Nonlinear Vibrations of Rotors in Systems with Magnetic Bearings

Abstract

A method is suggested for building mathematical models of dynamics of rotors in magnetic bearings of different types (passive and active ones). It is based on Lagrange-Maxwell differential equations in a form identical to that of Routh equations in mechanics. The main distinguishing feature of the model is the possibility to account for the following: the nonlinear dependence of magnetic forces on gaps between movable and stationary parts in PMBs and AMBs, and on currents in the windings of AMB electromagnets; current delay in the windings of AMB electromagnets, i. e. nonlinearity linked to the inductance of the coils; the geometric links between the electromagnets of one AMB and the links between all AMB of one rotor, which results in coupling of processes in orthogonal directions; practically any AMB control law; limitations on the control current caused by physical constraints in the control system controller; dissipation fluxes as well as magnetic resistances of AMB magnetic core sections, making the mathematical model insensitive as regards origination of "nonzero" gaps and currents. Numerical analysis has been performed for one of the possible variants of a complete rotor magnetic suspension realised in the form of the laboratory model. It includes two radial passive magnetic bearings with permanent annular magnets and one axial active magnetic bearing with a stator in the form of a shell core. Modelling validity is confirmed by comparing analytical and experimental data. Analysis of linear and nonlinear rotor dynamics phenomena observed in the laboratory model with magnetic bearings is described. It is known that analysis based on linearised models allows judging only the stability of equilibrium states with small deflections. The negligible nonlinear equation terms in this case, when investigating motion with increasing deflection, allow expanding the information content of the mathematical model about nonlinear effects occurring in the system. Estimated results revealed shortcomings of the linearised models. Based on this conclusion the need of using nonlinear models for adequate description of the dynamics of such systems is proved.