni ch ol ai de s

38

has shown qualitatively how

errors in the damping measurements can be made using this approach. In order to show

this,

Equation (18) will be used in the form

9

+ Dj0 + D

2

£ =

(25)

where

Dj = Dj(0.r?.t)

D

2

=

D

2

( 0 . 0 . t )

.

Multiplying Equation (25) through by 2#/ D

2

and integrating yiel ds

- 0 ' ( S

2

/ D

2

)

|t

2

P*

2 ..

n

J

t.

+ 2DjD

2

]((9/D

2

)

2

dt

Assuming th a t t j and t

2

a r e t i me s t h a t p eak s i n t h e mo t io n o ccu r , t h i s ex p r e s s i o n

r ed u ces t o

d \ - 0 \ - J '

2

[D

2

+ 2 D j D

2

] ( ^ / D

2

)

2

d t



29

A dynamic stability condition is obtained from this expression as

[ t

2

+ 2DjD

2

] > 0 .

For zero viscous damping (Dj = 0 ) , the convergence or divergence o f the motion is

dictated b y the rate of change o f D

2

with respect t o time. Thus the variable aero

dynamic stiffness term

D

2

supplies

a

mechanism

by

which energy may

be fed

into

or

taken away from the system. The linear theory, Equation (18), assumes that D

2

is

constant and therefore that the constant term D j determines the convergence or

divergence

o f

the system. One can readily see that

if

the linear theory

is

applied

t o

a situation where

D

2

is a

variable, the value

o f Dj

obtained will

b e

erroneous.

This is true even if the linear theory is applied t o the nonlinear oscillation over

small approximately exponentially damped intervals. The magnitude o f this error i s ,

of course, dependent

of

the magnitude

o f D

2

.

When the aerodynamic nonlinearities are large, Equation (25)

is

very difficult

to

solve. Fortunately, as mentioned previously, there are many aerodynamic configurations

of interest for which these nonlinearities are small. In addition, these shapes usually

have damping moments that are small compared

to

the restoring moment.

For

such shapes

Equation (25)

may be

written

as

6 + V ^ 0 , 0 . t )0 + [D

20

+ D

2 l

( d . e . t ) ] 6 = d . (26)

where

D

2

= D

2 0

+ D

2 1

D

2 0

# » D.e

D

20 » °2l •

This equation may

b e

reduced

t o

the van der Pol equation, which

is

given

as

6 + D

20

d = e $ ( 0 . 0 . t ) . (27)

where < p ( 0 , 0 , t ) contains the nonlinear terms and e is a small dimensionless parameter

which characterizes how close the system

is to a

linear conservative one. For

the

above example,

e $ ( 0 . 0 . t )

= -

\D . (6 . e . t )d + D

2 1

(0 ,0 . t )0 ]

The motion may

b e

approximated for short time intervals

by

the solution

for a

harmonic

oscillator (e = 0) which is given as

0 = 0

O

co s (/(D

20

)t + cp) .

Since the right-hand side of Equation (27) is small but not zero, 0 and cp are

actually slowly varying functions

of

time.

It i s shown in Reference 39 that a second-order nonlinear differential equation,

like Equation (27), may be reduced to an equivalent linear second-order equation with



30

v a r i a b l e c o e f f i c i e n t s w h ic h h a s a f i r s t - o r d e r s o l u t i o n i d e n t i c a l t o t h a t of E q u at io n

(27).

T h i s eq u i v a l en t l i n ea r eq u a t i o n may b e w r i t t e n a s

a 20

- r ^ 0

+

[/(D

20

)

+

c p

m

]

2

0 = 0 .

om

where the m subscripts indicate mean values over a cycle o f oscillation.

The equation above can

be

rewritten

as

0 + Cj(t)(9 + C

2

( t ) 0 = 0 . (28)

In order t o solve this equation, the following transformation will b e used:

0 ( t ) = u(t)A(t) . (29)

Differentiating this expression

and

substituting into Equation (28) yields

uA + 2uA + iiA + CjUA + CjUA + C

2

uA = 0 .

Choose u and A such that

2uA + CjUA = 0 . (30)

This reduces the previous equation to

uA

+ UA +

CjUA

+

C

2

uA

= 0

.

(31)

Dividing Equation (30)

by A and

integrating yie lds

-i/c dt

u = K e

1

, (32)

where K is the constant of integration.

Differentiating this expression and substituting into Equation (31) yields

A + B(t)A = 0 . (33)

where

B(t) = [C

2

- i p \ - jCj] .

According to a theorem by Winter

40

, the asymptotic solution of Equation (33) is

A(t) = [B(t)]'*[Kj cos Y(t ) + K

2

sin T(t )] , (34)

where T ( t ) = J"/B(t) dt .



31

This solution applies after a finite length of time has elapsed, such that the oscilla

tory motion approaches being periodic.

It is interesting to note that for certain forms of B(t) , Equation (33) may be

reduced to Bessel' s dif ferential equation.

Equations (29), (32), and (34) may be combined and manipulated algebraically to

yield

0( t) = 0(0)

B(0)

B(t)

-ijc,(t) dt

e cos [T(t) + cp] , (35)

where 9 ( 0 ) and B(0) are the values of envelope amplitude and C(t ) respectively

at t = 0 .

For the case being considered, where Cj varies slowly and is small compared to

C

2

, B(t) given in Equation (33) may be reduced to

B = co

2

~ C

2

.

The following form of Equation (35) may be applied over intervals of the d ata where

Cj is approximately constant:

l i i

— ' e '

T C l t

cos [Y(t) + cp] ,

co(t)j

where

T(t)

= f co ( t ) dt

J

o

Cj = C/I

Y

.

In order to use this solution to determine Cj . co and y must be known as functions

of time. •

3 . 2 . 1 . 3 F r ee O s c i l l a t i o n ( T hr ee D eg r ee s o f F r eed om)

Solutions of the equations of motion referenced to the nonrolling coordinates will

be obtained in this section*. For this development it is assumed that the model has

triagonal or greater symmetry and that the XYZ axes correspond to the principal

inertia axes of the model. The effects of slight asymmetries of the model will be

considered, i.e., asymmetries which have a negligib le effect on the above assumptions.

3 . 2 . 1 . 3 . 1 E q u a t i o n o f P i t ch i n g an d Y awing M o t io n s

For a model of the above type rotating about a point fixed in space, Equations (5)

describe the motion.

• These methods have been extracted from References 38 and 41.



32

Iq + I

Y

Pr

(SM)« =

p

V

2

Ad

(36)

Ir - I

Y

Pq = (2M)ar =

p

V„Ad

~

(37)

where I = Is = Ig .

For motion about

a

point fixed

in

space,

= a

* * - f l .

The angular rates in the nonrolling coordinate system are given as

q = q cos $ - r sin $

r = r cos $ + q sin $ ,

where

$

~ f P(t) dt .

Jo

Substituting

the

angular rate s gi ven in Equatio ns ( 4 ) , these expressions reduce

to

q

= e

r = y/ co s

d .

When the motion is restricted to small angular excursions

( d «

l ) , Equations (39)

reduce

t o

(38)

(39)

q =

0

(40)

Multiplying Equation (36) by i and subtracting Equation (37) from it, these equations

may

be

combined

to

form

a

single diffe rential equation

I(-r + iq) - iI

x

P(-r + iq) = iqo,Ad(C

m

+ i C

n

) .

Assuming small angular excursions

and

substituting

the

time derivatives

of

Equations

(38) and (40) into the equation above yields

1 0

+

la) - iI

x

PC§ + lot) = iqa,Ad(C

m

+ i C

n

)



33

The complex angle of attack may be reduced to

v + iw

S = ~ p + i a ,

V

c

w here i t i s a ss umed t h a t / 3 ~ v / V

c

and 6c ~ w/V

c

. S u b s t i t u t i n g t h e t im e d e r i v a t i v e s

of th i s complex angle in to the equat ion of mot ion ,

I < f - i I

x

P g - = iq^AdCC,,, + i C

n

) . (41 )

Assuming that first-order linear aerodynamics

are

adequate

and

that

t he

motion

is

described by oc , /3 , cc , /3

,

P , q , a nd r , t h e pitching- and yawing-raoment c o

efficients a r e given in th e body coordinates XYZ as

C

« =

C

mo

+

V

+

C

mq

g ©

+

c

ma

W

+

C

m p

/ ^

c

n = c

n0

+ V ? + c

nr

( ^ + c

n/

g (~J +

c

npa

a ( ^ J .

The terms C

m 0

and C

n 0

were Include d in this expansion o f t h e coefficients t o account

fo r th e assumed slight asymmetry o f t h e model*. A s outlined by Murphy"

1

, th e Magnus

coefficients,

C

n

• , C

A ,

C „

p

, and C ,

have

a

negligible effect

on the

motion;

therefore they were not included.

Fo r

a

missi le with triagonal

o r

greater symmetryt.

C

n/3

C

n £

C

n r

-

=

=

ma

ma

mq

-

=

=

~

C

M a

-

C

M a

C

M q

C

np a

C

»p/3 ~

C

np a •

Substitute these relations into t h e yawing-moment equation, then mult iply t he result

by

i and add it to the

pitchi ng-moment equation

t o

obtain

K C , + ic

n

) = i ( c

B 0

+ ic

n0

) + c

Ma

(/3 + ia ) +

+ (c

Mq

+

c

u 6

) 0 + id)

— +

c

Mpa

(/3

+

ia)p

,

•

This approach was first used

by

Nicolaides

42

in

his development

o f

the "Tricyclic" theory,

t See References 38 and 41 for a discussion of symmetry.



34

where

P = Pd/2V

C

.

Substituting

the

complex angle

o f

attack referenced

to the

body coordinates,

£ 2. fi +

ia ,

into t he above relation yields

1

<

C

.

+

i c

n> = i ^ m o

+

i c

no )

+

C

Ma

<f

+

_£d

2V„

+

<

C

Mq

+

< W

—

+

C

M p a

^ P

. (42)

It

now

becomes necessary

to

transform these coe fficie nts from

the

body coordinates

to t he nonrolling coordinates. Thi s transformation is given as

i(C

m

+ i C

n

) e

1 $

= i ( C

m

+ i C

n

) .

The t r a ns f orm at i on for the complex ang le of a t t a ck i s g iven as

T h i s i s e a s i l y p ro v ed b y ex p an d in g t h i s r e l a t i o n an d eq u a t i n g r ea l and i mag i n ar y

p a r t s a s f o l l o w s

/v + iw \ v + iw

(co s $ + 1 s in $) =

\ \ J

v = v co s $ + w si n $

w = w co s $ - v si n $ .

T h e s e r e l a t i o n s a r e t h e n o n r o l l i n g v e l o c i t y c o m p o n e n t s

4 1

Murphy

41

us ed t h e ab ov e t r an s f o r ma t i o n t o t ran s f o r m E q u a t i o n ( 4 2 ) t o t h e n o n r o l l i n g

co o r d i n a t e s . T he r e s u l t may b e w r i t t e n a s

i( C

m

+ C

n

) = i(C

m Q

+ i C

n 0

) e

1 $

+ C

M

J +

+

(

C

M q

+

<W j f

+

U *

D

JP •

S u b s t i t u t i n g t h i s r e l a t i o n i n t o E q u a ti o n (4 1 ) r e s u l t s i n

| + (G + 1 H ) | + ( J + 1 K ) | = L e

i p t

, (43)



35

where

^

M 2

/ x

G =

-

i n

(c

»*

+

^

H = - P

QooAd

J

= ~ —

C

M a

K - ^ C P

2 V . I

M p a

L

= i ~ <

C

mo

+ i C

no) •

3 . 2 . 1 . 3 . 2 S o l u t i o n f o r C o n s ta n t C o e f f i c i e n t s

Assuming constant coefficients, a solution to this sec ond-order linear diff erential

equation

is

given

a s

~ (X

+ l<a,)t (X ,

+

ico

)t iPt

<f = Kje

l 1

+ K

2

e

2 2

+ K

3

e . (44)

where

K

3 = "

X..,

+ i w,

{(P -

e o . ) ( P

- co

2

) + \ j \

2

+ i[Xj(P -

o>

2

)

+ X

2

( P -

c o . ) ] }

(G

+

iH)

±

/ [ G

2

+

2GHi

- H

2

- 4(J + IK)]

subscript

j = 1 or 2 .

Using boundary conditions, t he remaining constants are evaluated as

I ~ <

k

2 1

+ l a

V l ) ^ 0 -

R

3

)

K

1 ,2 X

l , 2

-

k

2 .

1

+

i K . 2

W

2 . 1>

The solution given

in

Equation

(44) is

"Tricyclic"

in

that

it is

described

by

three

two-dimensional vectors

or

arms

(Fig.35). For

mod els spinning

at a

constant rate,

t h e

first two rotating arms a re damped, and the third is a constant a mplitude rotating

vector which results from the forcing function given in Equation (43). Thi s rotating

trim vector,

d u e t o t h e

asymmetry

o f

the

model,

causes

t he

axis

o f

symmetry

of the

model

to rotate o n an imaginary conical surface. A s P increases, the magnitude of this

rotating trim is decreased. For a statically stable model with sufficiently small



36

damping, the total angle of attack may increase as P approaches co^ . In this case,

the possibility of resonance exists only with the nutational frequency

(co

x

)

since the

precessional frequency

(co

2

)

car ries a sign opposite that of P ( R e f . 3 8 ) . This will

be shown later in the text.

The restrictions on the above solution are:

(i) Configurat ions with triagonal or greater symmetry having only slight configura-

tional asymmetries.

(ii) First-order linear aerodynamics.

(iii) P = constant.

(iv) Small oscillation amplitudes.

If the relations for the modal damping rates and frequencies given in Equation (44)

for j = 1 and 2, respectively, are first added together, then multiplied together,

the following exact relationships result"

1

:

k

*

+

K =

"

G =

l £ j

( C

«

+

° ^

x

x

co, + com

= -H = — P

1 2

I

\A.

2

- co.co. = J = — c

M a

aa,Ad

V

2

+ K.CO. = K = - ^ — C

M p a

P

For most configurations of interest

I ( - H

2

- 4 J ) | » |(G

2

+ 21GH - 41K)

Therefore the expression for the modal frequencies and damping rates is approximated by

v/{-(H

2

+ 4J)} +

, (G + iH) 1

X-4 + ico. ~ ± -

J J o o

G

2

+ 2iGH - 4iK

2iv(H

2

+ 4J)

In order to derive the expression for the gyroscopic stability factor, only p r e

dominant aerodynamic effects are considered in this relation, which reduces it to

Xj + l&)j ~ |[-i H ± i/{-(H

2

+ 4J)}] .

For an aerodynamically statically unstable model, J < 0 . It is obvious then, for

this case, that unless - (H

2

+ 4J) < 0 the motion will diverge, since Xj > 0 . If



37

H

2

> - 4 J , t h en X , = 0 , t h e co n s t a n t - am p l i t u d e p e r i o d i c mo ti on r e s u l t s . T he g y r o

s c o p i c s t a b i l i t y f a c t o r i s d e f in e d a s

-H '

4J

n

x

V P

2

4q

C0

Ad \ I / C

(45)

'MO

The stability condition is given as

1/S

g

< 1 .

Static stability

is

insured

for all

aerodynamlcally statically stable models ( C

H a

< 0)

and for aerodynamically statically unstable models ( C

M a

> 0) when S > 1 .

If

we let

T

= (i -

i/ s

g

r*,

then T < i f o r C

u

„ < 0 , and T > 1 for C

u

„ > 0 when the roll rate of the model

Ma

Ma

is sufficiently high

to

insure static stability.

The expression for the modal damping rates and f requencies is given a s

Xj

+ i^j - -

G K r

- (1 + T ) ± _

2

H

H

f

1

- 1 ±

2 V T,

(46)

where

K

i ~

G

KT

- (1 ±

T)

±

—

2 H

2V

m

I

<

C

Mq

+ C

M a > d ±

T

>

±

-

C

H

P 0

7

(47)

H

2

J

I

1 1

; ;

=

n

p {

(48)

It

is

shown

in

Equation (47) that

t he

Magnus moment tends

to

increase

the

damping

of o ne modal vector and decrease the dampi ng of the other. A model will, therefore,

diverge when

I^Mpa^xl

(C

Mq

+

c

M&

)(i

+ r ) |

For a spinning model, the modal frequencies are unequal, as evidenced in Equation

( 4 8 ) .

The

largest angular rate (co^)

is

known

as the

nutation frequency

and the

smaller

angular rate

(c o

2

)

is the prec ess ion frequency. The nutation frequency

( c o j

always

carries t he same sign as the roll rate (P); therefore, K j rotates in the same direction

as

K

3

. For an

aerodynamically statically stable model ( r < \ ) ,

th e

frequencies

are

unequal

in

sign

and

therefore cause

t he

modal vectors

K

x

and K

2

to

rotate

in

opposite

directions (Fig.

3 5 ) .

When P = 0 , T = 0 , then ojj = _o;

2

, X., = X_ , and the motion

is described by lines, ellipses and circl es (Pig. 3 6 ) . A s P increases ( T > 0) the

motion becomes flowerlike

and

finally,

for

large

P

,

the

motion becomes like that

of

a fast spinning gyroscopic pendulum (Pig. 3 7 ) .

For an

aerodynamically statically unstable



38

model (r > l ) , th e frequencies are equal in sign and therefore t he modal vectors K j

and

K

2

rotate

in the

same direction. Motions characteristics

of a

fast spinning

aerodynamically statically unstable model

are

given

in

Figure

38.

In orde r

to

obtain

the

damping rates

and

frequencies

of the

modal vectors,

t h e

usual approach

is to fit the

solution given

in

Equati on (44)

to the

observed motion,

using a least squares differential correcti on method. For a description of this

method and its application, see References 41 and 42.

3 . 2 . 1 . 3 . 3 S o l u t i o n f o r V a ry in g R o l l R at e

In many cases

of

interest,

the

roll rate

is not

constant

as

assumed.

If the

region

of resonance is avoided and the roll rate is slowly varying, the solution given in

Equation (44) may be used to fit the data if it is applied over intervals of the motion

where

P is

nearly constant.

For roll rates near resonance,

P is

usually small enough such that

G » H and

J

» K

. Roll rates varying near resonance will therefore have

a

negligible effect

on

th e

homog eneous solution

of

Equation (43). This

is not

true

of the

particular

solution which undergoes drastic variations at this co ndition; therefore Equation (44)

is no t valid for thi s condition.

In orde r to investigate the problem of roll rates varying near resonance, we will

rewrite Equation

(43) as

1 + (G +

i H ) |

+ (J +

1 I ) |

s

L e

i $ ( t )

, (49)

where

$(t )

2 . / P ( t ) dt

"o

G

+ iH =

constant

J

+ IK =

constant

The complementary function

of the

diff erential equation (49) will

be the

same

as

that given

in

Equation (44).

T he

task

is now to

obtain

a

particul ar integral. This

will be accomplis hed using the method of variation of parameters, as Murphy did in

Reference

41.

Assume

the

particular solution

is of the

form

~ (X +lo. )t (X,+i<u)t

<f

p

= Aje

x J

+ A

2

e

2 2

. (50)

where th e coefficients Aj are varia ble unknowns and X . and

oo .

are known consta nts

(j

= 1 or

2).



39

A p p l y i n g t h e me t h o d o f v a r i a t i o n o f p a r ame t e r s r e s u l t s i n t h e f o l l o w i n g t w o s i mu l

t an eo u s eq u a t i o n s :

( X , * l « . ) t (X • lo> ) t

Aje

x 1

+ A

2

e

2 2

= 0 (5 1)

. (X • « ) t (X + 1« )t i<P(t) ,- n s

A j ( X j + i o ; j ) e

x J

+ A

2

(X

2

+ io;

2

)e

2

= Le

( 5 2 )

S o l v i n g E q u a t i o n s ( 5 1 ) an d ( 5 2 ) s i mu l t an eo u s l y r e s u l t s i n

- (X , + i<a ,) t + l $ ( t )

Le

1 :

1

(Xj - X

2

) + i (co. -c o

2

)

- (X, • ico ) t * i$( t )

Le

2 2

A

2

( X

2

- X j ) + 1 ( 0 J

2

- CDj)

T h es e ex p r e s s i o n s a r e i n t eg r a t e d and s u b s t i t u t ed i n t o E q u a t i o n ( 5 0 ) . T he s o l u t i o n

to Equat ion (49) is then given as

- ( X , + l « j ) t ( X , + l « . ) t

£ = Kje

x :

+ K

2

e

2 2

+

r -i i / n *

1 P ( t ) d t

r t | (Xj • l »

1

) ( t - t

1

) (X

2

• l «

2

) ( t - t

2

) l

' o

e

^ 1 -

x

2

+ K ^j - ^

2

)

In ord er to use t h i s so l u t io n , P must be known as a fun c t io n of t ime.

By a r b i t r a r i l y i n c l u d i n g t h e t er m e

3

in th e K

3

v ec t o r , t h e s o l u t i o n g i v en a s

Equat ion (44) can be made to appro xim ate ly compensa te for the changes in th e ro l l in g

t r im angle* . Thi s so lu t io n then does no t r e qu i re knowing P as a func t ion of t and

is given as

- (X. * l » . ) t ( \ , * i c o ) t ( X . + i o O t

<f = Kje

i 1

+ K

2

e

2 2

+ K

3

e

3 3

(53)

where co

3

= P .

I t s h o u ld be emp h asi zed t h a t t h e s o l u t i o n s o b t a i n ed t h u s f a r a r e f o r s l o w l y v a r y i n g

r o l l r a t e s . I f t h e r o l l r a t e i s l a r g e an d r a p i d l y v a r y i n g , t h e s e s o l u t i o n s a r e n o t

a p p l i c a b l e .

* This method was re la te d to Mr L.K.Ward in a p ri v a te communication with Pr ofe sso r Robert

S.Eikenberry of the Department of Aerospace Engineering, University of Notre Dame, Notre Dame,

Indiana.



40

3 . 2 . 1 . 3 . 4 S o l u t i o n f o r V a r i a b l e C o e f f i c i e n t s

The effects o f variable aerodynamic stiffness and damping were discussed previously

in Section 3.2.1.2.

In

that section,

the

need

for

correcti ng

the

damping measurements

for

t he

effects

o f

variable aerodynamic stiffness

was

emphasized.

It was

shown th at,

for slowly varying coefficients and small damping, a second-order nonlinear differe ntial

equation, like t h e homogeneous portion of Equation (49), can be reduced to a linear

equation with variable coefficients and retain the same f irst-order solution.

For the nonlinear case (J = J(£,s,t), etc.), the homogeneous portion of Equation (49)

may be written correct ly t o first or der a s fol lows:

£ + [G(t) + iH(t)]£ + [J(t) + l K ( t ) ] | = 0 . (54)

Murphy"

1

derived a correction to the linear compl ementary function b y assuming the

following solution to Equati on (54):

i*j i4 >

2

<f = K je

l

+ K

2

e

2

, (55)

where

K, is a

variable

and

c p

s

( t )

= j

o)j(t) dt (j = 1 or 2) .

T h i s s o l u t i o n i s s i m i l a r i n form t o l i n ea r ep i cy c l i c co mp lemen ta ry f u n c t i o n s o b t a i n e d

f o r c o n s t a n t c o e f f i c i e n t s .

T he f o l l o w i n g s e t o f eq u a t i o n s was o b t a i n ed b y s u b s t i t u t i n g t h e f i r s t and s eco n d

t i me d e r i v a t i v e s o f t h e s o l u t i o n ( 5 5 ) i n t o E q u a t i o n ( 5 4 ) and s ep a r a t i n g r ea l an d imag i n a r y

p a r t s :

—

[ K J / K J ]

+ [K j /K j ]

2

- c o

2

+ G[k j/Kj] -H ojj + J = 0 (56)

cJj + 2[k

j

/K

j

] a )

J

+ H [k j/ K j] + Goij + K = 0 . (57)

S in ce co »

KJ/KJ

and Kj/Kj i s s lowly va ryi ng , Equa t ion (56) may be reduc ed and

m a n i p u l a t e d a l g e b r a i c a l l y t o y i e l d

. 2

H

C O , = ±

J

2

H

+ J

2.'

(58)

This is essentially the same as the frequency relation obtained for constant coeff icients.

Equation (57) ca n be rearranged to obtain

c o , H

K./K, = X . (L + _ _ , ( 59)

2 <

%

4aJ

J



where

41

. 1 (Geo. + K)

2 (o)j + H /2)

J

Wj + H/2

This relation shows that variable aerodynamic stiffness and variable roll rate

both affect the convergence or divergence of the motion. If the linear theory were

applied directly to motion of this type, the complete right-hand side of Equation (59)

would be obtained as the damping rate.

Equation (59) is now integrated to obtai n

Kj(t) = Kj(0)

H ( 0 )

2

+ 4J(0)

11 J (Xj

+ H / 4 & J )

dt

H ( t )

z

+ 4J(t)

Using Equations (58) and (59), and t he condition of slowly varying para meter s, the

equation above can be rewritten as

Kj(t) = Kj(0)

c o ^ O ) + H(0)/2

«.(t) + H(t)/2

X.t

where it is assumed that

V, » H/{4(^j + H/2)} .

Substituting th is into Equation (55), the compl ementary function become s

Kj(0)

ojj(O) + H (0)/ 2

c o

x

( t ) + H(t) /2

X.t +l*(t )

e +

+ K

2

(0)

O J . ( O )

+ H(0)/2

- i '

c o

2

( t ) + H(t )/2

2

X

2

t + l *

2

( t )

e

This solution may be applied over portions of the motion history where X. is

approximately c onstant If c o . ( t ) and P(t ) are known.

3. 2 . 1 . 3 . 5 Graphic Data Red uct ion Method (Approximate)

A graphic data reduction method may b e used to determine the modal damping rates

(Xj) and f requencies (^j)- In order to apply thi s method, it is necessar y to convert

the tricyclic motion to epicyclic motion. This is accomplished using the roll rate

history as follows:

— ipt

f e

= Be

l(a>j-P)t l ( «

8

- P ) t

w

_ i $

0

+ Ce

+ K

3

e



42

where

K

3

e

B

C

1»„

Kj6

Xjt

x,t

K

2

e

= constant

The motion

is

transformed

t o

body fixed coor dinates

b y

rotating each angle through

Its

corresponding roll orientation ( P i g . 3 9 ) . This results i n epicycl ic mot ion about t h e

point K

3

e

1 $ 0

.

At maxima o f t h e epicyclic motio n, B and C add, and at minima they subtract

( P i g . 4 0 ) . Using th e maximum a nd minimum amplitude points a nd their corresp onding

times, t h e variation of B and C with respect t o time i s obtained (Fig. 4 1 ) . Taking

the ratio

o f t h e

natural logarithms

o f

these expressions yiel ds

K. =

K

=

t

2

- tj

logo

/ r *

t

2

- t j

log.

•?

(60)

The modal frequencies a r e obtained using F igure 40 as fol lows:

2TT

+ y

co =

t

2

- t j

( 2 - n

-

22

t

a

- t j

C

Ma

< °

Combi ning Equatio ns (45), (46), and (48) wit h t h e above frequency relations, t h e

pitching-moment slope

i s

given

a s

'MaE

4qooAd

• f ] P

2

- K - C

2

)

•

Assuming t h e roll rate i s suffici ently small that Magnus effect s c a n b e neglected.

Equations (47) a nd (60) c a n b e combined t o yield

F 1 2 Vq, i

LC

M q

+ C

M a

J

E

A A

2

1

[c

+

c •] = — ^

L

S l q

+ L

M a

J

E

A A

2

A d ' Oa, ( 1 + r ) ( t

2

- t j )

1 1

log. I T *

A d ' q , ( 1 - r ) ( t

2

- t j )

l o g

e

' - *



43

The effect of balance ta re damping on the total damping rates must be accounted for

before the aerodynamic coefficients are obtai ned. The use of a spherical gas bearing

pivot eliminates this difficulty for almost all cases of practical interest, due to

its negligibly small tare damping.

3.2.1. 4 Forced Osc ill ati on

The motion

of a

model balance system which

is

rigidly suspended

in an

airstream

and

forced

to

oscillate

in one

degree

of

freedom (pitch)

as a

sinusodial function

of

time

may be expressed as

1

Y

0 - (M^

a

+ Ug

t

)0 - (M

ea

+ H

e t

)0 = M cos cot , (61)

where the subscript a refers to the aerodynamic values and the subscript t refers

to the tare contribution of the mode l balance system.

Simplifying

the

nomenclatur e

by

letting

<

M

e

a

+

M

9t )

= c

<

62

>

and

Equation (35) beco mes

- (

M

e« + "st) =

K

. (63)

I

Y

(9 +

C0

+ K.0 = M cos

cot .

(64)

The complete solution

of

this equation consists

of the sum of the

complementary

and

particular solutions.

It can be

shown tha t, when

M = 0 , the

compleme ntary solution

is Equation (19) for the case of the free-os cillati on system.

To solve for the particu lar solution which is necessar y for the steady-state case

after the decay of the initial transients, it is assumed that the solution is of the

form

0 = 0

Q

cos (cot - cp) ,

which may be expanded to yield

0

= #

0

( c o s

cot

cos

cp

+

sin cot

sin

cp) .

The velocity and acceleration are then found as

0 = d

0

co(-sia cot

cos

cp

+ cos cot sin

cp)

d

=

-d

Q

co

2

(cos cot

cos

<P

+ sin

cot

sin

d) .

Substituting in Equat ion (64) and equating coefficients yields

*0

(K - I

y

oJ

2

) cos cp + Ceo sin cp

(



44

tan cp

Ceo

(65)

(K - I X )

Equa t ion (65) shows th a t fo r con s ta n t - am pl i tud e mot ion , when th e forc ing f r equency i s

equal to th e na tu ra l f r equency of th e sys tem (cp = 90 deg ) , th e in e r t i a t e rm b a la nce s

th e r es tor ing-mo me nt t e rm and the damping moment i s p re c i se ly eq ual to the forc ing

t o r q u e ,

C 0

Q

co

= M

(66)

Under these conditions, it is only necessary to measure the forcing moment M , t h e

angular frequency co and the system amplitude d

Q

t o measure the damping parameter C

It i s not always necessary t o operate at resonance, since electronic resolvers can

be used

t o

separate the moment into comp onents in-phase and

90

deg out-of-phase with

the displacement. Application o f resolvers i s covered later in this section.

When operating at resonance the displacement, torque, damping and frequency are

related by

C =

M/0

o

co

.

I n t r o d u c i n g t h e r e l a t i o n s fo r C an d K ( E q u a t i o n s (6 2 ) and ( 6 3 ) ) , t h e ae ro d y n ami c

damping moment and th e eq ua tio n for M ^ become

"0a

"0a

= (Mfl« + M fl

t

). - Olfe )

$t 'm

a

e t>v

(M„

B

+ M a

r

) . - O fe )

•0t 'w

et>\

(67)

(68)

where v and w in d ic a t e the vacuum and wind-on co nd i t i on s , r es pe c t iv e l y . In t e rms

of th e measured q u a n t i t i e s , moment, am pl i tude and f r equency

"0a

• M

o/w

.\r

The derivativ es when reduced

t o a

dimensionless form become

C

mq

+ C

m a = M ^ V a / q ^ A d

2

)

m a

= M f l a

/ a

-

A d

A further examination can b e made o f the forced oscillation system with viscou s

damping

b y

the use

o f

vector diagrams which are convenient for deriving the equations

used in a system having resolvers t o permit testing near a resonance condition.



46

The in-p ha se and ou t -o f -p ha se moments and an gle s , in d i ca te d by M

f

and Mj and 0 .

and <9j , r e sp ec t i ve ly , in F ig ure 42 , a re r e la te d by

s i n cot = Mj/M , sin y = 0^ 0.

co s cot = M,./M , co s y = 0

T

/ 0

O

•

(73)

Combining Equations (72) and (73) leads to the following relationship for the damping

function:

co(0

2

+ 0 1 )

•

Introducing

the

relationship

for C ,

Equation (72) becom es

.

+ M

,

t s

«i*r -

" A

co(0

2

+ 0f)

Since the structural damping moment parameter varies inversely with the frequency, the

equation

for the

aerodynamic damping. E quation

(67) is

given

as

"0a

where v and w in d ic a t e vacuum and wind-on co nd i t i on s .

From t h e v ec t o r r e p r e s en t a t i o n show n i n F i g u r e 4 2 ( a ) , t h e eq u a t i o n f o r t h e s t a t i c

moment may be expressed as

M

t

9

r

- M

T

9

1

co(0

2

+ 0 \ )

w

V r " "A

eo(0

2

T

+ 0

2

)

%

v

w

w

co s cp = co s (cot - y)

(K - I

Y

co

2

)0

c

(74)

and

lyco

2

+ — cos (cot - y )

Combining Equ at ion s (73) and (74) y ie ld s

K = I

Y

a/ +

Mj.e'j. + M

i

0

i

0\ + 0\

Using Equations (73) and (74), the aerodynamic moment derivativ e become s

"0a

2

+

/ "A

*

^

• I ei • 0\ j

I

Y

O J ' +

In a case where the phase angle between the reference and the model displacement is

zero ( y = 0, F igure

4 2 ) ,

the resolution of the forcing moment is somewhat simplified.



47

From th e phase diagram in Fi gu re 4 2( a) , we have

C

A

= M s i n

0

or

and

or

C = ( M /C j9

0

) s i n <p

Kd

Q

= l

Y

c o

2

d

Q

+ M c o s <p

K = I

Y

O J

2

+ — co s cp .

T h e a e r o d y n a m i c d e r i v a t i v e s

no w

b e c o m e

0 a

and

s i n <p

M

* a

= |

]

M

cod.

•

sin cp

v

M

,

I

y

O )

2

+ — C O S 0

2

HI

I

Y

O J + — c o s

cp

T h e s e s t a t i c a nd d y n a m i c d e r i v a t i v e s m a y b e e x p r e s s e d i n d i m e n s i o n l e s s f o r m a s

C

m q

+ C

m d =

M^

a

(2V

00

/qa,Ad

2

)

C

m a ' "Wl/Qa>Ad)

.

3 . 2 . 1 . 5 E f f ec t ive and Loca l Va lues o f Co ef f i c ie n ts

M e t h o d s h a v e b e e n d e s c r i b e d f o r o b t a i n i n g v a l u e s o f a e r o d y n a m i c d a m p i n g a nd a e r o

d y n a m i c s t i f f n e s s f r o m m o d e l s i n w i n d t u n n e l s . In t h e s t r i c t s e n s e o f t h e w o r d , a l l

c o e f f i c i e n t s t h a t

a r e

o b t a i n e d f r o m t h e s e m e a s u r e m e n t s s h o u l d b e c l a s s i f i e d a s e f f e c t i v e

c o e f f i c i e n t s ( s u b s c r i p t E ) ; h o w e v e r , i f

t h e

o s c i l l a t i o n a m p l i t u d e i s l o w

( 6 S t

1 d e g ) ,

t h e c o e f f i c i e n t s m a y

b e

t e r m e d t h e i n s t a n t a n e o u s

o r

l o c a l v a l u e ( s u b s c r i p t L ) .

E f f e c t i v e v a l u e s o f t h e c o e f f i c i e n t s a r e o b t a i n e d a s a f u n c t i o n o f t h e o s c i l l a t i o n

a m p l i t u d e

9

a s

t h e

m o d e l o s c i l l a t e s a b o u t

i t s

t r i m a t t i t u d e . L o c a l v a l u e s a r e

o b t a i n e d

a s a

f u n c t i o n

o f

a n g l e

o f

a t t a c k

a

, and th e d a t a a r e o b t a i n e d w i t h t h e m o d e l

o s c i l l a t i n g

a t

s m a l l a m p l i t u d e s ( u s u a l l y

b y

m e a n s

o f a

f o r c e d - o s c i l l a t i o n b a l a n c e

s y s t e m )

a t

v a r i o u s a n g l e s

o f

a t t a c k . T h e r e a p p e a r s , h o w e v e r ,

t o b e

s o m e q u e s t i o n

a s

t o t h e v a l i d i t y o f t h i s s o - c a l l e d l o c a l v a l u e t e c h n i q u e f o r o b t a i n i n g d a m p i n g d a t a and,

a t t h i s w r i t i n g , t h e q u e s t i o n h a s n o t b e e n s a t i s f a c t o r i l y r e s o l v e d .

R e d d

e t

a l .

"

3

h a v e d e v e l o p e d

a

m e t h o d w h i c h r e l a t e s t h e d a m p i n g d e r i v a t i v e s m e a s u r e d

b y t h e tw o m e t h o d s

s o

t h a t d e r i v a t i v e s c a n

b e

e x p r e s s e d

a s a

f u n c t i o n

o f

i n s t a n t a n e o u s

a n g u l a r d i s p l a c e m e n t w h i c h

i s

t h e

f o r m g e n e r a l l y u s e d

i n

t r a j e c t o r y p r o g r a m s .



48

If the local damping coefficient

f ( 0 )

can be expressed

1(9) = A + A

2

£

2

+

A „ 0

4

+ k

6

d

6

+ A

B

d

a

+ . . .

and

the

effective damping coefficient

f ( d . )

can be

expressed

as

t(d

0

) = c + c

2

0l

+

c

n

0 l

+

c

6

0

6

o +

c

e

0 l

+

... .

the coefficients are related by

C

= A

C

2 =

C

. =

C

6 =

C

8 =

;

(1/4)A

2

:

(1/8)A„

= (5/64)A

6

= (7/128)A

8

.

From these equations,

it can be

shown t hat

the

effective damping-in-pitch coefficient

over one full cycle of oscilla tion designated as (C

m

„ + C •)„ is related to the co-

efficient at an instantaneous angular d isplacement (C + C • ) , by

2V

K

Q

+ C

mA = ~ T I (A + A/

2

• A,* + Kf< • A

a

*« f ...)

q„,Ad

( c

B a +

c

n

j

>

= 3 » (

A +

h e

2

o +

^ 0 i

+

^ ^

+

A

i ^

+

. .

N

) .

mq

ma E ^ 2 ^ 4 ° 8 ° 64 ° 128 ° /

Therefore,

if

values

of (C

mq

+ C

m 6 t

)

L

have been determined

in

small amplitude

forced-oscillation tests,

the

amplitude time history

can be

determined

by

using

the

logarithmic decrement relation

( 9 ) = (9 )

e

<

C

mq

+ C

m a

)

(

(

l a

)

A d

2

/2Va

)

)(At/2I)

Additional details

on the

method

and the

derivation

of the

series coefficients

are

given

in

Reference

43.

3. 2. 1. 6 Transfer Equations

Aerodynamic analysis frequently requires

the

transfer

of

derivatives measured

at

one axis location

to

another

and may

involve

the

transfer

of

deriv ative s along

the Z

as well as the X

axis.

Although equations for transferring results along the X

axis are given in several texts and report s (Refs.4, 10, 44 and 4 5 ) , few references

include equations for the case involving a transfer of resul ts along the Z axis as

well. For

this reason,

the

derivation

of the

transfer equations

are

pre sented next

to illustrate

the

procedure

for the

case

of

longitudinal stability de rivat ives.



49

The quantities referred to the initial axes location do not have subscripts, and

the new axis locat ion is designated by the subsc ript 1.

X*-

(a)

Sketch (a) shows the position of axes XYZ and X J Y J Z J . The transfer dista nces x

and z are pos iti ve in the senses shown in the sketch.

The equations for transferring forces and mome nts from axes XYZ will be derived

by first writing the equations for the linear velocity components and the angular

velocities. Considering the linear velocity components of Oj . they become

U

= U + Qz

Vj = V - Pz + Rx

Wj = W - Qx •

(75)

Since the axes the data are being transferred to are parallel to the initial axes, the

angular velocities are

Pj = P

Qi = Q

R .



50

If

we let the

external forces

and

moments

for the

axes

XYZ be

designated

as (P„F P )

/ \ A y z

and ( M

X

H L M

Z

) , respectively, the moments are given as

M

X 1

= M

x

+ F

Y

Z

M

Y 1

= M

Y

+ F

z

x - F

x

z •

"Zi

M

z

- P

Y

x

(76)

The angular velocities

and

velocity components

of 0 and O j

parallel

to the

axes

for disturbed motion are given by

u

V

w

' l

-

=

=

=

U

0

+ u .

v .

W

0

+ w .

U

0 1

+ U

1 •

p

Q

R

P

:

=

=

=

=

P

0

+ P

Qo + q

R

0

+ r

p

o i

+ D

i

V

l = Vj

W

01

+

' l

Q

:

= V + Q,

Rj = R

0 1

+ rj

(77)

where

the

subscript

0

,

a s

used

for U

Q

, V

Q

, W

Q

, P

0

, 0 , . and R

Q

,

designates

the velocity components

o f th e

center

o f

gravity

and the

angular velocities

in a

reference steady-flight condition.

3 . 2 . 1 . 6 . 1 S t a b i l i t y D e r i v a t i v e s

The transformation equations

are

derived here

by

first expressing

the

force

and

moment component derivatives

in a

Taylor serie s expansion

and

employing

the

relation

ships derived for linear and angular velocities and forces and moments. It is con

venient here t o introduce the standard short form notation:

X,Y,Z, Components of the external forces

parallel to axes X , Y , Z .

L.M.N, Mome nts about axes

Y . Z

3 . 2 . 1 . 6 . 2 L o n g i t u d in a l S t a b i l i t y D e r i v a t i v e s

The general expansion of the aerodynamic force c omponents along the body X Z axes

and

the

pitchi ng moment

may be

expressed

as

follows:

X = X

0

+ X

u

u + X^u + X

q

q + X

q

q + X,w + Xj,w +

Z

= Z

Q

+ Z

u

u + Z-ii + Z

q

q + Z

q

q + Z,w + Z

;

w +

M

= M

Q

+ M

u

u + M^u + M

q

q + M^ q + M,w + M; w +

(78)



51

T h e t e r m s X

Q

, Z

Q

, M

Q

a r e t h e f o r c e s a n d m o m e n t s f o r t h e r e f e r e n c e s t e a d y - f l i g h t

c o n d i t i o n .

A f t e r n e g l e c t i n g a l l d e r i v a t i v e s w i t h r e s p e c t t o a c c e l e r a t i o n e x c e p t Z ^ a n d M ^

a n d n e g l e c t i n g t h e d e r i v a t i v e

X ,

E q u a t i o n ( 7 8 ) b e c o m e s

X = X

0

+ X

u

u + X

w

w

Z = Z

0

+ Z

u

u + Z

q

q + Z

w

w + Z ^ w [ (79)

M = M „ + M

u

u + M

q

q + M „ w + M

;

w .

I f w e n e x t c o n s i d e r t h e e q u a t i o n f o r t h e f o r c e a l o n g t h e b o d y X a x i s , t h e d e

r i v a t i v e s w i t h r e s p e c t t o v e l o c i t i e s u and w f o r s t a t i o n 1 a r e

V

U 1

B X j B P

X 1

B X J B U 3 X J BW

d u .

rtu

B u 3 u j B w B u j

S i nc e Xj = X ,

u I

Bu Bw

X u

l ) u T

+ X

* B u T

a n d , u s i n g E q u a t i o n s ( 7 5 ) a n d ( 7 7 ) ,

S i m i l a r l y ,

' u i

X

u

(80)

B X j B F

X 1

B X J Bu BX j Bw

X

w i

:

•*_ ~ ~

_

~ -

+

~

Bwj

Bw,

and

;

^u

Bu BWj Bw Bwj

ciw

X w l X u

B w

1

+ X

« B W j

a n d , u s i n g E q u a t i o n s ( 7 5 ) a n d ( 7 7 ) ,

x

w l

X

w

.

B y f o l l o w i n g t h e s a m e p r o c e d u r e , it c a n b e s h o w n t h a t , f o r t h e f o r c e d e r i v a t i v e s ,

Z

u i =

Z

u

Z

w 1

= Z

w

^ l

= Z

w

Z

q l =

Z

q

+

V "

Z

u

z

•

( 8 1 )

( 8 2 )



52

F o r t h e p i t ch in g - mo men t d e r i v a t i v e s , n eg l ec t i n g M w ,

BMj BM

X 1

B M J B U B M J B W B M J B q

M j . . - • — + — — - • + - -

B u, B U , B U B U , BW B U , B Q B U .

u B u j Bw Buj Bq Bu j

Us ing Equ at ion s (5 0) ,

"ui

BM Bu BZ Bu BX Bu

+ x - z ^ r — +

Bu Bu.

Bu Bu , Bu Bu ,

BM Bw BZ Bw BX Bw

+ + X z +

Bw B u, Bw Bu , Bw B u,

A f t e r s i mp l i f y i n g ,

BM Bq BX Bq

+ - - + x-

Bq Buj Bq Buj

M

ui

Bu Bu

+ x Z

u T - -

z X

u

Bu

i

A

w

Bu

Bu~

Bw

Bw Bw

+ M , — + x Z - z X

w

- — +

" Buj * Buj

w

Buj

Bq Bq

+ M

n

+ x Z

n

Q

B U j

q

B U j

Us ing Equat ions (77) ,

U l

% + ^ Z

u

"

z X

u

(83)

By fo l low ing th e same proc edu re , i t can be shown th a t t he o t he r p i t ch ing-moment de r iv a

t i v e s t r a n s f e r e q u a t i o n s a r e

M

wl

M

w i

M

w

+ x Z

w " zX

w

M*

+ x Z

w "

z X

w

M

q l

= M

q

+ x

<

Z

q

+

" * ) ~

z M

u

+ x

\ "

xz

<

X

w

+

*V

+ z

\ •

T he d i men s i o n a l f orm o f t h e d e r i v a t i v e s i n t h e t r a n s f e r e q u a t i o n s a r e

x

u = "CAPCOVA

* .

=

- C

A o t

^ V A / 2 j

(84)



53

Z

u

=

-C

N

PooVA

Z

w = -

C

N a ^ V A / 2

Z

w = -C

N 6 t

P » V A / 4

Z

q = - < W « , V A / 4

M

u

= C ^ V A d

"w =

C

m c A

V A d / 2

% = C

m q

Pa

)

V A d

2

/4

M

w = C

m

^ a , V A d

2

/ 4

(85)

After the equations are nondimensionalized, Equations <80) through (84) for trans

ferring resu lts at one station (XZ) to another station (XZ) become

'A;

C

A a i

C

A a

'Nl

'Nai ^Na

'Nai

C

N a

X z

:

N Q I

C

N q

+ 2

^

C

N a ~

2

d °

N

J

m

1

m

d

N

d

A

X Z

C

»a i

C

» a "

d

C

N a

+

d

C

A a

c

m d i =

c

mo.

J

Na

'nql

X X z

C

m q " Z

C

N q

+ 2

"1

C

m a "

4

Z

C

m "

- 2

x\

2

XZ

5 J

C

a

+

»

7

XZ

C

A a

+ 4

T J

C

N "

4

c

A

•

(86)

Following the same procedure, transfer equations for lateral stability derivatives may

be derived.



55

(

C

mq

+

C

md^ 2 " ^

C

m q

C

ma'

1

2

7 7

C

ma

C

Nq

+ C

N a

d d

where

ma

\ d ) Vd,

'Na

d d

r > _ c

mai

ma2

*i

x

i

The subscripts 2 and 1 on the damping derivatives refer t o t h e station where

the derivatives are measured, and the terms Xj and x

2

refer t o t h e dista nce between

moment centers. T h e constant 2 on the static terms in the equations is valid only when

the damping der ivatives a r e based on 2V

C

.

3.2.2 Represe ntative Syste ms for Tests o f Complete Models

3 . 2 . 2 . 1 F r e e - O s c i l l a t i o n T ec hn iq ue

The free-osc illation testing technique, by virtue o f i t s simplicity because complex

control systems are not required, provid es t h e experimenter with a wide la titude in the

approach

to

bala nce system design.

Although the merits o f t h e free-o scillat ion tec hnique are simplicity in construction

and use, which implies economy, i t h a s some basic deficie ncies. These consist of

(i ) th e difficulty in obtaining dat a under conditions o f dynamic instability, since

the model will diverge and may damage the crossed flexures if they are amplitude limited

without mechanical stops, (ii) a reduction in model angular displac ement d u e t o t h e

static restoring moment in the pitch o r y aw plane, and (iii) the inconvenience of re

ducing da ta influenced

by

nonlinear amplit ude

and

frequency effects.

Different techniques have been used for displaci ng mo dels mounted on a free-oscillation

balance system. These generally consist of some form of (a) mechanical release from a

displaced position,

(b)

application

of a

mechanical impulse,

and (c)

excitation

at

resonance and interrupti on o f t h e forcing source. Methods (a) and (b) provide positive

control over t h e initial amplitud e but have t h e disadvantage of requiring large f orces

to obtain high initial amplitudes when the elastic stiffness o f t h e flexure is large.

Method (c) overcomes th is difficulty but requires more complica ted equipment to force

the model

to

oscillate

at

resonance. Aside from using electrical excitation

for re

sonance,

excitation can be provided by using alternating air jets impinging on the

inside o f t he model.



56

Many free-oscillation dynamic stability balance systems have evolved in the United

States and in NATO countries to obtain information experimentally for support of vehicle

development and theoretical studies. To show the basic features of most of the systems

in current use, examples of some systems will be described which are designed with

crossed-flexure pivots, and gas bearing pivots.

To perform dynamic stability tests at Mach numbers from 18 to 21 in the AEDC-VKP

100 in. diameter hypervelocity tunnel (Tunnel F ) , which has a useful run duration of

about msec, a high frequency, fre e-oscillat ion balance system is used. To obtain up

to about 15 cycl es of data during a run, high mo del frequencies, mi nimum inertia models ,

and high crossed -flexure stif fness are required. A flexure system which provid es a

model frequency o f 150 c/s at oscill ation amplitude s up to ± 3 degrees is shown in

Figure 43 (Ref.47).

To displac e the pivot to an initial am plitude a couple rathe r than a moment is used

to minimize a beat oscillation that would be imposed on the oscillation envelope and

reduce the total stress experienced by the pivot.

In operation, the model and pivot system is displaced to a desired amplitude by the

pneumatic operation of two displacement arms which apply a couple about the pivot axis.

Once the desired amplitude is reached, two cams operated by the releasing piston make

contact with the displacement arms and instantaneously release the applied couple.

A low-frequency, free-oscillation, flexure pivot balance system also in use in the

AEDC-VKF is shown in Figure 44 to ill ustrate the sim plicity of the design and a method

for displacing the model. Two jets located 180 deg apart are mounted to the sting

just inside of the model base and are alternately supplied with high pressure air

through an oscillat ing servo valve.

Forces generated by impingment of the jets on the inside of the model cause the

model to oscillate. Amplitud e is controlled by the jet pressure, and frequency is

controlled by the servo valve. The model may be forced to oscillat e at its natural

frequency and when the required amplitude is achieved the air excitation is terminated

and the model is free to oscillate.

A free-oscillation flexure pivot system developed by ONE RA"

5

is shown in Figure 45

to illustrate a mechanical system for displacing the mode l. For most flexure pivot

balance systems, strain gages are bonded to the cross flexures to provide a signal

output for recording the amplitude time history and model oscillation frequency.

Free-oscillation balance systems, having either ball bearings or gas bearings for

the pivot , may not have mechanical s tiffness to provide a restoring moment. For these

systems the model is displaced by a mechanical system and released. If the tunnel has

a model injection system, the model can be preset to the required amplitude, locked,

and then released af ter injection into the airstrea m. An example of an AED C-VKP free-

oscillat ion ba lance system used in this mo de is shown in Figure 46.

Most of the free-oscillation balance systems described thus far have been developed

for use in relatively small wind tunnels. A free-oscillation balance developed by

the AE DC-PWT (Ref.49) for tests in the 16 ft transonic and supersonic t unnels is shown

in Figure 49. The bala nce has a single vertical flexure in the center and two longi

tudinal flexures at the sides for a pivot. A hydraulic mechanism is provided for



57

initially displacing the flexure and model , and a hydraulic cylinder provides a means

for locking the system. Strain gages on the flexures are used to obtain a measure of

the amplitude decay history.

A free-oscillation dynamic balance system has been developed by the University of

California Institute of Engineering Research to measure dynamic stability derivatives

in a low density wind tunnel

50

. Figure 50 shows the components of the balance system.

The model is mounted on a transverse rod supported in the vertical p lane by bearings.

The bearings have an electrically vibrated inner bearing race to minimize the friction

and provide repeat able bearing tares. System restoring moment is provided by a quartz

fiber attached to the tip of the transverse rod support. A cross shaft is mounted on

the end of the transverse rod support to displace the model by the use of an electro

magnet and also interrupt a beam to a photoce ll. One arm casts a shadow on a screen

to provide direct oscillation angle measurements.

A small scale, free-oscillation, dynamic balance developed by the AEDC-VKF for tests

in a small low density tunnel is shown in Fig ure s 51 and 52. The com merci al flexure

pivot is 0.125 in. in diameter. The fixed portion of the pivot is supported by two

supports, and the movable center portion of the flexure is clamped in the balance

centerpiece. As the movab le member is rotated, the air gap between it and the E-core

changes to produce a change in signal level proportional to the angle.

Angular position histo ries for the g as bearing pi vots are recorded with the E-core

transducer described in Section 3.2.2.1. Aerodynamic damping from a free-oscil lation

system is obtained thro ugh an analysis of the record of motion recorded on magnetic

tape,

film, or oscillo graph re cordings to obtain the logarithmic decrement. Reduct ion

of the data from a magnetic tape may be accomplished with a high speed computer to

curve fit the motion data, read the peak amplitudes, and finally curve fit data in the

form of log of the amplitude versus cycles of oscillation. If the data are recorded

on film or an oscillograph, many manual operations are involved which are tedious and

reduce the accuracy of the data reduction.

The ablation of a vehicle' s thermal protect ion system can result in motions which

have a significant effect on the vehicle performance. Study of these effects experi

mentally has been somewhat restricted because present capabilities of wind tunnel

facilities do not provid e a complete dupli cation of re-entry flow conditions. Ablation

affects dynamic stability primarily by changes in body shape, mass distribution, and

the mixing of gaseous produ cts of ablation with the boundary layer. At the present

time,

study of the effects of ablation have proceeded along the lines of using low

temperature ab lators and poro us model blowing techniques. This is desirabl e in a

study of shape change and its effects on stability; however, the results have limited

application for important paramet ers such as material proper ties and phase angle.

Mass addition, on the other hand, offers possibilities for simulating many of the

important par amete rs and enables measurement and systematic v ariation of these parameters.

Some of these parameters include mass injection rate, distribution, phase, wave form,

gas propert ies, and frequency of osci llation.

LTV has pioneered the development o f techniques and equipment to investigate the

effects that mass addition, through porous models, has on dynamic stability. These

investigations have been conducted in the LTV hypervelocity wind tunnel, which is a

modified h otshot -type facility with relat ively long run tim es (250 millis econds at



59

disc which allow light to pass to a photocell. The photocell triggers an electronic

counter (damping co unter) which giv es the total number of slot passa ges, n , the

vector passes in going from the outer radius r

:

to the inner radiu s r

2

, which

contains s number of radial slots.

Denoting the time at which the spot enters the slots tj and the time it leaves

ithmlc dec

(C/2I

y

)tj

by t

2

, the expression for the loarit hmic decrement A can be derived:

r

i =

E

o

e

-(C/2I

Y

)t

r

2

= E

0

e

and A = T T C / I ^ O = s/n log

e

(rj/r

2

) .

The frequency of oscillation is measured with a second electronic counter by cou nting

the number of cycles of a crystal oscilla tor during the period in which the damping

counter is in operat ion from a preset number n

x

to another preset number n

2

. If

the frequency of the crystal oscillator is f

c

and the reading of the counter n

c

,

n, — n,

f = - 2 - ^ - 4 ( f

c

/n

c

) .

Different di scs having a range of radius ratios and number of slots are used to

obtain good resolutio n for A and f . As noted in Refere nce 53, the dampometer can

be used to obtain an accuracy of < 2% for logarithmic decrements smaller than 2;

for higher va lues

of the

decrement,

the

accuracy decreases.

3 .2 .2 .1 .2 Tra vel Summation Method of

F r e e - O s c i l l a t i o n D a t a A n a l y s i s

There may be instances when performing wind tunnel dynamic stability studies where

very large amplitudes of oscilla tion will be of interest. In such cases, where it

may be difficult to obtain an analog o utput of the displacement time history, a light

source on the oscillating model may be used to indicate model positio n. Orlik-Ru ckemann

presents a data analysis tec hnique

5

" to obtain instantaneous r eduction of free-oscillation

data from this type of experimental arrangement.

In

the

simplest form,

the

experimental setup c onsists

of an

oscillating light source

seen through a stati onary window (Pig .57) which is positioned such that the light source

can be seen only duri ng a portion of a cycle. The total visible travel of the light

source

in

relation

to the

window

is

recorded

by

passing

the

light through

a

large

number of transverse narrow slots to a photomultiplier tube to record the numbe r of

pulses.

With the travel summation method, the logarithmic decrement can be expresse d as

A = (2/s)(l - b + b l o g

e

b)

m

(87)

where s = 2/x

0

S (

x

n

~

x

m ) (

x

o

x

m

)

n= o



60

and b is a constant with c onvenient va lue s arou nd 0.5, depending on the experimental

arrangement. Equation (86) thus shows that

the

logarithmic decrement

is

simply obtained

by dividing

a

constant

by the

dim ensionless

sum of the

travel segments.

By

proper

selection of b and a slotted window, the total error, including the approximat ions

involved

in

retaining te rms

in the

derivation

of

Equation (87),

can be

kept under

1%

for most cases.

An alternative method

for

obta ining

the

logarithmic decay involves

the

summation

of

the time intervals

for the

consecutive cycles

of

oscill ation d uring which time

the

light

source

is

between

x

0

and x

m

. This method

is

less accurate than

the

travel summation

method, because of approximations used in the derivation, but may be preferable since

it permits simpler experimental apparatus. These data reduction methods are described

in detail

in

Reference

54.

3. 2. 2. 1. 3 Logarithmic Cir cui t

A recording oscillograph

is

frequently used

to

record

the

decayi ng signal

of

model

motion from a displacement pickup. These data which appear as a harmonic oscillati on

with damping must

be

replotted

in

terms

of the

logarithm

of the

amplitude versus

a

time function

or

cycles

in

order

to

extract

the

damping constant from

the

results.

A

method which effects

a

considerab le saving

in

time

in the

analysi s

of the

data traces

consists of employing a logarithmic distorting circu it

6

, which converts the exponential

envelope of oscil lation into a linear e nvelope which is proportional to the number of

cycles

of

oscillation.

The

damping constant

is

then deter mined from

the

slope

of the

envelope.

3.2 .2. 1.4 In te gra ti on Method

Signals from displacement pic kups recorded

on

magnetic tape

may be

analyzed quickly

without

the aid of a

computer

by

making

use of the

integration meth od described

in

more

detail

in

Reference

6. To

employ

the

method,

a

series

of

complete cycles

is

selected

from the magnetic tape recording and the signal is rectified and fed into an analog

integrator which presents the integral as a reading Rj on a d.c. voltmeter. The

process

is

repeated

for a

second serie s

of

complete cycles

to

obtain

the

seco nd Integra l

represented

by R

2

and, if n is the

number

of

cycle s between sam ple

Rj and R

2

.

the logarithmic decrement is given by

The integration method

has the

advantage

of

giving weight

to

every cycl e

of

oscillation

in the range covered, as opposed to measuring the number of cycles to damp to a given

amplitude ratio. If noise exists on the displa cement signal the integration method

tends

to

reduce

the

effects

it may

have

on the

damping measurements.

3.2.2.1.5 Other Methods

A device called

a

damping trace reader

has

been d evelope d

by

L e a c h

55

to

measure

the

logarithmic decrement from oscillograph traces with about the same accuracy as for

hand data reduction.



61

3 . 2 . 2 . 2 F o r c e d - O s c i l l a t i o n T e ch n iq u e

Testing facilities have different requirements

for

dynamic stability balance systems.

This naturally

h a s

resulted

in the

development

o f a

variety

o f

different systems

now

in

use in the

United States

and in

Europe an countries. Space will

no t

permit

des

criptions

of all

systems

and

therefore only

a few

represe ntative ones will

be

discussed

to show important features

and

techniques.

3.2.2.2.1 AEDC-VKF

Forced-oscillation dynamic stability balance systems have been developed

at the

AEDC-VKF

for

operation

in

three c ontinuous flow tunnels, which cover

the

Mach number

range from

1.

5

to 12.

The dynamic balance system (Fig.58) consists

o f a

crosse d-flexure pi vot mounted

on

a sting

and

connected through

a

torque mem ber

and

linkage

to an

electromagnetic shaker

housed

in the

enlarged

aft

section

o f t h e

sting. Detai ls

o f t h e

crossed flexures,

input moment linkage,

and

model locking device

are

shown

in

Figure 59.

T h e

sting

and

shaker housing contain passages

for

water cooling which

is

necessary

for

tests

at

hypersonic speed. Cooling

o f t h e

crossed flexures

i s

accomplished through

t h e u s e o f

cool

air

sprays which

are

turned

off

while data

are

being recorded. Water-cooled

jackets which surrou nd

t he

crossed flexure

are

also used

in

some applications.

The

VK F

forced-o scillatio n balance

i s

equipped with

an

amplitude control system

which allows testing

o f

both dynamically stable

and

unstable model s. Testing

i s

normally

accomplished

a t t h e

resonant frequency

o f t h e

model balance system; however,

t h e

control

system does prov ide

a

means

o f

resolving

t he

in-phase

and

out-o f-pha se (with model

displacement) components

o f t h e

forcing torque. Because

o f t h e l ow

structural damping

level

o f t h e

balance

and the

aerodynamic damping levels present

at

supersonic

and

hyper

sonic speeds,

t he

balance system

i s

always operated

at

resonant

or

near-resonant

conditions.

It

was

shown

in

Section

3.2.1.4

that when

a

model

is

oscillated

at a

constant

amplitude

and at the

undamped natura l resonant frequency

o f t h e

system,

t h e

inertia

moment exactly balances

t h e

restoring moment

and the

forcing moment

is

equal

t o t h e

damping m oment.

A block diagram

o f t h e

control

and

data acquisition system utilized

at the VKF

system

is

given

in

Figu re 60.

T h e

control system

can be

operated

in

either

an

open-

look mode

for

testing dynamically stable configurations

or in a

closed-loop mode

for

testing either dynamically stable

o r

unstable configurations.

In

either mode

o f

opera

tion

the

frequency

o f

oscillation

is

controlled

by the

frequency

o f t h e

reference

oscillator.

Open-loop operation

is

obtained

by

using

the

amplified refere nce oscillator signal

to drive

t he

electromagnetic shaker motor. Resista nce strain-gage bridge s located

on

the input torque member

and on a

cross flexure (model displa ceme nt) provi de analog

signals

o f t h e

forcing torque

t o t h e

model

and the

model displace ment.

T h e

electrical

output

o f t h e

bridges

is

amplified

by a

carrier amplifier

t o

provide appropriate ampli

tudes

o f t h e

torque

and

displacement signals necessary

for the

dat a a cquisition system.



62

The closed-loop mode of operation operates on the amplified error principle. The

refere nce signal and the displace ment s ignal, which is phase -shif ted approximately

90 degr ees, are compared in a differ ence amplifier. This difference or error signal

is amplified and used to drive the shaker motor. The error signal will remain in

phas e with the refer ence signal for all conditio ns requiring a positi ve app lication

of torque. If the model is aerodynamically unstable, the error signal will be 180

degree s out of phase with the reference signal and a negative to rque will be supplied

to the model.

Due to h igh frequency vib rati on in the wind

tunnels,

signal co nditioning is required

to remove the noise from the torque and displacement signals before recording on the

dat a acquisition system. Two bandpass filters tuned to the frequency of the reference

oscillator are provided. The filters and other frequency-sensitive circuits track with

the frequency selected by the reference oscillator. The output of the bandpass filters

is amplified to provide the current necessary for the recording instruments.

The data acquisition system consists of an oscillograph, resolvers, on-line data

system, analog magnetic tape system, and Beckman 210 analog-to-digital converter system.

The torque and displacement signals are applied to the inputs of all the recording

instruments and to an oscilloscope displaying a Lissajous pattern. The Lissajous

pattern provides an indication of the 90 degree phase angle relationship between these

two signals necessary to the condition of resonance of the model balance system.

Under cer tain tunnel condit ions, and for certain model configu rati ons, it is bene

ficial to operate at a frequency displace d from the resonant frequency. The resolver

system is utilized to determine the in-phase and quadrature components of the forcing

torque with respect to the model displacement.

The in-phase component is obtained by multiplying the torque and displacement signal

with an analog multiplier. The quadrature component is obtained by phase-shifting the

torque signal 90 degrees and then multiplying the phase-shifted signal with the

dis

place ment signal. The output signals of the mult ipli ers contain a d.c. component and

a double frequency component. By filtering out the double frequency component, a d.c.

valu e is obtained which is prop orti onal to the in-phase and quadratu re torque components.

The In-phase torque component is zero at resonance, and the quadrature component is

a maximum value. A d.c. voltmete r is utilized to monitor the in-phase component and

serves as an indication of a reso nance condition. Both torque compo nents are recorded

on analog magnetic tape, the Beckman 210 converter, and the on-line data system.

Data recording systems utilized for forced-oscillation testing are the oscillograph,

analog magnetic ta pe, Beckman 210 converter, and on-line data system. The osci llogr aph

and analog magnetic tape p rovide permanent recordings of the dat a as obtained in the

wind tunnel. The Beckman 210 analog-to-dig ital converter provi des digital conversion

of the analog data and records the digital values on magnetic tape, which is then

processed by a high speed computer to obtain final data.

Reduced data are desirable, and in some cases necessary, while the test is in

progress.

The on-line data handling system provides immediate examination of the

damping derivatives. Here the term on-line signifies that the test data are fed

directly from test instruments into a computer. Therefore, the on-line data system

includes obtaining analog representation of par ameters to be measured, conversion of



64

3.2.2.2.3 NASA-LRC

The forced-oscillation balance system used by NASA-LRC is shown in a exploded and

assembled view

in

Figure 65. Sketches

of the

balance system

are

presented

in

Figures

66

a nd

67. Sinusoidal, constant-amplitude motion about

an

oscilla tion axis

is

produced

by

a

mechanical Scotch yoke

and

crank which

is

driven

by a

variable-f requency electric

motor in the downstrean porti on o f t h e sting. The stiff strai n-gaged beam located

between t he model attachment bulkhead and pivot axis i s used to measure the forcing

moment required

t o

oscillate

t he

model. Because

of its

location

the

signal output

i s

not Influenced

by the

friction

or

mechanical play

in th e

system. Model displaceme nt

is measured with a strain-gaged cantilever beam mounted be tween the fixed sting and

the oscillating model support. Springs of different stiffnesses are used to provide

a range

of

resonant frequencies.

Outputs from

the

displacement

and

forcing-moment strain gages pas s through coupl ed

sine-cosine resolvers which

are

geared

to the

drive shaft

and

thus rotate

at the

funda

mental drive frequency

(Pig.66).

Oscilla tion frequency

i s

measur ed with

a

signal genera

tor coupled to the drive shaft. In operation t h e resolvers transform the moment and

amplitude functions into orthogonal components which

are

measured with d.c. micro-

ammeters. From these components

t he

resultant applied moment, displacement,

and

phase

angle are found and, with th e oscil latio n frequency, t h e aerodynamic static and damping

moments are computed. A block diagram of the electro nic ci rcuit used to measure t h e

applied moment

is

shown

in

Figure

68 and a

similar circuit

is

used

to

measure

the

angular displacement.

Use is made of automatic digital equipment to acquire data quickly and eliminate

operator errors due to incorrect manual recordi ng or reading. All of the system var i

ables

are

automatically digitized,

and a

high speed computer

is

used

to

perform

the

data reduction

and

present results

in

coeffi cient form.

A complete description

o f t h e

system electrical relations

and

metho ds involved

in

the operation

and

data reduction

are

found

in

References

59 and 60.

3.2.2.2.4 RAE-Bedford

The forced-oscillation testing technique ha s been employed by the RAE, Bedford, i n

the development of a system for measuring stability de rivatives and damping derivatives

due

to

angular v elocity

in

roll

and ya w on

sting-mounted m od el s

62

. Derivatives

can be

obtained

for

motion

in

vertical

or

horizontal tr anslation (heave

or

sideslip); however,

it

is

noted

in

Reference

62

that

t he

apparatus

was not

designed

for

this purpose

and

the accuracy

of the

measurements

is

less than that required

for

most tests. Emphasis

in t he design h a s been toward a small s ize system that can be used for testing slender

wings

o f

fiber glass construction.

In some respects

the

balance system

is

similar

to

those described

in

Section

3.2.2.1

in that the model is mounted on a sting by means o f a flexure pivot allowing one degree

of freedom in rotation. Continuous excitation at or near the resonant frequency of

the system

is

provided

by a

movi ng-coil per manent-magnet vibr ation generator mounted

on

t h e

sting

and

coupled

to a

driving

rod

passing through

th e

sting.

A schematic arrangement

of

various p arts

of the

balance

for

yaw-sideslip oscillation

is shown

in

Figur e 69.

The

balance

is

mounted

on the end of a

sting

by

means

o f a



65

flexure pivot to allow rotation. A view of the flexure pivot and cylindrical section

of the balance which mounts inside of the model is shown in Figure 69, and photographs

are presented in Figure 70. Included in the flexure pivot as sembly is the drive linkage

to convert the longitudinal force in the driving rod into a couple acting between the

end of the sting and the model.

The balance for measuring oscillation in roll is illustrated schematically in Figure

69.

The unit is of one-piece constructio n and is instrumented with ya w and roll

accelerometer s. The system has one degree of freedom, and excitation in roll is

pro

vided by applying torque through the driving rod by means of linkage from the vibration

generator.

Although the flexure pivot i s located near the center of pressure of the wing models,

in order to avoid large deflections under steady aerodynamic

loads,

the flexibility of

the sting will allow some translation of the pivot axis which may be appreciable for

some cases. To allow for displacement of the pivot

axis,

for example, when measuring

pitching derivat ives, heaving derivati ves are measured by exciting a vertica l oscilla

tion (which will be accompanied by some angular movement) and measuring the changes

in natural frequency, excitation force, and angular movement.

The motion of the model on the balance system consists of a pitching component,

generated in part by the rotation of the model about the flexure pivot and in part by

the bending of the end of the sting. These two compo nents are present in different

proporti ons in the two natural mod es of the system. The pitching mode includes a

certain amount of heave motion of the pivot axis, and the heave mode includes an

appreciable amount of pitching motion and similarly for yaw and sideslip.

As was noted in Reference 62, it may not be ob vious ho w an exciting coupl e acting

between the model and the end of the sting can produ ce no deflection. The reason Is

that in the natural mode of oscillation in heave, for example, there is a certain

amount of rel ative angular movement between the model and the end of the sting (even

when there is no pitching component in the oscillation). It is thus possib le for the

excitation couple to feed energy into the system in this

model.

The analysis of the system is simplified by treating the system as having two degrees

of freedom with two natural modes of vibration - one predominantly pitching and the

other predominantly heaving. The heaving derivatives appear as a by-product of the

analysis to obtain the damping-in-pitch derivati ves. Yawing deriva tives are measured

in the same manner, with sideslip derivatives appearing as a by-product.

The basic measurem ents made with the balance system include (i) the excitation

frequency which is measured by an electronic counter, (ii) the excitation force which

is measured in ter ms of the exciter current, and (iii) the bala nce displace ments which

could be obtained as measurements of strains or relative displacements; accelerometers

were selected for the design since they are not affected by steady loads and are com

paratively free from interactions.

The balance is designed (Fig.

70)

with strain gages mounted on sections for measuring

static forces and moments, except for rolling moment which is more conveniently measure d

with the flexure unit designed for roll oscillations.



66

The automatic measuring and recordi ng equipment developed by the RAE for experim ents

concerned with oscillatory der ivative i s a digital amplitude and phase measuring system

(DAPHNE). It was

designed

to use the

wattmeter metho d

6

to

resolve

the

input signals

into in-phase

and

quadrature components,

and

accurate measurements

of

fr equency

are

made with

an

electro nic counter.

3 . 2 . 2 . 2 . 5

NASA-ARC

A method

and a

balance system

is

described

in

Reference

63 for

measuring

th e

lateral

and dynamic stability derivatives. The features o f t he system include single- degr ee-

of-freedom osc illations which are used to obtain comp onents o f pitch, yaw, and roll by

varying

the

axis

o f

oscillation. Amplitude control

i s

achieved with

a

velocity feedback

loop. T h e

system

is

equipped with

a

system

of

strain gage data proc essing

in

which

electronic analog computer elements are used in measuring the amplitude and phase

position. Moment derivatives which arise from angular motions can be measured with

the balance system. These include

t h e

rotary damping derivativ es

C , C + C • ,

and

C

n r

- C

n/

3 ; th e

cross derivatives

C a nd C j

r

- Cji ; and the

displacement

derivatives

C

m a

, C ^ and C

; / 3

.

The forced-oscillation system has two oscillation mechanisms - one for pure pitching

or yawing motion in which t he oscillation axis is perpendicular to the longitudinal

axis

of the

sting,

and one for

combined rolling

and

pitching

or

rolling

and

yawing

in

which

the

axis

o f

oscillation

is

inclined

45

degrees

t o t h e

longitudinal axis

of the

sting. T h e electromagnetic shaker is housed in an enlarged section o f t h e sting down

stream o f the model and connecte d to the flexure pivo ts b y push rods. Strain gages

were used

to

measure

t h e


and

supply Inputs

to the

analog com puter.

The theory of the system ope ration, feedback control, analog compu ting system,, and

description

of the

apparatus

is

presented

in

Reference

63.

3 . 2 . 2 . 2 . 6 U n i v e r s i t y o f M i ch ig a n

Results obtained at hypersonic and hypervelocity speeds have shown the importance

of the reduce d frequency of oscill ation. With the use of the free-o scillat ion testi ng

technique, the

frequency

of

oscillation

can be

varied only

by

changing either

the

moment

of

inertia

or the

flexure spring constant. Thus coverage

o f a

wide range

of

frequencies requires several time-consuming changes

in the

test equipment.

The University

of

Michigan

h a s

developed

an

electromagnetic, forced-oscillation

method which enables a range o f oscillation frequencies to be investigated with e ase

6

".

The method utilizes

the

principle

of the

moving coil galvanometer.

An

alternating

current,

passing through

a

coil embedded within

th e

test model, interacts with

a

static

magnetic field imposed

by a

coil system mo unted

in one

plane

of the

test section. This

causes

the

model

to

oscillate

in a

sinusoida l manner following

the

existing torque

but

out o f phase with it. From the phase lag of the motion relative to the existing moment

and wind-on and wind-off result s, it is possible to determine the aerody namic damping.

A schematic view o f t he test setup showing t h e external coil is presented in Figure

71. A

circular exciter coil

wa s

mounted coaxially within

t he

Plexlglas model which

was

mounted on a cruciform flexure pivot shown in Figure 72.



67

The methods have been used

for

measuring

t he

position

o f t h e

model.

One of the

methods utilized

t h e

output

of the

strain gage bridge shown

in

Figure

72 and the

other

was based

on the

response

of a

second circula r coil within

t he

exciter coi l located

in

the model.

A s t h e

model oscillates

in the

magnetic field,

an

electromotive force

is

induced

in the

secondary coil

and may be

used

for

sensing

the

cone position. Although

this technique

h a s

been success ful,

i t h a s t h e

drawback that

t h e

data reduction pr o

cedure

is

very lengthy, which

m ay

prec lude on-line reduction.

U s e o f

strain g ages

t o

measure

the

model position eliminates

t h e

difficulties associated with

t h e

two-coil

techniques.

The t echnique

o f

electrom agnetic excitation

and

using

t h e

phase angle

to

determine

the damping

h a s

limitati ons which

may

prohibit

its use for

some configurations. Although

the structural damping

wa s

small,

t h e

phase angle measurements

at

Mach number

2.5 on a

25 degree half-angle cone were less than

one

degree, which makes accurate measurements

of aerodynamic damping

at

this

and

higher Mach numbers very difficult

but not

impossible.

Improvements

in the

accuracy

o f t h e

damping me asure ments will result

i f t h e

model size

is increased,

t h e

pivot axis

is

moved

as far

forward

a s

possible,

and a gas

bearing

i s

used

to

eliminate amplitude re strictions

and

structural damping

d u e t o t h e

flexure.

In addition

t h e g a s

bearing would eliminate sting vibr atio ns that influence damping

measurements. Details

on the

instrumentation, data r eductio n procedu res,

and

sample

test results obtained with electromagnetic excitation

a re

presented

in

Reference

64.

3 . 2 . 2 . 3 Magne t ic Model Suspens ion Sys tem

Systems

for

capti ve model a erodynamic testing require support

o f t h e

model

by use

of

a

sting into

t h e

model base region,

a

side-mounted strut,

or

even thin wires. Each

of these techniques

m a y

introduce interference ef fects which

can in

some cases have

a

significant influence

on the

results. Support interference effects must

be

studied

experimentally

to

gain some idea

o f t h e

magnitude

o f t h e

influence, since

t h e

problem

has

no t

been r educed

to

theoretical analysis.

Magnetic suspension

o f

wind tunnel mode ls pr ovide s

one

means

for

elim inating

t h e

influence

o f t h e

support system.

T he

first successful suspension

o f a

wind tunnel

model that enabled

the

measurement

of

interference-free drag

wa s

accomplished

at

ONERA,

France,

in 1957

(Ref. 65). Since that time, several labora tories

in

Prance, England,

and

the

United States have been engaged

in the

development

of

magnetic suspension

systems

for

different purpos es.

T h e

potential

for

magnetic suspension

i s

very great

when

it is

considered that

no t

only

is it an

interference-free support system,

but it

is

a

balance system

for

static forces

and

moments

and

dynamic mea sureme nts, since model

motion

can be

controlled.

For example, control signals

may be

introduce d into

a

feedback control circuit

to

produce model moti on without alteration

o f t h e

basic system.

In

principle, models

can

be subjected

to

rigidly forced motion eithe r

in

sinusoidal

or

random mo des

or

rotational

motion with

t h e

translational degre es

o f

freedom fixed. With

t h e

latter mode

of

opera

tion, dynamic damping data

in

pitch

o r y aw a r e

obtained

by

either

th e

free-

o r

forced-

oscillation techniques. Roll damping measureme nts

m a y b e

made

on

axisymmetric

and

finned bodies

by

using

t h e

spin decay technique.

The development

of

magnetic suspension syst ems

h a s

progressed very rapidly

in the

past several years; however,

it has not

reached

th e

stage that

t h e

suspension systems

compete favorably with

t h e

more conventional techniques

for

obtaining static

and

dynamic



68

stability data. Magnetic suspension systems are, however, used routinely for wake

studies and some drag measurements- Although research to develop techniques for

making dynamic stab ility mea sure ments with a magnetic model suspension system is in

the formative stages, the status of the work accomplished to date is briefly summarized

to point out the major areas of development. Treatment of the equations of motion

and data reduction have been omitted, since each depends more or less on the specific

system; these topics are covered in the references.

The suspension system in use at the Gas Dynamics Laboratory at Princeton Univers ity

66

was developed to support models for experimental research on hypersonic wakes of axisym

metri c bodies. Since the configur ations of interest ranged from spheres to slender

bodies, a three-d egree- of-fre edom system and a one-point suspension was used (Fig. 7 3 ) .

The system is mounted vertically to use the gravity force to oppose drag, to help the

magnetic suspension. Models are made with ferrite spheres as the magnetic element to

be acted on by the magnetic field. A conical model with a ferrite sphere embedded in

the body is shown in Figu re 74. When only spheres are tested, the ferri te is used

without any other mater ial. Althou gh the Princeton system is not used for dynamic

stability measurements, such data can be obtained with the system and model shown in

Figure 74 by making use of the free-oscillation technique, provided position sensing

is incorporated in the system for model motion histories. Completely free rotational

model motion is also possible with the balance system developed by the University of

Virginia

67

, provided position sensing is incorporated.

The MIT magnetic model suspension system described in Reference 31 and shown in

Figure 75 has been used to obtain dynamic stability data from wind tunnel measurements

at supersonic speeds. The forced-oscillation technique was employed to measure damping-

in-pitch der ivati ves on a model u ndergoing a forced moti on by measur ing the change in

the phase of the motion. Use of this technique requires that the slope of the aero

dynamic moment curve,

U$

, be known and sinusoidal mo tion imposed by the forcing

function.

Although no attempt was m a d e

3 1

to measure damping due to plunging, a method

is described in the reference whereby such measurements could be obtained.

The RAE magnetic model suspension system in use at Mach 9 is shown in schematic

form in Figure 76, and a view of the system installed in the tunnel is shown in Figure

77.

The system, now in the final stage s of develo pment, was designed to measur e static

forces and moments and dynamic stability. Additional deta ils on the system may be

found in References 68 and 69.

The magnetic suspension system at Southampton University

70 ,71

makes use of three

horseshoe-shaped electromagnets and one hollow solenoid-like coil for control of the

model in six degr ees of freedom; t he model itself contains a permanent magnet. A

schematic arrangement of these magnets is shown in Figure 78. and a model subjected to

a forced oscillation in roll while suspended in the six-component balance is shown in

Figure 79. The system has been designed for the measurement of aerodynamic derivatives,

and emphasis has been placed on tests of winged models which requires proper control

over the model roll attitude. Although there are several different methods that have

been investigated for controlling model roll attitude, shaped cores within the model

have been investigated most extensively. The usual practice in magnetic suspension

systems is to have a magnetic core arranged to lie along the model axis. These cores

are generally circular in cross section and thus do not impart any resistance to

dis

turbances in roll. However, if the core is constructed with a noncircul ar cross section,

the core will adopt one or more preferred attitudes when suspended in the field of the



69

model supporting magnet

B

shown

in

Figure 78.

A s

pointed

out in

Reference 71,

a

model

containing

a

shaped core

can be

induced

to run as a

synchronous mo tor

by

varying curre nts

to

t he

lateral magnets; however,

t h e

model would have

t o b e

spun

u p t o t h e

synchronous

speed

to

allow lock on. Speed s grea ter than 1000

r.p.m.

have been produced

on a 0.75

in.

diameter body

of

revolution

at the

University

o f

Southampton. This technique could

be used

for

measuring roll derivatives.

Reference should

be

made

to

Reference

29 for

information

on

magnets, optics systems,

control circuits, operating techniques, equations

of

motion, data reduction,

and

problem

areas.

3.2.3 Repre sentative Systems

for

Tests

of

Control

and

Lifting Surfaces

Methods

for

experimentally o btaining dynamic stability data

o n

models

o f

missiles

and aircraft

may

also

be

applied

t o

investigations

o f t h e

aerodynamic characteristics

of control surfaces, such

as

rudders, elevators,

and

ailerons. Although aerodynamic

theory

h a s

been developed

to

analyze control surface behavior , r ecourse

t o

experi

mental verification

is

necessary becau se

o f

nonlinear effects which

may be a

function

not only

of

amplitude

and

frequency

but

also

of

Reynolds number

as

associated with

flow separation ahead

o f t h e

control surface.

Experimental investigations

to

date have been designed primaril y

to

measure

t h e

aerodynamic char acter istic s (hinge moment

and

damping)

o f

control surf aces o scillating

at transonic

and low

supersonic speeds where

the

theories show

a

strong influence

of

oscillation frequency

and

Mach numb er

72

.

Since

t h e

specific details

on the

design

o f t h e

model control surface

and

dynamic

balance system will depend

on the

ingenuity

o f t h e

designer,

no

attempt will

b e

made

to treat detailed problems peculiar

to a

particul ar system. Reference will, however,

be made

to

typical systems

t o

describe

t he

general techniques that

m a y b e

used

to

obtain control surface damping

and

hinge-moment data. Representative system s

for

obtaining damping data

o n

lifting surface s

are

also presented.

3 .2 .3 . 1 F r e e - an d F o r c e d - O s c i l l a t i o n T e c hn i qu e s

The methods employed

and the

data reduction,

in

general,

a r e t h e

same

as

those des

cribed

in

Section

3.2.1 for the

free-

and

forced-osc illation techniques applied

to

control surfaces

and

half models. Applications

of

these techniques

to a

control surface

mounted

on the

trail ing edge

o f a

tri angula r wing (Fig. 80)

a re

presented

in

Reference

73.

The

control surface

wa s

attached

to the

wing

by

means

of a

flat spring running

the full length

o f t h e

control surface. Excitation

wa s

supplie d with

a

torque motor

mounted

in the

model,

and

control surface p osition

wa s

measured with strain g ages

mounted

on

cros s flexures. These were used

to

increase

th e

natural frequency

o f t h e

system.

T o

operate

in the

forced-oscillation mode,

the

control surface

wa s

driven

at

the resonant frequency

of the

system where

t h e

torque required

to

sustain

a

given

amplitude

o f

oscillation

i s

directly proportional

t o t h e

damping

o f t h e

control surface.

Thus

t he

damping could

be

calculated from

a

measurement

o f t h e

armature current

o f t h e

torque motor.



70

For the free-oscillation tests, the system was driven at the resonant frequency of

the system to a given amplitude. The armature circuit of the torque motor was then

opened and the ensuing decay in amplitude was measured to obtain the damping from the

log decrement.

Although oscillating at resonance to obtain the initial displacement has the advantage

that the torque to deflect the control surface is small, it has the disadvantage of

increasing the mass moment of inertia of the system. Initial amplit udes for free-

oscillation motion may be obtained with a static or impulsive deflection of the surface

to the desired amplitude, but this technique has the disadvantage that the localized

loading may lead to some distortion of the control surface.

Forced-oscillation systems used for damping measurem ents, or as a means to achieve

initial amplitudes, generally require considerably mo re space compared to the require

ments for the free-oscillation system with a static displacement mechanism and therefore

are restricted to large models. Orlik -Ruc kemann

7

" has applied the forced oscillation

technique to obtain initial amplitudes for free-oscillation tests on small aerodynamic

control su rfaces by developing an experimental a ppara tus for use with half models.

The ap paratus for elevator oscillation te sts on a half-airplane model is shown in

Figu res 81 and 82. In operati on, the control surface Is oscillat ed by a pair of driving

col ls to a desired amplitude and at the resonance frequency of the oscillati ng system

and then the circuit is interrupted to allow a free oscillation. For this design, the

control surfac e motion is measure d by strain gages located on the control su rface

hinges which are, in this case, cantilever beams. The signal from the strain gages,

which is propor tional to the angular displace ment of the control surface, was process ed

with a dampometer to obtain the characteris tics of t he oscillation, and thus the aero

dynamic data.

M a r t z

75

has used the free-oscillation technique on rocket-powered free-flight models

to obtain hinge-moment characteristics for a full-span, tralling-edge control surface

on a delta wing. Thi s same technique can also be used in wind tunnel tests. The con

trol surface was mounted to the wing through a cantilever-type flexure hinge which

was instrumented with strain gage s to measur e the control surface deflection. Initial

displacement of the control surface was achieved by plucking periodically at the root

chord by means of a motor -driv en cam s hown in Figur e 83.

Orlik-Ruckemann and Laberge

25

have used electromagnetic excitation for free-oscilla

tion dynamic t ests on a series of delt a wings. An example of the system is shown in

Figure 84. This sys tem

76

has been developed to the stage that its operation is com

pletely automati c, that is, the output signal from the model suspension, which is a

strain-gaged cruciform elastic member, and the input signal to the electromagnetic

oscillator form a closed-loop circuit in which the frequency is automatically adjusted

to the natural frequency of the system. After the circuit is interrupted, t he model

oscillates freely; da ta are recorded on the dampometer, and the cycle is again repeated.

This automatic feature allows acquisition of a large amount of data within a short

period of ti me and is particul arly useful for blowdown tunnels which operate for very

short time perio ds where manual adjustment of i nstrumentation is not possible without

reducing the amount of data that may be acquired. A schematic block diagram of the

system control and da ta acquisition in shown in Figure 85.

A half-model, fixed-amplitude, der ivative measurement system used in the Aerodynamic

Research Laboratory of Hawker Siddeley Dynamics Ltd. i s shown in Figure 86. The system



71

is described

in

detail

in

Refe rence 23.

The

apparatus provid es measurements

o f t h e

wing

and

control surface derivatives

d u e t o

mode ls performi ng sinusoidal mot ion

in

pitch

and

verti cal translation.

A

nonresonant me thod

is

used

to

reduce

t h e

electronic

and mechanical system complexity.

The balance system

is

built

on a

nominally rigid platfo rm which

is

connecte d

to

earth

by

three supports

and

force detecting transducers. Angular, simple harmonic

motion

is

achieved with

a

variant

o f t h e

well-known Scotch Yoke mechanism which converts

constant angular motion into linear harmonic motion. Thi s motion

is

then imparted

through linkage

t o t h e

model shaft.

T h e

model

is

driven

by an

electric motor through

a variable speed gear

box so

that

t h e

fre quency

may be

varied between

5 and 25

cycles

per second

at

amplitudes

of 2

degrees.

The oscillating system

is

inertia balanced

so

that

t h e

difference between

th e

wind-on

and wind-off transducer outputs

in

terms

of

amplitude

and

phas e relationship with respect

to model mot ion

ar e

measured

for

oscil lations about

two

model

axes.

In-phase

and

quadrature c omponents

are

resolved from these meas urements

to

give

t h e

stiffness

and

damping derivatives

due to

pitch

and

plunge.

3. 3 D Y N A M I C R O L L D E R IV A T IV E S

As

a

vehicle having aerodynamic l ifting

and

control surfaces rolls about

i t s

longi

tudinal axis with

an

angular velocity

P ,

points

on the

surfaces follow

a

helix path.

Variations

in

spa nwise local a ngle

o f

attack

o f t h e

vehicle aerodynamic surfaces alter

the local normal

and

drag force loading di stribut ion

and

inclination

of the

lift vector.

In

t he

case

o f a

winged vehic le having tail surfaces, rolling

at

various rates produc es

a change

in

symmetry

o f t h e

trailing vorte x sheet,

and

thus influences

t h e

aerodynamic

characteristics

o f t h e

tail surface. These factors have

an

influence

on

derivatives

du e

t o

rolling.

3. 3.1 The ory

3 . 3 . 1 . 1 F r e e R o t a t i o n

The damping-in-roll derivative,

Cj , can be

measured using

the

free-rotation

method where

t h e

model

is

free

t o

rotate under

th e

influence

of an

initial rolling

velocity,

and the

subsequent decay

in the

roll rate

is

measured. Another version

of

the free-rotation technique consists

o f

forced co ntinuous rotation

by a

known m oment

imparted

by

deflecting

a

model

fin.

Although some

o f t h e

equations describing free rotat ion

are

include d

in

Section

3.2.1.3,

the

derivation

i s

included here

for

completeness

and in a

different format.

For

a

freely spinning axisymmetric body,

t h e

differential equation

o f t h e

free-rolling

motion

may be

derived

by

equating

th e

rate

of

change

o f t h e

angular momentum

t o t h e

applied damping mom ent:

I

X

P

=

M

X p

P

, (88)

where

th e

roll-dam ping moment,

M

x

P

, consists

o f t h e

contribution

d u e t o

aerodynamic

damping

and the

tare damping

due to

friction

in the

bearing system.

It is

assumed here

that these moments

a r e

linear functions

o f

angular velocity.

T h e

mechanical damping



75

3 .3 .2 . 2 F o r c e d Ro t a t i o n

An example o f a system

80

for forced, continuous-ro tatio n, damping-in-roll test s is

shown

in

Figure 90.

Th e

balance

h a s a n

electric motor which drives

t he

model through

a planetary gear system.

The

stationary ring gear mounted

in the

gear system

i s r e

strained from rotation by an instrumented cantilever beam. Torque to overcome model

aerodynamic damping and friction damping in the system is proportional to the restraint

provided by the ring gear. Test s in a vacuum provide tare values for correcting t h e

wind-on data.

A two-degree-of-freedom system,

in

which os cill ations

in

roll

are

forced,

h a s

been

used

by

NASA

t o

obtain rolling deri vatives

and

directi onal stability

and

dampi ng-in-

yaw data

77

. T h e system h a s been used f o r y aw amplitudes u p t o 2 degrees a nd forced

at oscillation frequencies up to 20 c/s. A schematic and pictorial representation of

the system i s shown in Figures 91 and 92 to illustrate t he modes o f model motion. By

use o f flexures and ball bearings, th e model is allowed two angular degre es o f freedom -

rolling

and

yawing

-

rolling being

a

forced oscillation

and

yawing being free. Harm onic,

forced oscillation in roll is produced by means o f rotary motion of an eccentric and

oscillatory motion

o f t h e

drive flexure indicated

in

Figu re 91. Variatio ns

in the

amplitude

o f t h e

oscillatory rolling motion

are

provided

by the

amount

o f

eccentricity

in t h e drive system. It is transmitte d t o t h e torque tube , which is supported a t t h e

front end by roll flexures and the aft end by a ball bearing. Thus t he only contact

between th e model and the support structure is through t he roll flexure system. Thi s

was necessary because o f t h e need to avoid the undesirable damping characteristics of

ball bearings.

Strain gages in the system at locations shown in Figure 92 are used to measure angular

roll and ya w positions o f t h e model, and the roll input to rque applied to the model to

maintain forced oscillations

in

roll

at a

constant amplitude.

The

amplitudes

and

phase

angles of the strain gage outputs are obtained by passing t he signals through a sine-

cosine resolver located at the drive motor and gear driven b y t h e motor shaft. T h e r e

solver outputs

are

directly proportional

t o t h e

in-phase

and

out-of-phase components

of

the

strain gage signals.

By

vectorial summation

of

these quantities, osci llatio n

amplitudes are obtained a s well a s t h e relati onships necessa ry to determine t he forcing

torque and yaw position pha se angles. An electronic co unter operate d by a portion o f

the oscillating, roll position, strain gage signal is used to measure oscillation

frequency.

With th e model rigidly restrained against motion in yaw, measurements o f t h e roll

position, torque required for constant amplitude roll os cillations and the roll-input-

torque phase angle

are

made

at

vario us frequencies

by

means

o f t h e

resolvers. These

data,

obtained for both wind-on and wind-off (near vacu um) conditions, pro vide

Cj + a C ^ a t zero angle o f attack and

C

l / 3

at angles o f attack.

With the model unrestrained against yawing motion, t he model is oscillated in roll

at t h e y aw natural frequency to increase the yawing amplitude. When the desired ampli

tude is attained, t he drive motor is stopped abruptly and the time history o f t h e d e

caying yawing motion is recorded. Wind-on and wind-off data pr ovide C

n r

- C

n

^ and

C

n/3-



76

Although the system can be operated in two degree s of freedom to obtain cross de

rivatives C

J r

- Cjo and C.g , the experience related in Refere nce 77 indicates that,

where the system was used in this mode, the results were unreliable.

The detailed derivations and data reduction equations used with this system for

measuri ng dynamic lateral stability deriv ative s are presented in Reference 77.

3. 4 M A G N U S E F F E C T

The "Magnus force and moment", "Magnus effect", and "Magnus characteristics" are

synonymous terms which are given to the aerodynamic forces and moments that result in

the plane perpendicular t o the flow when a body is spinning in the cross flow produced

at angles of attack. The name, Magnus, was given to the phenomenon in honor of the

Berlin physicist . Prof essor G.Magnus, who around 1850 prov ed with a series of experi

ments the existence of a Magnus force

81

. L u c h u k

82

g ives a thorough historical account

of the Magnus effect.

Magnus force da ta have become m ore Important with the more general use of spin to

produce stabilization and to overcome manufacturing asymmetries of finned projectiles.

Dynamic instability may result on a spinning projectile if the Magnus moment becomes

too large

83

. Also, the usual assumption that Magnus effects may be neglected because

the spin rate is small is sometimes erroneous

8

".

3.4.1 Typ es of Magnus For ces

There are several different types of Magnus forces and moments produced by var ious

aerodynamic mechanisms. The different Magnus effects are usually classified as those

caused by the body, by canted

fins,

by driven

fins,

by base pressure, and by vorticity

from forward wing panel s acting on the tail surfaces. These different types of Magnus

effects will be discussed briefly and, where practical, illustrated sketches are given

(Fig.93).

The body Magnus effect at low angles of attack is caused by the shifting of the

thicker boundary layer on the leeward side in the direction of the s pi n

85

(Pig.93(a)).

Thus the body shape is effectively altered and is asymmetric. At large angles of

attack where the leeward side flow Is separated, the spin shifts the separated portion

to an asymmetric shape (Fig.93(b)). The body Magnus force both at small and large

angles of attack is in the negative Y directio n for posit ive (clockwise looking

upstre am) spin rates. Generally, the force acts on the rear part of the body.

On positive-spinning canted fins (Fig.93(c)) at an angle of attack, each fin, as it

rotates through the leeward side, will lose part of its force because of the blanketing

of the fin by the leeward wake

86

. At small angles of attack there will be a net nega

tive side force which will tend back toward zero as the angle of attack is increased.

Also on canted fins, there is a Magnus moment, which ha s been pointed out by Bento n

87

,

caused by the axial component of fin normal force.

On driven fins at zero cant angle, each fin loses part of its lift distribution as

it rotates through the leeward wake

(Fig.93(d)).

Thus there is a net side force in

the positive Y direc tion

86

.



77

A Magnus moment can be established on the

fins,

caused by unequal fin base press ures.

This moment is small and can only exist if the boundary layer on the fins is laminar.

Platou

8

" gives a thorough explanation of this fin base pressure effect.

Benton

88

shows that wing-tall interference can cause large Magnus effects on a finned

missil e whose wings are deflected into an aileron setting. Basically, the side loads

are caused by the vortices from the wind interacting with the tail surfaces.

3.4.2 Test Proc edur e

Determining the Magnus effects by wind tunnel tests necessitates the measurement of

the loads on the model while it is sp inning at an angle of attack. It is a well-known

fact,

as illustrated in Figure 94

(Ref.89),

that the Magnus effects are dependent on

the spin parameter

(Pb/2V).

Therefore, in addition to the usual parameters such as

Mach number, Re ynolds number, and model attitud e, the spin parameter (Pb/2V) must a lso

be monitored.

The spin paramete r for the ground te sts must be matched with the full-scale flight

tests.

For most wind tunnel facili ties, and especially high Mach numbers , the model

size is small, and the velocity is much lower than flight velocity because enthalpy

is not simulated. There fore much higher roll rate s are required for the wind tunnel

model than for the full-scale vehicle to properly simulate the flight spin parameter.

These required high model roll rates, as will be noted later, often introduce special

problems in the experimental work.

Tests by Plato u

89

at Mach 2 at angles of attack up to 5 degr ees showed that the

Magnus force increased with grit size

(Fig.95).

Other investiga tions

8

"'

90

also have

shown that the Magnus effects are dependent on the nature of the boundary layer, i.e.

whether it is laminar, turbulent, or transitional. There fore, since it is usually

not possible to keep the Reynolds number of the ground te sts the same as the full-

scale flight test because of tunnel limitations, additional consideration of the in

fluence of Reynolds number and boundary layer transition, to aid in the application

of ground test results to flight data, is required.

The test method employed usually depends on the drive system of the test unit. The

different test units will be discus sed in a later section. For a motor dr ive system,

the data can be recorded at the various angles of attack while holding the rotational

speed constant, or the speed range can be covered for each angle of attack. With a

turbine drive system, the data can be recorded continuously as the spin decays while

holding other test conditions constant. In the blowdown tunnels, where the data re

cording time is limited, d ata have been recorded while increasing the rotational speed

with the turbine

91

. When using the "power-up" technique, careful examination should

be made to ensure that the data are not influenced by the air exhaust from the turbine.

Problems arise because of gyroscopic effects when data are taken continuously while

varying the angle of attack with the model spinning. Although atte mpts have been made

to eliminate the gyroscopic forces present by corrective data reduction

82

, it is best

to avoid thi s test technique if possib le because of the uncertainty involved.

3.4.3 Repre sentative System s and Drive Units

Basically, the majority of the Magnus force test systems have the model mounting

on ball bearings whose inner races are fixed to an inner shell which mounts directly



78

or indirectly to a four-com ponent balance. A drive unit, either an electrical motor

or an air turbine, is used to spin the model and normally is an integral part of the

system enclosed by the model. Tachometer units, consisting of permanent magnets mounted

on the rotating unit and a wire-wound pole mounted on the stationary portion of the

assembly, are usually used to measure the model spin rates. As the model rotates, the

combination provides a signal which is proportional to the spin rate. Typical s ystems

for conducting Magnus force tests are shown in Figures 96 to 100.

Rotational speeds on the order of 100,000

r.p.m.

are som etimes required to simulate

the full-scale flight spin rate parameter (Pb/2Vo,). Such speeds are very difficult,

if not impossible, to achieve with a reasonable size electric motor. Motors which are

available, consistent with model size and power requirements, are very expensive and

require positiv e internal cooling. Cooling provi des added complications to the system

because of routing the coolant lines. If the motor is the connecting link between the

balance and model, extreme care must be taken to ensure that the coolant and power

lines are not stiff enough to take some load off the balance or to create small inter

ference side loads that would influence the accuracy of the data. Also, with large

motors some unbalance in the rotor may create vibration problems. However, for large

models with fins which have high damping characteristics, the variable frequency motors

are often the best system and frequently simplify the design.

A typical Magnus force test installation

92

, using an electric drive motor, is shown

in Figure 96. The electr ic motor and gear box is an integral part of the mode l and

is the connecting link between the stationary b alance and the rotating model t hrough

a flexible coupling at the upstream end. The inner shell mounted to the balance

supports the two ball bearings on which the model rotates.

High model spin rates, not generally available with electric drive motors, have

been obtained by NOL, BRL, and WADD through the use of an air turbine housed within

the model. The air turbines can be made smaller than commercially available electric

motors,

and weights are a fraction of those for equivalent electric motors, thus

eliminating low resonant frequencies which can interfe re with strain gage measu rements.

Also, they are more versat ile for des igning in that there i s no direct l inkage required

as with an electric motor where there must be a direct drive between model and motor.

Procedures for design of small air turbines for Magnus force balances are given in

Refer ence 93.

Magnus force test systems utilizing air turbine drive units are shown in Figures 97

to 100. For the sy stems shown in Figur es 97 to 99, the model is mounted on ball bear ings

which transmit loading to a strain gage balance, whereas in the system shown in Figure

100 a gas bearing is used for model support. All of these have the air turbi ne near

the base of th e model, with the air exhausting out of the model base into the wake

region.

Some designs have been developed where the turbine is forward of the balance

and a return loop is used for the air exhaust. The return loop system has the adv antage

that tests can be made while increasing speed with the turbine without fear of an effect

of the air exhaust on the Magnus measur ements. However, the turbine, when enclosed

for the return loop, is out of necessity smaller for a given size model, and therefore

will produce less torque. In addition, problems may arise when air is passed through

the center of the balance becaus e of the changing balance temperature, thus reducing

the accuracy of the data.



79

The system shown

in

Figure

98 is a

design used

by

WADD

and is

similar

to one

deve

loped

by

BRL. Rotational speeds

up to

90,000

r.p.m.

have been obtained with this unit.

The rotating assembly, consisting

o f t h e

model shell, turbine assembly,

and

tachometer

permanent magnets, mounts

on two

ball bearings which support

t h e

model

on a

strain

gage balance. More detailed information

on

this system

m a y b e

found

in

Reference

94.

The system shown

in

Figure

99 is

used

in the

AEDC-VKF supersonic

and

hypersonic

tunnels. T h e

model mounts

o n

ball bearings

o n an

inner shell which fi ts over

t h e

balance.

T h e

turbine

i s a t t h e

rear

o f t h e

balance and, afte r

it has

powered

t h e

model

to the

required spin rate,

it can be

disconnected from

th e

model

by a

sliding

clutch. The balance is a four-component moment t ype incorporated in the system a s a

separate unit. Thus, different capacity balances

c an

easily

b e

used without changing

the other parts

o f t h e

system.

An

air

bearing

h a s

been used

by NOL in the

system shown

in

Figure

100 to

provide

either

a

Magnus force

or

roll-dampi ng test capability with

t h e

same balance equipment

without modifications.

T he

model

is

spun

up

with

an air

turbine drive,

and

when

t h e

required spin rate

h a s

been achieved,

th e

turbine drive

i s

disconnected from

th e

model

by

an

air-operated clutch. Magnus measurements

are

made with

a

four-component balance

which

is

interchangeable

to

provide different load capabilities. This balance

is

more complex than

one

normally used with

a

ball beari ng support, since supply

air for

the

g a s

bearing

and

balance coolant must

be

routed through

t he

balance.

3.4.3.1 Be a r i n g s

A critical factor

in the

rotating mo del Magnus force mea suring system

is the

bearings,

since they must

in

some cases

be

capable

of

withstanding rotational speeds

o f

100,000

r.p.m. or

more.

It is a

general p ract ice

to

mount

t h e

bearings

so

that they

are

sub

jected

to a

thrust load

at all

times. Thi s preload confines

t h e

ball rotation

to one

axis

and

prev ents brlnelling

o f t h e

race s that could occur from dynamic axial movement.

Temperature rises

in

bearings result from operation

at

high speeds

and

these should

be monitored

to

given

an

indication

o f t h e

bearing p erform ances.

Bearings used

in the

WADD M agnus force balance system

are run at

10,000

and

20,000

r.p.m.

and

temperatures

are

continuously monitored until

t h e

temperature levels

off.

A plateau

in

temperature

at the

forward

end of the

support strut

i s an

Indicat ion that

the bearings

are in

good condition

and

that

t h e

balls have worn

a

track cycle.

Temperatures

a re

continually monitored during

tests, and

excessive rises above about

150°P indicate that

t h e

bearings should

be

disassemb led, rinsed

in

solvent,

and re-

greased. Experience indicates that bearing temper atures much above 150°F signal

t h e

onset

of

freezing

o f t h e

bearing assembly.

As noted earlier,

th e

Magnus force system shown

in

Figure

98 was

designed

by

WADD

using much

o f t h e

experience from BRL.

A

series

o f

tests

wa s

made

by

WADD

to

extend

the

u s e o f t h e BR L

developed bearings

to

rotational speeds considerably above

t h e

40,000

t o

50,000

r.p.m.

specified

b y t h e

manufactur er

a s t h e

upper limit.

Bearing failures were generally found

t o

occur when

t h e

inner race temperat ures

reached

200 to

250°P. Because

o f t h e

heat generated, carbon parti cles from

the

lubricant

and phenolic retainer were produced,

and

under some conditions

t h e

carbon particles



80

scored the

races,

produced additional heat, and ultimately caused failure. This problem

was overcome by a conditioning procedure devised by WADD (Ref.94) which consisted of

bearing

run in to

seat

the

balls

and

also form

a

glazed surface

on the

phenolic retainer

at critical contact points.

Lubrication

of

bearings

is

dep endent

for the

most part

on the

type

o f

operation they

will be subjected to. Fo r intermittent service, where the bearings are allowed to

coo'l, lubrication consists

o f

light applications

o f

Aircraft

and

Instrument Grease ,

MIL-G-25760

and a few

drops

o f

Instrument L ubrica ting Oil, MIL-L-6085A.

N o

grease

seals are used, since a portion o f th e grease remains o n the bearings f or intermittent

operations.

Conditioning

o f t h e

bearing

in

this manner

h a s

resulted

in a

life span

of

10 to 50

runs with c leaning between every

10 to 20

spinups.

Continuous ope ration

at

high rotational speeds

o f

70,000

to

90,000

r.p.m.

requires

continuous bearing cooling to prevent excessive h eating and bearing freeze up. The

WADD system shown

in

Figure

98 was

designed with

a

bleed line from

the

main

air

supply

to provide cooling

air to the

bearings. Reduction

o f th e

front bearing temperature

was effectively achieved; however, c ooling of the rear bearing wa s ineffective. The

bearing temperatures

a re

shown

in

Figure

101 fo r

various throttle value settings

on

the

air

cooling

air

supply. Although this method

of

cooling

is

effective,

it

reduced

the number of runs between lubrica tions because t h e lubricant wa s carried away with

the cooling air. A final d evelop ment in the system included mist lubrication by an

alemite system to maintain lu brica tion while cool ing

t h e

bearings. Details

of the

lubrication system

are

shown

in

Figure

102.

Bearings with phenolic retainers were used exclusively in the WADD Magnus force

s y s t e m

98

, since these were found

to

possess

the

best proper ties

for

rapid acceler ations

and high speeds. Radial-type ball bearings provided t he best combinatio n with t h e

inner race spring-loaded.

A

preci sion bear ing with controlled minimum play

wa s

used

for

the

front bearing

and a

class

3

bearing

for the

rear bearing. Stainless steel

bearings were used at the rear station bec ause the rear bearing ope rated at higher

temperatures

a nd

greater dissipation

o f

heat

wa s

achieved with st ainless steel bear ings.

3 . 4 . 3 . 2 M od el C o n s t r u c t i o n

Dynamic conditions c an exist at high rotational speeds which render Magnus force

data unreliable, increase bearing wear, cause fouling in the air turbine dri ve system,

and also reduce turbine performance. These adverse conditions

may be

eliminated

or

minimized by dynamic balancing of the model. T h e balancing should be done carefully

and to a high degree of accuracy , since only a very small unbalance can significantly

influence

t h e

accuracy

o f t h e

data.

The ideal model

is one

with

lo w

weight

and

high strength

and

designed

to

maintain

good fits regardless

o f

temperature gradients, normal loading,

and

spin rates.

A

material combination used frequently i s 75ST alum inum and heat-treated stainless

steel. This combination h a s l o w weigh t and high strength, and good fits may be

maintained.

It is

particularly important that

a

minimum

of

parts

be

used

and

that

all

mating parts

be

made

to

close tolerances,

so

that dynamic balancing will

not be re

quired after each assembly.



81

3 . 4 . 3 .3 S t r a i n G age Ba l a n ce

Four-component strain gage balances are required to record fo rces a nd moments

generated in the pitch and yaw planes by model spin. Since Mag nus force s are at the

maximum

1/10 to 1/20 of t he

normal force

and can be

much smaller,

t h e

Magnus balance

must be stiff enough t o withstand l arge normal- forc e l oads while at the same time

maintain t h e sensitivity required to accurately measure the small side loads. N O L

developed a balance with eccentric side beam s so that when th e balance is subject to

yaw loads a secondary bending moment i s induced in the eccentric column which acts a s

a mechanical amplifier

79

. An AEDC- VKF ba lance of this type is shown in Figure 103(b).

The AEDC-VKF balance h a s a normal force t o side force capacity ratio o f 10. Balances

have also been developed which have extremely small outrigger side be ams that act as

pure tension members. Normal force t o side force capacities up to 25 have been

obtained with this design (Fig.103(a)).

Satisfactory results

can be

obtained with

t h e

forementioned meth ods utili zing co n

ventional wire strain gages. However,

an

attractive method

for

improveme nt

o f

Magnus

force measurements

is to use

semiconductor strain gages

a s

substitutes

for

wire-wound

gages. Wire gages have

a

gage factor

of

2, wherea s semi conductor ga ge fa ctors

are on

the order

of 60 to

200. Uti lizatio n

o f

semiconductor strain gages would produce

voltage outputs 30 to 100 times greater than t he conventional wire ga ges, since t h e

voltage output for a given stress is directly proportional t o t h e gage factor.

The feasibility of using semiconductor strain gages for Magnus force measurements

has been investigated by Platou o f BR L

96

. Resu lts from these investigatio ns show that

the semiconductor strain gage is very sensitive t o temperature. These adverse effects

can be diminished by temperature compensation provided factory matched gages are used

in each bridge. In additio n, th e usual precauti ons in mounting, bonding, and soldering

connections should

be

strictly adhered

to for

good performance.

An

example

o f t h e

increased sensitivity obtained using semiconductor strain gages is shown in Figure 104.

3. 5 P R E S S U R E M E A S U R E M E N T S

Most o f t h e interest in the last 20 years o r more h a s been in the development of

instrumentation for measuring total forces a nd moments o n a model rather than local

values.

However, some efforts have been directed toward measuring forces and moments

on segmented wings at supersonic s pe e ds

97

'

98

t o obtain s ection static aerodynamic

characteristics. Distributions o f forces and moments, as well a s t h e total loading,

are o f considerable value, especially to validate theories. Detailed study of aero

dynamic damping

m a y b e

enhanced

by

knowledge

of

local loading especiall y

for the

cases

of models having flow separations, boundary layer transition which varies with angle

of attack,

and

model s constructed with

low

temperature ablation simulators releasing

gas

to the

boundary layer.

Measurements o f detailed pres sure distributions on an oscillating model and integra

tion o f t h e distributi ons provi de a means to obtain local loading relativ e t o t h e

motion o f t h e model and in-phase and quadrature lift and moment.

With

the

exception

of

some unreported tests

at

AEDC-VKP

on an 8-degre e

cone

a t

Mach

4

and

measurements

o f

pressure distributions

on an

oscillating wedge

at

Mach

1.4 and

1.8 (Ref.99), most

o f t h e

experimental wo r k

1 0 0 - 1 0 2

h a s

been conducted

at low

subsonic



82

and transonic speeds. The auth ors reporting on these tests have described the trans

ducer installations, test techniques, and data reduction in detail, and therefore their

work has been used liberally in preparing the following sections.

3.5.1 Tra nsduce r Instal lati on

The most important considerations in developing a system for measuring pressures on

an oscillating model are the transducer characteristics and installation.

Some of the desirable charact eristics for a transducer to measure time-varying

pressures on oscillating models, in addition to resolving power a r e

1 0 3 , 1 0

":

(a) Transduce r outpu ts should be insensitive to the inertia loads imposed under

oscillatory conditions.

(b) Transducer outputs should be linear with applied pressure.

(c) The tra nsducer should be small to allo w installation of sever al units spaced

for adequate coverage and installation close to the measuring point to avoid

a lag in response.

(d) The resonant fr equencies of the transduce r com ponents should be well abo ve the

frequency range for pressure mea surements.

(e) The diameter of the pressure receiver should be smaller than the half wave

length of the press ure wave.

(f) The tra nsducer should be insensitive to environmental t emperat ure change.

In practice, there may not be one transducer with characteristics which embody all

of these requirements for a particular application, and therefore a practical design

must be developed with the sacrifice of some of the desirable features. In all cases,

however, the transducer should be selected giving major weight to both high resolving

power and high frequency response.

The next important consideration following selection of a transducer using these

guid elines is the installation. The method of installation is governed to a large

extent by the size and geometry of the model and the transducer response characteristics.

The transducer should be located near enough to the surface to accurately measure

rapidly varying pressures. Ideally, it is most desirable to have the transducer dia

phragm right at th e measuring point to have zero tube length and transducer volume.

This, however, is not very feasible for most installations, and, therefore, the trans

ducer must be connected to the pressure orifice by a connecting tube. Transducers

used are usually the differe ntial-pressure type which require a reference pressure

system on one side of the transducer diaphragm.

Connecting tubing between the orifice and transducer must be selected with proper

regard to the important variables - namely frequency response and phase shift.

3.5.2 Pressu re System Response

In the range of typical tubing lengths ranging from 0.1 to 6 in., tubing diameters

ranging from 0.015 to 0.125 in. and transducer volumes from 0.01 to 0.1 cubic in.,

D a v i s

10

"

notes that t he viscous and thermal effects predominate, with a dependency o n



83

the frequency o f oscillation. An example of the magnitude pre ssure response and phase

angle variation with frequency from Reference

104 is

presented

in

Figure

105 to

illus

trate these effects.

A method

is

presented

by

D a v i s

10

"

for

calculating

the

dynamic response

of a

system

and phase angle for a sinusoidal pressure input. The assumptions used in the formu

lation, based on the work of Ref er ences 105. 106 and 107, include

(a) constant press ure over the tube cross section,

(b) constant press ure over the gage volume,

(c) laminar flow throughout the system,

(d) small pressu re perturbations.

The expression for the pressure at the transducer or gage is given by

p

m ^ 1

P i V

e

l

2

co

2

V

E

8 p l

2

1

1 - - * — 5 - + i c o - Z -

V

t

c

2

V

t

p r

2

This equation should only be used as a rough g uide, since the pressure system has

been represented by a differential equation with frequency-dependent coeff icients.

Exact character istics

of a

pressure system should

be

determined

by

calibration.

3.5.3 Cal ibr ati on Tec hnique

Calibration of a transducer t ubing syste m which is to be used for unsteady pressure

measurements is necessary because of the numerous uncertainties which exist in the

analytical treatment.

The dynamic response calibration o f a pressure transducer and tubing system is

characterized by frequency response curves such as those in Figure 105 showing magnitude

and phase angle. Specialized calibrat ion equipment is necessary for calib ratio ns which

must be made over a wide range o f frequencies. Techniques for calibrating microphones

are directly applic able to calibration of pressure transducers and are used here to

outline general specifications for pressur e transducer calibration equi pm ent

10

"'

1

.

(i )

Th e

pressur e oscillation amplitude should

be

constant o ver long per iods

and

at different frequencies and easily variable over the range of interest.

(ii) The fr equency of oscill ation shoul d be continuously varia ble over the frequency

range of interest.

( H i ) Local conditions such as mean pres sure and temperature should be variable

over t he range of interest and known.

(iv) Th e wave form o f t h e pressur e o scillation should b e relatively free o f harmonics.

(v) Th e calibr ation equipment should ha ve a standard pickup near the point of

measurement.



84

For dynamic calibrations at f requencies of 25 c/s and lower, a system of the type

shown

in

Figure

106

will provide

a

sinusoidal, oscillating pressure. Oscillation

of

the pump plunger by motion of the motor drive cam pressurizes the diaphragm and pro

duces a n oscillat ing pres sure input to the transducer. The phase lag between the

pressure application

and

transducer response

is

determined from comp onents

of the

transducer output in phase and in quadrature with the motion of the plunger. The mean

pressure in the calibration may be adjusted for the pressure range of interest.

For frequencies between 10 and 100 c/s, a piston-type calibrator may be used;

however, at higher frequencies up to 5000, D a v i s

10

" recommends a cam-type pulsator of

the type shown in Fig ure 107. The pulsator chamber is supplied with high pre ssure air

which exits through an opening controlled by the rotating cam. The space between the

cam

and the

exit

is

adjusted until

the

wave form

i s

sinusoidal. Reference pressure

gages and the transducer and tube system to be calibrated are mounted to sense the

pressure within the chamber of the pulsator.

R E F E R E N C E S

1. Lanchester , F. W.

2.

Bryan, G.H.

3. Dura nd,

W.P.

(Editor)

A e r o d o n e t i c s . Constable, London, 1908.

S t a b i l i t y i n A v i a t i o n .

MacMillan, 1911.

A erodynam i c Theo ry .

Vol.V., Div.N, Springer, Berlin,

1934-1936.

4. Valensi, J.

A Review of the Techniques of Measur ing O sc i l l a to ry

Aerodynamic Fo rce s and Moments on Models O sc i l la t i n g in

Wind Tunn els in Use on the Co nt i ne nt .

AG 15/P6, Papers

presented at the Fifth Meeting of the AGARD Wind Tunnel

and Model Testing Panel, Scheveningen, Netherlands, May

3-7, 1954.

5.

6.

7.

8.

Arnold, Lee

Bratt, J.B.

Dayman,

B.

, Jr

Ames Research

Center Staff

Dynamic Me asurem ents in Wind Tu nn els .

AGARDograph 11,

August 1955.

Wind-Tunnel Techniques for the Measurement of O sc i l l a to ry

D e r i v a t i v e s . ARC R & M 3319, ARC 22 146, A ugu st 1960.

Fr ee -F l ig h t Te s t in g in H igh-Speed Wind Tun nel s . AGARDo

grap h 113. Mar ch 1966.

B al l i s t i c - R an g e T ech n o lo g y . AGARDograph in preparation.



85

9. Orlik-Ruckemann, K.

10.

P e rk ins, C D .

Hage,

R.P.

11. E tkin, B.

12. Babister, A. W.

13. Eastman, P. S.

14.

Eastman, P.S.

15.

Young, W.E.

16. Wittrick, W.H.

17. O'Neill, E. B.

Phillips, K. A.

18. Hodapp, A.E.

19.

Regan, P.J.

Horanoff, E.

20. Richardson, H. H.

21. Welsh, C.J.

et al.

22. Mechanical Technology,

Inc., (Edited by

N.P.Rieger)

23. Hall, G. Q.

.

Osborne, L.

A.

Methods of Measurement of Aircraf t Dynamic Stabi l i ty

D e r i v a t i v e s . National Research Council, Canada, LR-254,

July 1959.

A i r p l a n e P e rf o rm a n c e, S t a b i l i t y a nd C o n t r o l .

John Wiley,

New York, 1949.

D y n a m i c s o f F l i g h t . Joh n Wiley , New York, 1959.

A i r c r a f t S t a b i l i t y a nd C o n t r o l . Pergamon Press, 1961.

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101. Molyneux, W. G.

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Analy t i ca l Methods for Determin ing Spec i f i c Damping Energy

C o n s i d e r i n g S t r e s s D i s t r i b u t i o n .

WADC TR

56-44,

June 1957,

AD 130 777.

13.

Robertson, J.M.

Yorgladis, A. J.

I n t e r n a l F r i c t i o n i n E n g i n e e ri n g M a t e r i a l s . Journal of

Applied Mechanics, Vol.13, No .3, September 1946, pp.A173-

A182.

14.

Thorn,

R.P.

B u i l t - i n - D a m p i n g .

December 9, 1965.

Machine Design, November 25,

1965, and

15.

Ting-Sul Ke Exp er ime nta l Evidence of the V iscous Behavior o f G ra in

B o u n d a r i e s i n M e t a l s . Phy sics Review, Vol.71, 1947,

pp.533-546.

16.

Ting-Sui Ke I n t e r n a l F r i c t i o n o f M e t a l s a t V ery H i gh T emp er a t u r e s .

Journal o f Appl ied Ph ysi cs , Vol.21, 1950, pp.414-419.

17. Tsobkallo, S.O.

Chelnokov,

V.A.

New Method fo r the De ter m ina t io n of True Damping of

O s c i l l a t i o n s i n M e t a l s . Zhurnal Tekhnickeskoi Fiziki,

Vol.24, No .3, Mar ch 1954, pp. 499-510.

18. Ungar.

E.

E.

Hatch, D.K.

19.

von Heydekampf, G. S.

H i g h - D amp i n g M a t e r i a l s .

1961, p p . 4 2 - 5 6 .

Product Engineer ing , Apr i l 17 ,

Damping Ca pac i ty of Ma te r i a l s .

P a r t I I , 1 9 3 1, p p . 1 5 7 - 1 7 1 .

ASTM Proceedings, Vol.31,

20. Walsh,

D.P.

Jensen, J. W.

Rowland, J. A.

Vibra t ion Damping Capaci ty of Var ious Magnes ium Al loys .

US Department of the Interior, Bureau of Mines, Rolla,

Misso uri, 1962, Report o f Investigation 6116 (Unclassified).



95

Derivatives

Due to

Pitch

and

Rate

o f

Pitch-Theory

1.

A G e n e r a l I n v e s t i g a t i o n o f H y p e r s on i c S t a b i l i t y a n d

Co nt ro l Under T rimmed F l ig h t Co nd i t ion s .

Plight Sciences

Labora tory, Inc., Buffalo, New York, R-63-011-104, June

1964.

A D 434 804.

2. Acum,

W.

E.A.

Note on the Effec t of Thickn ess and Aspect Rat io on the

Damping o f P i tch ing Osc i l la t ions o f Rec tangu la r Wings

Moving a t Supersonic Speeds.

ARC CP 151, 1953.

3. Acum,

W. E.

A.

4. Alden,

L.H.

Schindel,

L.H.

The Comparison of Theory and Experiment for O sc i l la t i n g

Wings. ARC CP

681,

1963.

The L i f t, R ol li ng Moment, and P it c h in g Moment on

Wings

in Nonuniform Supersonic Flow.

Journal of the Aeronautical

Sciences, Vol.19, No. l, January 1952, p.7.

5. Allen,

H.J.

Es tim ati on of the For ces and Moments Act ing on I nc l i ne d

Bodies o f Re vo lu t ion o f High F in ene ss R a t io . NACA RM

A9126. November

1949.

6. Allen,

H.J.

Perkins, E. W.

A Study o f E f fec ts o f V isc os i ty on F low Over S len der

I n c l i n e d Bo d i e s o f Re v o l u t i o n . NACA Report 1048, 1951.

7.

Allen,

H.J.

M o ti on of a B a l l i s t i c M i s s i l e A n g u l a r l y M i s a l i n e d w i t h

the F l i g ht P ath Upon En ter ing the Atmosphere and I t s

Ef fe ct Upon Aerodynamic H ea tin g, Aerodynamic Loa ds, and

M i s s D i s t a n c e .

NACA

T N

4048, Oc tob er

1957.

8. Bobbitt,

P.J.

Malvestuto,

P.S. Jr

Es tim at io n of Force s and Moments Due to R ol l i ng fo r

S e v e r a l S l e n d e r - T a i l Co n f i g u r a t i o n s a t S u p e r s o n i c S p e e d s .

NACA

T N

2955. Jul y

1953.

9. Bourcier, M.

A pp lic at io n de la Methode Asymptotique au Ca lcu l de s

De r ivees Aerodynamiques de S ta b i l i t e en Hyperson ique pour

u n C o n e C i r c u l a i r e .

ONERA, Note Technique No.97,

1966.

10. Brown,

C.E.

Adams. M.

C

Damping in P i t ch and R oll of Tri an gu lar Wings a t Super

s o n i c S p e e d s .

NACA Report 892, 1948.

11. Charwat,

A.

P.

The S ta b i l i t y o f Bod ies o f Revo lu t ion a t Very High Mach

Nu m b e r s . Jet Propulsion, Vol.27, No. 8, Part 1, August

1957, pp.866-871.

12. Chester,

W.

Sup erso nic Flow Pa st Wing-Body Com binations.

Quarterly, Vol. IV, Aug ust 1953, p. 287.

Aeronautical

13. Coakley,

T.J.

Laitone, E. V.

Maas,

W.

L.

Fundamental Ana lys i s o f Var ious Dynamic S ta b i l i ty P roblems

f o r M i s s i l e s . Institute o f Engineering Research, Univer

sity

o f

Califor nia, Berkeley, C alifo rnia, Ser ies No.176,

Issue No .

1,

June 1961,

AD 259 127.



96

14. Corning, G.

15. Cunningham, H.J.

16.

Pink, M.R.

Carroll, J.B.

17. Dorrance,

W.H.

18.

Pink,

M.R.

19. Fink, M.R.

20. Pink, M.R.

21. Friedrich, H.R.

Dore,

P.J.

22. Garner,

H.

C

23. Garner,

H.

C

Acum, W.

E.

A.

24. Garner,

H.

C.

Acum, W.

E.

A.

Lehrian, D.E.

25. Garrick, I.E.

Rubinow, S.I.

26.

Gilmore, A.

Seredinsky, V.

MacKenzle,

D.

The P i t ch-Y aw-R ol l C oupl ing Problem of Guided Mi ss i l e s

a t H igh Angles of P i t ch . NAVWEPS 7383, 1961.

To ta l Li f t and P i t c h in g Moment on Thin Arrowhead

Wings

O s c i l l a t i n g i n S u p e r so n i c P o t e n t i a l F lo w .

NACA TN

3433,

May 1955.

H y p er s o n i c S t a t i c an d D yn amic S t a b i l i t y o f W i n g -F u se lag e

C o n f i g u r a t i o n s .

United Aircraft Corpor ation Research

Laboratory, C910188-17, January 1965.

Nons teady Sup er son ic F low About Poin ted Bodies of

R e v o l u t i o n .

Journal

of the

Aeronautical Sciences,

No.8,

August 1951, pp.505-511.

Vol.18,

H y p er s o n i c D yn amic S t a b i l i t y o f B i co n i c B o d i e s .

United

Aircraft Co rporation, Research Laboratory, Co nn., UAR- E20,

March 21, 1965.

H y p er s o n i c D yn amic S t a b i l i t y o f S h a r p an d B l u n t S l en d e r

C o n e s . United Aircraft Corporati on, Research Laboratory,

Conn.. UAR- C111, July 1964.

A Shock-Exp ans ion Method for Ca lc u l a t io n of Hy person ic

D y n a m i c S t a b i l i t y . United Aircraft Corporat ion, Research

Laboratory, Conn., C-910188-4, March 1964.

The Dynamic Motion of a M is s i le D escen ding Through the

A t m o s p h e r e .

Journal

of the

Aeronautical Sciences, Vol.22,

Septe mbe r 1955, pp.628-632.

Num er ica l Aspe c t s o f Uns teady L i f t i ng Sur fac e Theory a t

S u p e r s o n i c S p e e d s .

ARC CP 398.

P r o p o s ed A p p l i c a t i o n o f L i n ea r i zed T h e o r e t i c a l F o rmu l ae

t o G en e r a l M ech an i zed C a l cu l a t i o n s f o r O s c i l l a t i n g W ings

a t S u p e r s o n i c S p e e d s . NP L Aero. Repor t 388, 1959.

C o m p a ra t iv e C a l c u l a t i o n s o f S u p e r s o n i c P i t c h i n g D e r i v a t i v e s

Over a Range of Frequ ency Pa ram ete r .

ARC CP 591, 1962.

T h eo r e t i c a l S tu d y o f A i r F o r ce s on an O s c i l l a t i n g o r

St ea dy Th in Wing in a Su pe rso nic Main Stre am . NACA Report

872, 1947.

I n v e s t i g a t i o n o f D yn am ic S t a b i l i t y D e r i v a t i v e s o f V e h i c l e s

Flying at Hypersonic Veloci ty . Grumman Aircraft Engineer

ing Corporation, ASD-TDR-62-460, September 1963.

AD 421

874.



97

27. Harmon, S. M.

S t a b i l i t y D e r i v a t i v e s a t S u p e r s o n i c S p e e d s o f T h i n

Rectangular Wings with Diagonals Ahead of Tip Mach Lines.


28.

Harmon, S. M.

Jeffreys, I.

29. Harris, G. Z.

Th eo re t i ca l L if t and Damping I n Ro ll of Thin Wing with

A rb i t r a r y Sweep and Taper a t Superson ic Speeds , Sup erson ic

L e a d i n g a n d T r a i l i n g E d g e s . NACA TN 2114, May 1950.

The Ca l c u l a t i o n o f G e n e r a l i s e d F o r c e s on O s c i l l a t i n g

Wings in Supe rson ic F low by L i f t i ng Sur fa ce Theory .

Royal Aircraft Est ablishment, Technical Report

65078,

1965.

30. Henderson, A. Jr

P i t c h i n g - M o m e n t D e r i v a t i v e s

C

m

„

and

C-*.

a t Superson ic

° mq ma

r

Speeds for a Slen der-D elta-W ing and Slende r-Bod y Com

b i na t io n and Approx imate So lu t ion s fo r Broad-Del ta-Wing

and Sle nd er- Bo dy Co m bin atio ns. NACA TN

2553,

December

1951.

31 .

H j e l t e , P .

32.

Jones, A.L.

Alksne, A.

M e t h o d s f o r C a l c u l a t i n g P r e s s u r e D i s t r i b u t i o n s o n O s c i l l a

t ing Wings of Delta Type a t Supersonic and Transonic

S p e e d s .

KTH

Aero

TN 39 ,

S to c k h o lm ,

1956 .

The Damping Due to R o l l o f T r i a n g u l a r , T r a p e z o i d a l , a n d

R e l a t e d P l a n F o r m s

i n

Su pe rs on ic Flow . NACA

TN 15 48 ,

March

1948 .

33. Jones, W.P.

A e r o f o i l O s c i l l a t i o n s a t H ig h M ean I n c i d e n c e s . ARC R & M

2654,

1948.

34. Jones, W.P.

O sc i l la t i ng Wings in Compress ib le Subson ic F low. ARC

R & M

2855,

October 1951, (ARC

14.336).

35. Jones, W.P.

Summary o f

F o r m u l a e

a n d

N o t a t i o n s U se d

i n

Two-Dimensiona i

D e r i v a t i v e T h e o r y . ARC R & M 1958, August 1941.

36. Jones. W.P.

Supersonic Theory for Osci l la t ing Wings of Any Planform.

ARC R & M

2655,

1948.

37. Jones, W.P.

The Ca l c u l a t i o n o f A er od yn am i c D e r i v a t i v e Co e f f i c i e n t s

fo r Wings of Any Planf orm in Non-Uniform Mo tion.

ARC

R & M 2655, 1948.

38. Jones, W.P.

The In f luence o f Th ickness -Chord Ra t io on Superson ic

D e r i v a t i v e s f o r O s c i l l a t i n g A e r o f o i l s .

ARC R & M 2679,

1947.

39. Kelly. H.R.

The Es t im ati on of Norm al-Force, D rag, and Pitching-Moment

Co e f f i c i e n t s f o r B l u n t - Ba s e d Bo d i e s o f Re v o l u t i o n a t

L a r g e An g le s o f A t t a c k .

Journal

of the

Aeronautical

Sciences, Vol.21.

Augu st 1954, pp. 549-555.



98

40.

Kind, R.J.

Orl ik-Ruckemann, K.J .

41.

Labru jere , H .E .

S t ab i l i t y D e r i v a t i v e s o f S h a r p C o n es i n V i s co u s H y p e r s o n i c

Flow. N at i on al Rese arch Co unc i l , Canada, Repor t LR-427,

1965; a l so AIAA Jo ur na l . V ol .4 . No.8 , 1966, pp .14 69-14 71.

D e t e r m in a t io n o f th e S t a b i l i t y D e r i v a t i v e s o f an O s c i l l a t

ing Axisy mm etric Fu se la ge in Su pe rso nic Flo w. NLL TN W. 13,

Amsterdam, 1960.

42.

Laitone. E. V.

Effec t o f Acce lera t ion on the Longi tud ina l Dynamic

S t a b i l i t y o f a M i s s i l e .

ARS

Journal,

Vol.29, February

1959, p. 137.

43. Lawrence,

H.R.

Flax,

A.H.

44.

Lawrence,

H.R.

Gerger, E. H.

45. Lehrian, Doris E.

46. Lehrian, Doris

E.

Wing-Body I n te r f e r en ce a t Subsonic and Sup er son ic Speed s -

Surve y and New Dev elopm ents .


Sciences, Vol.21, May 1954, pp .289-324.

The Aerodynamic Fo rce s on Low Aspect R at io Wings O s c i l la t

ing in an In com pres s ib le F low.


Sciences, Vol.19, Novembe r 1952, pp.769-781.

C a l c u l a t i o n o f S t a b i l i t y D e r i v a t i v e s f o r O s c i l l a t i n g

Wings. ARC R & M

2922, Feb ruary 1953.

(ARC 15,695).

C a l c u l a t i o n o f S t a b i l i t y D e r i v a t i v e s f o r T a pe re d Wings

o f H ex ago n a l P l an f o rm O s c i l l a t i n g i n a S u p e r s o n i c S t r eam.

ARC

R & M

3298.

1963.

47. Lehrian. Doris E.

Smart,

G.

48. Lighthlll,

M.J.

49. Malvestuto,

P.

S.

Jr

Hoover, Dorothy M.

50. Malvestuto,

P.

S. Jr

Hoover, Dorothy M.

T h e o r e t i c a l S t a b i l i t y D e r i v a t i v e s f o r a S y m m e t r i c a ll y

Tapere d Wing at Low Su pe rso nic Sp eed s .

ARC CP 736, 1965.

O s c i l l a t i n g A i r f o i l s a t H ig h M ach N umber. Journal of the

Aeronautical Sciences, Vol.20, June 1953. pp.402-406.

L i f t an d P i t c h i n g D e r i v a t i v e s o f T h i n S w ept back T ap e r ed

Wings wit h Stream wise T ips and Sub sonic L eading Edges

a t S u p e r s o n i c S p e e d s .

NACA

TN

2294, Feb ruary

1951.

Su pe rso nic L if t and P i t c h in g Moment of Thin Sweptback

Tapered

Wings

P r o d u ced by C o n s t an t V e r t i c a l A c ce l e r a t i o n ,

Sub sonic Leading Edges and Sup er son ic Tr a i l i ng Edge s .

NACA

T N

2315, Mar ch

1951.

51. Malvestuto, P. S. J r

Margolis,

K.

52.

Malvestuto,

P. S. Jr

Margolis,

K.

Ribner,

H.

S.

T h eo r e t i c a l S t a b i l i t y D e r i v a t i v e s o f T hi n S w ept back W ings

Tapered to a Po in t w i th Sweptback or Swept forward Tr a i l i ng

Edges fo r a L im i ted Range of Su per s onic Sp eeds . NACA

Report 971, (Supersedes NACA TN 1761). 1950.

Th eo re t i c a l L i f t and Damping in Ro l l a t Supe r sonic Speeds

of Thin Sweptback Tapered Wings with Streamwise Tips ,

S u b s o n i c L ead i n g E d g es , and S u p e r s o n i c T r a i l i n g E d g es .

NACA Report 970,

1950.



100

66. Morikawa, G.

67.

Mueller. R.K.

68.

Munch, J.

69.

Platzer, M.F.

Hoffman,

G.

H.

70.

Neumark, S.

71.

Nielsen, J.N.

72.

N ielsen, J.N.

Kaattari,

G.

E.

73. Ohman, L.H.

74.

Ohman, L.H.




Crowe, C.

78.

Pitts, W. C

Nielsen, J.N.

Kaattari,

G.

E.

S u p e r s o n i c W i n g - B o d y - T a i l I n t e r f e r en ce . Journal of the

Aeronautical Sciences, Vol.19, May 1952, pp.333-340.

The G r a p h i c a l S o l u t i o n o f S t a b i l i t y P r o b le m s .

Journal

of the Aeronautical Sciences,

Vol.4,

June 1937, pp. 324-331.

An E xamp le f o r t h e P r a c t i c a l C a l cu l a t i o n o f S u p e r s o n i c

F lo w F i e l d s P as t O s c i l l a t i n g B o d ie s o f R ev o l u t i o n b y a

M e t h o d D u e t o S a u e r .

AFOSR TN 58-881, August 1958.

Q u as i - S l en d e r B o d y T h eo r y f o r S l o w l y O s c i l l a t i n g B o d i e s

o f R ev o l u t i o n i n S u p e r s o n i c F lo w .

NASA TN D-3440, June

1966.

Two-Dimens ional Theory of Osc i l l a t ing Aerofo i l s w i th

A p p l i c a ti o n t o S t a b i l i t y D e r i v a t i v e s .

RAE Report Aero

2449,

ARC 14,889. 1951.

S u p e r s o n i c W i n g - B o d y I n t e r f e r en ce .

of Technology Thesis, 1951.

California Institute

The Ef fe c t s o f Vor tex and Shock-Expans ion F ie ld s on

P i t c h a nd Y aw I n s t a b i l i t i e s o f S u p e r s o n i c A i r p l a n e s .

Institute o f the Aeronautical Sciences, Preprint 743, 1957.

A S u r f a c e F lo w S o l u t i o n a nd S t a b i l i t y D e r i v a t i v e s f o r

Bodies of Re vo lu t io n in Complex Sup er so nic F low. Part I:

T h eo ry an d Some R e p r e s en t a t i v e R es u l t s . National Research

Council, Canada, Report

LR-418,

1964.

A S u r f a c e F lo w S o l u t i o n a nd S t a b i l i t y D e r i v a t i v e s f o r

Bodies o f Re vo lu t i on in Complex Sup er son ic F low.

Part II;

Re su l t s fo r Two Fa m i l i e s o f Bodies of Re vo lu t io n for

1.

1 < M < 5 . National Research Council, Canada, Report

LR-419, 1964.

Eff ec t o f Wave R ef le c t io ns on the Uns teady Hyperson ic

Flow Over a Wedge .

AIAA Journal. Vol.4, No.10, 1966.

Stab i l i ty Der iva t ives of Sharp Wedges in V iscous Hyper

s o n i c F l o w .

National Research Council, Canada, Report

LR-431,

1965; also (slightly extended) AIAA Journal,

Vol.4, No .6, 1966, pp.1001-1007.

Compari son of Var ious Exp er ime nta l and Th eo re t i ca l R es u l t s

f o r S t a t i c a n d D yna mic L o n g i t u d i n a l S t a b i l i t y D e r i v a t i v e s

of Some Wing-Body Co nf ig ur at io ns at Su pe rson ic Spee ds .

NAE Laboratory Report (to be

published) .

Li f t and Cen ter o f Pr es s ur e of W ing-Body-Tai l Combina t ions

a t S u b s o n i c , T r an s o n i c an d S u p e r s o n i c S p eed s .

NACA

Rep ort 1307, 1957.



101

79.

Platzer. M.F.

A N o t e on t h e S o l u t i o n f o r t h e S l o w l y O s c i l l a t i n g Bo dy

o f R ev o l u t i o n i n S u p e r s o n i c F lo w . George C.Marshall

Space Flight Center, M TP-A ERO- 63-28, April 23, 1963.

80.

Quinn, B.P.

B la s t Wave Ef fec t s on the P i t c h i ng of B lunt Cones .

Paper

37, Vol.11, Proce edings of the 13th Annual Air Force

Science and Engineering Sympo sium, Arnold Engineering

Development Center, Arnold Air Force Station, Tennessee,

September 1966.

81. Ribner, H.S.

Malvestuto, F. S. Jr

82. Richardson, J. R.

S t a b i l i t y D e r i v a t i v e s o f T r i a n g u l a r W ings a t S u p e r s o n i c

S p e e d s . NAC A Repo rt 908. 1948.

A Method for Ca lcu la t ing the L i f t in g F orce s on Wings

( U n st ead y S u b s o n i c and S u p e r s o n i c L i f t i n g - S u r f ac e T h eo r y ) .

ARC R & M 3157, April 1955, ARC 18,622.

83. Sacks, A.H.

A ero d yn amic F o r ce s , M o ments an d S t a b i l i t y D e r i v a t i v e s

f o r S l en d e r B o d i e s o f G en e r a l C r o s s S ec t i o n .

NACA TN

3283,

November 1954.

84. Sa uerwein, H. A p p l i c a t i o n o f t h e P i s t o n A n a lo g y t o t h e C a l cu l a t i o n o f

S t a b i l i t y D e r i v a t i v e s f o r P o i n t e d A x i a l l y Sy m m etr ic

Bo die s a t H igh Mach Num bers. AVCO Corporation, RAD-TM-

61-40.

October 1961.

85. Seredinsky, V.

D e r i v a t i o n an d A p p l i c a t i o n o f H y p e r s o n i c N ew to n ian I mp ac t

Theory to Wedges, Cones and Sl ab Wings I n cl u di n g the

E f f e c t s o f A n g le o f A t t ack , S h i e l d i n g an d B l u n t n es s .

Grumm an Aircraft Engineering Corp oration Report XAR-A -42,

April 1962.

86.

Smith,

H.

B.

Beasley, J. A.

Stevens, A.

C alc u l a t io ns of the L i f t S lope and Aerodynamic Cen ter o f

Cropped De l ta Wings a t Su per so nic Sp eeds . RAE TN Aero

2697. 1960.

87. Spahr , J .R.

C on tr ib ut io n of the Wing Pa ne ls to the Fo rce s and Moments

of Sup ers on ic Wing-Body Com binat ions at Combined An gle s .

NACA TN 41 46, Ja nu ar y 1958.

88.

Stanbrook, A.

The L i f t -C urv e S lope and Aerodynamic Cen ter Po s i t io n of

Wings a t S upe r son ic and Subsonic Spee ds .

RAE TN Aero

2328,

1954.

89. S t a t l e r , I . e .

Dynamic Stabi l i ty at High Speeds f rom Unsteady Flow Theory.

Journa l o f the Aeronaut i ca l Sc iences , Vol .17 , Apr i l 1950 ,

p p . 2 3 2 - 2 4 2 .

90. Temple, G.

T he R e p r e s en t a t i o n o f A e ro d yn amic D e r i v a t i v e s .

2114, March 1945.

ARC R & M



102

91.

Thelander, J. A.

92.

Thomas,

H.

H.B. M.

A i r c r a f t M o ti on A n a l y s i s .

Laboratory. PDL-TDR-64-70,

US Air Force Plight Dynamics

Mar ch 1965.

E s t i m a t i o n of S t a b i l i t y D e r i v a t i v e s ( S t a t e o f th e A r t ) .

RAE TN Aero

2776.

1961; also ARC C P 664, and AGARD Repor t

339.

93. Thomas, H.H.B.M.

Ross, A.J.

94.

Thomas, H.H.B.M.

Spencer,

B. P.

R.

95. Tobak, M.

96. Tobak, M.

97.

Tobak, M.

98.

Tobak, M.

Allen, H.J.

99.

Tobak, M.

Pearson, W. E.

100.

Tobak, M.

Wehrend, W.R.

101.

Treadgold. D.A.

102.

van Dyke,

M.

D.

103. van Dyke,

M.

D.

104. W a t ki ns , C E .

The C a l c u l a t i o n of t h e R o t a r y L a t e r a l S t a b i l i t y D e r i v a t i v e s

o f a J e t - F l a p p e d W i ng .

ARC R & M 3277, Januar y 1958.

The C al cu la t io ns of the D er iv a t iv es I nvolv ed in the Damping

o f t h e L o n g i t u d i n a l S h o r t P e r i o d O s c i l l a t i o n s o f an A i r

c r a f t and C o r r e l a t i o n w i t h E x p e r i men t .

RAE Report Aero

2561, 1955.

Damping In Pi tch of Low-Aspect-Rat io Wings at Subsonic

a n d S u p e r s o n i c S p e e d s .

NAC A RM A52L04a, April 1953.

Oi

N o n l i n e a r L o n g i t u d i n a l D ynam ic S t a b i l i t y .

Presented

to the AGARD Flight Mechanics Panel. Cambridge, England,

September 20-23, 1966.

Oi the Use of the I n d ic ia l Fu nc t ion Concept in the A nal ys i s

of Unsteady Motions of Wings and Wing-Tai l Combinat ions .

NAC A Re por t 1188, 1954.

D yn amic S t a b i l i t y o f V eh i c l e s T r av e r s i n g A s cen d in g o r

Descending Pa ths Through the Atmosphere . NACA TN

4275,

July 1958.

A S t u d y o f N o n l i n ea r L o n g i t u d i n a l D yn amic S t a b i l i t y .

NAS A TR R-209. Septem ber, 1964.

S t a b i l i t y D e r i v a t i v e s o f C o n e s a t S u p e r s o n i c S p e e d s .

NACA TN 3788, September 1956.

A R evi ew o f M e th o ds o f E s t i m a t i o n o f A i r c r a f t L o n g i t u d i n a l

S t a b i l i t y D e r i v a t i v e s a t S u p e r s o n i c S p e e d s .

RAE TN Aero

2415, 1955.

Oi

S eco n d - O rd e r S u p e r s o n i c F lo w P a s t a S l o w ly O s c i l l a t i n g

A i r f o i l .

Readers' Forum, Journal of the Aeronautical

Sciences,

Vol.20,

J anuary 1953, p. 61.

S u p e r s o n i c Flo w P a s t O s c i l l a t i n g A i r f o i l s I n c l u d i n g Non

l i n e a r T h i c k n e s s E f f e c t s .

NACA TN 2982, July 1953.

Air F orc es and Moments on Tr ia ng ul ar and R el at ed Wings

w i t h S u b s o n i c L ead in g E d ges O s c i l l a t i n g i n S u p e r s o n i c

P o t e n t i a l F l ow .

NACA TN 2457, September 1951.



103

105. Watkins, C.E.

Ef fe ct of Aspect R at io on the Ai r Fo rce s and Moments of

Harm onica l ly O sc i l la t i ng Th in Rec tang u la r Wings in Super

s o n i c P o t e n t i a l F l ow . NACA TN

2064,

April 1950.

106. W a tk ins, C E .

Ef fe ct of. Aspe ct Ra ti o on the Air Fo rc es and Moments of

Harm onica l ly O sc i l la t i ng Th in Rec tang u la r Wings in Super

s o n i c P o t e n t i a l F l o w . NAC A Repo rt 1028. 1951.

107. Watkins, C.E.

Ef fec t o f Aspec t Ra t io on Undamped Tors iona l O sc i l la t i on s

of a Thin R ec ta ng ul ar Wing in Su pe rs on ic Flow. NACA TN

1895.

June 1949.

108.

Watkins, C.E.

Herman, J.H.

Ai r For ce s and Moments on T ri an gu la r and R el at ed Wings

wi th Subson ic Lead ing Edges Osc i l la t ing in Superson ic

P o t e n t ia l Flow. NACA Rep ort 1099, 1952.

109. Watts, P.E.

N o t e s on t h e E s t i m a t i o n o f A i r c r a f t L a t e r a l S t a b i l i t y

D e r i v a t i v e s a t S u p e r s o n i c S p e e d s . RAE TN Aero 2414, 1955.

110. Wood. R.M.

Murphy, C.H.

111. Wylly, A.

Aerodynamic Derivat ives for Both Steady and Non-Steady

M o t i o n o f S l e n d e r B o d i e s . BRL MR 880, April 1955.

A Second-Order So lu t ion fo r an O sc i l la t i ng , Two-Dimens iona l,

S u p e r s o n i c A i r f o i l . RAND Corpor ation Report, 1951.

Derivatives D ue to Pitch and Rate of Pitch-Experiment

1. Akat nov,

N.

I.

Sabaltls, lu. I.

Measurement o f P i tch ing-Moment Der iva t ives by a Self-

E x c i t e d O s c i l l a t i o n M e t h o d .

Zhurnal Tekhnickeskoi Piziki,

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No .9, 1956. Tra nslated in "Soviet Phys ics".

Vol.1,

No.

9,

1957, pp .1986-1993.

2.

Bablneaux, T.L.

et al.

The I nf lu en ce of Shape on Aerodynamic Damping of O s c il la

tory Motion During Mars Atmosphere Entry and Measurement

o f P i tch Damping a t Large Osc i l la t ion Ampl i tudes . TR-32-

380, F ebru ary 28. 1963.

3. Beam, B.H.

A Wind-Tunnel Test Technique f or Me asuring the Dynamic

R o t a ry S t a b i l i t y D e r i v a t i v e s I n c l u d i n g t h e C r o ss D e

r i v a t i v e s a t High Mach Numbers . NACA TN

3347,

January

1955.

4. Beam, B.H.

7?ie

E f fec ts o f Os c i l l a t io n Ampl i tude and F requency on

the Experi me ntal Damping in P i t c h of a Tria ng ula r Wing

Having an Aspect Ra tio of 4 . NACA RM A52G07, September

1952.



104

5. Bird,

K D.

6. Bratt, J.B.

7. Bratt, J.B.

Chinneck,

A.

8. Bratt, J.B.

9. Bratt, J.B.

Scruton, C

Dynamic S ta b i l i t y T es t i ng in a Hype rsonic Shock Tunnel .

July

16-17, 1963.

A Note on De r iv a t iv e A ppara tus f or the NPL 9) . In ch H igh

S p e e d T u n n e l .

ARC CP

269, Januar y

1956.

In form at io n Repo r t on Measurements of Mid-Chord P i t ch in g

Moment Der iva t ives a t H igh Speeds .

ARC 10,710, July 1947,

ATI

105 207.

77ie M easu remen t o f O s c i l l a t o r y A e r od y namic F o r ce s . AGARD

Wind Tunnel Panel, September

1953.

Measurements of P i t ch in g Moment D er iv a t iv es f or an Aero

f o i l O s c i l l a t i n g A b ou t t h e H a l f - C h o r d A x i s .

ARC R & M 1921,

November 1938.

10. Bratt.

J.B.

Wight, K . C

11. Bratt,

J.B.

Wight, K.C.

Chinneck,

A.

12. Broil, C.

13. Clay,

J.T.

Walchner,

0.

The E f f ec t o f M ean I n c i d e n ce , A m p l it u d e of O s c i l l a t i o n ,

P r o f i l e an d A s p ec t R a t i o o n P i t ch i n g Moment D e r i v a t i v e s .

ARC R & M 2064. June 1945.

F r ee O s c i l l a t i o n s o f an A e r o f o i l A bo ut t h e H a l f - C h o r d

A x i s a t H i g h I n c i d en ces , and P i t ch i n g Moment D e r i v a t i v e s

f o r D e ca y in g O s c i l l a t i o n . ARC R & M

2214, September

1940,

ARC

4711.

E s s a i s e n R eg im e I n s t a t i o n n a i r e E f f e c t u e s a u C e n t r e

d ' E s s a i s d e M o d an e- A v ri eux .

(Unsteady Flow Testing at

the ONERA Test Centre

at

Modane).

La

R ech erc he Ae'ro-

spatiale, No.100, 1964.

N ose B l u n t n e s s E f f e c t s on th e S t a b i l i t y D e r i v a t i v e s o f

Cones in Hy pers onic Flow. Vol.1, Paper 8, Transactions

of

t he

Second Technical Workshop

on

Dynamic Stability

Testing. Held at the Arnold Engineering Develop ment

Center, Arnold Air Force Station, Tennessee, April 20-22,

1965.

14.

Clunies,

D.A.

Stiles,

H.W. Jr

15. Colosimo, D.

D.

16. Dat, R.

Destuynder, R.

An I n v es t i g a t i o n o f t h e F e a s i b i l i t y o f a M e th od o f O b t a i n

i n g D a t a f ro m W hich t h e S t a b i l i t y D e r i v a t i v e s C and

C can b e S ep a r a t ed .

BSc Thesis, Aeronautical Engineer

ing Department, Massachus etts Institute o f Technology,

May 1957.

T he E f f ec t s o f M as s - T r an s f e r o n t h e D yn amic S t a b i l i t y o f

S l e n d e r C o n e s . CAL Rep ort 141, May 1965.

D e t e r m i n a t i o n d e s C o e f f i c i e n t s A e ro dy na m iq ue s I n s t a t i o n -

n a i r es sur une A i le De l t a de Mach 1 .35 .

ONERA NT 52,

1959.



105

17.

Daughaday,

H.

Duwaldt,

P.

Statler,

I.

Measurement of Dynamic S t a b i l i t y D er iv at iv es in the Wind

T u n n e l, P a r t I I , E v a l u a t i o n o f T e s t i n g T e c h n iq u e s .

WADC

TR 57-274.

May 1957.

18.

Pink,

M.R.

Carroll. J.B.

Hyperson ic S ta t i c and Dynamic S ta b i l i ty o f Wing-Fuselage

Co n f i g u r a t i o n s . U n i t e d A i r c r a f t Co r p o r a t i o n Re s ea r ch

Laboratory, C910188-17, January 1965.

19 Fink, M.R.

Hyperson ic Dynamic S tab i l i ty o f Bicon ic Bod ies . United

Aircra ft Corpor ation, Report UAR -E20, March 12, 1966.

20. Fink. M.R.

21. Fink,

M.R.

Hyperson ic Dynamic S ta b i l i t y o f Sharp and Blun t S lende r

C o n e s .

United Aircraft Corporation Research Laboratory,

Conn.,

UAR-C111, July

1964.

A Shock-Expans ion Method fo r Ca lcu la t ion o f H yperson ic

D y n a m i c S t a b i l i t y .

United Aircraft Corpor ation, Conn.,

C-910188-4, March 1964, AD 436 694.

22. Pink,

M.R.

Carroll,

J.B.

Ca l c u l a t i o n s o f H y p e r s o n ic Dynam ic S t a b i l i t y .

United

Aircraft Corporation, Conn., C-910188-8, June

1964,

AD

442 280.

23. Fischer.

L.R.

Exper im en ta l De te rmin a t ion o f the E f fec ts o f F requency

a nd A m p l it u de o f O s c i l l a t i o n on t h e Ro l l - S t a b i l i t y

D er iva t iv es fo r a 60° De l ta -Wing Airp lane Model . NASA

TN D-232, March

1960.

24. Poster, D.J.

Haynes,

G.

W.

R o t a r y S t a b i l i t y D e r i v a t i v e s f ro m D i s t o r t e d M o de ls .

Journal of the Royal Aeronautical Society, Vol.60,

Sep tem ber 1956, pp. 623-624.

25. Gilmore,

A.

Seredinsky,

V.

Mackenzie,

D.

I n v e s t i g a t i o n of D ynam ic S t a b i l i t y D e r i v a t i v e s of V e h i c l e s

F l y i n g a t H y p e r s o ni c V e l o c i t y .

Grumman Aircraft Engineer

ing Corp orat ion. ASD -TD R-62-460, Sept embe r 1963,

A D

421 874.

26. Grimes,

J.H. Jr

Casey, J. J.

Dynamic S ta b i l i t y Tes t ing w i th Ab la t io n a t Mach 14 in a

Long Duration Wind Tunnel.

Proceedings, AIAA Aerodynamic

Testing Conference, Washington,

DC,

March 9-10,

1964.

27. Hall, G.Q.

Osborne, L. A.

A Co n t r o l S u r f a c e O s c i l l a t i n g D e r i v a t i v e M e as ur em en t R i g

fo r Use wit h Half Wing Models Having Swept Hinge Li ne s.

Hawker-Siddeley Dynamics, Coventry,

ARL TN 205.

28. Hall, G.Q.

Osborne,

L.

A.

Superson ic Der iva t ive Measurements on the P lan fo rms o f

the MoA.

Flutter and Vibration Committee' s First Research

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Report 63/10.



106

29. Hall,

G.

W.

Osborne,

L.

A.

Tran sonic and Sup er son ic De r iv a t iv e Measurements on the

Pla nfo rm s of the MoA. Flutter and Vibration Committee's

First Research Programme. Hawker-Siddeley Dynamics,

Coventry, ARL Report 64/9.

30. Hoop,

H.H.

31. Holway, H.P.

Herrera, J. G.

Dayman,

B.

A Summary of Exper im enta l P i t c h Damping Co ef f i c ien t s fo r

the Per sh ing Re-Ent ry Body. AOMC RG-TM-61-49, December

19,

1961.

A Pneum at ic Model Launcher for F re e- F l ig h t Te s t in g in a

Co nv en t ion al Wind Tunn el .

TM 33-177, July 30, 1964.

32.

Jaffe,

P.

Prislin.

R.H.

E f f ec t o f B o u nd a ry L ay e r T r an s i t i o n o n D yn amic S t a b i l i t y

O v e r L a r g e A mp l i t u d es o f O s c i l l a t i o n .

TR 32-841, March

7,

1966. AIAA Paper 64-427, Vol.3, No. l, June 1964.

Spacecraft Journal, January

1966.

33. LaBerge,

J. G.

Dynamic Tests on a Cone in Hypersonic Helium Flow.

NAE

Laboratory Memorandum (unpublished), March

1963.

34. LaBerge, J. G.

Ef fec t of F la re on the Dynamic and S t a t i c Moment Cha rac ter

i s t i c s o f a H e m i s p h er e -C y l in d e r O s c i l l a t i n g i n P i t c h a t

Mach Numbers from 0 .3 to 2. 0 .

National Research Council,

Canada, Report LR-295, 1961.

35.

Maloy, H.E. Jr

36.

Molyneux, W. G.

A Method for Me asur ing Damping- in -P i t c h of Models in

S u p e r s o n i c F l o w . BR L

Rep ort 1078, July

1959.

Measurement of the Aerodynamic Fo rces on Os c i l l a t in g

A e r o f o i l s .

Aircraft Engineering, Vol.28, No. 323, January

1956, pp. 2-10.

37. Moore, J. A.

Exp er ime nta l De term ina t i on of Damping in P i t ch of Swept

and D el ta Wings at Su pe rso nic Mach Numbers .

NACA RM

L57G10a,

September

1957.

38. O'Neill, E. B.


A F re e- F l ig h t Method fo r S tudyin g the Dynamic Behavior

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Navy Symposium on Aeroballistics, 7-9 June 1966.

A Method fo r Ob tain ing the Damping in Pi tc h and Yaw in

a Sup er so nic Wind Tunnel , e t c .

KTH Fl 134, Stockholm,

1952.



A Method for Ob ta in ing the Damping in P i t ch D er iv a t iv es

of Wings a t Super sonic Speeds f rom Free Osc i l l a t ion Tes t s

w i t h H a l f M o d e l s .

PPA Ae 273, Sto ckhol m, 1952.

Measurement of Aerodynamic Damping and S t i f f n e ss De r iv a

t i v e s i n F r e e O s c i l l a t i o n w i t h A u t o m a t i c a l l y R e c y cl e d

F e e d b a c k E x c i t a t i o n .

National Research Council, Canada,

LR-246, June

1959.



107

42. Orlik-Ruckemann,

K.J.

Qu adra t ic E f fe c ts o f F requency on Aerodynamic D er iv a t i ve s .

AIAA J our nal, Vol.2, No.8, 1964. pp.1507-1509.


K.J.


Crowe, C

Some

A p p l i c a t i o n s of t h e F r e e - O s c i l l a t i o n T e c h n i qu e s f o r

M e a s u r i n g S t a b i l i t y D e r i v a t i v e s .

Seminar on Wind-Tunnel

Techniques

and

Aerodynamics,

KTH,

Stockholm,

1954.

Comparison of Various Experimental and Theoret ical

Re s u l t s f o r S t a t i c a n d D ynam ic L o n g i t u d i n a l S t a b i l i t y

D er iv at iv es of Some Wing-Body C on fig ura t io ns a t Sup erso nic

S p e e d s .

National Research Council, Canada, Laboratory

Report (to be published).

45.


LaBerge,

J.

G.

O sc il la to ry E xperim ents in a Helium Hyp ersonic Wind

T u n n e l .

National Research Cou ncil, Canada,, Report LR-335,

1962; also "Advances

in

Hyperve locity Techniques", Pl enum

Press,

N ew

York, 1962, pp.187-210.


K.J.

LaBerge,

J.G.

S t a t i c a nd D ynam ic L o n g i t u d i n al S t a b i l i t y C h a r a c t e r i s t i c s

of a S er ie s of D el ta and Sweptback

Wings

a t Supers on ic

S p e e d s .

National Research Council, Canada, LR-396,

January

1966.

47. Platzer, M.F.

Hoffman. G.H.

Quas i -S lender Body Theory fo r S lowly Osc i l la t ing Bod ies

o f Revo lu t ion in Superson ic F low. NASA TN D-3440, June

1966.

48. Pope,

T.C.

Stalmach,

C.J. Jr

P r e l i m i n a r y I n v e s t i g a t i o n o f t he E f f e c t s o f M a s s - I n j e c t i o n

o n D yn am ic S t a b i l i t y

a t

M = 17 .

Ling-Temco-Vought,

LTV

Report 2-59740-/3R-479, October

1963.

49. Pugh,

P.G.

Woodgate,

L.

Measurements of Pi tc hi ng Moment D er iv at iv es fo r B lunt

Nosed Ae rof o i l s O sc i l la t i ng in Two-Dimens ional Superso n ic

Flo w. A R C R & M

3315,

1963.

50. Scherer, M.

Mesure des Derivees Aerodynamiques en Ecoulement Trans

son ique e t Superson ique . Communication presente'e au

I Verne Co ngre s Ae'ronautlque Eur opeen a Cologne. ONERA

publication No.104, Septembre 1960.

51.

Scruton,

C.

Woodgate,

L.

Lapworth, K. C.

Measurement o f P i tc h i ng Moment D er iva t iv es fo r Ae rof o i l s

O sc i l la t i ng in Two-Dimensiona l Supe rson ic F low.

ARC

R & M 3234,

1962.

52. Scruton, C.

Woodgate, L.

Maybrey, J.F.

53. Scruton,

C

et

al.

Measurement of the Pi tc hi ng Moment D er iv at iv es fo r R ig id

Tapered Wings of Hexagonal Planform Oscil la t ing in Super

son ic F low. ARC R & M

3294,

1962.

Measurements o f the P i tc h i ng Moment D er iv a t i ve s fo r Ri g id

Wings of Rectangular Planform Oscil la t ing About the Wing-

Chord Axis in Supersonic Flow. ARC CP 594, 1962.



108

54.

Seredinsky, V.

55.

Shantz, I.

D e r i v a t i o n and A p p l i c a t i o n o f H y p e r s o n i c N ew to n ian I mp ac t

Theory t o Wedges, Cones and Sl ab Wings I nc lu di ng the

E f f ec t s o f A n g le o f A t t a ck , S h i e l d i n g an d B l u n t n e s s .

Grumman Aircraft Engineering Corporation, Report XAR-A-42,

Apr il 1962.

7?ie Mea surem ents of Damping- in -P it c h in the NOL Wind

T u n n e l s .

Presented at the Fourth Navy Symposium on Aero-

ballistics,

Naval Proving Ground, Dahlgren, Virginia,

1957.

56. Shantz, I.

Groves, R.

T.

D yn amic an d S t a t i c S t a b i l i t y M eas ur emen ts o f t h e B as i c

F i n n e r a t S u p e r s o n i c S p eed s . NAVORD Report 4516, January

1960.

57.

Sibley, G.O.

58.

Smith, T. L.


Cooksey, J.M.


Moore,

D.

R.

Lindsey, J. L.

61. Thompson, J.S.

Fail, R.A.

62.

Thompson, J. S.

Fail,

R.A.

63. Thompson, J.S.

Pail,

R.A.

64.

Tobak, M.

65. Tobak, M.

Reese, D.E. Jr

Mean,

B.H.

Dynamic S t a b i l i t y Te s ts of AGARD Models HB-1 and HB-2

in a Blowdown Tunn el a t Mach

5.4. VKI PR

65-141,

June

1965.

D y n ami c S t ab i l i t y M eas u r emen t s . BRL M 1352, June 1961.

S i m u l a t i o n o f R o ck e t P ow er E f f ec t o n V eh i c l e S t a b i l i t y

and Measurements of Aerodynamic Damping in a H yp erv elo ci t y

W i n d T u n n e l .

IAS Paper 62-24, presented at the 30th

Annual IAS Meeting, New York, January 1962. Also pub

lished in Aerospace Engineering, Vol.21, N o. 3, 1962.

S t a b i l i t y I n v e s t i g a t i o n D u r in g S i m u l a t e d A b l a t i o n i n a

H y p e r v e l o c i t y W ind T u n n e l , I n c l u d i n g P h as e C o n t r o l .

Sing-

Temco-Vought, LTV Report 2-59740/5R-2192, April 1965.

O s c i l l a t o r y D e r i v a t i v e M eas u r emen t s o n S t i n g - M o u n t ed

Wind-Tunnel Mo dels : Method of Tes t and R es u l ts fo r P i t ch

and Yaw on a Cam bered Ogee Wing a t Mach Numbers up to

2 . 6 .

RAE Report Aero

2668,

1962; also ARC R & M

3355.

Measurements of O sc i l l a to ry De r iv a t iv es a t Mach Numbers

up to 2.6 on a Model of a Su pe rso nic Tra nsp or t Design

S t u d y ( B r i s t o l T ype 1 9 8 ) .

RAE Technical Report 64,048.

1964; also A RC CP 815.

O s c i l l a t o r y D e r i v a t i v e M eas u r emen t s o n S t i n g - M o u n t ed

Wind Tunne l M odels at RAE Be dfo rd. RAE Technical Report

66,197, 1966.

Damping in P i t c h of Low-A spect-R at io Wings at S ubso nic

a n d S u p e r s o n i c S p e e d s .

NAC A RM A52L04a, April 1953.

Exp er im enta l Damping in P i t c h of

45°

Tr ia ng ula r Wings.

NA CA RM A50J26, 1950.



109

66.

van de Vooren, A.I.

67.

Vaucheret, X.

68. Vaucheret. X.

69.

Vaucheret, X.

Measurements

o f Aerodynamic Forces on Osc i l la t ing Aero

f o i l s . NLL Report MP 100. May 1954. Presented at the

5th Meeting of the AGARD Wind Tunnel and Model Testing

Panel, Scheveningen, Nethe rlands, May 1954.

D e p o u i ll e m e n t d 'O s c i l l a t i o n s L i b r e s d ' u n Co rp s d e Re n t r e e

P r e s e n t a n t u n Cy c le L i m i t e . (Processing the Free Oscilla

tions of a Re-Entry Body Presenting a Limit-Cycle .) La

Recherche Aerospatiale,

No .104,

1965.

M e th od e P r a t i q u e de D e p o u i l l e m e n t s d ' O s c i l l a t i o n s L i b r e s

a un Degre de L ib er ie , en Pre sen ce d 'u ne For ce de Rappel

D i s c o n t i n u e . (Practical Process ing Method for One-Degree -

of-Preedom Free-Oscillations with a Discontinuous Res

training Force .) La Recherche Aerospatiale,

No.89,

1962.

S t a b i l i t e Dynamique sur Maq uettes d 'Avions par la Methode

d e s O s c i l l a t i o n s L i b r e s . Presented at AGARD Fluid

Mechanics Panel, Cambridge, September 1966.

70. Walchner, 0.

Sawyer, P.M.

Koob, S.J.

71. Walker, H.J.

Ballantyne, Mary B.

72.

Wehrend. W.R. Jr

Dynamic S t a b i l i t y Te st in g in a Mach 14 Blowdown Wind

T u n n e l . Journal of Spacecraft and Rockets,

Vol.1,

No.4,

1964.

p p. 437-439.

Pr ess u r e D is t r ib u t io n and Damping in S teady P i tc h a t

Su pe rso nic Mach Numbers of Fl a t Sw ept-Back Wings Having

A l l E d g e s S u b s o n i c . NA CA TN 2197, Oct ober 1950.

An Ex per im ent al E va lu at io n of Aerodynamic Damping Moments

o f Cones wi th Di f fe ren t Cen te rs o f R o t a t i on . NASA TN

D-1768, March 1963.

73.

Welsh. C.J.

Clemens, P.L.

74. Wiley, H.G.

75. Woodgate, L.

Maybrey.

J.P.M.

Scruton, C

76. Wright, B.R.

Kilgore, R.A.

A Sm al l -S ca le F or ce d- O sc i l l a t i on Dynamic Ba lance Sys tem.

AEDC-TN-58-12. Ju ne 1958. AD 157 143.

The Si gn if ic an ce of No nlin ear Damping Trends Determined

f o r C u r r e n t A i r c r a f t C o n f i g u r a t i o n s . Presented at the

AGARD Plight Mechanics Panel, Cambridge, England, September

20-23,

1966.

Measurements of the Pi tc hi ng Moment D er iv at iv es fo r R ig id

Tapered Wings of Hexagonal Planform O sc il la t i n g in Sup er

s o n i c F l o w . ARC R & M

3294,

1963.

Aerodynamic Damping and O sc i l l a t o r y S ta b i l i ty in P i tc h

and Yaw of Gemini C on fi gu ra ti on s at Mach Numbers from

0 . 5 0 t o 4 . 6 3 . NAS A TN D-3334. March 1966.



110

Derivatives Due to Yaw and Rate of Yaw

1. Fishe r, L. R.

Fletcher, H.S.

2.

Fisher, L.R.

Wolhart, W. D.

3. Malvestuto, P.S. Jr

4.

Margolis , K.

Bobitt,

P.J.

5. Margolis, K.

Elliott,

M. H.

6. Margolis, K.

Sherman, W. L.

Hannah, M.E.

7.

Martin, J.C.

Malvestuto, P.S. Jr

8. Purser, P.E.

9. Queijo, M.J.

et al.

10.

Robinson, A.

Hunter-Tod, J.H.

E f f ec t o f L ag o f S i d ew as h o n t h e V e r t i c a l - T a i l C o n t r i b u t i o n

to Os c i l l a to ry Damping in Yaw of Ai rp la ne M odel s.

NACA

TN 3356. January 1955.

Some E ff ec ts of Ampli tude and Frequ ency on the Aerodynamic

Damping of a Model O sc i l l a t i n g C ont inu ous ly in Yaw.

NACA

TN 2766, September 1952.

Th eo re t i c a l Sup er son ic Force and Moment Co ef f i c ien t s on

a S i d e s l i p p i n g V e r t i c a l - a n d H o r i z o n t a l - T a i l C o m b in a ti on

wi th Subso nic Leading Edges and Sup er son ic T ra i l i ng

E d g e s .

NAC A TN 3071, Marc h 1954.

T h e o r e t i c a l C a l c u l a t i o n s of th e P r e s s u r e s , F o r c e s , a n d

Moments at Supersonic Speeds Due to Var ious Lateral

M o t io n s A c t i n g on T h in I s o l a t e d V e r t i c a l T a i l s . NACA

Rep ort 1268, 1956.

T h e o r e t i c a l C a l c u l a t i o n s o f th e P r e s s u r e s , F o r c e s , a nd

Moments Due to V ar ious L a te ra l Motions Ac t ing on Tapered

Sweptback V er t i c a l T a i l s w i th Sup er son ic Leading and

T r a i l i n g E d g e s .

NAS A TN D-383, August 1960.

T h e o r e t i c a l C a l c u l a t i o n o f t h e P r e s s u r e D i s t r i b u t i o n ,

Span Loa ding , and R o l l i ng Moment Due to S id e s l ip at

Su pe rso nic S peeds for Thin Sweptback Tapered Wings with

Sup er son ic T ra i l in g Edges and Wing T ips P a ra l l e l to the

A x i s o f W in g S y m m e t ry .

NACA TN

2898,

February 1953.

Th eo re t i ca l Fo rces and Moments Due to S id es l i p of a

Number of V er t i c a l Ta i l Co nf ig ura t ion s a t Sup er son ic

S p e e d s .

NAC A TN 2412, Sep temb er 1951.

An App roxim at ion to the Eff ec t of Geom etr ic Di he dra l

on the R o l l in g Moment Due to S id e s l i p fo r Wings at Tran

s o n i c a n d S u p e r s o n i c S p e e d s .

NAC A RM L52B01, April 1952.

Pr el im in ar y Measurements of the Aerodynamic Yawing De

r i v a t i v e s of a Tr ian gu lar , a Swept , and an Unswept

Wing

P er f o r mi n g P u r e Y aw ing O s c i l l a t i o n s , w i t h a D e s c r i p t i o n

o f t h e I n s t r u men t a t i o n E mp l o y ed .

NACA RM L55L14, April

1956.

T he A ero d yn amic D e r i v a t i v e s w i t h R es p ec t t o S i d e s l i p f o r

a De l ta Wing wi th Smal l D ih edra l a t Zero In c ide nce a t


ARC R & M 2410, December 1947, ARC

11,322.



I l l

11. Sherman, W. L.

Margolis, K.

T h e o r e t i c a l C a l c u l a t i o n s o f th e E f f e c t s o f F i n i t e S i d e

sl ip a t Supersonic Speeds on the Span Loading and Roll ing

Moment for Families of Thin Sweptback Tapered Wings at an

A n g l e o f A t t a c k . NACA TN 3046. November 1953.

Magnetic Suspension for Aerodynamic Testing

1.

Ma gnetic Susp en sio n of Wind Tunnel Mod els. The Engineer,

Vol.210.

N o. 5463, O cto ber 7, 1960, pp .607-608.

Magnetic Suspension Replaces Mechanical Support of Wind

T u n n e l M o d e l s . Electronic Design

News,

Vol.6, No.7,

July 1961, pp. 13-15.

3. Baron,

L.

A.

77ie Des ign and Co ns t r uc t io n o f an Automat ic C on t ro l

System fo r a Wind-Tunnel Mag netic Su spe nsi on System .

MIT Thesis, Mas sachusetts Institute of Technology, Cam

bridge,

Mass.,

J une 1960.

4.

Clemens,

P.

L.

Cortner, A. H.

5. Chr isinger, J.E.

B ib lio gr ap hy : The Magnetic Suspen sion of Wind Tunnel

M o d e l s . AED C-T DR- 63-20, Febr uary 1963. AD296400.

An I nv es t i ga t i on o f the Eng in ee r ing Aspec ts o f a Wind

Tunnel Magnetic Suspension System. MIT Thesis, MIT

TR 460, Ma y 1959, AD 221294.

6. Chrisinger, J.E.

et al.

Magn etic Susp ensi on and Balan ce System for Wind Tunnel

A p p l i c a t i o n . Journal of the Royal Aeronautical Society,

Vol.67,

No .635, Nov emb er 1963, pp. 717-724.

7.

Copeland, A. B.

Covert,

E.E.

Stephens, T.

Rec ent Advances in the Development of a Ma gnetic Su spe nsi on

and Ba lance System fo r Wind Tunne ls (Pa r t I I I ) . ARL-65-114,

June 1965.

8. Copeland, A. B.

Tilton, E.L. Ill

The Design of Magnetic Models for Use in a Magnetic

Sus pen sion and Balanc e System fo r Wind Tun nels . ARL-65-

113.

June 1965.

9. Covert, E.E.

et al.

Rec ent Advances in the Development of a Ma gnetic Sus pen sio n

and Bala nce System fo r Wind Tunnels (Pa rt I I ) .

ARL-64-36,

Marc h 1964.

10. Covert. E.E.

Tilton, E.L. Ill

11. Covert, E.E.

Tilton,

E.L. Ill

C al ib ra t i on o f a Magne t ic Ba lance System fo r Drag , L i f t ,

a n d P i t c h i n g M o m en t. Massachusetts Institute of Technology,

Aerophysics Laboratory, Cambridge,

Mass.,

May 1963.

Fu r th e r E va l ua t ion o f a Magne t ic Suspens ion and Ba lance

System for A pp lic at io n t o Wind Tun nels . ARL-63-226,

Dec emb er 1963, AD 427 810.



113

20. Laurenceau,

P.

Desmet, E.

Ad apt at io n de la Susp ension Magnetique des Ma quettes aux

Souff le r ies S.19 e t S .8 . (Adaptation of the Magnetic

Suspension

o f

Models

t o

Wind Tunnels

S.19 and S.8.)

ONERA

N T 7/1579

AP, Office National d'Etudes

et de

Recherches Aeronautiques, Chatillon-sous-Bagneux

(Seine),

Paris, Prance, Janvier

1957.

21. Matheson, L. R. Some Co n s i d e r a t i o n s f o r D e s i gn a nd U t i l i z a t i o n o f M a g ne ti c

S u s p e n s i o n .

Aerod ynamics Fundamental Memo No. 84, General

Electric Missile and Space Vehicle Department, Phila

delphia, Pennsylvania, May 1959.

22. Mirande, J.

Mesure de la R es is t an ce d 'un Corps de Re vo lut ion , a

M

Q

= 2. 4 , au Moyen de la Sus pe ns io n M agne tique ONERA.

(Measurement o f t h e Drag of a Body of Revolution, at

M = 2.4 , by Means of the ONERA Magnetic Suspension

System.)

No.70,

Office National

d'

Etudes

et de

Recherches

Aeronautiques, Mai-Juin 1959, p. 24.

23. Nelson,

W.

Trachtenberg, A.

Grundy, A. Jr

An Ap par atu s fo r the Stud y of High Speed Air Flow Over a

F r e e l y S u s p e n d e d Ro t a t i n g Cy l i n d e r . WADC TR 57-338,

AD 142127, December 1957.

24. Parker, H.M.

May,

J.E.

Nurre, G.S.

An Ele ctr om agn eti c Suspe nsion System for the Measurement

o f A e r o d y n a m i c C h a r a c t e r i s t i c s . OSR-2294, AST-4443-106-

62U, University of Virginia, Charlot tesvil le, Virginia,

March 1962.

25. Scott,

J.E. Jr

May, J.E.

Labo ra to ry In ve s t ig a t io n o f the Bas ic Na tu re of Low

D ens ity Gas Flow at High Spee ds, Annual Repo rt fo r the

P er io d 1 Ja nu ary 1960 to 31 December 1960.

University

of Virginia, Charlottesville, Virginia, January

1960,

AST-4435-109-61U.

26.

Stephens, T.

Methods of Control l ing the Roll Degree of Freedom in a

Wind Tunnel Magnetic Ba lanc e, P ar t I : P rod uc tio n of

R o l l i n g M o m e n t s .

ARL

65-242,

December 1965.

27. Tilton, E. L.

et

al.

The

Des ign and I n i t i a l Op era t ion o f a Magne t ic Mode l

Sus pen sion and Force Measurement System.

ARL-63-16,

August

1962.

28. Tilton, E. L.

Schwartz, S.

Sta t ic Tes ts on the Magne t ic Suspens ion Sys tem.

MIT AR

Memo 399, Massachusetts Institute

of

Technology, Naval

Supersonic Laboratory, Cambridge, Mass., July

1959.

29.

Tournier, M.

Dleulesaint,

E.

Laurenceau,

P.

Etude , R ea l i sa t i on e t Mise au Po i n t d 'une Suspens ion de

Maquette Aerodynamique par Action Magnetique pour une

P e t i t e S o u f f l e r i e .

(Design, Construction and Operation

of a Magnetic Suspension System for Aerodynamic Models

in a Small Wind Tunnel.) ONERA N T 1/1579 AP, Office

National

d'

Etudes et de Recherches Aeronautiques, Chatil

lon-sous-Bagneux

(Seine),

P aris, Prance.



114

30. Tournier, M.

Laurenceau, P.

Per fec t io nne me nts a l a Susp ens ion Magnet ique des Ma quet t es .

(Improvements in the Magnetic Suspension of Models.) ONERA

NT 5/1579 AP, O ffice National d'Etudes Reche rche s Ae'ro-

nautiques, Chatillon-sous-Bagneux

(Seine),

Paris, France,

Decembre

1956.

31. Tournier, M.

Laurenceau, P.

Dubois,

6.

La Suspension Magnet ique ONERA.

(The ONERA Magnetic

Suspension Syst em.) Conference on Premier Symposium

International sur 1'Aerodynamique et 1'Aerot hermique des

Gas Rarefies, Nice , Prance, 2-5 Juil let 1958. Off ice

National d'Etudes

et de

Rec her che s Ae'ronautiques, Chati l

lon-sous-Bagneux (Seine), Paris, Prance, Pergamon Press,

London.

32. Zapata, R.N.

Dukes, T.

An E l ec t r o ma g n e t i c S u s p en s i o n S y s tem f o r S p h e r i ca l M o d e ls

in a Hy pe rson ic Wind Tun nel . Report 682, Gas Dynamics

Laboratory, Princeton University, Princeton,

New

Jersey,

July 1964.

Derivatives Due to Rate of Roll

1. Al lwork , P.H.

Continuous Rotat ion Balance for Measurement of Yawing

and R o l l in g Moments in a Sp in.

ARC R & M 1579, November

1933.

2. Bennett , C. V.

Johnson, J.L.

Exp er im enta l De term ina t ion of the Damping in Ro l l and

Ai ler on Ro l l in g Ef fec t iv en es s of Three Wings Having 2° ,

42°, a n d 6 2 ° S weepback . NACA T N 1278. Ma y 1947.

3. Boat r igh t , W.B.

Wind-Tunnel Measurements of the Dynamic Cross Der ivat ive

C

l r -

Cja . (R ol li ng Moment Due to Yawing V el oc ity a nd

t o A c ce l e r a t i o n i n S i d e s l i p ) o f t h e D o u g la s D - 5 5 8 - I I

Ai rp lane and i t s Components a t Super sonic Speeds Inc lud ing

D es c r i p t i o n o f t h e T ech n i q u e . NACA RM L55H16, November

1955.

4.

Bobbitt.

P.J.

L i n e a r i z e d L i f t i n g - S u r f a c e a nd L i f t i n g - L i n e E v a l u a t i o n s

of S idewash Behind Ro l l in g Tr i an gu lar Wings a t S upe r son ic

S p e e d s . NAC A Repo rt 1301, 1957.

5. Bobbitt,

P.J.

Malvestuto,

P.S. Jr

6. Brown, C.E.

Adams,

M.C.

Es t im at i on of Forc es and Moments Due to R ol l in g fo r

S e v e r a l S l e n d e r - T a i l C o n f i g u r a t i o n s a t S u p e r s o n i c S p e e d s .

NACA TN 2955, Jul y 1953.

Damping in P i t c h and R ol l of Tr ia ng ul ar Wings at Sup er

s o n i c S p eed s .




115

7.

Brown, C.E .

Heinke, H.

S. Jr

Pre l iminary Wind-Tunne l Tes ts o f T r iangu la r and Rec tangu la r

Wings in St ea dy Ro ll a t Mach Numbers of 1 .62 and 1.92 .

NACA TN

3740,

June 1956.

8. Conlin, L.J.


9. Da rl ing , J .

Comparison of Some Exp erim enta l and Th eo re t ic al D ata on

Damping- In -Rol l o f a De l ta-Wing-Body Conf ig ura t io n a t

Super son ic Speeds . Na t iona l Resea rch Counc i l , Canada ,

LR-266, December 1959.

Roll Damping of the Spinner Finner Model at Mach 2.49.

NAVORD R epo rt 4502 , 1958.

10.

Evans, J.M.

Pink, P.T.

S t a b i l i t y D e r i v a t i v e s - D e t e r m i n a t io n o f I

O s c i l l a t i o n s .

ACA-34.

April 1947

by Free

Australian Council for Aeronautics, Report

11.

Evans, J.M.

Pink, P.T.

S t a b i l i t y D e r i v a t i v e s - D e t e r m i n a t i o n o f I by F r e e

R o l l i n g . Australian Council for Aeronautics, Report

ACA-35,

May 1947.

12. Franks. R.W. The Ap pl i ca t ion o f a S im pl i f ie d L i f t in g -S ur f ace Theory

t o th e P r e d i c t i o n o f t he Ro l l i n g E f f e c t i v e n e s s o f P l a i n

S p o i l e r A i l e r o n s a t S u b s o n i c S p e e d s . NACA RM A54H26a,

December 1954.

13.

Goodman, A.

Adair. G.H.

Es ti m at io n of th e Damping in R ol l of Wings Through the

Normal F l ig h t Range o f L i f t C oe f f i c ie n t .

NACA TN 1924,

July 1949.

14. Goodman, A.

Fisher, L. R.

15.

Greene, J.E.

16.

Hannah, M.E.

Margolis,

K.

17.

Harmon. S. M.

Martin, J.C.

18. Harmo n,

S.

M.

Jeffreys,

I.

I nv es t i ga t i on a t Low Speeds o f the E f fec t o f Aspec t Ra t io

a nd S weep on Ro l l i n g S t a b i l i t y D e r i v a t i v e s o f U n t a pe r ed

W in g s . NAC A Repor t 968, 1950.

An I nv es t i ga t i on o f the R o l l in g Mot ion o f C ruc i fo rm-F in

C o n f i g u r a t i o n s . NAVORD

6262.

1958.

Span Load D is t r ib u t io ns R es u l t in g from Cons tan t Angle o f

A t t a c k , S t e a d y Ro l l i n g V e l o c i t y , S t e a d y P i t c h i n g V e l o c i t y ,

and Cons tan t Ve r t i ca l Acc e le ra t ion fo r Tapered Sweptback

Wings with Streamwise Tips, Subsonic Leading Edges and

S u p e r s o n i c T r a i l i n g E d g e s .

NAC A TN 2831, Dece mber 1952.

T h e o r e t i c a l Ca l c u l a t i o n s o f t h e L a t e r a l F o r c e a nd Y a wing

Moment Due to R ol li ng at Su pe rso nic Speed s for Sweptback

Tapered Wings with Streamwise T ips , Su per son ic Leading

E d g e s . NAC A TN 2156, July 1950.

Th eo re t i ca l Li f t and Damping in R ol l of Thin Wings w ith

Ar b i t ra ry Sweep and Taper a t Supe rson ic Speeds , Supe rson ic

L e a d i n g an d T r a i l i n g E d g e s . NA CA TN 2114, May 1950.



116

19 .

Jo ne s, A. L.

Alksne, A.

The Damping Due to Ro l l o f Tr ian gu lar , T ra pe zo ida l , and

R e la te d Pla n Forms in Su pe rs on ic Flow. NACA TN 1548,

March 1948.

20 .

Lockwood, V.E.

21. Lomax, H.

H ea sl et , M. A.

D amp i n g - I n - R o l l C h a r ac t e r i s t i c s o f a 4 2 . 7 S w ept back Wing

as Determined from a Wind-Tunnel In v es t i ga t i o n of a

Tw is te d Se m isp an Wing. NACA RM L9P1 5. A ugust 1949.

Damping- In-Rol l Calcu la t ions for S lender Sweptback Wings

and S le n d e r Wing-Body Co m bi na tio ns . NACA TN 1950,

September 1949.

22.

MacLachlan, R.

Letko. W.

C or re la t io n of Two Exp er im enta l Methods of Determ in ing

the R o l l i ng C h a ra c te r i s t ic s o f Unswept Wings. NACA TN

1309,

May 1947.

23.

Margol is, K.

T h eo r e t i c a l C a l cu l a t i o n s o f t h e L a t e r a l F o r ce and Y awing

Moment Due to R o l l in g a t Sup ers on ic Speeds for Sweptback

Tapered Wings with Stream wise T ips , Subs onic L eading

E d g e s .

NAC A TN 2122, June 1950.

24.

Martin, J.C.

Gerber, N.

Oi

the Effe ct of Thic kne ss on the Damping in Ro l l of

A i r f o i l s a t S u p e r s o n i c S p e e d s . BRL Report 843, January

1953.

25. Martina, A. P.

26.

McDearmon, R.W.

Method fo r C al cu la t i n g the Ro l l in g and Yawing Moments

Due to R o ll in g fo r Unswept Wings With or W ithout Fl a p s

o r A i l e r o n s b y U se of N o n l i n ea r S e c t i o n L i f t D a t a .

NACA

TN

2937.

May 1953.

Wind-Tunnel In ve s t ig a t io n of the Damping in Ro l l o f the

Douglas D-5 58 - I I Rese arch Ai rp la ne and I t s Components a t


NAC A RM L56F07. Septe mber 1956.

27. McDearmon, R. W.

Jones, R. A.

Wind-Tunnel In ve s t ig a t io n of Damping in Ro l l a t S upe r son ic

Speeds of Tr iangular Wings a t Angles of At tack .

NACA RM

L56P13a, September 1956.

28.

Michael, W. H. Jr

A n a l y s i s o f t h e E f f e c t s o f W ing I n t e r f e r e n c e o n t h e T a i l

C o n t r i b u t i o n s t o t h e R o l l i n g D e r i v a t i v e s . NACA Report

1086.

1952.

29. Pinsker, W.J.G.

A S emi - E m p i r i ca l M eth od f o r E s t i m a t i n g t h e R o t a r y R o l l i n g

Moment D er iv a t iv es of Swept and S le nd er

Wings. ARC CP

524,

1959.

30. Polha mu s, E. C.

A Simple Method of Es t im at in g the Sub sonic L if t and Damping

in R o ll of Sw eptb ack Wings. NACA TN 1862, A p ri l 1949.

31. Queijo, M.J.

J a q u e t ,

B.

M.

Ca lcu la t ed Ef fe c t s o f Geomet r i c D ih edr a l on the Low-Speed

R o ll in g D er iv at iv es of Swept Wings . NACA TN 1732, O ctober

1948.



119

19. Purser,

P.E.

Stevens, J.E.

E x p l o r a t o r y Ro c ke t F l i g h t T e s t s t o I n v e s t i g a t e t h e U se

of a F r ee ly Sp inn ing Monoplane Ta i l fo r S t ab i l iz in g a

Body.

NACA

R M

L52I05a, Oct obe r

1952.

20. Regan,

P.J.

Horanoff. E. V.

21. Sieron,

T. R.

Wind Tunnel Magnus Me asurem ents at th e Na val Ordna nce

L a b o r a t o r y .

AIAA Paper 66-753, AIAA Aerodynamic Testing

Conference, September 21-23,

1966.

Oi th e "Magnu s E f f e c t s " o f a n I n c l i n e d S p i n n i n g S h e l l a t

Subson ic and Transon ic Speeds . WADD Technical Report

60-212,

1960.

22. Sieron,

T. R.

The Magnus C h a ra ct e r i s t ic s of the 20 mm and 30 mm Pro

j e c t i l e s i n t h e T r i s o n i c Sp e ed Ra ng e. WADC Technical

Note 59-520, October 1959.

23. Stone,

G.

W.

77ie

Magnus In s t a b i l i t y o f a Sound ing Rocke t .

AIAA Paper

No.66-62,

AIAA

3r d

Aerospace Sciences Meeting,

New

York,

January 24-26,

1966.

24.

Thorman,

H . C

Boundary Layer Measurements on an Axisymmetric Body with

S p i n an d Yaw.

Guggenheim Aeronautical Laboratory, Cali

fornia Institute

o f

Technology, Report

o n

Army Contract

DA-04-495-ORD-481, November

1957.

25.

Wolff, E.B.

Koning, C

Tes ts fo r De te rmin ing the E f fec t o f a Ro ta t i ng C y l inde r

F i t t e d in to th e Leading Edge of an Ai rpla ne Wing. NACA

TM 354, March

1926.



120

Methods of Measurement

of Dynamic Stability

Derivatives

Model Free to Move

in All Basic Degrees

of Freedom

Model Restricted in

Several Degrees of

Freedom

Free-Flight Model

Tests

Model Moving in the

Test Section

Measurement

of Forces

and Moments

Forced

Oscillation

Forced

Rotation

Measurement

of Model

Motion

Free

Oscillation

Free

Rotation

Cross

Oscillation

Pig.

1 Methods of measuring dynamic stability derivatives



121

POSITIVE SENSE OF AXES,

VEL OC ITIES , AND ANGLES

I S INDICATE D BY ARROWS.

c

N

(-c

z

)

Z W . Z T

C n , w O

NOTE:

1 . POSITIVE VALUES OF FORCES,

MOMENTS, AND ANGLES ARE

INDICATED BY ARROWS.

2 .

ORIGINS OF WIND AND STABILITY

AXES HAVE BEEN DISPLACED

FROM CENTER OF G RAVI TY , FOR

C L A R I TY .

C

L

Pig . 2 Axes system s



122

Note:

M o d e l O r i en t a t i on R e f e r en c e d t o

t h e S y m m e t r i c F l i g h t C on d i t i o n

N o t e:

Quantiti es given in

p a r en t h e s i s a r e v e c t o r s .

Pig. 3 Orientation of coordinate systems



S o u r c e AEDC-VKF

A E D C

1 5 5 6 4 - 6 6

Pig.4 Multlpiece crossed flexures

'ci



Cross-Flexure Pivot

Model-Pivot

Adapter

(Magnesium)

Torque Tube Pivot

Cruciform Pivot

Source AEDC-VKF

A E D C

44920-64

Pig.5 Cross-flexure, cruciform and torque tube pivots



125

Canti lever Support

Al l dimensions are in inches.

Moving

0.125 Diam

J

0.045

0.20

Fixed

Moving

Double End Support

Canti lever Support

Fig. 6 Small scale instrument flexure pivot



126

So ur ce DTMB, Ref . 17

Pig .7 Knife edge pivot

Source DTMB, Ref. 17

Typical Model

Sting.

Pig .8 Cone pivot - three degrees of freedom



Fig . 9 Bal l bear in g p iv o t

3



Model Mounting Bulkhead

Flexure

S

" I "

B e a m Se c t i on

O s c i l l a t i n g S h a f t

Source AEDC-VKF

(S ha de d E l em e n t s) 0 1 2 3

S c a l e i n I n c h e s

Pig.10 High amplitude forced osc il l at io n balance with bal l bearing pivot



129

" 0 "

Ring Seal

0.032-diam

Orifice, 8 Holes

Equally Spaced

0.00095—

Outer Ring

Shield

Shim

End Plate

2.965

Diam

Mounting Pads

Source AEDC-VKF, Ref. 19

All dimensions are in inches.

Transducer Housing

T

3.75

k

U —

'• Q B E

Eccentric Ring

•Outer Ring

^ S ^ - C o i l

1.87

\ - n c i

E" Core

Pig .11 Gas bearing pivo t



131

.-10.0

• a

i n

-5.0

X

. CD

•1 . 0

0.1

1— I I I I I

o

D

0

Q

e

p

Radial

Load,

lb

- 24

a 124

s 224

s 324

• ±11 .5 deg

~ 300 psig

0.5 1.0

Frequency, cps

J

i i I i i

5.0

10.0

Source

AEDC-VKF, Ref. 18

o

CD

T 3

ro

I

X

CD

150 200 250

Radial

Load, lb

Pig. 13 Damping charac teris tics

of

AEDC-VKP

g a s

bearing



132

Source AEOC-VKF

-Th ru st Gas Bearing

Fig.14 G as bearing for roll-dampi ng system

_i-

PPtch

T '

Core

(Tw 180* Apart*

Pltch-Y»w Eccentric

Model Mounting Surface

Bearing Pads

Y w

" t "

Cora

(Tm 180° Apart)

Source AEDC-YKF

Plenum

-Bearing Housing

-Bearing Core

P i g . 1 5 S p h e r i ca l g a s b ea r i n g



T e s t S e c t i o n

2

2

Z2gaZ^222£SSSSSSSSSSSSSfZ5

I p - X « K \ \ \ \ \ \ \ \ \ ^ f < H

Permanent Field Magnet

Oscillating Colls

Cruciform Spring

Source NAE

P i g . 1 6 T w o -d i me n si o na l o s c i l l a t o r y a p p a r a t u s

CO

'mm



136

-0.8

-0.4

Vd

od/2V

00

^ 0.75 0.010 to 0.015

0 1.50

0 2.25

D

3.00


Conical Flow

Theory

- a

E

o

E

O

-4.0

-2.0

• • . I

0.6 0.8 1.0

8

1.0

0.8

0.6 -

0.4 -

0.2

2.0

_ L

Firs t- and Second-

Order Potential Flow

Theory

l I

4.0 6.0 8.0 10

20

Re

t

x 10"

6

-«q T —

m

L - Laminar Flow at Base

T - Tu rb u lent Flow at Base

Solid symbols represent conditions

for which photographs are shown.

0.6 0,8 1.0

Re.x 10

I

(a ) M^ =

2.

5 , d_/d = 0.4 , a = ± 2 deg

Fig.19 Support interference effects



137

Source AEDC-VKF,

Ref. 28

-0.8

-

-0.4

V

d

ud/2V

03

XI

0

0

D

0.75

1.50

2.25

3.00

0.010 to 0.015

Conical Flow

Theory

e ° -4.0

E

o

-2.0

I I I I

0.6 0.8 1.0

0.6 0.8 1.0

Firs t- and Second-

Order Potential

Flow Theory

2.0

4.0 6.0 8.0 10

Re^xlO

6

20


for which photographs

are

s hown.

L

-

Laminar Flow

at

Base

T - Turbulent Flow at Base

Re.x

10

(b) M = 2.5 , d„/d =0.8 , a = ±1.5 deg

Fig.19 Support Interference effects



138

CD

E

-0.8

-0.4

i

s

/d ud^Vo)

^ 0.75 0.010 to 0.013

0 1.50

0 2.25

a 3.00

_a_

S ^ P

5

-


Conical Flow

Theory

.o -4.0 -

E

o

+

E "

U

-2.0

I I I I

0.6 0.8 1.0

8

•9 -

J I

J I 1_L

First- and Second-

Order Potential Flow

Theory

2.0 4.0 6.0 8.0 10

Re^ x

10 '

6



0.6 0.8 1.0

6.0 8.0 10

R e ,

x

1(

(a ) M^ = 3 . d

s

/ d = 0 . 4 , a = ± 2 d eg

F i g . 2 0 S u p po rt i n t e r f e r e n c e e f f e c t s



139

-0.8

-0.4

L i d ud/2V

co

^ 0.75 0.010 to 0.013

0 1.50

0 2.25

o 3.00


- & ~ ~ ^ = Z -

•

Conical Flow

Theory

fe

o -4.0

+

c

E

o

-2.0

I I I I

0.6 0.8 1.0

First- and Second-

Order Potential

Flow Theory

J L J i L -L

2.0

R e, x 10'

4,0 6.0 8.0 10



L -Laminar Flow at Base


0.6 0.8 1.0

6.0 8.0 10

Re.x 10

(b) M

ro

= 3 , d

s

/d = 0 .8 . CC = ±1.5 deg

P i g .2 0 S u p p o rt i n t e r f e r e n c e e f f e c t s



140

V

d

od/2V,

O


-0.8 -

-0.4 L

xi 0.75 0.009 to 0.010

0 1.50

0 2.25

o 3.00

o Free Flight

tf Free Flight with Boundary-Layer Trip

• Conical Flow

Theory

a ^ r j First- and Second-

'^"-•g Order Potential Flow

Theory

I I I I I I J. J

0.4 0.6 0.8 1.0

2.0

Re^

x 10

6

4.0 6.0 8.0 10


for which photographs are s hown.



0.4 0.6 0.8 1.0

6.0 8.0 10

Re.x 10

(a ) M ^ = 4 , d

s

/d = 0.4 . a = ± 2 deg

Pig.21 Support interference effects



Source AEDC-VKF. Ref. 28

E

-0.8

-0.4 L

XI

0

0

a

0

£

s

/d UOVZVQJ

0.75 0.009 to 0.010

1.50

2.25

3.00

Free F light

or Free Flight with Boundary-Layer Trip

Conical Flow

Theory

141

e

-4.0 -

8

1-4 r

1.2

1.0

0.8

0.6

0.4

0.?

-0.2

0.4

First- and Second-

Order Potential

Flow Theory

| ' | i I

0.8 1.0 8.0 10

R e ^ x l O "

6


(or which photographs are shown.


T - Tu rbu lent Flow at Base

1

0.6

i

i

0.8

1

1.0

1

2.0

Rej x 10 "

6

4.0

6.0 8.0 10

(b) Mo,

= 4 , d

g

/d = 0.8

,

a =

±1.5

deg

Pig.21 Support interference effect s



142

M

m

=

2.5

' C O

R e

i

x

-6

Moo-3.0

Re.

x 10"

o

A

1.0

6.0

D

0

1.5

6.0


Re^ x 10"

6

^J 2.8

0 5.4

& Free Flight

rf Free Flight with

Boundary-Layer Trip

Conical Flow Theory

E

7

•? 9

First- and Second-

Order Potential

Flow Theory

. . . i

8

a .

u . u

0.4

0.2

n

_ o

A

i

o

o

A

A

. 1 1

M * 3

j .

0.2 0.6 1.0 0.2 0.6 1.0 0.2 0.6 1.0

d

s

/d

Fig.22 Variation

of

static

and

dynamic stability coefficient

and

base pressure ratio

with sting diameter

for Z

g

/d = 3



143

Pig.23 Model bala nce and support assembly for Hotsho t (AEDC-VKF tunnel P) tunnel

Fig. 24 Free oscil latio n balance system



Source AEDC-VKF

Pig.25 Transverse rod and yoke system installed in test section tank



145

1 0

T r a n s v e r s e Rod D i a m e t e r - 1 .1 0 i n .

P i v o t A x i s - 0 . 6 0 i n .

O T r a n s v e r s e R od D a t a

A F r e e O s c i l l a t i o n S t i n g D a ta

— . _ T h e or y

Source AEDC-VKF

- 4 i -

- 2

0

- 1

- 2

• e

E

U 0

= 0 . 9 1 x 1 0

R e . - 0 . 6 9 x 1 0

cod _ T

-ryrj- « 1 . 0 4 X 1 0

- - ^ r * * * * * *

C T - 4

- 2

1.16 x 10

6

0

- 4

- 2 -

= 1.4 4 x 10

— i t t f r C

Re^ = 1 .8 3 x 1 0

6

% * - = 1 .7 0 x 1 0 "

3

1 1 1 1

R e

d

= 1 .8 3 x 1 0

6

'y^- = 1 .8 2 x 1 0 "

1 1 1

4 8 12 16 20 24 28

A m p l i t u d e o f O s c i l l a t i o n , ± 6 , d eg

F i g .2 6 D ynam ic s t a b i l i t y d e r i v a t i v e s v e r s u s a m p l i t u d e o f o s c i l l a t i o n , Mach 10



146

2.125 in.

0.30

in

Source AEDC-VKF.

Ref. 21

Pitch Axis,

41-percent

Body Length

-12

* -

8

8 -

+

o

E

-4

-

t

•12

•o

-8

e

Straight Afterbody

Conf igurat ion

Flared Afterbody

Conf igurat ion

&

0

8

9

o

D

a

Re

d

~ 3.9 x10

s

, f = 21 cps

Cy lindrica l Rod (with Roughnes s)

Double Wedge Rod (with Rou ghness)

Cylindrical Rod (No Roughness)

0.125 0.250

Rod Diameter

or

Thickness,

in.

Flared Afterbody Configu ration

0.375

0.560

Re

d

= 3 . 9 x 10

5

,

f~

20 cps

o Cylindrical

Rod

O Doub le Wedge Rod

I

- L

e

i

9

-L

0.125

0.375.250

Rod Diameter or Thickness, in.

Straight Afterbody Configuration (with Nose Roughness)

0.500

Pig.27 D a m p ing d e r i v a t i v e s

for the

flared

and

straight afterbody configurat ions

m e a s u r e d in conjunction with a s e r i e s of simulated transverse rod

configurations, forced oscilla tion

tests,

amplitude 1.5 de g , M a c h 5



147

Pig.28 Magnetic suspension system,

end

view

Suspension Electromagnets

y Model

Light Sou rce and Photocell

Drag Solenoid

Pig.29 Basic

"V"

configuration m agnetic suspension system



148

Source AEDC-VKF

15.5°

43

100

60

40

c

g 20

J 10

a 6

S 4

Cross -Flexure Pivot

Tare Damping

IKBalance Structural)

_ i _

6 8

Mach Number

10

Gas

Bearing

Piyot

V

^ A e r o

dynamic

Damping

Tare

Damping

« 1 %

10

20

F i g . 30 C o n t r i b u t i o n o f a e r o dy n ami c , s t i l l - a i r , and s t r u c t u r a l d amp in g t o

to ta l damping

Source AEDC-VKF

Tank in Closed Position

Pig.

31 Sket ch

of the

vacuum system used

to

evaluate still-air damping



149

1.6

0.4

© - 71.1

rad/sec

Source AEDC-VKF

6

8 10

Pressure,

psia

Mechanical

and Material

. 1

12

14 16

Fig. 32 Variat ion

o f

structural damping with static pressure

200

*-.

c

i .

loo

• a

so

E

s

i 20

e

fe 10

<

-I 5h

E

s

Zn

1

I I

Source AEDC-VKF

0 1

5 6 7

Mach Number

8 9 10 11

F ig .33 Va r ia t i on o f the r a t i o o f model s t i l l - a i r damping to mode l ae rodynamic

damping with Mach number



150

u

16

12 -

8 -

4 -

o Armco 17-4

D Ni-span-C —

A Bery llium Copper

0 Nickel

o Aluminum (2024-T4)

v Magnesium

A 7

*4%J04f2m ^ m T m m r t & ^ ' f ' ' L

8 12 16 20 24 28 32 36 40 44

M

ft-lb

9- rad

Pig .34 Va r ia t io n of the ang ular v i scous -damping-moment param eter w i th ang ular

r e s t o r i n g - mo men t p a r ame t e r

P i g . 3 5 T r i c y c l i c p i t ch i n g and y awi ng



151

S o u r c e R e f . 41

P i g . 36 Typica l free flight epic yclic pitc hing and yawing moti ons of a nonrolling

statically stable symmetrical missile

S o u r c e

R e f . 41

Pig.37 Typic al free flight epicyclic pitching and yawing moti ons of a rolling

statically stable symmetrical missile



152

Source Ref. 41

P i g . 3 8 T y p i c a l f r e e f l i g h t p i t c h i n g a nd y aw in g m o ti on o f r a p i d l y r o l l i n g s t a t i c a l l y

u n s t ab l e s y mmet r i ca l m i s s i l e

0.040

2 0.016

0.032

F i g . 3 9 M o tio n t r an s f o r m a t i o n



153

-1

B(t-t

2

»

/

T \

\

\ /

cft • t

2

> y

******mmj \

( C +

B

i f l

3

— - y - ^ ^ ^ — T '

Btt -t i»

y f

\ / C ( t - t j )

i

/

* = »

^ - ( C - B)

Fig .40 Complex ang ular motion

B - K , e

X

l

X

?

t

C - K 2 6

l

C+ B

Time, t

F ig .41 Expon en t ia l decay o f th e modal ve c to rs



154

Reference

Plane

pfo

Reference

Plane

i

M:

1

M

r

r

• "7

/

/ > "

^

e

r

M

3

^

\

v .

B

i

Forced Oscillation Off of Resonance

Reference

Plane -Cuft

Forced Vibration at Resonance

Fi g .4 2 Phase d iagram



155

Pressure Supply Tubes-

-Strain-Gage Bridge

-•—Cross-Flexure Pivot

S o u r c e

A E D C - V K F , R ef . 47

P i g .

43 High frequency free oscil lati on balance

for

Hotshot tunnel test s

Displacement Arms

Flexures

Releasing Pressure

Inlet

Displacing Pistons

Displacing Pressure Inlet

1 2 3

-Set Screw

_ i _

Scale, in.

Pig.43 Continued



A c t u a t i o n

S y s t e m

D i s p l a c e m e n t L e v e r

S o u r c e O NE RA , R e f . 4 7

Fig. 45 Free o sc il la ti o n balance and displacement mechanism

3



159

P i g . 4 7 F r e e - o s c i l l a t i o n t r a n s v e r s e r o d b a l an c e



o

A E D C

7 6 4 3 - 6 6

A E D C

7 6 4 4 - 6 6

Sou rce AEDC-VKF

Item No.

1

2

3

4

5

Description

Gas Bearing

Transverse Rod

"E "

Core

"E " Core Mo u nting Bracket

Eccentric

Item No.

6

7

8

9

10

Description

Gear Rack

Air Actuated Piston

Lock Housing for Air Piston

Lock Actuating Air Line

Model Mounting Pad

Pig.48 Photographs

of the

free-oscil lation (+35 deg) t ransverse

rod

balance



Vertical

Flexure

Forward

Support

Block

Tongue and V-Block

\Braking Mechanism

Hydraulic Motor

'/

Cam and Follower

Actuating Mechanism

Hydraulic Cylinder

%M*4

^•v*" 4« ^kfe

A x i a l F l e x u r e - ^

^ —H o u s in g \

A f t S u p p o r t B lo c k - ^

^ S t i n g

A d a p t e r

Source AEDC-PWT,

Ref . 49

Fig.49 Free-o scillatio n balance



162

Photo Cell

m m /

Light Beam

Quartz Fiber

Electromagnet

Transverse

Rod Su pport

Bearing (Electrically

Vibrated Inner Race

Source U

of C,

Ref.

50

Fig. 50 Tra nsverse rod balance for low density hyper sonic flow tunnel

A E D C

4 3 8 5 2 - 6 6

4 3 8 5 1 - 6 6

S ource A E D C -V K F

Fig.51 Photograph

o f t h e

assembled balance



163

j jSF

1

Source AEDC-VKF

Pig. 52 Small scale free-oscillati on balance

PERFORATED BULKHEAD-

PRESSURE TRANSDUCER-

DYNAMIC BALANCE-

INNER SKIN

OUTER SKIN

(POROUS)

INNER SKIN

DIVIDING PARTITION

LOWER PLENUM

OUTER (POROUS) SKIN

GAS SUPPLY HOLE

RTITI ON-

CROSS SECTION

AT

BALANCE ^

TRANSDUCER

UPPER PLENUM

GAS SUPPLY LINE

BALANCE FLEXURE

S o u r c e

LTV, Ref. 51

Fig.53 Basic mass in jection balance and porous model



S o u r c e L T V, R e f . 5 2

VZZZZm7Z7

A

Air Passages

Air Passages

J

0.01-in. Wall

Wall

0.0i-in^?2^0

:

09

izzzzz^J.

Crucifo rm Design Rectangular Design

Designs of Dynamic Balance Pivot Cross Section

£

Pig.54 Dynamic stability balance suitable

f o r g a s

injection



165

Pig. 55 Oltronix dampometer

Source Oltronix, Ref. 53

High Damping

Pig.56 Typic al calibrated slotted disks and principle o f dampometer operati on



166

Light Source

Travel Summation Method

A

• I

( 1 - b + b l n b )

s

?

m

s - f E ( x

n

- x

m

) - ( x

0

- x

m

)

x

o

o

Narrow Slots

Source NAE, Ref.

54

Pig.57 Pree-oscil lation data reduction

S h a k e r M o t o r -

m

o I

' - L o c k i n g D e v i c e

P u s h Rod

S o l e n o i d

f or

A c t u a t i n g M o d e l

L o c k i n g D e v i c e

S ource A E D C -V K F i

i I i I i l i i i i

0 2 4 6 8 10

S c a l e

In

I n c h e s

Pig. 58 Sting balance assembl y

of

the forced oscillation balance system



167

4$

Y o k e

L i n k a g e

S o u r c e A ED C -V K F, R e f . 5 6

[

Hinge Flexure

Strain-Gage Bridge (Displacement)

•Connecting Flexure

•Input Tor que Memb er

(Strain Gaged at Minimum

Cross Section Area)

Model Locking Device

0 1 2 3

• I I • I I I

S c a l e i n I n c h e s

Pig.59 Small amplitude (±3 deg), fo rce d- osc il la t io n balance



TORQUE

ANALOG SIGNAL INPUTS

DISPLACEMENT

IN-PHASE

TORQUE COMPONENT

LOW FREQUENCY

PEAK-TO-DC

CONVERTER

LOW FREQUENCY

PEAK-TO-DC

CONVERTER

FREQUENCY

COUNTER

ANALOG-TO-

DIGITAL CONVERTER

QUADRATURE

TORQUE COMPONENT

DC RATIO

DIGITAL VOLTMETER

ANALOG-TO-

DIGITAL CONVERTER

MANUAL INPUT

CONTROLS

TUNNEL

PARAMETER

INSTRUMENTS

I

SCANNER

(DIGITAL)

L & N

ANALOG-TO-DIGITAL

CONVERTERS

ERA 1102

DIGITAL COMPUTER

FLEXOWRITER

OR

PRINTER

Source AEOC-VKF

Pig.61 On- l ine data sys tem for force d-os ci l la t ion dynamic s ta bi l i ty tes t ing

a

to



o

P i g . 6 2 C o n t r o l p a n e l f o r f o r c e d - o s c i l l a t i o n sy s te m



171

S o u r c e A ED C -P W T, R e f . 5 7

^STMM-GMD) LEV sptme

FO* UUSUWG ANOJUUt DcrLECTirx

w FOB m e o u o c r v u u n o N

StCTiow A-|>

F i g . 6 3 F o r c e d - o s c i l l a t i o n b a l a n ce

Pig.63 Continued



TWO - PHASE

OSCILLATOR

GAGE

SUPPLY

PANEL

EXCITATION

PHASE

ADJUST

CHASSIS

DRIVE SIGNAL

i

SERVO

CONTROL

T

FEEDBACK SIGNAL

READOUT

EXCITATION

GAGE OUTPUT

ERROR

SIGNAL

BALANCE

POSITION

FEEDBACK

COMPUTER

TABULATORS

AND

PLOTTERS

Source

AEDC-PWT, Ref. 57

to

Fig.64 System block diagram for forc ed-osc il lation balanc e



ASSEMBLED OSCILLATION

BALANCE MECHANISM

DRIVE SHAFT

OPTING STING

DISPLACEMENT BRIDGE

MECHANICAL SPRING

Source NASA-LRC. Ref. 59

SCOTCH-YOKE

PIVOT

SHAFT

MOMENT BRIDGE

Pig.65 Photograph of

the

forward po rtion of

the

forced-oscilla tion bala nce mechanism



Moment ,—Scotch

yoke

Displacement beam

r

S o u rc e NASA-LRC, Re f . 61

Pig.66 Schemat ic view

o f

model

and

driving-syst em components



S o u r c e N A S A -L R C , R e f . 6 1

Displacement.

beam

_ Moment

beam

^ ^

Model pivot

axis

. \

j .

y y > / / / / / /

} / ' / ' * *

i

' / { ' / * * r * J *

' ' ' *—*

5-

-Scotch yoke

F i g . 6 7 D e t a i l s o f o s c i l l a t i n g m ec ha ni sm



Attenuator

.

Bridge

balance

circuit

Linear

amplifier

y>flre\

/

strain X

< gage y

Xbrldge

\S

(Real comp onent)

Resolver

Attenuator

(imaginary component)

Attenuator

Phase

shifter

Phase

shifter

Demodulator

and

filters

D-c

nlcroafflffleter

—

'

W Demodulator

and

filters

D-c

mlcroammeter

05

o

0-V 3K

C

Carrier

Source NASA LRC, Ref. 60

Fig.68 Block diagram of electronic circuits used to measure model displacement and

applied moment



177

FP tQ UC NC Y MOTSIJLATtO

VAW 1

r i j

j | ElClTCR CUPBENT

i

Schematic arrangement

for

yaw-sideslip oscillations.

S o u r c e RAE, Ref. 61

Pitch

or

yaw spring unit.

Roll spring unit.

P i g . 6 9 F o r c e d - o s c i l l a t i o n d yn am ic s t a b i l i t y b a l a n c e



- J

00

• r

Balance

t i i

O I 2

INCHES

C

Source RAE, Ref. 61

Accelerometers

Pig.70 Y aw-roll balance and accelerom eters



179

B

/ / / / / / /

.Mognet Coil

TOP

Ins t rumenta t ion

Leads

V I I I I I > I

mz

i i i i 11 1 11 11 11 1 n

11 J i / i

aM.

i ) i i i i i ) i ) i i ) - m

Pi vo t Ax i s

Source UM, Ref. 64

SIDE

Krr

Coi

Posit ion

Magnet Coir

\....L.7^^

/

-

/ / / / / / / > < / / / / 1 1 1 I /I i > 1 1 1 r r ,

F i g . 7 1 F o r c e d - o s c i l l a t i o n b a l a n ce w i th e l e c t r o m a g n e t i c e x c i t a t i o n

Pivot Axis

Exciter Coil Lead

Strain-Gage

Leads

Magnetic

. Field

Wind

Direction

Source

UM,

Ref. 64

P i g . 7 2 C utaway v iew o f t o r s i o n a l f l ex u r e p i v o t w i t h s t r a i n g ages f o r

f o r c e d - o s c i l l a t i o n b a l an c e w it h e l e c t r o m a g n e t i c e x c i t a t i o n



181

Source PU, Ref. 66

Nylon

Copper

Pig. 74 Cone model

for

magnetic suspension

Lift Electromagnets

Lateral

Electromagnets

Mirror

Model

^—Light Source

i ^ s ^ ; and Pho toce ll

Drag Solenoid

Pig.75 Basic

L

configuration magnetic suspension system



182

P H O T O C E L L S

N O Z Z L E OF

W I N D T U N N E L

D R A G S O L E N O I D

SIDE VIEW

L I F T

E L E T R O M A G N E T 5

U G H T S E A M S FOR

L A T E R A L C O N T R O L

L.1GMT

B E A M FOR

D R A G C O N T R O L

L I G H T B E A M S FOR

L I F T C O N T R O L

Sourc e RAE, Ref. 68

I F T E L E C T R O M A G N E T

L A T E R A L

E L E C T R O M A G N E T

S C H L I E R E N

BEA.M

D R A G C O i L

L I F T L I G H T

B E A M

D R A G L I G H T

B E A M

L A T E R A L

L I G H T B E A M

END VIEW

Fig.76 Magnetic suspension system



Sourc<

British Crown Copyrig ht. Reproduced

with the permi ssion of Her Britannic

Majesty's Stationary Office.

©CONICAL TEST MODEL

©FRONT LIFT ELECTROMAGNET

® REAR LIFT ELECTROMAGNET

® FRONT LATERAL LIGHT BEAM

® REAR LATERAL LIGHT BEAM

©QUARTZ WINDOW

©DRAG LIGHT BEAM

©REAR LIFT LIGHT BEAM

©DRAG PHOTOCELL

©REAR LATERAL ELECTROMAGNET

©FRONT LIFT PHOTOCELL

©R EA R LIFT PHOTOCELL

©UPPER SCHLIEREN WINDOW

©LOWER SCHLIEREN WINDOW

©DRAG COIL

©INLET NOZZLE (lVl -8-6)

©L I G H T SO U R C E U N I T

©MODEL LAUNCHING UNITS

©OBSERVATION WINDOW

@ SERVICES SUPPLY PANEL

Pig .77 Magnetic suspen s ion working sec t io n . 7 in . x 7 in . hyperson ic wind tunn e l

a >

to



184

Airstream

Direction

) Source SU

Ref. 70

\

Pig.78 Schematic diagram of the suspension magnet array of a six-component

magneti c wind tunnel balance and suspension system



185

Fig.79 Model subjected to a forced oscillatio n in roll while suspended in the

six-component balance



186

Pig.80 View of control surface drive



ATTACHED TO WIND TUNNE L STRUCTURE

VERTICAL SUSPENSION SPRING

(b)-

.-SRHCER

*•—REFLECTION PLATE

Sour c e NAE, Ref . 74

CONTROL SURFACE

WING

CONTROL

SURFACE

•y » m >> » r — B O P S

t

DETAIL OF HINGE ARRANGEMENT

(o) INPUT AND OUTPUT OF THE

STRAIN GAUGE BRIDGE

(b ) A.C. INPUT TO THE COILS

Pig .81 Apparatus for elevator osc il lation

CO



188

Fig. 82 Appa ratu s for elevator oscilla tion



PERMANENT MAGNET

CRUCIFORM SPRING

(WITH STRAIN GAUGES ONI

POLE PIECES

O F M A G N E T

COIL'

COIL HOLDER

THIS SPRING END ATTACKED

TO FRAME

MODEL CLAMPED TO

THIS SPRING END

ARM CARRYING COIL HOLDERS

THIS FACE OF FRAME ATTACHED

TO W I N D TU N N E L W A L L

S o u r c e NA E, R e f . 7 6

Pig.84 Elastic model suspension

a nd

electromagnetic oscillator

f o r

osclllation-in-pitcu



ELASTIC

MODEL

SUSPENSION

MODEL

DISPLACEMENT

TRANSDUCERS

PRINTER

I

MAX - MIN

TRIGGER

H

ELECTRO

MAGNETIC

OSCILLATOR

DAMPOMETER

COUNTERS

' ^ - c T

<

o I

W l

P H O TO C ELL

PHASE

SHIFT

MECHANICAL CONNECTIONS

ELECTRICAL CONNECTIONS

POWER

AMPLIFIER

LIMITER

Source NAE, Ref. 76

Pig.85 Free oscillat ion with automatically recycled feedback excitation



192

P i g . 8 6 H a l f- mo d el o s c i l l a t i n g r i g



193

Source NOL, Ref. 78

* * *

Pig.8 7 Roll-damping system

E l e c t r i c a l C l u t c h

ou rc e NOL, Ref . 79

Tachomete r

R a d i a l S u r f a c e s — . T h r u s t S u r f a c e -

T h r u s t S u r f a c e

4 0 0 - p s i A i r

S u p p l y f o r B e a r i n g

Air Motor

Air to Motor

Fig .88 Air bear ing roll-da mp ing mechanism



a

*.

Air Turb ine

-Thrust Gas Bearing

N

—Gas Bearing Supply Line

Pig.89

Ai r

bearing roll-damping balance

Wafer-Cooled Sting

Source AEDC-VKF



D y n a m i c S t e a d y - R o l l B a l a n c e , In t e r na l D e t a i l s

27.875-

S l e ev e A s s e m b l y — ,

S p i d e r B

B r a k e L i n k s —

\

/ _., V- t

A

\ / Piano \ —

t

Front Portion Outer Shell? \ / wire \

Brake Lever Support \ / \

-Tachometer

•Electrical I Rotor

\Condui -—Rear Portion

Outer Shell

Source OAL , Ref. 80

Sectio n B-B Section A-A

Pig .90 Dynamic s te ad y- ro l l ba lance

On



Pig.92 Pictor ial drawing, forced-osc illation balance



198

Sections

Boundary layer

is

shifted

to

asymmetric shape

b y

sp in.

(a) Magnus force

at low

angles

of

attack

( 0 *>>

Separated region is shifted

to asymmetric shape b y sp in.

Cy(-)

(b) Magnus force at large angles of attack

Lower lift is

lost f i rst.

Force on th is f i n i s b lanketed

off leaving

a net

side force.

Axial Components o(

Fin Normal Force

(-C

n

) Net Couple Results

(c) Magnus ef fe ct s on can ted f i n s

_ / — Force on th is f i n

" ^ f

is

blanketed

off.

\ Z . — Cy(+)

(d) Magnus effects

on

drive n fi ns

Pig.93 Illustrations of the various Magnus effects



M O Q - 2 . 0 , Re

d

=0.65

= 10

4

TD

OO

8

0.004

0.003

0.002

0.001

0

-0.001

-0.002

-0.003

-0.004

L--0-—9

o

j ^ ^ T o .

°y

/

'

Q 6

n r

1

Source BRL,

Ref. 89

No.

20 Emery Grit Transition Strip

</) <m>

< / • > on

o> co

> - -C

^

E

•S

£

S 0

D - Q

CL >

(J

u

£ c

^ CD

C o

C O

-2

\

r^///i

Data

for all

angles

of

attack

lie in

this region.

p ^

0 0.1 0.2 0.3 0.4 0.5

V

0D

0 0.1 0.2 0.3 0.4 0.5

Fig.94 Effect of spin p arameter on the Magnus characteristics

to

CD



200

a

T}\ 8

3 \ >

eg

CM

8

>

8

Q .

a

o

o

o

o

ml

o

o

S o

Cm

M = :

0 0

Re , =

a

0 . 0 1 0

0 . 6 5 x 1 0 '

S o u rc e BRL, Re f . 89

0 . 0 0 5

0

H _

a

Sym

O

D

A

O

d e g

Laminar Boundary Layer

No. 20 T r a n s i t i o n S t r i p ( L a r g e s t G r i t )

No.

40 T r a n s i t i o n S t r i p (M edium G r i t )

No. 80 T r a n s i t i o n S t r i p ( S m a l l e s t G r i t )

Fi g. 95 Eff ect of boundary la ye r on th e Magnus for ce



-Tachometer No. 1

Tachometer

No. 2 -

Pig.96 Electric motor drive. Magnus force system

Source Sandia Corp., Ref.

92

Magnetic

Tachometer

zzzzzzzzz

\\\w

/ / / /

Two-Stage

Air Turbine

Model Drive

z / / / / / z :

/

/ \ * / ; / / , > / / >

S-

Turbine

Air

Supply

Source NO L, Ref. 79

F o u r - C o m p o n e n t B a l a n c e

Fig .97 Magnus forc e system, turb in e dr iv e wi thin the model

w



NOZZLE a HOUSING ASSEMBLY

BALANCE

^^^mm^m

wmmzM

T—-TURBINE

URBINE LOCK RING

[ZZZZ3£

" " " { ' " " " " " " " - * •

MAGNET RING S REAR <— SLEEVE

BEARING RETAINER

Q= 90° TO 180° MODEL

PICKUP COIL

t. a a a a a

G

w^vvwr

J J t W / J J U f - M J j m - r - r

BEARING PRELOAD SPRING

BALANCE SHOWN ROTATED 9 0*

a =90° TO 180° MODEL

FORWARD BEARING

INNER RETAINER

Source

WADD, Ref. 94

to

Pig.98 Magnus force system, turbine drive within

the

model



Air Turbine

Model Drive

•Four-Component Balance

Fig.99 Magnus force balance

Water-Cooled Sting

Source AEDC-VKF

"E "

C O R E -

TRANSDUCER

-M A G N E T IC

TACHOMETER

AIR OPERATED

CLUTCH

-AIR EXHAUST

'mm-BALANCE COOLING

i - BEARING AIR

' ^ B A L A N C E C O O L I N G

^AIR INLET

Source NOL, Ref. 79

AIR BEARINGS

THRUST & RADIAL

MULTI-STAGE

AIR TURBINE

Pig .100 Magnus-ro l l damping ba lance wi th a i r bear i ngs

M



204

190

170

o" 150

S

u

3

4->

a

u

01

| 130

5

H

M

e

•

tl

CO

a

« 110

90

70

Valve Opening in Percent

J

L

J

L

Source WADD, Ref. 94

J

I I L

10 20 30

rpm x 10"

40

50

60

Pig.101 Variation of bearing temperature with spin rate for bypass air cooling



SCALE

OIL MIST

PATH

Source WADD.

Ref. 94

-TUNNEL STINO

STRAIN GAGE BRIDGE

(I

of 4)

€_.—k

RPM COIL

TYPICAL THERMOCOUPLE

®

(

>t4

'

T U N N E L

M

n u

&

D

»

CONTROL ROOM

POWER

SUPPLY

JK C 10V

, M f t f l TT 9 i » °

A

TRON ' H |» W-

SUPPLY ? r r r ;

NOBATRON

POWER SUPPLY

6 V DC

SKC CARRCR

AMPLIFIER

BROWN

POTENTIOMETER

X

r

J

IBM

SUMMARY

PUNCH AND

TABULATOR

SANBORN

OSCILLOGRAPH

HEWLETT

PACKARD

- | FREQ. METER

9** " *

V f 4

LINEAR

SWEEP

**

SKC VOLTAGE

DEMODULATOR

1 P

110 »

6 0 -

1R0WN

STRIP CHART

RECORDER

CEC CARRIER

AMPLIFIER

TIME

DELAY CIRCUT

DEMODULATOR

PREBEND

POSITIONING

B DRIVE

CONTROL

MO v

« 0 -

L 8 E E B

T^frj

ELECTRONIC

PLOTTER

ELECTRO

INSTRUMENT

X- Y PLOTTER

(TYPJ (I of 4 )

TO PEN DROP RELAY

r ^ « "

ELECTRONIC

FILTER

-o*-d

%

1200 PSI

PRESSURE

CM

TAPE

~W I RECORDER

OIL M IST REGULATOR

WATER TRAP

DEMODULATOR

OVERDRIVE

CHECK SCOPE

(TYP.) (I of 4)

5

1

^ " R E C O R D E R

0

AIR

REGULATOR

^H2f

ALEWTE

OIL HIST SYSTEH

MOISTURE [

MOISTURE FILTER

A PARTICLE

FILTER

PRESSURE

STATIC FORCE | UAONUS FO RCE

RECORDING

8 MONITORING SYSTEM

DRIVE

8 LUBRICATING SYSTEM

Fig.102

Oil mist lubricatin g system and other instrumentation



207

Cm

> 4

< U

bD

a

O

+ - >

a

o

u

0 . 5 0

1.00 -

Semic onductor Gage

Sourc e BRL, Ref. 96

—

1

_^22

— - * ^ - ,.

1 1

1 2

M o d e l S p i n R a t e , rpm x 10

C o n v e n t i o n a l G a g e

3

- 3

4

Fig.104 Comparison of semiconductor and conventional gage outputs



208

be

0)

X3

be

c

<

ca

S

a,

120

110

100

9 0

8 0

7 0

6 0

50

4 0

3 0

2 0

10

0

1

' I ' I I | I I I

S o u r c e , R e f . 1 0 4

1 0 0

I

. 1 . 1

I I M I I

2 0 0 4 0 0

F r e q u e n c y , f, c p s

7 0 0

Pig.105 Experimental response magnitude and phas e-ang le ca li br at io n for 1.38 in. of

0.030 in. nominal-diameter tubing connected to a gage volume of 0.02 cu.in.

fo r a i r a t sea- level condi t ions



209

Motor Driven Cam

Return Spring

Source Ref. 101

Plunger

Corrugated

Rubber Seal

To Manometer

Piston

/ / s / / / / / /

Source RAE, Ref. 101

F i g .1 0 6 Pump f o r s i n u s o i d a l o s c i l l a t i n g p r e s s u r e

P r e s s u r e G a g e t o

b e C a l i b r a t e d

O r i f i c e

- ^

y//////w/////////7rmr.

H ™

N

r

1

R e f e r e n c e P r e s s u r e G a g e

H i g h - P r e s s u r e

S u p p l y

So ur ce , Ref . 104

P i g .1 0 7 Cam -ty pe p u l s a t o r c a l i b r a t o r



D I S T R I B U T I O N

Copies o f AGARD publications m a y b e obtained in the

various countries

at the

addresse s given below.

On peut se procurer de s exemplaires des publications

de 1' AGA RD

aux

adresses suivantes.

BELGIUM

BELGIQUE

Centre National

d'

Etudes

et de

Recherches

Aeronautiques

11, rue

d'Egmont, Bruxelles

CANADA Director

of

Scientific Information S ervice

Defense Research Board

Department

of

National Defense

"A" Building, Ottawa, Ontario

DENMARK

DANEMARK

Danish Defence Research Board

0sterbrogades Kaseme

Copenhagen 0

FRANCE

O.N.E.R.A.

(Direction)

25 avenue d e l a Division Leclerc

92 C hatillon-sous-Bagneux

GERMANY

ALLEMAGNE

Zentralstelle

fur

Luftfahr tdokumentation

und Information

Maria-Theresia Str.21

8 MUnchen 27

At tn : Dr Ing . H . J .R au te nbe rg

GREECE

GRECE

Greek National Defence Staff

D.JSG,

Athens

ICELAND

ISLANDS

Director

of

Aviation

c/o Plugrad

Reykj

avik

ITALY

ITALIE

Ufflcio

del

Delegate Nazionale

all'

AGARD

Ministero Difesa

-

Aeronautica

Roma

LUXEMBURG

LUXEMBOURG

Obtainable through Belgium

NETHERLANDS

PAYS

BAS

Netherlands Delegation to AGARD

c/o National Aerospace Laboratory

NLR

Attn: M r

A.H.Geudeker

P.O.Box 126

Delft



NORWAY

NORVEGE

Norway Defence Research Establishment

P.O.Box 25

Kjeller

Attn:

M r O.Blichner

PORTUGAL Diraccao do Servigo de Material

da Porca Aerea

Rua d a Escola Politecnica 42

Lisboa

Attn:

Brig.

Gen. J.Pereira

do

Nascimento

TURKEY

TURQUIE

Turkish Delegation

to

NATO

North Atlantic Treaty Organisation

Brussels 39, Belgium

Attn: AGARD National Delegate

UNITED KINGDOM

ROYAUME UNI

Ministry of Technology

T.I.L.(E) Room 401

Block A, Station Square House

St.

Mary Cray

Orpington, Kent

UNITED STATES

ETATS UNIS

National Aeronautics and Space Administration

Langley Field, Virginia

Attn:

Report Distribution and Storage Unit

*

Pr in t ed by Techn ica l Ed i t i ng and Reproduc t ion L td

Harford House, 7-9 Ch ar l o t te S t . London , V. l .

techniques for measurement of dynamic stability derivatives in ground test facilities

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