11

1_

01

(

4

4

Figure

2.

Stable

(shaded)

and unstable

regions

for the X= 0

solution of Mathieu's

equation

d2X/dT2

+

(a

+

fP

cos

T)

X

=

0.

The

points

(2.6)

lie on the

straight

line

OP,

and for the inverted state

to be stable

they

must

all lie within the

shaded

region

with a <

0,

in

the manner indicated

(for

N=2).

Now,

the

stability

regions

of Mathieu's

equation

are

well

known

and indicated in

figure

2.

We

achieve

stability

of

our inverted

system

if,

by

choosing

e

and

ow

appropriately,

we

can

place

all the

points

2(-/2,

6g),

i

=

..

,N

(2.6)

in the shaded

part

of

the

region

a <

0.

To see

that this

can indeed

be done

it

is

simplest

to

pass directly

to the

high-

frequency

limit,

i.e. 02 > 02ax

so

that

w0)/02

is small for all i

=

1

...V,N.

Now,

when

lal

is

small the

stability boundary

which

passes

through

the

origin

is

given

asymptotically by

a

=

_fl2

(see,

for

example,

Jordan &

Smith

1987,

p.

257).

Moreover,

the

upper

boundary

to the

shaded

region

in

a

<

0 is known to meet

the

a

=

0

axis

at

I0fI

=

0.450.

Thus all

the

points

(2.6)

will be in the

stable

region

if

(2

1222/ 2 1

<

e/g

<

0.450,

i=

1,

...,N,

i.e.

if

/2g/oi

< e <

0.450g/0

(2.7)

for all

i

=

1,...,N.

This is

ensured

by

(2.1).

While

this

completes

the

proof,

it is worth

considering

a

more

general

and

geometric

approach

to the matter.

All

the

points (2.6)

lie on a line

through

the

origin

with

slope

-eo0/g.

Imagine,

then,

that we attach an elastic band

OP

to the

origin

in

figure

2,

stretch

it

out,

and mark

points

along

it

so that their distances from

the

origin

are in

the

proportion

o1'

:

:

...:

(o.

By choosing

e

and

w0

we

may

vary

the

length

of the

band,

and its

slope

-

eO2/g,

at

will.

Now,

if

)max

and

o)min

are

almost

equal,

so that the

marked

points

are

tightly

clustered

along

the

band,

we

may evidently

steer those

points

into

the

stable

region

in

figure

2

even

if

c1a

is not

particularly

small,

so that 02

is

not

large compared

with

2ax.

If,

on

the

other

hand,

COmax

is

substantially

greater

than

omin

-

as

will

typically

be the

case

-

we

may

still steer all the

points

into

the

stable

region,

but

only

by

moving

the

band

very

close to the a

=

0

axis in

figure

2

and

by shrinking

it

accordingly.

The

stability

criterion

(2.1)

will then

apply,

and the

region

of

stability

in

the

e-oo

plane

will

be as indicated in

figure

1.

Proc.

R. Soc.

Lond. A

(1993)

A

pendulum

theorem

241

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