Mohammad Faraji Mahyari - Assistant Professor, Department of Mechanical Engineering, Yadegar -e- Imam Khomeini (RAH) Shahre Rey Branch, Islamic Azad University, ۷th Kilometer, Khalij Fars Highway, ۱۸۱۵۵/۱۴۴, Tehran, Iran.
Khadijeh Faraji Mahyari - Ph.D. student, Mechanic of Bio Systems Engineering and Agricultural Mechanization, De-partment of agricultural machinery engineering, University of Tehran, Imam Hoseyn Sq., ۳۳۱۹۱۱۸۷۵۱, Karaj, Iran.
Zahra Faraji Mahyari - Ph.D. student, Mechanic of Bio Systems Engineering and Agricultural Mechanization, De-partment of agricultural machinery engineering, University of Tehran, Imam Hoseyn Sq., ۳۳۱۹۱۱۸۷۵۱, Karaj, Iran.

The main goal of this research is to analyze the effects of slenderness ratio and boundary con-ditions on natural frequencies and instability threshold of a Rayleigh rotating shaft. To this end, non-dimensional kinetic and potential energies are written while lateral vibration is considered. Finite element method is employed to discrete the formulations and Lagrange formula is em-ployed to find vibrational equations. The results represent the significant effects of rotational speed, moments of inertia and centrifugal forces on natural frequencies of the shaft. The natural frequencies are delivered for different boundary conditions. Dimensionless natural frequencies are smaller for a shaft with more rigid boundary conditions. Different effects of centrifugal force on the forward and backward natural frequencies are represented. Also the effects of slenderness ratio and rotational speed on the natural frequencies are delivered for different boundary conditions, which tend to decrease dimensionless natural frequencies. For specific value of rotational speed, the first backward natural frequency is vanished and the shaft be-comes unstable. This rotational speed is known as critical rotational speed. The effects of slen-derness ratio on the critical rotational speed are delivered for different boundary conditions. These formulations can be used in design and choose of the safe working conditions for a Rayleigh rotating shaft.

Rotating shaft; natural frequency; boundary condition; slenderness ratio; instabil-ity;.