Solving the Damped Harmonic Oscillator

Photo: Rhett Allain. Setting up the damped harmonic oscillator

Maybe you have a solution for the harmonic oscillator. One example would be a horizontal mass on a frictionless surface connected to an ideal spring. It’s super important physics problem that shows up in many different places. If you need a review, I have you covered in this previous post.

Three Solutions for the Simple Harmonic Oscillator

OK, but what if it’s a mass on a spring with a drag force where the magnitude of the drag is proportional to the velocity of the mass? This is what we call the “damped harmonic oscillator” since the drag force will decrease the oscillation amplitude over time.

If the mass moves in the x-direction, we can write this problem using Newton’s second law in 1D (don’t need vectors).

Here, I’m using the common notation for derivatives with respect to time where x-dot (x with a dot over it) is equal to dx/dt and x-double dot is the second derivative. Also, k is the spring constant (in N/m) and b is the drag coefficient (in kg/s). This differential equation is much more challenging to solve compared to the simple harmonic oscillator (that’s why it’s called “simple”) — but we can still do…