Finite-element analysis of CTS-like flexible spacecraft

) FINITE-ELEMENT ANALYSIS OF CTS-LIKE FLEXIBLE SPACEX:!RAFT

:i:J.t~ jt.)"; ,;~ ... { K:UyVt:;I'I1~ 9 1 - DE ... FT

by , \liC. \'.\1\

P. K. Nguyen and P. C. Hughes

June,

1976

,

tJrIAS Report No. 205

•

..

..

FINITE-ELEMENT ANALYSIS OF CTS

-

LIKE FLEXIBLE SPACECRAFT

by

P. K. Nguyen and P. C. Hughes

Date Submitted: April,

1976

•

•

..

- -~ - - - ---~---~---.

Acknowledgement

This work was sponsored by the Communications Research Centre,

Department of Communications, ottawa, under contraC't No. OlGR-36100-2-CE 10 •

•

•

•

Summary

The dynamics of a class of non-spinning flexible spacecraft is

. studied in general terms. The spacecraft consists of a relati vely rigid centre body to which a number of flexible appendages are rigidly attached. The centre body has pitch, roll and yaw attitude motion. Deformations of the appendages are assumed elastic, small in amplitude,· and are studied via a finite-element analysis.

Using a transformation from distributed displacements to finite-element displac.ements, the angular momenta and moments of inertia of the flexible appendages are expressed in terms of elemental inertia matrices. The spacecraft attitude motion is shown to have an effect similar to reducing the inertia of the appendages whereas an on-board momentum wheel can have

an

effect similar to stiffening the flexible appendages. Mutual interactions between centre body and appendage motions are illustrated. With a view to control system design and spacecraft motion simulation, expressions for appendage ~dal gains are provided.

A simplified CTS (ConmrunJ.cations Technology Satellite) model is used to illustrate a number of important points. The analysis is presented in detail to provide a means for interpreting the variables used in the general derivation. Numerical results show excellent agreement with results obtained in a similar study using distr;i.buted (modal) coordinates •

iii

1. 2.

3.

4.

5.

• TABLE OF CONTENTS Acknowledgement SUlIIIllary Table of Contents List of Symbols INTRODUCTION

MOTION EQUATION OF SLEXIBLE SPACECRAFT

2.1

Motion Equation of Spacecraft

2.2 Rigid and Flexible Angular Momenta

2.3

Motion Equation of Appendages

2.4

Torque-Free Motion With0ut Stored Angular Momentum

2.5 Torque-Free Motion With Stored Angular Momentum

2.6

Spacecraft Inertance and Modal Gains

ii Ui iv vi 1

4

4

8

11

13'

13

17

2.6.1

General Motions in Terms of Unconstrained Modes

17

2.6.2

General Motions in Terms of Constrained Modes

19

2.6.3

Some Properties of System Gain Matrix

21

DESCRIPTlON OF CTS AND lTS IDEALIZATION

3.1

ldealized Model

3.2

Finite Elements and Nodal Numbering Algorithm

DYNAMICS OF THE SIMPLIFIED CTS MJDEL 4.1 Basic Motions

4.2

Motion Equations

4.2.1 Array Twisting and Pitch Attitude 4.2.2 Array Bending and Roll-Yaw Attitude

4.3

Numerical Results and Discussions

4.3.1

Array TWisting and Pitch Attitude 4.3.2 Array Bending and Roll-Yaw Attitude CONCLUDING REMARKS

REFERENCES

APPENDIX A - FLEXIBLE ANGULAR MOMENTUM hF IN TERMS OF FINITE-ELEMENT COORDINATES

APPENDIX B - MOMENT OF lNERTlA OF FLEXIBLE APPENDAGES ABOUT THE SPACECRAFT CENTRE OF MASS

iv

23

23

25

27

29

32

.

32

35

42

43

46

47

49

APPENDIX C - NATURAL FREQUENCIESOF VIBRATION FOR TWO RELATED SYSTEMS

APPENDIX D - DEFINITIONS OF MODAL GAIN MATRICES Kn

APPENDIX E - STIFFNESS MD INERTIA MATRICES OF A PLATE ELEMENT WITH PRE-LOADING

APPENDIX F - MOTION EQUATIONS FOR A CONSERVATIVE SYSTEM WITH SIMPLE LINEAR HOLONOMIC CONSTRAINTS

APPENDIX G - EFFECT OF S'PORED ANGULAR MOMENTUM ON NATURAL FREQUENCIES

APPENDIX H - EFFECTS OF DISCRETIZATION AND DISPLACEMENT ÇONSTRAINTS ON SPACECRAFT NATURAL FREQUENCIES

'

.

- - -

-Upper Case Roman A B C D E E F G H I 1* b K L M N P P R T

u

List of Symbols Matrix defined by (18)

Constant matrix defined in (G-l)

Matrix relating nodal displacements to distributed dis-placements

Eigeruna:trix of d

Young' s modulus of elasticity

Elasticity matrix

Displacement distribution matrix defined by (A-2) Transfer function

Matrix defined by (36)

Moment of inertia of the undeformed spacecraft Dimensionless body inertia

Elemental stiffness matrix

Lagrangian

Elemental inertia matrix Matrixdefined by (D-2) Tension

Dimensionless tension

Matrix relating elemental nodal displacements to the displacements of assembled structure

Position vector expressed in local frame

Transformation matrix from datum to local frames Unit matrix

Matrix defined by (A-10)

v

Matrix defined by (D-l)

w

Matrix defined by (D"'l)

x

Output matrix

Y!-

Matrix defined by (A-2l)

y _{Input matrix}

Z Matrix defined in (A-3)

_"

Lower Case Roman

a Width of plate element

b Distance from array root to spacecraf't centre of mass

c Column matrix of constants, see (A-2)

d Nodal displacement matrix

e Length 0 f plate element

h Plate thickness

h Angular momentum vector

...

~s

stored angular momentum

1, _{Boom length, array length}

.

m Number of finite elements in 2~direction

m* Dimensionless mass of pressure plate

n Number of finite elements in 22 direction

p Mass of pressure plate

.9.

Column matrix of generalized coordinates

r

-

Position vector from the spacecraf't centre of mass

s Laplace transform variable

t Time

t

~ Torque

w Array width

u,.v, w Displacements in a plate element, to be used in Appendix

E only

x,y,z Greek '1 5

-e

6 ti: • -~

v

p

cr

cr

A

Cartesian coordinates in datwn frame

Angle defined by

(135)

Free parameter defined in

(c-4)

Off-set angle between local and datum frames Continuum displacement

strain

Dimensionless coordinate defined in (E-8) Pitch angle

Column matrix of attitude angles Modal gain matrix for mode i

Poisson I S ratio

Dimensionless coerdinate defined in (E-8) BeOID (linear) mas s density

Blanket (area) mass densi ty stress

Roll angle Yaw angle

Natural frequency

"Rigid" nutation frequency

Distributed displacement vector expressed in local frame Column matrix defined by (50)

Matrix defined by (D-ll)

Eigenmatrix for ~

Matrix defined by

(43)

Dimensionless natural frequencu. Dimensionless nutatien frequency

Diagonal matrix of eigenvalues~2

Script Letters

~ Matrix defined by (23)

tf,

Nodal d~splacement constraint matrix

~

Matrix defined by (A-26)

•

_X

_{Stiffness matrix of assembled structure}

0J(; Inertia matrix of assembled structure

~

Vector space

cu,

Matrix defined by (21)

Special Symbols

-t Column matrix whose elements are all unity

Superscripts

0 Preloading state

x IlCross" operator defined in (10)

*

Dimensionless parameter, except in Appendix G where it

denotes an eigenvector T Transposed matrix Subscript c Constrained

.

( ) d _{dt ( )} ix

A NarE ON DIMENSIONS OF CERTAIN MATRICES

if

₌

3 x 3n matrix (Equation A-l0)

r-

=

3 x 3n matrix (Equation A-21)

A.

₌

3 x 6n matrix (Equation 18)

-1

M.

₌

6n x 6n matrix (Equation A-8)

-1

P.

₌

6n x 6nN matrix (1 equation above 18)

-1

T.

₌

3 x 3 transformation ,matrix (between B-5 and B-6)

-1

d. :;;;;

-1 6n x 1 matrix (beginning of Appendix A)

d

₌

N(6n) x 1 matrix (beginning of Appendix A)

<U

:;;;; 6nN x N(6nN) _{(Equation 21)}

ä

₌

3 x 6nN (Equation 23)

3lG

=

6nN x 6nN matrix (Equation 23)

~

:;;;; 3 x 1 matrix (Equation 17)

where n :;;;; Number of nodes in the element under consideration

N = Number of elements in the structure

1. INTRODUCTION

The last two decades have seen an 'impressive increase in the sophis-tication of' spacecra:ft attit ude control systems

[1].

Of' particular interest in the context of' this report are the complications th at arise in connection with the attitude dynamics of'.the. spacecra:ft. Foremost among these complications is the nonrigidity of' much of' the spacecra:ft structure, particularly the append-ages that are deployed a:fter the vehicle has been inserted into orbit. Simul-taneous advances in computer technology and techniques f'or structural dynamical analysis have enabled even more detailed structural mödels to be generated leading, in turn, to (usually) rEüiable design of' the attitude control system. The level of' detail required in the dynamical model of' a particular spacecra:ft f'or a particular rnission is a matter of' technical judgement. If' ·the geonetry of' the f'lexible portions of' the spacecra:ft is not simpIe, or if' a detailed structural modtü is thought to be prudent, the :finite-element method is likely to be the most attractive approach.

Elastic def'ormations arf! most. of'ten amlysed using one of' two general f'amilies of' techniques: distributed coordinates, or discrete coordinates. Frequently these two approaches may be used in combination. Likins [2] has given a clear and inf'ormed discussion of' these two alternatives, and their cornbination ('hybrid r coordinates).

The f'inite-element method has an interesting history. Frustrated by the inability of' the classical approach (partial dif'f'erential equations, boundary conditions , f'ini te-dif'f'erence numeri cal techniques) to deal wi th complicated structures, imaginati ve engineering analysts began to adopt a strategy in which the structure was approximated by a large nurnber of' simple structures each of' which was amenable to relatively simple analysis. They then assembled these

simple substructures to f'orm a physical approximation taking due care1to sati~f'y

the required tcompatibilityt conditions in this assembly process. This new strategy, in cornbination wi th the availabili ty of' high-speed digi tal computers, led to a level of' success that it is no exaggeration to say was spe ct acular • In due course, the applied mathematicians got wind of' all this. They proceeded to place the f'inite-element technique on a f'irm mathematical basis. While .doing so, their ef'f'orts generated an interésting alternative view of' what the f'inite-element method in f'act was 0 They demonstrated that ·the method need not be. I.

viewed as a physical approximation but as a mathematical approximation in whicp.

the Ritz method was employed in conjunction with piecewise polynomial trial f'unctions

[3J.

In the f'inite-element method the def'ormation of' a structure' in the (small) domain of' the relement r may be thought of' as represented by the displacements at its 'nodes' • Def'ormations at points other than the nodes are related to the displacements at the nodes via a 'displacement f'unction r - the . Ritz trial f'unctions. As the nurnber of' these f'unctions becomes inf'inite arid they become 'complete'" the f'inite-element approximation can be shown to converge to the exact solution f'or the structural model used.

In spite of' its approximate nature, the f'inite-element method is, in practice, o:ften pref'erred to the distributed coordinate approach, mainly beçause

of' its versatility. This is particularly true when a general multi-purpose simulation computer program is desired. In both approaches, the dynarnics of' a class of' f'lexible spacecra:ft can be studied in general terms without specif'ying the details of' the f'lexible sub-bodies

[4,5,6].

The behaviour of' the f'lexible sub-bodies is studied in a separate 'module'. This is where the f'inite-element

FIG.

1

ARTIST'S

CONCEPT

OF

CTS

( Courtesy of SPAR Aerospace Prod ucts Ltd. )

method has i ts advantages • The needed characteristics (inertia, stiffness, da.II!Ping, etc.) of various elements c'an be specified in advance and stored in the 'flexibility module'. For different flexible structures, one needs only to specify the finite elements to this 'module'.

The motion equations of flexible structures in the finite-element method are ordinary differential matrix equations since the spatia!L dependence of the deformations has been absorbed in the discretization process. This is another advantage of the fini te-element method; it leads more directly to a set of ordinary differential equations. Realizing these points, the present report analyses the dynamics of a spacecraft of the CTS (Communications Technolcgy Satellite) class using the finite-element methode

The work presented herein is motivated by the CTS program. CTSis

a non-spinning, three-axis stabilized geostationary satellite. It consists of a relatively rigid centre body to which are appended two flexible solar arrays (Fig. 1). The solar arrays are sun oriented, thus must rotate about their own

(north-south) axes. The model used in this report is depicted by Fig.

4.

Similar to CTS, i t has a rigid centre body plus two major flexible appendages. In the first part of this report (Section 2) the spacecraft motion equations will be derived in gener al terms and the following points will be discussed in connection with the finite-element methode

(i) Mutual dynamical interaction of the centre body and the appendages.

(ii) Spacecraft motions in terms of 'constrained' and 'unconstrained' appendage vibration modeso

(iii) The effect of on-board angular momentum storage on the flexible spacecraft attitude motion.

Among a number of interesting results is the treatment of the transfarmation from distributed coordinates to finite-element coordinates which allaws the angular momenta, moments of inertia, etc. of the flexible appendages to be written in terms of nodal displacements • The details of this develapment are presented in Appendices A and B.

In the second part of this report (Sectiori

4)

asimplified CTS model

(Fig.

4)

taken from

[7]

will be used. The use of this sirnplified model is

twofold. Firstly, i t helps to compare the results with those in

[7]

where

distributed (modal) coordinates were used. Secondly, i t helps illustrate a

number of important steps taken in the derivation in Section 2. It is notep. that if the gener al analysis in Section 2 has been implemented in a computer

program, the derivation in Section

4

would no longer be necessary. The

simplified CTS model consists of a rigid body and 'two flexible solar arrays

(Fig.

4).

The solar cell array is mounted on a blanket which is in a state of

tension provided by a cantilevered support boom. The tension on the blanket

gives i t 'stiffness' and the compression on the boom lessens its stiffness. This is taken into account by including a derivation of the stiffness matrices for plate and beam elements using Martin' s method [12]. Finally, to be

consistent with

(7,8,9]

the spacecraft pitch and rOll/yaw motions will be

considered separately with numerical results being presented in terms of dimensionless parameters.

2. MarrON EQ,UATrON OF FLEXIBLE SPACECRAFT

The flexible spacecraft is represented by a 'topological tree' as

depicted in Fig. 2. The 'trunk' is a relatively rigid centre body and the

'bra..l1ches' are flexible appendages which might have 'sub-branches'. Although

the spacecraft may have both translational and rotational mot i ons , in this

report we are interested only in rOtational motions* particularly from an

attitude control standpoint • The spacecraft attitude is described in reference

frame~ The origin of frame(h)is the spacecraft centre of mass and the axes

ofG) are the principal axes of the undeformed spacecraft. Frame@also serves

as the datum reference frame for appendage displacements. Unless otherwise

specified, all vector quantities will be described in frameG) A number of

loc al or working reference frames~ will be used occasionallY, for example

when deriving the element stiffness and inertia matrices. Frames(2) are fixed

in the appendages and they will be defined where used. (See F.ig.

'

')f

.

2.1 Motion Equation of Spacecraft

The attitude dynamics of the spacecraft can be simply described by

.

h = t

...,. ...,.

(1)

where h

=

total angular momentum of the spacecraft ab out the spacecraft mass

..., centre

t

=

disturbing and control torques •

...,.

The angular momentum h can be decomposed into '7

(2)

where ~

=

stored angular momentum, e.g., from a momentum wheel,

~

=

momentum associated with the rotational dynamics of the spacecraft.

Let r denote the position vector of a point on the spaèecraft measured from the ...".

spacecraft centre of mass, then

MOre explicitly, let subscripts A and B denote the appendages and the centre

body respectively. Then (3) becomes

!;.n

=

J

!;B

x

~B

dm +

J

!;A

x

~

dm

(4)

B A

For small amplitude deformations, referring to Fig.

5,

~A can be-decomposed into

*We assume that the motions ofthe spacecraft centre of mass and attitude motions

are uncoupled. This assumption is of ten satisfied in practice with the aid of

certain symmetry properties •

4

'"

locol

reference frome

t

FLEXIBLE

<

APPENDAGE

RIGfD

CENTER BODY

...

FIG. 2

TOPOLOGICAL TREE OF FLEXIBLE

SPACECRAFT

>

-

_"C ~ Cl)

-

U :l "C

e

a..

Cl) )0- u 0 ~

C-a:

_Cl)

a:

e

~

a:

Cl) ~

«

..J 0

_a:

en en

_~

t-U Cl)

....

~ 0 (!)

>-

_Cl)

-

Cl) LL

-t... ::J 0 U

...

where 5

_....

=

elastic displacement vector

'rigid' cOIr.!Ponent of

;A

!,AR

=

Therefore the momentum!;n can be wri tten as

~

=

I:;B

x

!;B

dm +

I~AB

x

~ABdm

+

I~AB

x

~

dm

B A A

+I

~

x

(r

+ 5)dm

• "'TAB ... (6)

A

where the first two terms. on the right hand side clearly represent the angular

momentum of the rigid spacecraft. Equation (6) allows a further decomposition

of h via

where ~

=

'rigid' angular momentum

=

first two terms on the r~ght hand si~e

of Eq. (6)

=

'flexible' angular momentum = last two terms on the right hand side

of Eq. (6)

Equation (1) now becomes:

8 be the 'Then the

(8)

Let

.!$,

.!!R,

~

be the components of

~, ~, ~

in frame

(2),

and let

column matrix of attitude angles for the main body, i. e., of frame

G

torque-free motion is àescribed by

(~

+

~)

+

~x

(.!!s

+

~

+

~)

= 0

where the dot represents a differentiation with respect to time and

.

0 -83 82 'X

.

0

.

8

₌

83 .... 8l. (10)

.

.

₀ -:82 8~

The cross (x) operator defined by Eq. (10) will be frequently used in subsequent

sections... For non-spinning spacecraft, * we can neglect the second order term

*For spinning spacecraft,

ê

is no longer first order and the terms .~(.!!R + ~)

should be retained, where-w is the angular velocity of the main body.

_...

Under

present as sumption s , w ~ 8.

~x(ÈR +~) so that Eg. (9) reduces to

(11)

2.2

Rigid and Flexible Angular Momenta

that

From the definitions of ~ and ~ in Eg.

(7),

it can be demonstrated

(12)

h =

I

r X . 5 dm

.:...""}' -AR -

(13)

A

where I

=

inertia matrix ofthe entire undeformed spacecraft.

!AR and ~ are the co~onents of ~AR and ~ in

<D .

In Eg.

(13),

second and higher order tercrns have been neglected.* The integral

in Eg.

(13),

however, must be interpreted in terms of a finite-element

formula-tion, where only (discrete) nodal displacements are available instead of distribu

₁

ed

displacrments~. The fOllowing section is conseguently devoted to the transfor~­

tion that allows

.ÈP

to be expressed in terms of the elemental inertia matrices ,and

nodal displacements.

Assuming there are N finite elements in the appendages". we can wri te

N

~=LI !~~dm

(14)

i=l A.

~

where the subscript i denotes the ith element. In order to keep the analysis

uncluttered we are going to work with only ÈF. and eventually a stumnation will

be taken over N elements. Since the elemental inertia matrix is derived using

the coordinates in the local frame,

®,

it is more convenient to derive ÈF. in

terms of the coordinates of the local frame. To this end, we choose alocal

frame

®

i as exemplified in Fig.

5

and we let

.!i

denote the transformation

matrix that brings a vector in

Q)

into ® i ' e.g., !iE.AR gives the co~onents

of

~AR

in

®i·

Since (!i

~)x =

! i

~x

! iT for any column matrix

~,

we find that

J

x •

=

@

!AR

~

dm =

®

_~

·IT.

_-~

A.

~

A.

~

®iI

(!i !AR)x (!i

~)

dm

Ai

®J

~x ~

dm

A.

AR ~

(to first order)

*For spinning spacecraft, ~ will contain two additional terms

I!~ ~x ~

cm1

and

I

~x ~x

!AR dm

A

A

8

(15)

tip ma ss

blanket

tip mass (p)

FIG. 4 IDEALIZED CTS MODEL

SPACECRAFT CENTER OF MASS FIG. 5 After deformafion Before deformafion

LOCAL FRAME

®i

where ~ and ~ are the components of ~ and

.2

in

®

i • Referring to Fig. 5, (15) can be written as where~. = components 1. R = componen t s

=

®

i

J

(~Oi +~)

x

~

dm

~

=

®.

RX

J

b.

dm + 1. -0. -1. A. 1. of i in ® i ' -rO i . of R in

®.,

... 1.

r = position vector from the spacecraft centre of mass to the origin

-rO

i

0 of

®i'

R

....

=

position vector from 0 to the point in consideration.

(16)

Aquation (16) shows that the flexible angular momentum!!F. can be determined as

the sum of two terms. The first term is the moment of

th~

total momentum as if

it were located at O. The second term is the angular momentum of the element

relati ve to O. [ ..

In Appendix A, it is shown that (16) can be WIi tten as:

ÈF. =

[(~.

:t

+

re)

~t

+

:!

.

!:!r]~i

(17)

1. 1.

where !:!t and

!:!r

are defined in (A-8) , ~ is the nodal displacement vector (of

element i), and

QX

and!x are defined by (A-10) and (A-21) respectively.

Let

d

=

nodal displacement of the assembled structure

P.

=

matrix relating d. to d via d.

=

P. d

-1. -1. -1. 1 . -I

~i

=

(~

rf

+

re

::!)

i I

~

=

(-~~-

)

.

then hTi'l.. _.;..;;;.'

=

A. _{-1. -1. -1. -}M. P. d and in frame

'î',

_\&

~.

1.

=

<D

T~

h.

= 'î'

TT i A. M. P.

d

-1. ~""F. \::;;I - 1. 1. 1. -1.

Q)

T · T •

=

1 T. A. P. Pi M. P. d -1. -1. -1. - 1. 1.

-because

P.

P~

is a unit matrix.

-1. -1.

10

(18)

(19)

- "I

,

Taking the sum over N elements, we find:

N

h

=

rp\

(T: A. P

.

) (P: M

.

.

P.)cÏ

=

-?~ \,;V

L

-J. -J. -J. -J. -J. -J. -

(20)

1

By defining the ~ matrix by

(21)

(where U

=

a unit matrix which has the same dime;nsion as

P~

M p. and

<ti

is made

up by

N

such Q) we can replace

È:.F

by _1 - _J.

where

!?JrG

= assembled inertia (and mass) matrix of the entire flexible structure

••• : pT

~L

P )T

I -N;::"'"N-JIJ I

(22)

(23)

Equation

(22)

allQws

È.F

to be expressed in terms of the assembled inertia matrix

en;

and the nodal displacement vector d. The matrices ~ and 9l5will appear again

In the motion equation of the appendages. ....

2.3

Motion Equation of Appendages

In the fOllowing analysis, we will describe the appendage motiQns as 'unconstrained' in the sense that the spacecraft main body is allowed to have (attitude) motion. The kinetic energy*of the i th element is:

1

J .

(K.E·)i

=

2

(~

-A.

J.

No'ting fr om Appendix B that

*The natural structural vibrations are assumed to have much higher frequenciesthan

the orbital frequency. This assumption allows the orbiting frame of reference to

be taken as inertial in the appendage motion equation.

that is, the moment of inertia of the i th element about the spacecraft centre of mass, the above equation can be rewri tten as*

l·T • 'T T . l ' T T •

(K.E.).

=

-2 d. M. d. +

e

T. A. M. d. + -2

e

T. I. T.

e

(24)

~ -~ -1. -~ - -1. -1. -~ -~ - -~ -1. -~

-where the results in Appendix A have been used.

where K.

-~

The potential energy of element i is given by:

1 T

(P.E.).

=

-2 d. K. d.

1. -~ -~ -1.

T

!!

~ d(vol)

E

=

elasticity matrix, i.e., ~

=

E E

~ - stress

E

=

strain

b = matrix relating E to d., i.e.,' E = b d.

- -1. - - - 1 .

(25)

The appendage motion equations are now derivedwith the ai<i of a Lagrangian

formulation. TheLagrangian will not include the terms associated with stored

angular momentum because"it is being employed to derive only the appendage motion

equations and these do not contain È:s' Wi th l the inertia moment of the whole

(undeformed) spacecraft about the spacecraf't mass centre, the sYi>tem Lagrangian is N N

I

T ' 'T

L

T T ' " P. M. P. d + e T . A. P. P. M. P. d -1. -~ -1. - - 1. 1. 1. .1. 1. 1. -1 1 N +

~ ~T

!

~

-

~

i

T

"

L

R~ ~i

Ri

i

1 Let

K. P.

=

assembled stiffness matrix

"""'1. -~

Then L becomes

L

=

! ä.

T

9Jb

d

+

ë

T &.

~

d

+

!

ë

T I

ë -

!

dT ,% d

2 - - - 2 - - - 2 - - - (26)

*For sp inning spacecraf't, the kinetic energy has addi tional terms containing the

spin vector ~ which give rise to a 'Coriolis matrix'

J

2QT

~x

.Q.

dm,

a 'tangential stiffness matrix'

J

T

C

W

X

C

_ _ _ dm,

and a 'centrifugal stiffness matrix'

J

QT

!i

x

~x

C dm.

12

The Lagrange motion equation f'or ~ is simp1y

(27)

f'or a conservative system. The quantity &,T 8 is composed of' the displacements of' the nodal points induced by the rigid rotätion of' the spacecraf't. Examp1es of' these displacements wi11 be shown in Section 4.2 and in Figs. 12 and 14.

2.4 Torque-Free Motion Without Stored Angular Moment um

In the absence of' stored angular momentum, the spacecraf't motion equation is simply

(28)

as can be seen f'rom (11). From (12) and (22), equation (28) can be rewritten as:

This is also seen to be the Lagrange equation f'or 8 using (26). Equations (27) and (29) show the mutual interaction •• of' spacecraf't-attitude motion and the appendage f'lexibi1i ty. E1iminating 8, the appendage motion equation (27)

becomes

-9lG(u -

aT

r-

1

tL

~

is then the "e f'f'ecti ve" inertia of' the appendages in the presence of'-f'reë attitüde m.otion of' the main body. The "constrained" motion equation of' the

appendages is obtained f'rom (27) simvly by 1etting ~ =.Q:

Slbd

+$d

=

0

- -c - - c (31)

where the subscript c is used to denote the constrained disp1acements.

The matrix

~

El

r-

1

a~is

positive def'inite because 9mis positive def'inite and

r-l.

is posItive dëfinite and diagonal (recall that-the axes of' f'rame

CD

coincide wi th the spacecraf't principal axes of' inertia) • According to Appendix C, the constrained f'requency Wc of' (31) is always 1ess than the

uncon-strained f'requency w of' (30). Consequent1y, the "e f'f'ecti ve" inertia can be

also termed the "reduced" inertia of' the appendages since the attitude motion has the tendency to "reduce" their inertia.

2.5 Torque-Free Motion With Stored Angular Momentum

This type of' motion is studied separate1y because, as wi11 be i11ustrated, the appendage and spacecraf't motion equations are coupled via

a complex term. The stored angular momentum will be shown to introduce addi tional

vibration modes which degenerate to 'rigid-body' modes as

!:s

tends to zero.

The spacecraft motion equation (11) can now be written as

The substitution of into

(32)

gi'lrs: d == d -0

e

==

e

- 0 iwt e iwt e

which clearly indicates that do and ~ are in general complex.

Let"

then

w

2 det H == w2 det I -

~

!

.!!s

Note from equation (36) that

det H

<

det I

because

!

is positive definite. For det ~

r

0, we can show

where

det I H - I-l.

-S = det

H-Then sol~ng (34) for

e ,

we find

- 0

e

- 0 14 T

!;s.!!s

w

2 _{det H}

(32)

(33)

(34)

(35)

(36)

(38)

On the other hand, premultiplying (34) by

~

we obtain the following relation

eT I e + eT

S

9lGd

=

0

- 0 - - 0 0 - 7 " " - 0 (40)

which is independent of~. Substituting (39) into (40) and equating to zero

the real and imaginary parts, we obtain

(4lA)

(41B)

T T T

where the properties

!

=!,

lis

=

!is

and

&.

=

-&.

have been used. Note that

when

ÈS

=

Q,

the terms inside the brackets in equations (4lA) and (41B) vanish.

For

.!;s

=f Q, the ampli tude ~ of the free oscil1ations should be such that (4lA)

and ~41B) are satisfied.

Using (33) and (39), equation (27) can be written as

Premultiplying (42) by

~~

and making use of (41B) we have

which indicates that only the 'in-phase' co~onent of

e

name1y

- 0

':I.' = -

H 9.9YGd

- 0

.:.:s - -

- 0

(42)

(43)

affects the free osci11ation. For this reason, instead of (42), we only need to consider:

- w2

3JG

d - w2

9rG

él

':I.'

+

X

d

=

0

- - 0 _ .- - 0 - - 0 - (44)

Expanding (43)

(w

2 det I -

~

!

.!:s)

!o

= [- w2 det(!) I-l.

+

.!:s

.

~]

g

~

~

(45)

we can combine

(43)

and

(44)

into

(46)

The angular momentum!!s therefo]'e affects . the s~iffness matri~of the a~pendage.

Note that when !!s

=

2"

the augmented matr1ces

!!s

!

~ and

!!s

h;;

g.

~van1sh. As

a re sult, the lowest frequency of the system tends to zero aslk tends to zero. The effect of the on-board angular momentum!!s on the spacecraft natural frequency

W is, in general, quite complicated. We choose to discuss further the three

special cases,

(i) det H = 0

(ii)

W~

=

~

!

hs/det

!

»W~

where wl.

=

lowest elastic natural frequency,

(iii)

Case (iii) is appropriate to the CTS simplified model, sowe leave the discussion

to Section

4.3.2.

From equation

(36),

it can be seen that the special case det H

=

0

implies:

w2 _{= h}T _{I h /det I b,. w}2

₍₄₇₎

.::s -

.:::s

-

= N

where wN is usually known as the 'rigid' nutation frequency. For det

.!!

= 0,

equation

(34)

simply becomes

and the interaction between spacecraft and appendage motions no longer exists.

Some "p'eculiar things" are therefore expected to happen. Indeed, since

e

and

~ are not dependent on each other at det

.!!

=

0, the unconstrained frequegcy should

be the same as the constrained frequency. The unconstrained frequency is the spacecraft frequency when the centre body is not constrained to be stationary. When the centre body is stationary, the frequency is called the "constrained"

frequency. The "peculiar thing" was described by Hughes

(7]

as: "When this

[Le. det H

=

0] happens, a unique situation exists in t:tlat all the flexible

elements deflect, but the mode shapes are such that the spacecraft motion is the same as it would be without fle:idbility, Le. the flexible motion does not exert any net torque on the main body".

16

r - - - --- - -

-2 . 2)

When the angular momentum is very largeT(wN »w~

,

most of the spacecraft momentum is contributed by

Ès

c.!?~

:

Ès Ès).

As a re sult , when

wN ~ 00, the attitude motion ~ tends to

2.

The, effect of the attitude motion

~ on the appendage displacement ~ becomes negligible. The centre body behaves as if it were held stationary. As expected, the unconstrained frequency tends to the constrained frequency as wN ~ 00.

2.6 Spacecraft Inertance and Modal Gains*

The attitude motion

e

has been shown to affect the appendage motion d (see

(30)

et seq.~ The flexible appendage motion is likewise shown to inf1uence the spacecraft attitude motion via the spacecraft inertence. The free oscillation frequencies

w,

the mode shapes. ~o obtained above are now used to describe the general spacecraft motion.

2.6.1 General Motions in Terms of Unconstrained Modes

We shall first consider the case

Ès

=

2.

Equation (8) leads to

•• ••

l

§. + ~~ ~

=

i

(49~

Let

§.2

= diag

(wi, ••• ,

w~), where w~

< •..

<

w

_{n are n natural frequencies of the}

spacecraft, then from equations (27) and (29) we cen define two eigenmatrices <l>

end

Q

by:

(50)

(51) More over , let .9. contain the generalized coordinates associated wi th structural deformations, end set.

(52) Then (49) cen be rewritten:

..

..

e.

l

~ +

(l

~ +

g.

~

Q)

3.

= l

~ = ~ (

53)

where the term inside the brackets vanishes because of

(50).

~(t) is therefore the response the spacecraft would have if it were rigid.

Turning now to the appendage motion equation, insert (52) into (27).tö find (54)

* This section very much resembles the enalysis in

[4].

In fact, it is intended as another version of Hughesi _{analysis when the flexibility is studied by the}

finite-element technique.

T

Premultip1ying (54) by ~ , we can show

where the fOllowing identi ty has been used:

(55)

Equation (56) is obtained by premultiplying (51) by QT. Using (50), equation (55)

can be rearranged to yield

(57)

The matrix DT'.9!& D - <'PT I <'P is diagona1 since, using customary procedures, we can

show the orthogonality condition' dT

9rb

d - 0 . - - 0 . ~ J eT I e - 0 . - - 0 . ~ J

o

(i

{ j )

where i and j denote two different modes. The diagonal e1ements of QT

~

Q -

~T

l

~

are dT $ d - 0 . - - 0 . ~ ~ = dT :Jf, d

/w~

- 0 . - - 0 . ~ ~ ~ (59) - eT I e o. - - 0 . ~ ~

which, incidentally, suggests two possible normalizations. A possib1e normality

condition is

dT SR; d = I

- 0 . -:- - 0 . a

~ ~

where Ia is an inertia moment charaderistic of the appendages. Then dT 3rr; d - 0 . - - 0 . ~ ~ - eT I e o. -:--0. ~ ~ (60)

is positive definite. according to (59), ,because

X

is positive definite. Alternatively,

one Can de fine another normali'ty condition as - .

which is equivalent to dT

~d

- 0 . - - 0 . ~ ~ I

e

=

I o. a ~ dT

!JC

d ' = I

w~

- 0 . - - 0 . a ~ ~ ~ (61) (62) It is then c1ear that, in (60), the eigenvectors do. are norma,lized in terms of

the appendage inertia and in (62), they are normaÏi~ed by means of the appendage

stiffness. Using (62), equation (57) becomes

In the fOllowing, an overbar will be used to denote the Laplace-transformed variables • Taking the Laplace transforms of (52) and (63) with zero ini tial conditions , we obtairr

-

-

-.~

=

~ + ~

s,

where s = Laplace transform (complex) variable. Eliminating S" we get:

ë

=

(Q

+

~2 ~[s2

Q

+

n2r:L~T

l)

-:L

ë

a

(64)

However, taking the Laplace transform of the rigid motion equation,

(53),

we have

(66)

Combining

(65)

and

(66),

S2 I

(s)

ë

=

t

-e - -

(67)

where Ie(s) is the transfer function between angular acceleration of the main body and the torques applied to the main body. It will be termed the I inertance I ,

and is given by

Recall that the eigenmatrix ~ is made up by the eigenvectors ~Oi:

cp

=

[8 8 • •• 8₀ ]

- -0:L -02 - n

then according to Appendix D we can write ·the modal gain matrix K. as -:L

K.

=

I-:L 8 8T I

-:L ·a - 0 . - 0 .

:L :L

and the system gain matrix ~ is

, 2.6.2 General Motions in Terms of Constrained Modes

(68)

(70)

We still consider the spacecraft motion equation (49) and the appendage motion equation (27) except that we now expand the displacement d in terms of

"constrained" modes ~c" The subscript c will be used to denote the "constrained" modes.

Let

d(t) = D q (t)

- -e-e (71)

where .Qe is the eonstrained eigerunatrix ob'tained from (31); i.e • .Qe satisfies the re1ation

S1GD .122 =

X

D

- -e -e - - c (72)

where ~ = di.ag (wie, •.. , w~c)' W~e

< ...

<

_{wnc • Inserting (71) irrto (27)}

we get

G'IIrD 'qt +CIC'D q ar. TOl

"'" _{- -e -e} .1\1 _{- -e -e} = - v.:1 g;

e

T

Premultip1ying (73) by .Qc we get

DT

~

DT

q

+ DT

~

D q

=

DT

9ló

D

(q

+ .122 q )

-c - -c -e -e - -e -c -e - -e -e e -e

Sinee the orthogona1ity condition for (31) is

dT

~d =

0

-oei - _Oej

- DT

9r&e:

ë

-e - -

-T

.Qe ~.Qe is diagonal. Simi1ar1y to (60) we ean normalize do aeeording to

- e

This is equivalent

To bring in the simi1arity between (77) and (63) we define*

so that (77) beeomes I

(~i

+ .122 _{q ) = q,T' I}

ë

~":='e e -c e -(73) (75) (76) (77) (78) (79)

Taking the Lap1aeè transforms of (40) and (74) with zero initia1 eonditions, we get

S2 I

ë -

s2 I q,

q

=

t(s)

- - - ~e -c - (80)

(81)

* This definition of ~c is artificia1 because the eonstrained mode disp1aeemerrt

do is independent of attitude ang1e.

- e

(80) and (81) are combined to give:

The inertance in this case is

Similarly to (69) and

(70),

in this case the modal gain matrix ~c is

T

K;C =

I

8

_{c '}

8

_{c .}

I

-'-

ac-~-~-, and the system gain matrix .!:.c is

2.6.3

Some Properties of System Gain Matrix

As noted in Appendix D, we can write

which can be expanded into

n \ ' K.

L

-~ i=l n

I

.!:.i =

I~~ ~ ~T

!

i=l n \ ' I-~ <p <pT I

L

.!:.ic ac c c -i=l

Firstly, since I is a diagonal matrix

-T

I

I

=

U

Secondly, from

(58)

and

(62)

we have

21

(82)

(83) (84)

(85)

(86) (86)

(87)

or

Hence

~ ~Q

Il

~i

=

(,g - ,(l. fA)-l. Ia fA

where fA =

~~i ha~

been used (see Appendix B).

Thirdly, from (75) and (76) we have

or

Hence

Q,

~

D

n

T 3f;

ll~

=

I I

- - -c -c - ~ ac-A

We can now rewrite (86) as

n

\'.. =

I-l.(U _ I-l. I )-l. I L!::.i - - - -A -A i=l n \ ' K'

=

I-l. I

L

~c - -A i=l (88)

(91)

(92) (93)

A relationship between the system' gain ;matrices! and ~c can be

established by observing that the inertance s.hould be independent of the modal

expansions, i.~.

!e

=

fec. Using ~s2) as defined.bY

(D-ll),

this equality

can be written as'

[_u

'

-+ s2 _K

_li]

LU -

s2 K A ]

=

U

- -ç-c (all s)

which can be simplified to:

K A - K A - s2 KAK A = 0

.... c -ç - - -c -c (all s) (95)

Finally, in the presence of stored ansular momentum (Le,

!!s

f

2)

the spacecraft

motion equation becames

- - - -- - - -- --- - - ---

-\

•

where Ie can be replaced by Ie because of

(94).

Thus, if 'inertance' is to be

- c - -

-the transfer function between

e

and t, we see that the inertance of the overall

spacecraft, in the presence of

Es,

iS-lees) - s-~~.

3.

DESCRIPTION OF CTS AND ITS IDEALIZATION

The analysis presented in Section 2 is now applied to a model of the

Communications Technology Satellite (CTS). CTS is an experimental satellite

designed for testing advanced space technology and is scheduled for launch early in

1976

by the Department of Communications. An artist' s conceptual drawing of CTS is shown in Fig. 1. Since an overall description of CTS has been given by Franklin and Davison [lol, the description below is confined to those aspects that are

germane to the main subject under discussion.

Designed to operate in geostationary orbit, CTS consists of a (relatively)

rigid centre body pointing to the earth and two large solar' arrays oriented towards

the sun. lts power needs will be primarily supplied by the two 21.5' x 4.25'

solar arrays which generate approximately 1260 watts. The spacecraft flexibility

with which we are concerned originates in these solar panels •.

As depicted in Fig.

3,

each solar panel has three maj or structural

members: a support boom, a pressure plate at the tip, and a blanket supporting

asolar cell array as schematically shown in Fig.

4.

(i) The support boom is of bi-STEM design. lt extends out from the spacecraft

body and supports the pressure plate at the end.

(ii) The pressure plate is attached to the solar panel and i t is designed to

distribute uniform tension to the solar panel. The pressure plate transmits the tension acting on the solar panel to the support boom as a compression. At the time of writing, the pressure plate is designed to be connected to the boom through a bearing which allows the panels to rotate freely without transmitting any appreciable torque to the boom. The pressure plate also carries thin cables on which the longitudinal edges of the panels are supported.

(iii) The solar cell array is a very thin, flexible blanket kept in tension by

the pressure plate. , The solar cells are glued to the blanket and the

blanket is folded in 'accordion' fashion before being deployed.

The solar panel is offset from the support boom to prevent contact with it. The boom is in the shadow of the panel; this provides thermal protection. Other meIDbers of each solar array include the telescopic elevation arms (in a V-shape)

and the pallet at the root of the array. These two s tructures can be seen in

Figs. 1 and

3 •

3.1 Idealized Model

The application now made of the analysis' presented in Section 2 to the

CTS model is intended to be illustrati ve rather than a sophisticated finite-element analysis of the CTS solar array. A much more detailed model can, of course, be constructed at the cos t of m9re computer time and storage. We are

going to idealize CTS in su eh a way that the analysis is quite relatively simple, yet i t contains important features of the array. The idealization used is the one employed by Hughes

[7,8,9]

in his continuum mechanies study of the CTS solar array. The results presented in the following sections will therefore have a twofold function, namely to act as an illustration of finite-element techniques, and to provide a comparison with continuum mechanies results for the same struc-tural model.

The following assumptions will be made in the subsequent analysis:

(1.1) is rigidly cantilevered at Hs root and i ts a.xis passes through the spacecraft centre of mass;

(1.2) has a uniform linear mass density, P, and uniform flexural stiffness,

EI;

(1.3) aspect ratio. is very small so that rotary inertia is negligible;

(1.4)

does not take part in twi sting .

(2) Pressure Plate (2.1) is rigid;

(2.2) the tension, P; is uniform.

(3)

Solar Panel

(3.1) The array root structure is rigid, i.e., the solar panel is rigidly cantilevered at i ts root;

(3.2) The solar panel has the same length and lies in the same plane as the boom;

(3

.3)

The panel has uniform area mass densi ty, 0-;

(3.4)

Flexural bending stiffness of the panel is negligible, its stiffness is

provided only by the tension, P;

(3.5) Only out-of-plane deflections are considered. Other assumptions are:

.

(i) Thermal stresses in the blanket and boom are negligible;

(ii) The moment of inertia

.!A

varies with time on a quasi-statie basis (the panels track the sun).

Justifications for all ass1..ur~rtions above have been discussed by Hughes [7}, so are omi tted here except for the justification of assumption

(3.4).

A similar solar array has been analysed by Coyner and Ross [11] using finite elements. In their model, the blanket was kept in tension by two supporting booms running

"

.

along the longitudinal edges of the blanket. By including the blanket flexural bending stiffness, they found that its contribution to the blanket natural frequencies is negligible. In practice, the bending stiffness of the blanket is indeed very smalle

The idealized model of CTS is shown in Fig.

4.

3.2 Fini te Elements and Nodal Numbering Algorithm

Under the assUIr!Ptions mentioned above, the solar array can be modelled by plate and beam elements. Under the loading P, the plate and beam elements have additional 'stiffness' which has been referred to by different names in the literature: membrane stiffness, initial stress stiffness, geometrical

stiffness. In this report, we adopt the name 'initial stress stiffness' as used by Przemieniecki

[13]

because it brings to mind the geometrical nonlinearity in the structure (the de"flections are large enough to cause ~ignificant changes in the geometry of the structure ) •

Using the convention that a tension is a positive load, the initial stress stiffness of the plate element in Fig. 6is found to be:

2

pO _{-2 2} Synnnetric

~=be (97)

-1 1 2

1 -1 -2 2

where pO is the uniform tension on the element and e"is~tl1.e length of the element. Tt should be noted that pO (on the ele~t) is different from.P (on the entire blanket). A detailed deri vation of (97) is shown in Appendix E. In the same appendix it is shown that the-inertia matrix of the plate "element is

O"ae =

3b

4

2 1 2

4

2 1 Synnnetric

4

2

4

where 0"

=

(area) mass density of -the plate element, a

=

width of the plate element •

•

The stiffness and inertia matrices of a beam element in Fig.

7

have been given by PrZ'emieniecki

[13];

his results are quoted below:

12

6L

-12

6L

Syxnmetric

=

elastic stiffness

=

EI L3 25

'"

_pO d, ~

~31t:

L

_~2

d2i

d

4

L

po

I·

L

·1

g,

ELASTIC STIFFNESS INITIAL STRESS STIFFNESS INERTIA

INITIAL STRESS STIFFNESS _INERTlA

112 (OL -12 6L

l

I

36 3L -36 3L

l

r

156 22L 54 -13L

[

,

-2 -1

.: 1

M

.~[:

Z

~

1

I

6L 4L2 -6L 2L' pO

I

3L 4L 2 -3L L21 22L 4L2 13L _3L 2 po -2 2

_•

4 2 EI PL ~Eal' K a- _{M • 420} K a _ -6L -6L -G 30L -3L -22L - 6e -: 2 -2 - 36 1 2 4 -12 12 -36 36 -3L 54 13L 156 -1 -2 2 2 2

I

_{6L 2L}2 4L2 _L 2 4L2

I

-13L _3L2 -z2L 2 -6L 3L -3L 4L

FIG. 6 PLATE ELEMENT TOTAL STIFFNESS ~ = !h + ~G

FIG. 7 BEAM ELEMENT

36

3L 4L2 Symmetrie

pO

~

== initial stress stiffness = _{30L -36 -3L 36} (100)

3L _L2 -3L 4L2 156 22L 4L2 Symmetrie

!%

_ pL _{- 420} (101) 54 13L 156 -13L -3L2 -22L 4L2

where p

=

(linear) mass density of the beam,

L

=

length of the beam element,

EI

=

beam bending stiffness.

The total stiffness

!B

of the beam is therefrire ~ +

!E.

The blanket is' discretized into m elements in the 2J. direction and

n elements in 22 direction. We leave mand n arbitrary at tB'is point since we will perform munerical experiments on the accuracy of our results in terms

of n and m. The numbering algorithm shown in Fig.

8

is found convenient for

assembling the blanket stiffness and inertia matrices. A similar scheme is also shown in the same figure for the beam elements.

4 • DYNAMECS OF THE SIMPLIFIED CTS MODEL

The analysis in Section 2 deals with a gener al CTS-like flexible

spacecraft. The stored angular momentum vector

!;s

can be ef any magnitude and

can point to any direction • The appendage structure is specified simply by the

inertia distribution 9fG and the elastic stiffness jr. Thus, the appendage

struc-ture can be made up by different types of finite-elements (e.g. beam elements, triangular or rectangular elements, etc). Tfese finite elements can, in general, have their nodal displacements along or withJ.n their boundaries. They can also

have different kinds of elastic deformations , °e.g. in-plane ar out-of-plane

bending deformations . Therefore, as mentioned in the Introduction, a computer

program can be wri tten to implement the analysis in Section 2 accounting for

various possible kinds of elements. If such a program is available, then to

simulate the motion of the simplified CTS model, one needs only to specify its configuration in terms of:

- the spacecraft inertia characteristics (inertia distribution.!, stored

momentum ÈS) ;

- the spacecraft elastic structure (beam and rectangular plate elements, their geometrie and elastic properties, and how they are assembled); - the external (and control) torque t;

- the initial conditions for

e

and d:

i3

~2

•

•

m

+t.

2m+2 3m+3

(Js!

_{(n+u (m}

-

₊₁₎

d

k+

_J

+m . dki'J+m+1 ~

d

k+

_J-

_t

o

d

d

,spo'

cF

P

+'

d

_2P

d

2P +2

ELEMENT

N!! P

ELEMENT N! k

(p

>

m nl

k= (j-1)m

+

i

=

1, •••

,m

j

=

1,. ..

,n

FIG. 8

NUMBERING ALGORITHM

The spacecraft attitude control system can be described by a separate module in the program which can be connected to the dynamics module. The 'modal gains' described in Section 2 will then be useful in this control systero representation.

The direct application outlinedabove will not be made in this report; in fact, the general simulation program has not been wri tten. We will, therefore, reduce the generality of the analysis in Section 2 to the particular case of the

simplified CTS model. One advantage of so doing is that i t will be possible

to illustrate a nuIDber of important points which are either hardto visualize or difficult to analyse in the more general context. More precisely emphasis

will be placed on the derivation of the nodal induced disPlacementsELT

e

(caused

by the spacecraft attitude motion) and the effect of stored angular momentum

È.S on the spacec~aft natural frequencies. Consequently, instead of extracting

this special case from the general analysis in Section 2, we will analyse the simplified CTS model directly. Comparisons will be made wi th the equations in

Section 2 where applicable.

The fOllowing analysis is confined to the free motions of the

space-craft. Spacecraft response to external and' control torques - through the

spacecraft inertance - is just a direct application of Section 2.6 once the

free motions have been analysed. To assess the accuracy of the finite-element

model, the effect on natural frequencies of the nuIDber of elements will be discussed. A more extensive treatment of this matter is presented in Appendix

H.

4.1 Basic Motions

As stated earlier , our present interest centres on the rotational (attitude) motions of the spacecraft, and with those motions of the flexible

arrays which have interactions with the spacecraft attitude rootion. SynJIIletry

allows us to define four 'types of flexible motions which (in a linear analysis)

can be superimposed to give the resultant rootion. These four types of rootion

are shown in Fig.

9

(af ter Hughes and Garg

[7])

as:

(a) symmetrie twisting, pitch angle

8

f

0,

(b) skew-symmetric twisting,

e

=

0,

(c) skew-symmetric out-of-plane bending, roll angle ~

f

0,

(d) symmetrie out-öf-plane bending, ~

=

0.

It is evident that only types (a) and (c) are of concern here. In fact, not

all four types need be studied since, if desired, (b) can be deduced froro(a)

by letting the body inertia

!B

tend to infinity. Notice from Fig. 9 that

symmetrie twisting is associated with spacecraft pitch motion, and that

skew-symmetrie out-of-plane bending is associated with roll motion. In the latter

case, if a momentum wheel is present (the CTS roomenturo wheel is aligned with

the pitch axis) , or if the array (local) axes

®

do not coincide with the

spacecraft (datum) axes

CD,

coupling between roll and yaw occurs. We assume

this to be the case in the subsequent analysis and Fig. 10 defines the two

reference frames

<D

and

®.

Finally, assumption (1.1) in Section 3.1 allows

array twisting and bending to be uncoupled and we therefore will consider the two following cases separately:

(i) array twisting coupled to pitching,

(ii) array bending coupled to rolling and yawing.

symmetrie twisting,

e

p 0

---

---•

skew - symmetrie twisting t

e

= 0

---

---_ (c) skew - symmetrie

/

---out -

rA -

plane bending t ~ #-0

---;

---/

---

_--

_---

---

---out - of - plane bending t ~ • 0

FIG. 9

FOUR BASIC TYPES OF FLEXI BlE MOTIONS

--PITCH AXIS

lroll)

FIG. 10 REFERENCE FRAMES

FIG. 11 SOLAR ARRAYS IN TWISTING MODE

I spocecroft in pitch )

FIG. 13 SOLAR ARRAYS a BOOMS IN BENOtNG MODES

4.2 Motion Equations

In this section are derived the motion equations; numerical results and discussion are then presented in Section 4.3.

4.2.1 Array Twisting and Pitch Attitude

First note that twis ting of the array does not -transmi t any torque

to the boom because of a bearing mounted at the boom tip. In fact this is the justification for the assumption (1.4) in Section 3.1. Deformations

there-fore occur in the blanket only. The inertia of the array, however~ is the

inertia of the blanket and the pressure plate since the (rigid) pressure plate is in motion wi th the blanket.

Due to the symmetry of the blanket about 22' we will consider only one half of the blanket. Using the numbering algorfthm in Section 3.2, we can form the assembled matrices*

M

=

inertia matrix of one half-array including tip mass

and K = stiffness matrix of one half-array

as shown in Appendix I. The pressure plate is represented by beam elements whose inertia is determined by

I

[

156

2i-

54

54 ]

156

(102)

where p

=

pressure plate mass and m

=

number of eleIlEnts along

St:

(for one

half-array). Equation (102) is (101) af ter the two degrees of f'reedom associated with rotational nodal displacements having been eliminated.

To be compatible with the continuum m.echanics analysis

(7,8]

we impose

a cnndition on the nodal displacements that all displacements along 2). to be proportional to the distance from the 22 axis. For example, at constant y

(Fig. 12) ~ d. x. ). _l. di +1 xi +1 2d. l.+m 2x di+m (103) = = -w w di+m _{xi +}

m

where d. ,

l. ••• , di+m are the noda1 displacements at y = constant with di+m being

at the edge of the blanket,

xi' ••• , Xi +m are the x- coordinates of the nodes i, •.• i tm,

w

2'

=

half of the blanket width

=

Xi im •

* Rigid degrees of freedom along the blanket root are deleted. 32

nodol displocement

induced

displacement (due to pitch)

FIG. 12 INDUCED DIS PLACEMENT

a

CONSTRAINT ON NODAL DISPLACEMENTS

( soIor arrays in twisting mode)

nodal

displacement

induced

displacements ( due to roN. yaw)

FIG. 14 INDUCED DISPLACEMENT

a

CONSTRAINT ON NODAL DISPLACEMENTS