A first order theory for predicting the stability of cable towed and tethered bodies where the cable has a general curvature and tension variation

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• ~ .0

von

KARMAN INSTITUTE

FOR PLUID DYNAMICS

TECHNICAL NOTE

68

A FIRST ORDER THEORY FOR PREDICTING

THE STABILITY OF CABLE TOWED AND TETHERED

BODIES WHERE THE CABLE HAS A GENERAL CURVATURE

AND TENSION VARIATION

by

James D. DeLaurier

RHODE-SAINT-GENESE, BELGIUM

von KAR MAN INSTITUTE FOR FLUID DYNAMICS TECHNICAL NOTE

68

A FIRST ORDER THEORY FOR PREDICTING THE STABILITY OF CABLE TOWED AND TETHERED BODIES WHERE THE CABLE HAS A GENERAL CURVATURE

AND TENSION VARIATION

by

James D. DeLaurier

TABLE OF CONTENTS

page ABSTRACT

LIST OF SYMBOLS i

INTRODUCTION 1

1. THE CABLE EQUATIONS OF MOTION 4

1.1 The Complete Cable Equations 4

1.2 The First Order Cable Equations 8 1.3 The Nondimensional First Order Cable Equations 12

2. THE BODY EQUATIONS OF MOTION 14

The Force and Moment Equations 14 The First Order Force and Moment Equations 18 2.1

2.2

2.3 The Nondimensional Form of the First Order Force 2.4

and Moment Equations 21

The Force and Moment Terms 23

2.5 End and Auxiliary Conditions given by the Force

and Moment Equations 32

The Transformation of the End and Auxiliary

Conditions to the Cab1e Coordinates 38 2.6

3. THE SOLUTION OF THE CABLE-BODY EQUATIONS 44 The Method of Solution for the Case where the

Cab1e has a General Curvature and Tension

Variation 44

The Longitudinal Solution 46

The Lateral Solution 57

3.1

3.2 3.3

3.4 The Computer Technique for Finding the Roots of

the Characteristic Equations 70

4. A COMPARISON OF THE THEORY WITH EXPERIMENT 72

4.1 The Test System 72

4.2 The Stabi1ity Tests 80

4.3 The Theoretical Resu1ts and a Comparison with

the Experiments 80 4.4 Conc1usions 89 REFERENCES APPENDIX I APPENDIX 11 APPENDIX 111 APPENDIX IV 90 91 95 99 107

ABSTRACT

The objective of this research was to investigate the dynamics of cable-body sytems, and ~n particular, to develop an analysis for finding the stability of towed and tethered bodies in a fluid stream. Particular applications for which the analy-sis may be used include towed underwater devices, towed and te-thered finned balloons, towed reentry decelerators, and towed airborne devices.

The cable-body system is treated analytically by consi-dering it to be essentially a cable problem, where the body pro-vides end and auxiliary conditions. Moreover, the cable itself is considered to be composed of cable segments - each with its own mean tension and angle. These segments are then matched

-one to the next - by the end conditions of displacement and slope, thus yielding a physical model for a cable with a general shape and tension variation. The mathematical description of the first order form of this problem is a sequence of nonhomogeneous boun-dary value problems in linear partial differential wave equations, with linear ordinary differential end and auxiliary conditions. Further, the equations uncouple to give a "lateral" problem and a "longitudinal" problem - as in first order airplane dynamics. The solution of either problem takes the form of a transcenden-tal characteristic equation for the stability roots. These roots are extracted by using an electronic computer and a roots locus plot.

In order to provide a check on the theoretical analysis, a ser~es of tests we re performed on a cable-body system tethered in the V.K.I. open throat, low speed wind tunnel. The quantities measured were the system's longitudinal and lateral frequencies of oscillation and stability boundaries. Since this test system contained most of the essential features of the theory, and since the theoretical results and experimental results compared favo-rably, it is felt that this analysis provides a reasonable method for treating the first order motion of a large variety cable-body systems.

b b B

..

B c C C ( ) C a -+ F

-

~

-LIST OF SYMBOLS

- Acceleration of the body mass center with respect

t 0 fA'.

-+ -+ + - Acceleration components in the el, e2 and e3

directions respectively.

- Nondimensional acceleration components defined by equations (2.61) -+ (2.63).

- Functions of À defined in equations (3.67) and (3.68).

_ Body characteristic length.

_ Unit vectors defined in Fig. 1.

_ Cable buoyancy force per unit length.

- Body buoyancy force.

_ Nondimensional body buoyancy force defined by

(2.44).

_ Longitudinal characteristic length as defined in Ref.

4.

_ Nondimensional cable equation coefficient defined by (1.25).

_ Abbreviation for eosine )

.

_ Cable force coefficient defined by (1.6). _ Cable force coefficient defined by (1.6). _ Cable force coefficient defined by (1.7). _ Drag coefficient.

- Lift coefficient.

_ Moment stability derivatives defined by

(2.45).

_ Force stability derivatives defined by

(2.44).

_ Nondimensional time derivatives defined by

equations

(1.25), (2.50),

and

(2.51).

_ Space fixed unit vectors defined by Fig. 5. _ Functions of cr defined in equations (3.38) and

(3.39). _ Force term.

F , F. r J g G , G. r J (h 1 ) . 1. (Hl)" (H2)' 1. l. (H3)" (H .. ). l. l. i I I I xx' yy' ZZ I xZ J J, J k k1 ... k7 K L m mg M 1, M2' M3 ... ... ... n 1 , n2' n3 p, q, r - ii

-- Cab1e f1uid force components defined by (1.3) and (1. 4 ) •

... ...

- Body forces in the nl, n2 and n3 directions respective1y.

- Functions of À given by equations (3.67) and (3.68).

- Gravitationa1 constant.

- Functions of cr given by equations (3.38) and (3.39).

- Function of À defined by (3.60).

- Function of cr defined by (3.31)

-

Functions of À as defined by (3.63).

- Functions of cr as defined by (3.34).

-

Nondimensiona1 inertia term defined by (2.49). - Moments of inertia about the x, y and z axes

respective1y.

- Product of inertia with respect to the x, z axes. 1

_ (_1)2

- Nondimensiona1 cab1e terms defined in Eq. (1.25).

- Nondimensiona1 cab1e term defined in Eq. (1.25).

- Cab1e coefficients defined by equations (1.26), (1.27) and (1.28).

- Nondimensiona1 cab1e equation force coefficient defined by (1.6) and (1.7).

- Cab1e 1ength.

- Body mass.

- Nondimensiona1 body weight as detined in Eq. (2.44). ... ...

- Body moments about the n1, n2 and n3 axes res-pective1y •

- Body fixed unit vectors defined by Fig. 5.

... ... - Body angu1ar ve10cities about the n1, n2 and n3

p, q, r R R a R a R

(P.'

s s

s

s(

t

..

t T m T m u, v, w u r ' v r ' wr u ' , v ' , w '

u, v,

w + U - i i i

-_ Nondimensiona1 body angu1ar ve10cities defined by equation

(2.47).

- Cab1e cross-section radius.

_ Distance from the body mass center to the cab1e attachment point.

_ NODdimensiona1 form of R _a defined by Sq.

(2.46}.

- Distance from the body mass center to the center

of buoyancy.

- Nondimensiona1 form of RB defined by Eq.

(2.46).

. + + + .

- Reference frame flxed to el, e2 and e3, deflned 1n Fig.

5.

_ Reference frame fixed in the undisturbed f1uid stream.

- Cab1e 1ength coordinate.

_ Nondimensional cable length coordinate defined by

(1.25).

_ Body characteristic area. _ Abbreviation for sine ( ). - Time.

_ Nondimensional time defined by

(2.48).

_ Cab1e tension force.

_ Mean va1ue of the magnitude of the cab1e tension force.

_ Nondimensiona1 va1ue of T as defined by Eq.

(2.44).

m

_ Body+mas! centet

ve~ocit~

components.in

~

in the nl, n2 and n3 dlrectlons respectlve1y. _ Body+mas~ centet ve~ocit~ components.in

R'

in

the nl, n2 and D3 dlrect10ns respect1ve1y. _ Perturbation va1ues of u , v and w respect

i-ve1y as defined by (2.33~. r r

_ Nondimensiona1 va1ues of u', v' and w' respec-tively, as defined by

(2.46).

-+ v r -+ v s - ~v

-- Velocity of the body mass center with respect

to

dL

- Velocity of the body mass center with respect

t 0 (}.,'.

- Velocity of a cable element with respect to ~'.

- Velocity of a cable element with respect to ~.

x, y, z - Body coordinates fixed in ~ and defined by Fig.

5.

x, y, z - Nondimensional forms of x, y and z respectively

as defined by equation

(2.46).

" " .J

x, y, z - Cable coordinates fixed in ~ and defined by Fig. 1.

" , ... , ";z'

X , Y , - Perturbed values of x, y, z defined by

(1.13).

x, y, z _ Nondimensional forms of

x',

y',

Z'

defined by

(1.25).

Y, Z

Greek Symbols

- Functions of s defined by equations

(3.44)

and

(3.10).

a _ Wing angle of attack.

a _ Angle between the i coordinate axis and the f,; coordinate

axis as defined in Fig. 1.

a

_

Angle of the cable segment as defined in Fig. 2.

r.

_

Cable constant as defined by equation

(3.45).

~

6. _ Function of À as defined by equation

(3.45).

~

6( _ A finite difference in the ( ) quantity.

E _ An order of magnitude defined by equations (1.15) and

(2.22).

z; - Cable coordinate as defined in Fig.

1.

-

Body Eulerian angle with respect to 6\ as defined in Fig.

6.

9

9

_-

Perturbed value of 9 defined by equation

(2.19).

90 - Equilibrium value of

e

defined by equation

(2.19).

e

_

Constant defined by equation

(3.26).

À _ Stability root for lateral mot ion.

À _ Real part of the stability root, À.

À. -J ( A ) •

-1 ~

-( \I ) • -1 E;

-11'1+'lr60= p

-p

-a

-a _r -a.

-J t -( u ) . -1

,

-,

-t

-~

-~ -'I' -+ w -(n) .

-1 - v

-Imaginary part of the stability root, À •

Cable constant ae defined by equation

(3.11).

Nondimensional body maas defined by equation

(2.48).

Function of a det'ined by equation

(3.10).

Cable coordinate as defined in Fig.

1.

Constants of the end conditions and auxiliary conditions as defined by equations

(2.107), (2.108), (2.109), (2.110),

(2.111), (2.112), (2.126), (2.127), (2.128), (2.129)

and

(2.130).

Fluid density.

Cable density per unit length.

Stability root for longitudinal mot ion.

Real part of the stability root, a.

ImagiDary part of the stabiliry root, o.

Perturbed cable tension defined by equation

(1.14).

Function of À defined by equation

(3.31).

Body Eulerian angle with respect to ~ as defined 1n Fig.

6.

Perturbed value of

,

defined by equation

(2.19).

Constant defined by equation

(3.54).

Body Eulerian angle with respect to

0...

as defined 1n Fig.

6.

Perturbed value of Ijl defined by equation

(2.19).

Const ant defined by equation

(3.54).

Angular velocity of the body with respect to ~. Function of a as defined by equation

(3.11) •

Subscripts

_ With respect to the cable attachment point (except for F ) and C ). _{a a}

_ Derivative with respect to a.

_ Buoyancy force term.

-

vi

-)g - Gravity force term.

)

.

_- With respect to cable segment "i".

1

) 0 - Reference value.

)p - Derivative vith respect to p.

) _- Derivative vith respect to q.

q

)r - Derivative vith respect to r.

)u - Derivative vith respect to u.

)v - Derivative vith respect to v.

( _)v

_-

Derivative vith respect to v.

(

)al

-

Derivative vith respect to al·

)a2 - Derivative vith respect to a2·

)a3 - Derivative vith respect to a3·

) 6 - Derivative vith respect to 6 •

) ~ - Derivative vith respect to ~

.

)1JJ - Derivative vith respect to 1JJ. SU;EerscriEts

(

.

)

- 1

-INTRODUCTION

This researeh was direeted toward developing a teehnique

tor analyzing the stability ot eable-body systems. Specitic

examples ot such systems tor whieh the teehnique is applieable

are the tollowing :

1. Towed tlight vehicles sueh as gliders or sailplanes towed behind a powered aireraft.

2. Towed surveillance devices, both those towed through the air beneath a powered aireraft, and those towed through the water beneath a surtace vessel.

3. Tethered balloons, sueh as finned balloons used tor carrying instrumentation at an altitude.

4.

Re-entry decelerators, such as tlared cone and

para-balloon devices under current development.

Although the towing of gliders is an old skill, it is well known that it requires a pilot at the glider's controls in order tor the system to be stable. In the case of unmanned towed and tethered devices, th ere exists considerable evidence that

under certain combinatio~ot cable length and free stream

velo-city, a supposedly stable system goes totally unstable. Mettam

noted in Ref. 9 that during a slow speed pickup maneuver, an

axisymmetric device towed beneath an RAF helicopter went unstable

to destruction. Also, Etkin (Ref.

5)

described how he produced

total instability in a non-litting model tethered in a low speed wind tunnel. Further, the author, in Ref. 3, produced instability and measured critical cable lengths for a litting body tethered in a low speed vind tunnel.

However. cable body systems have a further interesting feature. It has been observed that even if a system is unstable "in the small". it may be stable "in the large". In other words, the system may undergo limit cycle oscillations, and not be to-tally unstable. This behaviour is common with tethered finned balloons as observed by current tests at NASA Langely. Such

- 2

-motion is also possible for other classes of cable-body systems, and the important point is that even though this mot ion is non-destructive, it may give rise to unreliable instrument readings. Thus, it is feIt that the technique presented in this report for predicting first order stability, i.e., stability "in the smalI", is of considerable value toward designing a solidly stabIe system.

All previous theoretical work on cable-body dynamics has been directed toward developing a first order stability ana- . lysis. And further, a very major portion of the work to date

has used the approach of treating the system as being a rigid body dynamics problem, where the cable is accounted for by some force condition at the attachment point. This force condition takes the form of cable "stability derivatives", and is derived from the assumption that the cable is in instantaneous equilibrium with respect to certain of its end conditions. For instanee,

Glauert (Ref.

6)

based his analysis of a towed body on the cable end conditions of displacement. Similarly, Bryant, et.a~. (Ref.2) used the same conditions for his analysis of tethered lifting bodies. The most sophisticated example of this approach is given by Maryniak (Ref.8) in his study of towed glider stability, where he considers cable end conditions of both displacement and velo-city. Using the cable "stability derivative" approach has the merit of giving a polynomial equation with constant coefficients for the stability roots, but such a physical model does not

contain the basic mechanical nature of a cable-body system. A more meaningful approach is to treat the system as being a cable problem, where the body supplies certain end conditions. This physical model is much more general, and contains the previous mode of analysis as a special case. Basing his analysis on this approach, the author (Ref. 3) obtained a solution for the first order stability of a cable-body system where the cable is nearly straight, and has nearly uniform tension along its length. The equation for the stability roots was complex and transcendental,

- 3

-but the roots vere readily obtained by a computer roots locus plot.

Also, to date, good experimental research on cable-body systems has been very sparse. There are few examples where the system's properties are weIl documented, and the system's behaviour is carefully measured. One example of good work is Etkin's tests on a tethered axisymmetric body in a wind tunnel.

(Ref.

5 ).

Also, Mettam (Ref.9 ), conducted a similar series of

experiments, obtaining a very complete line of data. Currently, Tracy Redd of NASA Langley is towing finned balloons with an

instrumented truck, and is obtaining data for very long cable

lengths. Further, the author in Ref. 3 ran a series of carefully

controlled experiments on a tethered lifting body in a wind tunnel.

The author has continued his previous work on cable-body systems by developing an analytical solution for the system's stability, where the cable has a general curvature and tension variation along its length. This solution is based on an

exten-sion of the theory of Ref. 3 , and to this end, the development

of the cable and body equations of motion in Chapters 1 and

2 closely follow the text of Ref. 3. Further, the author has

compared the theory with experiment by running a carefully

controlled series of tests with a tethered model ~n the V.K. I.

3 meter, low spee~ open throat wind tunnel, which is ideally

suited to such experiments. Although the theory is readily appli-cable to a vide range of appli-cable-body systems, for example, tethered balloons, towed re-entry decelerators, and towed underwater

devices, it is feIt that this test system contained all of the important features of the theory, and as such, was used to provide a check on the analysis

4

-1. THE CABLE EQUATIONS OF MOTION

1.1 The Complete Cable Equations

The physical model of the cable 18 subject to the

following assumptions :

a. The cable has uniform density and geometry along its entire ~ngth.

b. The cable is perfectly flexible and inextensible • c. The cable has a cross-seetion that ia ro~nd. or nearly so - such as stranded wire.

d. The cable is totally immersed in a homogeneous uniform stream •

e. The Reynolds number of the cable's crossflow is subcritical.

v

~---.X

Fig. 1 Cable coordinate system and nonfluid-dynamic forces acting

on a cable element, às

The dependent variables of the cable are the coordi-nates of a point on it, ~,

y,

and

t,

and the tension, T, at

that point. The independent variables are the distance along the cable, 8, and time t. The forces acting on an element of the

cable, ds, are now found. For the contribution of the cable tension, one has that

- 5

--..

-..

-..

Further, expressing this with respect to the bI, e2, b3 coor-dinate system, one obtains that

(1.1)

Assuming that buoyancy acts in the positive

z

direct ion and gravity acts in the negative

Z

direction, one obtains that the gravity force, dF

g, and the buoyancy force, dFB, on the element act together to give

Considering the fluid dynamic force dF_f , on the element, ds, note first that

-..

-..

-..

v _r

=

U - v _s

-..

where v is the cable element's velocity relative to the refe-s

rence frame ~ (see Fig. 1) and; is the relative velocity of

r

the fluid to the cable element. Upon introducing the cable force coefficients,.c

a and cb ' one obtains components of the fluid dynamic force in the plane of dF

f and ; r as follows

(note fig. 2)

dFb

~ I

V-!...r _ _ _

~~_-l~

df

Q

Fig. 2 The fluid dynamic forces acting on the cable element

- 6 -dF :I: e p i ;

I;

Rds a a r r and dF b -+

(;

-+

=

c pv )( )( n}Rds b r r -+ is given Nov, v by r -+ - l l - + 3" -+

-+ rt}b3 v _r

=

(UCa - _at }bl

_!Z

_3t eZ

-

(USa

Also, as described in page 3.9 ot Hoerner (Ref. 7),

- 3Iz c

=

c + K(l - CZa} a ao

-and c_b

=

K(l - CZa)Ca vhere

Ca =

-+ -+

(v .n)

r (1.3) (i.4) (1.5) (1.6) (1.8)

Thus, (1.5), (1.6), (1.7), and (1.8) into (1.3) and (l.4) give expressions for the t1uid dynamic forces on the element. Further, equating all of the forces on the cable element to its accele-rat ion vi th respect to @.., one has that

50, (1.1) through (1.8) into (1.9) give the complete cable

• -+ -+ -+ •

equations of motion expressed 1n the bl, eZ, b3 coord1nate

-+ •

7

-.

_a

2

_F;

_a

(T

-H)

+ + KS 3èi)[(UC; -

aF;

2 w 2 P =

-

pR(C

ät)

+ (ll) +

at

2

as

aO

at

-

2] 1/2

.

_{_ .ll) _}

..

[{ (UC; -

aF;

2

+ (USa +

ll)

(UCa KS2 a CapR x

ät)

₊

at

at

•

(1.10)

...

and the e2 component is

~.., 2 (~) +

at

(1.11)

Fina11y, the b3 component ~s

8

--

11

2}

ar; { -

af;

af;

a'"

l i

-

a a}

+ (USa +

at)

as

+ (UCa -

ät)

as -

(ft)

as -

(USa +

af)af

x

-

-(b - pg)sa (1.12)

1.2 The First Order Cab1e Equations

5

/ displaced position position v

X

Fig. 3 Coordinates of the displaced cable

A small perturbation analysis is performed on equations (1.10), (1.11), and (1.12) 80 as to obtain their first order

forms. Noting Fig. 3, the ~ axis is aligned through the end points of the cable's equilibrium configuration. Further, con-sider a perturbation from equilibrium such that

9

-where ~o(s) and ÇO(s) are the equilibrium va1uea and ~'(a,t), y'(s,t), and ~'(a.t) are the perturbation va1uea trom equili-brium. A1so, conaider the cab1e tension to be expreasib1e as

T(s,t)

=

T (a) + 'r(a,t)

eq

,

(1.14)

where T (s) is the equilibrium cab1e tension and T(S,t) is eq

the perturbation va1ue from equilibrium.

Now, assume sma11 perturbations trom the equilibrium position such that

where E « 1.

av'

and

at"-Note that it fo11ows that

ar'

and

TI-

=

A1so, assume that over the cab1e's 1ength

t(s,t)

=

O[E]T eq

And note that this direct1y givea that

aT

=o[e:]

~

•

(1.16)

Tm - 10

-

Taq-o

.,

I

I

f

I

I

L

Fig.

4

The assumed variation of T with s eq

s

Further, an important assumption is made that for the cable length considered, the equilibrium tension varies nearly linearly, and may be given by

T =T +.!!(s-~)

eq m fls 2

,

where T is the mean value of the equilibrium cable tension. m

Also, it lS assumed that the variation of T is small compared eq

with T , namely

m

(1.20)

Finally assume that the cable has a shallow curvature such that

- 11

-...

ao : a and : 1

Note that conditions (1.19), (1.20), and (1.21) may be readi1y met for most cables by considering a short enough segment, L.

Nov, taking (1.13) through (1.21) into the complete cab1e equations, (1.10), (1.11), and (1.12), and dropping

terms vith 0[&2] and higher, one obtains the first order cab1e equations of mot ion :

,

(1.22)

and

12

-Note that within the context ot assumptions (1.16), equations

(1.23) and (1.24) are the principa1 cab1e equations of motion.

1.3 The Nondimensiona1 First Order Cab1e Equations

Define the fo11owing factors

ç'.Y'.~'.s

L

Further, de fine the quant i tie s

, J

-Using these to nondimensiona1ize (1.22), one obtains

,

where

- 13· -Simi1ar1Yt (1.23) becomes

-2 - .. a 2v av D y - C2 ~ + k6Dy + k7 ~

=

0

aä

2

_as

where ,.

-k6

=

J(C + KS3a + KC2aS2a)

ao

and

And fina11Yt (1.24) becomes

where

and

t

-

1~

-2. THE BODY EQUATIONS OF MOTION

2.1 The Force and Moment Equations

The physical model of the body is subject to the follow-ing assumptions :

a. The body is rigid

b. The body is completely immersed in a homogeneous fluid stream

+ +

c. The body is symmetric with respect to the nl. n3 plane (see Fig. 5)

d. The cable is perfectly free to pivot at the attach-ment point

+

e. The center of buoyancy is on the nl axis.

... 15

-It is important to note that the body equations vill be expressed in terms of x, y, zand the Eulerian angles . ,

e,

and • relative to~. Although this is different from traditional airplane practice, these coordinates are necessary in order to relate the body equations to the cable equations. Hovever. the force and moment terms vill be derived relative to the body

-+ -+ -+

fixed axes, nl' nl, and n3. This is done because fluid dynamic effects on a flight vehicle are traditionally taken vith respect to body fixed axes, and thus this vill enable the introduction of the classical stability derivatives of airplane practLce. Eventually, br transformation equations. these force and moment terms vill be expressed in terms of x. y, zand "

e,

and

+.

The force-acceleration equations are, as in Etkin.

(Ref. 3 ) ,

Fl ~ m(~ + qw - rv)

,

F2

=

m(~ + ru - pv) ( 2 • 2 )

and

and the moment-angular acceleration equations are

MI

=

I _xx

P -

l _xz ~ + q(I _zz r - I _xz p) - rI _yy q

_•

(2.4)

=

I q. + reI p - I r) - p(l r - l p)

yy xx xz xx xz

,

and

•

(2.6)

Hote that u, v, and vare defined by the velocity of the mass center :

...

...

...

...

v _c

=

unI + vn2 + wn3

16

-•

Also, p, q, and rare defined by the body's angular velocity with respect to <R:

...

...

...

...

w

=

pnl + qn2 + rn3 _• (2.8)

Consider now the relations between : i and ~i (i=l,2.3) based on the Eulerian angles as defined in Fig. 6. These are

l

==========::::::t~~11

_e2

- 11

-•

... ...

(s1jIses, + ... (s1jIses+ - e,S');3

e2

=

SIjIeenl + e,e+)n2 +

•

(2.10)

and

... ... ... ... _(2.11)

e3

=

-senl + eeS+n2 + eee+n3 _•

...

Note that v may be expressed as

c

...

.

...

...

.

... _(2.12)

v

=

xel + y e 2 + ze3 c

!his and equations (2.1), (2.9), (2.10) and (2.11) give that

zse

_•

•

(2.14)

and

• (2.15)

Simi1ar1y. reso1ving (2.8) with (2.9), (2.10). and (2.11), one obtains • • p

=

•

-

IjISe

_•

(2.16) • • q

=

ee+ + Ijlees.

,

anji •

.

r

=

,eee.

-

es, _• (2.18)

Equations (2.13) through (2.18) into equations (2.1) through (2.6) give the force and moment equations, (2.1) through (2.6), in terms of x, y, z, Ijl, e, . , and their derivatives. The

resu1t 18 resu1t

-ing equations are nonlinear, but in the spirit of the stability analysis, a small perturbation analysis is performed. and linear, first order equations are derived.

2.2 The First Order Force and Moment Equations Consider a perturbation of the Eulerian angles and their derivatives such that

8

₌

80 + 9, ~

=

_'0 + ~

,

~

=

_"'0 +

'"

,

(2.19)

. .

!

•

.

! • • !

e

₌

90 + 9, ~

=

~O + ~

,

and ~

=

~O + ~

_•

(2.20)

where the "0" quantities are reference values, and the "." quan-tities are the perturbation values. Further, de fine the reference configuration of the body to be that of statie equilibrium, thus,

.

. .

~o

=

~o

=

~o

=

80

=

~o

=

0

_•

and 8 0 is a fixed value according to the condition that the ;1

ax~s passes through the attachment point and the mass center (see Fig. 5).

Now, assume small perturbat ions such that

! ! !

~

,

e, ~ =

o

[E]

_•

~, e, ~ =

o

[E]

(U/b)

• •

O[e:]U

and x, y, z = where e: « 1 (2.22)

When (2.13) through (2.22) are substituted into (2.1) through (2.6), and terms containing an

0[E

2] or higher are dropped, one obtains the first order form of the force and moment equations :

..

..

Fl

=

m(xCeo

-

zse 0 )

,

(2.23)

F 2

=

my

,

(2.24)

..

..

- 19

-Ml

=

lxx' -

(I seo

_xx +

I

_xz

ceo)' •

and I 6 yy

•

-=

(I

ceo

+

I

seo)~ zz xz (2.26) • (2.28)

Finally, note that whereas the dynamics of the body deals with its motion with respect to the inertial reference frame.~, the fluid dynamic effects on the body depend on its mot ion relative to the fluid stream, @..'. The velocity of the maBS center re1a-tive to the fluid stream, ~'. is givea by

-+

=

v _c

For motion subject to the small perturbation conditions. (2.19) through (2.22), equations (2.13) through (2.15) and (2.29) give that u

=

iceo

r and

zseo - UC60

_•

,

w

=

zC60

+

_{iS60 - US6 0 - UC606}

r

(2.30)

(2.31)

(2.32)

Now, considering again the equilibrium reference condition, one defines velocity perturbations by

u

20

-Thus, from equations (2.30) through (2.32), (2.33) gives that

.

u'

=

xC90 (2.34)

v'

=

U~ + Y - US90~

,

and

w'

=

~C90 + iS60 - _{UC9 0 9 •} (2.36)

Note, however, that the body's acceleration with respect to the

fl ui d stream,

{R',

is ident i cal to that vi th respect to

<R.

Thus.

@.

d;

=

~ _dt

where, for the small

(2.15) give that al = xC90 zS60

,

a2 = Y

,

and a3 = zC90 + xS90 •

,

perturbation case, equations (2.13) through

(2.38) (2.39)

(2.40)

Similarly, the angular acceleration of the body with

respect to (i\. , is identical to that with respect to 6\. Thus, for

the smal 1 perturbation case, equations (2.16) through (2.18)

give that • • _(2.41) p

₌

~

-

~S9o ! q

=

9

,

(2.42) ! and r

=

_{ceo' •} (2.43)

... 21

-2.3 The Nondimensiona1 Form of the First Order Force and Moment Equations

The factors used to nondimensiona1ize the force and moment equations are identica1 to those used in American airp1ane stabi1ity convent ion - except that no distinction is made between a "10ngitudina1" and "lateral" characteristic length. Now define

Fl. F2. F3, B, mg, T CX' Cy' Cz ' B, T m _(2.44) mg,

_-

,

m _(pU2S/2) MI' M2' M3 Ct' C _m' C _n

-

_•

(2.45) (pU 2Sb /2) A X, y, z, RB' R A u' v' w' 2R, 2R a •

•

,(2.46) x, y, z, (b/2 ) u, v, w a

-

,

-

U al, a2, a3

f~u7~

)

r al, a2' a3

-

,

p, q. r -

,

(2.47) (2U 2 /b) t _-

-

_b 2U t , lJ

-

4m _pSb

,

(2.48) I xx' I yy' I zz' I i ~ i i

-

xz (2.49) xx' yy' zz' xz pS(b/2)3

,

D( )

_-

_2ü

b d ( ) (2.50)

d t

•

and D2( ) _-

-

b2 d 2 ( )

_•

(2.51) 4U2 dt 2

- 22

-Introducing these into the force and moment equations, (2.23) through (2.28), gives :

•

,

A A C

_z

=

~S6oD2x + ~C8oD2z

•

-

-Cl

=

i _xx D2,

-

(i _xx S60 + i C60)D2. xz

,

C

=

i D26 m

_:r:r

•

and

-C _n

₌

(i C60 _zz + i S6o)D2.

-

i D2, xz xz

Similar1:r. equations (2.34) through (2.43) become

u

=

C60Dx - S80Dz

_•

_(2.58) v

=

D:r

+

,

-

_{S6 0}

+

,

_(2.59) v

=

S80Dx + C60Dz

-

C606

_•

(2.60)

..

A al

=

C8oD2x S80D2z

_•

_(2.61) A a2

=

D2:r

_•

(2.62)

..

A a3

=

_{C8 0D2z} + S60D2x

_•

_(2.63) p

₌

D~

-

_{S6 0D,}

_•

_(2.64) q

=

D8

23

-and

•

(2.66)

2.4 The Force and Moment Terms

Consider the forces F., and the moments M., (i=l,2,3)

1 1

on the body. Rewrite these as

F.

=

F. + AF.

1 10 1 and M. 1

=

M. 10 + AM. , (i=l,2,3) 1

where F. and M. are the reference values, and AF. and AM. are

10 10 1 1

the perturbed quantities. Noting that the reference condition was defined to be statie equilibrium, (2.21), one obtains that

F. = M. = 0 10 10

thus

F.

=

flF. and M. = flM. _•

(2.67)

1 1 1 1

Now, in the spirit of small perturbations, F. and M. are assumed

1 1

to vary linearly with the perturbed velocity, the acceleration, and the angular velocity of the body relative to

AI.

Also, accoun-ting for the body veight and buyoyancy, F. and M. are assumed to

1 1

vary linearly with the perturbed Eulerian angles. Finally, a

cable force and moment contribution is accounted for by the terms flF c . and flMc .• Thus, the general first order expressions tor F. 1 1 1 and M. are

àF. = 1

1

aF. aF. aF. aF.... aF ....

+---.!.p +--!.q +~r +--!.. +~

e

ap aq

ar

a;

a8

and

~M.

=

1

- 2~

-au'

aM. aM. aM. aM. aM. aM.

1

u'

+

--l

v'

+ ~ w' +

_äaï

1 1 1

al + aa2 a2 + aa3 a3 +

av' aw'

aM. aM. aM. aM. aM. _ aM._

~ p + ~ q + ~ r + --! ./, + ~

a

+

--l •

+ àM_c1.

ap aq ar 3~ ~ aa ai ~

Following Etkin (Ref.

4),

the cross derivative terms are droppe4 That is, fluid dynamic stability derivatives of symmetrical quan-tities with respect to unsymmetrical variables, and those of unsymmetrical quantities with respect to symmetrical variables, are considered to be equal to zero. This assumption is consis-tant with the small perturbation analysis about the equilibrium configuration. Thus, af ter nondimensionalizing according to equations (2.44) through (2.47), one obtains from equations

(2.68)

and

(2.69) :

,

+ C

_z

25

-(2.74) and C

=

C v + _{Cn a2} + _{Cn p} + _{Cn r} + Cn ~ + cnee + Cn~~ + Cn n nv a2 p r ~ ~ c

•

(2.75)

vhere the subscripts u, v, v, al. a2, a3. p, q, r, ~, e, and ~

denote the partia1 derivative of the coefficient vith respect to the nondimensiona1 form of that variab1e. Considering nov the gravity and buoyancy effects, one has that

F

+

B

=

g (see Fig. 7)

Using equation (2.11). and performing a sma11 perturbation ana-lysis by using equations (2.19) through (2.22), one obtains the

first order form of this expression. ,Upon nODdimensiona1izing

by equations (2.44) one then has that

26

-and

A A A A

CXe

=

-(B - mg)Ce

_{o ,}

CZe

=

-(B - mg)S60

and

... ...

C

y

,

=

(B -

ag)ceo

_•

,

Similarly, the moment about the mass center due to buoyancy

e~~ects is given by

(see Fig. 7) , (2.80)

~or which, again, by equations (2.11), (2.19) through (2.22),

and nondimensionalizing by (2.45), one obtains that

,

... RB

=

2

seo

(2.82) also •

21

-z

...

"B

....

_e,

Fig.

7

CabIe, buoyancy, and mass forces on the body

The cable terms in the force and moment equations provide the mathematical link between the body's motion and the cable's motion. The cable force is

T

(2.84)

a

Using equations (2.9) through (2.ll), one may resove this into

-+ -+ -+

the nl, n2, and n3 coordinate directions. Further, consider a perturbation of the c~ble from equilibrium such that

2&

-( a}é ) ₌

_(ti)

₊ ( a

x')

_(li)

₌

_(lL.)

as _a as _a

as

_a

•

as _a as

•

a (l!)

.,

( a ~ , ) and _as

=

(ll)

_as + _as

,

(2.85) a a a

where the

,,_tt

terms are the equilibrium values, and the primed

terms are the perturbation quantities. Consistant with the

pre-vious small perturbation assumptions. (1.15), the primed terms

are considered to be of order E. Thus, upon substituting (2.85)

into (2.84), and dropping terms of order E2 and higher, one

obtains the first order cable force. Now the equilibrium cable force is T v (~)

ce o -

as a

seo

+ (l!) as a (2.86)

and, substracting this fr om the first order cable force equation gives one an expression for the perturbation cable force. Nondi-mensionalizing by (2.44) and (1.25), one thus obtains that

Cy _c

and

"la""

a'"

=

To (a:) seo -

(a: )

a a

=

-To[(!L.)

_{a s a s} +

{(!!)

a a

seo

+ (az) as

•

a ;] • (2.88)

•

29

-In a similar ~ashion, the cable moment terms are derived. The moment on the body due to the cable ~orce is

M

c

-+ -+

=

T x R nl

a a

,

where

T

is given by (2.84). Again, when one uses equations (2.9)

a

through (2.11), (2.90) may be resolved into ;1' ;2' and

~3

components. Further, utilizing (2.85), a small perturbation ~rom equilibrium is taken, and one drops terms o~ order E2 and higher. This gives the first order form of the cable moment. Now, the equilibrium cable moment is

'"

seo + (!.!)

dS

,

a

and, substracting this from the first order cable moment equa-tion gives one an expression for the perturbaequa-tion cable moment. Nondimensionalizing by (2.45) and (1.25), one thus obtains that

,

(2.92)

(2.93)

- 30

-and

Fina11y, substituting (2.58) through (2.66). (2.77), (2.78). (2.79). (2.81) (2.82). and (2.83) into equations (2.70) through (2.75). one obtains the A ... A _ _ ~orce . . and moment expressions in terms of the x, Y. z, ~,

e.

and , coordinates :

,

(2.95)

Cz

=

[<cz ce o - Cz seo) D

2 ₊

_{(cz ceo - czuSeo)D];}

₊

a3 al W

- 31

-,

(2.98) C

= [(

Cm

ce

0 - Cm

se

0) D 2 + (Cm

ce

0 - Cm

se

0 ) DJ; + m a3 al w u

,

and (2.100)

Thus, these equations - along with (2.87), (2.88), (2.89),

(2.92), (2.93), and (2.94) - into equations (2.52) through

(2.57) give the complete force and moment equations for the

body. Note that the fluid dynamic force coefficient terms may be directly related to the "stability derivatives" of standard

airplane practice. The transformation equations to relate one to the other are given in Appendix I.

- 32

-2.5 End and Auxiliary Conditions given by the Force and Moment Equations

As mentioned ~n the introduction. the key to the solu-tion of the cable-body problem is to solve the cable equasolu-tions. where the body equations of motion provide end and auxiliary

conditions. To this purpose. the bedy equations of motion are nov rearranged and combined so as to be in a more convenient form fo~ their application. First. note that (2.88) and (2.94) combine to give an auxiliary condition :

• (2.101)

A second auxiliary condition is given by (2.89) and (2.93)

Cm _c - R _a Cz _c :. 0 _•

Also. a third auxiliary condition is given by (2.92)

Ct _c

=

0

Now. (2.87) x ceo + (2.89) x seo gives an end condition

(Cx ceo + Cz seo) +

(li)

è

c c as _a

(2.102)

(2.103)

(2.104)

Similarly. (2. 87) x se 0 - (2.89) x ce 0 gi yes a second end condi-tion : 1 (cx seo - _{Cz ce o ) -} (!!)

ë

c c as _a (2.105)

= ..

TO

= -

- -

Cy

..

c

TO

- 33

-• (2.106)

These conditions may be expanded into ru11 form by using equa. tions (2.95) through (2.100) and the force and moment equations,

(2.52) through (2.57). Doing Buch, one finds that the auxi1iary condition, (2.101), becomes (2.107) where

..

- R _a (lJ - Cy _a2 ) - _{Cna2 '}

..

1f23 - -(Cn + R Cy )

,

p a p ... ce o [ ~+

.. ..

- ;g)] 1f24

-

seo(C n + R Cy )

-

R (B

,

v a v 2 a - i _zz ceo + i _xz seo

,

..

1f26

-

Cn seo

-

Cn ceo

-

R (ce OCy

-

_{seocy )}

,

p r a r p

and

..

1f27 - - (C n + R Cy )

- 34 -Similarly, (2.102) becomes " "

-(~28D2 + ~29D)x + {~30D2 + ~31D)z + (i D2 + ~32D + ~33)a • 0 • yy (2.108) where ~28

,

,

,

•

~32 - R Cz _{a q} Cm _q

_•

and " se 0 [ ~+ " "

- m"g)]

~33

-

Ceo(Cmw

-

R Cz ) _{a w}

-

₂ R (B _{a •} Also, (2.103) becomes "

-(Ct

a2D2 + CtvD)y - (ixxD2 - CtpD + seOCtv)~ + (~19D2 + ~2oD +

.

+ Ct )1/1 = 0 v

where

35

-- i _xx

seo

+ i _xz

ce o

and

Further, the end condition, (2.104),becomes

... ...

..

_ (WID 2 + w2D)x + (W3D2 + W4P)Z + (wsD + w6)6 , (2.110) where

,

,

and

36

-Also, the end condition, (2.105), becomes

A A _

=

(W13 D2 + Wl~D)x + (W15D~ + W16 D )Z + (W17 D + wlS)e • (2.111) where

,

,

•

1

_(CSoC

Z

seocX

) w17 -

-

A

-

,

To

q q and 1

_CSO(S6oCx

_{ce oC}

Zw ) (ax) '!TIS

-

A

-

-w as

To

a

37

-,.

-

-=

{W7D2 + waD)y + (wgD + wIO)~ + (WIlD + w12)~ • (2.112)

where (Cy

-

~ ) _{Cy Cy} a2 _v

.-::.R.

w7

_-

,

'TI'8

-

-

,.

_•

lTg

-

..

•

TO TO TO 1 (C80Cy _r - S80Cy ) _p

_•

and

38

-2.6 The Transformation of the End and Auxi1iary Conditions to the Cab1e Coordinates

-

U

..

-...

~(L,t) Q

-

-

--

-

-

-,-

,-'

....

"...

"

... - 1\

...

.... I

x

Fig.

8

The re1ationship between the body

coordinates and the cab1e coordinates _I'

Note that the end and auxi1iary conditions, (2.107)

through (2.112), are expressed in terms of the

_....

_..

x',

y',

and

Z·

coordinates of the cab1e and the x, y, and z coordinates of the body's mass center. Thus, in order to app1y these conditions direct1y to the cab1e equations, (1.27) through (1.29), they

must be transformed to the cab1e coordinates, ~, y, and ç.

Consider now the fo11owing transformation equations (see Fig.

8)

-x

=

Ca~(L,t) Saç(L,t)

-

+ R c~ce

a

y

=

y(L,t)

+

R

s~ce

a

,

,

and

-

-z

=

Caç(L,t) + Sa~(L,t) - 39 -R

sa

a (2.115)

Using the sma11 perturbation assumptions for the cabie, (1.13) through (1.18), and the smal1 perturbation assumptions for the body, (2.19) through (2.22), one obtains the transformation equations for the first order prob1em. Further, upon nondimen-siona1izing by (1.25), (2.46), (2.50), and (2.51), one has that

Dx

,

(2.116)

,

(2.117) Dy

,

(2.118) (2.119) Dz (2.120) and (2.121)

Also, note that at the attachment point, one has the following re1ationships :

,.

.ti.

(L,t) - (1.!)

40

-and

..

I I

(L,t) + (l!.)

II

(L,t) as as as • a

--Mu1tip1ying (2.122) by (az/as) and (2.123) by (sx/as) , and

a a

substracting the two, one obtains a re1ationship for the cab1e slope in the two coordinate systems. Further, one obtains the first order form of this re1ationship by using the sma11 pertur-bation re1ations, (1.13) through (1.18) and (2.85), and nondimen-siona1izing by (1.25) :

•

Now, by means of equations (2.116) through (2.121) and (2.125), the end and auxi1iary conditions, (2.107) through (2.112), may

_.

_.

be expressed in the cab1e coordinates, ~ and y. First, these re1ationships into (2.108) give the auxi1iary condition :

-

...

.

(~45D2 + ~46D)~(1,t) + (~47D2 + ~48D + ~33)e

= 0 ,

where _ l. yy

..

~29Sa)

,

..

2R (~28SeO + ~30CeO) a

,

(2.126)

41

-and

•

Simi1ar1y, (2.107) transforms into the auxi1iary condition

-(WSID 2 + wS2D + w27)'

=

0 (2.127) where

,

and A

Further, (2.109) transforms into the auxi1iary condition

42

-where

(

~)C

- b Lv

•

and

Now, an end condition is given by (2.116), (2.117), (2.120), (2.121), (2.110) and (2.111) into (2.125) :

-

,.

-('!I'40 D2 + '!I'41D)~(1,t) + ('!I'42 D2 + '!I'43D + '!I'44)e

where

(

~)

[Cl!)

'!I'40 - b as '!I'42

,

a

,

('!I'1 8e O + '!I'3 Ce O) -

(~)

('!I'13 8e O + '!I'IS Ce O)] a

,

,

- 43

-and

•

Finally, an end condition is given by

(2.112)

(2.130)

where

,

,

44

-3. THE SOLUTION OF THE CABLE-BODY EQUATIONS

3.1 The Method of Solution for the Case where the Cable has a General Curvature and Tension Vari-ation

Note that two of the important assumptions for obtain-ing the first order cable equations, (1.26), (1.27), and (1.28), were based on the cable segment'e having a shallow curvature,

(1.15), and a small tension variation (1.18). As mentioned in section 1.2, these assumptions may, in general, be closely realized if the cable segment considered is short enough. With this thought in mind, consider the general situation where the cable has an arbitrary curvature and tension variation. The assumption is now made that, in order to treat the first order motion of such a cabIe, it may be considered to be suDdivided into short segments for which the first order cable equations apply. That is, each segment is treated as a separate cable

problem, each with its own mean angle, mean tension, and tension variation. Further, these segments are joined mathematically, one to the next, by matching their end conditions of displace-ment and slope. Finally, end conditions for the cable segdisplace-ment adjacent to the body are given by the end conditions derived from the body's equations of motion, (2.129) and (2.130).

45

-To consider the end conditions in detail, subdivide the cable into n segments. A general segment and its properties are assigned the number i, where the value of i depends on that segment's position from the origin of coordinates (see Fig. 9). For example, the segment adjacent to the body is the nth segment.

Now, the LI

=

°

end of the first segment is assumed to be fixed to the origin of coordinates, thus

-

..

l;dO,t) :=I

°

,

-

..

and YI(O,t)

=

°

•

For the intermediate segments, upon assuming that any given point on the cable displaces perpendicularly from its tangent on the equilibrium configuration, one has that

_ A _ ...

L(i_l)l;(i_l)(l,t) :=I Lil;i(O,t)

,

and

_ #lil, _ A

L(i_DY(i_l)(l,t)

=

LiYi(o,t)

( 3.4 )

Note that this last assumption is consistant with the small

perturbation assumptions, (1.15), applied to all of the segments. Further, upon matching the end slopes of the segments, one

obtains that for the perturbation coordinates,

and

az;.

..

:=I

-2.

(o,t)

a

ä.

1.

a

1" 1.

as.

_1.

...

(o,t) _• ( 3.6 )

- 46

-Finally, as mentioned before, end conditions for the L

=

1 end

n

of the nth segment are given by equations (1.129) and (1.130), vhere L in these equations nov equals L •

n

Note nov the very important fact that the complete set of cable-body equations uncouple into tvo separate problems. The cable equation, (1.28), applied to each segment, along vith the end conditions (3.1), (3.3), (3.5), (2.129), and the auxilia-ry condition, (2.126), constitutes a complete problem for the

. . . . A _ ".

general solution of r;(s,t) and e(t). Similarly, the cable equa-tion, (1.27), applied to each segment, along with the end condi-tions (3.2), (3.4), (3.6), (2.130), and the auxiliary condicondi-tions,

(2.121) and (2.128), constitutes a complete problem for the .. _ A _ A ~ A

general solution ot y(s,t), ~(t), and ~(t). Physically, this

means that the first order problem uncouples into tvo distinct modes: lateral and longitudinal motions. Such uncoupling is,

in fact, observed by experiment (Chap.

4

and Ref. 1). Finally.

-

-

..

note that consistant with conditions (1.16), the ~(s,t) variabIe

is of no significance in the first order problem.

3.2 The Longitudinal Solution

As stated before, the longitudinal problem is described by the folloving equations :

Cable Equation :

From equation (1.28), one obtains

vhere "2 C. 1

ar;.

1

as.

1

=

0

,

"2 C. -1. and (T ).b 2 m 1 J. -1

47

-,

k. _ 1

(i

=

I, 2, ••• , n) End Conditions

-

" c,;l(O,t)

=

°

,

. . .... . . A L(i_l)c,;(i_l)(l,t)

=

Lic,;i(o,t)

ar,;.

1

-as.

1. " (O,t)

,

,

where 1

=

2, 3, ••• , n; and trom (2.129),

,

,

k. 1.

ar,;

n "

-

"

-as

_n (l,t) where

=

(w~oD2 + W~lD)c,; (l,t) + (~~2D2 + ~~3D + W~4)e , n

-(wlSCa _n ( 3.8 )

48

-and

Auxiliary Condition

From (2.126), one obtains

~ ,. ~ (W4S D2 + W46D)~ (l,t) + (w D2 + w4SD + w33)e

=

0 n '+ 7

,

where 2L ~ w~S - (~)(w30Can and 2L ~

~46

-

(~)(w31Can

-

~29San)

_•

Observe that the cable equation of motion, (3.7), is a linear partial differential equation with constant coeffi-cients. The formal solution of such equations is usually based on the technique of separation of variables - as explained in Berg and McGregor (Ref. 1). Further, (3.7), along with its end and auxiliary conditions, defines only a boundary value problem; that is, initial values are unspecified. Thus, in the spirit of a stability analysis - along with noting that the end and auxiliary conditions are linear and have constant coefficients _ a perturbed harmonie mot ion is assumed :

~ ~ ,. ~.(s.,t)

=

l. l. • at Z.(s.)e l. l. 1.

=

1, 2, ••• , n (3.10)

Sueh harmonie motion is, in faet, observed in cable-body expe-riments (Refs. 2, 3, and 5). Lastly, note that (3.10) implies

49

that all of the cable segments displace similarly with time -with only a difference in amplitude and phase angle. This ia assumed so because the matching condition (3.3) requires that the ends of adjacent aegments must move together.

Now, substituting (3.10) into (3.7), one obtains that

-Z. ( a . ) 1. 1. =(Z.)e _{1. 1}

-

-(AiH1i)si (Ai-Oi)s + (Z.) e 1. 2

where (Z.) and (Z.) are constants, and 1. 1 1. 2 A. _ 1. and 0. 1. [ (02 + _{(k 3}).a) _ A2l.. + ________ --~l.--"2 C. 1.

Considering the first segment, i=l, one obtains from (3.11) and the end condition, (3.1) :

AIS 0IS1

=

(ZIl1e (e

-OIS I - e )

Further, (3.11) gives for the second segment that

(A 2 +0 2 )S2

= (Z2)le

{A_{2 - ( 2} )S2

+ (Z2)2e

Upon matching (3.12) and (3.l3) with (3.3) one obtains

- 50

-(3.14)

A1so, by matching (3.12) and (3.13) with (3.5) one obtains

-Ol

J

+ e )

=

(3.14) and (3.15) combine to give

,

(3.16)

where

~2

-Thus, (3.16) into (3.13) gives that

(3.18)

Continuing on to the third segment, i=3, one has trom (3.11) that

51

-•

Again, upon matching (3.18) and (3.19) by (3.3) and (3.5), one obtains

,

(3.20) where

{A,

n 2

_{-0, }}

0 3]

[::.:

L2 + n2 ( e n 2 + Q2-e -n2 ) - A3 -(e Q2e ) Q3 -(en2 + Q,e-O ')}

1

• (3.21)

[~:

{A,

+ _{n 2} n2 - n 2 - A3 + n3 (e _{- Q2}e )

So, (3.20) into (3.19) gives

Now, the general solution for any segment, i, may be found by continuing this matching process along the cable such as to include that segment. This then yields that

-z. (

s. ) 1 1 where A.5.

n.

5.

=

(Z.) e 1 l e e 1 1 1 1 Ql

=

1 and

-n.5.

Q.e 1 1) 1

52

-[ {

'1 ( . ) -'1 ( . )

L. ( e ~-l +Q. e l . - l ) __ -.;;.l._ A +'1 ( ~-l )

L(i_l) (i-l) (i-l) '1(i_l) -'1(i_l)

(e -Q(i_l)e ) Q. -~ [ { '1 ( . ) -'1 ( . ) L. ( e ~-l +Q. e . ~-l ) ~-.;;.~_ A +'1 (~-l)

L(i_l) (i-l) (i-l) '1(i_l) -'1(i_l)

(e -Q(i_l)e )

i = 2, 3, ••• , n _•

Considering the last segment, i=n, one has, from equations (3.10):

- ( .. A) Z (s- )eot z; s , t

=

n n n n where,

-Z (s ) n n A s '1 s = (Z ) e n n(e n n n 1

,

-'1 s Q e n n) n

Further, in the spirit of the harmonie analysis - as discussed earl ier in this section - assume that

(3.26)

where

e

is a constant. Substituting (3.25) and (3.26) into the end condition (3.6), one obtains

A

n

n ( n e e '1 -'1

1

.( e n + Q e n) - '1n '1 n .-'1 ( Zn ) 1 + (e n - ,Q e n) n

- 53

-Similarly, upon sUbstituting (3.25) and (3.26) into the auxiliary condition, (3.7), one obtains

(3.28) Equations (3.27) and (3.28) are two linear homogeneous equations in (Z) _n and 0. Thus, it follows that an equation for a

(charac-1

teristic equation) may be obtained by putting these two equations into a determinanct, and setting it equal to zero. Doing this, one has 11. - Q n n

,

Q (e n

Note that a ~s, in general, complex. That is,

a

=

a + ja. ,where J

=

(_1)1/2

r J (3.30)

=0.

Thus, the characteristic equation, (3.29), is a complex trans-cendental equation involving a complex variable. To facilitate finding the roots of this, it is expanded into two real charac-teristic equations in two real variables, a and a .• To this end,

r J

consider first

n .•

From (3.23), ~ n. = (n.) ~ ~ r + j(n.) ~

.

J

,

(3.31)

54 -",here (h2)· (h2) . (0. ) _- l. Cv.

,

(0. )

_-

l. Sv.

,

1 _{C .} 1 1 . C. 1 r _l. J ₁ 2 - o. J + (k3).0 l. r

}2

+ (20 r o. J + and v. -1 -1 tan 2 r J [ 20 o. +

Next, note that Q. is, in general, complex 1 Q. = (Q.) 1 1 r + j (Q. ) l. • J •

Thus, one obtains trom

(3.31)

and

(3.32)

that

o.

1

o.

-0. (e1+Q.e 1) 1

o.

-0. ( e 1 _ Q . e 1) l. ",here

,

1

2(0.) _ (Q.)2 _ 2(0.) e 1 r 1 . 1 . J J [ 2(0.) e 1 r_{C (2(0.) ) _} 1 . J

. .

.

.

2]1/2

(k3)·0.) , l. J

(3.32)

x

. .

.

• • • a.nd (O.) e ]. j 4 (0. ) 1. r - 55 -_ (Q.)2 -_ (Q.)2 1. r 1. j + 2(0.) e 1. r [ 2(0.)

e

1.

r

D ( 2 ( 0 .) _ (Q.)

r]

2 1. • ]. J [ ~(O./ 1. 2 e r S (2(0.) - (Q.) 1 . . 1. • J J

And further, from equa.tion (3.24),

,

[

L.

L(i~l)

(A(i_1)

+

(H3)(i_1»

-

Ai] 2 +

2 (0. ) ]. (Q. )

-1.

[

L.

(Oi)r1' +

r

L(i~l)

(A(i_1)

+

(H3)(i_1»

- A.

]. +

_ (0.)2 _ (0.)2 1. r ]. j • • • • • _L.

,

+ ].

L(i_1)

2 and

2[(Oi)r

L.

1.

(H4)(i_1)

-

(0. ) x

L(i_1)

1. • (Q. ) J 1 .

(

L.

_(Oi)

J

₂ J

L (

i~l

)

(A(i_1)

+

(H 3 )(i_1»

-A .

+ + 1 r • •

•

• • • • •

•

• • • • • _{L. 2}

,

i=2,3, ••• , n. + (L ]. (H_{4 ) ( .} 1) + (0.) )

(i-1)

1- ]. j •