Snow Fugitive

A couple weeks ago I got a really interesting question/story from my friend Garrett, relating an experience entirely foreign to me here in Florida:

Last weekend we got a huge blizzard in MA. In Boston we tied the record for the most snowfall in a single 24h period (~24''). I have street parking and my car gets absolutely buried. Snow up to the windows on both sides. The next day I dig out and am driving my car. Everything is fine until I get on the highway. Once I am above ~50mph, my car starts to shake violently. I pull over, thinking I may have a flat tire. Nope. The tires are fine and no snow is in the wheel wells. So I drive again and it is still happening. So I come up with the following hypothesis: When my car was parked and getting snowed in, the snow built up more-so on the lower side of my wheels/rims. This gave my wheels an uneven moment of inertia. Below a certain speed the force and oscillation period of the unbalanced wheels was low. But above ~50mph my suspension could no longer compensate, and I began to feel the vibration (similar to an unbalanced washing machine).

My initial thought was that at a certain speed he was hitting the resonant frequency of the snow/wheel combination. As the wheel turns, it has to exert a centripetal force on the snow to keep it moving with the wheel. According to Newton, that means the snow is exerting a centrifugal force on the wheel, which will be given by

where m_s is the mass of the snow, v is the car's speed, R is the radius of the tire, and r is the direction from the wheel hub to the snow. This will be a sinusoidal force, with frequency proportional to the speed of the car. That appears to be consistent with Garrett's experience, where the vibration didn't start until he got to a certain speed.

To check that, we need to come up with a model of the car's suspension. I found this paper promising: It suggests a setup where the tire and the car suspension each act as a damped spring.

The ks are the spring constants, the cs are the damping coefficients, and the zs are the height from the road. We can write the forces on the wheel and body as

Each of these springs will exhibit resonance at a frequency we can calculate:

However, if we plug in some of the values used in the paper above, we get frequencies that correspond to about 1 mph or less! Garrett pointed out that I shouldn't be so surprised:

I think it makes sense that the resonance peak you identified is at such a low speed. I'm assuming that car manufacturers design the suspension to NOT resonate (due to unbalanced wheels) at typical driving speeds. Moreover, what I experienced did not feel like a resonance peak. I did not notice the vibration at all until I was above ~40mph or so, but from then on the vibrations increased with increasing speed. There was no point at which I felt the vibrations decrease as I increased speed. 

Taking another look at the equation for the snow's centrifugal force, we see it's proportional to v^2, so the faster the car is moving, the more the snow unbalances the wheel.

I put together a simulation of the system (using SSMs again, since I've been spending too much time around engineers) and we can see exactly this effect in both the average displacement of the car body:

and in the maximum displacement:

I had such a great time thinking about this and discussing with Garrett, I decided to put together another HTML5 doodad that you can play with below. I had to tweak the parameters a little, and there are still some significant transient effects, so you may get some crazy results just after changing the inputs, but they'll settle down after a few seconds. Thanks for the idea, Garrett!