Free bending vibration of a multi-step rotor

The free bending vibration of a rotating shaft composed of multi-step segments with each segment having a uniform circular cross-section has been analyzed using the Timoshenko beam model. Torque has been applied at each end as in a power-transmission shaft and the gyroscopic effect due to the shaft rotation has also been considered. The separation of time and spatial variables has been performed for the differential equation of motion, and then the spatial solution in a segment of uniform cross-section has been expressed in a closed form. Application of boundary conditions has induced global equations of a rotor system, from which the natural frequencies and corresponding mode shapes have been obtained. The entire procedure is straightforward, and thus the resultant mode shapes are continuous unlike the result of existing methods such as the finite element method and transfer matrix method. Effects of the rotating speed and the applied torque at both ends on the free vibration of the rotating shaft have been discussed. The natural frequencies of the forward and backward modes have been compared with the results using the finite element method. The shear force and bending moment distributions along the shaft at each mode have been derived by differentiating the mode curves. (C) 1999 Academic Press.