Wavelace

Coupled Oscillators

Bodies · mechanical systems

springs
m=10^6  x=-2
m=1  x=-0.7
m=1  x=1
m=10^6  x=2
k=1  from=1  to=2  rest=1
k=c  from=2  to=3  rest=2
k=1  from=3  to=4  rest=1

Open in the app The dials and keys named below are the app's.

The setup

Two unit masses sit on a line between two walls. Each is tied to its wall by a spring of stiffness 1 and natural length 1, and the two are tied to each other by a third spring of stiffness c and natural length 2. The walls are masses of a million, so nothing moves them. At rest the masses sit at −1 and 1. The first starts 0.3 to the right of its place, the second exactly at its place, and neither is moving to begin with.

The exchange

On its own each mass would swing with period 2π. The coupling spring lets the moving one push on the still one, a little each swing, and the energy walks across. The first mass slows as the second grows, until the second has all of it and the first stands still; then it walks back. The time for one crossing is π / (ω₂ − ω₁), where ω₁ = 1 and ω₂ = √(1 + 2c) are the two normal modes, the masses swinging together and against each other. At c = 0.5 that is 7.6 s; measured, the second mass is fullest at 7.8 s.

The beat is the sum of those two modes at nearby frequencies, the same arithmetic as two close tones beating in the ear. Here it is a mechanism rather than a formula: no term is written for it, the springs do it.

Try

  • Drag c to 0: the second mass never moves, since nothing joins it to the first. At 1 the crossing takes 4.5 s. At 2 the two modes are far apart and the exchange is no longer a slow beat.
  • Start both masses shifted the same way, x=1.3 on the third line: that is the in-phase mode on its own, period 6.28 s, and nothing crosses. x=0.7 instead is the other mode, period 4.44 s.
  • Top in the deck: the picture is a line, and from above the four masses and three springs read as a diagram.

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Bodies · mechanical systems