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Unbalanced magnetic pull induced by arbitrary eccentric motion of cage rotor in transient operation. Part 1: Analytical model

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Abstract

Eccentric rotor motion of an electric motor induces an unbalanced magnetic pull. This eccentricity force may couple the electromagnetic system with the flexural vibration modes of the shaft. This paper presents an analytic model for this eccentricity force allowing an arbitrary rotor motion and transient operation. The general equations were simplified for the constant flux and steady-state operation. These simplified equations correspond well to the results presented earlier.

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Abbreviations

a p±1 :

model parameters related to harmonic components p±1

B :

magnetic flux density in the air gap

\( \underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{B}}} _{\upsilon } \) :

magnetic flux density in the air gap; harmonic component υ

c p±1 :

model parameters related to harmonic components p±1

d r :

outer diameter of the rotor

F e :

unbalanced magnetic pull

f δ :

magnetomotive force in the air gap

f s :

magnetomotive force of stator currents; fundamental component

f r,n :

magnetomotive force of rotor mesh n

\( \underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{f}}} \,_{{{\text{r}},\upsilon }} ,\underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{f}}} _{{\,{\text{s}},\upsilon }} \) :

magnetomotive force of component υ induced by rotor and stator windings

g p±1 :

force variable related to harmonic components p±1

ir,n, irb,m:

current of rotor cage-ring segment n and rotor cage bar m

\( \underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{i}}} _{{\,{\text{r}},\upsilon }} ,\underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{i}}} _{{{\text{s}},\upsilon }} \) :

harmonic rotor and stator current component υ

\( \underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{i}}} _{{{\text{ re}},p}} \) :

fundamental component of equivalent rotor current

krw, ksw:

winding factor of rotor and stator winding

K :

ratio between the effective angular width of the tooth and the slot pitch

k p±1 :

coupling factors of components p±1 due to leakage flux and saturation

k 0 :

parameter related to the negative spring constant

L :

self-inductance of one rotor-cage mesh

L bσ :

slot- and tooth-tip-leakage inductance of one rotor bar

L eσ :

end-leakage inductance of one ring segment between two rotor bars

L m,υ :

magnetising inductance of harmonic component υ

Lr,υ, Ls,υ:

rotor and stator inductance of harmonic component υ

Lrσ,υ, Lsσ,υ:

leakage inductance of rotor and stator for harmonic component υ

l e :

equivalent core length

Nr, Ns:

number of turns in series of rotor and stator winding

N se :

equivalent number of turns in series of stator winding

p :

number of pole pairs

p c :

centre point position of the rotor

p 0 :

constant whirling radius of the rotor

Qr, Qs:

number of rotor and stator slots

q p±1 :

mixed variables related to harmonic components p±1

Rb, Re:

resistance of one rotor bar and one end ring segment between rotor bars

Rr,υ, Rs,υ:

rotor and stator resistance of harmonic component υ

s :

slip

T e :

electromagnetic torque

t :

time

\( \underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{u}}} _{{{\text{r}}{\text{,}}p}} ,\underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{u}}} _{{{\text{s}}{\text{,}}p}} \) :

rotor and stator voltage of fundamental component

δ e :

equivalent air gap including slotting

θ :

angle of rotor rotation

ϑ b :

electrical angle of space vector of magnetic flux density

Λ 0 :

air-gap permeance per unit of area

µ 0 :

permeability of free space

σ r :

radial tensile stress

τ p±1 :

time constant of harmonic component p±1

υ :

ordinal number of a harmonic

ϕ :

angular coordinate

φ r,n :

magnetic flux through cage mesh n

\( \underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{\phi }}} _{{{\text{r}},\upsilon }} \) :

magnetic flux of rotor cage for harmonic component υ

\( \underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{\psi }}} _{{\text{r}}} ,\underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{\psi }}} _{{{\text{r}},p}} \) :

rotor flux linkage of fundamental component

\( \underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{\psi }}} _{{\text{s}}} ,\underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{\psi }}} _{{{\text{s}},p}} \) :

stator flux linkage of fundamental component

ω s :

angular frequency of stator field

ω b :

angular frequency of magnetic flux density of fundamental component

ω w :

whirling frequency

Ωm:

mechanical angular velocity of rotor

x :

complex number

x*:

complex conjugate of x

\( \underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{x}}} \) :

space vector

\( \underline {\tilde x} _m\) :

discrete Fourier transform component m of sequence x n

\( \underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{x}}} ^{{\text{r}}} ,\underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{x}}} ^{{\text{s}}} ,\underline{{\ifmmode\expandafter\hat\else\expandafter\^\fi{x}}} ^{{\text{k}}} \) :

space vector (or complex number) in rotor, stator or more general reference frame

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Acknowledgements

The authors gratefully acknowledge the financial support of the National Technology Agency of Finland (Tekes).

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Author notes

  1. T. P. Holopainen

    Present address: ABB Oy, Electrical Machines, P.O. Box 186, 00381, Helsinki, Finland

Authors and Affiliations

  1. VTT Industrial Systems, P.O. Box 13022, 02044 VTT, Finland

    T. P. Holopainen

  2. Laboratory of Electromechanics, Helsinki University of Technology, P.O. Box 3000, 02015 HUT, Finland

    A. Tenhunen & A. Arkkio

  3. High Speed Tech Ltd., Tekniikantie 4 D, 02150, Espoo, Finland

    E. Lantto

Authors
  1. T. P. Holopainen
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  2. A. Tenhunen
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  3. E. Lantto
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  4. A. Arkkio
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Correspondence to T. P. Holopainen.

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Holopainen, T.P., Tenhunen, A., Lantto, E. et al. Unbalanced magnetic pull induced by arbitrary eccentric motion of cage rotor in transient operation. Part 1: Analytical model. Electr Eng 88, 13–24 (2005). https://doi.org/10.1007/s00202-004-0257-z

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  • DOI: https://doi.org/10.1007/s00202-004-0257-z

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