Yes, but we need more detail. Are you interested in:
Bending vibration? Which modes?
Bearing housing vibration?
The frequency of all of these can be calculated, or at least estimated. But you first need to fully define what you are looking for. If you are interested in the vibration amplitude at a critical speed or natural frequency, that is a function of damping and the forcing function. Have you had the Theory of Vibration class yet?
I haven’t taken a vibrations class sadly. So the shaft is for a spindle designed for milling which is rotating at 8000 RPM. There is a force on one end of the spindle which makes contact with the material. What I have done so far in simulation software is calculate the resonance frequencies with 6 different mode shapes. The first mode had a frequency equating 180000 RPM which is pretty high. Ideally I would want to measure how the shaft vibrates at the point of contact and I assume this involves the bending of the shaft. So a bending vibration would be ideal right? Sorry I am not too familiar on what the correct bending analysis is. Could you point me in the right direction? Appropriate equations would be helpful.
Machine tool spindles are supported by several bearings. The spindle itself is short and stiff. The natural frequencies of the spindle / bearing system are strongly affected by the bearing stiffness. The result is a system that is too complex for simple hand calculations. There may well be one or more rigid body modes of vibration.
My 5th Edition of Formulas for Stress and Strain, by Roark has several pages of equations for calculating the natural frequency of beams with various support conditions, but nothing that would work for a machine tool spindle. But the 5th edition has only 624 pages, while the latest 8th edition has 1072 pages, so might be worth a look.
The short, stiff shaft with complex geometry and multiple bearings of a machine tool spindle is best analyzed using FEA. Even that is a challenge because it is difficult to properly model the stiffness of the various bearings. Note that both the radial and axial stiffness of each bearing must be specified.