Longitudinal-Torsional and Two Plane Transverse Vibrations of a Composite Timoshenko Rotor

Document Type: Research Paper


Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan


In this paper, two kinds of vibrations are considered for a composite Timoshenko rotor: longitudinal-torsional vibration and two plane transverse one. The kinetic and potential energies and virtual work due to the gyroscopic effects are calculated and the set of six governing equations and boundary conditions are derived using Hamilton principle. Differential quadrature method (DQM) is used as a strong numerical method and natural frequencies and mode shapes are derived. Effects of the rotating speed and the lamination angle on the natural frequencies are studied for various boundary conditions; meanwhile, critical speeds of the rotor are determined. Two kinds of critical speeds are considered for the rotor: the resonance speed, which happens as rotor rotates near one of the natural frequencies, and the instability speed, which occurs as value of the first natural frequency decreases to zero and rotor becomes instable.


[1] Zu J., Hans R.P., 1992, Natural frequencies and normal modes of a spinning beam with general boundary conditions, Journal of Applied Mechanics, Transactions ASME 59:197-204.
[2] Zu J., Melanson J., 1998, Natural frequencies and normal modes of for externally damped spinning Timoshenko beams with general boundary conditions, Journal of Applied Mechanics, Transactions ASME 65:770-772.
[3] Dos Reis H. L. M., Goldman R. B., Verstrate P. H., 1987, Thin-walled laminated composite cylindrical tubes—part III: critical speed analysis, Journal of Composites Technology and Research 9(2):58-62.
[4] Gupta K., Singh S. E., 1996, Dynamics of composite rotors, Proceedings of the Indo-US Symposium on Emerging Trends in Vibration and Noise Engineering, New Delhi, India.
[5] Bert C.W., 1992, The effects of bending twisting coupling on the critical speed of drive shafts, Composite Materials, 6th Japan/US Conference, Orlando Lancaster.
[6] Kim C.D., Bert C.W., 1993, Critical speed analysis of laminated composite, hollow drive shafts, Composites Engineering 3:633-643.
[7] Banerjee J.R., Su H., 2006, Dynamic stiffness formulation and free vibration analysis of a spinning composite beam, Computers and Structures 84: 1208-1214.
[8] Chang M.Y., Chen J.K., Chang C.Y., 2004, A simple spinning laminated composite shaft model, International Journal of Solids and Structures 41(4): 637-662.
[9] Boukhalfa A., Hadjoui A., 2010, Free vibration analysis of an embarked rotating composite shaft using the hp-version of the FEM, Latin American Journal of Solids and Structures 7: 105-141.
[10] Choi S.H., Pierre C., Ulsoy A.G., 1992, Consistent modeling of rotating Timoshenko shafts subject to axial loads, Journal of Vibration and Acoustics 114: 249-259.
[11] Bert C.W., Malik M., 1996, Differential quadrature method in computational mechanics: A review, Applied Mechanics Reviews 49: 1-28.
[12] Du H, Lim M.K., Lin N.R., 1994, Application of generalized differential quadrature method to structural problems, International Journal for Numerical Methods in Engineering 37:1881-1896.
[13] Sun J., Ruzicka M., 2006, A calculation method of hollow circular composite beam under general loadings, Bulletin of Applied Mechanics 3(12): 105-114.
[14] Reddy J.N., 2004, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, Boca Raton, FL.