A special type of equivalent circuit is developed that is particularly suited for the numerical calculation of the critical speeds of uniformly rotating shafts. The shaft may consist of several spans rotating in flexible bearings supported by a flexible foundation. Either the various spans may be asymmetrical along two principal axes or the bearings may have asymmetrical flexibilities. The same circuits are, of course, valid also for the study of the natural frequency of vibration of a skewed asymmetrical beam held by flexible supports at several points, such as a turbine bucket. The circuits may be solved by slide rule, by punch-card methods, or by a “network analyzer.” The procedure is self-checking as the correctness of the numbers may be checked at any stage of the calculations. For the symmetrical case, the results are identical with those found by the method of Prohl (1) or Myklestad (2), even though the circuit method requires fewer calculations. In general, the equivalent circuits for the various spans, bearings, overhung loads, and foundations are independent and are interchangeable for different combinations of shaft structure. Also, any variation in a component part leaves the circuits of the other parts unchanged. A punch-card study of a turbine generator shaft with fixed bearings carrying a high-pressure turbine, a low-pressure turbine, and a generator is given in a companion paper by A. W. Rankin (3).