© Institution of Mechanical Engineers. Rotating shafts often experience undesirable large-amplitude whirling oscillations associated with resonance at critical speeds. This paper further develops a nondestructive technique in which measured information about the growing nature of the response is used to predict an incipient critical speed. A number of models of varying degrees of sophistication are developed and tested using the new approach, but the main advantage of the method is that it is model-free and thus possesses considerable practical utility. In addition, further experimental results are presented for the case of two disks mounted on a shaft, and the technique is successfully demonstrated in predicting a critical speed associated with a higher mode.