Prediction of vibration and instability of rotating damped beams with an elastically restrained root

The vibration and instability of a rotating nonuniform beam with the frequency-dependent structural damping, the equivalent viscous damping and the root damping are investigated. The exact solution for the vibration of the system is derived. The complex characteristic governing equation is divided into two coupled real equations expressed in terms of the real and imaginary variables. The frequency equation is derived in terms of the eight normalized fundamental solutions of the two coupled differential equations. It can be shown that, if the coefficients of the coupled differential equations can be expressed in polynomial form, the exact fundamental solutions can be found by the matrix method of Frobenius. The complex frequency relations among different systems are revealed. It is revealed that the effects of the structural damping and the root damping on the decay rates of higher modes are greatly larger than those of lower modes. However, there are almost the same effects of the equivalent viscous damping on the decay rates of lower and higher modes.