Dynamical model for the CTS development model solar array under the influence of gravity

3 ArR. 1978

DYNAMICAL f()DEl FOR THE CTS DEVElOPMENT KlDEL SOLAR ARRAY

UNDER THE INFLUENCE OF GRAVITY

TECHNISCHE HOGESCHOOL OELFT

LUCHTVAART- EN RUIMTEVAARTTECHNIEK BIBLIOTHEEK Kluyverweg 1 - DELFT

3

~t?.

'919

by

G. B.

Sincarsin

DYNAMICAL MODEL FOR THE CTS DEVELOPMENT MODEL SOLAR ARRAY UNDER THE INFLUENCE OF GRAVITY

Submitted Octobèr,

1976

May,

1977

by

G. B. Sincarsin

UTIAS ~echnica1 Note No. 206 CN ~SSN 0082-5263

- - - -- -

-Acknowledgement

The author is deeply indebted to Dr. P. C. Hughes for his supervision and guidance during the preparation of this work. His patience and understanding of the problems involved in writing the manuscript are also appreciated.

The author is a1.so grateful to M. Jankovic who individually rederived

the slightly-modified zero-gravity model presented in this technical note. Ris work was performed simultaneously with that of the author and as such provided a

check on the final equations of motion for the zero-gravity case. Thanks are

also extended to 8. C. Garg for permitting the use of his matrix element subroutines in the computer program for the zero-gravity case.

The DMSA and test facility, from which the experimental results were

obtained, were graciously provided by the 8pace Mechanics Group of the Communications Research Centre in ottawa.

Financial assistance for this undertaking was provided by the National Research Council of Canada (Operating Grant No. 8-7309).

The author would also like to thank Mrs. L. Q,uintero and Mrs. W. Dillon for their preparation of the figures and the manuscript.

Summary

A detailed model for the CTS (Communications Technology Satelli te) development model solar array (DMSA) is presented, accompanied by models for the dynamics of the array in both normal-gravity and zero-gravity environments. The major objective of the mathematical derivations is' the development of the gravity-associated terms in the model; this is aimed at providing a basis for comparison with the author's ground testing of the solar array. The earlier (zero-gravity) model of [Hughes-l] is extended for this purpose. Two mathematical approaches, vectorial mechanics, and variational principles, are used to deri ve the equations of motion for the DM3A under the influence of gravity. r;rhe equations resulting from each method are identical, providing a mutual check. A comparison is made between the predicted natural frequencies in the one-gravity and zero-gravi ty environments. The latter values were about 5Cf1/o lower - an expected result.

Experimental results from ground tests are compared wi th theoretically predicted natural frequencies from the gravity-included model, and the agreement is found to be within

7%

for the lower modes of vibration, and within

16%

overall. The theory tends to underestimate the measured values. The relative ordering of the out-of-plane and in-plane boom fundamental frequencies is also successfully predicted by the analysis.

I. ' Ilo

III.

IV.

CON.rENl'S

Acknowledgement

Summary

Notation

INl'RODUCTION

I.l

Mbtivation

I.2

MOdelling Decisions

I.2.l

I.2.2

I.2.3

I.2.4

Constrained vs. Unconstrained MOdes

Array MOdel

Array Deflections

Mathematical Approaches

TENSION AND COMPRESSlON EXPRESSIONS

lI.l Variable Components

ll.2 Constant Components

lI.3 Final Expressions for Tension and Compression

VECTORIAL MECHANICS APPROACH

lII.l Derivation of Component Equations

IIl.l.l Boom Equations

IIl.l.2 Blanket Equations

IIl.l.3 Twist Control Line Equations

IIl.l.4 Pressure Plate Equations

lII.2 Derivation of Equations of MPtion

IIl.2.l Summary of the Component Equations

III.2.2 Reduction of Component Equations to Final

Equations of Motion

IIl.2.3 Summary of Equations of Motion

VARIATIONAL

PRINCIPLE APPROACH

IV.l Derivation of Kinetic Energy Expressions

IV.l.l

IV.l.2

Component Kinetic Energy Expressions

Final Kinetic Energy Expres si on

,

IV.2 Derivation of Potential Energy Expressions

ii

iii vi 1 1

3

3

3

8

9

'

9 11 11

13

17

17

17

19

25

25

29

29

31

36 38 38 38

39

40

IV.2.1

IV.2.2

IV.2.3

IV.2.4

IV.2.5

Boom Potential Energy Expressions

41

Blanket Potential Energy Expressions

43

Twist Control Lines' Potential Energy Expressions 47

Pressure Plate Potential Energy Expressions

49

Final Potential Energy Expression

50

V.

VI.

Dl.3

Derivation of Equations of Motion

Dl .3.1 Áw1ication of Variational Principles to Find

Page 51

Final Equations of Motion 51

Dl.3.2

Summary of Equations of Motion

57

SOLUTION OF THE EQUATIONS OF MOTION

V.l A-Mode Natural Frequencies

V.2 B-Mode Natural Frequencies

V.3 Discussion of Numerical Result(s with Comparison to

Values Obtained During Ground Testing

V.3.1 Numerical Procedure for Determining the Natural

59

59

61 63

Frequencies for the A- and B-Modes 63

V.3.2 Numerical Results and Discussion 64

CONCLUDING

REMARKS

REFERENCES TABLES

APP]!:NDIX A - DERDlATION OF INERTIAL AND EXTERNAL FORCE TERMS APPENDIX B - DERDlATION OF GENERAL KINErIC AND POTENTIAL ENERGY

EXPRESSIONS

APPENDIX C - TYPICAL DISPLACEMENTS IN THE Y-DIRECTION APPENDIX D - RElmCTION OF THE DIFFERENI'IAL EQUATIONS APPENDIX E - GRAVITY-INCLUDED COMPUI'ER PROORAM

APPENDIX F - ZERQ-GRAVITY COMPUTER PROORAM

v

,

upper Case Roman A AA BB C C{y) D

..

F{S)

" 7 7 No"tation

Mode shape for v (y,t) in nth constrained mode, see

x (v.14)

Cross-sectional area of the ith cqmponent

Coefficient matrix of unknown cQnstants relating the

A{Y) and A' (y) varia,bles evaluated at y = d and y = ~,

see Appendix D

Coefficient matrix for the A~modes, see (V.12)

Flexural stiffness {EIA)o of support boom with respect to out-of-pla,ne deflections

Flexural stiffness (EIA)I of support boom with respect to in-plane deflections

Coefficient matrix for the B-modes, see (v.26) Constant compression in support boom in absence of

gravity, C

=

P + 2p

Compression expression when array in presence of gravity, see (II.15)

Variable component in C{y) expression, see (II.ll)

Total constant component in C{y) expression, see (IL10) Constant component No. 1 in Cc expression, see (IL5) General coefficient matrix of unknown constants relating

!'(YI) and ~(YF)' see Appendix D

jth column of

Q

matrix, see (D.18)

Young' s mod\llus for the i th component

Lagrangian component functions, see

(rv.49,

50, 51)

x, y and

z

components of force of pressure plate on

support boom

F., F. -,l. l. I I , "'7 Ol. I",. ïlPl.

~z

L M xs M(e) ~ ""7

F or ce vec or ac t t ' l.ng on l. ,th componen t 0 f array, and

corresponding magnitude

Functional resulting from integration of Lagrangian with respect to time, see (IV

.54)

Area moment of inertia of support boom about x and z axes (Le., IAx :;:: IAz

=

:J:

_A)

Inertia dyadic of i th array component about the origin of the x" y" z, reference frame, see Fig. A-1

]. ].' ].

Inertia dyadic about centre of mass of the i th array component, see Fig. A-l

MOlllI:mts of inertia of pressure p1ate about the x t ' Yt and Zt axes re spe ctively , see Fig.

1-4

Moment of inertia of inboard-pal1et/e1evation-arms

sub-assembly about an ax:is parallel to the z-axis passing through its centre of mass, see (III.112)

Moment of inertia of blanket about an axis parallel to the z-axis passing through its centre of mass, see (III.112) Moment of inertia of array-minus-boom about z-axis, see (III.112)

The Lagrangian, L

=

T - V

s s

x, y and

z

components of the moment exerted by the pressure plate on the support boom

stat~c bending moment about the x-axis, see (II.16) In-plane and out-of-plane bending moments of support boom about an arbitrary point R, see Fig. 11I-1 Arbi trary moment vector depending on the rotation vector

e

~

~i'

Me

P Q R., R_. 7'1. "7"1t'~ R

s

T s T(y)

u (y), U(y)

n .th

Moment vector acting on ~ co;mponent of array, and corresponding magnitude

Constant tension in blanket in absence of gravity Arbitrary reference point

Position vectors from the origin of the inertial frame, th

to the origin of the i component x., y., z. :rotating

~ ~ ~

frame, to the infinitesimal mass element of the ith component, and to the mass centre of the ith component, re spe c'tively , see Fig. A-l

th

Position vectors from the origin of the i component xi' Y i' zi rotating frame to an infinitesimal mass

th

element of the i · component, and the mass centre of the th

i element, respectively, see Fig. A-l Arbitrary reference point

Arbitrary reference point

Total kinetic energy for the entire system (i

.e.,

the array), see

(IV.2)

Kinetic energy for the ith component of the array, see

(IV.4)

Tension expression when array in presenc~ of gravity, see

(II.14)

Variable component on T(y) expression, see

(II.8)

Total constant component in T(y) expression, see

(I1.7)

Constant components Nos. 1, 2 and

3

in the Tc expression,

see

(11.3,

4, 6)

Mode shape for u(y, t) in

n

th constrained A-mode, see (V.l)

UC

u

_s

v

w (y), w(y) n

wc

w

s

w.

~

Lower Case Roman

b

Coefficient matrix of unknown constants relating the U(y), U'(y), U"(y) and U"'(y) variables evaluated

at y

=

0 and y :: 1" see Appendix D

Tota.l strain energy fqr the entire array, see (B.15) strain energy for the i th component of the array, see

(IV.15)

M d o e s ape h for V (y, t)

~n

.... nth cons t · d A ra~ne -mo e-, see d

o

(V.l)

Coefficient matrix of unknown constants re lating the

V(y) and V'(y) variables eva.luated at y :: d and y :: 1"

see Appendix D

Tota.l potential energy for the entire array, see (IV.3) Potential energy for the i th component of the array,

see (rv.14)

. th

Mode shape for w(y, t) ~n n constrained B-mode, see

(v.14)

Coefficient matrix of unknown constants relating the

W(y), W' (y), Wt _{, (y) and W" t (y) variables evaluated}

at y = 0 and y :: 1" see Appendix D

Tota.l externa.l work done on entire array, see (B.15)

External work done on the ith component of the array,

see (rv.16)

th

Variable coefficients of a genera.l morder linear homogeneous ordinary differential equation, see (D.l) Thickness of blanket

th Arbitrary constants in the solution of the general m

order homogeneous ordinary differential equation, see (D.2) ix

d df.

(ei,

t)

"'7J. ? r e f s g h

Centre of mass of ith array component

Distance between s'UIlPort boom root and inboard pallet, see Fig. 1-3

m constants corresponding to the kth derivative form th

of the general sol~tion to the general morder

linear homogeneous ordinary differential equation, see (D.6)

nf · . t . mal 1 t f · th t

I J.m esJ. . mass e emen 0 J. array componen , see'

.'~

Fig. A-l

8 and r components of the differÈmtial pa.th· vector

dB, see Fig. B-l

...,

General inertialforce on an element of ma~s at position

th .

.!;,

of the i array component, see (A.l, 3,~

5)

..

i '

Magnitudes of the x, y and z components or' the inertial th

force acting on the i array component

Magnitude of external force, due to gravity acting in

the y-direction, on the ith component of the array

Distance between support boom and blanket, see Fig. 1-3

Constant negator-spring force

Acceleration due to gravity, g = 32.174 ft/sec2

Height of the pressure plate, see Fig. 1-3

Boom root stiffness with respectto out-of-plane defleC'tion

Boom root stiffnefis with respect to in-plane deflection stiffness of elevation arms with respect to in-plane deUection, see (III.63)

,ga,

dtl. m p A r, r, r ~ "7 r. ~l. S, ds, s, ds . , . "'7 s ...,0

stiffness of boom tip wi'th respect 'to in-plane deflection,

see (111.72)

Boom torsional s'tiffness, see (111.58)

Length of support boom

Arbi'trary reference leng'th and cor;responding .~fferential,

see F;ig.

c-

3

Order of 'the general linear hamogeneous ordinary differen-tial equation given by (D.l))

'th

Mass of the i componen't of the array

Lumped mass of inboard-pallet/ elevation-arms sub ... assembly Mass of pressure plate

Tension in each . ·twis't con'trol line, both in presence and absence of gravity (Le., lines assumed massless) A general vector in a rota'ting reference, see (A.4) Ma'trix of 12 generalized c~ordinates, see (IV .52)

General position vector, corresponding unit vector and magnitude

Undeflected port;ion of the posi'tion vec'tor~i' see

(A.6)

General displacement vector and corresponding differen-'tiill plus appropriate magnitudes, see Fig. B-l

General displacement vec'tor and correspond;ing differ-th

ential for the i array component, plus appropriate magnitudes

Arbitrary starting position of the gener al displacement vector ~

Magni tudes of two arbitrary pos i tions along the general

di splacement vec tor s

_-,

xi

t to' ti u(y,

t)

v(x, y,

t)

v

(y, t)

o v

(y, t)

x v~(y, t), va(y, t) va vb w w(y,

t)

x,

y,

z

x.,

y.,

z.

~ . ~ ~ y(;x:) , x(y)

~(YI)' ~(YF)

x(YI) ., x(YF)· - J - J

Out-of-plane, twist and total slopes for the number 1 twi-st control line, see (III.30, 31, 32)

Independent variable, time Arbitrary points in time

Out-of-plane deflection of boom

Dynamic out-of-plane deflection of blanket, i.e., deflection is taken relative to ~, Yb' zb axes

Component of v(x, y, t) due to out-of-plane motion of array, see (III.13)

Component of v(x, y, t) due to twist motion of array, see (III.13)

Dynamic out-of-plane deflections of the number 1 and 2 twist _{control lines, i. e., deflections are taken}

relative to ~, Yb' zb axes, see (lII.33, 34)

Matrix of mode shape variables, evaluated at y = 0, d and

t,

for the A-modes, see (V.ll)

Matrix of mode shape variables , evaluated at y = 0, d

and

t,

for the B-modes, see (V.25)

Width of blanket (pressure plate and inboard

1

pallet/ elevation-arIDS sub-assembly)

In-plane deflection of boom

Inertial reference frame, see Fig.

1-4

th

Rotating reference frame for the i component, see Fig.

1-4

Arbitrary general functions

Boundary value vectors for the function x(y), see (D.9) Special assumed boundary value vectors for the function x(y), see (D.17) and (D.18)

Zt

yPper Case Greek

Lower Case Greek a(t) ~(t) 8, de, 8, d8

...,

"

-'

8., dB., 8., de; "7~ . ..,~ ~ oL 8 -..,0 Arbitrary y vaJ.ues

Initial and finaJ. y values for the interval of integration used to reduce the differentiaJ. equations of motion, see Appendix D

Z component of pressure plate mass centre

th

ModaJ. ~litude for ~(t) in n constrained B-mode~

see (v.l4)

th

Natural frequency of the n constrained A-mode Natural frequency of the nth constrained B-mode

Twist angle of pressure plate wi th respect to the inboard pallet, about Yt axis

In-plane deflectio~ (rotation) of array-minus-boom Shearing strains in the xy, yz and xz directions, see

Deflection portiqn of the position vector R., see (A.6)

. . -?~ .

Strains in the x, y and Z directions , see (B.l. 7) Arbitrary dummy variable, linear or angu!ar

General angu!ar displacement vector and corresponding differential, plus appropriate magnitudes, see Fig. B-l

GeneraJ.. angu!ar displacemenb vector and corresponding differential for the ith array componE;lnt, plus appropriate magnitude

Arbi trary s tarting angle of the genera! angu!ar displace-ment vec tor ~

!let) p cr , cr "., cr x y Z T xy' ',_, T yz' T XZ

x(t)

w . ..,.~ Sub scripts (

.

) B ( . )b ( ')p (')t (')tc (. )pbt (.

\

( • ) j

Magnitudes of' two arbitrary angles f'or the general

angular displacement vector

e

" -::-0

Angular def'lection of' the pressure plate-boom "tip

torsional spring, see (III.50) Boom mass/length

Blanket mass/area

Normal stresses in the x, y and z directions , see (B

ol

?,

)

Shearing stresses in the xy, yz and xz directions , see

(B.:!:.7)

th

One independent solution to the general m -order

linear homogeneous ordinary ~f'f'erential equation given

by (D.l), see (D.2)

Rotation of' pressure plate about the Zt -axis

Angular velocity vector f'or the i th array component,

see Fig. A-l

Boom Blanket

Inboard-pallet/ elevation~.e.rms sUb-assembly

Pressure plate

Twist control lines

Arr ay-minus-b oom

.th t

~ array componen

j th sub component of' i th array component or dl.llIlDly

sub-script with range f'rom 0 to m

Dummy subscript, ranges f'rom 0 to m

Deri vat i yes

-(

.

)

(0:)

o other Notation 5(-) (.;)

CJ

Abbreviations AMB CTS DMSA

lP

OP

T Mass çentre

Rotating frame origin

Deriva.tive with respect to time

Time derivative of a vector as seen by an inertial observer

Time derivative of

a

vector as seen by a rotating observer

Derivative with respect to y

kth deri vati ve wi th respect to independent variable

lst variation of the given quant~ty

Vector or dyadic Matrix

Array-~nus-boom

CommuniGations Technology Satelli te Development Model Solar fV:ray

ln-plane Out- of -plane ';rwist

I. INTRODUCTION I.l Mot i vation

Versatile communications satellites capable of performing varied

tasks in conjunction with mobile ground stations~ of ten in remote and isolated:

areas of the world, are a present-day reality. The power requirements for such satellites are, however, very demanding. Due to the rather low efficiency of solar cells in converting the sun' s radiation into elec'trical power i t

becames necessary to design spacecraft with surface areas large enough tp

acco~date a great number of celIs. Since one of 'the foremost concerns in

spacecraft design is to minimi~e weight, in ~ attempt to limit launching cos'ts,

i t is no't surprising that many designers have opted for incorporating new

light-weight 'flexible I solar panels in their next generation of conununications

satellites, rather than using heavier traditional 'rigid' panels.

The CTS (Communications Technology Satellite), a joint space venture by Canada and the U.S.A., reflects this recent trend. As canbe seen by Fig. I-I, the spacecraft consists of a rigid central main body flanked by two symmetri cally deployed solar panels. The inherent struc'tural flexi bili ty in

such a design, howeyer, poses many problems, not the least of whicn is how to

conduct dynamical ground tests. One possible solution, and the one chosen in the case of CTS, was to test specific sections of 'the spacecraft in such a manner as to minimize 'the effect of gravity on that section. The results from the separate tests were then incorporated into a comprehensive dynamical model öf 'the entire satellite.

Another problem, of no less importance, is how to mathematically model such a spacecraft during these ground tests. Models ter the on-orbi t behaviour of such satellite designshave not been corroborated by extensive

flight data; it therefore becomes imperative to develop a.means for che~ng

these models prior to launch. Given a mathematical model of the sa'tellite

during ~round testing, and assuming that this model agrees favourably wi'th

results from such testing, it is reasonable to expect ·that the corresponding

on-orbit model, which is ob'tained by let'ting gravity go to zero in the gravity-included (ground-test) model, should predict the on-orbit behaviour of 'the spacecraft •

This 'technical note is based on the validi ty of this premise. At present, in the literature a zero-grayity model, and a gravity-includeq model which can be reduced to a corresponding zero-g;ravi'ty model. as sugges'ted above, exist and are given in [Hughes-l] and [Vigneron-l] respectively. Two sets of experiment al results from dynamical ground tests of 'the DMSA are also available and can be found in [I!arrison] and [Sincarsinl. While a preliminarycomparison between the ground test results from [Harrison], and the corresponding predicted

theoretical values from [Vigneron-l] has been ,performed for an early version of

the DMSA, the test results found using the final version of the DMSA [Sinearsin], have not yet been compared with theoretical predictions. To facilitate this

situation, it was decided to extend the zero-gravity model given by [Hughes-l} to include gravity. This provided a model with which to compare the results given by [Sincarsin1, as weIl as providing a mathematical model based on

continuum mechanics wi'th which to compare 'the discretiz.ed energy model given

FIG. 1-1 ARTIST'S CONCEPTION OF CTS

a slightly modified version of the zero-gravity model given in [Hughes-l}, the modified zero-gravi ty model itself, and a comparison between predicted and

experiment al ground test reS1D.ts would provide the background material necessary to support or deny the aforementioned premise, when a sub st ant i al amount of flight data finally becomes available. It is the prime motivation of this note to provide this background material.

Since sub st ant i al effort has been directed towards deriving a zero-gravity model [Hughes-l] and experimental results are readily available in

[Sincarsin], the majority of this note is concerned with the derivation of the gravity-included model.

1.2 Modelling Decisions

1.2.1 Constrained vs. Unconstrained Modes

It is possible to describe the motion of the CTS spacecraft in terms of two distinct types of modes. One type, the 'constrained' modes, require that the attachment point of the array to the spacecraft remains stationary, both in translation and rotation, while the other type, the 'unconstrained' modes, permit this point to move [Hughes and Garg}. While the choice of mode type is usually left to the discretion of the person modelling the satellite, in this work, the choice is governed by the restriction that a comparison of theoretically predicted values to their experimental counterparts is to be made. Hence, the modes chosen must be compatible with the ground-test situation. In the ground-test configuration the DMSA was hung vertically downward in a vacuum

chamber. In particular , the attachment point of the array was fixed to a mechan-ical shaker which was neither free to rotate nor free to translate when at rest. As a re sult i t is natural to model the DMSA in terms of constrained modes. The fact that during forced exci tation the shaker ei ther translates or rotates does not invalidate this conclusion, because the shaker itself is not free to respond to the forcing input. Therefore, the entire response to this excitation is observed as array deflections. Thus the constrained-mode analysis for free Vibration will yield natural frequencies which are theoretical predictions for the resonant frequencies found during ground testing.

1.2.2 Array Model

The model and individual parameters for the array are described below. The actual numerical values for parameters are provided in Table 1. Figures 1-2 and 1-3 show actual and schematic illustrations of the DMSA, respectively. Figure

1-4

gives the sign conventions used, defines the x-y-z axis frame (assumed inertial for present purposes), and indicates the three types of array motion, twist (T), in-plane (lP) and out-of-plane (Op). In fact, the array will be considered to undergo only two distinct mot i ons , namely, OP vibrations and TjIP coupled vibrations. The so-called A-modes of

Vibration correspond to OP motions, while B-modes correspond to TjIP motions . Much of what follows is paraphrased from [Hughes-l].

(i) Each array is thought to consist of five components, namely, a Bi-STEM boom (subscript 'B'), a blanket (subscript 'b'), a pressure plate at the outboard tip (subscript 't'), a palletjelevation-arms sub-assembly at the inboard end (subscript 'p') and a pair of twist control lines, one along each edge of the blanket (subscript 'tc'). A sub-system

( IN-PLANE)

ELEVATION ARMS

INBOARD PALLET

TORS ION LINE TAB - - - - 4 l I I

FOLD LlNE

TWIST CONTROL LlNES·

PRESSURE PLATE BLANKET CONNECTION LlNE (Hidden)

PRESSURE PLATE

NEGATOR-SPRING

TIP -TENSIONING MECHANISM

BI - STEM BOOM

INBOARD PALLET BLANKET CONNECTION LlNE

-I:N~---HINGE LlNE

BLANKET

DUMMY SOLAR CELLS

~---=:::::~~U---POWER

AND SIGNAL CABLES

TWIST CONTROL LINE

~~mt~\l:::::::::::::::~-ACCELEROMETERS

TWIST CON TROL LlNES TENSIONING MECHANISM

FIG. 1-2 THE CTS DE~LOPMENT MODEL SOLAR ARRAY (DHSA) (AFIJ.'ER SmCl\Rf~m)

MECHANICAL SHAKER (RIGID) MECHANICAL SHAKER (RIGIDl

k

_I

,k

₂

T-d

1~~=~

11

I1

11

11

11

11

11

I1

11

I1

I

IN BOARD PALL ARMS SUB-ASSEMBLY

(mp)

~-BLANKET--~ (CTW({-d») ...,,--- BOOM

(pt)

,----1H---1It--~ TWIST CONTROL---...

I~

LlNES

PRESSURE PLATE

(md

GRAVITY

FIG. 1-3 SCHEMATIC OF DHSA ~10DEL

Z, Ze,

zb,

Zp

Zt

t

Out-of-Plane (OP)

f=~-@

B~\ ~~::s;~~~,

.

,,~,~

4

(

~y

,

Ye

x,xe,Xb'X

p

x-y-z

Inertial Frame

Xj- yj-Zj Component Frames

FIG. 1-4 COORDINATE SY$TEMS (AFTER HUGHES-l)

which will prove usef'ul is called the array-minus-boom (AMB) sub-system consisting of' the inboard-pallet/elevation-arms sub-assembly, the blanket and the pressure plate (subscript 'pbt').

(ii) The boom is modelled as a long slender rod of' constant circular

cross-section and length ~; the blanket is modelled as a thin sheet of' length

(~-d) and width w; the pressure plate is modelled as a rigid plate of' height h and width w; the inboard-pallet/elevation-arms sub-assembly is modelled as a bar of' width wadistance d f'rom the boom root; the twist

control lines are modelled as f'ine wires of' length (~-d). The blanket

is of'f'set a distance e f'rom the boom.

(iii) The boom has a unif'orm mass per unit length,

p;

the blanket has a

unif'orm mass per unit area, rr; the pressure pLate has mass mt, unif'ormly distributed over its area hw; the inboard-pallet/elevation-arms

sub-assembly has a combined mass llIp distributed unif'ormly along i ts width

w; the twist control lines are assumed massless, as is the boom-tipi pressure-plate connection.

(iv) The boom has a f'lexural stif'f'ness Bo with respect to def'lections

parallel to the z-axis (OP), and a f'lexural stif'f'ness BI with respect

to def'lections parallel to the x-axis

(IP).

The root has rotational

stif'f'nesses ko and kI, corresponding respectively to OP and lP motions. The boom has a torsional stif'f'ness k4.

(v)

The inboard--pallet/elevation arms sub-assembly is considered rigid in

the OP direction. However, this sub-assembly can rotate during T/IP mot i on; the rotational stif'f'ness of' the elevation arms f'or this motion

is denoted by k2 •

(vi) A rotational stif'f'ness, k_{s ,} is assumed to exist betweenthe boom and

pressure plate at the tip of' the array where they join.

(vii) The blanket is in a state of' unif'orm tension across the width w of' the array with the total tensile f'orce, in the absence of' gravity, P, directed along the blanket parallel to the y-axis. The tension is supplied by a negator-spring tip-tensioning mechanism.

(viii) The two twist control lines are each under a constant tension, p, provided by a second negator-spring system.

(ix) As a consequence of' (vii) and (viii) the boom is under a compression

of' P

+

2p, in the absence of' gravity.

(x) The pressure plate centre of' mass is of'f'set a distance Zt f'rom the boom.

Itx, lty' and ltz are the inertias of' the pressure plate about an axis through its centre of' mass parallel to the x, y and z-axes, respectively.

(xi) _{The inboard-pallet/elevation-arms sub-assembly has an inertia I pz}

about an axis parallel to the z-axis and passing through its centre of'

mass.

(xii) The blanket has a moment of' inertia of' Ibz about an axis parallel to the z-axis and passing through its centre of' mass. The inertia of' the

(xiii) The small inplane offset of the mass cent re of the pressure plate is neglected in order to uncouple OP and T/IP motions.

I.2.3 Array Deflections

In the calculation of natural vibration modes, all deflections of the array are considered to be elastic and non-dissipative in nature. It is also assumed that the deflections are small (first-order) quantities, the products of which can be neglected in order that the final equations of motion

are in linearized form. An important consequence of this fact is that the principle of superposition can be applied.

This consequence permits the division of the array motions into A-and B-modes, the actual general deflection of the array being the superposition of these two uncoupled sets. Even more basic, however, is the ability to

isolate dynamic and static deflections by applying the principle of superposi .. tion. Obviously under the influence of gravity a complicated structure such as the DMSA will experience static unbalanced forces, and mommts which will result in initial deflections prior to vibration. Therefore, the total deflection when the array is in motion includes this static component plus a dynamic component. For present purposes, the dynamic component is the only one of interest, and as a re sult the static components are removed from the analysis by subtraction whenever they appear.

Now considering Fig.

1-4

it is possible to describe the various deflections of the array components. The boom is capable of deflecting in either the z or x-directions. Boom deflections parallel to the z-axis (OP) are denoted by u(y,t), while deflections parallel to the x-axis (lP) are denoted by w(y,t). The boom also twists through an angle aCt) about its central axis, which corresponds to the y-axis when the boom is not in motion. The blanket is capable of a combined deflection, due to OP and T motions, given by the quantity v(x,y;t). As part of the AMB the blanket also expe-riences a rigid rotation t3(t) about the z-axis, which represents an lP motion. The two twist control lines are assumed to deflect as straight lines, with the deflection of each line, v~(y,t) and V2(y,t), containing components which arise due to both OP and T actions. As can be seen the principle of super-position has been used in defining the blanket and twist control line deflec-tions. The pressure plate undergoes a rigid rotation aCt) about the Yt-axis, which is parallel to the y-axis and passes through the intersection point of the blanket connection line, which is a distance e above the y-axis, and the line which bisects the width of the pressure plate, and is parallel to the z-axis. The pressure plate also goes through a rotation

x.(

t) ab out the blanket connection line, or xt-axis. Since the plate is attached to both the boom and blanket the OP deflections of this component can be given in terms of the boom or blanket quantities. The AMB motion, a t3(t) rotation about the z-axis, provides the lP motion of the pressure plate, as weIl as the lP motion of the inboard-pallet/elevation arms sub-assembly. As stated previously this is the only motion permitted to this sub-assembly because i t is assumed rigid in OP and T.

Before leaving the topic of the array deflections, it is best to define the notation used in conjunction with these quantities to indicate different types of differentiation. Differentiation with respect to the spatial variabIe y.will be denoted as ( )', while temporal differentiation will be given by ( ).

1.2.4 Mathematical Approaches

"

Two different mathematical approaches are available for describing

the dynamics of DMSA as suggested by [Lanczos]: vectorial mechanics, and variational principles. He further summarizes ;four principal viewpoints in which vectorial and a.'Y1alytical (or variational) mechanics differ:

(1)

(2)

(3)

(4 )

"Vectorial mechanics isolates the particle or body and considers i t as

an individual; analytical mechanics considers the system as a whoIe. Vectorial mechanics constructs a separate acting force for each moving particIe; analytical mechanics considers only one single function (e.g., Lagrangian, Hamiltonian, work function). This one i'unction contains all the necessary information concerning forces.

If strong forces maintain adefinite relation between the coordinates of a system, and that relation is empirically given, the vectorial treatment has to consider the forces necessary to maintain it. The analytical treatment takes the given relation for granted, without requiring know-ledge of the forces which maintain it.

In the analytical method, the entire set of equations of motion can be developed from one unified principle which implicitly includes all these equations. This principle takes the form of minimizing a certain quantity, the 'action' • Since a minimum principle is independent of any special reference system, the equations of analytical mechanics hold for any set of coordinates. This permits one to adjust the coordinates employed to the specific nature of the problem.!I

While the Newtonian theory of vector mechanics is based on the two fundamental vectors, momentum and force, and variational principles are based on the two scalar quantities, kinetic and potential energy, the two nethods are equivalent for free moving bodies. [Lanczos] makes the point, however, that if constraints exist in the system the analytical treatment is simpIer and more economical because Newton's third law of motion, 'action equals reaction', does not embrace all the possible cases of forces which maintain the constraints on the system, but suffices only to describe the dynamics of rigid bodies or, for the purposes of this technical note, flexible bodies.

Since both methods can describe the motions of the DMSA, and are equivalent for the problem posed, it was decided th at the equations of motion for the array would be derived from both formulations. This serves as a check on the final equations obtained as weIl as detailing the differences between the two approaches, as s~gested above.

11. TENSION .AND COMPRESSION EXPRES SI ONS

The most obvious consequence of deploying the DMSA vertically downward, in a constant one-gravity environment is to alter the tension and compression in the system from that experienced in space. As such, the tension and compression in the array can be subdivided into variabIe and constant

force components. VariabIe' components are functions of the axial coordinate y, whereas constant components are invariant over the length of the array.

~

x

~---.r---T ~ •

Z

~l-

_y

y

0)

Blanket

R

a-wg(t-y)

~

T(y}+ve

t

Gravity

x

ti

~Z

y

y

~

IwIS

pg

(~-y)

t

C(y)+ve

b) Boom

FIG. 11-1 VARIABLE CO~1PONENTS IN THE BOOM COMPRESSION AND BLANKET TENSION DUE TO THE INFLUENCE

/

11.1 Variable COmponents

The variable components discussed here arise solely from the effect of gravity and are non-existent in space.

Consider Fig. II-la, where it can be seen that a length (~-y) of the blanket, which corresponds to a mass of ow(~.y), hangs below the point R. As a result a force equal to owgU-y) acts downward on this point. Therefore, the weight of the blanket itself below a given point introduces a tensile force at that point which, in general, is given by the expression

T (y) = + owg(~ - y)

v (11-1)

I t is apparent by referring to Fig. II-lb that a similar argument applies to the tensile force generated in the boom by its own weight below the arbitrary point S. Tbe boom, however, is nOrmally under compression so that a tensile force results in a variable campression component at a distance y aJ.ong the boom of

(11-2)

11.2 Constant Components

These components arise from both the structuraJ. design of the array and the effect of gravity on the pressure plate. Under zero-gravity condi tions those resulting from the structural design represent the actual tension and

compression in the system.

While in reality the constant force components are invariant with regar,d to how the array was assembled it is, however, easiest to understand their origin by considering~the step-by-step procedure used during assembly. The first step in assembling the arr~ was to mount the boom actuator and inboard-pallet/elevation-arm assembly so that the boom would be extended vertically downward. Tue booll1 was left retracted and the blanket hung freely from the inboard-pallet/elevation-arm assembly. The pressure plate was then attached to the bottom edge of the blanket. This caused a constant tensile force in the blanket given by

T =

m

g

c~ t

The boom was then extended and secured to the pressure ~late using the tip-tensioning mechanism. In this state, as represented in Fig. 1I-2a the boom had not yet been extended far enough to activate the negator coil springs of the tip-tensioning mechanism.. The boom was then extended further, activating the negator springs as shown in Fig. 1I-2b. A negator spring is capable of delivering a constant spring force (fs) over a large displacement range,

unlike a conventional spring which exerts a force proportional to the displace-ment of the spring. The spring force, applied by the action of the negator

RETRACTED NEGATOR SPRINGS

SIDE VIEWS

BOOM

1

EXTENSION DIRECTION

Q-

R-EXTENDED NEGATOR SPRINGS

a)

BEFORE ACTIVATION

of

np-

TENSIONING MECHANISM

t

f

Sb)

AFTER ACTIVATION

of

TIP- TENSIONING MECHANISM

-";i

I ~

tGI -

i B I

v

RETRACTED NEGATOR SPRINGS

r

BOOM EXTENSION DIRECTION

Q

R-BACK VIEWS

l

r-I~ :... I F

-~fs

a ~ EXTENDED N SPRIN EGATOR GS

FIG. II-2 CONSTANT COMPONENTS IN THE BOOH COMPRESSION AND BLANKET

TENSION DUE TO ACTIVATION OF THE TIP-TENSIONING

component equal to

T

=

f

C2 S (II-4)

in the blanket and a compressive component equal to

C

= f

c~ s (II-5)

in the boom, as shown in Fig. II-2b. To finally complete the array assembly the twist control lines were extended from the pressure plate along each edge of the blanket and attached to the inboard pallet. These 'lines were kept taut by a small secondary negator spring assembly, mounted on the bottom of

the pressure plate (Fig. I-2). The action of this springassembly was to .

attempt to close the gap between the inboard pallet and the pressure plate assembly. As a re sult the blanket "sagged", or in other words a compressive force was introduced which acted against the tension in the blanket. Since a force of p existed in each twist control line, the corresponding tensile

component resulting from this action was

T

= -

2p

CS (II-6)

No compressive load is transferred to the boom because the n.egator springs

merely change length slightly; that is, the gap between the points

Q

and

R

in Fig. 1I-2b widem.only marginally, and since the negator spring maintains a constant force with displacement over a reasonable displacement range the compression in the boom remains unchanged.

This completes the derivation of the constant components; however, an unknown quant i ty f s has been introduced in the analysi s • It will be

nec-essary to solve for fs before final expressions can be,written for the tension

in the blanket, T(y), and the compression in the boom, IC(y).

I

II.3 Final Expressions for Tension and COmpressio~

The total constant tensile force can be found by sunmting. (II-3),

(II-4) and (II-6), of the previous section, to obtain

T =T +T +T

c c~ C2 cs

= mt g

+

fs 2p (II-7)

and from (II-l) the total variable tensile force is

T

(y)

= + 5wg(~ -

y)

~+ve

x~z

y

t

T(t)

t

2p

C{

t)

mtg

e

Zt

FIG. 11-3 FREE-BODY DIAGRMl OF STATIC FaRCES ACTING ON THE PRESSURE

Hence the totaJ. tension in the blanket can be written as

T(y)

=

Tc + Tv(Y)

=

fs - 2p + ~tg + awg(t - y) (II-9)

Similarly (II-5) yields the totaJ. constant compressive force in the boom

(II-IO)

and (II"2) gives the totaJ. var~able compressive component as C

(y)

v pg(t -

y)

(II-ll)

which implies that the totaJ. compression in the boom is given by

C(y)

=

Cc + Cv(Y)

=

f - pg(t -

y)

s (II-12)

Now, in the zero-gravity environment the tension in the blanket is given by P. Therefore setting g equaJ. to zero in (II-9) ánd equating the result to

P

yields

f =P+2p

s (II-13)

FinaJ.lythe generaJ. expressions for the tension and the compression are obtained by substituting (II-13) into (I1-9) and (II"12) to get

T(y)

=

P

+ ~tg +

awg(t -

y)

(II-14 )

C(y)

=

P

+ 2p -

pg(t -

y) (II-15)

All the forces acting at y

=

t

are shown on a free body diagram of the pressure plate assembly in Fig. II-3. The purpose of doing this i·s to bring to the attention of the reader, that while the forces in the y direction are in equilibri~ there is an unbaJ.anced statie moment about the x-axis equal to

x,w

x,w

z,u

Ftz

a)

COMBINED MOTIONS of THE BOOM

z,u

Ftz

Mtx .A1IIr:+--~

F

ty

I I

dfB~

-

J(~J

,t)

~~~=~~~~~;;;;;<"-;;-~_-.1_ ~_L

_______

J_1

~========~~====~~~

_I---~---~t

OUT-OF-PLANE DEFLECTION of THE BOOM

Ftx

Mtz I I

~~~~~~~~~~~~~~

_{___}

~;~~~Y/1

t=======================~L-~lt

c) IN - PLANE DEFLECTION of THE BOOM

It Will be important to recaJ.l this fact later when the finaJ. equations of motion are derived.

1110 VECTORIAL MECHANICS APPROACH

In this section a vectoriaJ. mechanics approach is used to derive the equations of motion for the dynamics of the DMSA as ground tested, and hence, while under the influence of gravity.

111.1 Derivation of COmponent Equations

Prior to deriving the finaJ. equations of motion is is necessary to determine the equations which govern each constituent part of the array. In this regard, the fOllowing four components will be considered: the boom, the blanket, the twist control lines, and the pressure plate.

111.1.1 Boom Equations

A three-dimensionaJ. view of the combined motions of the boom is presented in Fig. lIl-la, while out-of-plane (OP) motion is highlighted in Fig. III-lb, and in-plane (IP) motion in Fig. III-lc. Considering Fig.

lIl-la, three differentiaJ. forces come into the analysis, two inertiaJ.

(IIL1)

df

Bz

= -

P ü(y, t)dy

(IIL2)

and Qne externaJ.

df

BE = g P dy (IIL3)

Appendix A gives a detailed derivation of aJ.l the inertiaJ. and externaJ. forces for each individuaJ. component.

Now talting the moment about the point R, shown in Fig. III-lb, and setting it equaJ. to ~ero for equilibrium,yields

- u(y,

t)] -

J[U('I'},

t)

y

t

+

J

('I'} -

y)df

Bz =

0

Y - u(y,

t)]df

BE (IIL4 )

To first order the bending moment of this slender beam is given by

Hence, using (IIL2), (lIl.

3)

and (lIL

5),

Eq. (II1.4) reduces to

. I

- Bo u"(y, t) + Mtx + Ftz(t - y) - Fty[U(.e, t) - u(y, t))

- flu

_{h ,}

t} - u(y, t}]gp

d~

-

fl~

-

y}p

ü(~, t}d~

=

0

(III.6)

y

Y

The presence of gravity in the model is explicitly shown in this equation. The corresponding boundary conditions for the above equation, which are also illustrated in Fig. lII-lb, are

u(o, t)

=

0 (lIL 7)

and

Bu"(O, t)

=

ko u' (0, t) (lIL8)

The lP boom equation of motion is deri ved in a totally analogous manner by referring to Fig. lIl-Ic. The resulting moment equation is

•

BI w"(y, t)

+

M_tz - Ftx(t .- y)

+

Fty[W(t, t) - w(y, t)}

+

t[W(~,

.

t} -

w(y, t}]gp

d~

+

r(~

-

y}p

'

l;(~, t)d~

= 0 (lI1.9)

Y Y

The extra term due to the inclusion of gravity in the model is again self-evident. The appropriate boundary conditionsfor the lP motion are

w(o,

t)

= 0

(IIL10)

and

BI w"(O, "t). = k

I w' (0, t) (IILll)

The boom is also modelled as having a torsional stiffness k4 , which must provide

the res,istive moment to balanee .Mt"y when the boom is in twist (T). It therefore

follows th at

(lII.12)

This completes the equations applicable to the boom.

III.l.2 Blanket Equations

The motion of the blanket in the z-direction is due in part to an OP deflection and in part to a T deflection. This fact is pictorially

illus-trated by Fig. 1II-2a, and algebraically stated 'as

v(x, y, t)

=

v (y, t) - x v (y, t)

o x . (IIL13)

where the principle of superposition has been applied. The first term of Eq. (III.13) arises due to OP motion and the second due to T.

The blanket is assumed to be under tension only in the y-direction.

Then at any x the situation shown in Fig. 1II-2b prevails, where T(y)/w

results fromthe assumption that the tension is uniformly distributed across the blanket. Now considering the sum of the forces in the z-direction, i t follows that for equilibrium

(III.14) Since the defections are f:!mall i t can be assumed that the stretching ofthe blanket during vibration does not cause atension variation nor does the total length of the array change app;reciably. Neglecting higher order terms this

equation reduces to I

T(y) v"(x, y, t)

+

T'(y) v'(x, y, t) - aw';'~(x, y, t) :;: 0 (III.15)

The fact that the tension is no longer constant in the blanket when gravity

is present results in the generatidn of a first derivative term in the governi,ng

differential equation.

The boundary conditions' which apply to the blanket can be seen in Fig.

1II-2a. They are

v(x, d,

if)

= 0 (IIL16)

and

v(x,

t,

t)

=

u(t,

t) - x

aCt)

(IIL17)

Fig~e 1II-2c illustrates the array-minus-boom (AMB) sub-assembly

d1iring T/IP motion. While this motion does not represent the action of the

blanket alone, the blanket does, however,. play an important part in the motion

and as such the derivatiQn of the equations governing the AMB are.included in

this sub-section. .

There are nine differential forces of interef:!t present during the

x

a)

COMBINED MOTIONS

of

BLANKET

z,v

z,v

( ) L

r

x

i

-

Ix(y,t)

xv.

Y,t

'--=:_~,"-l"!.é}

_

n U

n~

:O(y,t)

X'"

.y

View Across Blanket at a Distance Y Along the Array

b)

FORCES ACTING on a TYPICAL BLANKET ELEMENT

z,v

v( x,y,t)

X I~

T~)

_ _

te

'

.

y

. . . . I . . . .

-"

I \ 1 \ 1 \ I

'~~Ftx

I ,~

t

' ::

I1

\~

Mtz c!--:.:::.

-:::.==~~

dfty,dftE

I

Fty --

~~<t)

~I

raat!)

j:

w ( l , t ) ·

c)

IN -PLANE /TWIST MOnON

of

THE AMB

"elnáining three are in the x .. direction. Three of the forces in the y-direction are externaJ.. forces' and resul t .from the inclusion of gravi ty in the model. The forces always occur in groups of three because the AMB is composed of three distinct componerrts: (i) the inboard-pallet/elevation;..arma sub-.assembl.y, ("1i) the blanket, and (iii) the pressure plate. The nine forces, grouped according to these components are as follows:

(a) for the inboard-pallet/elevation-arms sub-assembly,

(b) for the blanket,

df'px

=

d Î3(t)

(! )

dx df'py = - x 13'(t)

(! )

dx

df'pz = g (

f )

dx

••

df'bx

=

y t'(t) cr dx dy .... -

.

...

df'bY

= -

x t'(t) cr dx dy df'bz = g cr dx dy

(c) for the pressure plate,},

df'tx =

[$

·

~·(t)

- (z -

e

)

ä(t)

1 (

~

Ix

dz df'ty

=

[-x

~·(t)

+

z lx(t)

J

dx dz df'tz = g ( : ;

)dx

(IIL18) (IIL19) (IIL20)

These forces are obtained (see Appendix A) by referr:i,.ng to Figs. III-2a, III-4a and III-4b. The existence of T/IP coupling and the complexity of the array structure necessi tated considering severaJ.. views of the AMB in order to completely define all the pertinent terms in the force expressions.

Now taking the moment about the point S and equating it to zero for equilibrium yields

(IIL2l) Contd.

(III.21) Contd.

+

J

=

d

~

( t ) di' pE +

J -

Y

~

( t) df tE +

J -

[t~

(

t) - (z ., e) a ( t ) ] df tE

=

0 where the appropriate limits for the integrals are:

(a) over the x variable - - t o -w ₂ w ₂ (III.22)

(b) over the y variable d to

t,

and (III.23)

( c) over the z variable _{Zt -} h

_'2

_{to Zt}

+~

(III.24 ) 2

The integration limits for the first two variables are self-explanatory. In the case of the Z variable ,.however, some clarification is required. First it is necessary to define the z coordinates corresponding to the top and bottom of the pressure plate, re spe ctively,. In order to do this it is assumed th at the cent re of mass of the pressure plate is coincident with the geometrical

centre of the structure. While this is not actually the case, however, the centre of mass differs from the geometrical centre by less than 3% in the z-direction, and 1% in the x-direction [Sincarsin]. Now defining the total height of the pressure plate as h, it follows that since zt is the distance to the centre of mass, the top of the pressure plate is given by the z-coordinate zt + h/2, while the bottom is gi ven by Zt - h/2. This, however, defines the entire z range of interest because for z from 0 to Zt - h/2, according to the assun.q:>tions made in modelling the array, no mass exists, and as such the inertial and external forces in this interval are equal to zero. Subsequently, the value of any integral over this range which involves these quantities is equal to zero. It should be recalled that the aCt) rotation of the pressure plate is defined about the Yt-axis. This axis is parallel to the y-axis and passes through a point defined by the intersection of the blanket co:p.nection line (the Xt-axis) and the line which bisects the width of the pressure plate (the Zt-axis). Simi-larly with respect to the pressure plate, X(t) is a rotation about the blanket connection line. These facts are important when determining the moment arms through which the gravitational forces act.

Now substituting (III.21), (III.22) and (III.23) into (III.24) and defining the limits of integration gives

w

- k

2

~(t)

-

'\z -

Ftyw(t, t)

+

Ft'! -

f

d

2

~(t)

(;!. )

dx w

-f

t

J2

_y2~(t)cr

_{dx dy} -d w 2 22 (III.25) Contd.

+

x(z - e)"X(t)] ( : ; ) dxdz'

(z -

e)a(t)]~

(

:! )

dxdz = 0 (ll1.25)

After eval~ting the integrals in the above equation and grouping terms, the

final expres sion is

where and J,.,

=

L +

O"W(

t -

d)

(t+d)2

+

I + m t

t 2

+ I + m dg e;. DZ 2 tz ' pz P 1 pz 1 = -₁₂ m _p wg _. 1 l tz

= -

m w 2 12 'ti (ll1.26) (ll1.27) (III.28)

The inertia definitions are consistent with the assumptions made in

modelling tlle array, as cited in Section l.2. In fact, the J 2 expression can

be found directly using Fig. IIl-2c and treating each component of the AMB sub-assembly as a rigid body, finding tlle moment of inertia of each body a'bout an axis parallel t0 the z-axis which passes through its centre of mass and then applying the parallel axis theorem to find the corresponding inertia for each

component about the point S. This is the reasoning behind the gr01,lping of the

terms in the J;i expression.

Figures llI-2c, Ill-4a and lll-4b also combine to show the importaht

geometrical relationship between the lP quantities w(.t, t) and ~(t), and the

T quantity aCt), namely

wet,

t)

= - t~(t)

-

ea(t)

(III.29) Since this is a purely geOlnetrical identity gravity does not explicitly enter the equation; however, from (lII .26) it is evident that gravi ty causes the

values' of the variables to be different from those expected in a space

a)

COMBINED MOTIONS

of

TWIST CONTROL L1NES

Á

~vx(l,t}

v

₂(y,t) _/ ~ :J; - -

/ 7

\1,,,(1, t) _ - , / / - v ,

/

----i-~T-7t

d

/

)

w-I

104--='--+l/ " . /

~

/ /

e

7 / /

~///

L:::-=-~~~==

=f/

. - - Y l - + t Yi. _{2 Vx} (

_y,

t)

-±- --

_{- -}

~-::;.-=-T

-VI (y,t) x~-'-""'=""--""

View Across Blanket Width at a Distance Y Along Twist Control Lines.

b)

MOTtON

of

TWIST CONTROL LlNE NUMBER 1

Z,V.

--

~~~~--==~-=-=~---14-~,_~_==-::: ___ --_ _--_ _ _ _ _ _ _ _ _

~---~---~~I

y

c)

MOTION

of

TWIST CONTROL LlNE NUMBER 2

z,v

₂

d

e

-r---~---~bl

y

I·

t

This concludes -uhe derivation of the equations necessary to describe the motion of the blanket component.

III.l.3 ~st Control Line Equations

The combined motion of the two -twist control lines is given in

Fig. III-3a, while 1II-3b and III-3c show the situation for each individual

line. As wi th the blanket, two components superimpose to gi ve the general equations for the twist lines, namely, an OP component and a T component.

Since the twist control lines are assumed to deflect as straight lines (i.e., their mass is neglected), the equation for the deflection of each

is simply found by multiplying the slope of the line by the distance-to any

point y along the line. In particular, from Fig. III-3b, the slope due to OP motion is

v

(t,

t)

o

SOP

=

Ct -

d) (III.30)

and the slope due to T motion is

(III.3l)

and therefore the total slope of line 1, assuming small angles, is .,,

-[v

(t,

t) -

!2

v

(t,

t))

o x

(III.32)

Since the line starts at y

=

d, the deflection at y is given by the expres sion

(III.33)

Using Fig. III-3c and an analogous argument the deflection for twist line 2 at any point y is given by

(III.34 )

As the twist lines were assumed massless, the presence of gravity does not alter their governing equations.

III.l.4 Pressure Plate Equations

Free-body diagrams for the two possible motions of the pressure plate, OP and T/IP, are shown in Figs. III-4a and III-4b, respectively. From these

figures i t can be seen that three inertial force~ and one external force are

z,u,v

;~\TJ\

_~-

Zt

\ -f

vp(~:t)

V l e ,

e

u(.e,

t)

y

--.-a) OUT-OF-PLANE MOTION

of

PRESSURE PLATE

END VIEW

Fty

ea(t)

wtf.,t) . ~l (Zt-e)a(t)

IN-PLANE/TWIST MOTION

of

PRESSURE PLATE

1'1 1'1 I I I I I I I I I I I I I !I

Blanket, Boom

C)

PRESSURE PLATE -BOOM TIP MOMENT

and df_{tx ::}

[t

~(t)

- (z - e)ä(t)]

(:! )

dx dz df_ty

= [-

x

~(t)

+ z X(t)]

(:! )

dx

d~

df tz - [-

v

0

U,

t) + x (i( t)]

(:! )

dx dz df_{tE :: g (}

:! )

dx dz (III.35)

Four relationships ean be found by referring to Fig. III-4a. The firs~

is obtained by summ.:tng the forces in the z-direction and setting them equaJ. to

zero for equilibrium. 'This process yields

w/2

- F

_{tz -}

J

~~)

VI

(x,

t,

t)dx - p

v~(t,

t)

-vi/2

-PVI(t,t)+Jdf t· =0 2 z (III.36)

Substituting in the appropriate (IIL35) e quati on , and evaluating the inertial

force i~tegral over the limits described in.the previous sub-section, results

in Eq. (IIL36) becoming

w/2

- Ft -

J

TI!l

v I (x,

t,

t) dx - P [ v

~

U,

t) + V I

(t,

t») - m.

v

(t,

t) = 0

z w . 2 '" 0 ( III.

37)

-vi/2

The seeond equation is found by requiring the sum of the forces inthe y-direetion to be in equilibrium, that is

w/2

J

-T.f!} dx - 2p - _Fty +

J

df_ty +

J

df

tE ;:;: 0

-w/2

. .

,

(IIL38)

Again substituting the appropriate (IIL35) equations and eyaluating the integrals gives

(IIL39)

Now, from (IL14) and (IL15) of Section IL3 it follows that

TU)

=

P + ~ g (III.40)

and

cU)

=

P + 2p (III.41)

and therefore·

Thus (III.39) can be written as

(III.43)

The third equation found using Fig. III-4a is obtained by taking the x-moment about poin,t Q and equating it to zero, for equilibri\llIl, to obtain

- M. - F e - F e X ( t )

~

J

(z - e)

df

-

J

(z - e) X ( t )

df

.

tz

-'tx ty tz . ty

-J

(z - e)df

_tE

=

0 (III.44)

Apply1ng (III.35), integrating and neglecting the second-order term, X(t)v(x,.t,t), results in the equation

(III.45) where

The fourth and final expres sion apparent f'rom Fig. III-4a is the geometrical identity

i{t)

=

·û'

(t,

t)

(III.46) The equation obtained :from Fig. 1II-4b governs only the T motion of the pressure plate, since the lP motion nas been included in the derivation of the AMB equation of motion (111.26), as described in sub-section 111.1.2. Now forming the equilibrium y-moment expression about the point Q, gives

- ,\y

+

_{Ftx e}

+

Ft,(t)e

+

1.

2

T~l)

v'

(x,

i"

t)x

dx +

P

vl.(t,

t)

(~)

-w/2

- P vi.

(i"

t)

(~)

+

J

(z - e)M

_{tx -}

J

_{x Mtz}

+

J

(z -

_e)/3(t)MtE

~

0

(rrr.47)

As before, substitute the appropriat~ (III.35) equations into {III.47) and integrate. The result is

w/2 - l\y + F

tx e +

Ft~(t)e

+

J

T~.t)

VI (x,.t,t)x dx +

p~

[vi

(.t,

t) -

v~(.t,t)]

-w/2

+

~t(Zt

-

e)~(t)

_{- [Ity} +

~(Zt

- e)2Jä(t) + mtg

~(t)(Zt

- e)

=

0 (III.48)