(21) { -(A + Ma^2)n^2 + Kn + gMa}
X {-(A1 + M1h1^2 + Mb^2)n^2 + K1n + g (M1h1 + Mb)} - M^2n^4a^2b^2 = 0,
a quartic for n, giving the frequency n/2[pi] of a fundamental
vibration.
Change the sign of g for the case of the gyrostats spinning upright,
one on the top of the other, and so realize the gyrostat on the top of
a gyrostat described by Maxwell.
In the gyrostatic chain of S 14, the tension T may change to a limited
pressure, and U^2 may still be positive, and the motion stable; and so
a motion is realized of a number of spinning tops, superposed in a
column.
16. _The Flexure Joint._--In Lord Kelvin's experiment the gyrostats
are joined up by equal light rods and short lengths of elastic wire
with rigid attachment to the rod and case of a gyrostat, so as to keep
the system still, and free from entanglement and twisting due to pivot
friction of the fly-wheels.
When this gyrostatic chain is made to revolve with angular velocity n
in relative equilibrium as a plane polygon passing through Oz the axis
of rotation, each gyrostatic case moves as if its axis produced was
attached to Oz by a flexure joint. The instantaneous axis of resultant
angular velocity bisects the angle [pi] - [theta], if the axis of the
case makes an angle [theta] with Oz, and, the components of angular
velocity being n about Oz, and -n about the axis, the resultant
angular velocity is 2n cos 1/2([pi] - [theta]) = 2n sin 1/2[theta];
and the components of this angular velocity are
(1) -2n sin 1/2[theta] sin 1/2[theta] = -n(1-cos[theta]), along the
axis, and
(2) -2n sin 1/2[theta] cos 1/2[theta] = -n sin [theta], perpendicular
to the axis of the case. The flexure joint behaves like a pair of
equal bevel wheels engaging.