Dynamic analysis of rotating damped beams with an elastically restrained root

The vibration of a rotating damped blade with an elastically restrained root is investigated. The effects of viscous damping and the translational and rotational damping at the root of blade are considered. The flow-indued force and moment between the tip of a blade and the casing are simulated by using the time-dependent boundary conditions. A simple and efficient algorithm for deriving the semi-analytical steady state solution of the general system is proposed. The governing equation is divided into two coupled real differential equations. The two coupled equations are uncoupled into an eighth-order characteristic differential equation. The eight corresponding boundary conditions are obtained. The eight linearly independent homogenous semi-analytical solutions of the eighth-order characteristic differential equation are derived. If the coefficients of the uncoupled governing differential equation are constant, the exact fundamental solutions are obtained. The exact steady state solution is obtained by using Greenʹs function in terms of the eight homogenous solutions. Moreover, the influence of the translational, rotational and viscous damping constants on the frequency response curves of a rotating beam are investigated. The opposite influence of the transverse viscous damping constant and the root damping constants on the frequency of resonance is revealed.