On the use of flexible strings in gravity-gradient stabilization systems

'9

~UG. \961

June,

1969

~(RNIS(HE

HOGESCROOl DELFT

VUEGTUIGIOUWKUNOI

itBlIOTHEEK

ON THE USE OF FLEXIBLE STRINGS

IN GRAVITY-GRADIENT STABILIZATION

SYSTEMS

by

S. C. Garg

IJ

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ON THE USE

qF

FLEXIBLE STRINGS IN GRAVITY-GRADIENT STABILIZATION

SYSTEMS

by S. C. Garg

Manuscript received January,1969 .

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ACKNOWLEDGEMENTS

~he author gratefully acknowledges the active guidance and helpful

dis-cussions contributed by Dr. P.C. Hughes, his thesis supervisor. The

enlighten-ing discourses with Prof. B. Et~in are also sincerely acknowledged. Facilities of the Institute of Computer Science, University of Toronto were used extensive-lYt Financial support was provided by the National Research Council of Canada

(Grant no. A-4183) and the U.S. Air Force Office of Scientific Research (Grant no. AF-AFOSR 68-1490). The author wishes to thank these organizations in

part-icular. Finally, thanks are due to Dr. G.N. Patterson, Director of the Institute

for Aerospace Studies, University of Toronto, for providing the opportunity to

carry out the research described herein .

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SUMMARY

A related family of three gravity-gradient-stabilized synchronous equa-torial satellites is analyzed considering small-angle attitude dynamics in the orbital plane. A characteristic feature is the use of long thin perfectly flexible strings to achieve the desired mass distribution, which eliminates the solar radiation pressure and heating effects inherent in conventional rigid-boom designs. Damping is achieved by articulated hinges and magnetic-ally-anchored viscous fluid dampers. The designs are optimized using a numeri cal gradient-search technique and considering the time to damp to half amplitude of the least-damped mode as the performance index. The pointing performance is found to be relatively insensitive to parameter variations. Optimized results are presented for a series of constrained satellite config-urations .

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TABLE OF CONTENTS Chapter I: Introduction 1.1 General Considerations 1.2 Configuration Selection 1.3 Scope of the Analysis Chapter 11: Equations of Motion

2.1 Assumptions

2.2 Derivation of the Equations

2.21 Coordinate System and Formulation 2.22 Force and Moment Expressions 2.23 Kinetic Energy Terms

2.24 Damping

&

Gravity-Gradient Terms 2.25 Forcing Functions

2.26 Complete Equations; Partitioning 2.27 Equations for the 5D/F System 2.28 Equations for the 3D/F System 2.3 Solution of the Equations

2.31 Homogeneous Equations 2.32 Steady-State Solution 2.33 Numerical Calculations Chapter 111: Qptimization Studies

3.1 Introduction

3.2 Optimization Methods 3.21 The Gradient Method

3.22 Description of the Actual Method 3.3 Application of the Method

Chapter IV: Results and Conclusions 4.1 Introduction

4.2 Numerical Results 4.21 3D/F System 4.22 5D/F System 4.23 7D/F System

4.3 Discussion of the Results 4.4 Conclusions 1 2

4

5

6

6

7 7

9

12 14 16 17 17 20 21 22 22 23 23 24 25 28 28 28 28

29

29

30

References

Tables of Numerical Results Appendices

I: Simple Pendulum in a Gravity-Gradient Force Field

11: Feasibility of Using a Rigid Wire

111: A Single-Axis Damper for Geostationary Orbits

Figures 1-13

32

A B C A,B,C B' F M M T W a b C. 1 d d e f k k' 1 m s x NOTATION

Rotational moment of inertia ab out roll axis, slug-ft 2 Rotational moment of inertia ab out pitch axis, slug-ft 2 Rotational moment of inertia ab out yaw axis, slug-ft 2 Coefficient matrices defined by Eqn. 2.01

Ratio of moment of inertia in pitch to mass of tip mass, ft2 Force acting on any body in linearised force field, Ibs. Pitch moment in linearised force field, ft.-lbs.

Total mass of the satellite, slugs or lb.-mass

2 2 Kinetic energy, used in Lagrange's equations, slug-ft /sec Work done in a virtual displacement, ft.-lbs.

Radius of main satellite body, ft.

Length of tip stub (see fig. 1 & 2), ft.

Coefficients used and defined in the text, ft. Length of hinge stub (see fig. 1), ft.

Center-of-pressure offset of the main body, ft.

Orbital eccentrici ty; also base of Ná:pi:eraan logari thms Forcing function vector defined in eqn. 2~01, Ibs. Hinge spring coefficient (see fig. 1), ft.-lb./radian

=

k/3mw2 , has dimensions of length, ft.

o

Length of flexible string or wire, ft.

Tip mass (see figs. 1,2,3,) or mass in general, slugs

Solar radiation pressure on perfectly absorbing surface, lb/ft2 Generalized coordinates used inrLagrange~s equations

Laplace Transform variabIe (not necessary)

x,z Co-ordinates a10ng axes defined in Fig.4, ft.

e

Ang1e of pitch of main body

a

Ang1e between hinge stub: and main body

~ Ang1e betwee~f1exib1e string and hinge stub

l Qribita1 anoma1y ang1e (see Fig.4)

5 Ang1e between tip stub and f1exib1e string ~ Area ra~io of tip sphere to main sate11ite body À Ratio of mass of main body to the tip mass ~1 Hinge damping coefficient, ft.-1b./radian/sec. ~2 w o 1 2 c x z i

Note:-Tip damping coefficient, ft.-1b./radian/sec.

Time-average angu1ar velocity in orbit, radian/ sec. Subscripts

Pertaining to the lower tip mass (see Figs 1,2,3) Pertaining to the upper tip mass (see Figs 1,2,3) Pertaining to the (centra1)sate11ite main body A10ng the x-co-ordinate direction

A10ng the z-co-ordinate direction \

General subscript when referring to co11ection of quant.i ties. Symbo1s not defined above are defined in the text where necessary. In particu1ar the Appendices have a different general notation, defined therein.

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CHAPl'ER 1. INTRODUCTION 1.1 General Considerations

For effective u~e of artificial satellites, control of their

atti-tude is essential, and With new developments increasingly stringent require-ments are being placed on it. Depending on mission lifetime, control may be

provided by active or passive means. Active systems, such as those involving

gyros, momentum wheels, or mass expulsion methods, require power and are therefore suited only to short-lived missions. These are also inherently more complicated and less reliable than passive systems which rely on environ-mental forces for stabilization. Passive attitude control is especially well-suited to earth-oriented satellites. Applications of these are many, notably for communication, navigation, reconnaisance, and for geodetic or meteorolo-gical surveys. The accuracies demanded also vary accordingly, from a fraction of a degree to three or four degrees (ref.l).

A widely and successfully used principle for earth-oriented passive

attitude stabilization is the gravity-gradient effect. This effect consists of restoring torques exerted on an elongated body in orbit by the invers

e-square gravitational field so as to stabilize it with its axis of least inertia

aligned with the local vertical (e.g. ref.2). But these torques are

conserva-tive, and even though for a rigid body this captured position is a stable one, for good response to disturbances and low characteristic damping times, some

artificial damping scheme is necessary. Various methods have been suggested

for this, and in fact the damping scheme used in a particular satellite

configuration is one of its prime characteristics. It might even appear that in the development of gravity-gradient technology, attention has been primarily centered on damper systems (ref.3) .

As a matter of fact, a common denominator of most gravity-gradient designs has been the use of long storable tubular booms, extended af ter orbi~

establishment. The length of the booms is used to increase the rotational

inertias, and to amplify the gravity-gradient torques. Synchronous satellites, for example, require excessively long booms since for a given satellite these torques go down inversely with the cube of the distance from the earth's center.

The solar radiation torques can become quite comparable to them at high

alti-tudes. Now the use of booms in fact worsens this problem by virtue of the considerable lateral areas presented to the oncoming solar radiation, and in

articulated configurations oscillations of the constituent bodies further increase the solar torques. Another problem is the thermal bending due to

solar heating which sets up circumferential temperature gradients in the

~ollow metallic booms, causing differential expansion. The resulting bending makes the geometry unsymmetrical, further aggravating the solar torques. There

are other effects , t,oo: the principal axes are slightly altered, étltangi~ the

stable orientation. Even the center of mass may shift, throwing station

keeping devices or microthrust engines out of alignment. Thus the use of booms, though q~ite successful, is not without problems.

The boom manufacturers have made considerable efforts to solve these

problems, but it would be even better, at least from the viewpoint of reducing solar ~Drqlles, to be able to do without booms at all. However, the length

afforded by the booms is still desirable. Therefore the replacement of booms with a string or thin fibre of negligible areas has been suggested (ref.4) . Some investigation of the idea has been done previously at UTIAS but no

numerical results were presented (ref.5). At the outset if might appear that the use of strings would lead to problems because of the complete flexibility of the strings. However, a mass may be mounted at the end of the string, so that the centrifugal and gravity-gradient forces acting outwards on it will keep the string always outstretched, and under normal circumstances it will behave just like, a rod except at the ends, where no torque can be sustained because of flexibility, and these may be idealized as pure hinges. Thus it is

seen that the concept is realizable, in so far as the strings stay taut. It is expected that the negligible area of the strings will help make solar torques negligible, and there will be a substantial weight saving because of the removal of the booms. Also, the strings can easily be made non-metallic so that thermal warping etc. of the string will be eliminated. Thus a study of this concept is desirable and has been conducted.

It is not proposed, however, at this stage to go into the deploy-ment and capture problems which will be associated with this configuration. It may be desirable, for example, to keep the strings unextended during initial tumbling, or the extension of the strings may itself be used to reduce tumbling rates. Extension will be assisted by the forces acting outwards on the tip masses and if this is not enough semiactive means such as gas microjets located in the tip masses may be used. The problems associated with this phase may well be a limiting factor of this concept, and for their analysis the full

large-angle nonlinear equations of motion are required. But in view of the complexity of the large-angle equations and in keeping with the nature of a preliminary study it is intended here to study only the small-angle oscillatory motion. This investigation also includes optimization studies, which would be unwieldy if the full nonlinear equations were to be used. +he linearized model has been used for optimization in quite important investigations (e.g. ref.6).

1.2 ·Configuration Selection

Once it has been decided to use booms only when the use of strings is impossible (e.g. as 'yaw stabilizers'), the configuration still remains

arbitrary to some extent. A quite successful configuration consists of two masses mounted at the end of long booms above and below the satellite body.

Damping can be achieved by articulation, or by dampers located in the tip masses, both of which may provide velocity-proportional damping. This has been studied previously at UTIAS with only articulated damping and was shown to be quite successful (ref.7). It is therefore selected as the basic configuration with the vertical booms replaced by strings. Some logical additions to the design are apparent at once on account of the flexibility of the strings. These are

described below.

I ... If articulation is still to be used as a means of damping, obviously the string alone is unsuitable because it has no rigidity and thus cannot pro-vide any moment arm. It is then necessary to introduce a short but rigid 'stub' at the joint between the string and the main satellite body (as in fig.l). Then a hinge damper and a spring can be used taking advantage of the relative motion

between the stub and the main body. The spring is expected to augment the gravi ty-gradient "stiffness" during oscillations of the articulated bodies . II ...•.• The flexibility of the strings keeps the tip mas~es quite free to rotate about their own axes without causing an appreciable torque on the main

that the damping torque will not be as effective in damping the body motion or even the string motion, as the motion of the tip mass alone.* This was confirmed by studying the motion of a simple pendulum with a velocity-proportional damper

located in the bob, in a gravity-gradient type force field (Appendix 1). It

was found that the "swinging" mode of the pend.ulum string is indeed very poorly

damped in comparison to the motion of the pendul~~ bob about its own axis. The

pendulum model also showed th at if a short rigid stub is attached to the tip

mass it increases the moment arm for the string tension to act upon; it

thereby increases the coupiing between the swinging and the rotational modes

which is responsible for good damping of the former. An optimization of the

pendulum parameters showed, as expected, that the longer the stUD is made,

the better is the performance. So a tip stub is also incorporated in 0 the

design. The most general configuration investigated therefore consists of

two short stubs, a string, and tip mass on both sides of the main body; i t has

springs and hinge dampers also in addition to the tip dampers located in the

tip masses. It has seven degrees of freeQom in the pitch plane and is

desig-nated the "7D/F System". This is shown in fig.l. An alternative configuration,

with which comparison will be interesting, results if the hinge subsystems

(damper, spring, and hinge stub) are dropped, reducing the pitch degrees of

freedom to five. This will therefore be referred to as the "5D/F System" and

is shown in fig.2. It still contains the tip stubs.

111 ....••.... It mayalso be possible to drop the tip stubs resulting in a much

simpler configuration if the string is replaced by a wire which has a finite

rigidity instead of being completely flexible, but also therefore will have a

finite thickness. To indicate how thick a wire might be required, the simple

pendulum model introduced above is re-examined in Appendix 2, allowing a steel

wire of circular cross-section to replaee the perfectly flexible string and

the stub.

Using values for the damping coefficient, tip mass and wire

length similar to the ones encountered in the actual satellite and a force

field identical to that at synchronous altitude, it is found tha-ç a wire of .01"

diameter will behave essentially like a rigid rod hinged to the main body; in

the sense that the wire shape due to bending deviates very slightly from a

rigid link, when the pendulum undergoes oscillations of near-orbital frequency.

Of course oscillations of like frequeney are expected in the satellite due to

disturbances, and the fact that the wire behaves like a rigid rod implies that

there is very little motion of the tip mass about its own axis and the 'swinging'

mode is well damped. All this is pos~ible by using only a .01" thick wire,

so that the radiation pressure torques will still be negligible compared to those

~due;,.o to other SOill'ces sueh as orbital eccentricity, and the solid cross-section

may help reduce thermal bending problems considerably. In any case, in order

to evaluate the effect of using strings, it is interesting to compare the

per-formance of the

1D/F

and 5D/F systems with that obtained by using a rigid rod

or boom in place of the strings; and the wire may be considered a good practical

approximation to this ideal, that does not violate toa much the assumptions

made in the analysis. Thus the configuration shown in fig.3 is also studied

and is designated the "3D/F System". It has only the tip dampers, and to

permit comparison with the other two systems as also in view of the small wire

thiekness, radiation pressure on the wire is neglected and its mass assumed to

be negligible. It is to be noted that even though the hinge subsystems have been dropped, the 'rigid' wire is still hinged to the main body instead of being rigidly attached to it. The use of a hinge is expected to alleviate bending stresses in the wire caused by the damper moments.

With three configurations it is intended to make a comparative study in order to assess the feasibility of the string concept.

1.3 Scope of the Analysis

It is proper at this point to briefly discuss some limitations of scope, inevitable in a preliminary study, that might be removed in a later

investiga~ion. Firstly, only a linearized analysis is attempted, and the

equations lof motion come out as simultaneous ordinary linear differential equ-ations which can'easily be expressed in matrix form and solved.

It is further assumed th at a separation is possible between the longitudinal or in-plane dynamics and the lateral dynamics. This is a very important assumption, since here are derived and solved only the longitudinal equations, the lateral dynamics being left to a fuller investigation. It should be mentioned in this context that all configurations studied involve velocity-proportional dampers located in the tip masses. These could con~eivably have been of the magnetically anchored viscous type developed by the spacecraft department of the General Electric Company (ref.8). It turns out, however, that in an equatorial synchronous orbit this type of damper provides poor pitch damping and moreoverrleads to coupling between the lateral and longitud-inal equations. Since the synchronous equatorial orbit has many practical applications it was already singled out for study, and so the General Electric damper cannot be used directly. The type of damping here assumed (i.e. one that produces no coupling) could still be provided by a modified GE damper which allows relative motion of the two spheres only about a single axis. This concept is slightly better explained in Appendix 3 which considers the appli-cation of this type of damper to the simple pendulum model considered earlier. Of course a single-axis damper will provide no damping about the other two axes. Even if the type of damper assumed remains an idealization with the present technology, this fact still does not obscure the basic aim of the present study, which is to demonstrate the feasibility or otherwise of using strings. It is, further, not considered impossible to develop a damper to meet the assumed requirement.

The negligible area of the string introduces much simplification if the configurations are made symmetrical and if all bodies are made spherical. In that case, solar radiation torques can be made to vanish if tbe area of the tip masses is chosen properly (see section 2.25). The only solar torques th en arise from the short stubs which are therefore to be kept as short as possible. Actually an order of magnitude estirrate of the solar torques due to the stubs

showed them to be much smaller than torques due to orbi t eccentridi ty (e = • Ol) . Since the solar torques also turn out to be periodic with orbital frequency it is hoped that good response to eccentricity will also entail good response to them and to other periodic disturbing torques. So the relatively large if unrealistic value of .01 was used for the eccentricity in all calculations.

A geostationary orbit also means that magnetic interactions with the

geomagnetic field will not produce any variable torques, neglecting random fluctuations of the field. A constant attitude error is possible, but for

simplicity the satellite is assumed to have no residual magnetic moment. With these restrictions a comparison is made of the three systems

described earlier. Due to the large number of parameters it is possible to

compare meaningfully only by optimization, which forms an important part of

the present study. The idealization of the physical problem leading to the equations of motion is next described and the equations of motion derived

and solved.

CHAPl'ER 11: EQUATIONS OF MOTION 2.1 Assumptions

In presenting the idealization of the physical problem, the basic

assumptions are summarized below, with a discussion of validity wherever

necessary.

(1) Gnly the small-angle motion about a steady state is considered,

neglecting terms of second and higher order in the libration

angles.

(2) The longitudinal equations of motion are assumed to be uncoupled

from the lateral ones and are therefore derived independently.

(3) The strings are assumed to be perfectly outstretched all the

time. Without librations the differential centrifugal forces

and gravity-gradient forces acting outwards on the tip masses

would keep the string from collapsing, and oscillations only

increase the centrifugal forces.

(4)

A geostationary (synchronous equatorial) orbit is assumed for

application and illustration purposes. This orbit has immediate

practical applications, and the assumption also simplifies the

torque expressions.

(5) Radiation pressure torques are assumed to be negligible, and for this the appropriate area ratio between the tip masses

and the main body must exist, as discussed in section 2.25.

Torques due to stubs are considered to be dominated by other

periodic 4isturbing torques.

(6) The satellite configuration is entirely symmetrical and all

bodies are spherical. The stubs and strings are assumed to be massless for simplicity since this does not affect the

results very much while simplifying the equations.

(7) No earth shadow effects are considered. In fact, at worst a

synchronous satellite spends only

4%

of its total orbital

time in the earth's shadow; and no time at all for a range of orientations of the sun. In any case radiation torques are

negligible even in the lighted portion of the orbit (see 5

above). Any self-shadow effects are also neglected.

(8) This is not really an assumption, but for calculation purposes the main body radius and the total satellite weight are kept constant.

(9) The tip dampers are assumed to provide velocity-proportional damping in pitch in the rotating co-ordinate system. For a single-axis GE damper this implies that the inner sphere does not move much (see Appendix

3).

2.2 Derivation of the Equations

First the equations of motion for the

1D/F

System are derived, and then those for the other two systems are deduced from these as special cases. The equations thus obtained agree, as expected, with those derived independently for the 5D/F and 3D/F Systems. Before giving further details of the derivation, however, it is worthwhile to indicate here some important general features. The equations of motion for all systems can be written

as:-where

A

(2.01)

is a square matrix containing inertias of the system and is designated the "inertia matrix" .

B is a square matrix containing damping terms exclusively and is designated the "damping matrix".

C is a square matrix containing the gravity-gradient If stiffness" terms and is designatedthe "spring matrix".

X is the unknown vector containing an ordered sequence of the librational angles chosen as generalized co-ordinates. f is a forcing vector which in this case contains sinusoidal

forcing terms due to orbit eccentricity.

In equation 2.01, dots denote differentiation with respect to time.

The matricies A, B, C are all symmetrie matrices. It is found that by changing

the dependent variables it is possible to partition all three matrices

simultan-eously to abtain twa sets of independent equatians, reducing camputatian time.

In that case, hawever, symmetry is lost. Details of this will be given later (see sectian 2.26).

2.21 Co-ordinate System and Formulation

In any dynamics problem the question of using the Eulerian or the Lagrangian formulation inevitably arises. In general Eulerts equatians are more

suited to mechanical problems where it is important ta know the details of internal

farces etc. They also give better physical insight into the system. But when overall system performance is of primary interest, as in the present problem, it is betber ta avoid the involved subst~tutions assaciated with elimination of the internal forces and to use the Lagrangian formulation instead. A canvenient form of Lagrangets equations is as

follows:-d

dt

[~k

J-oW

d<lk

k 1 ... n (2.02)

where T is the total kinetic energy relative to the chosen reference frame, qk

are the generalized co-ordinates, and W is the total work dorre by the entire

apparent -force field, including terms iarising from the motion of the reference

frame, if any; during a virtual displacement consistent with the system

con-straints, whenever they exist. In the above, relativistic effects are not

considered.

The basic frame of reference chosen is that attached to the satellite

center of mass with one axis aligned with the local vertical. This body-centered

frame is shown in fig.4, and is a rotating frame, causing apparent inertial terms

in the overall force field. The generalized co-ordinates chosen are the

relative angular displacements of the constituent bodies, i.e. the stubs, strings, tip masses and the main body.

2.22 Force and Moment Expressions

The forces and moments acting on the constituent bodies of a

multi-bodied satellite in a slightly elliptical orbit are now given. These are

taken from ref.9 and are derived on the assumption of small eccentricity and

a linearized gravity field since the satellite dimensions are much smaller than

the distance from the earth's center. The rotating co-ordinate system is used

and the centrifugal"force and gravity terms are combined to give an effective

total force field. The expressions

are:-F x F z 2 mw

o [(e cosr) x - (2 e sinr) z + 2

dz ] dr mw2 [(2 e sinr)x + (3 + 10 e sin;)z - 2 o M

w

2 [3(A-C)e + 2 e B sinr ] o dx

J

dr (2.03)

where x, y, z are the co-ordinate directions as in fig.4, e is the orbital

eccentricity, r denotes the orbital anomaly, and A, B, C are the principal

moments of inertia of the satellite about the x, y, and z axes respectively.

Also F(with the appropriate subscript) denotes the force components and M is

the moment in pitch, which is denoted by the angle e. The frame of reference

is also a principal-axes frame and the z axis, pointing vertically downwards,

is coincident with the principal axis of least inertia when all libration

angles are zero. This implies that the steady state is one in which there

are no oscillation angles and the satellite does not rotate with respect to

this frame. Notice in particular that for a spherical body A

=

B

=

C and the first term in the moment expression vanishes.

2.23 Kinetic Energy Terms

The equations of motion are derived using Lagrange's formulation. In this section are given the kinetic energy terms which will form the left hand side of equation 2.02. The total kinetic energy in the rotating frame

is:-T

=

~

Bb

ë

2 +

~

B

(é~1+~1+$1)2

+

~

B

(ê

+

ä2~2+

5

_{2 )2}

where the kinetic energy of rhe stubs and strings is neglected in comparison

to the larger terms and in keeping with the assumption of masslessness. The

mass-center co-ordinates in the above can be written

as:-x ₁

=

x _c + a sine + d sin( e + al) + .~:. sin( e + a ₁ +

13

₁)

+ b sin(e + al + ~l + 5₁) (2.05)

zl Z + a cose + d cos(e + a_l)+ ~ cos(e + al +

13

₁)

c

+ b cos(e + al + ~ +

1 51) Similar expressions apply for x

2 and z2 with subscripts 1 replaced by 2 and the

+ signs outside brackets replaced by '-'., In the above, greek letters are used

for ~he libratio~angles as defined by fig.5, and subscripts 'c', '1', and '2'

stand for the satellite main body and the lower and upper tip masses respectively.

From the definition of the mass center we also ha ve:-mx + m(x

l + x2 ) 0 c c

(2.06)'

m Z _{c c} + m(zl + Z2 ) 0 From these it... is directly obtained that

x (m/m ) _(xl + x 2 )

c c _(2.06A)

Z _c = -_' (m/m _c ) _(zl + z2 )

When the small-angle approximations are used (sinv ~ v, cos v ~ 1) and the above

equations are used to eliminate x and z explicitly fr om 2.05, the following

expressions are obtained for the ffiassLceRter

co-ordinates:-Xl = c e 1 + c a 2 1 + c

₃

13

1 + c4 51 + c5 a2 + c6 ~2 + c7 52 x₂ = -{ c 1 e + c5 al + c6

13

1 + c7 51 + c2 a2 + c

₃

13

2 + c4 52 x _(cc- _al + c6

13

₁ + c

7

51 + c5 a2 + c6

13

2 + c7 52) c

₅

zl - z ₂ = Cl z _c = 0

where the c. are constants defined as follows

:-1. c 2

=

(d + ~ + b)(l-m/M) c 4 = b (l-m/M) ) (2.07) Cl (a + d + ~ + b) c

3

= (~ + b)(l-m/M) c 5 (d + 1 + b)(m/M) c6

=

(~ + b)(m/M) c7 rnb/M . (2.08)

These expressions can now be substituted into the kinetic energy equation

2.04. The derivatives d/dt (dï/oqk) are next formed and arranged in matrix

order as in equation 2.01. Since T does not depend explici tly on qk these terms form the inertia matrix 'A' which multiplies terms of the type

qk'

These

are the only contributions to the inertia matrix, since the generalized forces

do not depend on

qk'

Only the final matrix is given here, the ordering

se-quence being in accordance with the x-vector defined as

follows:-~T

=

~

; ai;

fj.;

_{51 ;}

a

_{2 ;}

~2;

.

5~-.J

The matrix A appears in Equation 2.10.

2 B 2 _cl +~ _{cl (c 2+c5l)} cl (c 3 +c6), cl(c4+c

₇

)

cl (c 2+c 5)· m + 2 B' + B' + B' + B' + B' c l(c2+c₅) C2 2 +(À+l)C

0'

₅ c₃c2+ B' c2C

4

+B'

2cl5

+ B'

EEj

À+l)C 5C6 +(À-n)c

₅

c

₇

2 + B' -Àc 5· c l(c3+c6) c3C2+B' 2 cc+'B' . + ( c + B' 3 2 3 4 c3c5 c2c

6

+(À+l)c 5c6 +(À+l)C6c .

~Àc5.c6

+ B' +(À+l)c6

_.

₇

c l(c4+c

₇

)

c2c4 +B' c3c4 + B' c2 4 + B' c4c5+c2c

₇

+(À+:L)c c +(À+l)c 6c

₇

2 + B' +(À+l)C

7

-À c5c

7

5

7

c l(c2+c5) 2 2c 2c5 c3c5+c2c6 c4c5+c2c

₇

c 2 + .

13'

2 + B' _Àc 2 _-Àc 5c6 -Àc5c

7

+(À+l)c 5 5 c l (c4+c

7)

c 2c

₇

+c4c₅ c₃c

₇

+c4 c6 2 c4c

₇

c2c4· +B' + B' -Àc 5c

7

-Àc6c

₇

2 _+(~+l)c c₇ -Àc

7

_. 5'r c l(c3+c5) + B' C 2C6+C3C5 - Àc 5c6 2c 3c₂ 6 -Àc 6 c 4c6+c3c

7

-Àc 6c

₇

iC3c2+c5c6 (Ml)+ B' c 3c4 + B' +(Ml)C 6C

₇

EQUATION 2.10: INERTIA MATRIX FOR THE 7D

IF

SYSTEM.

-2.24 Damping and Gravity-Gradient Terms

(2.09) c l(c4+c

₇

)

+ B' C2C

₇

+C4 C5 Àc 5c

7

c 3c

7

+c4c6 Àc 6c

₇

2c 4c

₇

_Àc 2

7

2c4+ B' +(Ml)c 5c

7

~~

+ B' +(Ml)c2

7

These appear on the right-hand-side of Lagrange's equat.ions b\lt are later transposed to give the matrices 'B' and 'C' in equation 2.01.

~hese terms arise from the force field of equation 2.03. plus the torques due to the dampers and springs during articulated motions of the satellite.

(1) Dampi9g Terms

The virtual work done in small ,displacements of the angles by the damper torques may be written as:

5W damper

=

-I-l

l alool I-ll

ä

2 002

-1-l 2

(é

+

ä

_{l +}

~1+51)5(

9 + al +

~l

+ 5₁) -1-l2

(e

+ Q2+

~2

+

5

₂) 5(9 +

a

₂ +

~2

+ 5₂)

where the tip dampers are ideal:,i.zedoas producing damping moments proportional

to the an§Ular velocity of the tip mass relative to the rotating co-ordinate system. As stated earlier, this is an assumption at present satisfied only by

a General Electric-type magnetically-anchored-viscous damper constrained to a

single axis (for an a~alysis of this see Appendix

3).

If a~ all-axis damper of

the GE type is used the complete equations of m0tio~ should be considered.

When the derivatives

oW

/Oqk are formed and arranged in matrix form as dictated by the state vector as given in equation 2.09, and transposed to the

left-hand side of Lagrange's equations, the 'B' matrix of equation 2.01 is

ob-tained. There are no further contributions to the B-matrix since the gravity-gradient terms do not depend on the angular velocities. The resultant matrix appears in equation 2.11. 21-l_{2 1-l2 1-l2 1-l2 1-l2 1-l2 1-l2} 1-l2 I-l l +1-l2 1-l2 1-l2 0 0 0 1-l2 1-l2 1-l2 1-l₂ 0 0 0 1-l2, -~2 1-l2 1-l2 0 0 0 1-l2 0 0 0 I-l l +1-l2 1-l2 1-l2 1-l 2 0 0 0 _{1-l2 1-l2 1-l 2} 1-l 2 0 0 0 _{1-l2 1-l2 1-l2} EQUATION 2.11: DAMPING MATRIX POR THE

?DIp

SYSTEM .

(2) "Spring" Terms:

These involve the work done on the constituent bodies by the inertial

+ gravity force field by the equation 2.03, and the hinge springs. We have the

following expres sion for the virtual work do ne by these forces during a small

virtual

The last two terms in equation 2.12 are due to springs located at the hinges. The moments M. are the gravity-gradient torques exerted on th

₂

'i'th body and involve only terms due to orbit eccentricity of the type 2 ew B sin, if the bodies are spherical, as they are assumed to be in the present gase. The forces can be easily found by substituting the appropriate co-ordinates from 2.07 into the force expressions 2.03. For forming the derivatives of the type àW/àqk' however, derivatives of the form àXi/àqk and àzi/àqk are required, which are considered next.

Derivatives of the type àXi/àqk are easily seen to be constant from 2.07. However, equation 2.07 gives a false impression about derivatives of the type àzi/àqk since while zl and z2 are constant to a first approximation, their rates of change are not necessarily negligible. This is because, for example, although

cosv

~ 1 we have d/dV (cosv)~

v.

Consequently linearization should follow, no~ precede, differentiation and so àz~àqk

f

0 and can be shown to be as in equation 2.13, given in the table following:

~

8

_al

_{~l 51}

_a2

zl.

J_

c5al+c6~1 C

5(8+

al)

c 6 (

8+al

+~\) C7(8+a:l o~ 8+0:2) . z~ -c5a2-c6~2 +c6~1+c7 _{51 +c}

751

+t:\ +51)

i-'C6~2-C752

+c

7(51-52)

z2 -x₂ If

"

"

"

zl -x ₁ -C2(8+o:1) f-C§(8+o:1 -C4(8+a:1

"

-cl l -c451 ~1)-c451 +~1+51)

Xc 0 -c ₅ -c₆ -c ₇ c₅

iX₂ -c ₁ -c ₅ -c6 -c ₇ -c ₂

~l Cl c₂ c₃ c4 c₅

iEQUATION 2.13: TABLE OF DERIVATIVES àzi (7 D/F SYSTEM) àqk ~2 52 -c 6(8+o:2 -c₇(8+a:2

+~2}ç752

+~2+52)

"

"

"

"

c6 c₇ -c 3 -c4 c6 c₇

The derivatives in equation 2.13 can now be used with equation 2.12 to calculate the contribution of the force fields to the basic equation 2.01. Again, all terms of second or higher order in the angles are neglected in

keep-ine with the assumption of small-angle oscillations. Also the products like 8 8 are considered second-order terms and are dropped. Products of the small eccentricity 'e' and the small angles are also considered negligible and are not kept. There is, however, some explanation required here. In time-varying equations without damping of the type

the second term in the coefficient of x can cause instability even for

vanishingly small 'e'.* The present equations are of a similar type, except

that now damping is present. Since the unstable solutions without damping

grow with time exponentially, with damping it is possible to ensure stability

for large enough damping, and ~he linearization process is valid in this case. ~inearization of equation 2.12 then leads to a 'spring' matrix which multiplies

the x-vector, plus terms proportional to eccentricity of the type 2e.!w~~in'Y, where

!

is a constant column vector. The spring matrix C is transposed to the

left-hand-side of 2.01, but the non-homogeneous terms due to orbital eccentricity

are not transposed and the form the "forcing function due to eccentricity" on

the right-hand-side of equation 2.01. The matrix C is given in equation 2.14, divided by 3

mw;

for convenience in expression .

. ~I 2 Ac-Cc f-c + _{1 m} c l(c2+c₅) ci(c₃+c6) cl(c4+c7) cl (c2+c5) cl(c3+c6) cl(c4+c7) ~1(c2+c5} c l(c2+c₅) cl (c

3

+c6) Cl (c4 +c7) 0 0 0 2 + K/3

mw

0 ~1(c3+c6) c l(c3+c6) cl(c3+c6) cl(c4+c₇) 0 0 0 ~l (c4 +c₇) _{cl (c4+c7) cl (c4} +c 7) cl (c4+c7) 0

o

'

0 ~1(c2+c5) 0 0 0 cl (c2+c₅) c l(c3+c6) c l(c4+c7) +K/3mw2 0 c_l(c 3+c6) 0 0 0 cl(c3+c6) cl (c3+c6) cl(c4+c7) ~1(c4+c7) 0 0 0 c l (c4+c₇) cl (c4+c₇) cl(c4+c₇)

IEQUATION 2.14: SPRING MATRIX FOR THE

7

D/F SYSTEM (Divided by 3 mw~

2.25 Forcing Functions:

As explained above, sinusoidal forcing terms of orbital frequency proportional to eccentricity exist, the expressions for these have already

been derived in section 2.24 above. Physically the terms exist because in an elliptical orbit the local vertical does not rotate at a uniform rate but has a periodicity, while the satellite due to inertia tends to retain the same orientation. Thus the satellite oscillates about the local vertical at orbital frequency. The forcing functi.an due to orbital eccentricity is as follows * Pointed out to the author by Dr. P. C. Hughes.

2 :::>c + - 1 B c + 2 B' m c (c -'-c )+ B' 1 2

5

c_l(c

3

+c6) +

B'

cl(c4+c7) + B' C l(C2+C

₅

)

+ B' C l(c

3

+C6)+ B' cl(c4+c7) + B' (2.15)

O~dinarily there will also be a forcing function due to solar radiation pres ure on he tip masses. Actually this can be eliminated, as will now be shown.

Neglecting radiation pres~ure on the string and the stubs and ass~ng perfectly reflecting surfaces, the expression for the forcing function due to solar pressure

is found to be as follows. d/siny c CT -2 C

₅

(1 + CT) c CT

-3

cé l + CT) f d" t"

=

(TI

Y

2

P

)siny -ra la lon c a c 4 CT - C7(1 + CT) _(2.16) c

5

(1 + CT)-C2 CT C

6

(1 + CT)-C

3

CT C 7(1 + CT)-C4 CT

The first term forcing 0 arises if there is a center-of-pressure offset of the main body relative to the satellite mas~ center. FoX' a spherically symmetric main body this is negligible and in any case it leads only to a constant attitude

error which can be predicted. Thus the assumption of a completely symmetrical satellite is made. Now referring back to the expressions for the c. (equation

2.08) it is seen that all other co~ponen+-s of 2.16 vanish identically if

( 1 - ~ ) CT

=

~ (1 + CT)

M M

On simplification this condition becomes:

CT

=

m/m

c (2.17)

'l'hü, is really the condition for equal accelerations due to sol.ar pres ure im

-parted to the tip mas es and the main body; since the solar force is directly

proportional to the area. The area ratio wil.l henceforth be assumed to satisfy

2017, and no solar t orques will be considered. Of course the torques due to

stubs and strings are assumed negligible. There also the torques consist of

calculation shows that the amplit~de will normally be at least anc order of magnitude below that of the eccentrieity torques, the response to which is being eomputed.

2.26 Complete Equations; Partitioning

All terms in the basie equations 2.01 are now available, and the equations ean now be stated in their final form. However, before doing so some transformations are carried out for ease of eomputation and solutinn of the e quati ons , and these will presently be deseril;>ed . .

(1) The independent variabIe is ehanged from time to the orbital anomaly. For this the average veloeity Wo of an equivalent eircular orbit is used sinee the eccentricity is small. We have:

d

dt

-

w

o d

dl (2.19)

This will cause the eharacteristic roots to come out directly in orbits rather than in seeonds. The matriees A and Bare multiplied by w2 and Wo respeetively as a result of this substitution. The whole equatiog is then divided bY' . .m.u~ so that the w~ multiplying C and f eancels out.

(2) By changing the dependent variables it is possible to partition the matrices A, B, C and thereby obtain two independent, un-coupled sets of equations whieh require less computation to solve; these are called the 'symmetrie' and 'anti symmetrie , equ-ations respeetively. The new dependent variabIe vector x is now gi ven

by:-It is now seen that the first four elements of x represent the symmetrie motions of the satellite constituents; i.e. are-the angular displacements of corresponding bodies above and below the satellite main body relative to each

other. Similarly the last thre~ elements of x are 'antisymmetrie' displa ce-ments. Hence the names for the two sets of equations, sinee the first set contains the symmetrie displacements exclusively and the second one, a nti-symmetrie.

The partitioned sets of equations for the 7D/F system are finally given in equations 2.20 and 2.21; first the symm~trie ortes and then the antisymmetrie ones. It will be observed that the matrices are no longer symmetrie, but of eourse the form is still like equation 2.01.

f--' \.J1 ...---2 Bc+ 2B C l (C2+C

₅

)

+ B' 2 cl + _m 2 c 1(c2+C5)+2B' 2 (c 2+c₅) +B' . 2 c 1(c/c

6

)+2B' ( c 2 +c 5)( c

3

+c

6)

+B' 2 cl (c

4

+C7}+2B' (c +c 2 5 7 -:)( c4 +c ) +B' I . . -~ 2 2 + A -C c c c 1(c2+c

₅

)

cl _m 2 c 1(c2+c5) c1(c2+c5) - I ~ cl (c

3

+c6)+B' cl (c4 +C7)+B

I

e"

I (c

3

+c

6

)(c2+ ( c4 +c7)( c~+

cx"

1 +cx" 2 c )+B' 5 c )+B' 5 2 (c

4

+c7)( c

3

+ ~"+~" (c/c

₆₎

+B' _{1 2} C

6

)+B' : (c

3

+c

6

)(

c

4

+ 2 5"+5" (c

4

+c7) 1 2 C 7)+B' +B'

)

-c 1(c3+c6) cl (C

4

+c

7

)

e

cl (c 3 +c6) Cl (c4 +c₇) 11

cx

1 +CX2 ~2 + -IJlJJ o 2 1 2 iI₁ l~ ~2 2 1 2 1 2 cl + 1 1 1 1 1 1 1 1 B c 2m + B' C 1(c2+C₅) + B' + 3

1 - - - 1 1 J

+ K' (=4e sin'YI _ _ _ _ _ --4 2 c 1(c3+c6) c1(c

3

+c6) cl (c

3

+c

6 )

Cl (c

4

+c

7

)

I I

~1 +~2 cl (c3+c6) + B' 2 c 1(c

4

+c7) c1(c

4

+c7) Cl (c

4

+c7) cl (c

4

+c7)

I 1

51+52 c1(c

4

+c7) + B'

EQ,UATION 2.20: "SYMMEI'RIC" EQUATIONS FOR THE 7D!F SYSTEM.,

1 I

e

Icx₁ '+(X' ₂

I

~'+~'

_{1 2} 5'+5' 1 2 ~

It-" lOl\l

-( _c )2 2 2-c

₅

+ 2Àc

₅

+ BI (C 2-C

₅

)(

C

3

-C6) + 2Àc

5

c6 + BI (C 2-C

₅

)(C4- C

7)

+ 2Àc

5

c

7

+ BI

-(C 2-C

₅

)(c

3

-C6) + 2ÀC

5

C6 + BI 2 2 (c

3

-C6) + 2Àc6 + BI (C

₃

-C6)(C4- C

7)

+ 2Àc 6c

₇

+ BI + 1-12 IIlJJ o 1 + 1-11 , 11 1 1-1₂ 1 I 11 1 1 _{1 11}

lJ

_l5i~

...

·3

I

cl (c 2+c

5

)+ KI Cl(C

3

+C6) cl (C 4 +C

7

)

I.

C_l(C

3

+C6) cl (C/C6) cl (C4 +C

7

)1

Cl (C4+C7) _C1(C4+C

7)

cl _(C4 +c

7

)1

L--(C₂-C

5

)(

c4 -c

7

)

+ 2ÀC

5

C

7

+ BI (C

3

-C6) (C4-C

₇

)

+ 2Àc 6c

₇

+ BI 2 (C4- C

7)

.

, I I 2 I + 2Àc + BI I

,

7

--.J

(Ci

_l

-Ci

₂ \ 10 ~

t'1-t'2

I =1 0

\5

₁

-5

₂ 1 I 0

EQUATION 2.21: "ANTISYMMErRIC" EQUA'TIIONS FOR THE 7D/F SYSTEM

x ...

.

+3

2.27, Equations for the

5D/F

system

These equations were in fact derived independently but a different approach is presented here. The

1DIF

system reduces to the

5D/F

system if the following changes are recognized, (compare figs. 1 and 2).

(1)

(2)

Put d = ~ = k = 0, eliminating the hinge subsystems.

1

Elimination of the hinge stubs also eliminates two degrees of freedom and therefore al - a

2

=

0 .

Since d = 0 all ci in equations 2.08 are no longer distinct because c 2 = c3 and c5 = c6. Therefore the ci arè refined as follows: Cl = (a + 1 + b) c

3

= b(l - m/M) c

5

= mb/M c 2 = (1 + b) (

t

-

m/M); (1 + b) (m/M); (2.22)

With these changes it is seen that the second and third equations in 2.20 and the first and second in equation 2.21 become identical and thus one equation can be dropped from each of the two sets. In view of (2) above, two variables are also dropped so that the resulting system of equations is self-consistent~

The equations thus obtained are valid for the

5D/F

system and agree with those derived independently as expected. These equations are given in equations 2.23 and 2.24 respectively. The changed ~ - vector is given by:

-2=

L

8 , _{131 +} 13 2, 51 + 52; 131-132, 51-52

J

-B 2 I 2 _Cl +~ + 2B' c 1(c2+c4)+B' c (c +c )+B' 8 m 1 3 5

-

-I

8

,

2 1 1 c l (c2+c4)+ "2B' '2 (c 2 +c4)( c3+c5 )3"+13" ~2 2 (c₂+c4'" +B' + -1 2 mw +R' 0 2 1 1 13'+13' 1 2 ) 2 Cl (c 3+c5)+ 2B' (c2 +c4)( c₃ +c₅) (c +c -3 5 )2+B' 5"+5" 1 2 2 1 1 5'+51 2 ' +B'

-

' - -

-

_'- .J 2 A -C 2 B 2 cl +~ c l(c2+c4) Cl(C3+C5) 8 cl +~ + B·' m 2m 2 C l(C2+C4) cl (C2+C4) Cl(C3+C5) 131+132 . =4esini cl(c2+c4) + B' 2 C_l (C/C 5) Cl (C3+C5) c (c +C1 3 5 '" ) 51+52 cl(C3+c5) + B'

Again it is observed that the matricès are no longer symmetric, af ter partition-i:r..g.

2.28 Equations for the 3D/F System

These were again derived independently, but are here deduced from the equations of section 2.27 to present a different approach. In order to ob-tain the 3D/F system the following changes are required in the 5D/F system (figs.

2 and 3): (1 )

(2 )

Since in the 3D/F system the string (or wire) behaves essentially lQke a rigid rod, the tip stub is no long er required and we must put b = 0, eliminating two more degrees of freedom i.e., 5

=

5

=

0.

1. 2

From equations 2.22 it is evident that with b

=

0, c

=

c

5

=

°

and

so the c. are again re-defined as follows:-

3

1.

(2.25) Now from theequations 2.23 and 2.24 two variables can be dropped, and from (2)

above the last equations of 2.23 and 2.24 must also be dropped. Thus a

self-consistent set of equations is obtained for the 3D/F system. The symmetrie and

antisymmetrie equations are given in equations 2.26 and 2.27. Equation 2.23 is

seen on page 1~ and equations 2.24, 29

&

27 are seen on page 19. The new x-vector is given by:

It is seen that symmetry is lost in the system matrices af ter parti tioning.

Thus finally all equations of motion have been derived and presented,

and attention must be directed to their solution. It is to be noted here th at

the assumption: 'of decoupling between the later al and longi tudinal' equations is

reflected here by the absence of any terms depending on the lateral variables.

The method of derivation also serves to illustrate the close relationship between

the three configurations.

2.3 SOLurION

A complete formal solution of the equations of motion for given initial conditions would be somewhat involved and would yield little extra useful information, since the system is linear and so important information abou~ the

performance characteristics can be readily found using linear system theory. It

would be best to convert the equations to first-order form and to obbain the

eigenvalues and eigenvectors of the resulting system matrix, which characterizes

the system (e.g. ref. 10). However, with the present choice of variables this

is not possible, for reasons to be explained later. It was decided to find only

the eigenvalues of the system which give information about relative stabiLity and

the characteristic settling times af ter a disturbance. The steady-state response

to orbital eccentricity is also obtained to give an estimate of maximum pointing

error. In fact this information gives sufficient insight into the system. It is

expected that if the characteristic damping times are short the system as a whole

will perform weil and the characteristic modes of oscillation need not be computed

t-' \0 2

(C

2

-C

4)

+

2I\C~+B'

(C

₂

-C4

)

(C

₃

-c

₅₎

f

\

+ 21\c c

+B'

~"-~"

4 5

1 2

.

-(c

2

-c4)(

C

₃

-C

₅

)

+21\

c4c5 + B'

2 (c -c )

+B'

3

5

+21\ _c2

5

+ ~c Il1J.J o 11 11 /5' -5' 1 2

+

3

)/

'

(c +c )

C

l (C

2

-C4

cl

3

5

11

~1-~

2

c

l

(c

3

+c

5

)

cl (C/C

5

51-52

EQ,UATION 2,24:

"ANTISYMMEI'RIC" EQ,UATIONS FOR THE 5D

/

F SYSTEM

2Ci+

:i>

+2B'

I

cl

(C

2+

~( 9"

I

r

1

~f

9') /

2

ci +

\C

C

/ cl (C

2

+

V

₃

)/f

9

J

+ B'

,

~2

l+

3

,

1

2

Cl(C2+C3)

(c +c ) +B'

2

l~î+~~)

_Lll

j

l~l

_"fl2]

₂

_{c1 (C}

_{2 +}

_C

₃

_{) / cl (C}

₂

_+C

₃

_1t1

+~2

2

3

+ 2B'

EQ,UATION

2.2

6:

"SYMMETRIC" EQ,UATIONS FOR THE

3

D

/

F SYSTEM

[(C 2-C

3

)2+

2À~

~

+ B']

(~i-~

2

)+ ~O (

~

1-~

2

)+

[

3

C

1(C

2

+C

3

)

J(~

~2

)

=

0

EQ,UATION 2.27:

"ANTISYMMETRIC" EQ,UATION FOR THE

3

D

/

F SYSTEM

o

o

I

_c2+B'+~

2

B

1

m

cl (C 2+C

3

)

I

+B'

,

- _I i

~.31 Homogeneous Equations

The homogeneous part of the equations of the form 2.01 is known to have a solution of the form x

=

a est So tbis assumed solution is substituted into equa~ion 2.01. The charactëristic equation of the system is obtained from the fact that since a is not identical1y zero, the determinant of the remaining matrix must be zero tor an unforced solution to exist. Thus we

get:-s2 A + s B + C

I

=

0 (2.28)

On expansion of 2.28 the characteristic polynomial, the roots of which are the characteristic transient roots of the system, is obtained. Computer programs for both these steps were available and so the calculations could be easily made.

However, a problem of some fundamental importance ari ses here. Systems with equations of motion of the form 2.01 come up rather frequently in satellite dynamics. Stability theorems exist for such systems provided the inertia matrix is positive definite (e.g. ref. 11). In the present case, however, the mathe-matical model of the systems involves some 'degrees of freedom' with which no mass is associated. For example the inertia matrix in equation 2.20 has a rank of 2 instead of

4,

since there is no mass associated with the hinge stub and string co-ordinates. ~he singularity of the inertia matrix also causes the leading coefficient of the characteristic polynomial and the first two leading

coefficien~s to be zero for the 5D/F and 7D/F systems respectively. This occurs in both the symmetric and the antisymmetric sets. Here matrix-form stability theorems (e.g. ref. 11) are no longer applicable. The Routh-Hurwitz criteria can still be used af ter expansion of the characteristic polynomial, but the computation may be involved for large matrices. Further and perhaps more im-portant, the equations cannot Qe directly put into first-order form since the process involves inversion of the inertia matrix, like in ref.12.

\

Of course all ~his can in principle be avoided by proper choice of the generalized co-ordinates and by deriving the equations in the first order form to start with (Hamilton's formularion may be useful) . But the proper co-ordinates may not always be readily apparent, and in view of the importance of having the equations in }he standard form, it was sought to find a method for converting the equations from a second-order to a first-order form even with a singular inertia matrix. This was indeed show~ to be possible by a transformation of co-ordinates. The appropriate transformation turns out to be the solution of a set of homogeneous algebraic equations with a zero determinant. Applying it to the original system of equations leads to a new set of second-order equations with a new non-s.ingular inertia matrix, plus some first-order equations. From this all equations can be put into first-order form. It was found that if Lagrange's equations are applied to a system with 'n' generalized co-ordinates, and the resulting inertia ma~rix has a rank 'r', the whole system of equations can be made equivalent to (n + r) first-order equations in the canonical form ~

=

A~, and thus the system has (n + r) eigenvalues.

However, the eigenvalues as such are not affected by the above transformation which only serves to make calculation of the characteristic

modes easier (these are simply the eigenvectors of the system matrix) . But since for the present case the eigenvectors were not felt to be very important, the transformation was not actually carried out. The eigenvalues were obtained

zero leading coefficients. By using the rule stated above it is seen that the 7D/F system will have

6

'symmetrie' eigenvaltles and

4

'antisymmetric' ones since in both of eqvations 2.20 and 2.21 the inertia matrices have a rank of 2. Thus the first two leading coefficients of both the symmetric and the antisymmetric characteristic polynomials should be zero. This fact is

im-portant to know since in the numerical computation, due to round-off errors, these leading coefficeints were of ten erroneously cî~culated to be as high as

10

3

when other coefficients were of the order of 10 ; and a false idea of the roots could have been obtained. Moreover, sometimes the erroneous coefficients were negative, apparently indicating instability. Therefore the order of each characteristic polynomial was reduced accordingly by setting these coefficients equal to zero and the eigenvalues were found from the reduced equations. There are 10,

8

and

6

eigenvalues respectively for the 7D/F, 5D/F systems.

During computations unstable roots were never in fact encountered so the systems were stable at least in the parameter space investigated.

Instability will be expected, for instance, if negative damping were to exist Stability was also confirmed when the mass of the stubs was approximately taken into account, then very large negative (i.e. "..':".stable) roots were ob-tained in addi tion to the roots of the reduced equations; this indicated very heavy convergence of the stubs in the actual system. Thus stability of the configgnations is assured within the stated assumptions, and no fundamental information is compromised by neglecting the mass of the stubs and strings. 2.32 Steady-State Solutimn,

The equations of motion have the

form:-!

sin,

Thus due to orbital eccentricity a steadysstàte oscillation at orbital fre-quency exists. A particular solution is assumed of the form

x

=

a sin, + b cos, (2.29)

Substituting this into the equations and comparing coefficients of sin, and

cosy on both sides leads to the following equations:

-(c - A) ~ + B a

=

0

(2.30)

- B ~ + (C - A) ~ = 0

So a and b can be easily solved for by a simple matrix invers ion and then equation 2.29 gives~. The magnitude and phase relative to the input of x are given by the foll

owing:-I

x.

I

=

J

a~

+

fs~

- 1 1 1

x.

- 1 tan -1 (b. 1 / a. ) 1

This completely specifies the ~teadW-state response.

21

It is to be noted here that the antisymmetric equations contain no lorcing terms due to eccentricity so that a2

=

al' ~2

=

~l' etc. Also, a center-of-pressure offset from the center of mass in the main body would cause an error in the body-pointing amplitude also; this can be found similarly by superimposition, ie, by including a constant term in the assumed steady-state solution 2.29. It mayalso be mentioned that radiation pressure torques if present would force the antisymmetric modes exclusively.

2.33 Numerical Calculations

Calculations were carried out on the IBM 1130 computer at the Institute for Aerospace Studies and the IBM 709411 at the Institute of Com-puter Science, University of Toronto. Programming, except for some very help-ful subroutines already available, was done by the author. Brief comments on the programs

follow:-DET2, DET3,

&

DET4 (available beforehand) calculate the coefficients of the characteristic equation given the matrices A,B ,C. ZERPOL SOLN7 SOLN5 SOLN3 ROOTS FF ,GRAD, & 0Pl'2

(library subroutine) calculates the roots of the characteristic polynomial.

solves the equations of motion for the 7D/F system to give characteristic roots and pointing error ampli tude s .

solves the 5D/F system and is similar to SOLN7. solves the 3D/F system and is similar to SOLN7. is an output subroutine and calculates period in orbits and orbits to half amplitude given the transient roots.

are optiffiization subroutines, described in greater detail in the next chapter. These wer also avail-able beforehand but required some modifications, also described later.

The programs all worked satisfactorily. DET4 was somewhat less accurate. CHAPI'ER UI: OPI'IMIZATION STUDIES

3.1 Introduction

Due to the large number of parameters and differences of configurations in t he three systems a meanillgful comparison of performance cannot be easily made. However, if the best possible performance (in some sense) is available, with the same performance index in each case, the task is relatively simple. Thus optimization is required and it was decided upon. With linearized

equ-ations of motion, finding a meaningful solution is not so difficult with high-speed computers a search for the optimum can be qu:i..te successfuli'Y made by using direct numerical methods. The performance index chosen was the time-to-damp to half amplitude required by the least-damped mode. This is

found by experience not to have smooth mlnlma (ref.13) but was the only practical one; since the pointing error amplitudes are not very sensitive to parameter variations as compared to the transien~ performance.. Moreover the pointing error is quite acceptable for many applications. (about 1 degree for e

=

.01) and so a need for minimization was not very acute. The maximum time-to-damp, on the other hand, can easily change by a factor of 10 or even 100 and is positively intolerable in many cases.

3.2 Optimization Methods

In complex engineering problems like tme present one where no algebraic description of the. performance exists in the sense thai; the per-formance index is best defined in terms of numbers, classical methods such as the theory of maxima and minima cannot be applied. Direct numerical methods are to be used, of which the methodof steepest descent (also called the method of gradients - ref.14) is very promising for many applications. Many modifi-cations of the gradient method have be.en reported in the li terature, and the method has been used for quite complicated engineering problems successfully. The modifications usually affect convergence rates and in some case.s stability of the computation. Here an essentially unmodified gradient method is used except for some programming features. It has been successfully used in a very similar application at UTIAS, and a not unimportant factor in its choice was the availability of computer programs, which however had to be modified con-siderably.

3.21 The Gradient Method

Consider a scalar point function in parameter space f(~) which undergoes a change 'df' with a small change dx in the parameter vector, of fixed magnitude 'ds'. Then we

have:-df

=

(V f) . d x (3.01)

If the magnitude of d ~ is fixed, df is greatest if d x is parallel to Vf, or

dx + ds

*

(Vfj

Iv fl )

(3.02)

where ds is the scalar step magnitude~ The minus sign isused for minimization and the plus sign for maximization. In t~is method steps are continually taken which provide the greatest possible change in the functiön for a given step si ze at a given point. The components of Vf are easily found numerically, thus the method is well-suited to computer program applications.

Constraints: If some constraint exists on the system it can also be handled. In the present case, however, inequalitY .constraints (e.g. stub length less than a pre-specified value) were encountered which are not amenable to analy-tical treatment so easily. Tb impose these constraints a 'cut-off' technique was used; i.e. prohibiting any increase of. the constrained parameters beyond the limits imposed, even if indicated by the gradient method (equation 3.02). The optimization process then proceeds in the space defined by the remaining parameters. ~his may not be analytically desirable but was found to work well.