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The Swing Spring: A Search for Classical Monodromy

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The Swing Spring: A Search for Classical Monodromy
The Swing Spring: A Search for Classical Monodromy
Carrie A. Weidner
University of Colorado at Boulder
Physics REU
Abstract
This experiment was designed to demonstrate classical monodromy in the swing
spring system. A swing spring in 1:2 resonance shows a stepwise precession in swinging
planes. Monodromy is manifested as a discontinuity in the precession angles. We have
been successful in showing the stepwise precession of the swing plane; however, we have
been unable to show agreement between the experimental and theoretical step angles.
Work on the experiment continues, and we soon hope to demonstrate theoretical and
experimental agreement, as well as to show monodromy.
1
phase space. A successful demonstration of
monodromy would show the discontinuity in
step angle in the first circle but the smooth,
continuous nature of the step angle in the
latter.
3
3.1
Experiment
Experimental Setup
The experimental setup was mostly constructed
and functional when I began this project, so
not much description will be given to the
construction of the setup. Essentially what
was necessary was a black enclosure, constructed from foam board and felt, to provide a dark background against which the
cameras could capture the moving ball. Two
cameras were included in the setup, one for
the xy-plane, another for the yz-plane. This
way, the motion of the illuminated ball could
be captured in three dimensions. Due to a
noticeable blur in the motion of the ball, the
shutter speed on the cameras was set high,
about 1/60 to 1/100 of a second. This allows
less light to get into the camera, reducing
blur but also brightness of the ball. For this
reason, bright halogen lights were mounted
near each camera, with the intention of saturating the image captured by the camera.
This renders the ball as bright as possible
on the black background. This high contrast is necessary for the computerized image processing tools to recognize the ball.
Gain and saturation can also be adjusted
with the computer by adding light to each
pixel in the grayscale image (a higher pixel
value gives a brighter picture). The swing
spring itself is by far the most important part
of the whole setup, given that the resulting
behavior depends highly on the ν = 2 condition. To construct this, brass swivels are
glued to the ends of an appropriately loose
spring. A spring must be chosen such that
it is loose enough to provide long enough
transition times to be noticeable, but not so
loose that damping becomes an issue due to
the long times necessary for data taking. A
stiffer spring is generally to be avoided, how-
b
Energy
(scaled)
a
a.
Angular Momentum (scaled)
b.
b
pi
2pi
pi/4
pi
-pi/4
Figure 1: (a) Energy and Angular Momentum Phase Space (b) A circle without the
singularity shows continuity in precession
angles (c) A circle surrounding the singularity shows a clear jump discontinuity in precession angles
ever, since it is difficult to tell when the mass
is in pure swinging motion, since the energy transfer between swinging and springing modes happens on such a short timescale.
The spring constant k is measured by finding
the amount of time it takes a known mass on
the end of the spring to oscillate ten times.
With this value, the spring is then calibrated
to resonance by tuning the length, such that
mg
= 14 , giving the 1:2 resonance necessary
kl
to observe monodromy. Once the necessary
length has been found, fishing line is added
to the spring to give the extra length necessary. Then, the spring is tested to make sure
the energy transfer between swinging and
springing modes is observed. The current
swing spring setup in use has k = 6.5 N/m,
l = 1 m, m = 0.2246 kg. The mass in the
setup is a stainless steel ball painted white.
To maintain the contrast, the experimenter
wears a black long-sleeved shirt, black gloves
and a black ski cap. For a diagram of the experiment, see Figure 2.
spring
lights
ball
z camera
lights
xy camera
3.2
Data Taking
Once the ball is hanging in the enclosure,
both cameras are turned on and manually
focused on the ball. An automatic focus is
not desirable because the changing focus in
the camera lenses changes the pixels to meters calibration factor. Once the cameras are
properly focused, footage of the still ball is
recorded in all dimensions. From this, the
radius of the ball as captured by the camera
(in pixels) can be measured and compared
to the known radius of the ball. This allows
the experimenter to set the pixels to meters
calibration, as well as to find the center of
the still ball (which corresponds to the origin). Then, the ball is started moving by
hand, with ideally low energy and low angular momentum. This is necessary since the
scaled energy and angular momentum variables are series expansions about the origin
in phase space–remaining close to the origin gives much better approximations. The
cameras then capture the motion of the ball,
and the resulting movies are input into the
computer, where they are deinterlaced (for
Figure 2: The Experimental Setup
clarity) and synchronized (so that the z motion and corresponding xy motion line up)
in Adobe Premiere Pro CS3. This allows
the experimenter to analyze the z motion of
the ball and pick out the swinging motion
in the xy-plane much more easily. With a
non-zero angular momentum, the swinging
motion strikes out an ellipse in xy-space, although the ellipse narrows out significantly
with decreasing angular momentum. A video
of this motion is then fed into MATLAB,
where each frame in the video is rendered
grayscale, then changed to simply black and
white by setting a contrast limit. Pixels above
this limit are rendered white, where those at
or below the limit are black. This gives the
image of a white ball on a black background.
Then, the edges of the ball are found using
the gradient function in MATLAB, where a
nonzero gradient corresponds to the edge of
the ball. These points are then fit to a circle,
and the center of this circle gives the position of the ball. A sample fit to a ball is
shown in Figure 3. In this way, the motion
of the ball can be tracked through the video.
The data are then fit to ellipses using a fitting scheme given by Halı́r̆ [1]. Examples of
typical ellipse plots showing a constant precession angle are shown in Figure 4. The
axes are in units of pixels, with (0,0) at the
lower left-hand corner.
To find the position of the data run in
phase space, it is necessary to measure the
instantaneous conditions of the ball. That is,
we must measure the position and velocity of
the ball at any time in the motion. For ease,
we choose a point on the ellipse with low curvature, and thus the least acceleration. The
position is measured relative to the center of
the still ball, and the instantaneous velocity is found by taking the central difference
approximation. That is, the position of the
ball is taken at the frame before and after
the frame at which we measure the velocity.
We then divide the distance traveled in these
three frames by the time taken to shoot three
frames (at frame rates of 29.97 fps). These
values are then fed into a Mathematica script
that computes the scaled energy and angular momentum of the system and places the
run at a point in phase space. The script
then calculates the theoretical precession angle based on the experimentally-determined
scaled variables. Theory was provided by
Holger Dullin in [2].
4
4.1
a
b
c
Conclusion
Future Work
The experiment is fully constructed and running. However, we are unable to present
any good data because there is little agreement of experiment with theory. This is
most likely due to imprecision when mea-
Figure 3: (a) The white ball on a black background (b) Points on a circle and fitted line
(axes in pixels) (c) The actual image of the
ball and the fitted circle
References
Experimental Ellipse Fitting Data--Precession Angles
[1] Radim Halı́r̆ and Jan Flusser, Numerically Stable Direct Least Squares Fitting
of Ellipses. Department of Software Engineering, Charles University, Czech Republic, 2000.
[2] Holger Dullin, Andrea Giacobbe and
Richard Cushman Monodromy in the
Resonant Swing Spring Physica D 190,
pp. 15-37, 2004.
y position
(pixels)
x position (pixels)
Figure 4: Ellipses Showing a Precession Angle of 5 Degrees
suring the position and velocity of the ball
combined with the sensitivity of the theoretical precession angle to the experimental
measurements. We must work to measure
the position and velocity better. One cause
of imprecision is the camera’s depth of field.
We must work to quantify how the pixels
to meters calibration value changes as one
moves in the x, y and z directions. Once
this is well-quantified and accounted for in
the image processing, we hope to have better agreement between the experimental and
theoretical precession angles. When we have
this agreement, we will proceed in traversing
the two circles needed to adequately show
the presence of monodromy–that is, the circle around the singularity at the origin and
one without this singularity within it. After this has been demonstrated, the resulting paper will be submitted to the American
Journal of Physics.
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