A New Methodology for Modeling and Free Vibrations Analysis of Rotating Shaft Based on the Timoshenko Beam Theory

The linear lateral free vibration analysis of the rotor is performed based on a new insight on the Timoshenko beam theory. Rotary inertia, gyroscopic effects, and shear deformations are included, but the torsion is neglected and a new dynamic model is presented. It is shown that if the total rotation angle of the beam cross section is considered as one of the degrees-of-freedom of the Timoshenko rotor, as is common in the literature, some terms are missing in the modeling of the global dynamics of the system. The total deflection of the beam cross section is divided into two steps, first the Euler angles relations are employed to establish the curved geometry of the beam due to the elastic deformation of the beam centerline and then the shear deformations was superposed on it. As a result of this methodology and the mutual interaction of shear and Euler angles some variable coefficient terms appeared in the kinetic energy of the system which makes the problem be classified as the parametrically excited systems. A linear coupled variable coefficient system of differential equations is derived while the variable coefficient terms have been missing in all previous studies in the literature. The free vibration behavior of parametrically excited system is investigated by perturbation method and compared with the common Rayleigh, Timoshenko, and higher-order shear deformable spinning beam models in the rotordynamics. The effects of rotating speed and slenderness ratio are studied on the forward and backward natural frequencies and the critical speeds of the system are examined. The study demonstrates that the shear and Euler angles interaction affects the high-frequency free vibrations behavior of the spinning beam especially for higher slenderness ratio and rotating speeds of the rotor.