A rigid body rotation with 3 joints

Question

I was playing around a physics simulation and I saw this. (One of my first questions , forgive if this question is too simple or has grammatical errors)

Here it is in action.

I wanted to figure out its time period as it is periodic motion, i also figured that they are performing rotation along the joints and I can use transition from 1 rotation frame from other, Also something more important is this image.So the rod 1 performs motion in a circle where as rod 2 performs motion in an ellipse and rod 3 performs some other periodic motion which i do not know.

I need help in understanding this motion and figuring out its time period. Thanks for the help in advance. (You can use Physion where User named Dimitri made this)

Answer 1

True double or triple pendulum would give chaotical motion (non-periodic), because if for example we take double pendulum :

then $$ \begin{aligned} \frac {d \theta_1}{dt} &= f(\theta_1,\theta_2) \\ \\ \frac {d \theta_2}{dt} &= f(\theta_1,\theta_2) \end{aligned} $$

i.e. both angle changes depends on the current state of angles, though dependance is through different functions. We can say that many systems which are defined with "inter-dependant" differential equation system , like in the form $$ \frac {dx}{dt}=y+C\\ \frac {dy}{dt}=x+C $$

will exhibit chaotic behavior. One of those systems - double or triple pendulums - has no concept of "period", because pendulum end moves in chaotic way :

The thing that your given simulation gives periodic motion of pendulum end is made artificially by script,- forcing pendulum first stick to move in strict circle. It's like some "engine" would rotate first stick periodically. Hence period of this artificial movement is simply period of first stick. You can extrapolate it from the fact by looking at the given physion.net simulation, specifically, when it reaches the same angle $\theta_1$ from which it has started from.