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with non

of energy [345", (4), with Q=0] that the motion is stable what- Quadruply free cycloiever be the values of the gyrostatic coefficients 12, 34, 13, etc. dal system and therefore in this case each of the four roots A2 of the biquad- dominant gyrostatic ratic is real and negative, a proposition included in the general influences. theorem of § 345xx below. To illustrate the interesting questions which occur when the 's are not all positive put

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where 12, 34, 13, etc. denote any numerics whatever subject only to the condition that they do not make zero of

12. 34 +13. 42 + 14. 23.

When w1,,, wz, w49
are all negative each root A of the bi-
quadratic is as we have seen in § 345 real and negative when
the gyrostatic influences dominate. It becomes an interesting
question to be answered by treatment of the biquadratic, how
small may g be to keep all the roots λ real and negative, and
how large may g be to render them other than real and positive
as they are when g=0? Similar questions occur in connexion
with the case of two of the 's negative and two positive,
when the gyrostatic influences are so proportioned as to fulfil
345 (41), so that when g is infinitely great there is complete
gyrostatic stability, though when g=0 there are two instabilities
and two stabilities.

system

number of

345x Returning now to 345 and 345, 345, and 345, Gyrostatic for a gyrostatic system with any number of freedoms, we see by with any 345 that the roots A2 of the determinantal equation (14) or (17) freedoms. are necessarily real and negative when w1, w,, wz, w49 etc. are all positive. This conclusion is founded on the reasoning of § 345" regarding the equation of energy (4) applied to the case Q = 0, for which it becomes 7+ V=E,, or the same as for the case of no motional forces. It is easy of course to eliminate dynamical considerations from the reasoning and to give a purely algebraic proof that the roots A of the determinantal equation (14) of 345vii are necessarily real and negative, provided both of the two quadratic functions (11) a,2 + 2 (12) a ̧a, + etc., and 11a,2+2 12a,a,+ etc. are positive for all real values of a,, a, etc. But the equations (14) of § 343 (k), which we obtained and used in the course of the corresponding demonstration for the case of no motional forces, do not hold in our present case of gyrostatic motional forces. Still for this present case we have the con

2

Case of

equal roots with stability.

Application of Routh's theorem.

clusion of § 343 (m) that equality among the roots falls essentially under the case of § 343 (e) above. For we know from the consideration of energy, as in § 345", that no particular solution can be of the form tet or t sin ot, when the potential energy is positive for all displacements: yet [though there cannot be equal roots for the gyrostatic system of two freedoms (§ 345*) as we see from the solution (25) of the determinantal equation for this case] there obviously may be equality of roots in a quadruply free gyrostatic system, or in one with more than four freedoms. Hence, if both the quadratic functions have the same sign for all real values of a,, etc., all the first minors

* Examples of this may be invented ad libitum by commencing with pairs of equations such as (23) and altering the variables by (generalized) orthogonal transformations. For one very simple example put = and take (23) as one pair of equations of motion, and as a second pair take

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The second of (23) and the first of these multiplied respectively by cos a and
sin
a, and again by sin a and cos a, and added and subtracted, give

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Eliminating and by these last equations, from the first and fourth of the equations of motion, and for symmetry putting, instead of §, and 4 instead of ŋ', and for simplicity putting y cos a=g, and y sin a=h, and collecting the equations of motion in order, we have the following,—

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for the equations of motion of a quadruply free gyrostatic system having two equalities among its four fundamental periods. The two different periods are the two values of the expression

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When these two values are unequal the equalities among the roots do not give rise to terms of the form teλt or t cos at in the solution. But if − (4g2+1h2), which makes these two values equal, and therefore all four roots equal, terms of the form t cos ot do appear in the solution, and the equilibrium is unstable in the transitional case though it is stable if - be less than g+h2 by ever so small a difference.

of the determinantal equation (14), §345, must vanish for each Application double, triple, or multiple root of the equation, if it has any theorem. such roots.

of Routh's

with insta

transitional

tween sta

It will be interesting to find a purely algebraic proof of this theorem, and we leave it as an exercise to the student; remarking only that, when the quadratic functions have contrary signs for some real values of a,, a,, etc., there may be equality among the roots without the evanescence of all the first minors; or, in Equal roots dynamical language, there may be terms of the form text, or bility in t sin ot, in the solution expressing the motion of a gyrostatic cases besystem, in transitional cases between stability and instability. bility and It is easy to invent examples of such cases, taking for instance the quadruply free gyrostatic system, whether gyrostatically dominated as in § 345x, but in this case with some of the four quantities negative, and some positive; or, as in § 315xx, not gyrostatically dominated, with either some or all of the quantities ,,..., negative. All this we recommend to the student as interesting and instructive exercise.

instability.

of gyro

345xxi When all the quantities w1, w,, ..., are of the Conditions same sign it is easy to find the conditions that must be fulfilled static doin order that the system may be gyrostatically dominated. For 1 P2, ..., P, are the roots of the equation

if

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mination.

+ + +

=

Pa

Hence if-p,, -P...-P. be each positive, c,/nc, is their arithmetic
mean, and nc/C, is their harmonic mean.
greater than nc,/ © „−19
с and the greatest of P1,

Hence c/nc. is

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A

greater than c/nc, and the least of them is less than nc / C-1
Take now the two following equations:

is

λ'+λ-2Σ 122 +λΣ (Σ12. 34) +λ(12. 34. 56)+ etc. 0........(71),

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1-6

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Σ (≥1 2'. 34')2 + (3) Σ (Σ12'. 34'′. 56')*+ etc. = 0 (72),

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Cycloidal motion. Conditions

of gyrostatic domination.

Suppose for simplicity i to be even.

are (§ 345

All the roots A3 of (71) below) essentially real and negative. So are those of π,,,,..., w, are all of one sign as we now suppose them to be. Hence the smallest root -λ of (71) is less than

(72) provided

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Gyrostatic links ex

plained.

li Σ (12'. 34′. 56',..., i — 1, ¿')

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Hence the conditions for gyrostatic domination are that (74) must
be much greater than the greatest of the positive quantities
+ W.,..., ± Wis
and that (75) must be very much less than the
least of these positive quantities. When these conditions are
fulfilled the i roots of (18) § 345x equated to zero are separable
into two groups of i roots which are infinitely nearly equal to
the roots of equations (71) and (72) respectively, conditions
of reality of which are investigated in § 345xxvi below. The
interpretation leads to the following interesting conclusions:—

345xxiv. Consider a cycloidal system provided with nonrotating flywheels mounted on frames so connected with the moving parts as to give infinitesimal angular motions to the axes of the flywheels proportional to the motions of the system. Let the number of freedoms of the system exclusive of the ignored co-ordinates [§ 319, Ex. (G)] of the flywheels relatively to their frames be even. Let the forces of the system be such that when the flywheels are given at rest, when the system is at rest, the equilibrium is either stable for all the freedoms, or unstable for all the freedoms. Let the number and connexions of the gyrostatic links be such as to permit gyrostatic domination (§ 345) when each of the flywheels is set into sufficiently Gyrostati rapid rotation. Now let the flywheels be set each into sufficiently rapid rotation to fulfil the conditions of gyrostatic domination (§ 345): the equilibrium of the system becomes stable with half the whole number i of its modes of vibration exceedingly rapid, with frequencies equal to the roots of a certain algebraic equation of the degree i; and the other half of

cally dominated system:

cally do.

system:

mic oscil

rapid);

its modes of vibration very slow, with frequencies given by the Gyrostatiroots of another algebraic equation of degree . The first class minated of fundamental modes may be called adynamic because they are the same as if no forces were applied to the system, or acted between its moving parts, except actions and reactions in its adynathe normals between mutually pressing parts (depending on the lations (very inertias of the moving parts). The second class of fundamental modes may be called precessional because the precession of the and preequinoxes, and the slow precession of a rapidly spinning top oscillations supported on a very fine point, are familiar instances of it. Remark however that the obliquity of the ecliptic should be infinitely small to bring the precession of the equinoxes precisely within the scope of the equations of our “cycloidal system."

cessional

(very slow).

between

frequencies,

frequencies

345. If the angular velocities of all the flywheels be altered in the same proportion the frequencies of the adynamic oscillations will be altered in the same proportion directly, and those of the precessional modes in the same proportion inversely. Now suppose there to be either no inertia in the system except that of the flywheels round their pivoted axes. and round their equatorial diameters, or suppose the effective inertia of the connecting parts to be comparable with that of the flywheels when given without rotation. The period of each Comparison of the adynamic modes is comparable with the periods of the adynamic flywheels. And the periods of the precessional modes are com- rotational parable with a third proportional to a mean of the periods of of the flythe flywheels and a mean of the irrotational periods of the sys- precessional tem, if the system be stable when the flywheels are deprived of rotation. For the last mentioned term of the proportion we quencies may, in the case of irrotational instability, substitute the time of ities of the increasing a displacement a thousandfold, supposing the system with Byto be falling away from its configuration of equilibrium prived of according to one of its fundamental modes of motion (et). The reciprocal of this time we shall call, for brevity, the rapidity of the system, for convenience of comparison with the frequency of a vibrator or of a rotator, which is the name commonly given to the reciprocal of its period.

wheels.

quencies

of the system,

and fre

or rapid

system,

wheels de

rotation.

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