# Behavior of triple pendulum with heavy end mass

1. Dec 8, 2015

### ellipsis

Is a triple pendulum with a significantly heavy end-mass supposed to spaz around?

Using the Euler-Lagrange formula in Mathematica, I've found (and simulated here) a triple pendulum system with arbitrary masses and lengths. The rods are massless (so no moment of inertia's), and I've generalized the state equations in terms of angular coordinates. There is no friction.

There's a sanity check in the simulation: The total energy is quite constant.

When you make the end mass very heavy though (and decrease the time-step to compensate for the numerical stiffness), the system has a very odd twitching, oscillating behavior. Is this correct? Has this phenomenon been seen elsewhere, on other simulations? Some links would be appreciated.

I tried googling "behavior of triple pendulum with heavy end-mass", but nothing came up. Go to that link, set "Mass 3" to 100 or so, and "steps per frame" to 1000 or so to see what I mean.

2. Dec 9, 2015

Staff Emeritus
I think it would help for you to define the term "spaz around".

3. Dec 9, 2015

### ellipsis

I was going to attach a .gif animation of the simulation, but I couldn't get it to work. The trajectory of the end mass is fairly smooth, but the first and second masses bounce around wildly relative to each other, and attain quite high velocities (but total energy is constant).

When I get some more time (Exam week for me), I'm going to figure out how to do .gif screen captures.

4. Dec 9, 2015

### cjl

I suspect that's an artifact of the fact that you are assuming massless rods and frictionless joints. This makes the system not really physically realizable, since you'd never see those kinds of accelerations of the middle of the rod or the joints due to the friction in the joints and the inertia of the rods, as well as inherent damping from things like air resistance. When you ignore all of these other inertias and loss terms, you do tend to get a lot of movement from the middle masses, since the end mass (being rather massive) can induce very high accelerations in the intermediate masses, but I suspect you'd have a very hard time making a system behave like that in reality.