# Triple Pendulum

Bodies · mechanical systems

```
pendulum
l=1  m=1  a=2
l=1  m=1  a=2
l=1  m=1  a=2
```

[Open in the app](https://www.wavelace.com/app#p=87) · [This page](https://www.wavelace.com/presets/triple-pendulum)

### The setup

Three rods hang in a chain, all of length 1 and mass 1, all released from rest at `a = 2` radians. That last column is the starting angle, measured from straight down, so `2` radians is `114.6°`. The chain hangs from a pivot at the height of its total length, 3, so stretched out it just touches the floor.

### Three modes, then none

Give the chain small angles and it is a linear instrument with three normal modes. All three rods can swing in step, with a period of `≈ 3.11` seconds. The ends can swing opposed, at `≈ 1.32` seconds. The middle rod can fight both its neighbours, at `≈ 0.80` seconds. Any small motion is a fixed mixture of the three.

At 2 radians none of that survives. The governing equation, `M(θ) θ̈ + C(θ, θ̇) + G(θ) = 0`, couples the three angles through cosines and squared angular speeds. There is far more than enough energy for a joint to pass over the pivot, and the top rod alone turns eight full circles in the first minute.

Start a second run a thousandth of a radian away and the two are a radian apart by `t ≈ 6.5`. Make the difference a hundred times smaller and the parting is delayed only to `t ≈ 8.2`. The wildness on screen belongs to the system itself, and nothing in the chain is damped or driven.

### Try

- `Front` in the deck: the swing plane seen face on.
- Tame it with `a=0.2` on all three lines, and the three modes beat against one another instead.
- Disturb the top rod alone, `a=0.2` on the first line and `a=0` on the other two. Within half a second the swing has reached the bottom joint, which ends up moving further than the rod that started it.
- Shorten the bottom rod, `l=0.4` on the third line: a quicker tip on the same wild arm, and a pivot that drops to 2.4 to keep the chain on the floor.
- `Rewind` runs the identical path again, every time.

### Read more

- [Double pendulum](https://en.wikipedia.org/wiki/Double_pendulum)
- [Chaos theory](https://en.wikipedia.org/wiki/Chaos_theory)
- [Normal mode](https://en.wikipedia.org/wiki/Normal_mode)
- [Lagrangian mechanics](https://en.wikipedia.org/wiki/Lagrangian_mechanics)
- [Runge–Kutta methods](https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods)