Modeling of Electrodynamic Bearings

Short-circuited rigid coil in motion into a constant magnetic field B. The resistive and inductive parameters of the moving conductor, R and L, respectively, are due to the conductor itself (subscript c) and an additional part (subscript add). If a force F is applied on the coil, its resulting velocity ẋ induces eddy currents that create a force opposed and equal in magnitude to F. The dynamics of the system is equivalent to a spring-damper in series, whose parameters are referred to as k and c.

Simplified scheme of a conductor rotating in a magnetic field. Ω is the rotation speed of the conductor. The synchronous whirling is presented as a particular configuration of rotation: The precession speed is equal to the rotation speed, and the part of the conductor that crosses the magnetic field is always the same (darkened area).

Variables and parameters of the electrodynamic bearing model

Evolution of the electromagnetic force Fz generated by the electrodynamic bearing for increasing rotation speeds. F∥ and F⊥ are the contributions of the force parallel and perpendicular to the eccentricity, respectively. It can be shown that ∣F∥(ωRL)∣=∣F⊥(ωRL)∣.

Forces generated by the electrodynamic bearing (Eq. 19). F⊥ and F∥ are perpendicular and parallel to the constant eccentricity, respectively. They meet the same value of k∣z0∣∕2=cωRL∣z0∣∕2 for Ω=ωRL. For high rotation speeds, ∣F∥∣ and ∣F⊥∣ tend to k∣z0∣ and 0, respectively.

Poles of the Jeffcott rotor on an electrodynamic bearing. The poles at Ω=0 are inside the boxes. s1 is always positive and is always in the right-hand part of the complex plane, while s2 and s3 are always in the left-hand part of the complex plane.

Basic principle of electrodynamic bearings stabilized with a nonrotating eddy current damper. The nonrotating damping effect takes place in the conductor fixed to the stator (bottom of the figure).

Evolution of the poles of the system with nonrotating damping added. Like before, the real part of s1 is positive for low values of Ω, which means that the rotating damping is still preponderant. Nevertheless, s1 crosses the imaginary axis as soon as Ω is higher than a threshold value, referred to as Ωs.

Adimensional study of the stability properties of a generic electrodynamic bearing combined with an additive nonrotating viscous damper. The plots have been computed for various values of δ. It can be noticed that for higher values of δ (lower values of m), lower rotation speeds are necessary to stabilize the bearing.

Comparison on the forces parallel and perpendicular to the eccentricity. (◇) and (○) correspond to F∥ and F⊥, respectively, and are obtained from Fig. 57 in Ref. 6. The solid lines are obtained through the model (the thick and thin lines correspond to F∥ and F⊥, respectively).

Comparison of the magnitude and angle of Fz∕z0 (left and right axes, respectively). Angle θ is the angle of Fz∕z0 as defined in Fig. 4. (○) and (◻) correspond to ∣Fz∕z0∣ and θ, respectively, obtained from Figs. 11 and 12 in Ref. 11. The lines are obtained using the presented model (the solid and dashed lines correspond to ∣Fz∕z0∣ and θ, respectively).

Comparison between stabilizing rotation speeds versus nonrotating viscous damping. (◇) corresponds to the data obtained from Fig. 15 in Ref. 11, while the solid line corresponds to the stabilizing rotation speed obtained from Eq. 38.