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Investigations on the nonlinear dynamic characteristics of a rotor supported by hybrid foil magnetic bearings

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Abstract

Hybrid foil magnetic bearings (HFMB) are highly suitable for oil-free turbomachinery and high-speed compressors under variable conditions due to their advantages such as frictionless operation at low speeds, reliable high-speed operation and adjustable dynamic performance. By adjusting the working mode and load sharing ratio, HFMB can optimize the dynamic performance and improve the stability of the rotor system. This paper presents investigations on the rotordynamics of a rigid rotor supported by two HFMBs. The coastdown response of the rotor supported by HFMBs and gas foil bearings from 65 krpm to rest is recorded experimentally and used to validate the calculated results of the rotordynamic model. Computational methods are used to predict the effect of load sharing ratios and working modes on rotordynamics as a function of HFMB operating conditions. The effects of load and equilibrium position on the rotordynamics are also predicted. Orbit simulations, Fast Fourier Transform, Poincaré maps and bifurcation diagrams are used for the theoretical analysis. The results show that an appropriate load sharing ratio with hybrid mode can effectively improve the rotordynamic performance of the HFMBs-rotor system, while increasing the load can also significantly improve the stability of the system.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.

Abbreviations

\(A_{0}\) :

Cross-sectional area of the magnetic circuit

\({\text{C}}\) :

Radial clearance

\({\mathbf{D}}\) :

Damping matrix

\({\text{E}}\) :

Modulus of elasticity

\({\text{EI}}\) :

Bending stiffness of the top foil segment

\(e\) :

Eccentricity

\({\mathbf{F}}\) :

Vector of forces

\(F_{mx} ,F_{my}\) :

Electromagnetic force

\(F_{1} ,F_{2}\) :

Electromagnetic forces generated by bias currents

\(F_{bx} ,F_{by}\) :

Radial force generated by the bearing

\(F_{ux} ,F_{uy}\) :

Radial force generated by unbalanced mass

\(F_{AMBrx} ,F_{AMBlx} ,F_{AMBry} ,F_{AMBly}\) :

Radial force of the AMB

\(F_{GFBlx} ,F_{GFBly}\) :

Radial force of the GFB

\(h\) :

Film thickness

\(\overline{h}\) :

Dimensionless film thickness \({( = }h{\text{/C)}}\)

\(I_{T}\) :

Translational moment of inertia

\(I_{P}\) :

Polar moment of inertia

\(i_{0x1} ,i_{0x2} ,i_{0y1} ,i_{0y2}\) :

Equilibrium current

\(i_{0} ,i_{x1} ,i_{x2} ,i_{y1} ,i_{y2}\) :

Coil current

\(k_{i} ,k_{ix1} ,k_{ix2} ,k_{iy1} ,k_{iy2}\) :

Current stiffness

\(k_{x} ,k_{xx1} ,k_{xx2} ,k_{xy1} ,k_{xy2}\) :

Displacement stiffness

\(L\) :

Axial length of bearing

\(l_{0}\) :

Half-length of bump

\({\mathbf{M}}\) :

Mass matrix

\(M_{xb} ,M_{yb}\) :

Rotational moments generated by bearing forces

\(M_{xu} ,M_{yu}\) :

Rotational moments generated by rotor imbalance mass

\(m\) :

Rotor mass

\(mg\) :

Gravity of the rotor

\(m_{ul} ,m_{ur}\) :

Unbalanced mass

\(N\) :

Number of turns of coils

\(p,p_{l} ,p_{r}\) :

Pressure

\(\overline{p}\) :

Dimensionless pressure \({( = }p{\text{/Pa)}}\)

\({\text{p}}_{{\text{a}}}\) :

Ambient pressure

\(p_{i,j}\) :

Pressure at different nodes

\(R\) :

Radius of bearing or rotor

\(s\) :

Distance between grid points in the circumferential direction

\(t\) :

Time

\(t_{b}\) :

Thickness of bump foil

\(\Delta t\) :

Time step

\(u_{l}\), \(u_{r}\) :

Radius of the unbalance mass

\(v\) :

Poisson ratio

\(x,y\) :

Displacement of the rotor mass center in Cartesian coordinates

\(x_{0x1} ,x_{0x2} ,x_{0y1} ,x_{0y2}\) :

Equilibrium position

\(x_{0} ,x_{x1} ,x_{x2} ,x_{y1} ,x_{y2}\) :

Rotor center displacement

\(z,z_{l} ,z_{r}\) :

Axial coordinate

\(\overline{z}\) :

Dimensionless axial coordinate \({( = }z{\text{/R)}}\)

\(\Delta z\) :

Distance between grid points in the axial direction

\(z_{bl} ,z_{br}\) :

Distance from the bearings to the rotor mass center

\(z_{ul} ,z_{ur}\) :

Distance from the unbalance mass to the rotor mass center

\(\delta\) :

Vector of the displacement of the rotor

\(\varepsilon\) :

Local top foil deflection

\(\varepsilon_{i,j}\) :

Deflection of the top foil at different nodes

\(\phi\) :

Initial phase

\(\gamma\) :

\(( = \frac{\upsilon }{\omega })\)

\(\mu\) :

Gas viscosity

\(\mu_{0}\) :

Permeability of air

\(\theta ,\theta_{l} ,\theta_{r}\) :

Angular coordinate

\(\theta_{0}\) :

Attitude angle

\(\theta_{x} ,\theta_{y}\) :

Rotation angle of the rotor along different axes

\(\tau\) :

Anytime

\(\upsilon\) :

Excitation frequency

\(\omega\) :

Angular velocity of rotor

\(\Lambda\) :

Bearing number \(\left( {{ = }\frac{{{6}\mu {\Omega }}}{{{\text{p}}_{{\text{a}}} }}\left( {\frac{{\text{R}}}{\omega }} \right)^{{2}} } \right)\)

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Funding

This work was supported by the National Natural Science Foundation of China (U22A20214), National Key Research and Development Program of China (2021YFF0600208), and the Science and Technology Innovation Program of Hunan Province (2020RC4018, 2020GK2069).

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Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, China

    Hang Zhang, Libo Feng & Kai Feng

  2. College of Electrical and Information Engineering, Hunan University, Changsha, 410082, China

    Miaomiao Cheng & Xiang Zhou

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  1. Hang Zhang
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  2. Miaomiao Cheng
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  3. Xiang Zhou
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  4. Libo Feng
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Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by HZ, MC, XZ, LF and KF. The first draft of the manuscript was written by HZ, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Miaomiao Cheng.

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Zhang, H., Cheng, M., Zhou, X. et al. Investigations on the nonlinear dynamic characteristics of a rotor supported by hybrid foil magnetic bearings. Nonlinear Dyn 111, 14879–14899 (2023). https://doi.org/10.1007/s11071-023-08635-z

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  • DOI: https://doi.org/10.1007/s11071-023-08635-z

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