The Physics of Coupled Oscillators

Rhett Allain
19 min readMay 3
Photo: Rhett Allain. Coupled oscillators

I know you’ve looked at a mass on a spring, but have you looked at 2 masses with THREE springs? It looks like this:

There are two masses (the yellow and cyan blocks) with a spring connecting between them. Then there are two more springs connecting each mass to the fixed points on the outside (the red balls). We call this a coupled oscillator since there are two oscillating masses and they are coupled together. See — physics doesn’t have to be so confusing. At least the name makes sense.

Although we are going to look at masses moving in one dimension, the physics associated with this problem also appear in real life. If you have something oscillating (which is about half of everything in the modern world), it can have an influence on another nearby oscillator.

If both masses can only move back and forth, then we can describe the whole system with just two coordinates — the x-position of mass 1 and the x-position of mass 2. We don’t even have to use the same origin. With that, we have a setup like this.

If the x-values are measured from the point of equilibrium, then the spring forces will be the the spring constant (k) multiplied by the position (x). Of course each mass will have two spring forces — which might be in the negative or positive direction. Oh, but there’s no friction (that’s important). Also, I’m going to start off with two equal masses. For the springs, the two outer springs have equal stiffness (k1 = k3 = k) and I’m just going to call that spring constant k. The inner spring is different and I’ll call it b instead of k2. But call it whatever makes you happy.

I’m going to solve this problem several different ways. I’m going to start off with the methods that are the simplest to understand and then move to more complicated (but also more useful) solutions.

Numerical Solution Using Newton’s Second Law.

You might think this solution is advanced since it requires a computer program, but the math and physics is very straight forward. In fact, using this method a first semester student could solve it. Note: I

Rhett Allain

Physics faculty, science blogger of all things geek. Technical Consultant for CBS MacGyver and MythBusters. WIRED blogger.