Wiesenborn, Robert Kyle (2006-12). Circular sensor array and nonlinear analysis of homopolar magnetic bearings. Master's Thesis.
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.