When the rotating speed of a rotor varies periodically with a frequency νω(ω=mean rotating speed of the rotor), the rotor is governed by differential equations having variable coefficients with time. In such a system, it is usually expected that unstable vibrations take place. It is found, houwever, that there occurs no unstable vibration in rotating shaft systems with a variable rotating speed.Variable inertia terms induced by the variable rotating speed result in forced vibrations with frewuencies νω+ω₀, νω-ω₀ as well as ω₀, where ω₀ is the frequency of an external force. It follows that at resonance ω₀≒νω+p and ω₀≒p-νω (p=a natural frequency), forced vibrations of frequencies ω₀-νω and ω₀+νω occur respectively. Furthermore, the external force of frequency ω caused by unbalances of the rotor yields three forced vibrations of frequencies ω, (1+ν)ω, (1-ν)ω.