Modern analysis uses —plots of natural frequencies versus rotor speed—accounting for gyroscopic effects and bearing stiffness changes.
If you don't have a PDF ready, you can:
(NATO/STO): An educational note that explains the mathematical modeling of bearing and seal reaction forces and includes examples for troubleshooting high-performance pump rotordynamics. Rotordynamics of Semi-Rigid and Overhung Turbomachinery
—the rotational speeds at which a system's natural frequencies are excited, leading to potentially catastrophic resonance. Core Concepts and Modeling
I’ve just compiled a (PDF format) that walks through: