The prediction of critical speeds of a rotating shaft is a crucial issue in a variety of industrial applications ranging from turbomachinery to disk storage systems. The modeling and analysis of rotordynamic systems is subject to a number of complications, but perhaps the most important characteristic is to pass through a critical speed under spin-up conditions. This is associated with classical resonance phenomena and high amplitudes, and is often a highly undesirable situation. However, given uncertainties in the modeling of such systems, it can be very difficult to predict critical speeds based on purely theoretical considerations. Thus, it is clearly useful to gain knowledge of the critical speeds of rotordynamic systems under in situ conditions. The present study describes a relatively simple method to predict the first critical speed using data from low rotational speeds. The method is shown to work well for two standard rotordynamic models, and with data from experiments conducted during this study.