Abstract: In this paper, a general case of vibration of a shaft with multiple rotating masses is discussed. For finding frequencies, we use the generalized Newton's method which is, proposed, by F.H.Chao. A difference equation method is also used to determine the principal modes of, vibration of the system. Accurate numerical results can be obtained directly by using the tabulation method given in this paper. It is a convenient, practical and general one which can be applied to find the frequencies and amplitudes of torsional vibration in electric machinery, Diesil engines and propeller shaft.