Attitude stability and performance of a dual-spin satellite with large flexible appendages

October,

1971.

ATTITUDE STABILITY AND

PERFORMANCE OF A

DUAL-SPIN SATELLITE WITH LARGE

FLEXIBLE APPENDAGES

by

D. B. Cherchas

,

9

'U

4

ATTITUDE STABILITY AND PERFORMANCE OF A DUAL-SPIN SATELLITE WITH LARGE

FLEXIBLE APPENDAGES

by

D. B. Cherchas

Submitted September, 1971.

ACKNOWLEDGEMENT

The author wishes to express appreciation to Dr. P. Co Hughes for his expert guidance throughout the course of this study.

The cooperation and interest of Mr. G. J. Cloutier and Mro R. Hornidge of the Hughes Aircraft Company in providing sate1lite configurationsand parameters is gratefully acknow1edged.

Appreciation is a1so expressed to Mrs. B. Wadde1l for typing the manu-script and to Mr. N. K. Phung for drawing the mathematica1 symbo1s and figures.

Financia1 support has been provided by a 1967 NRC Science Scholarship, NRC Grant No. A4183 and USAF Grant No. AFOSR 68-1490.

SUMMARY

The attitude stability criteria and nutation decay times of a dual-spin'

satellite with a large flexible solar array are found in terms of significant

satellite parameters. The equations of motion are developed from Hamilton~

prin-ciple and ~hen linearized, with solar array natural modes utilized as array degrees

of freedom. It is shown that with proper mass balancing, the equations for spin

variables can be decoupled from those for transverse variables. The linearized

equations are numerically studied through the system eigenvalues thus yielding

direct statements of stability and nutation damping time. These eigenvalue studies

are done first assuming the array to be rigid and then with array flexibility in-cluded. A complementary analysis for the solar array develops practical

informa-tion regarding array natural mode frequencies and shapes and boom buckling con~

siderations •

Numerical studies of the satellite attitude stability and nutation

damp-ing time assumdamp-ing the solar array is rigid indicate that the ratio of spin inertia

to the geometrie mean of the combined transverse inertias is a significant ratio.

The value of unity marks the minimum value of this ratio before energy

dissi-pation in the despun section is required for stability. It is shown that

stab-ility is unachievable below a certain value of this inertia ratio for a given

damper mass. Upper and lower stability boundaries exist for the nutation damper

damping constant and the damper performance is best at high nutation frequencies.

Results of the same studies aliowing array flexibility but na array damping show

little ~hange in stability boundaries and putation damping times from the rigid

array results except when the nutation frequency becomes close to an array natural

frequency. In this circumstance, stability criteria and performance are signifi

-cantly and undesirably affected. _I~is shown thatthe ~b~trary use of ~ery smal~

-amou~ts of array damping in the flexibility studies could lead to many important

I

.

Ilo 111. IV.

v

.

TABLE OF CONTENTS Notation INTRGDOCTION 1.1 Dua1-Spin Sate11ites 1.2 Spin Stabilization 1.3 Research Needed

SOLAR ARRAY ANALYSIS

2.1 Analysis Requirements

2.2 Array Description

2.3 Related Analyses

.2.4 Differential Equations

2.5 Linearized Equations with Boundary Conditions

2.6 Natural Vibrations

2.7 Results and Discussion

EQUATIONS OF SATELLITE MarION

ABour

THE-~.C.ENTRE 'OF3MASS

3.1 Deve10pment of Equations

3.2 Generalized Coordinates

3.3 Kinetic Energy

3.4 P0tential Energy

3.5 Generalized Forces

3.6 Linear Equations

3.7 Discussion

STABILITY ANB PERFORMANCE ANALYSIS

4.1 Stability Conditions and Nutation Decay Time

4.2 Description of Eigenvalue Study

4.3 Case Studies

4.4 Motion of Coro~lete1y Rigid Dual-Spin Systems

CQNCLUSIONS

5.1 Solar Array

5.2 Dual-Spin Satellites without Flexible Appendages

5.3 Dual-Spin Sate11ites with Large F1exib1e Appendages on

Despun Portion

5.4 Despin Control and Unbalance

REFERENCES FIGURES

APPENDIX A~ Attitude Motion of a Single Spinning Body with a Two

Degree of Freedom Nllitation Damper

PAGE 1 1 1

6

6

6

7 7

7

9

13

19

20 20 20 22 23 23 24 32 33 33 33 34 38

40

40

40

41 41 42

APPENDIX B: Satellite Kinetic Energy with Large Flexible Solar Array

APPENDIX C: Component Angular and Centroidal Veloeities APPENDIX D: Solar Array Mass Integrals

APPENDIX E: Solar Radiation and Gravity-Gradient Torques i J . '

A B b d d -xp d s E. 1

E

s e N0rATION

y coefficient matrix in decoupled nutation variable equations

antenna reference axes

satellite combined moment of inertia in ~ direction

through total center of mass

satellite combined moment of inertia in ~2 direction

through total center of mass

y coefficie~t matrix in decoupled nutation variable equations

solar array sheet half width

nutation damper bob wire length nutation damper damping coefficient

vector from reference point on array mount to mass element on flexible array

vector from center of array end piece to mass element on end piece

distance along ~l from center of array mount to base of array boom 1

solar array symmetrie mode shape Young's modulus of solar array sheet

solar array width

antenna centroidal body axes; ~l' ~2' ~3

axes fixed to base of nutation damper bob wire fuel centroidal bodyaxes

total satellite centroidal inertial attitude axes

array mount centroidal body axes

rotor centroidal body fixed axes

axes fixed at mounting point of array boom 1;

~l~ ~2' ~3

I I I aU, a22, a33, 1a12,la13,123 1 1'll, etc. , I pll, etc. , 1 rl1, etc. , I s J b J s Kd

t

M

M.

l. m m a m_d m_f m p m r m spa m xp N Q. J q. J

9t

Rf r_d

force field acting on array cross-piece at sheet attachment

moments and products of inertia of antenna in

~

moments and products of inertia of

moments and products of inertia of moments and products of inertia of total spin inertia

=

1r33 + 1f33 flexural stiffness of array boom

fuel in

~

array mount rotor in

~

flexural stiffness of sheet in lateral plane nutation damper spring stiffness

solar array length

in

~

decoupled nutation variable equations system matrix solar array torsional mode shape

total satellite mass antenna mass

damper bob mass fuel mass

array mount mass rotor mass

solar array mass array end piece mass

eigenvalue of satellite nutational mode generalized force

generalized coordinate satellite inertia ratio radius of fuel tank

r f r p r r ~l' T T e -t t s U

V

.

l W x

al'

13-13

2, )'

)'3

5 1, E . , al ~2' ~3

a

2,

a

3

~3'

~:t 5 2 Ebi

distance, along spin axis, from antepna reference point

to fuel mass centre

distance, along spin axis, from antenna reference point to array mount reference point

distance, along spin axis, from antenna reference point

to rotor referenee point

solar array reference axes

satellite kinetic energy about mass centre

forcejwidth across array sheet

time

array sheet thickness

satellite potential energy function

solar array lateral mode shape

distance from centre line of sheet, measured

perpendiu-lar to centre line, to sheet mass element, positive in

-

~3 dii'ection

position vector of array mass element as seen in

~

position of antenna mass centre relative to antenna

reference point

position of array mount mass centre relative array

mount reference point

decoupled nutation variable equations state vector where lT =

(à

1

a

₂

8

₁

,82.'

Gai. • Ébi •

)lat. ' Pb;'"

~àt

• ;>bt J

ai •

a2 • &1 ,

8

2 ' Eat ' Ebi' ).Lat' P-bi ' Jat' "bi) Euler angles taking

~-9i

generalized coordinate for rotor spin

àrray lateral mode functions

array sheet massjlength

rotation taking

~--~

rotation angles taking ~A~9b

I-l_ai, I-l_bi I-l v _{. ,vb'} al. l. v s 'f3 IT _l.J •. kt Pb Pbp Pf Pxp 0-S

.

0-angles taking

%

~~

angle'; taking

~ ~~,

to zero th order; also used modal amplifier time function in array analysis array torsional mode generalized coordinates fuel viscosity

array lateral mode generalized coordinates array sheet Poisson ratio

rotation angle taking

~ ~ ~

solar array mass integrals solar array boom mass/length array base-piece mass/length fuel mass/volume

array end-piece mass/length array sheet mass/area

rotor nominal spin rate

frequency of solar array natural mode angular velocity of jth body in satellite

fuel motion variables transformed from ~1'~2'~3

I. IN'I'RODUCTION

1.1 Dual-Spin Satellites

<

The abili ty to confidently design and build a satelli te that meet.s the the mission requirements for attitude stability and controllis of fundamental importance in ensuring mission success. The classification of attitude control systems ranges from the passive systems such as gravity-gradient,stabilized space-craft to cQmple~, _{I .} fully active systems such as OAO.

A class of currentcommunications satellites might wel~ be placed in the semi-active category. These are the dual-spin variety, cemprised of a rapidly spinning rotor intended to giv~ gyroscopic stiffness, and one or two ,slowly spinn-ing bodies rotating ab out the rotor spin axis and acting typically as communi ca-tions antennae or solar energy gathering surfaces. The attitude of thespin,axis

and rotation rates of the bodies'can be adjusted by mass expulsion or momentum wheel control. Figure 1 indicates .a prototype· (courtesy of Hughes A~rcraft

Company) of such a satellite.

It is the intention of this analysis to find and study the torque-free stability 'criteria and nutation damping time of such a dual~spin spacecraft.

1.2 Spin Stabilization

The ccmcept of spin: stabilization is of fundament al importance ,in under-standing dual-spin spacecraft attitude motion, so a review of bot:q. thebasic pri' n-cipals and advanced techniques of gyroscopic stabilization is in order.

a) single rigid body

The Euler equations of motion for a single rigid body in·a torque-free environment are I_l ,J;2,I 3 principalmoments of

l

1

w

1 W 2 W 3

( 1

2 -

1

3 ) inertia of body -I~W2. W3 w

_{1 (}

I~

_-

_{I, )}

(1.1) ~'W2'W3 comp?nents of body

1

₃

w?,

W_{1 lU 2 (}

1

_{1 -}

1

_{2 )}

angular velocity in

_i1,12.,

_-

_i

_a

where, referring to Fig. 2, axes

i

,

~,

1

are body fixed principal.axes. It is clear that there are three immediate

s

olut~ons

to these equations, i .e., pure spin about any of the three axes. However, if we examine the stability of rotation close to a principal, axis, say

1

3, by assuming <.03) Wl , <.02 and dif:ferentiating and intersubstituting the first two equations of (1.1) we see that

,

(L2) Therefore, for stabIe .periodic solutions for W

l and W2' the mass must be distributed such that 1

3) 11, 12 or 13

<

11, 12. Rotation about the axis of inte r-mediate inertia is unstable.

b) stngle body wit~ energy dissipation

A heuristic ,analysis of a single body witll internal energy dissipation* induced by nutational motion cau be done from an 'energy-sink' approach, i,e"

by recognizing that the kinetic energy decreases, but ,the total angular momentum remains constant, A spinning body which is stabIe in this situation then is one which decreases its nutational amplitude as the kinetic energy ~ecreases,

An

energy-sink argument for a single "spinning body, is, as follows ., For a fixed

magnitude of angular momentum h, the kinetic energy T of a body spinning about an axis with moment of inertia I is

T

=

i

h

2

2 1 (1,3)

Therefore, the axis of maximumr, I is the axis of mJ.nJ.mum T 0 Thus, rotation about a body axis near the ax~s of maximum I would decay to pure spin about the maximum inertia axis .when

T<O

.

Note that in pure spin, L e., when nutation and precess.ion no longer exist,it i s assumed that the energy dissipation disappears. Reference 2 is one of the early analyses identifying the aboye

stateà 'major-axis' rule·for stability. An energy-sink analysis of course mu~t

be complemente~ with a more rigorous stability analysiso Appendix A analyzes the motion of a spinning body with a two degree of freedom internal damper, The

requir~ents for nutational stability are shöwn to be the same as disclosed by the energy~sink. argument, The nutational instability of Explorer I in spin about i ts minor axis of inertia (the hoped for axis of stabili zation) is an expensi ve illustration of the major axis rule. The energy dissipation occurred in small flexible antennas.

c) dual-spin systems

A dual-spin,satellite (see Fig. 3) is also capableof being spin stabi-lized. The gyrosc9pic stiffness is developed by revolving the bodies about their common spin axis in such a way that a significant amount of angular momentum exists

along the spin axis. A practical realization of this is one body rapidly spinning and one or more bodies almost c9mpletely despun for solar power gathering or

communications purposes, It ,is the combination of the simple spin stabilization mechanism and a despun platform that is the essence of duai-spin desirability. Since .it is known that for a single spinning body any internal energy dissi pa-tion will cause a nutation buildup unless the spin is about the maximum moment of inertia axis the question of stability in the presence of internal energy dissipation for dual-spin systems must be investigated.

At this point, a note on meeting inertial geometry constraints in satel-lite manufacture is in order. The inertia ratio of a satellite is a function of its geometry and mass density distribution. For· a spin-stabilized body, the axis of spin is generally an axis of revolution i,e., allmoments of inertia per-pendicular to this axis tbrough the centre of mass are the same, To make maxi -mum use .of space in the launch vehicle compartment, the satelli te is mounted for launch with the spin axis 'along the rocket centerline. To accommodate larger satellites, it is relatively easy and inexpensive to make the compartment longer along the centre line but,much moreexpensive to 'increase the payload compartmen~ diameter. Thus, satellites which have a narrow cylindrical shape (minimum moment of inertia) can be much less expensive to launch than dis~ shaped (maximum moment * herein energy dissipation shal.). refer only to dissipation induce.d by- nutational

of inertia spacecraft). Also, having to make a rotor of a certain diameter for

no functional reason except to satisfy inertia ratio requirements is an

undesir-able design restriction.

A considerable amount .of research into dual-spin stabili ty theory has

been developed in recent years. The most significant advance was made ip

1964-1965 when it was noted independently by 1orillo3 , and Landon and Stewart4 , that

by ensuring sufficient energy dissipation takes place on the despun bodie(s),

spin stabilization can be maintained despite rotor internal energy dissipation

at any ratio of rotor spin inertia to satellite transverse inertiao These

analyses assume both bodies to be bodies of revolution about the common spin axis.

As in the single spinning bodyanalysis, the stability conditions have been

developed from an energy-sink approach and by stability analysis of the .satellite equations of motion with an explicit damper model. Reference 3 develops the

cri teria from the equations of motion while Ref . .

4

is an energy-sink development 0

An energy-sink analysis similar to that of Ref. 5 is given below.

Referring to Fig.3, the angular momentum h and ki~etic energy

the dual-spin system can be expressed through I

where

h

2 _(1.0.

1)2 +

(C.aIl)~

+

(Aw

r

)2

2

T

Aw~

1"

Hl~

"I-

C.Q~I

I

=

moment of inertia about spin axis of body I C

=

moment of inertia about spin.axis of body 11

A

=

total satellite transverse inertia through satellite centre of mass

W

T

=

satellite transverse angular velocity

n

I

=

spin rate of body I

nu

=

spin rate of body II T of (104) (I. 5)

Following the energy sink concept , it is recognized that h2 will be a

constant and

T

will be negative in the presence of energy dissipation. Combining

the expressions for

T

and

h

2 we have

2 • ₂

_fL

2 •

0

=

I .0,1

UI

+ C aU 11"1-

A

WTW_{T ( 1.6)}

.

AWTWT

1-0.

1

6.

1 "I-

C.{111.ö.

U (I. 7)

T

=

"I-.

.

.

.

.

T

=

_{Tl +}

_Tu

= _{- IÀ.}

1 O,I

- C

À.

u

n.

u

( I.8)

.

.

where TI and TIl represent the rates of energy dissipation in bodi~s land 11 respectively and

Rewriting,

thus, from the

.

TI - ~l statement

.

cnu

= for T, Eq c (1.7), and

Tu

~1I substi tuting (I.9 j from Eq. (1.9) we have

( Tl

+

TIl )

À₁ Àll

=

(LIO)

where

TI' T

_{rr (}

0, À T ) 0

Now, utilizing Lyapunov stability theory, since

Y2

AW

T 2

is a positive

definite function, if A~T

W

T is negative definite, the nutational,motion isasymp~

totically stabIe about W

T

=

O.

or.

or

Thus; for asymptoticstability

ÀT

(Tl

oT

T

n )

<

0

ÀI Àu

Therefore, the system is stabIe when

À1 ' Àn )

À

₁) 0

,

Àu<Q

and À 1

<

Q , À. 11

>

0 and 0

TI

') Àl Tl

<

TIl ÀI Àll

.

Tu

À_II

For the case of one body despun i.e.,

n

I

=

0, the condition becomes

+

<

0

( LIl)

( L12)

(L13)

Thus, if body I has no dissipation i.e., TI

=

0 then

(ciA

-

1» 0 i.e.,

spin about maximum inertia axis is required for stab~lity. The presence of a

significant amount of dissipation in body I can modify this requirement

signifi-cantly, allowing values of

cl

A <. L The energy -sink analysis of Ref.

4

is similar

to the preceeding development except that damping is allowed in only one body.

A more definitive argument can be.advanced by writing the satellite

equations of.motion with an explicit damper model. References 3 and 5 give

analyses of this type with dashpots as the damper modeIs. The stability

require-ments developed through these analyses are the same as yielded in the

energy-sink approach.

The development of the dual-spin theory led to a symposium

6

devot~d

to the subject. Some of the papers are described below. A persistent problem

in.dual-spin equations is the presence of periodic coefficients in the equations

of motion. These coefficients appear because of the large rotation rate.between

the despun and spinning parts. If damping appears in only one of two symmètric

bodies, the periodicity can be avoided. However, for a general configuration,

an initial development of the equations yields periodic coefficients. Mingori!

handled the periodicity by utilizing Floguet theory which essentially gives the

conditions for stability of a linear set of equations if the system matrix is

8

this can be a lengthy and cestly precedure. Barba, Heeker and Leliakev were

able_ te remeve the periedic ceefficients by a suitable variable .transfermatien.

The transfermatien is pessible because ef the symmetry ef the roter damping medel

abeut the spin axis. ~e, remaining censtant ceefficient linearized equatiens are

easily handled. Velman develeped a tuIl scale simulatien heweyer, ne deeper

understanding ef the stability criteria is interpreted frem it. In Ref. 10,

Ierille discussed the perfermance ef the Hughes Aircraft dual-spin cenfiguratien. Any ef the papers in this sympesium whiah yield stability criteria.are develeped

enly fer c9nfiguratiens in which beth spinning parts are bedies ef revelutien

abeut the spin axis,

Likinsll developed an analysis fer damping in the platferm enly. This

paper utilized Reuthian analysis to state the stapility criteria.

More recently, Pringle12 applied Lyapunov stability theory te a general

class of dual-spin -satellites. Vigneren13 used the methedef.averaging to develop

appreximate stability ctiteria for a dual-spin system with a dashpet damper in

beth bedies. Cloutierl eptimized platferm damper design; this paper allewed a

platferm with different transverse inertias but assumed no damping in the reter,

hence stability criteria are net developed.

Several dual-spin satellites have been placed inte erbit. The first

dual-spin satellite to be stabilized by spinning abeu~ a minimum mement ef

inertia axis 'was the TACSAT I , a U.S. military cemmunicatiens satelli te launched en Feb

9

,

1

969

.

This satellite exhibited seme occasienal perieds ef marginal

stability during which a nutation angle ef abeut ene degree .persistêd for several

days. It is recognized15 that flexibility in the spacecraft spin-despin bearing (and net the spin stabilizatien mechanism) was responsible for this marginally

stable .behavieur . On Jan

26,1971

,

the Intelsat IV was launched. This spacecraft

is a large duel-spin cemmunicatiens satellite. Cempar~es frem ten natiens have

been enrel led as majer subcentractors assisting Hughes Aircraft Ce. in the

fabri-cation and testing of the satellite. Ne reperts 6n the nutational stability

have been available.

Dual-spin satellites with large flexible appendages are beceming mere

impertant with the need for greater en-beard pewer. The pewer requirement indi

-cates the usefulness ef large selar cell arrays which are compactly stored during

launch and then deployed when the satellite is in its eperatienal pesitian. The

presence of large flexible appendages on a dual-spin spacecraft necessitates

re-determining the fundament al dual-spin criteria with damping in both spinning

members and, as weIl, in the flexible structure. The effect af flexipility on

the performance of attitude and despin cantral systems alsarequires careful

study.

References

16

and

17

appraach the fermulation af the flexible satellite

equatians by using a cambinatian ef discrete variables fer the essentially rigid

campanents and modal defarmatian caardinates far flexible appendages. This allo

-cation ef variables is termed the hybrid ceordinate methadlB. The emphasis in these papers is en the influence ef flexibility en the attitude and despin control

systems. Since na damping is included in the rator, the effect ef flexibility on

the basic dual-spin criteria is not shawn. Also, the use ef smal1 amounts of

viscous damping in the appendage models may serve te mask seme important effects.

1.3 Research Needed

The analysis to date of,dual-spin satel~ites has revealed the stability criteria for a dual-spin system in which both the spinning and despun portions are

bodies of revolution about the spin axis. By limiting the energy dissipation to be present in only one body, analytic stabil.i ty criteria have been obtained. How-ever, when damping is present in both bodies, the periodicity of the system has restricted (due to the computational requirements of Floquet analysis) the stabi -lity analysis to statements of stability at a small number of points in parameter space. Referenc~ 13 has advanced this somewhat by applying the method of avera-ging. It has been indicated, e.g. Ref. 3, that the requirement for an internal

energy dissipation devieeto act to damp out nutation is that the nutation fre-quency be positive in the body containing the dissipation. More explicitly, the transverse angular velocity vector must rotate positively, relative to fixed body axes, about the bodyaxis along which spin is positive. For bodies of revol u-tion, the only cases analyzed, this requirement has been met by the major axis

rule. In systems that have unequal transverse inertias, it must be determined whether the inertia ratios necessary to ensure a positive nutation frequency in· a body containing energy dissipation will ensure the removal of nutational angu-lar mOPlentum.

The studies of dual-spin satellites have not developed a statement of the effect of flexibility on basic dual-spin criteria.

Evidently, basic dual-spin theory can be advancedp Some of the research

requirements which motivated the present contribution are:

a) development of a clear statement of the equations of motion and solution for

a dual-spin system with no damping in either body and no despin motor torques, bearing friction, environmental torques and with one body not necessarily a body of revolution.

b) analysis to determine the stability criteri a of a dual-spin system with representative models of damping in both the spinning and despun members but no large flexible appendages. The despun members should not neces

-sarily be·. bodies of revolution about the spin axis.

c) repeat of b),but with large flexible appendages on the despun damping in the flexible appendages should not be included for of·this study to determine the effects of flexibility alone. tions including appendage damping should also be done.

section. The

the majori ty Some calcul

a-d) a complementary analysis of the solar array will also be required to facili -tate incorporating array deformations into the satellite equations of motion.

The solar array is discussed in more detail in section 11. 11. SOLAR ARRAY ANALYSIS

2.1 Analysis Requirements

Deflection modes of the solar array are required for array degrees of freedom in the satellite equations of motion. The deflection of an array (where

an array here is meant to be one.solar cell sheet with its two supporting booms) relative to its base which is fixed to the rigid components of the satellite is,

vibration modes of an array as it vibrates with a fixed base are ideal for this

purpose in that the natural modes, for small deflections, are orthogonal with

respect to an integral over the entire array mass i.e"

where X. is an array natural mode shape. Thus, werequire the natural modes of a solar~array as it vibrates with a fixed base,

2,2 Array Description

The array analyzed is the Hughes Aircraft Co. FRUSA* variety as depicted

in Fig. 4. The array is stored during launch and initial maneuvers by rolling the

sheet of solar cells onto a cylindrical roller. The supporting booms are the weIl

known STEM** or BISTEM booms,which can be rolled into a compact compartment until

required. When the satellite is in its operational posture, booms and sheet are

extended outward into th~ final configuration shown in Fig,

4

with thesheet held

in tension, Reference.19 explains the array construction in ,more detail, Another array configuration is the T - boom type as described in Ref. 20.

2.3 Related Analyses

Analysis of the dynamics of large roll-out arrays for spacecraft

application is a relativelY new field. Yao 21 has performed a finite element

analysis for the CTS*** satellite array. Hughes22 has analyzed the same array

from a continuum mechanics approach, Coyner and Roas 20 have analyzeä a similar

T - boom roll-up array from fini te element considerations. Hughes Aircraft Co,

have both finite element and continuum mechanica analyses of FRUSA arrays,

al-though no publications of extensive analyses have been made in the open

liter-ature 0

2.4 Differential Equations

The analysis technique is to write the differential equations of motion

of the separate arrayelements i.e., booms, sheet and cross-piece and then to establish the boundary conditions to be met for a periodic vibration. It is

anticipated that there will be three types of modal vibrations possible i.e"

symmetrie, torsional and lateral modes (see Fig.

5

)

.

In the symmetrie mode,

both booms and the sheet vibrate in the same direction out of the nominal plane.

The deflection of each boom is the same and the sheet's deflection is not a

func-tion of dist~ce across its width. The torsional mode is a rota~ion about the

array centre line. The deflection of one boom is the negative of the other and

the sheet deflection function is reflected on either side of the centre line, In

the lateral mode, the array deflects only in thenominal plane and both booms

have the,same deflection.

Referring to Fig.

6,

we can write the differential equations for each

array element as follows:

a) boom differential equation

* Llexible ~olled-~ ~olar ~ray

** extendible,booms manufactured by Spar Aerospace Products Ltd,

The boom is modelled as a thin beam cantilevered at its base. The

differential equation is formed by balancing, at an arbitrary.cross-section, the

boom's elastic resisting moment with the applied moment fr om inertial forces on

the rest of the boom and boom tip forces. This balance is stated by an

integral-differential equation, i.e.,

where T.

=

'G

₌

'àx

oS

aI

=

OS

t

1

Pb [(

~(s')

s

+

(~(e)

-

~(S))

x

Pb

=

boom mass/length

iK

a

_{= ot2.}

-~(S))x~J

fi

or 2

J

b

=

boom flexural stiffness

( 11.1)

ds'

~ is the sum of the moments acting at s due to acceleration of a boom

element at-s' and the tip force on the boom. b) s~eet differential equation

(i) symmetric,and torsional modes

The sheet is modelled as a membrane with tension in one direction, along

ox/

dS.

The differential equation which balances the inertial force on a sheet

element with the restoring force on the element due to sheet tension is

( 11.2)

where

0;

=

sheet mass/area

T _e

₌

sheet. tension, force/width

(ii) lateral mode (see Fig.

7)

In the lateral mode, the sheet and booms move only in the nominal array plane. If wrinkling does not occur, the sheet bends as a large flat beam.

Assum-ing the sheet does not wrinkle, the centre lin~ equation can be approximated as

?lX3

_Mb

()2.M 1

ox

l : : _ _ b

- -

_(11.3) Ss é)

x

2

_KG

s

1 ₁ where

e

~

(

- x

_{1) dx'1}

f

(i-x

1) -

p(x~(t)

-

X3(X1))

Mb.

~

-1

~

o

X2> I

=

_àt2.

x,

+

x

1

~

₌

mass/length of sheet

J

S

=

flexural stiffness of sheet in lateral plane

G

=

shear modulus of sheet s

K

₌

4

t b

3

s

Equation (II.3)represents.the beam bending equation including the effects of shear. ~ is the bending moment at xl from the inertial forces due to accelera-tion at xl and the end forces on the sheet.

c) cross-piece

The cross-piece must move according to .Newton's laws in translation of its mass centre and rotation about its mass centre. Therefore, the following equa~

tions must be true: For translation

1 2

r

b 3

- f -

f

+

J ..

f

dw

-b

and.for rotation

where

m

=

cross-piece mass xp

Pxp

=

cross-piece mass/length

2.5 Linearized Equations with Boundary Conditions

(11. 4)

The next step is to assume.the correct boundary co~ditions for tbe three array mode types, i.e., symmetrie, torsional and lateral, and to state the

differential equations for eaeh mode type. Assuming small deflections, the diff-erential equations for the three array vibration types with the appropriate

boundary conditions are: a) symmetrie mode

(i)

boom (either one)

(11. 6) Now differentiating Eq. (11.6) twic~ witb respect to xl yields the fourth

order partial differential equation

(11. 7)

with boundary conditions

Xl (0,

t,)

0 ?lx2. (t,

t)

0 .::: ()X2.

=

t '3X_a (0, t, ) ~ 0

J

i/'x,.

(t,

t)

',2. Ox2. (t,

t)

1,

z

- f

_ft ';)(.1 _b

₀

₃ + X₁ 2

_d

_{X 1} where fl

=

1 f2 1 fl 2

=

i

2

These boundary conditions describe a cantilever boom base. The tip

con-ditions are found fr om Eq. (11.6) at xl

=

e

and from the first differentiation of

Eq. (11.6) at xl

=

t.

(ii) sheet

For tbe symmetrie mode, we use the x

2 component of Eq. (11.2) i.e.,

32.

_X2 (x, ,

t)

_Te

_é)\2.(x~,t)

öt

2

OS

ox2.

1 (11. 8)

with the following boundary conditions x 2 (O,t)

=

0 x 2 (~,t)

=

x2(

,e

,t)l where

(

t

,t)l means x 2(.f"t) x 2 of boom 1 etc.

The boundary conditions state that the sheet is unqeflected at Xl

=

0

and follows the boom motion at Xl

=

t(the sheet and booms are connected at Xl

=

t

by the cross-piece).

(iii) cross-piece

In the symmetrie; mode, the cross-piece translates , without rotation, in

the x

2 direction. The vector translational equation, Eq. (11.4), has two

com-ponents. -2 fl,2 - T 2b

=

0 1 e b -2 fl,2 -

T

J

O)l.z

(t,

t) d ... 2 e --b

,(h,

(11.10)

ten-•

the booms and the component of sheet tension in the .x

2 direction act to accelerate .

the cross-pieceo

b) torsi onal mode

(i) boom 1

The boom equation and boundary conditions are the same as Eq. (II.7) i.e.,

o

lb

_{?J4-X 2} (x.₁,

t) _

f

,

'0

2 )(.2. ()I., ,

t)

~ 4-

dX

2 oX, 1 ( IL ll)

with boundary conditions

()2X2

U.,

t)

- 0

ox2-

_,

Xz(o,t)

=

0 ë)x2 (o,t)

₌

0 OX , Jb ~\2 (.t,t)

=

ox?>

_ f' ₊

ox,(t,t)

_f1

2.

_ox,

.,

1 where

Boom 2 will have a deflection of opposite sign, same magnitude. (ii) sheet

For the tor.sional mode, Eq. (II.2), is again applicable i.e.,

Te.

'0\2 (X" IN,

t)

o-s ox;

( IL12)

with boundary conditions x 2(O,w,t)

=

0 x 2(e,w,t)

=

x2

(

~,t)1

+

~

(1 +

~W)(

X2

(t,t)' -

X2.

ei,

t)1) where x 2(i , t)1

=

-x2(t,t )2

The boundary condition at xl

=

t

states that the end deflection of the

sheet follows the motion of the cross-piece as the booms deflect in opposite directions.

(iii) cross-piece

In the torsional mode, the cross-piece rotates about its mass centre. The only component of the translational equation, Eq. (II.4), is

1

-2 f - T 2b

=

0

1 e

and the rotational equation is

(n .14)

Note h0W, in theintegral term of Eq. (II.14), the sheet tension adds

torsional stiffness to the array.

c). lateral.mode

(i) boom (either one)

The boom equation and b0undary conditions are the same as Eq. (II.7)

ex-cept st~ted in the x

3

direction i.e.~

I

o4X~

(x"t) _

f'

O"X3(X "

t)

0\3 (X,

,t)

b ax4

1 ax"

~

Pb

2~

-

0

1 \

with boundary conditions

where X 3 (0, t)

=

0, OX?I

(o,t)

o

oX 1

J

~~X3

(.ft,

t) b

ox

l 1 -e for boom 2 ~2X~

(.f,t)

é)x~

_

f~

+ dlt3

(i,

t)

f1

oX 1 1 (ii) sheet

The sheet bending equation

a2.x~

( x, ,

t) Mb _ ()2 Mb 1

ox~

J

_s

é)X~

KG

s

o

M

""-l

tt

()'X3(X;,t)(x'_X )dx' +

fel-x,) -

p(x

3

ct,t)-

Xl

(xot))

b

at

2 1 1 1 X, (IL15) ( IL16) . t

•

can be reduced to a differential equation by differentiating twice wit4 ~espect

to xl' yielding

a

4x.₃

(x"

t)

=

_1

r

-1

è)2

x!,

(XI' t) + P

~2X3 (x,

,t)

1

~x{

J

_s

l

at'

ox~

J

with boundary conditions

X 3( 0, t ) = 0 02)(?> (

t,

t) Ox.

,

2

__ 1_(_ '(

iXl

ct

t) KG_S

ot'

-f ...

P

aX3«(t) ()X, + àx?> (0, t)

ox,

+ P

à2X~

Ct,

t) ) () X2 1

o

O~ )(

>

(t..

t) p ~l X ~ (t,

t)

~X, àV -

äxt

The boundary eonditions at xl

=

t

eome, from Eq. (II .17) at' Xl

=

e

and i ts first derivative at Xl

=

t.

(iii) eross-pieee

In the lateral mode, the eross-pieee translates in the x

3 direetion. The two eqmponents of the translationa1 equation, Eq. (11.4), are

where

2.6

Natural Vibrations P

=

2 T b e (11.18) (11.19)

For the purpose of deseribing boom defleetions, we are interested in the array natura1 modes, i.e., in vibrations whieh are periodie in the three modal direetions • For any array moti,on, i t is neeessary that the differentia1 equations and boundary eonditions derived above be satisfied. However, to deseribe periodie motion, we assume separati0n of variables in the defleetionsi.e.~

where

X₂ ("'"

t) '"

E(x,) ~(t) (symmetrie mode)

=

M

(x,

,'1.1)

~(t) (torsiona.;I.. mode) X 3

(x,

,t) = V(x.\)~(t) (lateral mode)

..

~ (11.20) (11.21) (11. 22)

E, M, and

V

are the mode shape funetions for the symmetrie, torsional and lateral modes respeeti vely. èp (t) deseribes the amplitude of the ,mode.

Now, substituting these forms for x

2 and x3 into Eqs. (11.7-11.19), we have for the modal differential equations

(i) boom

The mode shape eCluation f'ound from ECl.

(11.7)

is

with the boundary conditions

E(O)

=

0 E'(O)

=

0 and À is def'ined by

..

~ (ii) sheet E"(l)

=

0 J b E 111 (

t )

~

= -

f

~

+ E' ( t)

f

~ ~

The mode shape eCluation f'ound from Eq.

(11

.

8)

is

(11.23)

(rr.2

4

)

E"

+

À. E

=

0

(11

.

25

)

o with boundary conditions

E( 0) = 0 E '

Cl )

~

=

x

2 (

t,

t) 1

and ~o is def'ined f'rom

..

~

o

(iii) cross-piece

The translation eCluations, f'rom Eqs.

(11.9)

and

(11.10)

are

1 ,..-f 1

=

Teb

- 2

f

~

-

Te

Jb

EI ( t)

~

d

w

-b

•

b) torsional mode (i) boom

(11. 26)

(

11. 27

)

(11.28

)

One boom the same as Eq.

(11.23),

the other boom, same magnitude opposite

sign.

(ii) sheet

The sheet mode shape equat10n is the same as Eq.

(11.25)

except f'or the

(iii) cross-piece.

(i) boom

The rotationa1 equations are

1 - f = Tb 1 e b

ef~

-

T~

f

VI

M'(t,

w)~

dw

-b c) lateral mode (n .29) (11.30)

The boom mode shape equation for the lateral mode is the same as Eq.

(II.23) with E-V and subscript 2-3.

(ii) sheet

The mode shape equation is, from Eq. (11.17),

where

v""

+ BV" + CV

=

0 B

=

C

=

KG_{s J s} "P 1 + -KG_s

-

a

2 '[ IJ's 1 +

.-L

KG_s

with the fo11owing boundary conditions

V(o)

=

0

yl/(t)~=

1

l-'(V(t)~ +PVII(t)~]

KG_s

(n .31)

VI(O)

=

0

V"'(e)

~

=

~l-f

_~ +

?V/(t)~1+_1

_KG

f'(v(Oi -

pv/ll(t)~l

s

L

(iii) cross-piece

The trans1ationa1 equations in terms of mode shape are, from Eqs. (11.18)

and (n .19)

b

Te.

f

V/(O

~

dw

-b (11.32)

The equations for mode shape Q~ the boom or sheet are seen to be, ordi-nary linear differential equations of either seeond or fourth order.

The general solution form of the seeond order sheet equation i.e.,

Eq. (II.25) is

1 +

HCÀ~

_X1

thecoefficients G and H to be determined from boundary conditions.

(I1.33)

The general solution form of the fourth order boom equation, , i .e., Eq. (II.23) is

(I1.34 )

where

The general solution form of the fourth order sheet equation i.e., Eq. (II.3l) is

v

=

l

S ~t){' + V! C~a)l.1 -+- t-lsh~3)t,1 + Q Ch~3"'1 (I1.35)

where

_~2

₌

l

~

(

1 _{+ (1 -}

~~)l)J

~?l

•

l

~

((

1 -

~)a

1 )1

-

1) a

Now by writing the boundary eonditions for the three modes (i.e., the boundary conditions in Eqs. (II.23 - (II.3l) ) in terms of the general solution

forms just.given, the eondition for a periodic so~ution to exist can be expressed.

The fact tbat a set of linear homogeneous algebraie equations for the eoeffieients of the general solutionforms must have a non-trivial solution is utilized.

This in turn means that the determinant of a certain matrix must be zero. For all

three modes, the algebraic equations for the coefficients in the general solution ean be redueed to ones involving only two unknowns.

a) symmetrie mode

The mode shape is of the form

E ... _{(II .36)}

for each boom and

A

1

=

À~2.

S À,a

t

+

À~

À3

sn

À!I

t

A -

.À~ A2. ::

À.~

C À₂

.f.

+ À2.3 eh

À3~

A2.

and A, G are solutions of

[Ds(~)]

[;]

-

[~]

( 11.38)

Therefore, for a non-trivia1 solution for A and G we require À such that

ID

(À)\

=

0 where

s

\ D

s (À,) \ =

1 -

À 3 S

À~

t

l ( -

J

b À

~

+-

Te.

b

~

2.) (

A

2 c

À,

2.

t

+ 1\ 1 S À?.t )

+

Jb>'~ (-X~À,

eh

À

₃

e

+ À\

À.~

sh

À~i)

+

Teb

(-oh,

À, eh

À~.e

.

.

+ À

1 À.3

sh

À~t

))

+

(À.

3

1.

2 S

À,2.l -

Áa

À2

sn

À.~ ~

(11.39)

+ À

~

A., (- c

À

2.

~

T

eh

À. 3

t )) (

Te.

b

À

~

c

>-.

to.f. -

S À

~ ~

f1.p

e

J

b À /

(2

f

b)} } / À

~

Á

₂

\ D s (,,) \ is zero when i ts numerator is zero, providing the denominator ~3 Jt 2 '1= O. The advantage of Eq. (11.39) as the form for

ID

(~)\ is that the roots of the numerator can be used for /D

(À)\

=

0 where thesnumerator invo1ves no possib1e

zero divisors. s.

b) torsional mode

The mode shape for the torsiona1 mode is

(II.40)

for a boom and

(11. 41)

for the sheet.

For the re1ated determinant to be zero we require

\ DT

p.)\ '"

0 where \ DT ("-) \

= {-

Àl S À!

t

l (-

J

_b

À'.

+

T,b

À.)(

11..

c~,t

+

1'., ""t)

+J

bÀ3 (- À2. À.,

eh

À!I

t

+

A~

X23 sh

À~~)

+ Teb

(-Ä

2 }..2

eh

À~t

+

Ä

~

h.

a

sh

À

~

.f. )

J

+

(À

~

A 2. S À

~

t -

Ä 2. À 2 S h À!I

t

(11.42) +

>.~À\

(-

C>-'z!

+

en

X~t))(1 ;~~~ X!

cÀ

1

0

i -

s>.ttrltpeJbÀ

)\1

À?,Àa

c) lateral mode

The lateral mode shape can bestated as

(II.43) for ei ther boom.

and

v

=

L

(ShX1

+

~r

e

~2)1.,

-

~: sh~~x.~

-

~r

eh

~aX1)

for the sheet.

(II.44)

For the related determinant to be zero, we require \

DL., (

À') \ ...

0

where \ DL (À.) \

=

l

s~2.~

+

~r (C~2t

-

eh

~3

i) -

~

sn

~3

e ][

J_{b (}-

À

3 2

cÀ/, -

À~

1\sÀJ and wher.e _

À2À~

eh À₃

t

+

AÀ~ shÀ~t)

+

t

(À2 cÀ2.

t

+

À~.ÀsÀ2.t

- À2chÀs

i

+ À

3

ÀsnÀ?,l)

+

~(SÀ2t -Ac~at- (~:)ShÀ3t

+ .i\ehÀ

3t)]

+ :IS

r

(1

+

:G )

l-

p!

Cf2

e

+

~r

(

~:

s

P2

t -

p~

sn

~2 ~

) -

f~?2 ch~3

e]

2

L

s

(II.45)

-(is -

~:: )(~2CPtt

-

pr(~2SPzt

+

f3sn~3e)

-

P2

ch

P3

t)]

x

lSÀzt -

ACÀ2

t -

~: ShÀ3~

+

A

ch~3t1

-~~ S~2.t

-

P2P3Sh~3t

- K

1sP2

t

+

K,-i:-

sh~?,l

~~Cf2i

+

p~chp"&t

+

K1C~2,e

-

K,ch~3i

The nat~ral frequencies of the solar arr~ in the three types of modal

vibrations are found numerically by starting ~ at zero and increasing it ,until the ,value of the critical determinant changes sign. The À at_lthis point represents

the existence of a natural vibration of frequency SJ,

=

(J

_b

'AI

fb

)V2.

The mode shapes

are now available to withi~ a constant multiplier

i To complete their specification, the mode shapes are later normalized so that

J

EI dm z:: 1 etc. All parts of the

computer calculations in which roundoff error can be critical are performed in

2.7

Results and Discussion

\

Figures

8

to

11

indicate typical results of the calculations. The

ten-.sion in the substrate* can be varied for a given array without changing other array

parameters. Therefore the dynamic characteristics of the array can easily be

modi-fied by changing the tension. From a design standpoint, the variation of natural

frequencies with this parameter is an important behavioural pattern. The array parameters selected below are representative of a Hughes FRUSA.

l

=

18

.

2

ft. _~

₌

0

.

25

')

lb/ft.2 e

=

6

ft. E

₌

7

x

10

6

psi s fb

=

0.1949

lb/ft.

Y

_s

=

0

.

4

J b

3.32

x

105

lb-in.

2

=

A plot of the first three modal frequencies versus sheet tension for the symmetric and torsional modes is shown in Fig.

8

.

As an interesting compari-son, Fig.

9

shows the frequency versus tension plot for the sheet stretched

be-tween two fixed points. The symmetric and torsional mode frequencies are close

to those of the sheet alone. Figure

10,

a plot of mode sha~es at Te

=

3.12

lb./ft. also indicates the closeness to pure sheet modes in the symmetric and

torsional vibrations.

Figure

11

shows that for lateral vibrations, thefrequencies are

con-siderably higher than in the symmetric or torsional modes and that they are

not strongly dependent on tension. The first mode, see Fig.

10,

is primaritythat of the booms alone, while mode 2 is a combination of both sheet and boom motion

in closer relative proportions. Calculations made using a boom stiffness and

sheet tension four times those of the array just described (allother parameters the same) gave frequencies roughly twice as large, as would be expected.

It is clear that increasing the tension to too high a value will re sult in bOGm buckling. An indication of some Euler critical load values would be useful here. The Euler critical loads (see Ref.

23)

in terms of sheet tension

for various boundary conditions are:

a) pinned-pinned c) T

=

24

.

95

lb/ft e fixed-fixed

1

T

=

99.4

lb/ft e b) d) pinned-fixed

1

T =

6.24

lb/ft e sliding-fixed

~

T =

49.75

lb/ft. e

These are the statie critical loads for common boundary conditions. The appropriate static critical buckling load for the array is case a). It is note-worthy that the first radical frequency drop in Fig.

8

occurs at approximately T

=

6.3

lb/ft. i.e., very close to the Euler load for case b). Therefore, for

e

practical buckling considerations it would be wisest to consider case b) as the critical load if Euler loads are being used as a basis for design. Case b) represents the closest of the four cases to the actual buckling conditions in

that the booms are cantilever mounted and have the tip force transmitted to them from the sheet which deflects as the booms deflect.

111. EQ,UATIONS OF SATELLITE MOTION

ABour

THE CENTRE OF MASS 3.1 Development of Equations

Th~ equations of satellite motion about the combined mass centre are written in ~erms of the kinetic energy T, pote4tial energy U and generalized coordinates q. from Lagrange's equations.

J

(11101)

An important feature of the analysis is the modelling of energy dissi-pating devices within the spacecraft. The sources of energy dissipation are an

antenna-mounted nutation damper, fuel slosh in the rotor (allother damping

mecha-nisms within the rotor have an effect at least an order of magnitude below that of the fuel slosh, see Ref. 10) and structural damping in the flexible appendage. The nutation damper modelled is a pendulum type damper (see Fig. 12) consisting of

a mass attached to the end of a flexible wire, the mass being free to move through a viscous fluid. This type of damper is considered to be an excellent nutation

damper and is in current use. See Ref.

6

for a detailed discussion of this and

other nutation dampers.

The energy dissipation due to fuel slosh occurs primarily in the fuel boundary layer as indicated in Fig. 13. The fuel model is that of a rigid bulk cont.ained in a ~pherical cavity located on the rotor spin axis. Section

3.5

describes the generalized force representation of the boundary layer damping. Energy dissipation in the flexible solar array is very complex~ coming

from, among other things, friction due to boom seam connections and structural

damping within the array material. A linear viscous representation of damping

in structures is not a good model of the actual damping mechanism. However~ it is used to represent the presence of non-viscous damping in many situations be-cause it is a particularly easy form of damping to incorporate into the equations of motion. In actual fact, structural damping in the array booms is better

represented by using complex number theory. See Ref.

24

for a further discussion of this point. The frictional damping in the boom seam is not well represented

by viscous damping in that it is not a linear function of deflection velocity and could be a non-linear mechanism changing with the degree of seam contact. In

this study, some calculations will be done with the commonly used viscous

model-ling but only for the purpose of showing how the presence of an array damping

model will affect the stability results. 3.2 Generalized Coordinates

i .e., discrete coordinates representing the motions of rigid parts of

thesatel-lite and modal deform~tion coordinates representing motion of the flexible appen-dages relative to a reference point on the rigid satellite ·body.

The flexible appendage (solar array) deformation modes are the array

natural modes as found in section 11 normalized so that

J

X:

_t.. dm

₌

1.

,

X·

"

being a general mode shape. arra~

Figure 14 shows the reference axes used to contain the mathematical

simulation. The coordinates relate the axis sets as follows:

Axes Transformations Generalized Coordinates

~

~

~

_a

2

ai

3, 2, 1 Euler Angles

_a

_{3 '}

,

~

•

g;»

0

°2

6

1

"

"

"

)

~

..

g:p-"

"

"

'f~

0

,

0

~

..

~

"

"

"

cP?,

1>2.

•

1>1

~

..

~

"

"

"

'(3 0 0

~

...

_~

"

"

"

0 0

,

Àt

The vector d is described relative to

~s

by array natural mode

defor-mations, The deflections of the array relative to the

~

set are described by

modal deformation coordinates i.e.,

X₂

₌

_L

E . (t) EQ' (X₁₎ +

L:

}1

ai. (t)

M

ai

(X 1 ,

w)

Ol. , or =

E

E b~ (t)

E

b~

t){

1 ) +

L

)L bi.

(t)

M

bi. (X 1 '

\11)

(III. 2) X3

E

))Qi (t)

V

_al

.

(><1)

(III. 3) or

L

y

b~ (t)

V

bI ( )( 1 )

where E., M" V, are the symmetrie, torsional and lateral mode shapes respectivelyo

l. l. l.

The subscript 'a' refers to the array on the positive dl side and 'bI to that on the negati ve -dl side. E ai and Ebi represent symmetrie mode deflections in the same

direction. ~ai and )Ubi both represent torsional rotations positive along the Si

axis and

Y

.

,

v.

b, are both deflections in the same direction for the lateral mode.

al. l.

These mode shapes .represent deflections of boom, sheet and crosspiece and have a different functional form depending on which element is being represented.

The generalized coordinate ~3 is redescribed by ~3 =,~t + ~ wher~ ~ is

3

.

3

Kinetie Energy

Referring to Fig. 15, the kinetie energy of a group of rigid bodies about the total eentre of mass eau be written as

where

r

=

1 ~

m.

p.

~

m

L.."

~_~

Z;

over all bodies exeept referenee body m , ~ over all bodies . Therefore,

" 0 J

or

(III.4)

(III.6)

Now for a system with flexible appendages, the above formulation eau be used by breaking the deforming appendage into infinitesmal elements and ineluding eaeh element as rigid body.

The total kinetie energy for the satellite we are studying ean then be expressed as (see App.

B),

where

m (m-m ) spa spa

2m

~

is over appendage mass

~ is over all satelli te parts

J exeept appendage. m 2

=

1 - m spa /m m-m _spa (III.7)

P

=

r - d

- ~ -e

and ~j'

i j

are as stated in App. C.

3

.

4

Potential Energy

The internally stored potential energy is due to the elastic strain energy in the booms and in the deformed tensioned sheet, i.e.,

(111. 8)

where n

=

number of booms I

m

=

number of sheets.

3.5

Generalized Forces

The generalized forces originate from the damper spring resistance, the resistance to damper ball motion offered by the damper fluid and the viscous resistance generated in the boundary layer of the liquid propellant. The spring resistance of the damper could equally well have been included as part of the potential energy.

The damper spring and viscous generalized forces are

(III.9) wherè the first terms represent spring resistance and the second terms represent the viscous resistance of the damper fluid.

The fuel viscous resistance generated in the boundary layer of the

fluid in contact with tank wall, causes a torque ~, to appear as the fuel bulk

moves in its cavity. ~ can be found by integrating,over the entire boundary layer,

the force times the appropriate moment arm ab out the fuel mass centre. This

model-ling is similar to th at used in Ref.

8

and yields, in fuel axes, to first order in

~l' ~2'

+3

and derivatives • Is, cf 0 0 ~1

o

Is,

Cf 0

~2.

_{(111.10 )} o 0 lS~cf

;P'!J

where see Ref. 25

Pf

=

mass density of fuel

jJ.

₌

viscosity of fuel

-Cl

₌

oscillation frequency of relative motion of fuel to cavity.

=

;

TT R: (4 _

c2) ( 1 _ c2) Y'è

~

1l'

R~

(2

+ C 2 ) ( 1 - C 2 / 2

moments of area about each axis I =

s2>

c Rf = radius of spherical cavity r

f

=

radius of cylindrical hollow

Therefore the corresponding generalized forces are~ to first order

for q j

=

CP" </>, and

4>3

respecti vely .

( rILll)

Solar radiation and gravity-gradient generalized forces are formulated

in Appendix E.

3.6 Linear Equations

The equations, of motion for the satellite are written fr om Lagrange's

equations on the basis of the derived T, U and Qj and then linearized. The linearization is done by.negl:cting quantities second Grder or higher in~,

a

2,

a3,öl,ó2'~'

1>1'

C:P2'

~3'

'1'3 + '(3'

Eti,Et,

,'i:f~M~, 'E~i,Vi.and

t:g.eir derivatives.

A set of linear equations which allows a more direct investigation of the basic dual-spin criteria is one in which it is assumed that

a) The rotor is perfectly balanced - In the actual manufacture of a

rotor an attempt is made to ~~lance the rotor exactly. Some typical remaining

imbalance figures are 2 x 10 slug-ft2 of unbalance for rotors less than 3000 pounds. See Ref. 10 for further details,

b) Environmental torques are neglected - The torque-free stability

criteria and performance are of significant value in that they represent the

satellitès motion about an inertially fixed satellite angular momentum vector. Environmental torques will change the direction of the angular momentum vector

and as well alter the sate1lites motion about the angu1ar momentum vector.

The fermer wil1 be accounted for by control torques and the latter is negligibleo c) The nominal spin rates of the antenna and array are approximated as zero. - The non-zero nominal angular momenta of the antenna (spin-rate to stay earth-pointing is 7028 x 10-

5

rad./sec) and the ,array (spin rate to stay

sun-pointing is 2.0 x 10-7 rad./sec) would on1y slightly change the basic nutation

frequency of the satellite and are negligible quantities in the satellite equations.

d) y

=

0 i.e., the antenna is symmetric about the ~l - ~ plane (see Fig.~ ~CSymmetry about the a~ - ~2 plane is a typical design geometry for a dual-spin communications satelIite antenna (see Ref. 10).