New Approach to Instability Threshold of a Simply Supported Rayleigh Shaft

Document Type : Research Paper


1 Department of Mechanical Engineering, Islamic Azad University, Shahr-e Rey Branch

2 Department of Industrial Design, Islamic Azad University, Shahr-e Rey Branch


The main goal of this research is to analyse the effect of angular velocity on stability and vibration of a simply supported 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 Linear method is applied to analyse instability threshold of a rotating shaft. These results represent the significant effects of mass moment of inertia, centrifugal forces and rotational speed. Also, the differences between Rayleigh and Euler-Bernoulli modelling are delivered. Furthermore, the effect of slenderness ratio on instability threshold and the natural frequencies are illustrated. Increasing rotational speed leads to decreasing of instability threshold and forward and backward natural frequencies. These formulations can be used to choose the safe working conditions for a shaft.


[1] Grybos R., 1991, Effect of shear and rotary inertia of a rotor at its critical speeds, Archive of Applied Mechanics 61 (2): 104-109.
[2] Choi S.H., Pierre C., Ulsoy A. G., 1992, Consistent modelling of rotating timoshenko shafts subject to axial loads, Journal of Vibration, Acoustics, Stress, and Reliability in Design 114 (2):249-259.
[3] Jei Y. G., Leh C. W., 1992, Modal analysis of continuous asymmetrical rotor-bearing systems, Journal of Sound and Vibration 152 (2):245-262.
[4] Singh S. P., Gupta K., 1994, Free damped flexural vibration analysis of composite cylindrical tubes using beam and shell theories, Journal of Sound and Vibration 172 (2):171-190.
[5] Kang B., Tan C. A., 1998, Elastic wave motions in an axially strained, infinitely long rotating timoshenko shaft, Journal of Sound and Vibration 213(3):467-482.
[6] Jun O. S., Kim J. O., 1999, Free bending vibration of a multi-step rotor, Journal of Sound and Vibration 224(4):625-642.
[7] Mohiuddin M. A., Khulief Y. A., 1999, Coupled bending torsional vibration of rotors using finite element, Journal of Sound and Vibration 223(2):297-316.
[8] Gu U. C., Cheng C. C., 2004, Vibration analysis of a high-speed spindle under the action of a moving mass, Journal of Sound and Vibration 278:1131-1146.
[9] Behzad M., Bastami A. R., 2004, Effect of centrifugal force on natural frequency of lateral vibration of rotating shafts, Journal of Sound and Vibration 274:985-995.
[10] Banerjee J. R., Su H., 2006, Dynamic stiffness formulation and free vibration analysis of a spinning composite beam, Computers and Structures 84:1208-1214.
[11] Hosseini S.A., Khadem S. E., 2009, Free vibrations analysis of a rotating shaft with nonlinearities in curvature and inertia, Mechanism and Machine Theory 44:272-288.
[12] Yigit A. S., Christoforou A. P., 1996, Coupled axial and transverse vibration of oil well drill string, Journal of sound and vibration 195 (4):617-627.
[13] Yigit A. S., Christoforou A. P., 1997, Dynamic modelling of rotating drill strings with borehole interactions, Journal of sound and vibration 206(2):243-260.
[14] Timoshenko G., 1963, Theory of Elastic Stability, Mc Graw Hill, United state.
[15] Hildebrand F. B., 1984, Methods of Applied Mathematics, Prentice Hall Ind, United state.
[16] Rao J. S., 1983, Rotor Dynamics, New York, John Wiley & Sons, United state.