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Four freedoms, reducible to three if desired by a third thread in each case, diagonal in the first and second, lateral in the third and fourth, the freedom thus annulled being in each case stable and independent of the rotation of the fly-wheel. Three modes essentially involved in the gyrostatic system in each case, two inclinational and one azimuthal.

No. 1.-Azimuthally stable without rotation; with rotation all three modes stable.

No. 2.-Azimuthally stable, one inclinational mode unstable the other stable without rotation; with rotation two unstable, one stable.

No. 3.-The azimuthal mode unstable, two inclinational modes stable the other unstable, without rotation; with rotation one azimuthal mode and one inclinational mode unstable, and one inclinational mode stable.

No. 4.-Azimuthally and one inclinational mode unstable, one inclinational mode stable, without rotation; with rotation all three stable.

system with

345. Take for another example a system having three Gyrostatic freedoms (that is to say, three independent co-ordinates three freeY12, Y1), (16) become

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doms.

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3

where 91, 9, 9, denote the values of the three pairs of equals 23 or 32, 31 or 13, 12 or -21. Imagine 4,,,, to be rectangular co-ordinates of a material point, and let the coordinates be transformed to other axes OX, OY, OZ, so chosen that OZ coincides with the line whose direction cosines relatively to the -, -, - axes are proportional to g,, 9, 9. The equations become

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Gyrostatic system with three free

doms:

reduced to

a mere

rotating system.

√(9,2+9,2+9,"), and the force-components parallel

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3

to the fresh axes are denoted by X, Y, Z (instead of

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dV dV because the present transformation is clearly indy dz' dependent of the assumption we have been making latterly that the positional forces are conservative). These (28) are simply the equations [§ 319, Ex. (E)] of the motion of a particle relatively to co-ordinates revolving with angular velocity round the axis OZ, if we suppose X, Y, Z to include the components of the centrifugal force due to this rotation.

Hence the influence of the gyroscopic terms however originating in any system with three freedoms (and therefore also in any system with only two freedoms) may be represented by the motion of a material particle supported by massless springs attached to a rigid body revolving uniformly round a fixed axis. It is an interesting and instructive exercise to imagine or to actually construct mechanical arrangements for the motion of a material particle to illustrate the experiments described in § 345*.

345xii. Consider next the case of a system with four freedoms. The equations are

Ÿ‚ + 12¢ ̧ + 134 ̧ + 14¢ ̧ + π‚Ý‚ = 0
Ÿ ̧ + 214, +234 ̧ + 24¢ ̧ + ш ̧¥ ̧ = 0

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Denoting by D the determinant we have, by (18),

D= (x2+ w ̧) (\3 + w ̧) (λ2+wg) (\2 + w1)

+λ3 {34°(λ3+@ ̧)(λ2+w ̧)+122(X2+@3)(d3+w ̧)+422(λ2+w,)(λ2+w ̧)
+ 132(\2+w ̧)(\2+w ̧)+23°(\3+w ̧) (\2+w ̧)+ 14°(λ2+ @ ̧) (λ2+@2)}

+λ* (12 342+ 13 42 + 14 23)3

(29).

.(30).

If w1, wq, w ̧, to s be each zero, D becomes

λ3 + (122 + 133 + 14a + 233+422 + 34°) d® + (12 34+ 13 42 + 14 23)3λ*.

This equated to zero and viewed as an equation for λ2 has two

roots each equal to 0, and two others given by the residual Quadruply quadratic

free gyro

static

system without

d* + ( 1 22+ 1 32+ 1 42+23° +242+34°) λ® + (12 34+13 42 +14 23)=0...(31). force.

Now remarking that the solution of +pz+q2=0 may be
written

− 2 = } { p ± √ (p + 2q) (p − 2g)} = } { √(p + 2q) ± √(p − 2q)}',

we have from (31)

where

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− λ2 = 4 ( 122 + 13° + 14′ + 23′ + 24′+ 34′ ± √§) } .....(32),

= (r = 8)3

r = √ {(12 +34)3 + (13+ 42)2 + (14+23)"}

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(33).

case of fail

static pre

As 12, 34, 13, etc. are essentially real, r and s are real, and (unless 12 43 + 13 42 + 14 23 = 0, when one of the values of λ' is Excepted zero, a case which must be considered specially, but is excluded ing gyrofor the present,) they are unequal. Hence the two values of dominance. -λ given by (32) are real and positive. Hence two of the four freedoms are stable. The other two (corresponding to -λ=0) are neutral.

345x1 Now suppose 1, w, w, w1 to be not zero, but each Quadruply free cycloidvery small. The determinantal equation will be a biquadratic al system, in A, of which two roots (the two which vanish when a,, etc. cally domi vanish) are approximately equal to the roots of the quadratic

(12 34 + 13 42 + 14 23)2 λ* + (122π ̧ ̧+ 133w ̧ ̧+ 143w ̧ ̧

4

2 4

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and the other two roots are approximately equal to those of the
previous residual quadratic (31).

To solve equation (34), first write it thus :

1

=

(+-)'+ (12"+13"+14"+23"+24"+34'") }'; +(12'34'+ 13′42′+ 14′23')" = 0

.(35),

gyrostati

nated.

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Quadruply free cycloidal system, gyrostatically dominated.

Thus, taken as a quadratic for X-3, it has the same form as (31) for λ2, and so, as before in (32) and (33), we find

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Four irrotational stabilities confirmed, four irrotational instabilities rendered stable, by gyrostatic links.

where r√(12′+34′)+(13′+42′)3+ (14′+ 23')"} } .(38). 8' = √(12'-34')+(13'-42')+(14'-23')"}

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and

Now if w,, w, w, w, be all four positive or all four negative, 12', 34', 13', etc. are all real, and therefore both the values of

1

እo

given by (37) are real and positive (the excluded case referred to at the end of § 345, which makes

12'34' 13'42' + 14'23' = 0,

1

and therefore the smaller value of = 0, being still excluded).

4

Hence the corresponding freedoms are stable. But it is not
necessary for stability that a,,,, w, w, be all four of one
sign: it is necessary that their product be positive: since if it
were negative the values of λ given by (34) would both be
real, but one only negative and the other positive. Suppose two
of them, w, w1 for example, be negative, and the other two,
☎,,,,
,,,, positive: this makes w,,,,,, ww,, and ww, negative,
and therefore 13', 14', 23', and 24′ imaginary. Instead of four
of the six equations (36), put therefore

4

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Combined dynamic and gyrostatic sta

Thus 13′′ etc. are real, and 13′ 13′′ √-1 etc., and (38) become

r' = √/{(12' + 34')2 − (1 3′′ + 42′′)3 − (14′′ + 23′′)3})
8' = √{(12' — 34')2 - (13′′ – 42′′)3 — (14′′ – 23′′)°})

-

Hence for stability it is necessary and sufficient that

and

(12'34')> (13" + 42") + (14"+23")")
(12'34')> (13" - 42") + (14" - 23")")

.(40).

(41).

If these inequalities are reversed, the stabilities due to w,, to, and 34′ are undone by the gyrostatic connexions 13′′, 42′′, 14′′ and 23".

345. Going back to (29) we see that for the particular bility gyrosolution 1 = a,, = a ̧‹11, etc., given by the first pair of roots counter

1

of (32), they become approximately

statically acted.

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solution.

being in fact the linear algebraic equations for the solution in Completed of the simple simultaneous differential equations And if we take

the form

(53) below.

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for either particular approximate solution of (29) corresponding to (37), we find from (29) approximately

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Remark that in (42) the coefficients of the first terms are imaginary and those of all the others real. Hence the ratios a,a,, a/a ̧, etc., are imaginary. To realize the equations put

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Eliminating 1,,,, etc. from the seconds by the firsts of these

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and by eliminating P,, P., etc. similarly we find similar equations

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