Vibrations of Rotors Supported on Nonlinear Bearings

Abstract

Rotors in electrical machines are supported by various types of bearings. In general, the rotor bearings have nonlinear stiffness properties and they influence the rotor vibrations significantly. In this work, this influence of these nonlinearities is investigated. A simplified finite element model using Timoshenko beam elements is set up for the heterogeneous structure of the rotor. A transversally isotropic material model is adopted for the rotor core stack. Imposing the nonlinear bearing stiffnesses on the model, the Newton-Raphson procedure is used to carry out a run up simulation. The spectral content of these results shows nonlinear effects due to the bearings. The rotor vibrations are further investigated in detail for various constant speeds. These results show non-harmonic vibrations of the rotor in a section of the investigated speed range. The nodal displacements are stored in the vectorfor both element types. The indices k and l denote the corresponding nodes of the element e. Different geometrical cross section properties and material properties can be allocated for each element. The dynamic behavior of the rotor is described by the equations of motion of a multi degree-of-freedom systemThe stiffness matrix K is obtained from the the structural stiffness matrix K s and the bearing stiffness K b . The mass matrix M consists of the structural mass matrix M s and additional point masses M p . The structural stiffness and mass matrices are derived from assembling the local element stiffness and mass matrices. The damping matrix D is governed from the Rayleigh damping model. For the rotating system, the gyroscopic effects are included and taken into account by the gyroscopic matrix G. The force vector F considers the linear distribution of the rotor unbalance and the gravity in an element e. The force vector is given by