In this paper, an analytical solution for whirling analysis of axial-loaded Timoshenko rotor is presented and corresponding basic functions are derived. The set of governing equations for whirling analysis of the rotor consists of four coupled partial differential equations; using complex displacements, these equations can be reduced to two coupled partial differential equations. The versatility of the proposed solution is confirmed using published results and the effect of angular velocity of spin, axial load, slenderness and Poisson & apos; s ratio on the natural frequencies of the rotor are investigated.

In this paper, a differential quadrature element method (DQEM) is developed for free transverse vibration analysis of a non-uniform cantilever Timoshenko beam with multiple concentrated masses. Governing equations, compatibility and boundary conditions are formulated according to the differential quadrature rules. The compatibility conditions at the position of each concentrated mass are assumed as the continuity in the vertical displacement, rotation and bending moment and discontinuity in the transverse force due to acceleration of the concentrated mass. The effects of number, magnitude and position of the masses on the value of the natural frequencies are investigated. The accuracy, convergence and efficiency of the proposed method are confirmed by comparing the obtained numerical results with the analytical solutions of other researchers. The two main advantages of the proposed method in comparison with the exact solutions available in the literature are: 1) it is less time-consuming and subsequently moreefficient; 2) it is able to analyze the free vibration of the beams whose section varies as an arbitrary function which is difficult or sometimes impossible to solve with analytical methods.