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http://doi.org/10.1007/s00202-007-0066-2
http://hdl.handle.net/10251/103495
Springer-Verlag
Jover Rodríguez, PV.; Belahcen, A.; Arkkio, A.; Laiho, A.; Antonino-Daviu, J. (2008). Air-gap
force distribution and vibration pattern of Induction motors under dynamic eccentricity.
Electrical Engineering. 90(3):209-218. doi:10.1007/s00202-007-0066-2
1
Air-gap force distribution and vibration pattern of
Induction motors under dynamic eccentricity
P
EDRO JOVER RODRÍGUEZ*, ANOUAR BELAHCEN*, ANTERO ARKKIO*, ANTTI LAHIO**, JOSÉ
A. ANTONINO-DAVIU***
*Laboratory of Electromechanics,
Department of Electrical Engineering
Helsinki University of Technology
P.O. Box 3000, 02015 HUT, Finland
Fax: +358 9 451 2991
E-mail: vicent@cc.hut.fi
**VTT Technical Research Finland
***Universidad Politécnica de Valencia
Department of Electrical Engineering
P.O. Box 22012, 46071
Fax: +34 96 3877599
E-mail: joanda@die.upv.es
Abstract: A method for determining the signatures
of dynamic eccentricity in the airgap force
distribution and vibration pattern of induction
machine is presented. The radial electromagnetic
force distribution along the airgap, which is the
main source of vibration, is calculated and
developed into a double Fourier series in space and
time. Finite element simulations of faulty and
healthy machines are performed. They show that
the electromagnetic force distribution is a sensible
parameter to the changes in the machine condition.
The computations show the existence of low
frequency and low order force distributions, which
can be used as identifiable signatures of the motor
condition by measuring the corresponding low
order vibration components. These findings are
supported by vibration measurements and modal
testing. The low frequency components offer an
alternative way to the monitoring of slot passing
frequencies, bringing new components that allow to
discriminate between dynamic eccentricity and
rotor mechanical unbalance. The method also
revealed a non linear relationship between loading,
stress waves and vibration during dynamic
eccentricity.
Keywords: dynamic eccentricity, vibration,
stress, FEM, Fourier analysis, induction
motor
INTRODUCTION
Condition monitoring of electrical
machines is becoming increasingly
essential for both industrial and academic
sectors. It plays a very important role for
the safe operation of industrial plants and
enables to avoid heavy production losses,
whereas the choice of adequate monitoring
2
methods is a challenging task for the
academic world.
The most used indicators for monitoring
electrical machines are currents,
temperatures, voltages, chemical debris
and vibrations. In many cases, the overall
vibration level of the machine is sufficient
to diagnose mechanical failures [1], [2]. In
contrast, the effect of electrical faults on
the vibrations is still under investigation.
Airgap eccentricity is one of the main
faulty conditions of induction machines. It
causes excessive stressing of the machine,
increasing bearing wear and producing
harmful vibrations and noise. In the worst
case, it could produce rotor-stator rub, with
consequential damage to the stator core
and winding. Thus, the online monitoring
of rotor eccentricity is highly desirable to
prevent serious operational problems.
Pöyhönen et al. [3] showed that the
electromagnetic force is the most sensitive
indicator of airgap eccentricity. The only
drawback of this indicator is its low
accessibility. Nevertheless, since
vibrations are the consequences of the
forces on the machine structure,
identifiable signatures should be found in
the vibration pattern. Finley et al. [4]
compiled a resume table with a
comprehensive list of electrically and
mechanically induced components in the
vibration pattern. Their analysis is based
on analytical formulas. The conclusion
from this paper is that with solid
knowledge of motor fundamental it is
possible to ascertain the root cause of a
vibration problem.
Cameron et al. [5] developed a monitoring
strategy based on monitoring high
frequency vibration components (slot
passing frequencies). They presented a
theoretical analysis based on rotating wave
approach whereby the magnetic flux waves
in the airgap are taken as the product of
permeance and magnetomotive force
(mmf). This monitoring strategy has the
drawbacks that detailed motor information
is needed and the monitored frequencies
my be close to the resonance frequencies
of the machine.
Dorrell and Smith [6] described an
analytical model to study induction motor
with a static eccentricity based on airgap
permeance approach including the stator
and rotor mmf. The model examine the
interaction between the harmonics that
produce unbalance magnetic pull (UMP).
It is verified by experimental investigation
carried out in [7]. They obtained good
agreement at low slip. They concluded that
the effects of the higher order winding
harmonics and rotor skew can influence on
the magnitude of the UMP.
3
Dorell et al. [8] analysed the airgap flux
and vibration signals as a function of the
airgap eccentricity. This paper put forward
a theoretical analysis of the interaction
between harmonic field components due to
eccentricity. It is illustrated how
eccentricity faults can be identified from
vibration analysis using condition
monitoring techniques. However, the paper
does not make clear the dependence of the
vibration with the machine loading, an
important fact to take into account in a
monitoring system, and the possible modal
pattern of the stress waves are not
calculated.
Based on analytical methods, Verma and
Balan [9] presented an analysis of the
radial force distribution in squirrel cage
induction motors; they were concerned
with the noise problem. The analytical
approaches have drawbacks; they do not
take into account the slotting and
saturation effects.
Vandevelde and Melkebeek [10]
developed a method for numerical analysis
of vibration based on magnetic equivalent
circuits and structural finite element
models. From the combined
electromechanical analysis the vibration
and noise are predicted. This investigation
overcomes the drawbacks from the
analytical models and develops the
calculation of the radial forces in the
(frequency, spatial order)-domain but in
this work, the effects of the low order
forces due to eccentricity were neglected.
Based on finite element method (FEM),
Belahcen et al. [11] presented a similar
analysis for a mid-size synchronous
generator. The agreement between the
simulations and measurements of noise
and vibrations was rather good.
In this work, the method used in [11] is
applied to predict the excited vibration
frequencies due to dynamic eccentricity in
the stator of an induction machine fed from
a sinusoidal voltage source. The method
takes into account the possible mode
shapes of the stress waves distribution, as
well as the machine slotting and saturation.
The simulations and measurements show
that this fault has recognisable signatures
in the stress waves, and the magnitudes of
lower frequency stress waves during a fault
event increase greatly, producing forced
vibrations [12].
METHODS
Analytical, numerical and experimental
methods are used in this work. The
experimental methods are used for the
4
measurements of the vibrations of the test
machine under healthy and eccentric
conditions as well as to obtain the
frequency response functions (FRF) of the
stator and the mechanical system. The
numerical methods are used to solve the
magnetic problem in the cross-section area
of the test machine, whereas the analytical
methods are used to analyse the stress
waves and the spectra of the measured
vibrations and to compute magnetic force
distribution.
Theoretical analysis
In an ideal concentric rotor and stator the
radial forces are cancelled out and the
resultant net force acting between the two
cylindrical bodies is zero. If any
abnormality exists, UMP occurs. During
dynamic eccentricity events the position of
the minimum airgap length rotates with the
rotor position making the UMP direction
coincide with the position of the minimum
airgap, see Fig. 1. During this phenomena
significant forces are produced that try to
pull the rotor even further from the
concentric position.
Figure 1
The equivalent airgap is usually constant
with angular position
ϕ
, but if dynamic
eccentricity is present (see Fig. 1), the
airgap length can be expressed as
(,) [1 cos( )]
er
gtg d t
ϕ
ω
ϕ
=− − (1)
where
e
d is the degree of dynamic
eccentricity,
r
ω
is the rotational speed,
ϕ
is the angular position from some base line
and g is the average airgap. If low value
of dynamic eccentricity is assumed, the
airgap permeance can be represented as [8]
1
(,) (1 cos( ))
er
tdt
g
ϕ
ω
ϕ
Λ=+ − (2)
Assuming that the flux crosses the airgap
normally even during eccentricity event,
the airgap field can be derived as
0
(,) (,) (,)
s
bt t j td
ϕϕµϕϕ
=Λ
∫
(3)
where ( , )
s
jt
ϕ
is the linear current density
of the stator given by
( , ) sin( )
s
jtJ tp
ϕ
ω
ϕ
=− (4)
where
p
is the fundamental pole-pair
number of the motor and
ω
is the