Dynamic Analysis Of Rigid-flexible Coupling System Of A Rotating Blade

Based on the Hamilton principle, the dynamic problem of a rotating flexible blade is studied in this paper. A dynamic model of rigid-flexible coupling system consists of a flexible blade fixed on the edge of a rigid disc is built using a Hybrid set of Coordinates. Effects of rotation of the flexible blade on its deformation displacement are discussed. And variations of pre-twisted angle and dimensions of cross-sections along the axis of blade are studied. Both trasversal deformations resulted from bending and shearing in the rotating plane and its perpendicular plane are considered in the description of deformation displacement of the flexible blade. And centrifugal force potential energy resulted from the rotation of blade is considered.Based on Hamilton variation principle, continuous dynamic equations of the rigid-flexible coupling system of rotating blade are derived. The equations are a set of non-linear, partial differential and integral equations. The equations are discreted by finite element method. A set of discreted dynamic equations are given considering factors of pre-twisted angle, variable cross-section, shearing deformation, rotation of cross-section, and centrifugal stiffening effect. Effects of variations of pre-twisted angle, length of blade, radius of disc, dimensions of cross-section and rotating speed on dynamic behavior of blade are studied in the process of acceleration. Effects of variations of pre-twisted angle on dynamic behavior of blade subjected to impacting action are studied in the process of constant rotating speed. Periodical exciting force acting on blade at a constant rotating speed is introduced in the non-linear, time-varying and coupling dynamic equations. A general and compatible computational procedure is programmed. Based on state-space method, frequency and stability analysis of the rigid-flexible coupling system are carried out. Effects of rotation of the flexible blade on its deformation displacement frequency are studied. Effects of rotating speed on displacement response stability of blade are studied. And numerical results are given in the paper.Results show that variation of pre-twisted angle and cross-section have little influences on the displacement response of blade tip in the process of acceleration. When the blade is rotating at a constant speed and subjected to an impacting action, the variation of pre-twisted angle has great influences on transversal displacement of blade tip on the plane perpendicular to the rotating plane. And a tiny low frequency periodical exciting force will cause great transversal displacement or divergence on the plane perpendicular to the rotating plane for pre-twisted blade. The tansversal displacement on the plane perpendicular to the rotating plane will make the blade touch inside surface and result in fracture failure in the operation. Results also show that frequencies increase with increasing rotating speed for blade with high mass density, which case is called dynamic stiffening. Meanwhile, the first order frequencies decrease with increasing rotating speed for blade with low mass density, which case is called dynamic softening. Frequencies decrease with the dimension recucing of cross-section along the axis of blade and the effect of dynamic softening occurs earlier, which case will also make vibration amplitude of the blade increase and may touch inside surface.Considering the effects of many factors of an rotating blade, dynamic model and equations of a rigid-flexible coupling system are given in this paper. The model and equations can be used to solve dynamic problems of a rigid-flexible coupling system of a rotating blade in the noninertial system. Methods presented in the paper have certain theoretical significance and engineering application value. Results in the paper have some referrence value for theoretical analysis and design of blade in the engineering.