Zeszyty Problemowe – Maszyny Elektryczne Nr 80/2008
76
Arkadiusz Lewicki, Jarosław Guziński
Gdansk University of Technology, Gdańsk
DETECTION AND COMPENSATION OF TORQUE HARMONICS
IN DRIVE WITH INDUCTION MOTOR AND GEAR
WYKRYWANIE I KOMPENSACJA SKŁADOWEJ HARMONICZNEJ MOMENTU
W NAPĘDACH Z SILNIKIEM INDUKCYJNYM I PRZEKŁADNIĄ
Abstract: In many solutions of drives, the torque transmission systems are used for torque converting from the
motor shaft to the load. Properties of torque transmission system are a reason of mechanical vibration occurring in drive system. Analysis of drive vibration, usually measured by the accelerometer, makes it possible to
create diagnostic system for detection of mechanical faults of drive. In many applications, an introducing of
vibrations to drive system is undesirable. Mechanical vibrations of drive can be reduced by generation of appropriate torque components in electric machine. Proper compensation of mechanical vibration requires information about amplitudes and frequencies of vibration harmonics. In drive system with gear, the reduction of
mechanical vibration and early detection of gear damages require information about mutual position of toothed
wheel and the torque transmitted by gear. Analysis of state variables of induction motor makes it possible to
indicate the instants of toothed wheels meshing. It makes it possible to eliminate both: vibration and optical
sensors in systems of detection of gear damages and compensation of mechanical vibration. In this paper a
method of sensorless detection of torque ripples, caused by gear transmission, is proposed. Presented method
can be use in systems for early detection of gear damages and for active compensation of mechanical vibration. The results of simulation and experimental researches on proposed detection and compensation method
are presented in the paper. The experiments were provided for a drive with 10kW induction motor with gear
and for a high-speed train drive with gear and 1.2MW asynchronous motor.
1. Introduction
The main task of drive systems is stabilization
of speed of load shaft. Every change of load
torque on motor shaft by generation of appropriate value of electromagnetic torque can be
compensated. In many solutions of electric
drive systems, the electromagnetic torque is
transferred to load using mechanical torque
transmitters like: gear transmission, belt transmissions, shafts, coupling elements. In drive
systems with gear, the transmission error of
gear introduces ripples to electromagnetic
torque transmitted to load, and to load torque
transmitted to motor. The transmission error of
gear is an effect of variable stiffness of teeth
and inaccuracy of mechanical work of toothed
wheels [14, 15]. It is defined as difference between the actual position of the output gear, and
the position it would occupy if the gear were
perfect:
TE = ϕr − ϕo
(1)
where TE is a transmission error of gear, ϕr , ϕo
are positions of both shafts of gear.
The mutual position of gear shafts can be measured using special optical system or accurate
position encoders [4,11]. Mechanical oscillation
of drive system, caused by gear, can be mea-
sured by using accelerometers [5,9,10,12]. The
analysis of mutual shaft positions of gear or
analysis of mechanical oscillation of drive system gives possibility to early detect of gear (and
drive) damages [1,10]
The transmission error of gear has an influence
on speed and torque oscillations occurring on
gear output shaft. These oscillations can be reduced by modification of teeth profile [2,3]. In
drive systems with electric motor and gear, the
mechanical oscillations can be actively compensated by generation of appropriate torque
harmonics in electric motor [4,8,15]. Proper
compensation of torque ripples on gear output
shaft requires information of mutual position of
gear shafts. In this paper, the sensorless method
of torque ripples detection, caused by gear
transmission, is proposed. Presented method
can be used in detection system for gear damages and in control system of electric motor
with active compensation of transmission error.
2. Mathematical model of drive system
with gear
In drive system with gear, the spectrum of
torque oscillation consist only two harmonics
Zeszyty Problemowe – Maszyny Elektryczne Nr 80/2008
76
with known frequencies. The first of them is an
effect of unbalancing of mechanical elements of
drive and the second is an effect of the transmission error created by the mesh [1, 10] :
2
T r [p.u.]
1
0
0.5
0
20
40
20
40
∆ω [%]
0
-0.5
0.8
0
T em [p.u.]
0.6
0.4
0
time[ms]
20
40
Fig. 1. The transients of electromagnetic torque
Tem, load torque on motor shaft Tr and ripples
of rotor speed of induction motor, worked in
drive with gear. Results of simulation researches
fu =
f TE =
nr
60
(2)
z ⋅ nr
60
(3)
where: nr - speed of motor shaft, z – the number of teeth, oscillation frequencies: fu, fTE –
are connected with unbalance of mechanical
elements of drive and transmission error respectively.
In [8] the simplified model of gear was proposed:
dωr 1
= ( Tem − Tr )
dt
Jr
dωo 1
= ( Tr − To )
dt
Jo
dϕ r
= ωr
dt
(5)
dϕo
= ωo
dt
(7)
Tr = K ⋅ ( ϕr − ϕo ) + D ⋅ ( ωr − ωo )
(8)
(4)
3. Sensorless detection of gear transmission error
Analysis of speed or torque transients makes it
possible to indicate the instant of meshing and
makes it possible to detect mutual position of
gear shafts (fig.1). In sensorless control system
of induction motor, the rotor speed and rotor
flux components are estimated in observer system. The electromagnetic torque, generated
AC motor shaft position
o
o
180
360
0.45
0
o
0.4
0.35
0
20
40
60
80
0.01
100
time[ms]
meshing
frequency
0.005
0
0
50
100
150
200
250
300
350
400
frequency[Hz]
0.5
meshing
frequency
0.25
0
(6)
0
50
100
150
200
250
300
350
400
frequency[Hz]
where: ωr, ωo are angular speeds of driven
shaft and output shaft of gear, , Tem is the torque
of driving motor, Tr is the load torque on motor
shaft, T0 is the load torque of drive, D is
damping coefficient and K is the stiffness coefficient, described as the function of position of
driven wheel [13]:
K = K S + K D ⋅ sin z ⋅ ϕ ( t ) ,
(9)
(
where: Ks is mean value and KD maximum
value of stiffness coefficient.
The simplified gear model was used in simulation research of drive with induction machine.
In drive system with electric motor and gear,
the change of stiffness coefficient of toothed
wheels meshing introduces ripples to electromagnetic torque, transmitted to load, and to
load torque, transmitted to electric motor.
These changes of load torque on induction motor shaft cause oscillation of rotor speed and
electromagnetic torque generated in induction
motor (fig. 1). The amplitudes of rotor speed
oscillations and torque oscillations are too small
to compensate the load torque ripples, introduced by gear, in speed or torque control loop.
)
Fig. 2. The transients of rotor speed of 10kW
induction motor, estimated in speed observer
(ωr), the results of FFT analysis of estimated
speed (FFT(ωr)), FFT analysis of mechanical
vibration (FFT(a)) of drive system with gear.
Results of experimental researches.
in AC motor, can be obtained from:
Tem =
Lm
( ψˆ rx isy − ψˆ ry isx )
Lr J
(10)
where ψ̂ rx and ψ̂ ry are estimated rotor flux
components, Lm is mutual inductance, Lr is rotor inductance, J – motor intera.
Zeszyty Problemowe – Maszyny Elektryczne Nr 80/2008
Using of speed observer, proposed in [7] make
it possible to estimate the mechanical oscillations, caused by gear. The transient of rotor
speed of 10 kW induction motor, estimated in
speed observer and the mechanical vibrations of
drive, measured by accelerometer, are presented
on fig 2. The electromagnetic torque of electric
motor was transmitted to load using gear transmission. The reduction ratio of gear: n1/n2
=27/54. Because of damping of high frequency
speed oscillation in speed observer, analysis of
mechanical vibration in high-speed drive can be
done using electromagnetic torque, estimated in
speed observer. The transient of estimated value
of electromagnetic torque of 1.2MW AC motor
are presented on fig 3. The reduction ratio of
gear used in drive system: n1/n2 =25/75.
0.35
0.3
0.25
0.2
0.15
0
0.05
time[ms]
25
50
unbalancing
frequency
0.025
frequency[Hz]
0
0
100
200
300
400
500
600
700
800
meshing
frequency
0.001
0.0005
0
1200
1300
1400
1500
1600
1700
1800
frequency[Hz]
Fig. 3. The transients of electromagnetic torque
of 1.2MW induction motor Tem, and the results
of FFT analysis of estimated torque FFT(Tem).
Results of experimental researches
cracked tooth
1.015
76
cracked tooth
1.01
4. Sensorless detection of gear faults
1.005
1
0
10
time[ms]
20
In the paper [1,10] the method for detection of
gear damages is proposed. Presented solutions
bases on analysis of mechanical oscillation
spectrum and decomposition of measured vibration signal in angular position domain. Presented method makes it possible to indicate the
cracked teeth of gear. It is possible to replace
the signal from vibration sensor by the electromagnetic torque or speed estimated in speed
observer. The transients of speed and electromagnetic torque of induction motor and the results of FFT analysis of electromagnetic torque
are presented on fig.4. In presented case, one of
teeth of toothed wheel was cracked. Analysis
of oscillation of estimated value of torque, generated in AC motor, make it possible to indicate
the cracked teeth of gear without any additional
sensor.
30
1.2
1
cracked tooth
cracked tooth
0.8
0
10
20
0.02
time[ms] 30
meshing
frequency
0.01
0
1000
1100
1200
1300
1400
1500
frequency[Hz]
0.1
unbalancing
frequency
0.05
0
0
100
200
300
400
500
600
700
frequency[Hz]
Fig. 4. The transients of speed (ωr), electromagnetic torque (Tem) of AC motor worked in
drive with gear with one cracked tooth,. FFT
analysis
of
electromagnetic
torque
(FFT(Tem)).Results of simulation researches.
x 11z
x 12z
-
m
1
m
2
-
x 21z
x 22z
-
u sα
u1
D ecou p li n g
u2
T r a n fo r m
- a ti o n
I n v e r te r
u sβ
ψˆ
raα
ψˆ
ra β
ψˆ r α
ψˆ r β
x22
x 12
x 21
T r a n fo r m
- a ti o n
S peed
o b s e rv e r
isα
is β
is α
isβ
ω̂ r
x 11z
PLL
d2
dt2
Load
G ear
AC
m o to r
g e n e ra to r
Fig. 5. Multiscalar control system of induction motor with active compensation of mechanical
vibrations
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Zeszyty Problemowe – Maszyny Elektryczne Nr 80/2008
5. Compensation of mechanical vibration
in drive with gear
The change of meshing stiffness of toothed
wheels is one of reasons of mechanical vibration of drive system with gear transmission. The
transmission error of gear increase in case if
only one pair of teeth is transforms the torque
and decreases if two or more pairs transform the
torque. Compensation of mechanical vibrations
in drive with gear require information about
mutual position of both wheels of gear. The
mutual position of toothed wheels can be measured using optical sensors or position encoder.
Analysis of mechanical variables, estimated in
speed observer (fig. 2, fig. 3) makes it possible
to detect the changes of stiffness coefficient of
teeth. The components of electromagnetic
torque generated in induction machine, used for
compensation of mechanical vibrations, should
be synchronized with changes of stiffness coefficient of toothed wheels. On fig.5 the control
system of induction motor, with active compensation of mechanical vibration, is proposed. The
control system is based on multiscalar control
system of induction motor proposed in [6]. In
presented solution, the second derivative of estimated speed is used for elimination of constant variable from signal. The phase locked
loop is used for synchronization of compensation torque with signal from speed observer.
The results of compensation of mechanical vibration in drive with 10kW AC motor and gear
are presented on fig .6.
6. Summary
In this paper a method of sensorless detection of
mechanical vibration of drive with gear is proposed. Presented method bases on analysis of
speed and electromagnetic torque. Both variables are estimated in speed observer, used in
sensorless control system of induction motor.
Proposed method can be used in systems for
detection of gear damages and for active compensation of mechanical vibration in drive system with gear. The results of simulation and
experimental researches on proposed detection
and compensation method confirm its correctness.
7. References
[1]. Ail S, Li H.: Application of Order Cepstrum and
Neural Network to Gear Fault Detection, Multiconference on "Computational Engineering in Systems
Applications"(CESA), October 4-6, 2006, Beijing,
China,
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(France), June18-21, 2007,
[3]. Beghini M., Presicce F.: A method to define profile modification of spur gear and minimize the
transmission Error, AGMA Fall Technical Meeting,
Milwaukee, 2004,
[4]. Godler I, Ohnishi K., Yamashita T.: Repetitive
Control to Reduce Speed Ripple Caused by Strain
Wave Gearing, 20th International Conference on Industrial Electronics, Control and Instrumentation,
1994. IECON '94., Volume 2, 5-9 Sept. 1994
Page(s):1034 - 1038 vol.2,
[5]. He Q. Yi-bing L., Peng L.: Kernel Principal
Components Analysis for Early Identification of
Gear Tooth crack, Proceedings of the 6th World
Congress on Intelligent Control and Automation,
June 21 - 23, 2006, Dalian, China,
[6]. Krzeminski Z.: Nonlinear control of induction
motor. Proceedings of the 10th IFAC World Congress, Munich, 1987, pp. 349-354,
[7]. Krzemiński Z: A new speed observer for
control system of induction motor, Proc. of IEEE Int.
Conf. on Power Electronics and Drive Systems,
PESC’99, Hong Kong, pp. 555 – 560,
[8]. Krzemiński Z. Szmyła M.: Generation of high
frequency torque component in drive with induction
motor, 3rd Int. Conf. UEES 1997, Alushta, Ukraine,
[9]. Li C, J. , Limmer J, Yoo J.: Gear pitting assessment via time-frequency analysis and demodulation
- a case study, Proc. 49th Meet. of MFPT Society,
vol. 49, Vibration Inst., Willowbrook, IL, Apr.
1995,
[10]. Li H. Gear Fault Monitoring Based on Order
Tracking and Bi-spectrum Under Running-up Condition, 4th International conference on Fuzzy Systems and Knowledge Discovery, FSKD 2007,
[11]. Oswald F.B.: Gear Transmission Error Measurement System Made Operational , NASA Technical Reports Server, 2002,
[12]. Sadeghi M.H.,. Rafiee J., Harifi A.: A Fault
Detection and Identification System for Gearboxes
using Neural Networks”, International Conference
on Neural Networks and Brain, 2005. ICNN&B '05.
Volume 2, 13-15 Oct. 2005 Page(s):964 – 969,
[13]. Świtoński E, Mężyk A: Dynamical analysis of
optimisation of electromechanical machinery drives,
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2000,
[14]. Takeuchi T., Togai K.: Gear Whine Analysis
with Virtual Power Train, Mitsubishi Motors Technical Review, no. 16, 2004,
[15]. Tonshoff H. K., Kummetz J.: Active Compensation of Kinematic Transmission Errors in Servo
Zeszyty Problemowe – Maszyny Elektryczne Nr 80/2008
Drives for Machine Tools and Robots. Proceedings
of the American Control Conference, ACC 1999.
Authors
A.Lewicki: phone: +48 58 347-11-76, email
alewicki@ely.pg.gda.pl,
J.Guzinski: phone: +48 58 347-29-60, email
jarguz@ely.pg.gda.pl,
Gdańsk University of Technology,
Faculty of Electrical and Control Engineering
This work was supported in part by the European Commission under contract MTKI-CT2005-029986 (FP6) and by the Ministry of Science and Higher Education under contract
3050/T02/2007/32.
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