longitudinal-torsional and two plane transverse vibrations of a composite timoshenko rotor

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.