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- I

HYPERSO.\'/C RESEARCH LABORATORY

", ,1

I,

-,,I J

Al'{'{(>y~J for public release; distribution Unlimited.~ .l.. ' \

PROJEcr 7064

\ .

KEVIS E.YELMGREN, CAPTAIN, USAF\

AIRH"RCESYSTEMS COMMAND'~~»..w: ~6

AfU 72wOl00

JUNE 1972

/ /t"

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r }

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'~'l'?irJ';~~,

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r "

, I', i\

DYNAMIC BALANCING OF A NONROLUNG, STING SUPPORTED, WIND TUNNEL MODEL

PERfOJtMING NONPLANAR MOTIONS

KEVIN E. YELMGREN, CAPTAIN, USAF

HYPERSONIC RESEARCH LABORATORY

JUNE 1972

PROJECT 7064

AEROSPACE RESEARCH LABORATORIES AIR FORCE SYSTEMS COMMAND

UNITED STATES AIR FORCE WRIGHT-PATTERSON AIR FORCE BASE, OHIO

FOREWORD

This report was prepared by Capt Kevin Yelmgren of the Aerospace

Research Laboratories, Air Force Systems Command, under Project 7064,

/ entitled" High Velocity Fluid Mechanic s. "

The author acknowledges the advice given by Mr. Otto Walchner and

Mr. Frank Sawyer, as well as the technical support provided under contract

by Technology Incorpor ated.

ii.

ABSTRACT

Wind- off, bench tests on a wind tunnel model that had fr eedom to pitch,

and yaw, but no freedom to roll indicated that the pitching and yawing motions

were inertially coupled. An order of magnitude analys is showed that this

coupling was due to the product of inertia term I yz • The inertially coupled

equations of motion are solved, and a method is pres ented for determining

the magnitude and sign of Iyz from recordings of the model motion. The

method used to dynamically balance the model is described, and experimental

results are pr es ented that verify this method.

iii

TABLE OF CONTENTS

SECTION

I INT RODUCTION •••••• 0 • .. . . . . II THEORETICAL ANAL YSIS • • 0 • • • 0 • •

III EXPERIMENTAL INVESTIGATION. o • •

PROCEDURE •• o • • . . . . . RESULTS • • • • • • • • • • • • • • • • • • 0

IV DYNAMIC BALANCING METHOD • • • • • 0 • • " " 0

V CONCL USION. • · . . . . . . . · . . VI REFERENCES, · .. ,. . . . • 0 • 0 •

VII APPENDIX A •• · . . . . . . . · . . . .

iv

PAGE

1

3

11

11

12

14

15

16

23

!

I I i ,

FIGURE

1

2

3

4

5

6

II

. LIST OF FIGURES

Coordinate system ...• III • • III • • III • • • • • • • • • •

Effect of I on circular frequency in pitch yz

Effect of I on circular frequency in yaw. yz

Pitch and yaw components for original model plus an added I of -5.25 x 10- 5 slug-ftz

yz

a. Pitch component vs time

b. Yaw cOInponent vs time.

Pitch and yaw components for original model plus an added I of +4.27 x 10- 5 slug-HZ

a.

b.

yz

Pitch component vs time

Yaw component vs time.

Pitch and yaw components for dynamically balanced rrlOdel

a. Pitch c OlTlpOnent vs time

b. Yaw component vs tilTle.

v

PAGE

17

18

19

20

21

Ae, Be' Al/J' Bl/J

Ce, Cl/J

12

1 ,1 ,1 x Y z

1 ; I ,I xy xz yz

M,M,M x Y z

p,q,r

t

x,y,z

0l/J' ° e l/J,e,cp

NOMENCLATURE

An1.plitude components

Flexure darnping constants

Cons tant defined by Eq. (19)

Moments of inertia about body axes

Products of inertia about body axes

Moments about body axes

Angular velocity components on body axes system

Normal coordinates, Eq. (A-27) and (A-2S)

Time

Body axes system (Fig. 1)

Space fixed axes

Constants defined by Eqs. (A-38) and (A-39)

Eulerian angles

Constants defined by Eqs. (14) and (15)

Constants defined by Eqs. (16), (17) and (IS)

Circular frequency in pitch and yaw

vi

I. INTRODUCTION

In order to investigate possible aerodynamic interactions between

the pitching and yawing motions of a nonrolling body undergoing non-

planar motions, a flexure was designed to give a wind tunnel model

freedom to either pitch, yaw or pitch and yaw simultaneously. The model,

a 10 0 half angle cone, was weighted so that the body axes of rotation

were intended to be the principal axes of inertia, and yet Iy was not equal

to I z . Releasing a model balanced in this manner from a combined pitch

and yaw attitude with zero initial velocities results in a nonplanar

Lissajous Dl.otion, that is, a motion with different frequencies in pitch

and yaw. A s the model is not rolling, the pitch and yaw components of the

motion should be uncoupled, dmuped, sinusoidal oscillations.

Wind-off bench tests with the model in vacuum did not give the uncoupled

motion that was expected. The pitching and yawing motion envelopes were

not smoothly damped exponentials but rather 11 sinusoidally" damped expo-

nentials. The sketches below are an exaggeration of the motion obtained .

. The initial velocities in both sets of sketches are zero ( 8

0 = t/J 0 = 0), with

t/J 0 = 80

in the first set, and t/J 0 = 0, in the second set. This beating

type of motion indicates that the pitch and yaw motions are coupled due to a

slight inertia asymmetry, that is, the principal axe s of inertia do not coincide

with the axes of rotation. The purpos e of this report is to determine which

product of inertia causes the coupling, to determine its magnitude from the

1

/

observed coupled nlOtions, and finally to show how the luodel lnust be

. balanced to eliminate the inertial coupling.

~ In

0'" tlJ~ o~

Ct I-w~ :eN t-7

~

"' ", <-10 . 00

o o

'"

[.')0 we> 0":

~Iir 1I1I1 I\'IIII!I~ 1\1 !il'! \ ,Iii. Ii!

I 'Ii IIJ lilli!,!ij~I~~~flj\ ~ J 1111111lllll! I 1111,1i' I

~

, , , , , 1. 00 2.00 3.00 LOO 5.00

lIMfISfC)

a: 1+lll++ll+fll++\j++tl++t++!+++ec++8· .... wo Io .... .,; ,

o

, 6.00

'" ~~~-~-~-~~ '0.00 1. 00 2.00 3. GD 4.00 5.00

11ME(SEC)

2

o <>

'"

Dg lLJ •

1.00 2.00 3.00 1.00 5.00 6.00 11 MEl SEC)

o ~ !IAIIfIfHtitftltfl!1Mt\t\f\1AAf-.A.AAAft ~

(0 . o...g

'" ,

o o r-+-------.----,-----r---,----. '0.00 ]'00 2.00 3.00 4.00 5.00

T1ME~SEC)

II. THEORETICAL ANAL YSIS

The monlent equations of motion for a body fixed coordinate system

(Fig. 1), with its origin located at the c. g. of the body, can be written as(1)

M :; 1xp + (I -I )rq - I (r + qp) +1 (rp-q) x z y xz . xy

+ I (rz_ z) y7 :

M :; 1y ci + (I - I ) p r - I ( P + q r) + I ( P q - r) y x z xy yz

M =. 1zr + (I - I )qp - I (q + pr) + I (qr - p)

z y x yz xz

where p, q, r are the components of the angular velocity along the body

axes, x, y, z respectively. The angular velocity components can be written

as .

p = cf> t/J sin e . .

q = e cos ¢ + t/J cos e sin cp

. r = t/J cos e cos cp - e sin cp

(1)

( 2)

(3 )

(4a)

(4b)

(4c)

The terms M , M , M are the moments acting on the body. For a nonrolling x y z

vehicle in a vacuum, without wind, these terms represent the flexure moments

acting on the system. The pitch and yaw mOlnents can be written as

M = y

M = z

3

(Sa)

(5b)

For ¢:=~= 0 and asosuming sHlall angles, Eq. (4) reduces to

(6a)

q ":: e (6b)

_ r ;:~ _ tJ; ( 6c)

SubstitutingEq. (6) into Eqs. (-Z) and (3) gives:

:M :::. I e + (I - I )(-tJ;tfJ8)- I (-~e) + I (- ~iJ e - ~) .y y x z xy yz

-:+ I (_:/~e2 _.;F) "XZ 0/ 0/

(7a)

f. • ••

}\Mz ::= :lz l/t+ (I -I )(-tJ;e e)-1 (e -tJ;tJ;e) L " Y X yz

0; 7: I (iJ J; + ;j;e + ~ e) + I (e 2 - ~ e2)

: xz xy

(7b)

"JT.heS6:.:c'Sq1::l?o.tiQns ~an be sinlplified due to the snlall angle assunlption. Assunle

tithat:.p.itth Co C\nd·yaw, angle s can be approxinlated as

'~Tb~n,

. . , e~.=- -'We eo sin wet, tJ; = - WtJ;tJ;o sin WtJ;t

j.An:prd~r.9f magp.itude analysis can now be nlade for the individual ternlS of

4

ljJ 0 '" ljJ 0 wG ,...., o 0 ljJ . .

ljJO "-ljJo °0 WljJWO '"

e ~ 0 ~ ljJo O~WljJWO ljJG 0 '" ~GO WG

'" o 00/

ljJG ::::; ~GWG o 1{.1 .

02. ~ OGWG o 0

0 ::::; o WG o 0

ljJ '" 0/. WG '" 01{.l

~G 02 ::::; o/.G OGWG o 0 ljJ

All the product terITlS are sITlall cOITlpared to either the 0 or ljJ terITl if

the aITlplitudes 0/. and 0 are :3ITlall. Thus for sITlall angles, Eqs. (7a) o 0

and (7b) reduc e to

.. iJ;' M = I 0 - I

y y yz

.. . . M = I ljJ I 0 z z yz

As can be seen froITl Eqs. (8a) and (8b), the product of inertia terITl I yz

causes the coupling between the pitch and yaw ITlotions. Although I and xz

I may be of the saITle order of ITlagnitude as I ,they are ITlultiplied by xy yz

relatively sITlall quantities so that they can be ignored cOITlpared to the terITlS

. . I 0 yz

and I iti . yz

The effect of I on the pitch and yaw ITlotion and the value of I yz . yz

causing the coupling can be deterITlined froITl the solution of Eqs. (Sa) and

5

(8a)

(8b)

Substituting Eq. (5) into Eq. (8) gives

.. . I 8 -I t/J =-C 8-k 8 y yz 8 8

(9a)

.. . I t/J - I 8 - - Ct/J t/J - k t/J t/J z yz

(9b)

or', in matrix form,

(10)

Equation (10) is a .s econd order, coupled, differential equation. A s the

coefficient ITlatricesar€ sYITlmetric, this equation can he solved using a

1 -l • '(2., 3) Tl ... 1 d" . d 1 no:nna c.oor'uln.ate .sy:ste:m ',. ,1,e lnltla con ltions us e to so ve Eq. (10)

are

£J{O) ::: B ,t/J(O) == ~ ,0 .. ,. (11 ) 8(0,) ::: B ,l/I ((:0) - J/J '

,0 0

Following ;a 'ro·ethod identical to that given in Ref. 3, the solution to Eq. (II)

it) iieri-v·ed in .Append±xA. 'The :re.sultsar e

+ "I'" .1l.L (ll r/, ,.,,) ,..1ll. 0 'f'12 0 -;A., J t 'j ffi ffi [ (e. _ r/, ~)

,(e ,,';);' v6-'f' ,0/, cos.w.,F -+ '~ !.'jJ , 12 0 .'f' 'f'. Wt/J

( 12)

+ ~ (8 - ¢ tk)] sin~'Jt l wt/J 0 12 0 'f' ~

;and,

(13 )

6

where

Atj; :::

q>u :::

flJl Z :::

q>ll.2 Ce + CtJ;

flJl2l. ke + ktj;

~lCe+cy!' q>: 1 ke + ktj;

I WZ

y:z lj,; :::

IyW~ - ke

I WZ

y:z e :::

\W~-~

w2

::J!... '" '" 2

1zW~ - klj,;

I WZ

yz t/J

IzW~ -k~ I w2 yz e

12 = I I - (I )l. Y z yz

2 Ctj; w'1! 2ktj;

_ ke :0/ I

The sign in front of the square root term in Eq. (20) is determined from the

case of zero Iyz

' where wJ = ktj; lIz' and w~= kel\, If ktj;IIz is greater

than ke I I Y then the plus sign in Eg. (20) will be as s ociated with w~, and

the minus sign with w~. If ke 1\ is greater than ktj; lIz' then the reverse

will be true.

The effect of I on the model motion can be seen from Eqs. (12), yz

and (13). As there are two, nearly equal, frequency components pres ent in

each equation, the resulting motion for both pitch and yaw will be a beating

type of motion. That is, the envelopes will have a sinusoidal type pattern.

7

(14)

(15 )

(16 )

(17)

( 18)

(19 )

(20)

The effect of I on the frequencies can be deterITlined froITl Eq. (20), ' yz

which can be rewritten as

·(:;1) (l z /\ + \/!/~) .~I 1/\ + ky/kO ' ky; /1'8 :::

2(1 / I - 12 /12) . 2(1 /1 - 12 /12 (I /1 _ 12 /12) Z Y yz Y z Y yz Y z Y yz Y

and,

~. I z2 ke I z

2 2 kelz r )' / --z- I II + TY2

} kt,IJII .~, ::: Jz Iy+ I'f.; Iy

+'~ : -k/l -2 (I / I - I 2 I I 2 ) (I II - 12 I 12 - (I /1 - I 2 /12 ) .' .,\jJ . cz - ,z y yz y z y yz z z y yz y

Equa,tio!ls (21) .and (22) are plotted in Figs. 2 and 3, as the frequency ratios " " ',.'~ ~ _::; '0-- ~ 1,._ _ _ _ .- -.

,Wenkep/1~2 lP~ ,W1/-' l(ktf;IIz )1/2 versus \z 1\ for ktf;/ke equal to 1. 014, I, t- ". ....

These curves show that I yz

has a sITlall

effect on the frequency, causing it to increase or decrease dcr:;cnding on the •. : iI' (.'·l (. L :. ;-. t.~ = 1" !~ ,:.~. • - - -

value of I II . . '.J",. C~ ... z, _yo

.L ,+

The ITlOITlents of inertia, I and I , as well as the pr oduct of inertia, ~ L (C ;~. ~: • : y z

I , ~?-tl ~e obtained froITl Eqs. (16) to (18) as yz, \~ c . • ;

ke ¢w2 ¢w2

I 11 e 12 y!

::: -qi w2 w2 -'( ::

tf; e

I ~ ¢u w~ - ¢l~ ~ :::

i' w2 w2 .z .. /. tf; e

5l! (w2 _ w2 )

I y; e

::: -q; w2 w2 ;r~ ::

tf; e

8

(21)

(22)

( 23)

(24)

(25)

!

Ii 'I !

In order to deterrnine the Iyz of a system, values of kt/J' ¢ , Wt/J

and We are required. While the flexure stiffness,. kt/J' is kno,"vn from

static bench tests, We' Wt/J' and ¢ must be determined fron1 dynal11ic tests,

i. e., from recordings of the pitching and yawing motions.

Values for ¢ll and ¢12 can be determined as follows. Assuming that

the initial angular velocities are zero, Eqs. (12) and (13) reduce to

and,

or,

e(t) = -Aet -AttJ Ae e cos (Wet - Yee ) + Be e cos (Wt/Jt - Yet/J)

t/J(t) = -At/Jt -Aet

At/Je cos(Wt/Jt - Yt/Jt/J) + Bt/Je cos(Wet - Yt/Je)

where

Ae ¢lZ (¢ll t/Jo - eo) a = 1+ -q; W2.

e

Be == ~ (e 0 - ¢l 2. t/J 0) ~1+ ;~ Wt/J

9

( 26)

(27)

{28a)

(28b)

(29a)

(29b)

therefore, from Eqs. (28) and (29)

(30)

,...., The sign of I is associated with the sign of <p which depends upon

yz

the signs of the AI sand BI s. If the envelope of the motion initially decreases,

. then A and B in Eq. (26) or (27) will have the same sign. However, if tLle

envelope initially increases, then A and B will have opposite signs. The

sign of A always has the same sign as the initial release angle.

10

III. EXPERIMENTAL INVESTIGATION

The objective of the experimental investigation was to determine the

-I of the model, and then to eliminate it by properly balancing the model. . yz

-The I of the original model was small, and therefore it was difficult to yz

determine the amplitude components of the motion. In order to overcome

~his difficulty, a larger, known I was added to the model. The resulting yz

~otion was more easily reducible. The original I of the model, plus tht yz

added I was determined by fitting the data to Eqs. '- yz (26) and (27) by the

!.east squares differential corrections technique(4, 5). The added I was yz

then subtracted to give the I of the model. This was repeated with t... yz

another known I , and the model I was taken as the average of the two c.. yz yz

n~sults.

PROCEDURE

The two model configurations tested were the basic model plus:

Configuration

(1 )

(2)

\z added sl-ftz

-5.25 x 10- 5

+4.27 x 10- 5

I c;l-ft Z y added

4.843 x 10-4

4.843 x 10-4

Iz added sl-£tZ

5.393 x 10-4

5.393 x 10-4

For each model configuration the model was first balanced in the xZ'-o and

¥~ planes to insure that the center of gravity was located at the pivot point.

The model was mounted in a test cabin which was evacuated to less than

~rnm of mercury to eliminate aerodynamic effects. Th.e model was set at a

predetermined pitch and yaw angle, and released.

11

The resulting angular motion was recorded by strain gauges mounted

on the flexure. The analogue output from the gauges was digitized, and

stored on magnetic tapes. Calibration runs for the conversion of the digitized

data to degrees were made before and after each run. The pitch data was

fitted to Eq. (26), and the yaw data to Equation (27). The section length fitted

was one beat period. The amplitude components determined from the least

squares fit were then used to calculate ¢ll' and ¢ 12' Eq (30), and then (f,

Equation (18). The value of '¢, along with we' and W'if.!' which were also

obtained from the least squares fit, were used to calculate I ,Eq. (25), yz

as well as I, Eq. (23), and I ,Equation (24). Three consecutive sections were y . z

fitted, and the average values resulting from these three fits were taken as

the I ,I, and I of the configuration.' The added values of I ,I and I yz y z yz y z

Were subtracted from the measured values to give I ,I and I . of the yz y z

original model.

RESULTS

The motions that were obtained from the runs are shown in Figs. 4,

5, and 6. As can be seen from run #1, Figs. 4a and 4b, the e envelope

initially increases while the 'if.! envelope initially decreases. The I of yz

this configuration should be negative. This is in accord with the

I = -5.25 x 10- 5 sl-ftz, that was added to the model. For run #2, Figs. 5a, yz .

and 5b, the situation is reversed. The e envelope initially decreases,

while the 'if.! envelope initially increases. The Iyz of this configuration

12

should be positive. This is again in agreement with the I· = +4.27 x 10- 5 yz

sl-ftZ that was added to the model for this run.

The results of the least squares fitting procedure are summarized in

Table 1.

TABLE I: Results From Least'Squares Fit to Data

Configur ation

1

2

Iy model sl-ftz

0.002732

0.002737

0.002397

0.002405

-5 O. 30 x 10

0.37 x 10- 5

The average value for the I of the model is 0.34 x 10- 5 sl-ftZ with a yz

probable error of ±O. 1 i x 10- 5 .sl-ftZ, or a.bout 32% of the average value.

The average values for I and I are 0.002735 sl-ftz , and 0.002401 sl-ft z , y z .

respectively. The probable error of these terms is estimated to be about

0.1%.

The angle between the principal axes, and the body axes is given by

1 . -1 (2IYZ) ~ 2' tan I -I ,which for small angles reduced to I -I • For runs 1 z y, z y

and 2, with intentionally added I 's (Figs. 4 and 5), this angle is yz

10.2 degrees and 9. 6 degrees respectively. For the original model, the

angle is 0.6 degrees. Even this small angle of less than one degree was

sufficient to caus e inertial coupling of the pitching and yawing motion that

could be detected on oscillograph records.

13

IV. DYNAMIC BALANCING METHOD

The I of the model was determined to be O. 34 x 10- 5 sl-ftz. yz

An I of -0.32 x 10- 5 sl-ftz was then added to the model by placing equal yz

masses at 45° and 225° on the model (see sketch). This should reduce the net Iyz

0°

z

of the model to approximately zero. The model was statically balanced by

?lacb.g masses on the geouietriC, or p:dncipal axes, i. e., 0°, 90°, 180°,

° or 270. This prevented any additional I being added to the model. This yz

configuration was run, and the resulting motion, Figs. 6a and 6b, showed

that the model is in near perfect trim.

14

V. CONCLUSION

The present study of the motion in vacuum of a nonrolling body with

freedom to pitch and yaw shows that the product of inertia I is the yz

dominant term causing inertial coupling of the pitching and yawing motion.

The presented analysis, and the described method of eliminating the inertial

coupling have been verified by experiments. The model is now ready for

wind tunnel tests, the results of which will be reported later.

15

VI. REFERENCES

1. Rauscher, M., "Introduction to Aeronautical Dynamics," John Wiley

& Sons, Inc., New York, 1953.

2. . Foss, K. A., "Coordinates Which Uncouple the Equations of Motion of

Damped Linear Systems," Journal of Applied Mechanics, September 1958,

pp.361-364.

3. Anderson, R. A., "Fundarnentals of Vibration, II The MacMillan

Company, New York, 1967.

4. Nielsen, K. L., "Methods in Numerical Analysis, II The MacMillan

Company, New York, 1960.

5. Eikenberry, R. S., "Analysis of the Angular Motion of Missiles,"

Sandia Report SC-CR-70-6051, February 1970.

16

• X,cp,p

• 8

y,q

Z, r

FICURE 1 Coordinate system

17

N ~

>-H ...... Cb ~

...... Cb

3

1.08

1.06

1.04

1.02 -

1.00

0.98

0.96

0.94

0.92 0.00 0.02

FIGURE 2

1z Ily .. 1.05

I Z IIy • 0.95

1Z IIy • 1.00

0.04 0.06 0.08 0.10 0.12

I yZ II y

Effect of I on circular frequency in pitch yz

18

1.08

1.06

IZ/Iy • 1.00

1.04 IZI Iy • 0.95

1.02

~ N

H ..... ~ 1.00 ~

..... ~

3 0.98

0.96 IZ/Iy • 1.05

0.94

L-__ L-__ L-__ ~ __ ~ __ ~ __ ~ 0.920 . 00 0.02 0.04 0.06 0.08 0.10 0.12

I yZ II y

FIGURE 3 Effect of Iyz on circular frequency in yaw

19

a:

o o IJ)

I­W-Jo ~o I-~

I

o p •

~~. --------~---------r--------~--------r_------_. 10 . DO 1. DO 2.00 3.00

TIME(SEC) 4.00 5.00

a. Pitch component vs time

""'0 go W· Cl .... _ ~~~~~~4HUH~~+H-f~H#HHHH~+~~~4H~~~~ t-I

(j) Cb-g .

(17 . I

p o . ~~--------~--------.-------~---------.--------~ 10 . 00 1. DO 2.00 3.00

TIME(SEC) 4.00 5.00

h. Yaw component vs time

FIGURE 4 Pitch and yaw components for original model plus an added I of -5.25 X 10- 5 slug_itz yz

20

a:

a a . ID

I­Wo Ia I- •

(T) ,

o a ~~----~----~------'-----II-----' '0. 00 1 . 00 2. 00 3. 00 4- • 00 5. 00

t-I

a o . ID

(f)

n..g . (T) ,

a a .

TIME(SEC) a. Pitch component vs time

~~---------'f--------~fr--------'f---------rf --------" '0. 00 1 . 00 2. 00 3. DO 4- • 00 5. 00

TIME(SEC)

FIGURE 5

b. Yaw component vs time

Pitch and yaw components for original model plus an added Iyz of + 4.27 X 10- 5 slug-ftz

21

a: I--W l -It.? I-- •

N ,

o o . W;----------T.---------r.---------.r-----__ -.. ________ -.. 'e.oo 1.00 2.00 3.00 4.00 5.00

TIME(SEC)

. "<I-

. N ,

o o .

a. Pitch component vs time .

w~--------.---------~--------r_--------r--------l '0. 00 1 . 00 2. 00 3. 00 4. 00 5. 00

TIME(SEC)

FIGURE 6

b. Yaw component vs time

Pitch and yaw components for dynamically balanced model

22

VII, APPENDIX A

The governing equations of motion in a body fixed coordinate system

with its origin located at the c. g. are

I '0 - I ij; + C()e + kQ() = ° Y yz 17

- I fi + I iii + C",ip + ~,}f; = ° yz z 't" 't"

Consider first the unda:nped equations of motion

, . I()-I t/J+~()= ° y yz 17

= ° In matrix form, Eqs. (A-3) and {A-4),are written as

One possible solution has the form

Substituting Eq. (A- 6) into Eq. (A- 5) gives

or,

!~AI WZ sinwt

-AzWz sinwt

23

Al Sinwtl = {a} A z sinwt °

(A-I)

(A-2)

(A-3)

(A-4)

(A-5)

(A-6)

matrix addition then gives

:\ 1

The characteristic equations results from setting the determinant of the

.coefficient matrix to zero

(k - I Wz) (k - I wZ) - 1z w4 = 0 o y ~ z yz

(l/z - 1;)W4 - {ke1z + ~\)WZ + ke~ = 0

let

then, 'the natural frequencies of the system are

. kektJ; - 1z

The amplitude ratio for the frequency Wt/I is obtained by substituting

Wt/I into Eq. (A -7)

and,

then,

.Z4

(A-7)

(A- 8)

(A-9)

(A-IO)

The second subscript on A is used to indicate that the amplitudes are

associated with the frequency Wt/I' Thulj"one possible solution for the

equations of motion is given by the tri'al solution, Eq. (A-6), in which the

frequency is Wt/I and the amplitudes are related by Eq. (A;"10,). Another

solution is possible if sin wt is replaced by cos wt, i. e. ,

A procedure similar to that for Eq. (A-6) gives frequencies identical to that

of Eq. (1\-9), and an amplitude ra~io for

=

The complete solution for a free vibration at the first natural frequency,

Wt/I' is given by

() =

A similar procedure for the second natural frequency, W(} , yields

in which

() = AI(} ,sinw(}t + Bl(} cos w(} t

t/I = Az(} sinw(}t + Bz(} cos wet

(A-II)

(A-12)

(A-13)

(A-14)

(A-IS)

(A-16) ..

(A-17)

The most general solution for the free vibrations of the system is given

by the superposition of the motions in the two principal mode s of vibration,

given by Eqs. (A-,l3) to (A-16).

= . A 1tJ; sinWtJ;t + B1tJ; COSWtJ;t + Alesinwet + BleCOsWet

= A2.tJ; sinWl/} + B2.tJ; COSWtJ;t +A2.esinwet + B2.ecosWet

Imaking use of Eqs. (A-12) and (A-17)

(A-IS)

(A -19)

e = ~~ 1 AVJinw",t + Bzv, cosw",t ! + ~:: lAzeSinwet + B Ze COSWe+A-20)

'" =~zv, sinw", t + Bzv, cosw'" t! + lA zesinwet + BzeCOSWet! (A- 21)

. Eqs. (A-20) and (A-21) can be written as

where,

C/> = C/>2.2. = 1 2.1

The constants A~, B2.tJ;' Az,e ' and B2.e can be determined from the

initial conditions

2J6

(A-22)

(A-23)

(A- 24)

(A- 25)

(A-26)

(A-27)

(A-2S)

0(0) = 0 o

"/(0) = ,11 'f' 'f'0

The quantities ql and qz represent the amplitudes of the motion in

the two normal modes and are called the normal coordinates. A work-and

-energy approach to the equations of motion shows that the multiplication of

the equations of motion in the normal coordinate system by the constant

[:~ ~] will uncouple the equations of motion.

Now consider the complete equations of motion, Eqs. (A-I) and (A-2),

which can be written as

[

I - I J y yz

-I I yz z

fi

From Eqs. (A-22) and (A-23)

Substituting Eq. (A-32) into Eq. (A-31) and multiplying by [:~~ ~] will give

uncoupled equations in the normal coordinate system

[

m 11 [I 't'll Y

.: 1 - I f/J1Z- yz

+ [fj)u fj)12

matrix multiplication gives

2..7

(A - 29)

(A-30)

(A - 31)

(A-32)

: ~l~ l'g fly! 0 ql ¢l: Co + CtJ; 0 • 2 ql

WtJ; +

0 <Pl~ kO +\

0 q} C +C W2 qz 1Z e l/J qz

0

o +

where it was assumed tha: (¢ll ff lZ Ce + CtJ;) is much smaller than either

(~:1 Co + CtJ;) or (fj)lzz Ce

+ CtJ;)' and We and WtJ; are given by Eq. (A-9)

Equation (A-33)is uncoupled and can be rewritten as

Equations (A-34) and (A-35) can be solved in the usual manner to give

where

and,

The results, Eqs. (A-36) and (A-37) can now be transformed into the e and tJ;

coordinates using Equation (A-32).

28

(A - 33)

(A-34)

(A-35)

(A-36)

(A- 37)

(A - 3 8)

(A- 39)

(A-40)

(A-41)

where 'AtfJ= 0tfJWtfJ and \)=OeWe (A-42)

The constants in Eqs. (A-40) and(A-41) can be determined from the initial

conditions, Eqs. (A-29) and (A-30).

e -cjJ l/J e-cpl/J A - o .p 0

B = o 12 0

+ 0t/J~ i/I - q; if! q; Wt/J

. . cp l/J-e (p tf;-e

A - 11 0 0 B - II 0 0 + °eAe e - f e- '?)We

where "" cf> = 1>11 - ~2 {JI,-<13}

The solution for the damped motion, coupled by Iyz is

e(t)

(A-44)

. . (A - 45)

-'Aet 11 0 0 'I' 0 0 e 0 0 I (II> t/J -e ) [ril l/J -e 'A ( cp t/J -e ) ] ! + e ~' coswet+ ll;PWe + We 11 (J; sinwet

where (/, ffi and 1 are given by Eqs. (A-24) , (A-2S) and (A-43) respectively. 't'u' 't'12

29

UNCLASSIFIED SeC'urity_ Classification

DOCUMENT CONTROL DATA· R&D (Security dlls.!Hlcntion 01 tltlo. body 01 nb.'ftrnct and Indexing IInnotntion mu.o;t be enterod when the ovorall reporl Is classified)

I. ORIGINATING ACTIVITY (Corporate author) 20. REPORT SECURITY CLASSIFICATION

Aerospace Research Laboratories Unclas sified Hypersonic Research Laboratory 2b. GROUP

IWrh,ht-Patterson AFB Ohio 45433 3. REPORT TITLE

DYNAMIC BALANCING OF A NONROLLING, STING SUPPOR TED, WIND TUNNEL

MODEL PERFORMING NONPLANAR MOTIONS

•. DESCRIPTIVE NOTES (Type otreport and inclusive dates)

Scientific Interill1 5. AUTHOR(S) (First name, middle initial, last name)

Kevin E. Yelll1gren, Capt, USAF

6· REPORT OATE 7e. TOTAL NO. OF PAGES r' NO. OF REF~ June 1972

ea. RX~X=X~~X~X In- house Research 9a. ORIGINATOR'S REPORT NUMBER(S)

b. PROJECT NO. 7064-00-01 ARL 72-0100

c. DoD Elell1ent: 61102F Db. OTHER REPORT NO{S) (Any other numbers that may bo ltlJttl~ed thlo report)

d. DoD Subelell1ent: 681307 Code II Til to. DISTRIBUTION STATEMENT

Approved for public release; distribution unlill1ited.

II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

TECI:I OTHER Aerospace Research Laboratories (LH) Wright-Patterson AFB, Ohio 45433

13. ABSTRACT

Wind-off. bench tests on a wind tunnel ll10del that had freedoll1 to pitch, and yaw,

but no freedoll1 to roll indicated that the pitching and yawing ll1otions were inertially

coupled. An order of ll1agnitude analysis showed that this coupling was due to the

pr oduct of ine rtia te rll1 Iyz , The inertially coupled equations of ll1otion are solved.

and a ll1ethod is presented for deterll1ining the ll1agnitude and sign of Iyz froll1

recordings of the ll10del ll1otion. The ll1ethod used to dynall1ically balance the

ll10del is described, and experill1ental results are presented that verify this ll1ethod.

UNCLASSIFIED Security Classification

UNCLASSIFIED Security Classification

I., LINK !I. LINK B L.JNK C KEY WORDS

ROLE WT ROLE WT ROLE WT

Inertial coupling

.' It ,

1" U:¥' • n;,c modellbalancing

~f LC-! 't -- " , Two ~ree DLfre.e.d.om ITIot'lon

, ------

,.

"'U.S.Government Printing Office: 1972 - 759-485/90 Security Classification