Project Description

The goal of this project was to observe and understand oscillating patterns in a large array of pendulums. This system provides a simple experimental model in which solitons, or something much like them, can be observed. This system was first studied in the Ph.D. dissertation of Denardo[1] and in several followup articles, from whom some of our images are borrowed. Prof. Goodman learned about this experimental system in a talk by Victor Sánchez-Morcillo at the LENCOS 2012 conference. It has since been published in [2].

Professor Goodman's group learned about nonlinear waves, perturbation methods in mechanics, and dynamical systems.

2013 Group, Marlon de la Rosa, David Cunningham, Richard Haff, Prof. Roy Goodman, and Hamza Ahmad, with their experimental apparatus

The pendulum is a fundamental object of study in mechanics. Students in their first semester of physics learn that a pendulum of length ll subject to gravitational acceleration gg moves according to the differential equation

d2dt2θ=glsinθ\frac{d^2}{dt^2} \theta = -\frac{g}{l} \sin{\theta}

where θ\theta is the angular displacement from its stable rest position. They then learn to analyze this system when θ\theta is small by using sinθθ\sin\theta \approx \theta. This approximation linearizes the equation, and has the solution θ(t)=Acosω(tt0)\theta(t) = A \cos{\omega (t-t_0)}, where ω2=gl\omega^2 = \tfrac{g}{l}.

Most students stop there, but there's way more to the pendulum than that! In the 1950's, the Russian physicist Piotr Kapitza demonstrated that rapid vertical oscillation of the pendulum's support stabilizes the pendulum in the upright direction, a fact which we learned to analyze with the method of averaging, and which we demonstrated by connecting a pendulum to an electric jigsaw, as shown in this video by Mark Levi from Penn State University

See this youtube video

Nonlinear effects---those ignored by linearizing the linearized assumption made above---are responsible for much of the interesting behavior seen in pendulums.

Experimental Setup

The pendulums consist simply of beads set on V-shaped strings arranged around a circular metal frame. Each "V" overlaps with, and is attached to, its two nearest neighbors; see schematic below. The entire frame is mounted to a speaker, which oscillates vertically at a frequency specified by an alternating current pattern generator.

Schematic of experimental apparatus

Mathematical Analysis

The damped and parametically-driven pendulum

A single pendulum is driven by vertically oscillating its point of support at about twice the pendulum's natural frequency. After scaling θ=O(ϵ)\theta=O(\epsilon) and truncating the Taylor series for the sine function at cubic degree, it satisfies a nonlinear version of the Mathieu equation:

θ¨+ϵμθ˙+((1+ϵδ+ϵcos(2t))θϵ6θ3=0.\ddot \theta + \epsilon \mu \dot \theta + \left( (1 + \epsilon \delta + \epsilon\cos{(2t)} \right) \theta -\frac{ \epsilon}{6} \theta^3 = 0.

The frequency of the driver is chosen to excite the pendulum's natural frequency of oscillation, with a slight detuning.
The students analyzed this system using the method of multiple scales.
If the pendulum's natural frequency is sufficiently detuned from that of the driver, specifically if δ>1\delta>1, then the system is bistable, with both the solution θ=0\theta=0 and an oscillating solution being possible and stable. This condition is necessary for the existence of "localized pulse" solutions to a large chain of pendulums.

Array of pendulums

Now we consider an array of pendulums, each with angular displacement θn(t)\theta_n(t). Ignoring damping and driving, these satisfy:
θ¨n+θnϵ6θn3ν(θn+12θn+θn1)=0.\ddot \theta_n + \theta_n -\frac{ \epsilon}{6} \theta_n^3 - \nu( \theta_{n+1} - 2\theta_n + \theta_{n-1})= 0.

Schrödinger equation

Following [3], we derive an envelope equation which looks for slow variations in the terms

θn=A(X,T1,T2)ei(δnωt)+c.c.\theta_n = A(X,T_1,T_2) e^{i (\delta n -\omega t)} + c.c.

where X,T1, and T2X,T_1,\text{ and } T_2 are slow variables and δ\delta is the discrete wave number. We find in the end that this system has dispersion relation ω2=ω02+2ν(1cosδ)\omega^2 = \omega_0^2 + 2 \nu ( 1 - \cos{\delta} ) and that AA satisfies the Nonlinear Schrödinger equation (NLS).

iddT2Aτ+ω2Aξξ+ω024ωA2A=0.i \frac{d}{dT_2}A_{\tau} + \frac{\omega''}{2} A_{\xi \xi} + \frac{\omega_0^2}{4 \omega} |A|^2 A = 0 .

For δ\delta near zero, the pendulums are all approximately in phase, and ω>0\omega''>0, and AA satisfies the defocusing NLS equation. This system is known to have pulse-like solutions, in which AA is only nonzero in a small area of space. When δπ\delta\approx\pi, so that adjacent pendulums are approximately 180o out of phase it satisfies the focusing NLS equation, which has kink-like solutions.

A pulse shape and a kink shape.

In our experiments, we try to excite various shapes by oscillating the system at twice their oscillatory frequency. We succeeded in finding the kink solutions, but not the pulse solutions.

Experiments

Here is a movie of one experiment. This is at the so-called cutoff frequency, with δ=π\delta = \pi, and consecutive pendulums 180o out of phase. It is kind of mesmerizing!

See this youtube video.

And here is the movie after video-processing.

We also show similar movies in which two kinks are present, one near the top, and one on the bottom right. Note that the kinks have a definite width.

See this youtube video

And this one.

We processed data from these videos using the MATLAB Image Processing Toolbox to find the width and see how it compared with theoretical predictions in [4].

Maximum angular deviation for runs at three frequencies.

Finally, we performed numerical simulations of the system. We computed kinks at the upper cutoff δ=π\delta=\pi

Numerical simulation of the "upper cutoff kink."

We also foundnd kinks with a wavelength-4 carrier
wavelength 4 carrier


  1. B. C. Denardo, Observations of Nonpropagating Oscillatory Solitons. Ph.D. Thesis, UCLA, Aug. 1990. ↩︎

  2. V. Sánchez-Morcillo, N. Jiménez, N. González, S. Dos Santos, A. Bouakaz, and J. Chaline, “Modeling Acoustically Driven Microbubbles by Macroscopic Discrete-Mechanical Analogues,” Modelling in Science Education and Learning, vol. 6, pp. 75–87, 2013. ↩︎

  3. P. D. Miller. Applied Asymptotic Analysis. American Mathematical Society, 2006. ↩︎

  4. B. Denardo, B. Galvin, A. Greenfield, A. Larraza, S. Putterman, and W. Wright, Observations of localized structures in nonlinear lattices: Domain walls and kinks, Phys. Rev. Lett. 68, pp. 1730–1733, 1992. ↩︎