A colleague of mine wrote a critical speed computer program based on the transfer matrix method. I checked his results and found they were no different from mine (within the engineering accuracy) using the beam theory. With the rotor's weight as the known mass, elastic modulus and second moments of area of the shaft cross sections to obtain the stiffness and the bearings as the pinned or spring supports in a multispan beam system the natural frequencies can be determined very quickly by simple calculation. With the ability to compute it from first principle you could at least do a sanity check if you do get numerical answers from the readers.