The relationship between RPM and end load??

The normally accepted answer is that to avoid whirling, the safe RPM would be 80% of the natural frequency of the overhung shaft+disk in cycles per minute. The best way to get a rough idea of the natural frequency would be to measure it in the non rotating condition. In addition to the stiffening effect of the moment of inertia of the disk about a transverse axis, the flexibility of the chuck is an unknown and may have the opposite effect of reducing the natural frequency. A measurement would incorporate both these effects. However, gyroscopic stiffening effects will also influence the behavior in the rotating case. If you want a book which covers the theory of a thin cantilevered rotor, you could try "handbook of rotordynamics F.F. Ehrich". A detailed analysis is pretty complex.