In brief:
Some considerations and thoughts
Few
proven methods to reduce ellipse forming.
As
good as all authors I’ve seen mention the problem of the elliptical
path and account it to imperfections in the construction of the
pendulum. Some autors describe methods to prevent or limit ellipse
production, but none of them report actual success, with the
exception of the Charron ring and Eddy-current damping with a magnet
in the bob hovering over a metal ring.
Straight
line is an exception.
View
the straight line as a special case of the ellipse, namely with a
short axis = 0.
Give
me one good reason why the pendulum should follow this single
exception, in the presence of an infinite amount of elliptical
alternatives. It would need a mechanism which forces the pendulum to
the straight line.
Obviously
there is no such mechanism. Gravity is such a force, as are
attracting magnets in the center as proposed by several authors.
These mechanisms, although intuitive, seem not to work.
To
the contrary, there appears to be a mechanism by which the elepticity
increases. So there is energy transfered from the long axis to the
short axis. I have not seen any author who described such a
mechanism, or it was Kamerlingh-Onnes, but the math he uses goes far
beyond my capabilities.
Schumacher
method works.
Schumacher
describes a method of driving a pendulum such that the intrinsic
precession of the ellipse (not the elliptical path itself) is almost
pefectly eliminated. My experiments with this methods confirm this.In
the practial experiment which Schumacher descibes the ellipticity is
suppressed by a magnet in the bob which produces eddy currents in a
metal ring below the bob. He does not describe an experiment without
such damping.
My
experiments, without any form of damping, suggest that the
Schumacher method also gives some suppression or limitation of the
ellipse forming. If I switch off the drive pulses I see the
ellipticity increase rapidly.
Direction
changes periodically.
The
ellipticity and its direction of rotation change periodically. In my
setup this happens quite precise when the swing plane crosses the
north-south or the east-west direction. In the first and third
quadrants, south-west to north-east the rotation is CW, in the 2nd
and 4th quadrants it is CCW.
When
I force the ellipse to rotate the other way by slightly knocking the
bob it returns to the original direction within some 2 hours. The
speed of return looks more asymptotic than linear.This
periodic changing of direction is also described by Kamerlingh-Onnes,
but his math goes far beyond my capabilities.
An
interresting question is if such directional changes also would
appear on a non-rotating planet, thus without Foucault precession or
other Coriolis effect.
Pendulum
period is dependent on amplitude.
The
well known formula for the period of a pendulum T= 2
*
pi
*
sqrt
(L
/
g)
is an approximation, exact only for amplitude 0. For larger
amplitudes a complicated extra term must be added which makes T a bit
longer. The elliptical path of a pendulum can be dissected in two
perpendicular movements, nearly sine shaped, with periods T_long
and T_short
where always holds: T_long
> T_short.
Because of the different periods the phase difference will grow and
the ellipse will rotate or gradually change shape. Remember
the figures of Lissajous.
In
a simulation I’ve initially seen a CW precession in a CW rotating
ellipse, when the short axis goes slightly faster. After some time
the phase difference becomes to large to produce a convincing
ellipse.
For
some time I had the idea that this period
difference was
the cause of the intrinsic precession of the elliptical path. But
that does not fit with the formula for the ellipse precession, which
tells that the precesson increases with a longer short axis. The
reasoning above tells that the precession should decrease when the
periods of the long and short axes come closer.
Make
period independent of amplitude.
A
possibility to make the swing time (more) independent of the
amplitude can be found at Christiaan Huygens, a Dutch 17th century
phycisist who made some important improvements in pendulum clocks,
where the simple versions also suffered from amplitude dependency.
Huygens added a pair of “cheeks” with a cycloidal shape near the
pivoting point, This shape is (now) known for its ability to make
the pendulum swing time independent of the amplitude. I do not know
if Huygens knew about this property of the cycloid or that he just
used
his
intuition.For
a foucault pendulum this could be realized by suspending the wire in
a small tube with a cycloidaly varying diameter, the surface of
revolution of the cycloid, resembling a trumpet shape. With this the
swing time can be made amplitude independent in all directions.
But,
seen the reasoning above, it is questionable if it really contributes
to the desired pendulum operation.
Look on the "Not Understood" page for pictures of the Huygens clock and a cycloid pendulum.
Exact
period, example calculation.
My
pendulum has L= 4.3078 m, A = 230 mm. Suppose the ellipticity 23 mm.Swing
time according to the classical formula is 4.163640 sec. After a
3-term correction 4.164382 sec for the long axis and 4.163647 sec for
the short axis. This is a difference of 0.000735 sec. So after 1 /
0.000735 = 1360 periods the phase is shifted over 360 degrees. That
is after 5659 sec, or ca. 1.5 hour
So
this is a substantial effect that asks for questions, but it goes
much to fast to explain the observed phenomena.
One of those new
questions should be: Why don’t we see phenomena related to this
period difference? and: Does this difference play a role in the
transfer of energy from the long into the short axis?