Attitude stability of articulated gravity-oriented satellites. Part I. General theory, and motion in orbital plane

ATTITUDE ST ABILITY OF ARTIC ULATED GRA VITY -ORIENTED SATELLITES Part I - General theory, and motion in orbital plane

.

_.,

,

by B. Etkin

ACKNOWLEDGEMENT

The work reported herein was sponsored by the United States Air Force under Grant No. AFOSR-62-40 monitored by the Air Force Office of Scientific Research.

The author wishes to express his thanks to Prof. H. Maeda and Mr. J. H. Fine for checking the analysis J and to Mr. Fine for carrying out the numerical cornputations. The latter were performed on the IBM 7090 at the University of Toronto Institute of Computer Science.

Thanks are also due to Dr. P. A. Lapp of deHavilland AircraftJ who inspired the author's interest in the problem and took a continuing interest in the development of the solution.

,

SUMMARY

1. A theoretical framework is presented for analyzing the rotational and relative motions of compound satellite systems. It consists essentially of expressions derived for the forces and moments acting on the constituent bodies, and of their utilization in Lagrange's equation to find the equations governing the motion of the system.

2. The method is applied to a specific system intended for passive attitude stabilization, and numerical examples are calculated. The design is found to provide dam ping to 1/2 amplitude in as little as one-third of an orbit, and to have small response to orbit ellipticity.

I. Il. III TABLE OF CONTENTS SYMBOLS lNTRODUCTlON

DERlVATlON OF THE EQUATlONS OF MOTION 2. 1 The Gravitational Force Field

2. 2 The Inertia Force Field 2. 3 The Total Force Field

Page v 1 2 3 4 5 2.4 Force and Moment on a Finite Body 5 2.5 Forces and Moments for Small Eccentricity 6

2.6 Reduction for Near-Principal Axes 8

2.7 Equations for the Symmetric or Longitudinal Forces 8 2.8 Equations for the Asymmetric, or Lateral Forces 9 2. 9 Longitudinal Equations of Motion for the Particular

System 10

SOLUTlONS OF THE EQUATlONS OF MOTION 3. 1 Characteristic Equation and Characteristic

Modes (Longitudinal)

3. 2 Steady State Longitudinal Oscillations Caused by Ellipticity of the Orbit

3. 3 Oscillations Caused by Radiation Pressure REFERENCES TABLE I FlGURES 16 16 19 20 25 26

al> a2, a3 Al, A2, A3 a, b (A, B, C) b 1 B1, B2, B3 Cl' C 2, C 3 c c d e

.!g

~ (F x' F Y' F z) G (L, M, N) .. h J~"l etc.

J.

m m₁ · SYMBOLS coefficients (Eqs. 2.15) coefficients (Eqs. 2.17)

stabilizer dimensions (Fig. 2. 3)

principal moments of inertia of a body coefficient (Eqs. 2.15)

c oefficients (Eqs. 2.1 7) coefficients (Eqs. 2. 17)

resistance coefficieI).t of hinges

differential operator

dl

d 'Ç

eccentricity of ellipse

generalized force in Lagrangels equation gravitational force per unit mass

total force acting on a body

total moment about the m. c. of a body

space-fixed coordinate (Fig. 3.9) constants (Eqs. 2.39, 2.43)

moment of inertia (Eq. 2. 17) unit vectors along x, y, z. products of inertia (Eq. 2. 17) length of rod

total mass of satellite

p Pn

(P,

Q,

R)

r R s t T

w

Oxyz

f

~,?)~

~p

>

'?\:l,

ç~

"\1',9)

cp

solar radiation pressure norm al radiation pressure

principal angular velocities of a body relative to <Uxyz generalized coordinate

radius vector of particle (Fig. 2. 1) radius vector of origin (Fig. 2.1) earth mean radius

distance along orbit time

kinetic energy, or period of orbit constants in Eq. 3.4

work done in a virtual dis placement

gravity oriented reference frame (Fig. 2. 1)

angular displacements of stabilizer rods (Fig. 2. 3) gravitational constant

.

angular velocity, 0 average value of ~

position vector of particle (Fig. 2. 2)

centroidal coordinate axes for body (Fig. 2.2) coordinates relative to principal axes

Euler angle,s, defining rotation of body relative to

C

~?~

polar angle or true anomaly (Fig. 2. 1)

b

angle of stabilizer (Fig. 3. 9)

,)

_{angle of incidence}

( _{)0< pertaining to upper stabilizer}

( _)~ pertaining to lower stabilizer

.... ₍

....

1. INTRODUCTION

This report presents an analysis of compound satellite systems which are gravity-oriented, i. e. which maintain one principal axis approxi-mately vertical. The term 'compound' describes systems composed of more than one body, coupled byelastic and/or frictional elements.

The analysis was motivated by the problem of achieving effec-tive passive stabilization of earth satellites by the use of the gravity-gradient principle, which is now well known. However, the equations of motion de-rived would also be useful in the analysis of active control systerns (i. e. they can provide the forward-loop transfer function). Although distortional

elastic modes of motion were considered, no analysis of these is presented, since it was found that (i) only for extremely long and slender cornponents could the elastic-mode periods be long enough to be comparable to those of the rigid body modes, and (ii) theI'e is no coupling between elastic and rigid-body normal modes in the first-order theory.

The first part of the analysis (Secs. 2. 1 to 2.8) is completely general, leading to equations for the total force and moment acting on any constituent body of the system, regardless of shape, . orientation or position. The expressions are next linearized with respect to orbit eccentricity and rotations of the constituent bodies and finally, by us ing conditions of symmetry, they are separated into 'longitudinal' and 'lateral' systems of forces and moments.

The force system obtained can be applied, for exarnple via Lagrange's equation, to obtain the equations of motion of any given system. However, since satellites of interest may vary so widely in design, it be-comes necessary at this point to specialize the analysis to a particular con-figuration. This is done in Sec. 2. 9 for the stabilization systern which pro-vided the principal motivation for this study. Only the equations for longi-tudinal motion (in the orbital plane) are given. The analysis of the lateral modes is left for the future Part II of this report.

In Sec. 3 are given two sets of solutlons of the equations derived, for

(1) Characteristic mode analysis (transients)

(2) Steady state oscillations produced by orbit ellipticity

Finally, the disturbances produced by solar radiation pressure are evaluated in Sec. 3.3.

Il. DERIV ATION OF THE EQUATIONS OF MOTION

The equations of motion are derived by applying Lagrange's equation in a non-Newtonian frame of reference Oxyz with origin at the vehicle mass centre (R,

0 )

as illustrated in Fig. 2. 1. This is a oonvenient choice for describing the rotational and distortional motion of the satellite. It is easily shown that to the first-order* there is no back-coupling of the rotational and distortional motions of the vehicle on the trajectory of the mass-centre. The latter therefore describes a Keplerian orbit, and the motion of the re-ference frame used can be assumed known a priori. Oblateness of the earth is neglected, as is the presence of other astronomical bodies, and hence the orbit is an ellipse lying in a fixed plane.

We shall make use of Lagrange's equation of motion. which is

(2. 1)

where T is the tot al kinetic energy of the dynamic system relative to the moving reference frame, W is the work do ne by the entire effective force field (including inertia forces) during a virtual displacement compatible with the constraints. and the qk are the generalized coordinates. The kinetic energy is computed exactly as in an inertial reference frame, viz. :

T=

~ \(:it+~'.d)d'"

(2. 2)

the integral being over all elements of mass of the system. For an individual component rigid body of mass mi with mass centre at ei (Xi' Ji' zi) the kinetic energy is

(2. 3)

where (Ai' Bi' ei) are the principal moments of inertia, and (Pi> Qi' Ri) are the angular velocity components' (relative to Oxyz) about the principal axes..

In any application, the kinetic energy must be expressed in terms of the particular generalized coordinates.

* i. e., in which the gravitational force field is assumed linear over the satellite - see Eqs. 2.8. The integral of these equations, when the origin is the mass centre. yields F = m lio, the condition for a Keplerian orbit.

'.'

The foregoing is routine, and involves no departure from con-ventional applications of Lagrange's equation. The novel features of this application are bound up with the generalizèd force term -OW

/'0

'bt,

which must include the conservative gravitational field, the conservative inertia force field, and dissipative terms arising from friction.

2. 1 The Gravitational Force Field

The gravitational force per unit mass is

with components

f

=_)J.

f

-~ r~

-f~ -=

-)l

X-x. f~

f~~

-=- -

7~ ~

t~~ ~

1.

~R-~)

The radius vector has the length

'2. '/" f

=

[x

4

+

~l.-+ (~-1J) ] whence '2. '2. -~z. (2.4) (2. 6)

f-~= R~L(~)-Tl~)-T(\-~)l1

The quantities

(t \

i.

I

l )

are of order 10- 6 and hence it is a very good approximation to linea,rize the force field. Thus

(2. 7)

and

(2. 8) "

2. 2 The Inertia Force Field

The angular velocity af the reference frame is

- 3

~

and the acceleration of the origin is

a

_{- 0} =..Q.~ _{- 6} 0 (2. 9) 2. ..

~

=

Rw -

R.

o where

It follows that the inertia force field per unit mass due to motian of the re-ference frame is given by (see Ref. 1)

(2. 10)

These relations, which are linear in the coordinates, are )unlike Eqs. 2.8, exact.

We find it convenient to transform from t as the independent variable to ~ We have that

whence and d~ ~ ~dt

ft

=

w.~

fi:c

=

~

x:

+

t

~ !~1-

+

1.

~ !~

t· ':.

_t 0

1

t. '::. -

~

-\-

~'l.1t

-.1.

~ ~2.

_ 2 '!;)"2..

~)C

L~ 0 2. cl~

.!'lS'

(2. 11)

2.3 The Total Force Field

The total force per unit mass is the sum of Eqs. (2.8) and (2.11) i. e.

(2. 12)

2.4 Force and Moment on a Finite Body

It is required subsequently to have expressions for the total force and moment acting on a finite body, viz:

f

-=

~! ~W\

(2. 13)

~ ~

_\

~)(!

á"M

where

f

is the position vector of

d'M

relative to the body mass centre C (Fig. 2.2) and hence G is the moment about the mass centre.

Integrating Eqs. 2.12 for the resultant force, we get

(2. 14) where

wl._

~o ó'W1.. Q, :: _R Cl ': _1 l. J'( Ql.. =. I áw"1.

_{k::_~o}

Cl.: w1.;. 2. ~o ï:. ~ I

R

_R (2. 15) Q~ =. ~ "(;)7- _.c.~

_':

_-2.(:;) 2.

The integral for the moment yields

L:

~

I

J~'2

+

A

₂

J?~

+

À.~

\

~ ~

dW\

J

tL dl,?

tf\:

BI

~ç

ot

B'l.

l~

l'

u~

d't

(2. 16)

~:

Cl

J~'2

-\-

Cl.

J,\

-t

C

3 \

'?

1;

ch'fl

where (2. 17) and etc. ,

2. 5 Forces and Moments for Small Eccentricity

Wb,en the eccentricity of the ellipse is small, such that e

<<.

I, then the radius Rand the angular velocity ~ are given by (see Appendix of Ref. 2)

~

: ';:)0 (\

-t 7.

e

c.o5.

~

)

R

~

Ra (\ -

e

U)~ ~

' )

(2. 18)

where. Ro and ~o are the mean values of R and ~ ,and are related by

~7,._ ~

o -

R

~

o

ACKNOWLEDGEMENT

The work reported herein was sponsored by the United States Air Force under Grant No. AFOSR-62-40 monitored by the Air Force Office of Scientific Research.

The author wishes to express his thanks to Prof. H. Maeda and Mr. J. H. Fine for checking the q.nalysis , and to Mr. Fine for carrying out the numerical computations. The latter were performed on the IBM 7090 at the University of Toronto Institute of Computer Science.

Thanks are also due to Dr. P. A. Lapp of deHavilland Aircraft, who inspired the author's interest in the problem and took a continuing interest in the development of the solution.

,

SUMMARY

1. A theoretical framework is presented for analyzing the rotational and relative motions of compound satellite systems. It consists essentially of expressions derived for the forces and moments acting on the constituent bodies, and of their utilization in Lagrange's equation to find the equations governing the motion of the system.

2. The method is applied to a specific system intended for passive attitude stabilization, and numerical examples are calculated. The design is found to provide dam ping to 1/2 amplitude in as little as one-third of an orbit, and to have small response to orbit ellipticity.

1. Il. lil T ABLE OF CONTENTS SYMBOLS INTRODUCTlON

DERlVATION OF THE EQUATIONS OF MOTION 2. 1 The Gravitational Force Field

2.2 The Inertia Force Field 2.3 The Total Force Field

Page v 1 2 3 4 5

2.4 Force and Moment on a Finite Body 5

2.5 Forces and Moments for Small Eccentricity 6

2.6 Reduction for Near-Principal Axes 8

2.7 Equations for the Symmetrie or Longitudinal Forces 8 2.8 Equations for the Asymmetrie, or Lateral Forces 9 2.9 Longitudinal Equations of Motion for the Particular

System 10

SOLUTlONS OF THE EQUATIONS OF MOTION 3. 1 Characteristic Equation and Characteristic

Modes (Longitudinal)

3. 2 Steady State Longitudinal Oscillations Caused by Ellipticity of the Orbit

3.3 Oscillations Caused by Radiation Pressure REFERENCES TABLE I FIGURES 16 16 19 20 25 26

SYMBOLS al, a2' a3 coefficients (Eqs. 2 .. 15)

Al, A2' A3 coefficients (Eqs. 2.17)

a, b stabilizer dimensions (Fig. 2.3)

(A, B, C)

..

principal moments of inertia of a body

b 1 coefficient (Eqs. 2.15)

B1' B2' B3 coefficients (Eqs. 2.17)

Cl' C 2, C 3 coefficients (Eqs. 2. 17)

c resistance coefficieI).t of hinges

c

d differential operator d/ d ~

e eccentricity of ellipse

~

generalized force in Lagrange's equation

19

gravitational force per unit mass

~ (Fx' Fy, F z) total force acting on a body

G (L, M, N) total moment about the m. c. of a body go

h space-fixed coordinate (Fig. 3.9)

I, 11,12,13,14 constants (Eqs. 2.39, 2.43)

17 moment of inertia (Eq. 2.17)

~

.t,

k unit vectors along x, y, z.

J~? etc. products of inertia (Eq. 2. 17)

1.

length of rod

m tot al mass of satellite

p Pn

(P,

Q,

R)

r R s t T

w

Oxyz

f

~,?) l;

~p,

?p,

~p

'Y)8)cp

r;

~

solar radiation pressure norm al radiation pressure

principal angular velocities of a body relative to <Uxyz generalized coordinate

radius vector of particle (Fig. 2. 1) radius vector of origin (Fig. 2.1) earth mean radius

distance along orbit time

kinetic energy, or period of orbit constants in Eq. 3.4

work done in a virtual displacement

gravity oriented reference frame (Fig. 2. 1)

angular displacements of stabilizer rods (Fig. 2.3) gravitational constant

angular velocity ,

i'

average value of"

~

position vector of particle (Fig. 2.2)

centroidal coordinate axes for body (Fig. 2. 2) coordinates relative to principal axes

Euler angle,s, defining rotation of body relative to

C

~,?ç

polar angle or true anomaly (Fig. 2. 1)

angle of stabilizer (Fig. 3. 9) angle of incidence

pertaining to upper stabilizer pertaining to lower stabilizer pertaining to satellite body

.

'

1. INTRODUCTION

This report presents an analysis of compound satellite systems

which are gravity-oriented, i. e. which maintain one principal axis

approxi-mately vertical. The term 'compound' describes systems composed of more than one body, coupled byelastic and/ or frictional elements.

The analysis was motivated by the problem of achieving effec-tive passive stabilization of earth satellites by the use of the gravity-gradient principle, which is now well known. However, the equations of motion

de-rived would also be useful in the analysis of active control systems (i. e. they

can provide the forward-loop transfer function). Although distortional elastic modes of motion were considered, no analysis of these is presented,

since it was found that (i) only for extremely long and slender components

could the elastic-mode periods be long enough to be comparable to those of

the rigid body modes, and (ii) there is no coupling between elastic and

rigid-body normal modes in the first-order theory.

The first part of the analysis (Secs. 2. 1 to 2.8) is completely general, leading to equations for the total force and moment acting on any

constituent body of the system, regardless of shape, . orientation or position.

The expressions are next linearized with respect to orbit eccentricity and rotations of the constituent bodies and finally, by using conditions of symmetry, they are separated into 'longitudinal' and 'lateral' systems of

forces and moments.

The force system obtained can be applied; for example via Lagrange's equation, to obtain the equations of motion of any given system. However, since satellites of iriterest may vary so widely in design, it be-comes necessary at this point to specialize the analysis to a particular con-figuration. This is done in Sec. 2.9 for the stabilization system which

pro-vided the principal motivation for this study. Only the equations for

longi-tudinal motion (in the orbital plane) are given. The analysis of the lateral

modes is left for the future Part II of this report.

In Sec. 3 are given two sets of solutlons of the equations derived, for

(1) Characteristic mode analysis(transients)

(2) Steady state oscillations produced by orbit ellipticity

Finally, the disturbances produced by solar radiation pressure are evaluated in Sec. 3. 3.

Il. DERIV A TION OF THE EQUATIONS OF MOTION

The equations of motion are derived by applying Lagrange's equation in a non-Newtonian frame of refere'nce Oxyz with.origin at the vehicle mass centre (R,

0' )

as illustrated in Fig. 2. 1. This is a convenient choice for describing the rotational and distortional motion of the satellite. It is easily shown that to the first-order* there is no back-coupling of the rotational and distortional motions of the vehicle on the trajectory of the mass-centre. The latter therefore describes a Keplerian orbit, and the motion of the re-ference frame used can be assumed known a priori. Oblateness of the earth is neglected, as is the presence of other astronomical bodies, and hence the orbit is an ellipse lying in a fixed plane.

We shall make use of Lagrange's equation of motion. which is

(2. 1)

where T is the total kinetic energy of the dynamic system relative to the moving reference frame, W is the work done by the entire effective force field (including inertia forces) during a virtual displacement compatible with the constraints. and the qk are the generalized coordinates. The kinetic energy is computed exactly as in an inertial reference frame, viz. :

T

=

~

\ (

ei:'

+

~

, .. }')

cl

V<\

(2. 2)

the integral being over all elements of mass of the system. For an individual component rigid body of mass mi with mass centre at Ci \xi' Yi' zi) the kinetic energy is

(2. 3)

where (Ai' Bi' Ci) are the principal moments of inertia, and (Pi.· Qi' Ri) are the angular velocity components· (relative to Oxyz) about the principal axes.. In any application. the kinetic energy must be expressed in terms of the particular generalized coordinates.

* i. e .• in which the gravitational force field is assumed linear over the satellite - see Eqs. 2.8. The integral of these equations. when the origin is the maaS centre, yields F = m go' the condition for a Keplerian orbit.

"

'.'

The foregoing is routine. and involves no departure from con-ventional applications of Lagrange's equation. The novel features of this application are bound up with the generaliz~d force term "OW

/'0

Cfr .... , which must include the conservative gravitational field. the conservative inertia force field. and dissipative terms arising from friction.

2. 1 The Gravitational Force Field

The gravitational force per unit mass is

with cornponents

f

=_)J.

f

-~ r~

-f~

-=

-)l.

X 6~ f~

t~~ ~

-

7~ ~

+~~ ~ ~~R-~)

The radius vector has the length

1.

'/2-f

-=

L~4+ ~2..-+ (~-~)

]

whence 'l.. 1. -~t (2.4) (2.6)

f-~

=

R~

L

(~)

-T

l~) ~

(\-

~

)l

1

The quantities

(t '

-i

I { ) are of order 10- 6 and hence it is a very good approximation to linearize the force field. Thus

(2. 7)

and

2. 2 The Inertia Force Field

The angular velocity of the reference frame is

- 3

~

and the acceleration of the origin is

Q.o ::

!

~o

(2. 9)

where

_~o

=

_R~

l

_-

..

_R.

It follows that the inertia force field per unit mass due to motion of the re-ference frame is given by (see Ref. 1)

(2. 10)

These relations, which are linear in the coordinates, are)unlike Eqs. 2.8, exact.

variable to

'6'

whence

and

We find it convenient to transform from t as the independent We have that

d'C '::

~át cl "

d

di:

wdi'

2. 3 The Total Force Field

The total force per unit mass is the sum of Eqs. (2.8) and (2.11) i. e.

(2. 12)

2.4 Force and Moment on a Finite Body

It is required subsequently to have expressions for the total force and moment acting on a finite body, viz:

f

~

\!

dW\

(2. 13)

~ ~

\

î~! d~

where ~ is the position vector of

Ó'('V\

relative to the body mass centre C (Fig. 2. 2) and hence G is the moment about the mass centre.

Integrating Eqs. 2.12 for the resultant force, we get

(2. 14) where (;)1.._ ~o ó-W1.. Q\ :: _{C, :} _1 R- l. c:\)( Ql. =. I át;)"l.

_{-t::_~o}

Cl.: w1..-t '2.

~

1:. ;R- I R. _R (2. 15) Q~= 'Iv -W7.. 2. C~: -~<::;)

The integral for the moment yields

L=

À,

J~'1

-\-

Ä.2

J?~ +À~ ~~ii.

dWl

J

t2.

dIl?

r'I\;

BI

~ç

*

B2, I,?

-T

u~ d~

(2. 16)

N

=

C,

J~"2

-\-

Cl.

J,\

*

C

3 \

'?

~; à~

where 'l.

-~ ~

_ -W'

p... ::

_1.

ál::)

_.B,::

_{C\ ::}

\

2,

d'f

_R

Ä.z..=

';;)'"+3 ~o

,

d (:;)'

(1.'" \

ó"W

"I. (2. 17) R- el.'::.

ï

d'f

-1

~ and etc. J

2.5 Forces and Moments for Small Eccentricity

When the eccentricity of the ellipse is small, such that e

<<.

1, then the radius Rand the angular velocity ~ are given by (see Appendix of Ref. 2)

~

'" 0C) (\'\"

le "Os.

'Ir)

R::.

RC) (\ -

e

c...o~ ~

')

(2. 18)

where Ro and ~o are the mean values of R and ~ J and are related by

(2. 19)

'

Hence

(2. 20) The value of go occurring in Eqs. 2. 15, 17 is

~O::. ~2..

hence

(2. 21)

Af ter substituting Eqs. 2.20, 21 into Eqs. 2.15, 17 we get the following expressions, periodic in ~ , for the coefficients of the force and moment equations:

'2..

_0'1.

_.

Q\ -=.

~o

ec.os

'C C : 1 _{, 0}

e

,Ó).M, ' (

~

-=. _-2.

:t .

_e~'t C~:

Z0:

(~T _{\oe c..oc; 'lS ')} 1 ':;:;0'1. (\ -\-4-

e,Q)S

'6" ')

,

-

't

-

(z.

2.

'2.)

U₃

=

_~~ = -l,~o (\-\-4e ~'g)

2.6 Reduction for Near-Principal Axes

The moment equations (2.16) contain moments and products of inertia of the constitutent body with respect to its gravity-oriented centroidal axes

C

~

'?

I>

•

Since the bodies rotate with respect to these axes, the

inertia coefficients in general vary with time. Let A, B, C be the moments of inertia of the constitutent body with respect to a set of principal axes, which are located by the small Euler angles

't

9 J

4>

relative to the frame

C

1\ "2 ~ These angles are successive rotations about body axes initially coincident with

C

~~~ following conventional aeronautical practice (e. g. as in Ref. 1). The inertia coefficients which appear in Eqs. 2. 16 are then given by:

(2. 24)

Jc:-~ ::.(~-c)e

2. 7 Equations for the Symmetric or Longitudinal Forces

lf Oxz is a plane of symmetry of the satellite, and if there are no unsymmetric external forces (such as solar radiation), a motion of the system is possible in which

'3

=

1"

=

~:. 0 The forces and moments of interest (the 'longitudinal' ones) are then

(2. 25)

in which the variables are (X. z, e). A final step of linearization is now made, in which we neglect products of e with the small angle

e

and with the small quantities

d~/d'('

,

d}/cl'f (X, z may not be small for individual constituent bodies). The first order expressions for the forces and moments are finally

F::c. ::

'tY)

~o'l. [le~

",)

i -

üe4iM.

't)~

-t2

~~

)

f.~:: 'Y'rI~o2. ~l.e~'t)

i:

-t

(~-t \oe~"')~-'2.i~

1

I't\:: -

0!

[~t ~-

c. ')

e

-t 2.

e

~ .ivA

y

1

The ellipticity of the orbit enters the forces in the periodic terms which multiply x and

z,

and in the inhomogeneous term

_ 2e~~ 'f

0

₀" in the pitching moment equation. It may be anticipated

that these terms will in generallead to a steady state oscillation of the system at orbital frequency.

2.8 Equations for the Asymmetrie, or Lateral Forces

The other three forces and moments, to the same degree of approximation, are

F-a

-

-

'M~o2.

(\;.

3e~~) ~

L

=

c0

o'

~

4(C-S)

cp -

2 \

Yj

~

dW\

1

'"

':.

~:

l

u~

-

~

)

~

-2. \

Y2

:~

d'"

1

The integral expressions in Land N can be simplified at this point in view of the linearization with respect to the Euler angles. Let subscript p denote the coordinates of a mass element with respect to body-fixed principal axes. Then whence

t:

~ ~p-Vl~t-t~pe

"1-:

~~'t-\-\(p-~p<\,

ç

~ -~pe

+

'(pc?

;-~~

\

d~dm: \(~pt" ~p-çpq,

)(-qp

~t ,,~\> i~

)<1'"

(2. 27)

Neglecting produets and squares of the Euler angles, and recalling that the products of inertia are zero, we get

(2. 28)

and similarly

Thus the lateral and longitudinal force systems are uncoupledJ in the sense that displacements of one set do not introduce forces in the other. The

integrals, are:

l -:

;:U\)'2.

~

4-

(C-Bl

ct> -\-

(~+C

-B

)!~

1

\4::

-~:

t

~-~

')

t

~ (~+

(-

E:, ')

~

1

(2.29)

2. 9 Longitudinal Equations of Motion for the Particular System

The foregoing theory is general, and is applicable to any com-pound satellite system. From this point on however, it is no longer pro-fitable to maintain a general viewpoint, since the details of the analysis are so strongly affected by the particulars of the system to be studied. We therefore turn now to the system shown in Fig. 2.3, which is to be analyzed from the standpoint of passive attitude stabilization. It consists of the

satellite body, the two identical universally-hinged stabilizer rods at top and bottom (subscripts 0( , ~ ), and the two yaw stabilizers front and back which are hinged so th at they can rotate only in yaw. Subscript b is used to denote the body plus the two yaw stabilizers. There is friction in the stabilizer hinges, so th at the rods are acted on by couples - ~

èI.

and -,< ~ whenever

they are in motion relative to the satellite body. If c = 00 , the rods are rigidly locked, and no damping (energy dissipation) can occur. If c

=

0, there is again no dam ping possible. Thus it is evident that there will be an optimum value of the hinge friction between zero and infinity.

As generalized coordinates we take c( , ~

êb,

defined as shown in the figure.

Ca( ) Cp)

eb

are the mass-centres of the three con-stitutent bodies. Their positions are connected by the following relations, which express the fact that 0 is the mass-centre of the system.

(2. 30)

Kinetic Energy

The kinetic energies of the three constitutent bodies are given by

\ ..!. '2. ..!. 'L. \ •

2-Tb ::

~

mb (Xp

-t)b ')

+

2

Bb eb

,

(,'2. .:."')

,

('

2..

Tg( :.

1:

'N\ 0{ 'f.. rJ..

+

110(

..,.

1

Ba(

e

b 1-

~

') (2. 31)

T~

::

t

'M~

(;:; -t \ ; )

~

I

B~

(

ë

_b

-+

~

')'J.

The mass-centre coordinates are given in terms of the generalized coordinates

by -whence

)(0( ::

~b

+

Q,~eb

-t

1,

~

(eb+ol)

~o(

' :

~b

+

(l

~

eb

-\0

.t

~

(eb

1-0{ ')

~f--: ~'P- ~llÀNv~-Jr~ (~+0)

}-f.. ::

~

b - Q

~

f\ -

~.c.o-ó.

(eb

+

~

')

)( _{ol.. -}

-

_X

.

b

+

ct

~eb

G

b

t..e,.~(~

hl) (&b

.,.~)

. . . .

.

.

~oI. ~ ~b -Q~~~ -.t.~ (Gbtol~U~bt

ot')

:i~

-:

~b-<L~~~-..e,~(~t~)(~~~)

~~:. ~b +a.~~~-T..tAÀM.,~T~)(èb-t~)

For convenience. we rewrite Eqs. 2.30 as

(2.32)

(2. 33)

we find th at

From Eqs. 2. 32 and 2.34, to the first order in 8, 0( , ~

';{. = _ 'N\oI

.t.

(0{ -

r<. )

b YY'I

,-= total mass of satellite.

It then follows that, again to the first order, the partial derivatives of T required for Eq. 2. 1 are:

.

.

~b

":

Bb

~b'"

Ba(

(gb-t~) -\'

_

E~(E:\-t ~)+'MoI.~o(~b +",~X.fi dc,~b

.

.

.

.!. ~~b (" " .!.. ~ )(C'. ..:... 0 x.~

:: mb:::C

b ~~ 't

BQ(

~

t

6{ ) .,. \"1\0( ~ol ~ .,. w.~:l(! dë( d\ _ ~~

-where

.

"

mb

~b ~b

-I-

I?:.(>

U;lb

'<

~)

-\-

NI ot

i

0(

~

0( -\- ....

~ ÎI.~ ~~

• 'VY\ • • Xo(

-= -

-tr

~ (~-~) ~

Cl

eb

+1,.

(ë

_b

+~)

x(=>": -

t

~ (~-0)

-

a.~

-.t-

(G-b

+

~)

• ~

di

_{~ ~}

- (Q.tt) ;

~9b

..

cl'X.ot

:.e.

'MoI~"'tt\b

d~ 'M and _{~T /è)Qb ': dT/~~ :: ~T /~f-.}

=-

_{0 •}

The equations of motion 2. 1 therefore become

..

..

..

(,;)w

1

ê

_b

-r

I. I 0(

+

I \

~

::

~ Qb

(2. 35)

where

-=

the total moment of inertia about eb

1.,:.

BQ(

-t-

'Mol

.t.

(Q

-+.t. )

It

=

~ol

t

'M~

b'"

(l-

~)

I.~= 'N\oI

('M

al

..t. '\.)

W\ Generalized Forces (2.39)

The total work do ne by the conservative and dissipative forces during a virtual displacement is given by

bW

=

F~b ~~b

;.

ç:

1f

b

b

~b

;- }V\\.

~

eb

.;.

ç:~o{

b

'f..«;.

f1(~ b~

t

Mo((

b~t ~O{) -..c~ b~

(2.40)

-t

~'l'f.> ~

Xp> ;.

F~r~ b}~

oT

tt\f.>

(hE\

+

b~)

-,c

~

b

~

The values of Fx' Fz. M to be used in Eq. 2.40 are those given by Eqs. 2.26 i. e. they are the forces and moments exerted by the gravitational and inertia force fields. The couples -,<:.~ and -tC.~ are those produced by the friction in the hinges. In applying Eqs. 2.26 we nQte that

90(

=

Etb~~ Q~ ~ ~b-tf->

The three generalized forces are next obtained from {he virtual work ~

W •

i. e.

From Eqs. 2.32 and 2. 35 for the mass-centre coordinates we find the derivatives occurring above to be as given in the following tabie:

d/~eb

0/'00(

O/d~

-

_{- ~.e,:} 'mo(,f,.

:tb

0 _{m m}

-

_{~ ~cI. (0< -}

_{f-> ')}

_.t,~

₍

_eb

_t

_~)

_{-~~ (eb~~')}

~b

·

-

_~-<

_a.tb

_~

_(\-

_~~)

_~d.

_J;.

m

-

_~o(

_-.Qr(

_\

_-~)a(

_-

_A~~

_-

_(a.t!)t\

_-.t

₍

_\

_-

~

')

(&b+~i)

-,t.

'M~

(f\*

~

')

m

-

_-(

_I

_-~)ir

X~

- (Q ""

J,.)

-

~ 'Y't'\

J,.

-

_.t

_~

0( '\

.t (

, -

~)~

-\-

~

t.t-)

~

k

~

(9b

+cl)

.t.(\-

~)

(

~+~)

. ~ ~ (2.41)

,

..

With Eqs. 2.26 for the forces and moments. Equations 2.41 ean now be reduced after some routine work to the following linear equations:

Je

= -

~o2.l1e1~'t'

+

~

(A-C

_b

)9

_b

+

31,0(

-t

~ I,~]

100(

= -

Wo"

~ ~

I, S

'

b

-t

~

1,0(

t

le

!,4Lw'f

+

~o ~~

]

&~

:. -

~o2:l~

1\

/;b

+

~~,~+ ~eI,~~

+

~o ~

1

where A

=

Ab

+

2A 0( +2mC»( (a + b)2

(2. 42)

The equations of motion are finally obtained by combining Eqs. 2.42 and 2.38. with the result

(reil.

-T

~

1.4-)

J:,C~\~)

I,C~\~)

(l"á\Zd

+31.,)

In this form. as a result of the assumption e« 1. the

ellipticity of the orbit appears entirely in the inhomogeneous 'forcing' terms on the r. h. s. of the equations. The 1. h. s. then defines a homogeneous linear system of the sixth degree.

It should be noted that the independent variabIe of the above equations is the true ànomaly

't' .

not time or distance. and hence that periods and dampings come out naturally with 'orbits' as the unit of time or distance.

lIl. SOLUTlONS OF THE EQUATlONS OF MOTION

3. 1 Characteristic Equation and Characteristic Modes (Longitudinal) The characteristic equation of Eqs. 2.43 is

=0

It has the expansion

().2.

(11.

-1. \)

+

Z

À

+

~

1.,1 )(

t(1

À\

n ... )

lÀ~(Il~

I,)

~

Z"

Bl,1-lI,'

(À~H)1. ~

,.

0

The eigen values are then the roots of the quadratic

and of the quartic

where

T

4

:.

t.(I..2.~~~)-2.

1,2.

T~

=

!.

Z

Tl.:'

~!

4

(~l.

-\

!.?>) -\-

~

1:

1.\ -

\'2.

r

,2.

(3. 1) (3. 3) (3.4)

The quadratic equation (3.3) can be identified as the characteristic equation associated with the symmetrical mode)for which 9 = 0 and ~ - -0( • as in Fig. 3. 1. (If we set 9 = 0 and ~ =. -0( in Eqs. 2.43. the first equation is identically satisfied. and either of the remaining two yields the characteristic Eq. 3.3). The quartic can likewise be identified as the characteristic

equation associated with the two antisymmetrie modes,for which

0/.:.

~

•

(Fig. 3. 1). For these modes we can derive. from the first of Eqs. 2.43. with.

ol=.

(:3 that

=

2.1\

_r

(A'\.-It~)

À'2. -T

a

1.4-(3. 5)

where

01.

₀ and

9

₀ are the complex amplitudes of 0{ • Q in the anti-symmetrie modes.

Numerical Example

A two-parameter family of solutions has been calculated. with the following numerical data~

**

~

- .0015'& ~b

:.

eb

mb - Q,

,

.t

_-

_~

_\\

_{~b':. O.Ó'rt\b~2.}

eh

-0

~

r=

~

4 \

~b

~ot

:.

~

N\!I(

t.

1-i. e .• the length of one stabilizer is up to 11 times the satellite radius, and the mass of a length 'a' of each stabilizer is 1/2% of the mass of the satellite body. The formula for Bb implies that the radius of gyration of the satellite body is . 775a. and that for Bo( implies that the stabilizer is a uniform slender rod.

The formulae for the various coefficients which occur* in the characteristic equation are then

1.

~

\

~b":

\-\-

~ ~

~

'ö.'rO

;.\ ":

~ ~

L

'ft\çJ.

Bb

~~ t;).bO

Y1\h

* During the actual solution. it is convenient to make the characteristic equation (3.2) _{non-dimensional by dividing through by (Bb )3}

** This choiee implies that the moment of inertia of the yaw-stabilizer rods is not included in eb' If it were. a small increase in b/a would produce the

~-

~

-

-\

Bb-

_~ Y'f\ol _ 'tf.c(

jW\

b lY\ -

_{\ -\ l.l}

_~

_/'tf\

_{b ')} B~_

_{\ m}

(.e,)1.

'&b

\·80

m~

a.

The values of b/ a and

r

used were as follows

-b/a

=

1.1.5.2. 0.2.5.3.0.3.5.4.0.4.5.5.0, 6.0.8.0.11.0

r

= 0. 20 n, where n = O. 1/4, 1/2. 3/4. 1. 2, 3, 4. 5

The characteristic equation was solved on the IBM 7090 at the U of T. lnstitute of Computer Science for each of the above cases. For each

root of the equation, the characteristic decay time (time to 1/2 amplitude)

and the period were calculated, both in orbits. For the antisymmetric modes,

the ratio 9₀ /0(0 ' which defines the mode shape. was also calculated.

The principal results are presented in Table I. and representative

trends are shown in the plots of Figs. 3. 2 and 3. 3. Both osciUatory and

non-oscillatory modes were obtained. depending on the values of b/a and

r .

The best performance, from the standpoint of the number of orbits required

to damp an initial disturbance to 1/2 amplitude. was obtained for values of

.

b /

a about 5 and

r

about O. 4. Figure 3. 2 shows plots of the

least-damped antisymmetric modes for b/a values of 1. 5. 8. The value at b/a

=

5.

r

=

0.4 is seen to be only 0.34 orbits. Thus at this condition. a transient oscillation would damp during one orbit to approximately 1/8 of its initial

amplitude. Also shown as a matter of interest for b/ a = 5 only, is the damping

of the 'staggering' (symmetric) mode; it is not so rapid as that of the

anti-symmetric mode for

P

<

.57. but is greater for

r

larger than this value.

This mode is the less important one of the two. since it does not involve angular motion of the satellite body. Figure 3.2 shows quite clearly that it

would be important in design to optimize with respect to (b/a) and

r

Figure 3. 3 presents the periods of the same antisymmetric

modes. The period of the symmetric mode for b/a

=

5 is not shown - it is

nearly constant over the fuU range of \' at approximately 0.53 orbits. For

the least-damped mode at lower values of

r

is the one with longer period. and vice-versa for larger

r .

As a matter of interest. the ratio of the amplitude. of the satellite body to that of the stabilizer \

9

0

/11('0\'

is shown at

r

=

0.4 for the two modes.

3.2 Steady State Longitudinal Oscillations Caused by Ellipticity of the Orbit To calculate the steady periodic motions which are solutions of Eqs. 2.43. we !et

e

=

~

l

9

₀ ei,lf

1

0 ( ;

~

l

~o

e

t~]

~

::

~

L~o el~J

~'(:.. ~let~)

(3.6)

The complex-variable equations that correspond to Eqs. 2.43. (i. e. the equations of which 2.43 are the imaginary part) then become

(-I

oT

~4

')

'2.

1:.,

_2.

_1:,

2.1\ (- 1:1. oT

i

z -\-

~

1:.,)

-I.

_~

l

t,

-1.

₃

(-l~

_{-t ik}

+~)

The solution of these equations yields

a(o _ ~ :.

e

e

~

e

0<0

e

~o Q

;. 2l4-I ;'" -

1. ( "- -

1.~

)]

8

I ,i

+

(~-1.

1

)

(t -

~ ~)

where k =

L

Z, ~ ~ 'I. ,

'1z. •

21

21,

(~.7) ~

I,

(3. 8) (3. 9)

Numerical Example

The values of \

0<01

eland \

e

o

Ie \

were calculatedfor the same range of parameters as were used in Sec. 3. 1, and the results are pre-sented in the plots of Figs. 3.4 and 3.5. For example. at b/a = 5, and

t'

= .4, the values which. were near optimum from the standpoint of transient decay, we get from Fig. 3.4 that

\~\

=4.2

Hence for a low-altitude orbit,say R = 5000 mi, with Rapogee - Rperigee = 100 mi, we get

100

e = 2x5000= . 010

and

9₀

=

4. 2.x . 010

=

. 042 rad = 2. 40

Figure 3.4 shows that it is important to design the stabilizers correctly in order to avoid the resonances evident at b/a

==

2. 3 and 4.2. At these stabilizer lengths, one of the normal periods in pitch is the same as the orbital period, resulting in large steady state amplitudes. It mayalso be noted that the optimum design from the .standpoints of transient decay and response to orbital ellipticity may be appreciably different, requiring a practical com prom ise.

3.3 Oscillations Causedby Radiation Pressure

Solar radiation pressure can produce a moment about the mass centre of any component element when either the element or the radiation is not symmetrical with respect to the mass centre. The latter situation exists when the satellite is in the penumbra, see Fig. 3.6, where there is a gradient

in the radiation intensity. In this section we examine the magnitude of the disturbing torques which arise from this source relative to those from orbit ellipticity. We consider a satellite describing a circular orbit; with z axis oriented along the local vertical at all times.

The solar radiation pressure in the vicinity of the earth on a perfectly reflecting* surface normal to the incident radiation is approximately

P = 9. 1 x 10- 5 dynes / sq. cm.

=

1. 902· x 10-7 psf

* For surfaces which fully or partially absorb the incident energy, the pressure is less. We use the maximum value here in order to obtain a conservative

If the surface is inclined so that the normal makes an angle ~ with the radiation, as in Fig. 3.7, then the pressure aCts along the normal and has the magnitude

(3. 10) When integrated over the surface of a circular cylinder of diameter d with axis inclined at al) angle

& ,

as in Fig. 3.8, the normal force per unit length is*

(3. 11) The influence of radiation in producing rotations of the satellite will be largest when the orbital plane contains the radiation vector, as in

Fig. 3. 9. Let us consider for that case the couple exerteq on a rod of diameter d and length

l

.

Let the radia~ioI]. pressure

P

be given by

(3. 12)

The normal force per unit length is then (in the -x direction)

(3.13)

(3. 14)

which yields af ter integration

~

tv\:O.b1d~"1.b ~ ~

(3.15)

lf we define a 'momental impulse' by

(IY\.

I.

.\"ad.

= \

tv\.

c! '(

(3. 16)

Then the total impulse on passing through the penumbra is

~z

(M.

1..

}raJ.

~ ~

MJ

1S'

')SI. ~ '(2.

;O.<Olcl~2.b~ ~ !~d,(

(3. 17)

t.

where

"6',

and

"'1.

are the values of ~ at the edges of the penumbra. The included angle of the penumbra of the Earth is less than 1/20, hence we may assume its edges parallel for values of

b

greater than say 100, or RE/R = cos ~

<

.99. Let h be alocal space-fixed coordinate normal to the radiation, as shown in Figs. 3.6 and 3.9.

Then

and the relation between hand position on the orbit is

_\

-

~~

R -

'Ke.

Finally

(3. 18)

Assuming

&

to be constant on passing through the penumbra,

.

~

i

₂

I \

ei.e

~+b~·

c!p \

\.M.

L.

)rad.

= -

·0'5'55

RE

~

b

~ d~

ei'( "tI

(3.19)

The integral is (P_{2 -} PI)' Since~, corresponds to darkness, since

'lS'2

corresponds to full illumination P2

=

1. 90 x 10- 7 psf

PI = 0, and

Using the minimum vàlue of

b

quoted above, i. e.

&

=

100 , and taking RE

=

21 x 10 6 ft. we get

-1'.\)

IJ~

(M·l.)fad.

= -

2.1 \

(10

d...(.

ft. lb. rad.

For comparison, we consider the driving moment due to

ellipticity which acts on a rod. This is obtained from ~ ,of Eqs. 2.42,

by setting 9

=

~

-:

f->=

0) 9

i. e.

_{1 ::}

-è:::J:

r

'J.elkv'(

e

where I is the moment of inertia of the rod, i. e.

I

~

The impulse of this moment during one half cycle is 't\'

(1'1..1 \ '"

~

1-9

d't '" -

~

.0

0

1

'rn

.te

o

the ratio of the two momental impulses is

("I...:r..

\'dJ,

=

( "".I;)e

The value of to be used in the comparison is for R':' RE

i. e.

~Q2.

=

~E

_{IR; '::}

1.'5~(\(~-b')

whence

(~.l..)r~.

::.

S.

~2 (\D-~) ~

lt'\·

1.) e

(~J

Im ')

is larges,t if the cylinder is a thin-walled tube - then

where

d~

_

.l-M -

1I'ft

~

=

density of the material

t

=

wall thickness

(3. 21)

(3. 22)

For a Magnesium tube (to choose a light metal) ~ = 3. ~8 slugs/cu. ft. and for a very thin wall, t = . 005 in. ,

we get

_{dl ::}

_21b

m

and

_{( M. L')}

_ya.l.

=

lM.

I.)

e

-b

,.~

x

\0

e

T.qe value of e fOF which the two

effec~s

are equq.l, i. e.

l

M. r'\dd

=

(M. I.)e.

is

tn.~r~fore

l

e=

~.

2 x

~0-6.

T!lUS the perturbation due to radiation gr8;dient is sev,eral orders, of ~agr;lit,ude smaller than t~at due to elliptic;ity for aU orbits other 'tq.an thosE; irhi?'W are

near~y

perfect

~ircles.

This effE7ct 'can

th~~efore

be ignored in considerations of attitude control fo~ symmetrical configura,-tions. It must be noted, however, that this m~y not b!= f?o if the satellite is very unsymmetrical, a:n~ Flat the cOnîtuSion stated applies only to rotations I

of the syste~, not to the I?erturbation of the orbit. It lis known that the lattet effec~ can i

bT 'significq.nt fbr low density satel\i~es. , , '

, ; , I

, I , ' 1 ' I , At/high alt~ttll'dEiSl~ ' (e.g~i for ~ syh'chroJ;1,óus sate'l:~ite) wh~~ 'the

'èf.l=ctfng' ü>r'que due to gravity

gradie~t

is

relq.tiv~lymuch s~aller,

:

H~e

'

solari _{ra,diati,on preSsure may} it~~l~ _,pr:oduce apprebab~e _{asymmetry in the}

configuration, (see Fig. 3. ~O.>.

*

Pitèi:ping'motion of the satellHe body may then result. ' This phenom~ilOn has not!been st~died in ihiS report.

*

This was pointed out to the author by ;Mr. L.~. Davis of the General

Eleçtri~ ~o., MissUe & 8r:ace Vehiele :qept,

1. Etkin, B.

2. Etkin, B.

REFERENCES

Dynamics of Flight, John Wiley and Sons, 1959-Eqs. 4.8, 6.

Longitudinal Dynamics of a Lifting Vehicle in a Circular Orbit, UTIA Report No. 65, 1960.

TABLE I

SOL UTIONS FOR CHARACTERISTIC MODES

Note: Where no period is shown - the mode is non-periodic

Hinge Orbits to Period, Damping

damping 1/2 ampl. Orbits, ratio,

r

0.1 2 T

ç

b/a=1.0 0 Inf. 0.4348 0 Inf. 3. 1889 0 __ Inf. _{0.4337 0} 0.050 0.0485 2.5772 O. 9857 2.3028 3.0809 O. 1456 0.0721 --

-

-0.0345

--

-

-O. 100 O. 1755

--

--0.0130

--

-

-0.2674 --

--0.0122 --

--1.3335 2.7329 0.2199 O. 150 0.2745

_--

--0.0083

--

--0.5063

--

--0.0078

--

-

-1. 4310 2.4585 O. 1857 0.200 0.3710

_--

_-

-0.0062 --

--0.7365

--

--0.0058

--

--1. 7499 2.3517 O. 1462 0.400 0.7512 --

--0.0030

--

--1. 5941

--

--0.0029 --

-

-3. 2797 2.2549 0.0754

Hinge Orbits to damping 1/2 ampl

r

01 "2 0.600 1. 1294 0.0020 2.4247 0.0019 4.8694 0.800 1. 5071 0.0015 3.2485 0.0014 6.4701 1.000 1. 8845 0.0012 4.0697 0.0011 8.0748 0 Inf. Inf. Inf. 0.050 O. 1632 2.3293 O. 1504 O. 100 0.0816 1. 1285 0.0754 O. 150 0.0544 0.1174 0.0320 0.7477 TABLE I (continued) Period, Orbits, T b/a=1.0

--2.2379

- --2.2320 --2.2293 b/a=1.5 0.4688 2. 3390 0.4669 0.4941 2.3157 0.5058' 0. 6049 2.2327 0.7331 1.4745 - --2.0473 Damping ratio,

'ç

- --0.0505

-

--0.0379

--0,0304 0 0 0 0.3160 O. 1087 0.3469 0.6321 0.2127 0.7306 0.9481

--0.2884

Hinge Orbits to dam ping 1/2 ampl.

r

0.1 2 0.200 0.1051 0.0253 0.2374 0.0202 0.6773 0.400 0.2501 0.0106 0.6708 0.0091 I. 1414 0.600 0.3842 0.0069 1. 0564 0.0060 1. 6918 0.800 0.5163 0.0051 1. 4313 0.0044 2.2483 1.000 0.6477 0 .. 0041 1.8022 0.0035 2.8064 0 Inf. I Inf. Inf. 0.050 0.3853 2. 3970 0.3342 TABLE I (continued) Period. Orbits. T b/a=1.5

--1.8067

--1.5554

--1~5210

--1. 5098

--1.5048 b/a = 2.0 0.4887 1. 8576 0.4859 0.4935 1. 8499 0.4961 Damping ra-tio.

~

--0.2815

--O. 1482

--0.0984

--0.0737

--0.0589 0 0 0 O. 1395 0.0846 O. 1612

Hinge Orbits to damping 1/2 ampl.

\'

Ol 2 O. 100 0.1927 1. 1684 O. 1677 O. 150 O. 1284 0.7458 0.1125 0.200 0.0963 0.5302 0.0851 0.400 0.0867 0.0333 0.3350 0.0207 0.5959 0.600 O. 1622 0.0178 0.5712 0.0128 O. 9039 0.800 0.2273 0.0127 0.7908 0.0094 1. 2123 1. 000 0.2900 0.0100 1. 0049 0.0075

\

1. 5198 TABLE I (continued) Period, Orbits, T b/a

=

2.0 O. 5089 1.8245 0.5316 0.5381 1. 7722 0.6140 0.5889 1. 6681 0.8516

-

--1. 2210

--1. 1710

--1. 1568

--1.1508 Damping ratio,

ç-0.2790 0.1693 0.3293 0.4186 0.2529 0.5147 0.5581 0.3271 0.7402

--0.2199

--O. 1411

--O. 1044

--0.0830

Hinge Orbits to damping 1/2 ampl.

r

01. 2 0 Inf. Inf. Inf. 0.050 O. 7497 2.4709 0.6005 O. 100 0.3749 1. 2137 0.3015 0.150 0.2499 0.7842 0.2026 . -0.200 O. 1874 0.5608 0.1539 0.400 0.0937 O. 1569 0.0466 0.3787 . 0.600 0.0625 0.3404 0.0230 0.6190 0.800 0.0993 0.0307 0.4903 0.0162 0.8487 L 000 O. 1408 0.0216 0.6325 0.0126 1. 0740 TABLE I (continued) Period. Orbits. T b/a

=

2.5 0.5017 1. 5458 0.4977 0.5030 1.5424 0.5021 0.5072 1. 5317 0.5161 0.5144 1.5114 0.5425 0.5249 1. 4763 0.5886 0.6207

--1. 0335 1. 0697

--0.9710

--0.9578

---,

--O. 9525 Damping ratio.

ç

0 0 0 0.0736 0.0685 0.0916 O. 1472 O. 1375 O. 1850 0.2208 0.2074 0.2825 0.2944 0.2782 0.3878 0.5888

--0.2875 0.8832

--0.1700

--O. 1232

--0.0971

Hinge Orbits to damping 1/2 ampl.

r

.

Ol. ₂ 0 Inf. Inf. Inf. 0.050 1. 2907 I 2.5549 0.9358 O. 100 0.6454 1.2608 0. 4702 · 0.150 0.4302 0.8215 0.3162 0.200 0.3227 0.5951 0.2404 0.400 0.1613 " _{0. 2478} 0. 1308 0.600 O. 1076 0.2073 0.0381 0.5009 0.800 0. 0807 0.3236 0.0245 0. 7080 1.000 0.0645 0.4276 0.0186 0.9064 , TABLE I (continued) Period . , Orbits , T b/a

=

3.0 0.5106 1. 3270 0.5054 0. 5111 1. 3251 O. 5078 0.5126 1. 3193 0. 5154 0. 5150 1.3088 0.5289 0. 5186 1. 2921 0. 5503 0. 5447 0.9170 1.0559 0.5988 - -- -0. 8438 0. 7114 -0. 8353 1. 0370 - --0. 8321 Damping ratio,

ç

0 0 0 0.0435 0.0570 0.0596 0. 0870 0.1143 0.1197 O. 1306 0.1726 O. 1810 0.1741 0. 2323 0.2441 0. 3481 0. 3770 0.6639 0.5222 --O. 1822 0. 6963

--0. 1287 0.8704 --- -0.1005

Hinge Orbits to

. dam ping 1 /2 ampL

r

01. 2 0 Inf. Inf. Inf. 0. 050 2.0420 2.6557 1. 3118 O. 100 1. 0210 1. 3142 O. 6593 O. 150 I 0.6807 0.8605 0.4436 0.200 0.5105 0.6282 0.3374 0. 400 0.2553 0.2303 0.2097 0.600 O. 1702 O. 0985 0.0778 0. 4609 0. 800 O. 1276 0.2192 0.0346 0.6·694 1.000 0.1021 0.3019 0.0250 0.8648 TABLE I (continued) Period " Orbits , T b/a

=

3.5 0.5171 1. 1651 0.5101 0.5173 1. 1639 0.5118 0.5179 1. 1600 0.5167 0.5189 1. 1532 0.5254 0.5204 1. .1428 0.5385 0.5305 1. 0240 0.6868 0.5487

--0.7586 0.5777

--0.7564 0.6227

--0.7554 Damping ratio,

')

0 0 0 0.0279 0.0482 0.0429 0.0557 0.0966 0.0859 0.0836 O. 1458 O. 1292 0.1114 O. 1962 0.1729 0.2229 0 .. 4394 0.3389 0.3343

--0.1782 0.4457

--0.1233 0.5571

--0.0956

Hinge Orbits to damping 1/2 ampl.

\1

0.1. 2 0 Inf. Inf. Inf. 0.050 3.0368 2.7828 1. 6882 0.100 1. 5184 1. 3794 0.8486 O. 150 1. 0123 O. 9060 0.5710 0.200 0.7592 0.6645 0.4344 0.400 0.3796 0.2645 0.2609 0.600 0.2531 0.4692 O. 1077 0.800 0.1898 0, 1452 0.0484 0.6906 1. 000 0.1518 0.2181 0.0320 0.8962 TABLE I (continued) Period. Orbits _, T b/a

=

4.0 0.5220 1. 0409 0.5128 0.5221 1. 0399 0.5140 0.5223 1. 0369 0.5177 0.5228 1. 0316 0.5241 0.5235 1. 0239 0.5335 0.5280 O. 9513 0.6239 0.5359 0.7026 1. 2473 0.5476

--0.7050 0.5638

--0.7055 Damping ratio, ~ ~ 0 0 0 0.0189 0.0411 0.0335 0.0378 0.0824 0.0670 0.0567 O. 1243 O. 1004 0.0756 0.1671 0.1339 0.1513 0.3679 0.2544 0.2269 0.1625 0.7867 0.3025

--0.1116 0.3781

--0.0863

Hinge Orbits to damping 1/2 ampl.

f'

Ol 2 0 Inf. Inf. Inf. 0.050 4.3079 2.9507 2~0197 0 .. 100 2.1539 1.4644 1.0151 O. 150 1.4360 O. 9637 0.6829 0. 200 1. 0770 0.7091 0.5193 0.400 0.5385 0.3069 0.2930 0.600 0.3590 0.5098 0.1243 0.800 0.2692 0.7530 0.0832 1. 000 0.2154 O. 1577 0:0400 0: 9786 TABLE I (continued) Period. Orbits • T b/a

=

4.5 0.5256 O. 9430 0.5136 0.5257 0.9421 0.5146 0.5258 0.9395 0.5176 0.5260 O. 9·350 0.5228 0.5264 0.9284 0.5304 0.5287 0.5992 0.8697 0. 5326 0.6678 0. 9627 0.5382 0. 6716 2.5217 0.5456

--0.6724 Damping ratio;

ç

0 0 0 0.0134 0.0351 0. 0280 0.0268 0.0704 OL0560 0.0403 0.1061 0. 0839 0.0537 0.1425 0.1117 O. 1074 0.2100 0.3104 0.1611 0.1426 0.6486 0.2147 0.0976 0.9578 0.2684

--0.0754

Hinge Orbits to

dam ping 1/2 ampl.

r

Ol 2" 0 Inf. Inf. Inf. 0.050 5.8874 3.1838 2.2649 O. 100 2. 9437 1. 5816 1.1377 O. 150 1. 9625 1.0427 0.7647 0.200 1. 4719 0.7692 0.5806 0.400 0.7359 0.3365 0.3254 0.600 0.4906 0.5733 O. 1366 0.800 0.3680 0.8481 0.0917 1. 000 0.2944 O. 1082 0. 0522 1. 1022 TABLE I (continued) Period, Orbits _, T b/a=5. 0 0.5284 0.8644 0.5127 0. 5285 0.8637 0. 5136 0.5285 0.8612 0. 5162 0.5287 0.8571 0.5207 0.5288 0.8510 0.5273 0.5301 0.5872 0.7960 0.5322 0.6474 0.8299 0.5352 0.6498 1. 2008 O. 5390

--0.6503 Damping ratio,

S

0 0 0 0.0099 0.0298 0.0249 0.0197 0.0598 0.0498 0.0296 0.0900 0.0747 0.0395 O. 1208 0.0994 0.0790 O. 1885 0.2598 0.1185 O. 1233 0.5557 O. 1580 0.0840 0.8216 O. 1975

--0.0648

Hinge Orbits to damping 1/2 ampl.

r

01. 2 0 Inf. Inf. Inf. 0.050 10.0982, 4.0486 2.4109 O. 100 5. 0491 2.0182 1.2076 O. 150 3.3661 1. 3387 0.8075 0.200 2.5246 O. 9970 0.6082 0.400 L 2623 0.4852 0.3093 0.600 0.8415 0.7823 O. 1501 0.800 0. 6311 1. 1365 O. 1030 1.000 0.5049 1. 4685 0.0797 TABLE I (continued) Period, Orbits, T b/a

=

6.0 0.5323 0.7494 0.5052 0.5323 0.7487 0.5060 0.5323 0.7465 0.5082 0.5324 0.7427 0.5120 0.5324 0.7372 0.5175 0.5329 0.6839 0.5717 0.5336 0.6291 0.6738 0.5346 0.6256 0.7956 0.5359 0.6245 1. 1130 Damping ratio, ~ 0 0 0 0.0058 0.0203 0.~0231 0.0116 0.0407 0.0462 0.0174 0.0609 O. 0696 0.0232 0.0811 0.0932 0.0464 O. 1532 0.1993 0.0696 0.0881 0.4428 0.0928 0.0604 0.6475 O. 1160 0.0467 0.8382

Hinge Orbits to damping 1/2 ampl. . . _Ol

r

2 ..

_-

.

.

0 Inf. Inf. Inf. 0.050 23.5849 .. 12. 9073 1. 9720 O. 100 11. 7924 6.5420 O. 9839 O. 150 7.8616 4.4639 0. 6537 0.200 5.8962 3.4627 0.4879 0.400 2.9481 2. 2318 0. 2365 0.600 1. 9654 2.2385 O. 1522 0.800 1. 4741 2.5826 0. 1115 1.000 1. 1792 3. 0374 0. 0880 TABLE I (continued) b/a

=

8. 0 Period" Orbits , T 0.5359 0.6326 0. 4601 0. 5359 0. 6323 0. 4605 0. 5359 0.6315 0. 4615 0. 5359 0. 6303 0. 4633 0.5359 0.6285 0. 4657 0. 5360 0. 6190 0.4822 0. 5361 0.6104 0.5082 0. 5363 0.6055 0.5462 0.5365 0.6030 0. 6056 Damping ratio,

~

0 0 0 0. 0025 0.0054 0. 0257 0. 0050 0. 0106 0.0515 0. 0075 0. 0155 0.0777 0. 0100 0. 0200 0.1044 0. 0200 0.0305 0.2188 ·0.0300 '0.0300 0.3447 0.0400 0.0258 0.4743 0.0500 0. 0218 0.6035

Hinge Orbits to damping 1/2 ampl.

r

01.. 2 0 Inf. Inf. Inf. 0.050 59. 9687 170.8237 1. 7068 O. 100 29.9844 85.8295 0.8533 0.150 19.9896 57.6842 0.5688 0.200 1~ 9922 43.7516 0.4266 0.400 7.4961 23.5611 0.2131 0.600 4.9974 17.6025 O. 1420 0.800 3.7480 15.2176 O. 1064 1. 000 2. 9984 14.2713 0.0850 TABLE I (continued) Period, Orbits, T b/a

=

11.0 0.5363 0.5912 0.3496 0.5363 0.5912 0.3497 0.5363 0.5912 0.3500 0.5363 0.5911 . 0.3505 0.5363 0.5911 0.3511 0.5363 0.5907 0,3558 0.5363 0.5901 0.3639 0.q363 0: 5895 0.3762 0.5364 0.5890 0.3939 Damping ratio,

t'

0 0 0 0.0010 0.0004 0.0225 0.0020 0.0008 0.0451 0.0030 0.0011 0.0676 0.0039 0.0015 0.0902 0.0079 0.0028 O. 1806 0.0118 0~0037 0.2714 0.0157 0.0043 0.3626 _.. " O. 0197 0.0045 0.4542

o

z

x

E

FIG. 2.2 _{BODY -CENTRED COORDINATES}

C is mass centre of constituent body