(4) N = An^2 sin [theta] - [mu].CK = (A^n^2 - KM.KN) sin [theta]/A,
so that N is negative or positive, and the axis tends to rise or fall
according as K moves to the inside or outside of the hyperbola of free
motion. Thus a tap on the axis tending to hurry the precession is
equivalent to an impulse couple giving an increase to C'K, and will
make K move to the interior of the hyperbola and cause the axis to
rise; the steering of a bicycle may be explained in this way; but K1
will move to the exterior of the hyperbola, and so the axis will fall
in this second more violent motion.
Friction on the point of the top may be supposed to act like a tap in
the direction opposite to the precession; and so the axis of a top
spun violently rises at first and up to the vertical position, but
falls away again as the motion dies out. Friction considered as acting
in retarding the rotation may be compared to an impulse couple tending
to reduce OC', and so make K and K1 both move to the exterior of the
hyperbola, and the axis falls in both cases. The axis may rise or fall
according to the direction of the frictional couple, depending on the
shape of the point; an analytical treatment of the varying motion is
very intractable; a memoir by E. G. Gallop may be consulted in the
_Trans. Camb. Phil. Soc._, 1903.
The earth behaves in precession like a large spinning top, of which
the axis describes a circle round the pole of the ecliptic of mean
angular radius [theta], about 23-1/2 deg., in a period of 26,000
years, so that R/[mu] = 26000 X 365; and the mean couple producing
precession is
(5) CR[mu] sin [theta] = CR^2 sin 23-1/2 deg./(26000 X 365),
one 12 millionth part of 1/2CR^2, the rotation energy of the earth.
5. If the preponderance is absent, by making the C.G coincide with O,
and if A[mu] is insensible compared with G',
(1) N = -G'[mu] sin [theta],
the formula which suffices to explain most gyroscopic action.
Gyroscopic action of railway wheels.
Thus a carriage running round a curve experiences, in consequence of
the rotation of the wheels, an increase of pressure Z on the outer
track, and a diminution Z on the inner, giving a couple, if a is the
gauge,
(2) Za = G'[mu],
tending to help the centrifugal force to upset the train; and if c is
the radius of the curve, b of the wheels, C their moment of inertia,
and v the velocity of the train,
(3) [mu] = v/c, G' = Cv/b,
(4) Z = Cv^2/abc (dynes),
so that Z is the fraction C/Mab of the centrifugal force Mv^2/c, or
the fraction C/Mh of its transference of weight, with h the height of
the centre of gravity of the carriage above the road. A Brennan
carriage on a monorail would lean over to the inside of the curve at
an angle [alpha], given by
(6) tan [alpha] = G'[mu]/gMh = G'v/gMhc.
The gyroscopic action of a dynamo, turbine, and other rotating
machinery on a steamer, paddle or screw, due to its rolling and
pitching, can be evaluated in a similar elementary manner
(Worthington, _Dynamics of Rotation_), and Schlick's gyroscopic
apparatus is intended to mitigate the oscillation.
6. If the axis OC in fig. 4 is inclined at an angle [alpha] to the
vertical, the equation (2) S 4 becomes
(1) N = sin [theta] (A[mu]^2 cos [theta] - G'[mu]) + gMh sin ([alpha] -
[theta]).
Suppose, for instance, that OC is parallel to the earth's axis, and
that the frame is fixed in the meridian; then [alpha] is the
co-latitude, and [mu] is the angular velocity of the earth, the square
of which may be neglected; so that, putting N = 0, [alpha] - [theta] =
E,
(2) gMh sin E - G'[mu] sin ([alpha] - E) = 0,
G'[mu] sin [alpha] G'[mu]
(3) tan E = ------------------------ [~=] ------ sin [alpha].
gMh + G'[mu] cos [alpha] gMh
The barogyroscope.
This is the theory of Gilbert's barogyroscope, described in Appell's
_Mecanique rationnelle_, ii. 387: it consists essentially of a rapidly
rotated fly-wheel, mounted on knife-edges by an axis perpendicular to
its axis of rotation and pointing east and west; spun with
considerable angular momentum G', and provided with a slight
preponderance Mh, it should tilt to an angle E with the vertical, and
thus demonstrate experimentally the rotation of the earth.
Foucault's gyroscope.
In Foucault's gyroscope (_Comptes rendus_, 1852; Perry, p. 105) the
preponderance is made zero, and the axis points to the pole, when free
to move in the meridian.
Generally, if constrained to move in any other plane, the axis seeks
the position nearest to the polar axis, like a dipping needle with
respect to the magnetic pole. (_A gyrostatic working model of the
magnetic compass_, by Sir W. Thomson. British Association Report,
Montreal, 1884. A. S. Chessin, St Louis Academy of Science, January
1902.)
Gyroscopic horizon.
A spinning top with a polished upper plane surface will provide an
artificial horizon at sea, when the real horizon is obscured. The
first instrument of this kind was constructed by Serson, and is
described in the _Gentleman's Magazine_, vol. xxiv., 1754; also by
Segner in his _Specimen theoriae turbinum_ (Halae, 1755). The inventor
was sent to sea by the Admiralty to test his instrument, but he was
lost in the wreck of the "Victory," 1744. A copy of the Serson top,
from the royal collection, is now in the Museum of King's College,
London. Troughton's Nautical Top (1819) is intended for the same
purpose.
The instrument is in favour with French navigators, perfected by
Admiral Fleuriais (fig. 9); but it must be noticed that the horizon
given by the top is inclined to the true horizon at the angle E given
by equation (3) above; and if [mu]1 is the precessional angular
velocity as given by (3) S 4, and T = 2[pi]/[mu], its period in
seconds,
[mu] T cos lat T cos lat
(4) tan E = ----- cos lat = ---------, or E = ---------,
[mu]1 86400 8[pi]
if E is expressed in minutes, taking [mu] = 2[pi]/86400; thus making
the true latitude E nautical miles to the south of that given by the
top (_Revue maritime_, 1890; _Comptes rendus_, 1896).
This can be seen by elementary consideration of the theory above, for
the velocity of the vector OC' of the top due to the rotation of the
earth is
(5) [mu].OC' cos lat = gMh sin E = [mu]1.OC' sin E,
[mu] T cos lat
sin E = ----- cos lat, E = ---------,
[mu]1 8[pi]
in which 8[pi] can be replaced by 25, in practice; so that the
Fleuriais gyroscopic horizon is an illustration of the influence of
the rotation of the earth and of the need for its allowance.
[Illustration: FIG. 11.]
Euler's coordinate angles.
7. In the ordinary treatment of the general theory of the gyroscope,
the motion is referred to two sets of rectangular axes; the one Ox,
Oy, Oz fixed in space, with Oz vertically upward and the other OX, OY,
OZ fixed in the rotating wheel with OZ in the axis of figure OC.
The relative position of the two sets of axes is given by means of
Euler's unsymmetrical angles [theta], [phi], [psi], such that the
successive turning of the axes Ox, Oy, Oz through the angles (i.)
[psi] about Oz, (ii.) [theta] about OE, (iii.) [phi] about OZ, brings
them into coincidence with OX, OY, OZ, as shown in fig. 11,
representing the _concave_ side of a spherical surface.