On the vibration of a rotating shaft passing through the critical speed

Abstract

Several papers have been published concerning the vibration of a rotating shaft carrying a rotor at the time of passing through the critical speed. But most of them treat the rotor as a concentrated mass. In this paper, the equations of motion of the rotor when the shaft speed varies arbitrarily are introduced using a rotating coordinate system and taking into account the gyroscopic effect of the rotor. The equations are solved by Runge-Kutta-Gill's numerical method in various cases where the rotor and the bending rigidity of the shaft are symmetrical or asymmetrical and the shaft passes through the critical speed with a constant or a variable angular acceleration. The exact solutions expressed with Fresnel's integrals in the case when the rotor is a concentrated mass are compared with the results of calculation by the numerical method in order to check the accuracy of this method. In addition, the results of simulation by an analogue computer are shown.