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Elimination of Magnetically Induced Vibration Instability of Ring-Shaped Stator of PM Motors by Mirror-Symmetric Magnets

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Abstract

Purpose

Permanent magnet (PM) motors are widely used due to the progress of PM materials. Smooth operation is one of the most crucial requirements in precision applications, but vibration instability can be induced by fluctuating magnetic forces. This work is aimed at the elimination of magnetically induced vibration instability.

Methods

Three types of mirror-symmetric topologies are proposed to eliminate the magnetically induced vibration instability. All magnets in these topologies are circumferentially shifted into two groups and the magnets in each group are arranged into equally, increasingly and generally spaced topologies. An analytical model is developed by Hamilton’s principle to examine the elimination effect of vibration instability by the proposed topologies.

Results and conclusions

Based on the analytical model, the eigensolutions for the three topologies are formulated, and conditions for vibration instability elimination are determined as closed form expressions in basic parameters. The eliminations are demonstrated through several examples. Major analytical results are verified through numerical calculations.

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Abbreviations

\( o{ - }\rho \varphi \) :

Rotor-fixed coordinates

ku, kv :

Stiffnesses of uniform tangential and radial supports

u, v :

Tangential and radial displacements of a point on neutral plane in rotor-fixed coordinates

\( \varphi \) :

Positioning angle

t :

Time

h :

Radial thickness

b :

Axial height

d :

Density

E :

Young’s modulus

R r :

Radius of rotor

R s :

Neutral circle radius of stator

N :

Magnet count

\( \varphi_{j} \) :

Angle between jth (j = 1, 2, … N/2) magnet and polar axis in a group

\( \varphi_{ij,1} \), \( \varphi_{ij,2} \) :

The two positioning angles of the jth (j = 1, 2, … N/2) PM in the ith (i = 1, 2) group

\( \gamma \) :

Span angle of a magnet

T 0 :

Kinetic energy

\( \varepsilon_{\theta } \) :

Tangential strain

\( \varepsilon_{\theta 0} \) :

Tangential strain in neutral plane

\( \varepsilon_{\theta 1} \) :

Membrane strain

\( \hat{U}_{0} \) :

Potential energy from bending deformation of stator

A :

Cross-sectional area of stator

I :

Sectional moment of inertia of stator

\( p_{\text{m}} \) :

Density of magnetic energy

\( B_{\text{ag}} \) :

Air gap magnetic density

\( P \) :

Radial magnetic force per unit area

\( \hat{U}_{1} \) :

Work done by magnetic force

\( d_{0} \) :

Air gap length

\( \mu_{0} \) :

Vacuum permeability

\( h_{\text{m}} \) :

Magnetization thickness

B r :

Remanence

\( H( \cdot ) \) :

Heaviside step function

\( \hat{U}_{2} \) :

Potential energy of elastic uniform foundation

T :

Time period

\( \bar{t} \) :

Dimensionless time

\( \bar{u} \) :

Dimensionless tangential displacement of stator

\( \bar{v} \) :

Dimensionless radial displacement of stator

\( \bar{k}_{u} \) :

Dimensionless tangential support stiffness on stator

\( \bar{k}_{v} \) :

Dimensionless radial support stiffness on stator

\( \delta ( \cdot ) \) :

Dirac delta function

\( U(t) \) :

Complex function of time

x, y :

Real functions of time

x0, y0 :

Plural variables

~:

Complex conjugate operation

i:

Imaginary unit, \( \text{i} = \sqrt { - 1} \)

n :

Wavenumber

Q m :

Eigenvector

λ m :

Eigenvalue

\( \lambda_{\text{mRe}} \) :

Real part of eigenvalue

\( \lambda_{\text{mIm}} \) :

Imaginary part of eigenvalue

\( \alpha \) :

Angle between adjacent magnets

k1 ~ k6 :

Positive integers

\( \phi \) :

Positioning angle of the second magnet

\( \vartheta \) :

Increased angle between adjacent magnets

\( \text{sgn} ( \cdot ) \) :

Sign function

FTW:

Forward traveling wave

BTW:

Backward traveling wave

u, v :

Tangential and radial displacements

m:

Mirror symmetric

s:

Stator

r:

Rotor

C:

Cosine operation

S:

Sine operation

ag:

Air gap

F:

Forward traveling wave

B:

Backward traveling wave

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Acknowledgements

The authors are grateful to the National Natural Science Foundation of China (Grant Nos. 51675368 and 51705519) for supporting this research.

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Authors and Affiliations

  1. School of Mechanical Engineering, Tianjin University, Tianjin, 300350, People’s Republic of China

    Jinlong Liu, Shiyu Wang & Zheren Wang

  2. Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin, 300350, People’s Republic of China

    Jinlong Liu, Shiyu Wang & Zheren Wang

  3. Tianjin Key Laboratory of Nonlinear Dynamics and Control, Tianjin, 300350, People’s Republic of China

    Jinlong Liu, Shiyu Wang & Zheren Wang

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  1. Jinlong Liu
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  2. Shiyu Wang
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Correspondence to Shiyu Wang.

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Liu, J., Wang, S. & Wang, Z. Elimination of Magnetically Induced Vibration Instability of Ring-Shaped Stator of PM Motors by Mirror-Symmetric Magnets. J. Vib. Eng. Technol. 8, 695–711 (2020). https://doi.org/10.1007/s42417-019-00169-2

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  • DOI: https://doi.org/10.1007/s42417-019-00169-2

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