CR + K /A \
(12) - ------ + 2[mu]( -- + c^2) - Rcp = 0,
M \M /
/A \ CR + K /A \
(13) ( -- + c^2) n^2 = [mu] ------ - [mu]^2( -- + c^2) + [mu]R[rho]c - g(c - [rho]),
\M / M \M /
/A \^2 /CK + R \^2 /A \
(14) ( -- + c^2) n^2 = 1/4( ------ + Rc[rho] ) - g( -- + c^2) (c - [rho]),
\M / \ M / \M /
Thus for a top spinning upright on a rounded point, with K = 0, the
stability requires that
(15) R > 2k' [root]{g(c - [rho])}/(k^2 + c[rho]),
where k, k' are the radii of gyration about the axis Gz, and a
perpendicular axis at a distance c from G; this reduces to the
preceding case of S 3 (7) when [rho] = 0.
Generally, with [alpha] = 0, but a [+-] 0, the condition (A) and (B)
becomes
/C \ Q
(16) ( - + a^2) -- = 2[mu]ac - Ra[rho],
\M / L
Q CR + K /A \
-ac -- = ------ + Rc[rho] - 2[mu]( -- + a^2),
L M \M /
so that, eliminating Q/L,
_ _
| /A \ /C \ | /C \ /CR + K\ C
(17) 2 | ( -- + c^2)( -- + a^2) - a^2c^2| [mu] = ( -- + a^2)( ------ ) + -- Rc[rho],
|_ \M / \M / _| \M / \ M / M
the condition when a coin or platter is rolling nearly flat on the
table.
Rolling along in a straight path, with [alpha] = 1/2[pi], c = 0, [mu]
= 0, E = 0; and
(18) N/L = (CR + K)/A,
/A \
(19) D = ( -- + a^2) n^2 + g(a - [rho]),
\M /
CR + K
F = - ------ - Ra^2,
M
/A \
( -- + a^2) n^2 + g(a - [rho])
N D \M /
(20) -- = - -- = ----------------------------,
L F /C \ K
( -- + a^2) R + -
\M / M
_ _
/A \ (CR + K) | /C \ K |
(21) ( -- + a^2) n^2 = -------- | ( -- + a^2) R + -- | - g(a - [rho]).
\M / A |_ \M / M _|
Thus with K = 0, and rolling with velocity V = Ra, stability
requires
V^2 a - [rho] A a - [rho]
(22) --- > ---------------- > 1/2 -- ---------,
2g C / C \ C C
2 -- ( ---- + 1 ) ---- + 1
A \Ma^2 / Ma^2
or the body must have acquired velocity greater than attained by
rolling down a plane through a vertical height 1/2(a - [rho])A/C.
On a sharp edge, with [rho] = 0, a thin uniform disk or a thin ring
requires
(23) V^2/2g > a/6 or a/8.
The gyrostat can hold itself upright on the plane without advance when
R = 0, provided
(24) K^2/AM - g(a - [rho]) is positive.
For the stability of the monorail carriage of S 5 (6), ignoring the
rotary inertia of the wheels by putting C = 0, and replacing K by G'
the theory above would require
G' / G'\
(25) -- ( aV + -- ) > gh.
A \ A /
For further theory and experiments consult Routh, _Advanced Rigid
Dynamics_, chap. v., and Thomson and Tait, _Natural Philosophy_, S
345; also Bourlet, _Traite des bicycles_ (analysed in Appell,
_Mecanique rationnelle_, ii. 297, and Carvallo, _Journal de l'ecole
polytechnique_, 1900); Whipple, _Quarterly Journal of Mathematics_,
vol. xxx., for mathematical theories of the bicycle, and other bodies.
Gyrostatic chain.
14. Lord Kelvin has studied theoretically and experimentally the
vibration of a chain of stretched gyrostats (_Proc. London Math.
Soc._, 1875; J. Perry, _Spinning Tops_, for a diagram). Suppose each
gyrostat to be equivalent dynamically to a fly-wheel of axial length
2a, and that each connecting link is a light cord or steel wire of
length 2l, stretched to a tension T.
Denote by x, y the components of the slight displacement from the
central straight line of the centre of a fly-wheel; and let p, q, 1
denote the direction cosines of the axis of a fly-wheel, and r, s, 1
the direction cosines of a link, distinguishing the different bodies
by a suffix.
Then with the previous notation and to the order of approximation
required,
(1) [theta]1 = -dq/dt, [theta]2 = dp/dt,
(2) h1 = A[theta]1, h2 = A[theta]2, h3 = K,
to be employed in the dynamical equations
(3) dh1/dt - [theta]3 h2 + [theta]2 h3 = L, ...
in which [theta]3 h1 and [theta]3 h2 can be omitted.