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Research Papers

Modeling of Electrodynamic Bearings

[+] Author and Article Information
Nicola Amati

Department of Mechanics, Mechatronics Laboratory, Politecnico di Torino, Duca degli Abruzzi 24 10129 Torino, Italynicola.amati@polito.it

Xavier De Lépine

Department of Mechanics, Mechatronics Laboratory, Politecnico di Torino, Duca degli Abruzzi 24 10129 Torino, Italyxavier.delepine@polito.it

Andrea Tonoli

Department of Mechanics, Mechatronics Laboratory, Politecnico di Torino, Duca degli Abruzzi 24 10129 Torino, Italyandrea.tonoli@polito.it

J. Vib. Acoust 130(6), 061007 (Oct 15, 2008) (9 pages) doi:10.1115/1.2981170 History: Received February 07, 2008; Revised July 12, 2008; Published October 15, 2008

Abstract

A new model of electrodynamic bearings is presented in this paper. The model takes into account the $R-L$ dynamics of the eddy currents on which this type of bearing is based, making it valid for both quasistatic and dynamic analyses. In the quasistatic case, the model is used to obtain the force generated by an off-centered shaft rotating at a fixed speed in a constant magnetic field. The model is then used to analyze the dynamic stability of a Jeffcott rotor supported by electrodynamic bearings. The essential role played by nonrotating damping in ensuring a stable operating range to the rotor is studied by means of root loci. Comparison with literature results is finally used to validate the model.

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Figures

Figure 1

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.

Figure 2

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).

Figure 3

Variables and parameters of the electrodynamic bearing model

Figure 4

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)∣.

Figure 5

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.

Figure 6

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.

Figure 7

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).

Figure 8

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.

Figure 9

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.

Figure 10

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).

Figure 11

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).

Figure 12

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.

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