Often the best way is to get the person you need to convince to stand by the machine with you and if it's making a huge noise and vibrating badly, and he feels a bit scared he will agree it's time to stop it.
Now the interesting thing is that trend is important, if it's getting worse by the week or day or hour, common sense tells you to act ever more quickly.
Having strong management resistance to stopping a piece of equipment can be a wonderful learning experience because someone else is taking the responsibility of letting it go, and you get to see just how it turns out, which is a real jewel not to be missed.
Acceleration of high speed gear meshing vibration can often be quite high, like 15 g rms, but if you intergrate the spectrum to Velocity and then look at the tmf peaks you may find they are quite small, like 3 mm/s. Depending on the mass and stiffness of the gear case the levels can be very different.
Impulse demodulation is very useful, especially the CSI Peak Vue which does trap the true peaks and if you look at the waveform from a spectrum of say 100x shaft speed and 3200 lines you will see a pattern of metal to metal contact of some or all of the teeth for many turns of the shaft. This is bad as gears should not be impacting but they often do.
Hey you are on steep learning curve, just make sure you don't put yourself in a position beyond your ability and start making definite predictions when just starting out.
Cheers
John