433
VIBRATION ANALYSIS OF AN INDUCTION MOTOR
UNDER ELECTROMAGNETIC FORCE
Takuzo IWATSUBO, Shozo KAWAMURA,
Department of Mechanical Engineering, Kobe University
Yoshiyuki HAYASHI,
Graduate School of Science and Technology, Kobe University
presently, Mitsubishi Heavy Industries, Ltd.
Koki SHIOHATA
Department of Mechanical Engineering, Ibaraki University
Three-phase induction motor causes an unstable vibration because of electromagnetic force. In
this paper, a criterion for unstable vibration was introduced. To confirm this criterion, numerical
integration was carried out and the results showed a good agreement with the unstable region
obtained by the criterion. This criterion gives us the complete story about the stability for the case
when the parameters e.g., phase current, air gap length etc. are changed. It was cleared that the
natural frequency changes owing to the variation of phase current, and that the unstable vibration
occurs when the frequency of power supply and the natural frequency (changed by current) are
nearly the same. And it was also showed that the parameters can be classified into the categories
whether they have influences on the stability or not.
Keywords : Electromagnetic-Induced Vibration, Induction Motor, Parametric Excitation, Unstable Vibration.
INTRODUCTION
In this paper, the forced vibration response and the
unstable vibration of an induction motor (driven by a
variable operating frequency) taking into account the
effect of electromagnetic force on the rotor are analyzed.
First, to obtain an equation of motion, the induction
motor was modeled by using a finite element method,
where the unbalance force and electromagnetic force
used in the equation. Next, the forced vibration response was determined by direct numerical integration.
Then the unstable vibration was found to occur under
certain conditions, which agree with the instability criteria of Mathieu’s equation with multiple degrees of freedom. And range of instability was found to vary with
the electrical and mechanical parameters. The relationship between the variation of these parameters and the
instability was also derived, and it allows us to avoid
unstable vibration precisely.
SUMMARY
We analyzed real 2-pole and 4-pole induction motors
and the experiment was performed in order to verify the
theory by using induction motor-generator system. The
results are summarized below.
1. Natural frequency is decreased by phase current increases and air gap length decreases.
2. Unstable vibration may occur when the frequency
of power supply is equal to or near the natural frequency. The theoretical result of the range of instability agrees well with the one obtained by the
numerical integration as shown in Fig.1.
3. Though the unstable vibration scarcely occurs in a
4-pole induction motor, it occurs very often in a 2pole induction motor even if the unbalance phase
current does not exist.
4. Time varying term which is occured by the eccentricity of the rotor is experimentally obtained.
5. The results which is obtained by theoretical analysis
is verified by experiment and these two results agree
well as shown in Fig.2. Relation between power
supply frequency and phase current and instability
is represented.
1st natural frequency
10
stable
unstable
8
Unstable
5
1st natural frequency without
the effect of electromagnetic force
0
boundary between
stable and unstable region
natural frequency
6
phase current [A]
electric phase current [A]
boundary between
stable and unstable region
running curve
282[W]
163[W]
52[W]
Stabl
able
4
2 1st natural frequency without
the effect of unbalanced magnetic pull
40
60
80
100
frequency of power supply [Hz]
120
0
Fig. 1: Comparison of Numerical Integration and
Range of Instability
0
20
40
power supply frequency
20
60
80
0[Hz]
40
natural frequency [Hz]
60
80
Fig. 2: Operation Curve and Unstable Region
日本機械学会〔№00-6〕Dynamics and Design Conference 2000 CD-ROM論文集〔2000.9.5-8,東京〕
NOMENCLATURE
A : magnetomotive force (m.m.f.), A
b : flux density, T
C : damping matrix
e : mass eccentricity, m
f : force, N
I : current per phase, A
K : stiffness matrix
Kd : distribution factor
Kp : pitch factor
L : stator core length, m
M : mass matrix
O−XS YS : coordinate fixed to the center of stator bore
P : number of poles
p : number of pole-pairs, p = P/2
rR : radius of rotor, m
rS : radius of inner stator bore, m
s : slip
t : time, s
u : shaft deflection, m
xRS : rotor displacement in XS direction with respect
to O−XS YS coordinate, m
Z : number of stator conductors in slot per pole per
phase
α : angle between OS a and XR -axis, rad
∆I : quantity of unbalance current per unit phase, A
δm : mean air gap length, m
ε : small parameter, m
Λ : permeance, Wb/A
µ0 : permeability of free space, H/m
ρ : bearing eccentricity with respect to the stator bore,
m
σ : electromagnetic force per unit area, N/m2
φ : angle between OOS and XR -axis, rad
ω : frequency of periodic excitation, rad/s
ω0 : angular frequency of power supply, rad/s
ωe : angular velocity of rotor, rad/s
ωi : i-th natural frequency, rad/s
subscripts
i, j, k : any integer
R : rotor
S : stator
superscripts
e : electromagnetic
u : unbalance
1. INTRODUCTION
Three-phase induction motors have been used widely
in many industries for a number of years. Recently,
since the applications of induction motors and demands
for silent motors have increased, further detailed studies
on noise and vibration from induction motors have been
urged.
The investigation of the criteria of instability and consideration of the whirling motion is important when the
induction motor is fed by an inverter and operated at
variable operating frequency. Therefore, in the present
work, we analyzed the forced vibration response and the
unstable vibration of an induction motor (driven by a
variable operating frequency) taking into account the
effect of electromagnetic force on the rotor.
First, to obtain an equation of motion, the induction
motor was modeled by using a finite element method,
where the unbalance force and electromagnetic force
&% !"# $ Fig. 3: Analytical Model
used in the equation. Next, the forced vibration response was determined by direct numerical integration,
and the relationship between the electromagnetic force
and the frequency of the power supply (which causes resonance) are derived. Then the unstable vibration was
found to occur under certain conditions, which agree
with the instability criteria of Mathieu’s equation with
multiple degrees of freedom. And range of instability
was found to vary with the electrical and mechanical
parameters. The relationship between the variation of
these parameters and the instability was also derived,
and it allows us to avoid unstable vibration precisely.
2. MODELING OF INDUCTION MOTOR
2.1 Rotor, stator, and bearings
Figure 3 illustrates an analytical model of an induction motor, which consists of the rotor, stator, and bearings. The rotor is linked to the stator by two bearings
at the end of the rotor. For simplicity, it assumed that
the rotor and the stator, which includes both the core
and housing, are represented by beam elements. In the
elements, equivalent values of Young’s modulus (Iwata,
1991) are used for their cores. This model takes account
of the effects of multilayered steel sheets. The ball bearings are modeled by using spring elements arranged in
both radial and rotational directions (Matsubara, 1996).
This modeling is expected to yield quantitatively good
natural frequencies. Special matrices, i.e., mass, damping, and stiffness matrices, are obtained by using a finite
element method.
2.2 Excitation forces
In the induction motor, both a mechanical exciting
force, i.e., unbalance force, and an electromagnetic force
act on the rotor. The mechanical exciting force is induced by a residual unbalance of mass and excites the
rotor at a slip frequency, which is investigated here.
The electromagnetic force acting on the rotor and
stator can be calculated from integration of Maxwell’s
force per unit area. Maxwell’s stress is proportional to
the square of the flux density, which is found by multiplying magnetomotive force (m.m.f.) and permeance.
Thus, for calculating the electromagnetic force, both the
m.m.f. and permeance must be estimated. Kanzaki et
al.(1994) calculated the m.m.f. and permeance taking
account of the unbalance of phase current and the rotor
eccentricity with respect to the stator. The electromagnetic force is derived following their procedure as shown
below.
The cross section of the rotor and stator is shown
in Fig.4. It is assumed that there is a misalignment
between the stator and the rotor (i.e., a straight line
between the bearing centers is not at the center of the
? @&@ 7895 8 : ;=<=7>4 :96 ;6 48<=487
K2 =
. /
π · µ0 · L · rS
A1 A2 ,
2δm 3
(6)
and
*
)
(1
,
π · µ0 · L · rS
(A1 + A2 )2 [εu cos ωe t + ερ cos φ]
2δm 2
A1 A2 +
εu [cos(ωe t − 2ω0 t) + cos(ωe t + 2ω0 t)]
2
+ ερ [cos(φ − 2ω0 t) + cos(φ + 2ω0 t)] . (7)
f (t) =
0 /
-
, +
23
'
From Eqs. (1), (2)-(4), and (5)-(7), the electromagnetic force is found to be proportional to the displacement and to have components of 2ω0 , 2ω0 ± ωe , and ωe .
45 6 78#9 5 8: ;=<=7>4 :846 48<=487
2.3 Equation of motion
We assume that the mechanical exciting force acts on
the rotor and that the electromagnetic force acts both
on the stator and on the rotor, so the following equation
of motion can be obtained in matrix form:
"
#(
) "
#(
)
MR
0
ẍR
CR 0
ẋR
+
0
MS
ẍS
0 CS
ẋS
"
#(
) (
)
u
e
KR 0
xR
f +f
+
=
, (8)
0 KS
xS
−f e
D E
O
V
N
UW
P
F
A
B
S
QR
H
IKJ
L
M
U
G
A C
T
E
Fig. 4: Model of Rotor
OS : center of stator bore, O : center of bearing, S : geometric center of rotor, G : center of gravity of rotor, h :
distance from OS to a
stator bore) and that the rotor shaft is deflected due to
a mass eccentricity. The air gap and the permeance is
thus, nonuniform at different angular positions.
We can write the electromagnetic force as
f e = {K1 + K2 cos(2ω0 t)} xRS + f (t),
where, for the 2-poles induction motor,
π · µ0 · L · rS
A1 A2
2
K1 =
(A1 + A2 ) +
,
2δm 3
2
(1)
where f u is the mechanical exciting force vector, f e
is the electromagnetic force vector whose elements are
given in Eq. (1), and xR and xS are the displacement
vectors of the rotor and the stator, respectively. In
Eq. (8), the stiffness of the bearings and supports are
not written out explicitly for simplicity, and the proportional damping in the rotor, stator, and bearings is
adopted.
Substituting the right-hand side of Eq. (1) into Eq. (8)
and transposing the first two terms in Eq. (1) to the lefthand side gives
M ẍ + C ẋ + {K + K(t)}x = f (t).
(2)
(9)
This equation of motion is an ordinary differential equation and has a periodically time-varying coefficient K(t).
3. METHODOLOGY OF ANALYSIS
π · µ0 · L · rS
K2 =
2δm 3
A2
A2
A1 A2 + 1 + 2 ,
2
2
(3)
π · µ0 · L · rS
(A1 + A2 )2 [εu cos ωe t + ερ cos φ]
2δm 2
A1 A2 +
εu [cos(ωe t − 2ω0 t) + cos(ωe t + 2ω0 t)]
2
+ ερ [cos(φ − 2ω0 t) + cos(φ + 2ω0 t)]
f (t) =
A21
[εu cos(ωe t + 2ω0 t) + ερ cos(φ + 2ω0 t)
2
A2
+ 2 [εu cos(ωe t − 2ω0 t) + ερ cos(φ − 2ω0 t)]
2
A1 A2
+
[εu cos ωe t + ερ cos φ] ,
2
+
(4)
and for the multi-poles induction motor,
K1 =
π · µ0 · L · rS
2
(A1 + A2 ) ,
2δm 3
3.1 Digital simulation
To investigate the motor’s motion expressed by
Eq. (9), direct numerical integration (the Wilson-θ
method) is applied. This integration can find a solution of a dynamical system under parametric excitation.
In this study, to save computation time, a substructure
synthesis method is used to derive the equation of motion Eq. (9).
And a Fast-Fourier transform algorithm is applied to
obtain the frequency response.
3.2 Instability criteria
The right-hand side of the equation of motion Eq. (9)
is the function of independent variable t. Thus, the stability criteria of the system under parametric excitation
can be determined by solving an equation of motion of
the form
M ẍ + C ẋ + {K + K(t)}x = 0.
(5)
(10)
This equation of motion is called a coupled Mathieu’s
equation. Hsu(1963) developed explicitly the instability
Mean air gap length δm [µm]
400
Mass eccentricity e [µm]
10
Bearing eccentricity with respect to the
stator bore ρ [µm]
10
Distribution factor Kd
0.951
Pitch factor Kp
0.785
Stator core length L [m]
0.150
Radius of rotor rR [m]
0.150
Radius of stator bore rS [m]
0.151
Number of stator conductors in slot per
boundary between
stable and unstable region
electric phase current [A]
Table 1: Mechanical and Electrical Parameters of Analytical Model
1st natural frequency
10
stable
unstable
5
1st natural frequency without
the effect of electromagnetic force
0
40
60
80
100
frequency of power supply [Hz]
120
50
Fig. 5: Comparison of Numerical Integration and Range
pole per phase Z
Phase current I [A]
5.0
γ = ∆I/I
0.20
Slip s
0.30
of Instability
Table 3: Parameters Changed in order to Investivate the
Variation of Instability
Table 2: Natural Frequencies [Hz]
δm [×10
1st
2nd
Rotor (free-free)
3071.0
3708.5
γ = δI/I
Rotor with bearings
138.2
447.3
P
Whole system of motor
101.8
292.2
criteria for this equation of motion. He carried out a
first-approximation analysis and derived criteria for determining instability. His method combines variation of
parameters and the series expansion of the perturbation
method.
4. NUMERICAL EXAMPLES
The mechanical and electrical parameters used in the
following analyses are listed in Table 1, and the natural
frequencies of the system are listed in Table 2.
The instability criteria are calculated, and the range
of current and frequency at which the dynamic instability occurs in the motors is shown in Fig. 5. The
corresponding results of the direct numerical integrations and the variation of the first natural frequency are
also shown. The unstable range shows a good agreement with the the direct numerical integration. Thus
the range of instability can be calculated with sufficient
accuracy by the method described in this paper. This
figure confirms that the unstable vibration occurs when
ω0 is nearly equal to the natural frequency, which is decreased by phase current. K1 and K2 are determined by
three parameters (c, I, γ). Consequently, the instability
criteria are governed by these parameters and the frequency of excitation (2ω0 ). Since the instability criteria
are derived from Eq. (10), other parameters which only
have effects on f (t) in Eq. (9) (i.e., εu , ερ , and ε) are
not responsible for the instability. Thus a consideration
of these four parameters (c, I, γ, and 2ω0 ) is sufficient
to determine the instability criteria. The relationships
between these effective four parameters and the instability range are thus derived by changing these parameters
−4
m]
case 1
case 2
case 3
case 4
4.0
2.0-8.0
4.0
2.0-4.0
0.0-0.6
0.2
0.8-1.2
0.8
2
4
(Table.3).
Figure 6 shows the effect of the variation of γ on the
unstable range in a 2-pole induction motor (case 1).
This figure shows that the unstable range vary slightly
when γ is changed considerably. The effect of the variation of δm (substitute for c) on the unstable range for
2-pole machine is shown in Fig. 7 (case 2). This figure indicates that the variation of δm gives much larger
changes in unstable range that those in Fig. 6. Consequently, the air gap length influences instability more
than the unbalance of phase current. To further this
study, a 4-pole induction motor was investigated. And
to determine the unstable range in a 4-pole machine, γ
and δm were varied are shown in Figs. 8 (case 3) and 9
(case 4), respectively.
These figures show that the unstable range for 2-pole
and for the 4-pole induction motors is also found to exist. However, regarding the 4-pole motor, no unstable
range can be found unless γ is larger than 0.6. These
results strikingly contrast with those obtained from a 2pole motor, because the unstable vibration occurs even
if γ = 0 (Fig. 6). It can thus be concluded that unstable
vibration rarely occurs in a 4-pole induction motor.
5. EXPERIMENTAL RESULTS
It is difficult directly to obtain an instability phenomenon of the induction motor by experiment, so time
varying coefficient which is a cause of instability is measured and instability region is calculated by using the
coefficient. The time varying coefficient K2 in Eq.(9) is
10
30
= (unbalance
of phase current Y I) / (phase current I)
X =0
X =0.2
X =0.6
Electric phase current [A]
Electric phase current [A]
X
unstable region
5
0
40
60
80
100
Frequency of power supply [Hz]
20
unstable region
10
0
120
Fig. 6: Variation of Unstable Range (CASE 1)
[ = (unbalance of phase current \ I) / (phase current I)
[ =0.8
[ =1.0
[ =1.2
40
60
80
100
Frequency of power supply [Hz]
120
Fig. 8: Variation of Unstable Range (CASE 3)
Electric phase current [A]
Electric phase current [A]
30
Mean air gap[ ] m]
200
300
400
10
unstable region
5
Mean air gap[ Z m]
200
400
600
800
0
40
60
80
100
Frequency of power supply [Hz]
40
60
80
100
Frequency of power supply [Hz]
120
Fig. 9: Variation of Unstable Range (CASE 4)
(11)
and the force is
f2ω0 = K2 · ρ
10
0
obtained by setting φ=0 and εu =0 as,
πµ0 Lrs
A21
A22
(A
A
+
+
)
1
2
3
2δm
2
2
unstable region
120
Fig. 7: Variation of Unstable Range (CASE 2)
K2 =
20
(12)
From this relation K2 is obtained by measuring the
static eccentricity ρ and magnetic force at the bearing
f2ω0 .
The test apparatus is composed of 3 phase the 2 pole
induction motor and generator and resistance to exhaust
power of the motor. The magnetic force due to static
eccentricity is measured by loadcell which is attached on
the bearing of the rotor. The eccentricity of the bearing
and load of the motor are set to 0, 50 and 100 µm and
52, 163 and 282W, respectively.
By using the experimental result, the time varying
coefficient K2 is obtained as shown in Fig.10. From
this figure incline K2 becomes large, as the current and
torque increase.
Fig.11 shows the magnetic force and K2 against power
supply frequency in order to compare the experimental
result and theoretical one.
It is known from this figure that the experimental and
theoretical results and coincide in high frequency but are
some discrepancies in low frequency.
In stability region occurred by the eccentricity of rotor is obtained by using the coefficient K2 as shown in
Fig.12. As shown in the theoretical results, eigenfrequency decrease by the static magnetic force and the
instability is occurred at the region where the power supply frequency coincide with the eigenfrequency. Therefore the instability is occurred at the heavy load and low
speed motor condition.
6. SUMMARY
The rotor vibration in an induction motor was analyzed by taking into account the effect of electromagnetic force. This analysis involves direct numerical integration and determining instability criteria. Amplitude
and frequency of stationary oscillation were obtained by
numerical integration. And the unstable range derived
from the instability criteria show quantitatively good
agreements with numerical integrations. We analyzed
real 2-pole and 4-pole induction motors and the experiment was performed in order to verify the theory by
using induction motor-generator system. The results
are summarized below.
1. Natural frequency is decreased by phase current increases and air gap length decreases.
2. Unstable vibration may occur when the frequency
8
8
Unstable
boundary between
stable and unstable region
natural frequency
[N]
^
phase current [A]
f2
0
6
6
4
running curve
282[W]
163[W]
52[W]
Stabl
able
4
2 1st natural frequency without
the effect of unbalanced magnetic pull
40[Hz] _
50[Hz] _
60[Hz] _
70[Hz] _
80[Hz] _
2
-4
ave. : -1.00 ` 10 [m]
-1
0
2.93[A] _
2.81[A] _
2.30[A] _
2.10[A] _
2.01[A] _
34818[N/m]
38224[N/m]
31491[N/m]
26393[N/m]
22856[N/m]
0
1
[×10-4]
eccentricity [m]
0
Fig. 10: Relation of K2 and eccentricity
20
40
power supply frequency
20
60
80
0[Hz]
40
natural frequency [Hz]
60
80
Fig. 12: Operation Curve and Unstable Region
Table 4: Specification of induction motor
Experiment
12
eccentricity: 0[ a m](1-aE)
eccentricity: 50[ a m](1-bE)
eccentricity: 100[ a m](1-cE)
force [N]
10
8
6
Analysis
eccentricity: 69[ a m] (1-aA)
eccentricity: 119[ a m] (1-bA)
eccentricity: 169[ a m] (1-cA)
4
2
2
K2 [N/m]
1
60000
phase current[A]
3
Experiment
Analysis
50000
TYPE TFOAb FORM K-DK
Number of poles
2
Rated power
[kW]
0.75
Mass of rotor
[kg]
2.30
Rotor and stator stack length
[mm]
50.3
Outside diameter of rotor core
[mm]
67.45
Diameter of stator bore
[mm]
67.97
Air gap length
[mm]
0.26
40000
30000
30
30
40
40
50
60
70
80
power supply frequency [Hz]
50
60
70
synchronous speed [Hz]
80
90
90
Fig. 11: Unbalanced Magnetic Pull Aacting on Motor
of power supply is equal to or near the natural frequency.
3. Though the unstable vibration scarcely occurs in a
4-pole induction motor, it occurs very often in a 2pole induction motor even if the unbalance phase
current does not exist.
4. Time varying term which is occured by the eccentricity of the rotor is experimentally obtained.
5. The results which is obtained by theoretical analysis
is verified by experiment and these two results agree
well. Relation between power supply frequency and
phase current and instability is represented.
References
Hsu, C. H. , 1963, “On the Parametric Excitation of a
Dynamic System Having Multiple Degrees of Freedom”,
Trans. ASME, 30-3, pp.367-372.
Iwata, Y., et al., 1996, “Vibration of Rotor in Induction Motor”, Proc. of Dynamic and Design Conf. (in
Japanese), Fukuoka, Vol.B, pp.327-330.
Iwata, Y. and Sato, H., 1997, “Vibration of Rotor
in Induction Motor”, Proc. of Dynamic and Design
Conf. (in Japanese), Tokyo, Vol.B, pp.25-27.
Iwata, Y., et al., 1991, “Natural Frequency of Rotor
with Rotor Core”, Bulletin of the JSME (in Japanese),
Series C, Vol.57, No.544, pp.3748-3753.
Kanzaki, H., et al., 1994, “Unstable Electrical Vibration of an Induction Motor”, Bulletin of the JSME (in
Japanese), Series C, Vol.60, No.578, pp.3238-3244.
Matsubara, K., et al., 1996, “Study on Vibration and
Noise Reduction for Motors”, Proc. of Dynamic and Design Conf. (in Japanese), Fukuoka, Vol.A, pp.137-140.