Whirling analysis of multi-span multi-stepped rotating shafts

In this paper, differential quadrature element method is used to analyze free transverse vibration of multi-stepped rotors resting on multiple bearings. Timoshenko beam theory is used to show the gyroscopic effects. Also each bearing is replaced with four springs: two translational and two rotational acting on two perpendicular directions. Governing equations, compatibility conditions at the each step and each bearing and external boundary conditions are derived and formulated by the differential quadrature rules. First, convergence, accuracy and efficiency of the proposed method are tested by the presented exact solutions. Then, Campbell diagram is derived for a desired case study and variation of natural frequencies is investigated versus angular velocity of spin. This problem is solved by some authors using analytical methods which are time-consuming, especially for rotors with high number of steps and bearings. But, as shown in this paper, proposed solution is more effective and less time-consuming and is applicable for rotors with any number of steps and bearings without losing accuracy.