A Continuous Model for the Dynamical Analysis of Mistuned Bladed Rotors

Ideal bladed rotors are rotationally symmetric, as a consequence they exhibit couples of degenerate eigenmodes at coinciding frequencies. When even small imperfections are present destroying the periodicity of the structure (disorder or mistuning), each couple of degenerate eigenfrequencies splits into two distinct values (frequency split) and the corresponding modal shapes exhibit vibration amplitude peaks concentrated around few blades (localization phenomenon). In this paper a continuous model describing the in-plane vibrations of a mistuned bladed rotor is derived via the homogenization theory. Imperfections are accounted for as deviations of the mass and/or stiffness of some blades from the design value; a perturbation approach is adopted in order to investigate the frequency split and localization phenomena arising in the imperfect structure. Numerical simulations show the effectiveness of the proposed model, requiring much lower computational effort than classical finite element schemes. 1. Introduction Bladed rotors used in turbomachinery possess rotational symmetry provided that no imperfections are present, that is, if all the blades and associated disk sectors are identical to each other. Under this ideal condition, their dynamical behavior is characterized by the presence of couples of degenerate eigenmodes at coinciding frequencies, as shown in [1, 2]. That behavior can be studied in detail by using the finite element technique [1, 3, 4], and the computational cost may be significatively reduced by exploiting the periodicity properties characterizing these structures. In fact, for a given structural eigenmode, all the blades exhibit the same vibration amplitude with a constant phase shift between adjacent ones; thus only a single sector of the structure may be considered in the analysis, by enforcing suitable constraints depending on the admissible phase shifts [1]. As a matter of fact, in practice the symmetry is destroyed by the presence of unavoidable manufacturing defects, material tolerances, or damage arising during service. The loss of symmetry, called disorder or mistuning, may significantly alter the system dynamics even at small disorder level. In particular, the eigenmode degeneracy is removed, and the coinciding modal frequencies of a degenerate eigenmode are split into two distinct values (frequency split phenomenon). Moreover, the vibration localization effect may appear [5–7], consisting of a vibrational-energy confinement in small regions of the rotor, rapidly decaying far away. As a consequence, some blades may vibrate with