The Science of …Vibration Harmonics

Roush Fenway Racing came out with a statement about the missing oil tank lid on Wednesday that just begs to be deconstructed scientifically.

“The bolt holding the lid in place failed in its purpose as a result of vibration harmonics generated by the car and the race track during the race. The bolt was secure enough to survive 225 miles of practice, perhaps up to 399 of 400 miles of the race and the scrutiny of numerous inspections. It’s a tough business for any race team to have to pledge $100,000, 100 points and a six race crew chief suspension as an indemnity payment to NASCAR against a promise forced from us by NASCAR that no bolt will ever fail its purpose under race conditions.”

As Shawn Courchesne said in his blog, aptly titled, “Vibration Harmonics Seem To Be Shaking Up More Than The Cars At Roush Fenway” (which I’m nominating for best blog title).

“Vibration harmonics?” Are we talking about a racecar or the string section of the local symphony orchestra?”

Mr. Courchesne is actually closer to the truth than he might have intended, since vibration harmonics do occur in music and motor vehicles. Anything that vibrates can have harmonics.

It all comes down to waves. Regardless of whether they are sound waves, light waves or mechanical vibrations, the simplest form of a wave is something like the picture below, which is graphed with displacement (which is just distance) on the vertical axis vs. time on the horizontal axis. I should have changed the axis title, but I wanted to get this post up quickly in response to all the questions I’m getting this morning about vibration harmonics!

You can think of this graph as showing something (oh, let’s say…I don’t know…an oil tank cover) bouncing up and down in a regular motion. As you move horizontally across the graph, you’re watching the corner of the cover go up and down and up and down. To make it simple, I’ve picked a wave with a single frequency. The frequency of the cover’s vibration is just how many times the cover goes up and down every second. We measure this in units called hertz (which is abbreviated Hz). If the oil tank cover bounces up and down two times each second, we say that it is oscillating at 2 Hz.

A harmonic is a multiple of the base frequency. The harmonics of the 2 Hz vibration mentioned above would be 4 Hz, 6 Hz, 8 Hz, etc. If a vibration occurs at a certain frequency, the vibration harmonics would be the integer multiples of that vibration frequency.

The higher the frequency, the more rapidly the object vibrates. Vibrations and vibration harmonics are important issues in car design. If you strike a tuning fork next to an identical tuning fork (meaning both tuning forks are designed to be the same frequency), the tuning fork that wasn’t struck can start to vibrate. That’s because the tuning fork was designed to like vibrating at that particular frequency. Many objects have frequencies at which they like to vibrate, which we call resonant frequencies. Sort of like how you have a favorite driver, an object like a car chassis can have a favorite vibration frequency. Low-frequency vibrations (on the order of a few to tens of Hz) tend to make people feel nauseous, much like waves rocking a boat can make people seasick.

There are vibration harmonics in racecars. Wikipedia shows a noise spectrum for Michael Schumacher’s engine running at 16680 rpm, which I’ve reproduced below.

Don’t be put off by the sort of busy graph. The vertical axis shows the amplitude of the noise. It’s a logarithmic axis, which means that each tic mark is ten times more then the tic mark below it. (They use the log axis to be able to fit more detail on the graph. Otherwise the smaller details get lost.) The horizontal axis is the frequency, which is given in rpm. (rpm is an angular frequency, but that’s another blog. Bear with me here.)

The important thing to get out of this graph is that there are sharp peaks at 1500 rpm, 2000 rpm, 2500 rpm, etc. Those frequencies are harmonics of some fundamental (probably 500 rpm) in the engine. The peaks mean that there is a lot more vibration at those particular frequencies. This is an example for an engine because it’s what I could find quickly on the web, but the same phenomenon happens for a vehicle chassis.

What Mr. Smith is suggesting (I think) is that there just happened to be a vibration harmonic with a large amplitude that corresponded exactly to the frequency necessary to jiggle loose the oil tank cover, which normally would have been prevented by the bolts holding down the oil tank cover;however, a bolt failed and the combination led to the cover jiggling off.

David Poole reports (and I have independent confirmation from similar ‘people who should know’) that teams are required to have more than one bolt holding on the oil tank lid. It would take one heck of a coincidence to have just the right vibration harmonic and the concomitant failure of four bolts.  I have been informed that there is no firm rule requiring a minimum number of bolts to hold on the cover; however, teams are expected to use enough bolts that the cover doesn’t come off. NASCAR can argue that the team was well within its rights to use one bolt, but they also accepted the risk that if that one bolt failed and the lid came off, they were going to be penalized.

I think the most telling comment comes from Buddy Baker who, on Sirius radio Wednesday night said, “I’ve seen the pictures. If I were them… I think I’d just sit down and take the penalty.” The quote is inexact, but the general idea and the tone of his voice suggested that Roush Fenway Racing should probably just leave sleeping dogs lie. The rest of the RFR statement sort of suggests the same thing:

“We are currently evaluating whether or not the circumstances justify submitting to the unpleasantness of the appeal process, but from a competitive standpoint, we believe it is in our best interest to have Bob Osborne begin serving the penalty now. Chris Andrews, our chief engineer, will be at the track on Friday and Saturday to lead the No. 99 team and general manager Robbie Reiser will join Chris on the pit box Sunday during the race. We are confident that we have the depth of talent in our organization and the strength in Carl Edwards and the No. 99 team to overcome this penalty, should that end up being the final outcome of this unhappy episode.”

So all of a sudden, everything becomes crystal clear to the NASCAR conspiracy theorists: Robbie done it. Robbie Reiser rigged the oil tank cover to come off because he saw an opportunity to get back on the pit box. The only “vibration harmonics” involved were Robbie shifting uncomfortably in his executive chair back in Concord, pushing a pencil and missing being a crew chief.

Note added 3/7/08 7:00 am. Nate Ryan at USA Today has an interesting follow-up article about the oil tank controversy.