# Normal modes of the 2D double pendulum

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)

• +1 This is a really good question, since the motion is part periodic and part chaotic. – ja72 Sep 7 '14 at 14:29

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.

For a double pendulum there should be two normal modes. It sounds like you have done the analysis to determine the frequencies and mode shapes. If so, every motion will be of the form $x(t) =A \cos (\omega_1 t+\phi_1)+B \cos (\omega_2 t+\phi_2)$. If you fit the data to this form, you are in a normal mode if $A$ or $B$ is zero.

• I'm quite restricted with use of software. I can see the graph of the motion, but this is a university experiment with prescribed equipment and I am to find a way to spot a normal mode being excited without fitting the data (I don't know why, that's just the way it's to be done apparently). So that's why I was wondering about phase portraits of the motion, since I can easily generate this, but I need to know what a normal mode will look like. – user41208 Aug 3 '14 at 15:06
• If it is a normal mode the amplitude of every point should decrease steadily and smoothly. If it is not a normal mode, some amplitudes will increase as energy flows around. Think about the coupled pendulum. If you measure the location of one pendulum and start by exciting just one, the observed amplitude goes from 0 to max to 0 to max ... If you excite a normal mode, the amplitude of a pendulum just decreases smoothly due to friction – Ross Millikan Aug 3 '14 at 22:13

When you have excited a normal mode ,both of the pendulumns will oscillate with the same frequency and will be either in phase or out of phase.I think a better method to see the normal mode will be to oscillate the point of suspension with one of the normal mode frequencies.