The present program had several goals. Primary among these was an inquiry into the effect of gyroscopic and rotatory inertial effects on the whirling of rotating shafts. The secondary goals included the development of a computer program for the prediction of whirling of stepped shafts, the completion of an inquiry into an approximate method of analysis which is used widely in Naval predictions of shaft whirling phenomena, and the development of a very versatile and dependable test rig for the experimental investigation of primary and secondary effects in Rotor Dynamics. An appendix contains a complete derivation of equations of motion for shafts which rotate at constant speed. Sections II and III contain analyses of specific shaft systems which were performed with the use of the basic differential equations. The critical speeds and mode shapes were found for a uniform undamped rigidly supported cylindrical rotor with gyroscopic and rotatory inertia effects included. The cylindrical rotor was analyzed in fixed short bearings as well as in fixed long bearings. A second analysis inquired into the critical speeds of a system consisting of a single disk loaded at an arbitrary location on a uniform continuous shaft which was mounted on short fixed bearings. In general, it was found that there are two critical speeds for each order of flexible mode shape.