Triple Pendulum
pendulum l=1 m=1 a=2 l=1 m=1 a=2 l=1 m=1 a=2
Open in the app The dials and keys named below are the app's.
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.