Rotating Saddle Shape
parametric excitation - particle confinement - Paul trap - Mathieu equations
What it shows:
A ball is not stable when placed on a saddle shape -- it will roll down on either side.
Surprisingly, the ball does become stable when the saddle shape is rotated (about
its point of inflection) above some threshold angular velocity. Stability is maintained
in a limited range of speeds. This is a mechanical analog of confining a particle in a
"Paul trap."
How it works:
A large saddle shape (attached to a plywood disk) is mounted on a multi-purpose turntable.
The saddle shape is essentially a quadrupole gravitational potential. Rotation of this
potential subjects the ball to an alternating repulsive and attractive potential, much
like the time-varying electric quadrupole potential of a Paul trap used in trapping
single ions or electrons.
A 22 cm diameter rubber ball represents the particle we wish to confine. Rotational speeds
up to 236 RPM are possible with the variable-speed motor drive. At low speeds the ball
simply rolls off the saddle. Starting around 135 RPM and up to about 160 RPM, stability is
achieved; as the ball begins to roll away from the center in any direction, it experiences
a periodic force back toward the center as it encounters a "hill" twice every revolution of
the turntable. As the ball rolls back and forth through the center of the saddle, it
experiences a torque about its vertical axis and begins to spin about that axis, quickly
approaching the saddle rotational speed (a little less due to slippage). The actual
excursions about the center are typically one or two centimeters with occasional larger
perturbations. Ultimately it spins like a top in the center of the saddle accompanied by
small oscillations. The record so far is 15 minutes. The reason why, after being stable
for so long, the ball very suddenly (catastrophically) loses it, is not clear.
The phenomenon of parametric excitation is one in which an oscillating system is influenced
by periodically varying one of its parameters (the gravitational potential in this demonstration);
this is not the same as an oscillator being driven by a periodic external force. In the
case of parametric excitation, the equations of motion take on the form of the Mathieu
differential equation. Unlike the driven harmonic oscillator, parametric resonance takes place
at a frequency twice the natural frequency and within a certain range of frequencies
about that value. The linear Mathieu equation does not have a closed form solution. The
following empirical specifications for the stability of a solution are in lieu of an exact
functional dependence. The stability of the ball in this varying potential depends on (1)
the rotational speed, (2) the relative curvatures (of the ball and the saddle shape), and (3)
the ball's moment of inertia. The three variables are not independent. For a given saddle
shape, the ball's radius must have the same order of magnitude as the radius of curvature of
the saddle.
1
The moment of inertia of the ball is proportional to the square of its radius. The ball's
moment of inertia and saddle curvature determine the period of oscillation about equilibrium;
the 22 cm dia rubber ball has a frequency of oscillation of 1.22±0.05 Hz.
2
The interval of rotational speeds leading to stability is centered about 147RPM (2.45 Hz) which
is indeed twice the ball's natural frequency.
At higher rotational speeds the encounters with the hills impart a greater impulse to the ball,
knocking it well beyond the center, only to have it encounter another hill at a higher level
accompanied by a more violent interaction and very soon the ball is literally whacked out of
the saddle. The nonlinear Mathieu equation reigns here with chaotic motion the solution.
Setting it up:
Being rather large, it is best to have the turntable on the floor. Demonstrate the lack of
stability at low and high rotational speeds and then try your hand in the frequency range of
stability.
3
Center the ball on the saddle and release it with a slight twist of the wrist so that it starts
with a modest spin -- this technique enhances the probability of success. Long-term stability
seems to be quite sensitive to the initial conditions. Practice before the lecture! If the ball
remains in the saddle for one to two minutes, the chances that it will remain ten to fifteen
minutes are excellent.
Comments:
Like the Inverted Pendulum,
this is a marvelous demonstration of parametric resonance and a wonderful visual mechanical
analog of particle confinement, whether in Paul traps or cyclic particle accelerators. The reader
is encouraged to look at Pinto and Winter/Ortjohann below for more examples and many useful
references. We would like to acknowledge and thank Justin Georgi for solving the topological
problem and making the saddle shape. Rating ****
Related Stability Demonstrations:
P.N. Murgatroyd, Am J Phys 62, 281-282 (1994).
The magnetic analogue of the inverted pendulum
C. Sacket, E. Cornell, C. Monroe, and C. Wieman, Am J Phys 61, 304-309 (1993).
A magnetic suspension system for atoms and bar magnets
F. Pinto, The Physics Teacher 31, 336-346 (1993).
Parametric Resonance: An Introductory Experiment
H. Winter and H.W. Ortjohann, Am J Phys 59, 807 (1991).
Simple demonstration of storing macroscopic particles in a "Paul trap"
L.W. Alvarez, R. Smits, and G. Senecal, Am J Phys 43, 293 (1975).
Mechanical analog of the synchrotron, illustrating phase stability and two-dimensional focusing
1
The reason for this is simple. If the ball is small compared to the radius of
curvature of the saddle, it must roll quite far (relative to its own size) from
the center before it experiences any appreciable restoring force back to the
center. Generally speaking, displacements far from equilibrium do not effect
simple harmonic motion -- an approximation one must secure here.
2
The only "natural" frequency in this system is that of the ball rolling back
and forth in the saddle. This frequency was determined by fitting a sheet of
posterboard into the saddle (so that the ball sits in a trough of the same
curvature), and timing 10 oscillations of the ball in the trough.
3
The interval of rotational speeds (135-160 RPM) corresponds to turntable speed settings of 50-60; 55 = 147 RPM.