Vibrations vs Resonant Frequency

First of all, let me say that I’m not an Engineer. Rather, I’ve worked in the maintenance field for over 20 years, although most of that was related to aviation. The company I work with now manufactures electrical wiring and I’ve recently encountered a problem I’ve not seen before. Picture a 15 ft high 10x10 ft square tube that feeds finished wire vertically from the top through a rotating flyer that layers it into a holding basket below. This is a very basic description of one of our finished product coilers that has developed a noticeable vibration when we attempt to operate it at production speed. Many co-workers are saying it’s out of balance because of an oil-misting system we recently installed on the top of the machine. This is indeed, when the vibrations became noticeable but I’m not convinced it’s a balance problem. If it were truly out of balance, wouldn’t the vibration be present from start up through production speed? Please refer to my first statement again.
I’ve done some early research and I believe we may have some resonant frequency issues. On start up, the machine runs rock solid and smooth as silk. It’s only when we approach 95% speed that the tremors start. I’ve tried to have the Production Operators push past the initial vibes to see if we can “plane out” the machine but from the Manager on down, they are terrified of producing scrap wire.
Now, the company I work for is quite larges and has six maintenance departments in the local area and unfortunately the vibration analysis technicians work in a different department. Our Manager and theirs seem to be unable to get along and help looks to be too far down the road for my liking. All of that to get to these two questions; am I on the right track, and if so, what can I do about it?

The phrase "balance problem" is sort of a wastebasket term where a lot of things get tossed in, so take it with the proverbial grain of salt.

However, it does sound to me as if, in this case, balance really is the problem. The behavior you describe is fairly common. For example, I have a wheel on the front of my car that has been knocked out of balance from hitting a pothole (Here in western PA, we grow them in the summer also so we don't have such long waits to hit one in the winter.). I feel absolutely nothing until about 58 mph. From there through about 64 mph, there is vibration. Then it stops. This does involve resonance, but it's a combination of the forced input coupled with a natural frequency. Thus, the Q is pretty low, as evidenced by the fairly large range of speeds and the fact that I can drive it OK without the wheel disintegrating.

Can you get a spectral analysis on this beast? That should tell you the tale. Otherwise, you might try putting a strobe on it; sometimes you can tell what's happening with a strobe.

On the off chance it is just resonance, trying to push through will likely cost you a whole lot of scrap metal.