Torsional Vibrations of Finite Isotropic Composite Circular Cylinders

The frequency equation for the torsional vibrations of composite cylinders, either concentric or bounded end to end, have been obtained. In the former case, the variation of the resonant frequency with inner cylinder thickness has been studied for the lowest symmetric and antisymmetric modes. In the latter case, the variation of the resonant frequency with change in the ratio of the lengths of the cylinders has been studied for the fundamental mode and its first harmonic in the frequency range where propagation constant for one part of cylinder is real whereas the propagation constant for the other part is imaginary. Vibration patterns for some particular cases for the latter type have been drawn