Spin decay of a class of satellites caused by solar radiation. Part II. Arbitrary attitude of the spin axis

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KUNOE SPIN DECAY OF A CLASS OF SATELLITES 'iLIEGTUlGBOlJW

Q\~lI01\11ttK _{u u CAUSED BY SOLAR RADIA['ION}

PART 11: ARBI~RARY ATTITUDE OF THE SPIN AXIS

by

D. B. Cherchas

'.'

SPIN DECAY OF A CLASS OF SATELLITES CAUSED BY SOLAR RADIATION

PART- 11: ARBITRARY ATTITUDE OF THE SPIN AXIS

by

D. B. Cherchas

Manuscript received April,

1969.

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ACKNOWLEDGEMENT

The material presented herein is based on work done under the super-vision of Dr. P. C. Rughes of the Institute for Aerospace Studies at the University of Toronto. This research was sponsored in part by the Air Force of Scientific Research, Office of Aerospace Research, United States Air Force under AFOSR Grant No. AFOSR-68-l490, and in part by the National Research Council under Grant No. 67-3.

Data on the spin rate, spin axis atti tudlè.~~and percent sun histories for the Alouette 1 and Explorer XX were obtainedl from SPAR Aero space Company of Canada and the Defense Research Telecommunications Establishment (D.R.T.E) of D.R.B •

•

SUMMARY

The analysis studies the marmer in which solar radiation heating and pressure acting on the long flexible antermae of a spin stabilized satellite can cause the spin rate to significantly decrease. The problem is formulated by finding a differential equation for the shape of a boom antenna under heating

effects and inertial farces and deriving the expression for torque acting on the satellite due to salar radiation pressure on booms of known shape. By assuming small boom deflections and incorporating, for the first time, tpe spin axis attitude and salar exposure histories, the spin rate versus time histories of two example satellites, Alouette 1 and Explorer XX, are found, and excellent agreement obtained with actual flight data .

1. 2.

3.

4.

5.

6.

TABLE OF CONTENTS NOTATION INTRODOCTION

CALCULATION OF TORQUES THAT INFLUENCE BOOM SHAPE

2.1 Torques Due to Inertia1 Forces 2.2 Torques Due to Solar Heating

2.3 Differentia1 Equation for Boom Shape SOLAR RADIATION TORQUE

SOLurION OF EQUATIONS OF SHAPE AND MOTION

4.1 Simp1ifying Assumptions

4.2 Details of Torque Calcu1ations 4.3 Ca1culations App1ied to A10uette 1 4.4 Ca1culations App1ied to Exp10rer XX

DISCUSSION C ONCLUS I ONS REFERENCES APPENDIX A B C D j-"E F G FIGURES 1-16 1 1 2

4

7

8

9

9

13

15

16 16

19

20

D d d e E G !,J.,~

X,Y,Z

!* ,.si*

,.!?-x, y, z I ,I ,I x Y z I I P L M p r s b.S n T

e

T a: NarAT10l\T

All symbols not listed below are defined in the text. unit vector in direction of thermal torque

position vectcr of point on boom,written in satellite co-ordinate system

diameter of boom

emissivity of boom surface modulus of rigidity

shear modulus of boom

axes and parameters of a co-ordinate system fixed in space

axes and parameters of a co-ordinate system fixed to the satellite

non-twisting axes located at a boom cross-section

moments of inertia of satellite about satellite axes

!

*

,

.si*,

~*

bending moment of inertia of boom x-section polar moment of inertia of boom cross section

length of boom

torque acting on satellite due to solar radiation pressure

on one boom

solar radiation pressure reflectivity of boom

distance along boom axis measured from boom base

unit vector, normal to boom surface, pointing into the boom unit vector tangent to boom axis

temperature at surface of boom thermal time constant

absorptivity of boom

angle of a diameter lin~ at any given cross-section measured

counter-clockwise from .si or

.si*

)', n

E p

.!.l

'_. T ₁

*

T w

angle to diameter line having greatest positive temperature difference across it

two angles which make up ~; )' is fixed for any given diameter,

n

varies with the amount of boom twisting

unit vector in direction of solar radiation

unit vector in direction of force due to solar radiation reflection

solar radiation vector

thermal curvature

mass per unit length of boom

moment at a point on the boom due to inertial forces

component of !l along boom axis

thermal moment at a point on the boom

total moment acting at a point on the boom

total bending moment acting at a point on the boom

Euler angles relating

!*,

~*, ~ to

!'

~, ~

•

"

1. INTRODUCTION

This note continues the study of the interaction of a spinning

sate-llite with long flexible booms (Fig.l) with solar radiation. It was shown in

Ref.l that this interaction can result in a torque on the satellite which, when

taken over a long time period, will effect a significant change in spin rate.

The original work on this mechanism was prompted by the strange

spin-down of the Alouette 1, l~nched in 1962, which exhibited a spin decay of

approximately one rpm per year. The study (Ref.l) was done by B. Etkin and P.

C. Hughes at the Institute for Aerospace Studies (University of Toronto) in

1965. In this analysis, the spin axis of the satellite was held a~ ninety

degrees to the solar radiation vector and radiation was assumed to fall on the

satellite twenty-four hours per day; then an exposure factor was introd~ced to

fit the theoretical spin decay with the data. This fitting did not invalidate

the theory since it was the only theoretical decay which, with rational fitting,

could be made to coincide in magnitude and curve shape with the actllal decay. The work described herein is aimed at removing the exposMTe factor,

i.e. at finding the boom shape and satellite torque using the actual solar

exposure history ~nd for an arbitrary orientation of the spin axis relative to

the solar radiation vector. Therefore, much of the original work is left as it

was and an application of it is made to a more general case.

A differential equation for the shape of a boom subject to heating

effects and inertial forces is found first. Then, on the assumption that the

shape solution could be found, an analysis is made to find the torque acting on the satellite due to the interception of the solar radiation momentum flux by

the booms. The equations for boom shape and torque which have evolved to this

point in the study are complicated integral-differential equations. In order

to make the problem tractable, the equations are simplified by assuming small

boom deflections. The simplified equations are solved for an arbitrary direction

of spin axis and the solution applied to two example satellites using their

spin axis attitude and solar exposure histories. The Canadian Alouette 1 and

U.S. Explorer XX are the example satellites chosen.

2. CALCULA'tION OF TORQ,UES THAT INFLUENCE BOOM SHAPE

T~e following material will derive the torques which determine the

boom shape and the differential equation for shape. No attempt will be made

to solve the equation in this section.

In this analysis, both the boom bending and the mass of the booms are

taken to be small enough so that the satellite centre of mass is and stays at

the origin of the

!*, J*,

~ or body axes system (Fig.2). Also the booms are

taken to start at the origin of the above axis system. These simplifying

assumptions ca~ qe justified by the small boom mass/body mass ratio and the

body radius/boom length ratio for the example satellite studied. For example,

for Alouette 1,

boom mass

body maRS 0.0188 (.--for longest boom)

body radius

boom length = 0.04 (for shortest boom)

The i*, ~*, ~* system has its ~*, ~* axes coincide~t with the unbent

boom directio~s for the Alouette 1 configuration (Fig.3) and for small boom

deflections can be taken to be the princip~l axes of that configuration. For

the Explorer XX, the body axes are as shown in Fig.4 and also for small boom

deflections can be taken to be principal axes of that satelli te. The assumption

of the body axes to be principal axes is made throughout the remainder of this

section and in all other sections. The principal moments of inertia are assumed

to be constant. The~,~, ~ or space axes system is a system ofaxes fiNed in

space with the ~ axis pointing ~owards the sun. The origin cf the body axes is

at the origi~ of the space axes and the body axes are located with respect to

the space axes by three Euler a~gles and their derivatives as shown in Fig.2. The boom shape is determined by the torques which exist along the

length of the boom. In this case the important torques are the thermal torque

due to a temperature difference across the boom cross-section, the inertial

torque due to the acceleration of the boom and the elastic torque due to the

structural rigidity of the boom itself. The next two sections will derive

ex-pressions for the inertial torq e and the thermal torque.

2.1 Torques Due to Inertial Forces

The first moment to be evaluated is ~l' the moment at an arbitrary

point

P

due to inertial forces on the boom. The motion of the satellite about

its center of mass is the only motion we shall consider in finding the inertial

force on a boom element. Referring to Fig.2, we shall find the acceleration of

an element of the boom at P' in order to obtain the inertial force at that

point. The acceleraticn of P in t,he ~,

;1,

~ ref'erence frame can be calculated

fr om

a = a* + 2w x V* + w x ~ + ~ x(~ x ~) (II.l)

a is the acceleration in the i, j , k reference frame and V*and a* are the velocity

and acceleration in the i*,

i*,

k*reference frame respectively.- By writing the

right hand side of Eq. (11.1) in-body axes system co-ordinates, ~ will be written

in the axes set in which we want to work. The right hand side terms are written

in the

!*,

J*, ~ system as follows:

d

₌

xi* + y;i* + zk*

.

_•

_•

V-'I.-

₌

xi* + y/J.* + zk*

..

y~-*

..

a*

₌

xi* + + zk* w w i* + w j* + wk* x- y-

z-.

w = ~ _x-i* + ~ _y-j* + ~ _z-k*

where wx,w ,w_z' ~x'~y and ~z are as given in Appendix A. Therefore, expanding

a

=

[

x + 2 (w Z -w y) + (~ z-~ y)+(w W - W2X_w2x +w W z)] i* Y z Y z x y y z x y -+ [

y

+ 2 (w x-w

z)

+

(t

x-~ z)~(w w X _w2y _w2y + W W z)] J* z x z

x

x y x z y z . + LZ + 2 (w

Y

-w x)+(t y +

t

x)+(w W x_w2z_w2z + W W y)] k* x Y x Y xz x Y yz

-The inertial reaction force on the boom due to this acceleration is

f -P!:;

f i* + fj* + f k*

x-

r

z-where p

=

mass per unit length of boom, and

!

is the inertial force per unit

length at a pomnt P on the boom. The point P' will be identified by co-ordinates

x', y', x'. If these co-ordinates are used in f, the force per unit length at

P', identified as

!', is obtained. The distance vector between Pand

P'

is

~sep = (x' - x)!* + (y' - y)J* + (z' - z)~

The cross produc~ of the inertial force at P' with this distance vector gives

the moment at P due to the inertial force per unit length at P' ie.

T. d x f' -ln -sep = [(y'-y)f' (z'-z) f' ] i* z _Y + [(z'-z)f' - (x'-x) f' ] _J* x z + [(x'-x)f' - (y'-y) f' ] k* Y x

T. is the moment at P due ~o the inertial force per unit length at P'.

Mult--ln

iplying by ds' will give the moment at P due to inertial force on an element

at P' and integration of this torque for P' going from P to the end of the

boom will give the total inertial moment at P, i.e.

2:₁ (s)

~

lL

2:in(s' ,s) ds' (II.2)

Later it will qe necessary to know the component of ~l along the boom axis.

~his component can be fcund by taking the dot prcduct of ~~ with a unit vector

in the direction of the boom axis at the particular point ln question. The

unit vector tangent to the boom is

T

=

d

.2: i* +

ds

-and the axial component of ~l at P is

d d

.J.

.* + ~ k*

ds

.J.

ds

-r*'

=

T • T

·1 -1

-Thus3 we have found the inertial torque and its axial component at P.

2.2 Torques Due to Solar Heating

The other torque which acts to bend the boom is athermal torque

generated by a temperature difference across the boom cross section as effected by solar heating. We will derive an expression of this torque in the material

below.

Reference 1 proposes th at a first order differential equation closely

describes the temperature difference established across a diameter, ie.

where

è

~

e

(t ) + ~

e

(t,s, 1)

dt

'

s,

1

T

e

1

e

2

.

(see Fig.3)

Referring to Fig.5,

e ,

e~ are temperatures of elements~l and ~2 respectively, T is a thermal time cOnstant, ~l is the rate of heat absorption from solar

radiati_{on per unit area at 68 1 and q2 is the same quantity at} ~2; cl is a

constant based on the mass density and specific heat of the boom.

Now we define E as the solar radiation vector, E

= -

j E, where

E = solar energy flux in thermal units. Then q 1 and q'2 iiiay be -wri tten as

<1'1

=

P (ex ~. 68 1) where p(x) = x x

>

O.

=

0 x

<

O. now 'a.. - ~2 = P (0: E· 6S ) - p( 0: E • ~ ) ~.L - - 1 - - 2 and, since ~2 = -

ts

_{l '} =

ex

.§.. 68 1

Thus the differential equation for the temperature difference across a diameter of a given cross section can be written as

d

A 0. (t ) 6 El (t,s,r)

~ D Ö ,s,r + T

(11. 3)

It wi11 be usefu1 to expressL~ in the i*, j*, k* _{system. 681 can}

easi1y be wri tten in terms of the

1,;,"

r,

!-

system (see Fig.

5) ,

T,

T,

rare

axes at the cross-sectioQ which move with the boom but do not rotate as the

boom twists and if P was at the base of the boom,

'1,

T,

'k

wou1d be coincident

wi th

.?:.

*, .J.*,

.!?

We can write the inward normal to

-.6Sî

as

68 1

A ~ '~A-k

COSf-'

..J -

Sluf-' _

To write ~ in the i*, j*, k* system we need the trans~IDrmation between the

untwisted i,

T,

'k

system and-the i*, j* , k*' .system. This transformation can

be effected by two rotations. ReferrIng to Fig.6 and noting that

-1 dx TJ 1 cos ds -1 dz TJ ₃

=

cos ds cosTJ 1

=

cosCP1 cos(90 -TJ ) 3

we can easi1y write matrices to describe the rotations. The first rotation

abqut the

!

axis by (90-TJ3) i.e.

sinTJ₃ 0 -cosTJ

3 0 sinÇ3 cosTJ3

0 1 0 -cost) -cost)

cosTJ

3 0 sinTJ3 -sint) -sint) sinTJ3

-The next rotation is about the k axis by - CP1 _{and transforming 68 1 from the}

previous stage lIS 1 = = 68 2 is just the

o

o

o

1

sir$ cosTJ3_ COSCP1 + cost)

sint) cosTJ 3 sinCP1

-

cost)

-

sinÇ3 sinTJ₃ m i*+ m j* + m k* x- _J z-negative of this.

5

sint) cosTJ 3 -cost) (11.4 ) is

It is important to note that ~ is not constant for a given diameter

through changes in the Euler angles. It will be helpful to write

~ (s,t)

=

/

+

n

(s,t)

n=o@s=o

n

is due to torque acting along the boom axis and boom is not subjected to a torque along its axis.

an axial component which is, as found earlier,

Using

dn

ds

T!

=.!l . T

=

~ ',-I GI p

/ is the value of ~ if the

Only the inertial moment has

and ignoring any time lag in twisting (see Ap~ed~ix G) we can write

n

Therefore, s

J

* (s, t)

₌

.Tl ds 0 GI

~

(s,t)

= /

+

r

o

p T

*

ds 1 GI p

Eq. (11.3) is the differential equation for t:,

e

across a given diameter at a

cross-section. We are interested in the maximum t:,

e

which will'occur at a

given point P. The value of ~ which gives t:,

e

max is denoted ~ and determines

the direction of thermal torque. The thermal curvature vector is then KTQ

where

K

=c

t:,e

T

4

max (11. 5)

and D is a unit vector at right angles to the diameter of~ in the plane of

the cross-section of interest (see Fig.7). In the

r,

r,

~ set it is

Q

=

sin

~

r -

cos

~

r

and transforming to the ~*, ~*, k* axis set

cos r:J>1 cos 1)3 cos ~ sill<pl sin ~

D

=

_{sin 1>1 cos 1)3 cos} ~ cos</J

l sin ~

-sin1)3 cos ~

D i* + D .* + D k*

x - y.J. z

For small boom deflections it can be shown that (Ref.4) where M EI 1 r

=

K r

=

radius of curvature K = curvatu;re M

=

bending moment

Thus the thermal bending moment is given by

T

=

EI

-2

The tota1 torque at a point

P

is

T

= inertia1 torque

+ thermal torque

where ~1 and ~2 are given by (11.2) and (11.6) respective1y. 2.3 Differential Equation for Boom Shape

(n.6)

We now wish to use the tota1 torque va1ue at

P

to find a differantia1 equation for boom shape. A curve in space can be described by three unit

vectors which change their direction as one moves a10ng the curve (e.g. Ref.

5,

pp. 311, 312), Fig.8 i1lustrates these vectors. It is true that

but K can be written in

K

Tb EI

terms of the bending moment bending moment

EI

By taking ~ x

!

we have (see Fig.8)

therefore T X T = K N EI EI and T X

T

=

CIS

EI 7 i.e. (11. 7) !

Equation (11.7) is the differential equation for the boom shape.

3.

SOLAR RADIATION TORQUE

Now that the differential equation for boom shape has been derived,

the next step is the calculation of the torque on the satellite due to solar

radiation flux being intercepted by the satellite booms. Due to the time lag

in the temperature difference across a boom cross-section as effected by

solar radiation heating (see Eq. (11.3)), there will be a time lag in the boom

bending relative to the solar radiation input history. It is expected that this

time lag in boom bending will lead to a net solar radiation torque per spin

cycle on the satellite.

The force due to solar radiation pressure can be calculated from

(Ref.l). where dF A := ds

.

d ex p

.

cos v dF R

4

ds d 2 - 3

,

. .

r

.

p

.

cos v

dF is the differential element of force due to solar radiation momentum flux,

p, striking a boom element with lengths ds and diameter d. ex and rare the

absorptivity and reflectivity respectively of the boom and

21

and

.

Q2

are unit

vectors in the direction of the force due to absorption and ~he force due to

reflection respectively as shown in the sketch below,

9.

₂

T

E

~l

-

-

_si.

and

_2.

2 can be found from

T x

_C!:.

x

.J. )

5 :=

-2

I

!x,Si.

I

and cosv can be found from

The torque on the satellite due to radiation forces on all elements of the boom is

L L

!i=

J

~=J

o

0

where ~ = x!* ~ YJ* + z~ is the position vector of a point on the boom written

in the satellite co-ordinate system. For the tot al torque on the satellite a

similar integration would have to be performed for each satellite boom and the sum of the torques from each boom taken.

The equations of motion for the satellite can now be written. They

are L n I w - w w _{(I - 1 ) =I}

J

dM :. x x y Z _Y Z Xl i=l 0 n L • (I - 1 ) =I

J

1 w - w w dM . Y Y Z x Z X yl i=l 0 I

W

- w w (I - 1 )

=t

.

J

L dM . Z Z x y x Y Zl i=l 0

for an n-boomed satellite.

4.

SOLUTION OF EQUAT-IONS OF SHAPE AND MOTION

4.1 Simplifying Assumptions

In this section the equations of shape and motion are simplified to

tractable farm and solved. In thëssimplification the following assumptions are

made:

(i )

e

and ~ and ~ are quasi-steady. Forces more significant

than solar radiation pressure influence the attitude of the spin axis, i.e. gravity gradient and magnetic moment effects. Appendix D gives the spin axis attitude histories

for the two example satellites chosen. It is clear from

these histories that for short intervals of timeC9 and ~

can be assumed const~t. Figures 15 and 16 show the slow

spin decay of Alouette 1 and Explorer XX respectively.

(ii) Terms which are second orde~ or higher in boom deflection

or slope are negligible in comparison to first or zero

orderi3terms provided j;hat other factors do not cause terms

in which the higher order terms appear to fall in the same magnitude range as.lower order terms.

(iii) The component of torque along the boom axis is neglected. i.e., the boom is considered to be untwisted. For boom

deflections which are small, the axial component ofT~l' the

inertial torque, will be small and ~2 contributes no ~orque

along the boom axis.

As a result of assumptions (ii) and (iii) it can be shown that, referring to Eq. (11.4) cos epl 1 ~ sin ep

=

0 1 cos TJ

3

o

sin TJ

3

= 1

i.e. ~l has the same components in the][, ~ ~axes system as it has in the

bodyaxes set. This means that for purposes of calculating the thermal torque

on the boom, the boom is considered to lie along its nominal line.

In connection with assumption (ii), it should be noted that the small

deflection assumption is only useful for purposes of simplifying the boom shape

and solar radiation equations if the boom lies nominally along i*, j* or k*.

For any booms which do not lie nominally along one of these axes thë procëdure

is to change the !*, j*, ~* system temporarily so that the simplification can

be made and then transform the shape and torque values to the principal

!*, J*,

.!?

system.

This procedure is particularly easy for our example satellites since

the booms are all nominally in the same plane normal to the spin axis and the

co-ordinate transformation is a rotation about the k* axis until the i* axis

is collinear with the nominal boom line. Since we are analyzing the motion

about the k* axis only, k* is the only axis which has to remain as a principal

axis and the return transformation would not have to be made. This procedure

is implicit in the analysis which follows where any boom analyzed is taken

nominally along the i*,

Using these assumptions, the equations for findin~ boom shape become:

(i) ~hermal bending equation

C

3 [-cos~sin~sinep + cose cosep cos~ cos~ - sin~sine cosep ]

C

3

F( 8,

c:p,

~, ~) (IV.l)

(ii) total torque (acting on boom) components T _x ==

9

L T Y

pJ[

T Z == s P

J[

L s

z

(s'-s)~ '2 s'(z'-z)ds' + EI KTsinp ~ '2 (' ·2

(iii) boom shape equations

o

EI

d2y T ds2 z

EI

d2z ds2 = T Y

S~nce

dx/ds = constant and

(~)s=o=l,

it is clear that x

tl.ons.

(IV.3)

s under these

assump-The boom shape equations are solved for the steady-state y and z deflection values by assuming the following forms for y and z:

for 8,~,7/J constant. That is

z 0

y _ ~2 Y

This choice of solution forms is compatible with the sinusoidally varying thermal

torque component leading to be~ding in the ffi*, j* plane and the constant component

leading to bending out of this plane. For further discussion of this reasoning

see section

5.

become:

Using the assumed solution forms, the equations for ~inging boom shape

(i) thermal bending

~ (~8)

+

i

(~

8)

=

C

3

F(8,~,7/J,~

)

(ii) boom torques

T

=

0 X T

=

Y L

J

'2 p 7/J s L T Z P

J

[_~2

s'(y'-y)+ 2?Î;2Y'(s'-s)]dS:!: s

lJ.

(IV.5)

(Iv.6)

(Hi) boom shape EI d 2 x 0

2

== ds d2 EI J T (IV.7) ds2 z EI d 2 z ds2 -T y

It should be noted that T depends on, in the inertial term, and effect's only

the z deflection and similarly with Tand the y deflection. In other words

z

the y and z deflections are uncoupled when second order and higher deflection

terms are dropped.

The validity of the assumed solutioa form is established in the

following way. The thermal bending equation is solved (see Appendix F) for

given ~,8,~ and ~ to find the ~ value

(€)

which gives _{68max from which KT}

can be calculated. First values for Tand T_Z are found by neglecting the

integral terms in the boom torque equations i.e. using only the thermal

torque terms. These values for Tand T .are used in the boom shape

equa-tions to give the shape for the bgom. Th~t shape is used to evaluate the

inertial, i. e. integral, terms in T and T . These, added to the thermal

torque terms, give new values for T

Y

and TZ• These new values are used to

find a new shape. The iterative prgcess or

fin~ing

better values of Tand

T from each new shape, and then finding a better shape, is repeated

u~til

s~cceeding shapes agree within a specified limit. The above described

con-vergence on a shape is performed for values of ~ varying between 0 and 2 H.

Th~s a history of y and z for one satellite revolution is obtained and y is

indeed found to vary sinusoidally with ~ and z is found to be independent of

~(see Fig.9). Figure 10 shows a typical boom shape. The assumed forms for

y and z in this way are shown to be solutions of Eqs. (IV.l), (IV.2) and

(IV.3) and since they are compatible with the radiation input which drives

the shape, we take them to be the desired steady-state solutions for shape.

Therefore, for given e,~ and ~, the y and z values at any point in

the revolution could be found by converging to a shape at some value of ~

and then establishing Y(s) from

Y(s) == _sin(~ y( s, 7jJ)

+

'l/J )

o

and Z(s) from the found values. Y(sj, 'l/J and Z(s) are all that are needed

o

to give y and z at any value of ~ or s.

Now that the boom shape is solved for, the torque about the k* axis

due to solar radiation pressure can be foudn. Af ter reduction to first order

for'm, (the

.!?

torque component is somewhat simplér_'and for· 8, ~ constant the

only significant equation of motion, that about the k* axis becomes, for

+ where,

4

3 drp

..

"/J ! silÎllil.! L dap

J

(-a 22x + a12y) COSV dx

o

dz ds

(Iv.8)

In the above, a

12, a22,a~2 are components of the transformation matrix which expresses the space set 1n the body set. From Appendix A,

a

12 cos"/Jsin~ + cos8 C08~ sin"/J

a

22 sin?jJsi~ + cos8cos~cos"/J

a

32 - sin8cos~

M then is the torque about the ~* axis as effected by solar radiation strtking one boom at a given "/J. Therefore, knowing the boom shape for given 8, ~, "/J it is possible to find the torque about the k* axis from one

boom by performing the integration from 0 to L.

The average torque acting in ~he k* direction per revolution for an n-boomed satellite (all booms nominally in-the same plane) could then be found by integrating the k* torque from "/J = 0 to "/J

=

2~ for each boom, dividing

the result by 2v and sumffiing these average torques for each boom, ie •.

I

Mz

L

;1T

Î

(:

~

i

(1/J)

d1/J

i=l av i=l 0

4.2 getails of Torque Calculations

The actual solution procedure used is now described.

Since exact integration to find the boom shape, to find the:'k* torque for given 8,~,"/J and boom shape and to find the average torque per revo-lution is difficult, the integrations are done numerically.

To find the boom shape for a given e,~,~ and ~ the thermal torque

magnitude and direction at any point along the boom is found from KT and ~.

rhe thermal values of Tand T_Z are used in Eqs. (IV.7) and since the Tyand

T_Z

~alues,

if they are

~hermal

contributions only, will not be functions of s

Eqs. (IV.7) can be integrated as

2 s Z

=

_2" 1 2 -T S Y EI

to find the boom shape from thermal torque only. From the thermal torque boom

shape a first approximation to the inertial torques on the boom can be found

from Eqs. (rv.6) where tbe integral terms are the inertial torque components.

The sum of these inertial torques and the thermal torques give a new boom shape

through integration of Eqs. (rV.7) using the inertial and thermal total torque

components as Tand T z . The new shape will give a better approximation to the

inertial

torque~,

hence an improved shape will re sult from using these torques

and the process is continued until there is no appreciable change in shape

through one iteration. The integration of Eqs. (IV.7) is done numerically by

dividing the boom into one-half foot segments. Once a boom shape has been

fOQ~d in this manner for a particular ~ the shape at any other ~, for the

same! e,~ can be found by using Eqs. (rv.4).

The value of ~ is determined by first finding.the value of ~ where

the thermal deflection

o~

the

~*

direction is zero when

~

=

O. This is

accom-plished, for the boom nominally along the ~* axis, by finding ~ such that

;i

x ~*

-

~*

The value of ~ to satisfy the above relation gives the ~ value where y is zero

changing from negative to positive if ~

=

O. An additional term is added to

this value of ~ to account for the time laî in boom bending which would be

needed when ~

f

O. The term added is tan- wT. The reason for adding this term

is as follows. In the case where the radiation vector is in the boom plane the

differential equation for the curvature is (see Ref.l)

OKT 1

w

"è!iifJ

+ T KT

=

K c os ~ which has, as its steady-state periodic solution

KT

=

KT cos (~- ~) 0 KT KT 0 .Jl + w2 T2

P

tan -1 w'] w -if;

In other words, the boom bending lags the radiation input which gives that

bend.ing by tan-lwT. In the more general case with which we are dealing, where

O,~

f

0, the radiation input leading to bending in the ~*,

;i*

plane also varies

sinusoidally with ?/J and the thermal bending along the i* axis might be expected to lag the radiation input, for

~

f

0, by tan- l wT. Thus

J

x i*

[ ('/1 f . * ) + tan- l ';1 T ]

- 'I' or -~

=

I~ x ~*

I

'I'

(rv.9)

wo~ld be a sensible proposal to find ?/J. This theoretical value of ?/J is . _

checked finding the thermal shape as aOfunction of ?/J for different 8,

8

and ~ values. Thermal sinusoidal y variation has a phase lag closely agreeing with that found from Eq. (IV.9) (see Figs.ll and 12). The thermal and actual shapes have the same frequency and phase (see Fig.9) therefore ?/Jo found from Eq. (IV.9)

can e used for the lag of the actual shape.

M is found at a giveQ point in the satellite spin revolution by numericallyZintegrating Eq.

(rv.8).

~he integration to find the average torque per revolution is performed by dividing the 0 to 2~ range of ~ equivalent to one revolution into sixty equal sectors, finding M at these sixty different points in the revolution, summing these and dividiftg by sixty. This gives the average torque along the k* axis through one revolution as contributed by one boom. To account for the-satellite torque from a boom diametrically opposite to a boom whose average torque per revolution is known, the latter boorn's torque need only be multiplied by two. This is tmpossible because of the sinusoidal nature of boom shape with respect to ?/J.

I~ the above described manner, the average torque per revolution for any boom can be found. Thus by summing the average torques per revolution for all the booms the total k* average torque per revolution can be fourrl.

•

?/J,8

of a

4.3

Once the average torque per revolut~n for a given boom configuration, and ~ values can be found, all that is required to calculàte the despin given satellite is the 8, ~ and percent sun history of the sateltite.

Calculations Applied to Alouette 1

The Alouette 1 is the first example chosen. Thee,~ history is found by transforming the given spin axis attitude history (Ref.2) from co-declination

and right ascension values into the co-ordinate values of a unit vector in

the spin axis direction in the ~, ~,

!

set. Appendix D depicts the co-declination and right ascension history of Alouette 1. The steps in the transformation are to write the spin axis unit vector in the ~', ~',

!'

system from the right as-cension and co-declination, transform from this set to the ~o' ~o'

!a

set and then to the ~, ~,

!

set (see Fig.13). The co-declination is approximated by an absolute value sine function.

codec

=

_'2

TT _s~n• _' ( TTxt

40

(IV.10)

~he right ascension values are approximately constant over a co-declination nodding from 0 to ~/2 to 0 so the constant right ascension value can easily be selected knowing which part of the approximately two hundred day cycle the

spin axis is in. Appendix C gives the orbit characteristics of Alouette 1. The sateèlite is not immersed in solar radiation for t~enty-four

hours per day and Fig.15 shows the percent sunlight history for the Alouette

1 which is used in computing the spin decay. The percent sun was approximated

by fitting the given data with straight lines for the one hundred percent sun

plateaux and with sine curves for the variations below the one hundred percent

levels. The change in ~ over a five day interval is computed by holding ~,

e

,cp

and percent sun constant over that interval and thus a spin rate versus

time from 1aunch plot is made by dividing the flight time into five day

periods (sec Fig.l5).

4.4

Ca1culations Applied to Explorer

XX

The Explorer

XX

despin history is solved for in a manner exactly

the same as that used for the Alouette 1 except for the implementation of the

spin axis attitude and percent sun histories shown in Fig.l6. The spin axis

attitude history was approximated by fitting the curve given in Appendix D

with straight lines and the percent sun was fol~owed in the same manner as

for the Alouette 1. Five day intervals were again chosen as periods of

constant e,cp,~ and percent sun. Fig. 16 shows the despin plot for the

Ex-plorer

XX.

5. DISCUSSION

Ma~y assumptions and approximations are made in the foregoing

analysis. A summary and explanation of these practical measures which

have not been sufficiently discussed in previous sections is presented here.

The booms for Alouette 1 are actually Stored Tubular Extendible

Merribers rnanufactured by the SPAR Aerospace Company of Canada. This analysis

assumes the boom to be seamless in structure, however, it uses the flexural

stiffness (EI) for bending analysis and modulus of rigidity (G) for twmsting

analysis characteristic of the STEM booms. The error in not including the

overlap at the seam is introduced when the axis of bending is taken to be in

the direction of applied bending moment and the extent of this flaw remains

to be investigated. ~he assumptions made in using the first order

differen-tial equation for KT advanced in Ref.l are clearly stated in that reference.

'I'here is; however, a deviation from the intended application of thi.s equation

encountered in the way it is used in this work. The thermal bending equation'

proposed in Ref. l i s derived for the radiation vector lying in the boom

plane i.e., one diameter, always the same one, at any cross section always

has an extremum temperature difference across it compared to other diameters.

In our case, where the spin axis direct ion is arbitrary, the diameter with

the extremum 69 at a cross-section changes as the boom rotates. So the

relationship between 6 8max and KT in our case and in Ref.l is different

primarily because of the difference in the temperature distribution around

a

cross-section caused by a changing extremum 6 8 diameter. Equation (II.5)

relates _{68max and KT Dy a slightly higher} constant than used in Ref.l. In sections 2 and

3

it was recognized that since there wilL be a

component of inertial torque lying along the boom axis, the boom is subject

to a twisting torque. The angle of any diameter line relative to a

non-twisting reference line on a cross-section will therefore change as the axial component of torque changes. In computing the amount of twist,

n ,

due to

the axial torque, the time taken for the boom to twist tö its equilibrium

frequency of boom twisting is found to be forty-five to three hundred and

forty-five times larger than the greatest expected frequency of torque application

for the example satellites and therefore does not seriously jeopardize the

strength of the formulation in sections 2 and3 anilljnsection

4,

where the

twist-ing is neglected, has no influence.

The formula used to relate applied bending moment to curvature at

any point on the boom i .e.,

M 1

=

- K

EI p

is yalid if the material being bent obeys Rooke's law and can be easily used if

the value of I, the bending moment of inertia of the cross-section does not

change during the bending. These conditions exist for small curvatures which

is the case for the satellite booms studied.

The most obvious approximations of the analysis are made in section

4

where the equations of shape and motion are solved. These approximations are

discussed below.

In order to make the equations formulated in 2 and 3 tractable, terms

which are second order or higher in deflection or slope values are dropped in

comparison to first order terms and boom twisting is not considered.

In section

4

,

when solving the equations for boom shape, a solution

form in which the z deflection is assumed to be_independent of ~ and the y

deflection sinusoidally depending on ~ is taken to be the boom shape solution.

The closeness between a pure dinusoid and the y deflection versus ~ is shown

in Fig.9. For purposes of comparison, the amplitude of the compared sine

wave is slightly greater than the amplitude of the deflection curve but has

exactly the same frequency and phase as the deflection curve.

A solution for a boom shape is found by converging to that shape

through a n~ber of iterations. Some criterion has to be established to

de-cide when the shape has converged closely enough to the solution shape. If

succeeding y and z deflection values at the end of a boom agree to wit~in

one-hundredth foot, th en the shape is accepted for an Alouette 1 boom. Succeeding

deflection values must agree to within two-hundredths for an Explorer XX beam.

The numerical solution for boom shape divides the boom into one half

foot segments but the numerical evalvation of satellite torque about the k*

axis divides the boom into one foot segments. The one foot intevvals

are-considered to be small enough to give an accurate value for the torque from a

boom of known shape since that shape has small curvature values everywhere on

the boom and a very slight change in deflection stope will occur over: a one

foot interval. Evaluation of the despin over a five day interval where the k*

torque is calculated with a one foot interval and with a one-half foot interval

shows agreement to the fourth decimal place in the value of ~ ~ in radians/

second.

In finding the attitude history of Alouette 1 in terms of 8 ,~ an

approximation is made of the co-declination and right ascension history shown

in Appendix~. The approximation is made by using Eq. (IV.10) to find the

co-declination, and the right ascension is chosen from 5.6, 1.71, 4.18, 0.384,

or 2.79 radians depending on which time region of the five co-declination

noddings cycle is wanted. The co-declination and right ascension found in this

WBnner give a reasonable approximation to the true history in Appendix D and

are certainly close enough for comparing the predicted spin decay with the

flight data. The errors in co-declination will not flignificantly change the

spin decay rate. The right ascension values listed earlier are mean values for

any given nod~ing from the earth's axis and therefore, on the average, are close

to the actual values. It should be noted that the spin axis attitude history

in Appendix D is :n..ot really accepted as an accurate account of attitude due to the lack of rneasur'ing devices on board Alouette 1 for spin axis attitude

measurements. The percent sun history of Alouette 1 given in Fig.15 is well

approximated by the sine waves and plateaux used.

The E-xplorer XX attitude history and percent sun history are well

approximated by the fittings made.

The earth's orbit about the sun is taken to be circular which develops

errors in rWo ways.

(i) the variation in radiation intensity as the earth~to-sun distance

varies is neglected

(ii) the rate at which the earth-to-sun line sweeps around the ecliptic

plane is taken to be constant which is not true for an ecliptic

path

Neither of these approximations seriously effect the validity of the match

between the predicted spin decay and flight data.

ll'he main results of this study are the boom shape and satellite

de-spin. Fligures 9, 10, I I and 12 depict the boom shape resul ts . In Fig. 9

it is clear that the z deflection remains constant throughout a revolution and

the y deflection varies sinusoidally with~. Both of these features are

ex-pected results. 'rlhe shape is determined, for the most part, by the temperature

gradients across the boom due to solar radiation heating, so the constant z

and the periodic y should be explainable from the way in which the radiation

ef~ects a shape. Referring to Fig. 14 a study of the radiation vector-boom

a~is orientation history will explain the boom shape variation with~. To

I

analyze hOVl the curvature, as effected by solar ;radiatio~, varies wi th ~ we

analyze how the component of € at right angles to the boom axis varies wit h

~ since it is the only componënt which causes heating. The first step is to

break the radiation vector into three components, one along" the boom axis, one

at ninety degrees to the boom axis and the boom plane and the third at ninety

dègrees to the boom axis but in the boom plane (see Fig .14) . The two components

which effect a ~

e

across a diameter of a cross-section and hence effect a

bending are the two at ninety degrees to the boom axis. Taking

-Ij;

= 0 we shalJ..

now see how the two deflection values, y and z, should vary with ~ from thermal

effects. The radiation component normal to the boom axis multiplied by a

constant gives ~ 8 as stated in Eq. (11.3) and" mul~iplied by further constants

I'\:T and the thermal bending moment as stated in Eqs. (II.5) and

(n.6)respecti-vely. The thermal bending moment divided by EI and multiplied by the eosine of

the angle between the ~ 8_{max diameter line and} the boom plane gives the y

z curvature i.e., referring to Fig.14.

and we see that, taking

t

=

0, the y thermal deflec~ion should be sinusoidal

with ~ and the z thermal deflection constant. For ~

f

0, the.y and z thermal

deflections will behave the same as ~ varies as they did for ~ =

°

except that

the magnitude of the bending will be different and the bending will lag the

radiation input. Thus we have shown that the thermal boom deflection should

be sinusoidal in the y direction and constant in the z direction. This forms

the basis for the assumptioq of the solution forms given by Eq. (Iv.4).

The other major result, the satellite despin, is plotted in Figs.

15 and 16. It is clear that the despin rate changes through the plot. The

three influences on despin rate aside from the satellite configuration are

the fraction of ap orbit that the satellite is in sun-light, the spin axis

attitude and the spin rate.

Upon examining the despin plots for both satellites, it is clear

that the plateaux in the curve i.e., periods of small spin rate decay are

coincident with low percent sun and a near parallel condition between the

spin axis and solar vector.

6.

-CONCLUSIONS

The described work analyzes ~,I, ~.i how solar radiation interacts wi th

the long flexible booms of a long-boomed satel~ite to iqduce a torque which

effectively changes the spin rate of the satellites. The study is done for

an arbitrary orientation of the spin axis.

The results for first order boom deflections show that the boom

deflection parallel to the nominal boom plane is approximately sinusoidal with

the rotation angle aboMt the spin axis and that the boom deflection out of

the nominal boom plane is approximately constant with respect to the rotation

angle. The rate of despin has been shown to depend on the attitMde of the

spin axis, the fraction of the orbit that is sunlit, the spin rate and the

satellite configuration.

The predicted spin rate history matches well with the flight data

for Alouette 1 aRd Explorer XX which is reason to believe that the theory is

a good one.

With further verification, the program used to predict the spin

decay could be applied to other satellites of this class to foresee serious

spin~downs or spin-ups before launching. Spin-up behaviour analyzed for an

arbitrary spin axis attitude to the solar radiation vector could be

investi-gated by using higher initial spin rates.

1. Etkin, B. Hughes, P.C. 2. Graham, J. D. 3. Goldstein, H.

1

4.

Timoshenko,

s.

Young,

D. H.

5. Soko1nikoff,

I.

S. Redheffer,

:a.

M. 6. King-He1e, D.G. Ei1een, Quinn REF'ERENCES

"Spin Decay of a C1ass of Satellites Caused by Solar Radiation". Ul'TAS Report 107, Ju1y 1965. Conàensed and extended in "Explanation e:f the Anomalous Spin Behaviour of Sate11ites with Long F1exib1e Antennae". JSR VoL

4 ,

No.9, Sept. 1967.

"Dynamics of Sate11ites Having Long F1exib1e Extend-ib 1e Member s" • De Havi11and Aircraft Co. Toronto. "C1assica1 Mechanics". Addison-Wes1ey, 1965. "E1ements of Steength of Materials" • D. Van Nostrand, 1962.

"Mathematics of Physics and Modern Engineering". McGraw-Hi11, 1958.

"Tab1e of Earth Sate11ites Launched in 1957-1966" R.A.E. Technica1 Report 67039, February 1967.

APPENDIX A: TIME RATE OF CHANGE OF w

To find the time rate of change of angular velocfuty,

w,

we shall need

to describe the Euler angle rotations by three orthogonal transformations.

Referring to Fig.2 these transformations are

! '

,J.,

~

t

_{! l ' .J.l ,}

~l

about the k axis

!l' .J.l ,

~l

ct

!2' ~,

~2

ab out the !l axis

!2 .J.2 ~2

t

i* .J.* k* ab out the ~2 axis

where the matrices of rotation are

cos cp sin cp 0 1 0 0

-sin cos 0 D 0 cos 8 sin 8 C

0 0 1 0 -sin 8 cos 8

cos?jJ sin?jJ 0

-sin 7jJ cos 7jJ

o

B

o

o

1

The matrix which transforms from

l,

.J., ~ to l*, 1*, k* is the product of these

three matrices i.e.,

A

=

BCD ~s shown in Ref.3: c7jJccp -c8scps7jJ c7jJscp + c8ccps7jJ s7jJs8 A -s?jJccp -c8scpc?jJ -s7jJscp + c8ccpc?jJ c7jJs8 s8scp -s8ccp c8 all a₁₂ aU = a 2l a22 a23 a 3l a32 a33 where,

c7jJ = cos7jJ c8 cos8 ccp coscp

s7jJ simjJ s8 sin8 scp sincp Al

The angular veloei ty can be wri tten as

.

.

.

~

=

ep"!.

+

8~1

+

?jJ"!.2 As shown in Ref,3: w

₌

Sc ?jJ

+

cps ?jJs 8 x

.

w

₌

és

'l/J

+

epc ?jJs 8 Y

_.

w

₌

epc 8

+

?jJ z and since w

=

(~)bOdY

+

~~

=

0 w

=

w i* +

W

J.*

+

w

k* x - y

z--

..

w

=

[ -?jJ8s?jJ + 8c?jJ + ep8s?jJc8 + cp?fJs8c?jJ + eps?jJc8 ] i* "

,

.,

.

"

,

.

--[ -?jJ8c?fJ - 8s?jJ

+

ep8e7fJe8 - cp?fJs8s?jJ

+

epe?jJs8 ]

.J.*

•• I . [ - 8cps8

+

epe8

+

?jJ] ~

S

i"*+

S

i* +

S

k* x- y""- z-A2

APPENDIX B: MOMENTS OF INERT IA OF SATELLITE We wish to know I1' I

2 and I~, the rotationa1 moments of inertiaLbbout

i*, j*, k*, the three bodyaxes of thé satellite (Fig.2). Referring to the

sketëh bë1ow, the moment of inertia bout an axis of a body is in general given by

where the integration is over the entire body.

The moments of inertia of the sate11ite are thus found from n

L

i=l n

L

i=l n I3

=L

Pi

J

i=l 0

o

L. 1. L. 1. 2

(Yi

+ z.) 2 ds + I 1. sl 2 2 (z. + x.) ds + I 1. 1. s2 2 2 (x. + y.) ds + I 1. _{1. s3}

for a sate11ite with no booms. are the moments of inertia of the sate11ite body about the ~t ~*, and k* axes respective1y.

APPENDIX C: SATELLITE ORBIT CHARACTERISTICS

(i) Alouette 1 (from Ref.6)

orbital inclination nodal period semi-major axis perigee height apogee height eccentricity argument of perigee

(ii) Explorer XX (from Ref.6)

orbital inclination nodal period semi-major axis perigee height apogee height eccentricity argument of perigee 80.460 105.42 min. 7392 km. 1998 km. 1032 km. 0.002

8

0 79.870 103.97 min. 7323 km. 871 km. 1018 km. 0.01 2890

APPENDIX D: SPIN AXIS AT.rITUDE HISTORY FOR ALOUErTE 1 2~ 191(

1jO

rr

160 210 1 150 \ / 220 1ÇKl 140

"

/ 230 ll.O "- Assumed satellitte permant magnetic moment alon~he ~ spin axis =- !i microw~r metres -lO-A ouettä I - observe

91 ---10--Al~l1ette ca. cLlla.ted I

-100 ~ -QJ -70 gGO -d '60 I c:: o _ .~ 80 '50 32b :§ 0 '40 / ~ ₀ 100 \ \ 30 ,~~~ 330 _/ u 340 I

L

' 20 ~~\.r:p 350 10 Ihg\'lt OSç:J,

J::rj t-'

APPENDIX E

Satelli te Data

Alouette 1 Explorer XX Alouette 11

Boom lengths

2

booms @

75

ft.

2

booms @

60

ft.

2

booms @

120

ft.

2

boems @

37.5

ft.

4

booms @

30

ft.

2

booms @

37.5

ft.

Boom material steel Berylium-Copper Berylium-Copper

BoOf!1 diameter cl:

0.95

inches

0.5

inches

0.5

inches

Absorptivity,

a

_0·9

0.45

-

0.45

Reflectivit;y-, P!>

0.1

0.55

0.55

Mass/length, P

0.00212

slugs

0.000441

slugs

0.000441

slugs

ft. ft. ft.

Bending Stiffness EI

351

lbs - ft.

2

15.49

lb _ ft.

2

.

15. 9

4

lb. - ft.

2

Time constant, T

17

sec.

2.7

sec.

2.7

sec.

KT

o

.0064

ft.-l

_*

same as EKp~orer XX

Polar moment of

2

2

2

inertia I _m

681

slugs-ft.

84.4

slugs. -ft.

577

slugs-ft.

**

'"

Solar pressure

.95

x

10-7

psf

=-95

x

1

0-7

psf

.

95

x

10-7

psf

constant, p

APPENDIX F: SOLUTION OF FIRST ORDER THERMAL BENDING EQ,UATION

;

It is neèessary to khow the steady-state solution of Eq. (IV.l) i.e.,

[-cos~sin~sin~ + cosecos~cos~cos~ - sin~sin8cos~]

Changing the independent variable from t to ~ we have

or

now by holding e,~ and ~ constant and co~sidering ~ to be constant the same

equation becomes

C

~ cos~sin~sin ~

C

+

~ cosecos~cos~ cos~

_(E.l)

C

;:

i

.

sinecos~sir$

The steady-state solution of Eq. (E.l) is

(E.2)

+

C 3

t

_ _

-

3r

...J_"--_ _ _ cos~sin~

--=-_ _

+T

C~

3

cosecos~cos~

_ _ _ _ _ _

'

_

I

J

cos ~

(T-q; )2 + 1

The above equation is solved to find the ~ value for a maximum 68

and'the value of 68

max itself by setting

~

(68)equal to zero andfinding which of the two

~

values which satisfy this condition cause

~2 ~~~

to be less than zero. , That ~ value (~) substituted into Eq. (E.2) ,gives 68 • The value .of KT can be found from Eq. (11.5). The product of C3 and C4

~the

value 'K' given in Appendix E. A slight increase in'K' was necessary to fit the data; see

APPENDIX G: TIME LAG IN BOOM TWISTING

There will be a time interval between the time when a twisting torque

is applied on the boom and when the boom reaches and stays at an angle of twist

determined by that torque. If the natural torsional vibration frequency of

the boom is much greater than the frequency at which the twisting torque is

applied then the time lag can be neglected. Analyzing the motion of a

canti-lever boom and using the following notation

P

d mass per unit volume

I = free rotational moment of inertia of boom I' fixed end rotational moment of inertia of boom

1'

-P polar moment of inertia of boom

GI_p L

T twis~ing to~qae

n

angle of twist

we can say th at for a boom which is fixed at one end, straight, of constant

cross-section and has only its own structural rigidity giving a twisting torque

I' d 2

_n

+l1t

n

=

0 dt2 pr d2

n

₊

l1t

_n

₀

=

dt2 I'

I' ~! I however we shall use I for I' and the frequency calculated will be

3

lower than the true natural frequency. The natural frequency of torsional

vibration is Remembering that I P I W R

~~

~

radians!second

!+

element of cross-sectional area

we can say for a hollow shaft of constant cr0ss-section and therefore, I so dm=

J

· 2 P d L r dA G1 P LPdLI P LdA Pd L1 P G

Computing the value of ~ for Alouette 1, Alouette 2 and Explorer XX.

(i) Alouette 1 = (ii) Alouette 2 G 12 x 10

6

psi. L 75 ft. P d = 0.283 lb/irt 3

6

12 x 10 x 144 142.5 ·radians/second

= 22.7

cycles/second.

G

=

6

x 10

6

psi (for copper)

L = 120 ft. P d = 0.322 lb/ in3. 6 x 10 x 144 (120)2 0.322 x 1728 x 32.174 58.9 radians/second

=

9.38 cycles/sec0nd (iii) Explorer XX

G

=

6

x 10

6

psi (for copper)

L = 60 ft.

P

6

x 10 x 144 x 0.322 x 1728

32.174

117.8 radians/second

=

18.7 cycles/second.

The frequency ofaxial torque application is the same as the spin fre-quency of the satellite since the boom shape is periodic. Taking four cycles per minute or one-fifteenth of a cycle per second as the maximum frequency of torque application we can, by comparing this frequency to the ~ values for Alouette 1, Alouette 2 and Explorer XX, neglect the time lag in boom twisting for these

satel1ites.

x

FIG. 2.

k

Z

CO-ORDINATE SYSTEMS

y

1

1,

j,

k fixed in space

FIG.

3.

PRINCIPAL AND BODY AXES OF ALOUEl'TE 1

o.

1

I

_xp

o.

I

Izp

FIG.

4.

PRINCIPAL AND BODY AXES OF EXPLORER XX

••

J

Ixp

••

1

,....

k

~--~--+-~---

-.

_J

":"""

I

.*

J

1