This paper deals with problems of a vibration characteristics of a mechanical system With a shaft which is supported elastically. Such a problem, for instance, occurrs in a jet engine put on a wing of an aeroplane. Fundamental equations and their solutions are obtained and solutions for special cases are also given. The author shows how the natural frequencies of the machine are influenced by the rotating shaft inside them and how the natural frequencies of the shaft vary in response to conditions of the elastic supports. The cross section of the shaft is uniform.