Group 3 2014

Group members: Colin Campbell, Steven Harrington, and Andy Karsai

We seek to investigate the properties of a generalized double pendulum. A vast majority of previous investigations into the dynamics of double pendula constrain the motion of the pendula in question to motion in only two dimensions and keep the lengths of both pendula fixed. This still allows the viewing of chaos to be sure, but the dynamics of allowing the system motion in three dimensions are rich and quite intriguing. The Astrojax is notable in that, while the length of the entire system is constrained, the lengths of the two individual pendula are not. These lengths vary dynamically as the system experiences forcing and thus provides a good starting point for analysis of a general case. We intend to measure the Astrojax's response to well quantified forcing using a perpindicular array of cameras to track the position of each mass in the system. From there, the data for velocity and acceleration can be generated, analyzed, and put to use.

Introduction

An oscillation is defined, in physics, as a regular oscillation in magnitude or position, 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>\small{\theta_1}</math> and <math>\small{\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 legnth 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. <math>\small{\frac{d}{dt}\left(\frac{dL}{d\dot q}\right)-\left(\frac{dL}{dq}\right)=0}</math>