Group 3 2014
Group members: Andras Karsai, Steven Harrington, and Colin Campbell
We seek to investigate the properties of a generalized double pendulum. Previous investigations on double pendulums often limit it to a planar case, and fix the lengths of each stage of the pendulum. The Astrojax is notable in that the pendulum stage lengths vary dynamically. We measure its behaviors in response to well quantified forcing using a perpendicular camera array to track position, and from that generate three-dimensional trajectories for each pendulum bob. The trajectories' mean chaotic lifetime and dominant oscillation frequencies are analyzed, and we find that this chaotic pendulum's oscillation frequencies are linearly correlated with the forcing frequency in the low-frequency regime. We also conclude that a naive periodic forcing of this system without a feedback mechanism is not sufficient to create stable, long-lived orbits of the Astrojax pendulum.
Introduction
An oscillation <ref>http://en.wikipedia.org/wiki/Oscillation</ref> is defined, in physics, as a regular variation in magnitude, position, etc. around a central point. Oscillatory mechanical systems are a vast subset of dynamical systems. They can be used to describe the evolution of states for nearly all physical phenomena. Oscillations of various types occur in real world mechanical systems, nearly all of which are thermodynamically irreversible (via damping, friction, energy due to heat loss, etc.). Even the light you see and the sound you hear are the results of the oscillations of a medium or a field. Here, we specifically study a set of coupled, chaotic oscillations that form from the forcing of a double pendulum.
As any good physics enthusiast knows, student, instructor, or otherwise, pendula are one of the most fundamental and ubiquitous systems studied, and while they may be common, they are only ideal and easily solvable in the most simple of cases. While the equations of motion for a single pendulum are easily attainable, especially for the low amplitude/small angle limit, a "double" pendulum is an example of the effects of coupling and gives rise to chaotic behavior.
Traditionally, a double pendulum has rigid axes, thereby defining the distance the two masses that can be from each other and also from the pivot. Also, more oft than not, a double pendulum is constrained to move in two dimensions via a constant, fixed polar angle. As such, the system has only two degrees of freedom: <math>\scriptsize{\theta_1}</math> and <math>\scriptsize{\theta_2}</math>, these being the azimuthal angles the axes of their respective masses form with the vertical.
It is well documented that such a system demonstrates chaotic behavior. However, we wish to examine the kinds of behavior that arise when we increase the degrees of freedom from two to five. How does one go about doing this? The answer is found very simply in a child's toy: the Astrojax.
The Astrojax Pendulum
The Astrojax is an assemblage of three weighted spheres on a string. Some versions of the toy allow all three spheres to move along the length of the string, but the case studied here does not. Two of the spheres are located at the ends of the string, and the third sphere is allowed to move along the length of the string between them. Some small amounts of damping are of course present. For the Astrojax, the polar angles are not constrained, allowing for motion in a three dimensional plane. This gives the Astrojax five degrees of freedom:
- <math>\small{\lambda}</math> - The distance between the first and middle sphere
- <math>\small{\theta_1}</math> - The first axis' azimuthal angle
- <math>\small{\theta_2}</math> - The second axis' azimuthal angle
- <math>\small{\phi_1}</math> - The first axis' polar angle
- <math>\small{\phi_2}</math> - The second axis' polar angle
The lengths of the two pendula are defined as <math>\scriptsize{L_1=\lambda}</math> and <math>\scriptsize{L_2 = L- \lambda}</math> where <math>\scriptsize{L_1}</math> and <math>\scriptsize{L_2}</math> are the lengths of the first and second pendula, respectively, and where <math>\scriptsize{L}</math> is the total length of the string. It is important to note that because the length of the string is held constant and the length of the second pendulum depends on length of the first and the total length of the string, the length of the second pendulum is not an additional degree of freedom; it can be obtained through other lengths and variables already inherent in the system.
Hopefully, the addition of these three degrees of freedom will give rise to even more interesting chaotic behavior; if a rigid, double pendulum already possess chaotic properties, surely an unconstrained one will exhibit even more chaos, especially under forcing.
Previous Work
Surprisingly, the dynamics of three dimensional double pendulum are not well studied, not to mention those of a three dimensional variable length pendulum. Most studies of such systems constrain the lengths of the two pendula to be constant, giving them only four degrees of freedom as opposed to five for the Astrojax. Prior work is scarce, but there are some examples by Philip du Toit and Daniel Dichter and Kate Maschan.
The first differs from our experimental approach in that Dr. Du Toit simulated the system as whole. He simulated all three bobs, without any forcing, starting from some initial conditions. While his mathematical approach is still useful, his experimental process offers little in the way of the investigation of chaos, as he merely examined the dynamics of the system starting from some initial conditions, and did not infinitesimally vary those conditions to investigate how much differently the system evolved over time, which is one of the definitions of chaos.
Daniel Dichter and Kate Maschan completed much the same work as our group, however, they considered the string on which the Astrojax bobs were attached/moved to be a spring with some spring constant <math>\scriptsize{k}</math>, and they also took into account all viscous (i.e., damping) forces. Their paper is a qualitative approach to the system in much the same way ours ended up being a qualitative approach.
Methods
The motion of the Astrojax is, by nature, extremely complex. The motion of a double pendulum constrained to move in only two dimensions is already chaotic, and the sheer complexity of allowing motion in three dimensions can easily be extrapolated. It also bears stating that Astrojax is not purely a double pendulum. The center mass is free to move along the string, allowing the length of the two coupled pendulums to vary, but it is, fortunately, constrained to stay on the string itself, between the two end masses.
In order to proprerly observe, quantify, and analyze the data we made use of the Optitrack motion capture system and its proprietary software MOTIVE. A DENSO robotic arm held one of the end masses of the Astrojax and forced it, resulting in oscillations.
We forced the Astrojax in the vertical, or <math>\small{z}</math> direction in a trianglar wave rhythm using various speeds and amplitudes of the robotic arm. The software WINCAPS III was used in order to interface and program the robotic arm. The Astrojax were covered in infrared reflective tape for purposes of motion capture, along with a point on the robot arm. Thus, we could capture the three-dimensional dynamics of the two free bobs along with the forcing trajectories. Once captured and trajectorized, Optitrack's interpretation of marker disappearances and swaps in vertical position resulted in data gaps, marker identity loss, and marker switching. A tracking program in MATLAB was implemented (credit to M. Kingsbury) which used a ternary search algorithm which searched for least position differences in order to resolve each marker into a continuous path. Therefore we could achieve full 3-D trajectories for the middle & end bobs, along with the forcing marker.
In addition, we investigated the forcing motions used by a human in creating stable orbits with the Astrojax. To do this, we attached an infrared reflective marker to the human forcer's hand via a glove, and implemented the same methodology as with the robotic forcing, simply replacing the forcing marker.
Physical Interpretation of the System
Previous analysis of the Lagrangian<ref>http://www.cds.caltech.edu/~marsden/wiki/uploads/projects/geomech/Dutoit2005.pdf Philip du Toit</ref> and the Newtonian equations of motion have been derived in previous works. We provide a summary of these equations below. The generalized cooridnates are given as: <math> \scriptsize{q = [x_1,y_1,z_1,x_2,y_2,z_2]} </math>
This allows us to write the Lagrangian of the system as: <math> \scriptsize{L = \frac{1}{2}\dot{q}^TM\dot{q}-V(q)} </math> where <math>\scriptsize{V(q)=g(m_1z_1 + m_2z_2)}</math> and
<math> \scriptsize{M = \begin{bmatrix} m_1 & 0 & 0 & 0 & 0 & 0 \\ 0 & m_1 & 0 & 0 & 0 & 0\\ 0 & 0 & m_1 & 0 & 0 & 0\\ 0 & 0 & 0 & m_2 & 0 & 0\\ 0 & 0 & 0 & 0 & m_2 & 0\\ 0 & 0 & 0 & 0 & 0 & m_2\\ \end{bmatrix}} </math>