Three Coupled Oscillators

quasar_4

Hello all.

I am having a substantially difficult time with what should be, actually, a very simple problem.

I have three masses, each with a spring on each side (so three masses and four springs total in the system). My problem is writing down the equations of motion. I can do it when there are two masses and three springs, but I'm not sure what's happening with this many. I tried writing down the Lagrangian to compare to my Newtonian equations of motion, but they weren't the same, so somethings definitely wrong with one (or both) of them.

Basically, the problem is - assuming mass one is displaced to the right, we have a term with k2(x2-x1) where k2 is the second spring constant, and x2-x1 is the separation distance between mass 1 and 2. But now we also have another spring on the other side of mass 2, so do I need another term for k3(x3-x2)?

What happens with the middle spring?

Also, I prefer the Lagrangian method.. it just may not be right because of the same problem. Would the potential energy spring extensions be something like x1, x1-x2,x2-x3 and -x3? That's what I used for constructing my potential energies, but not sure these separations are correct...

Thanks!

mda

I'm not familiar with the Lagrangian representation, but yes, the force on each mass depends on the displacement of its neighbours. This is the definition of coupling.
Essentially the acceleration and displacement are related by a matrix that is tri?diagonal.

If I remember correctly it is reasonably easy to find some special solutions for an infinite chain, eg harmonic waves. A terminated chain will be slightly more complicated.

Not sure what you mean by middle spring... for a 3mass/4spring system there is a middle mass but not middle spring. The middle mass has force k(x1-2x2+x3).