[Physics] Normal modes of the 2D double pendulum

experimental-physicsnewtonian-mechanics

I'm performing an experiment with a 2D double pendulum, and in part of it I want to investigate the normal modes of the double pendulum, where the pendula are not of equal length or of equal mass. My question is – how will I actually know when I've successfully excited a normal mode?

I start by setting the initial angles to be in (roughly) the correct proportion to one another in order for the initial setup to be an eigenvector, but of course once I release the pendulum there is a slight 'jolt' which means I can't be sure the initial conditions were exactly an eigenvector (of course, realistically I'm only going to be closely approximating one). Then, once data is recorded I can generate a phase portrait with the computer, and the phase portrait I see when I get quite close to an eigenvector looks kind of like a tilted cylinder (sorry, don't know how to post a Matlab graph here). Is there a way to tell from this phase portrait (with the angles $\theta_1$ and $\theta_2$) whether I've hit a normal mode or not? Would appreciate some help.

(ie. if someone could post a picture of a phase portrait of the normal modes of a double pendulum, so that I know what I'm looking for, that would be very appreciated)

Best Answer

You will generally have three types of trajectories, periodic, quasiperiodic and chaotic. Plot the $\theta_1, \theta_2$ and these three will manifest as

  1. A Lisajouss-like curve, i.e. a curve which closes after a finite amount of time. This would be a periodic trajectory.
  2. A "box curve" shifting every once in a while, filling regularly a certain area and never closing. This would be a quasiperiodic trajectory.
  3. A never closing curve without any regularity - this would be a chaotic trajectory.

You can tell you are getting close to a periodic trajectory by the corresponding curve "almost closing" and shifting by a very small amount after every "almost close".

Now it depends how do you define a normal mode - it could be either any periodic trajectory, because in that case the two masses oscillate with rational periods which can be quotiented out into a common period (i.e. the curve-closing time). On the other hand, if you define a normal mode as the mode in which both masses move strictly with the same unquotiented frequency, you would have to choose a circle-like curve. In the extreme case of synchronized swinging (i.e. basically a simple pendulum) this curve would get flattened into a line.

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