Eigenvalues of the Free Rotation Mode of the Multi-bladed Rotor

The eigenvalues of the free rotation mode of multi-bladed rotor systems are of crucial importance when designing controllers. An unstable free rotation mode of the rotor is unfavorable. In this paper, the Differential Algebraic Equations (DAEs) of the multi-flexible-body system are linearized after having introduced a corotational formulation with respect to the floating frame of reference of the rotating center. The Multi-Blade Coordinate (MBC) transformation is performed to obtain the time-invariant system. The proposed formulations are verified through the numerical experiments carried out on the rotors of increasing complexity: a single rigid body, a rigid rotor, and a flexible rotor. Results reveal that the geometric stiffness matrix and the tangent stiffness matrix of constraints exhibit a centrifugal stiffening effect, moving the eigenvalues toward pure imaginary numbers; in contrast, the inertial stiffness matrix introduces a centrifugal softening effect, pushing the eigenvalues toward real numbers. Generally one has to discard the geometric stiffness matrix, the inertial stiffness matrix, and the tangent stiffness matrix of constraints in the linearized DAEs to obtain the zero eigenvalues for the free rotation mode, whereas the inertial damping matrix can be involved.