The Alan Sondheim Mail Archive

Ship and Bell

"We therefore infer that the _effects of different permanent disturbing
forces acting under similar conditions on the same coordinate are not
simply proportional to their respective magnitudes but depend on their

"We therefore infer that _a disturbing force whose period and real expo-
nential are nearly the same as those of any one free vibration produces a
large forced vibration._"

"We conclude that _one effect of the resistances on a disturbing permanent
force, which would otherwise produce a magnified forced oscillation, is to
modify that oscillation and to keep it within bounds._"

"In the same way _heavy church bells_ can _be easily set in motion_ pro-
vided the pulls are properly timed. To increase the oscillation each rope
should be pulled only when it is descending. A large heavy ship can be
made to roll, when its natural time of oscillation is required, by running
a gang of men to and fro across the deck at the proper time; the men run

(from Advanced Dynamics of a System of Rigid Bodies, Edward John Routh)

But if the men run uphill, surely the oscillation will ultimately be
damped since their effect in total would be to reduce the roll? Or does
the natural time depend on the minimization of the roll? Suppose on the
other hand the men run to and fro at the proper time - running downhill.
Then the oscillations will possibly increase as energy is added on the
downswing. With the descent of the bell, a pull raises it further,
increasing the potential energy. With the rolling of the ship, potential
energy is increased as the men running downward add to the slope. However
if upon running downward, the ship is rising, then the damping effect
might slow the rising of the ship, just as, if running upward, the ship is
sinking (sloping downward), the running will become first flat, then down-
ward, carrying the energy downward and pulling the roll. If the roll is
pushed, the roll is damped, if the men are running upward; if the roll is
pulled, the roll increases, if the men are running downward. The men run
first upward, then level, then downward, or first downward, then level,
then upward. Perhaps damping in the second case, increasing in the second.
Suppose the men _always_ run upward, then as the ship levels, the men stop
running; as it continues downward, the men turn around and run upward; the
damping is doubled. Suppose the men _always_ run downward, then as the
ship levels, the men stop running; as it continues upward, the men turn
around and run downward, the increase is doubled. By "doubled" one means
the force is applied twice on the swing, depending on the tilt of the
ship. In both cases, of the bell and the ship, gravity is paramount; one
runs or pulls against it, or one runs towards it, in which case pull is
impossible; however, a second cord might in fact pull the bell in an
opposite direction, just as the men might jostle from one side of the ship
to the other. Is this resonance, or the orders given to men, the acts of
men? The men follow the pull and push of the bell, slope of the deck of
the ship; there is no natural swing or pull here, only culture, commands,
actions in unison. The ship stops rolling or rolls over; the bell stops
ringing or turns full on its pivot. In any case, the dynamics are both
evident and peculiar; the men must experiment, over and over again, to
find the proper resonance, response. It is a difficult problem, settled
only in church, on the high seas, even though the dynamics are evident.

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