Abstract
A research report has been submitted. It deals with implementing a method for a mathematical description of the nonlinear dynamics of rotors in magnetic bearings of different types (passive and active). The method is based on Lagrange-Maxwell differential equations in a form similar to that of Routh equations in mechanics. The mathematical models account for such nonlinearities as the nonlinear dependencies of magnetic forces on gaps in passive and active magnetic bearings and on currents in the windings of electromagnets; nonlinearities related to the inductances in coils; the geometric link between the electromagnets in one AMB and the link between all AMBs in one rotor, which results in relatedness of processes in orthogonal directions, and other factors. The suggested approach made it possible to detect and investigate different phenomena in nonlinear rotor dynamics. The method adequacy has been confirmed experimentally on a laboratory setup, which is a prototype of a complete combined magnetic-electromagnetic suspension in small-size rotor machinery. Different variants of linearizing the equations of motion have been considered. They provide for both linearization of restoring magnetic or electromagnetic forces in passive and active magnetic bearings, and exclusion of nonlinear motion equation terms. Calculation results for several linearization variants have been obtained. An appraisal of results identified the drawbacks of linearized mathematical models and allowed drawing a conclusion on the necessity of applying nonlinear models for a well-defined description of the dynamics of rotor systems with magnetic bearings.
Highlights
A magnetic bearing (MB) is one of the variants of elastic-damping bearings
It shows radial and axial active magnetic bearings (AMBs) with electromagnets (Fig. 1, a, b) and a radial passive magnetic bearings (PMBs) with permanent annular magnets (Fig. 1, c, d), and the following notations are introduced: 1 – rotor; 2 – stators; 3 – AMB windings; 4 – AMB position sensors; 5 – comparator in AMB control system; 6 – AMB control device; 7 – amplifiers feeding control voltages to AMB windings, which are formed according to the accepted control algorithm; 8 and 9 – movable and stationary annular permanent magnets
Based on the possibilities of practical implementation of complete magnetic bearings of rotors, this study considers the options of using either radial or axial AMBs for stabilising a rotor over all five degrees of freedom or one AMB jointly with several PMBs in different design versions
Summary
A magnetic bearing (MB) is one of the variants of elastic-damping bearings. Its feature is the use of magnetic fields to provide stable rotor levitation. Uc2(x1, x2, y1, y2,z3 ), where f′′qr(x1,...,z3) and f′′′qr(x1,...,z3) are nonlinear terms of the equations of motion due to inertia forces and the second and third-order potential field; bx1,...,z3 are viscosity coefficients; rc 1,...,N are active resistances in winding circuits; uc 1,...,N are control voltages applied across AMB windings whose magnitude is formed according to the accepted control law depending on the rotor current position; mks, j are inertial and gyroscopic coefficients with the following values: m11. It allows evaluating the presence of resonant modes in the area being investigated and the kind of rotor motion corresponding to different rotational speeds. A detailed description of the results is given in the following in comparison with the results of numerical modelling
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