On the vibrations of a shaft passing through its critical speed

In this paper, we analyse whirling vibration and torsional vibration of a rotating shaft after passing through its critical speed for whirling. First, equations of motion for a rotor-shaft system are derived with accuracy to the second-order of shaft deformations, and it is demonstrated that Euler's equations of motion are equivalent to Lagrange's equations of motion. Then we deals with the passage problem of a rotor-shaft system under a constant driving torque. And analyses for our equations of motion show that the torsional vibration of a rotating shaft can be caused after passing its critical speed for whirling, and the vibration is discussed in both cases of symmetrical rotor and asymmetrical rotor.