Whirling analysis of unbalanced rotor using the vectors of solution coefficients

A prevalent issue in rotors is mass unbalance, which leads to synchronous lateral vibrations. In this paper, the critical whirling speeds and global responses of an unbalanced rotor-bearing system with viscoelastic boundary condition are computed using the relationships between the vectors of solution coefficients of differential equations governing the transverse displacements. The mathematical formulation is based on the matrix form of the aforementioned relationships and boundary conditions in the resonance case. The Timoshenko beam model, including the effects of gyroscopic moments is employed. It was found that the presented approach in comparison with the transfer matrix method reduces the matrix product when shaft segments have the same geometric and mechanical properties. A good agreement is found between the results of this study and the findings of studies available in the literature. Furthermore, the eigenvectors for free vibrations of the system were calculated by combining Natanson’s technique with the presented approach while considering the derived rotor’s characteristic equation after removing the forced term in the equation that describes the forced vibrations. Each obtained forward whirling mode shape describes perfectly the global response in the vicinity of the corresponding forward critical speed. The presented approach can, therefore, be applied to obtain the unbalance responses, whirling speeds, and associated mode shapes of rotor-bearing systems with complex boundary conditions.