fredt, all I can say is man ease up a bit. Are you sure you are not confusing over speed with critical speed? There are many turbine generator sets in industry that run right on their critical speed and have for years with no problem. If there is sufficient damping a critical speed is not critical. In addition even if the damping is not great, if it is well balanced, it still is not a problem. The class of machines I speak of are a bit balance sensitive, but it is not a big deal. We run high speed compressors that have enough damping and are so well balanced that the critical speed amplitude is less than 1 mil pp (25 microns), I could run those machines on their critical for hours or days with no problem.
Over speed protection is important, an overspeed will disintigrate a machine and can kill, and overspeed protection could be related to a pressure limit in a vessel, but not a critical speed. Just to make it clear overspeed= running the machine faster than it was designed to run, this will cause it to come apart dangerously. Critical speed= the speed at which a rotor natural frequncy is equal to the rotor turning speed. This can result in high vibration, but if well balance, and well damped, is usually not a problem.
As for the other question, I think it is a bit confused as well. I like to avoid having a critical at half the normal operating speed, since whirl instability comes in at a nominal 0.42 to 0.48X running speed. Whirl is typically manageable, but when the whirl instability excites the first critical, often refered to as whip, then a resonance condition occurs, and the very nature of the whirl instability is to negate the damping. Theoretically if whip occurs then the vibration amplitude should approach infinity, and fredt's concern is then founded. But practically, the rotor contacts the stationary elements and gains damping from that, limiting the vibration, but damaging the machine. If the first crtical is say 60% of normal running speed, there is no way for a whirl instability to excite it, and the problem of whip is avoided. The flip side is that you would like to avoid having a critical (ususally the second) at twice the normal running speed. There are certain machine malfunctions like severe misalignment, and shaft or support stiffness assymetry that can generate a 2X running speed forcing function. If there is a critical at 2X running speed it will then be excited. Again this may not be much of a problem if it is nicely damped, but second criticals rarely are so a little bit of 2X forcing function gets greatly magnified and may pose a problem. As a result, from a design stand point it is good practice to avoid haveing a crtical at twice running speed.