Circular sensor array and nonlinear analysis of homopolar magnetic bearings

Abstract

Magnetic bearings use variable attractive forces generated by electromagnetic control coils to support rotating shafts with low friction and no material wear while providing variable stiffness and damping. Rotor deflections are stabilized by position feedback control along two axes using non-contacting displacement sensors. These sensor signals contain sensor runout error which can be represented by a Fourier series composed of harmonics of the spin frequency. While many methods have been proposed to compensate for these runout harmonics, most are computationally intensive and can destabilize the feedback loop. One attractive alternative is to increase the number of displacement sensors and map individual probe voltages to the two independent control signals. This approach is implemented using a circular sensor array and single weighting gain matrix in the present work. Analysis and simulations show that this method eliminates runout harmonics from 2 to n-2 when all sensors in an ideal n-sensor array are operational. Sensor failures result in reduced synchronous amplitude and increased harmonic amplitudes after failure. These amplitudes are predicted using derived expressions and synchronous measurement error can be corrected using an adjustment factor for single failures. A prototype 8-sensor array shows substantial runout reduction and bandwidth and sensitivity comparable to commercial systems. Nonlinear behavior in homopolar magnetic bearings is caused primarily by the quadratic relationship between coil currents and magnetic support forces. Governing equations for a permanent magnet biased homopolar magnetic bearing are derived using magnetic circuit equations and linearized using voltage and position stiffness terms. Nonlinear hardening and softening spring behavior is achieved by varying proportional control gain and frequency response is determined for one case using numerical integration and a shooting algorithm. Maximum amplitudes and phase reversal for this nonlinear system occur at lower frequencies than the linearized system. Rotor oscillations exhibit amplitude jumps by cyclic fold bifurcations, creating a region of hysteresis where multiple stable equilibrium states exist. One of these equilibrium states contains subharmonic frequency components resulting in quasiperiodic rotor motion. This nonlinear analysis shows how nonlinear rotor oscillations can be avoided for a wide range of operation by careful selection of design parameters and operating conditions.