Angular Misalignment in Induction Motors with Flexible Coupling José M. Bossio Guillermo R. Bossio Cristian H. De Angelo Grupo de Electrónica Aplicada, Fac. de Ingeniería, Universidad Nacional de Río Cuarto, Ruta Nac. #36 Km. 601, (X5804BYA) Río Cuarto, Córdoba, Argentina.
[email protected] Abstract The problem of angular shaft misalignment in motors – load systems coupled through flexible couplings is analyzed in this work. A model for the analysis and diagnosis of angular misalignment in induction motors is presented. It allows studying the angular shaft misalignment effects over the motor torque, instantaneous power and currents through dynamic simulation. Additional effects introduced by the mixed eccentricity produced by the shaft misalignment are also analyzed through experiments. The results show that angular misalignment can be detected from electrical motor variables, but its correct diagnosis is difficult. The study is completed by vibration and thermography analysis.
I.
Fig. 1. Types of shaft misalignment.
I NTRODUCTION
Misalignment phenomenon is one of main causes for eco-
For machines driven by electric motors, misalignment
nomic losses in industry. That is due to the fact that it reduces the machine useful life [1]. To overcome such weakness, the correct alignment of the rotating machines is really important so that all those machine components more sensitive to faults due to misalignment can operate within their acceptable design limits. Those components are: rolling and sealing parts as well as couplings and shafts. An accurate alignment allows reducing axial and radial forces, and in consequence extending the machine useful life while keeping the rotor stability under dynamic operation conditions [7]. This does indeed reduce faults on the shaft, due to cyclic fatigue, and couplings wear and tear. Flexible couplings are widely used in industry to transmit power between the motor and load. Most couplings transmit torque through an elastomer or a metal spring, producing low vibration levels within an admissible range of misalignment.
shows to the havedynamic influence on the of stator spectrum, it affects behavior the current motor [1][2]. Thissince produces in consequence an increase in the amplitude of the components which are associated to the static and dynamic air gap eccentricity [6] As for the dynamic air gap eccentricity as for the mechanical rotor unbalance and misalignment, they both produce components in the current spectrum at frequencies given by f ± nf r r , which makes it possible to detect these phenomena. However, since similar components are produced by all them, their diagnosis is still difficult [4][5][6]. To overcome this weakness, it becomes necessary to identify misalignment effects over currents and other motor variables. Once identified, they may be separated from those produced by mechanical unbalance of the rotor and eccentricity. A further analysis on angular misalignment allows identifying not only the increase of the components at frequencies f + f r r and f − f f r r , produced by mixed eccentricity, but also an increase in the sidebands at twice the rotation frequency ( f + 2 f r r and and f f − 2 f r r ) [1][2][3]. The study of those components associated to angular misalignment is proposed in the present effort. Numerical simulation and experimental tests are part of such study. A dynamic model for the system that includes coupling as a universal flexible joint is also proposed. This model allows studying the effects of different levels of misalignment on currents, power and the motor torque through simulation. This study is completed through different laboratory tests. Vibration and thermographic analyses are also included to support the results.
However, most manufacturers recommend align flexible couplings as if they were rigid, either by to using alignment clocks or laser equipment [7]. It is truth that even misaligned flexible couplings are able to transmit torque producing quite low vibration levels. However, it is also truth that those levels may be eventually as high as to produce damage on the shaft bearings. Misalignment between shafts is a common phenomenon in the motion transmission process. There are three types of shaft misalignment: parallel, angular and combined (Fig. 1). Depending on its type, misalignment phenomenon may produce and/or increase vibrations at different frequencies. In particular, angular misalignment excites vibrations at rotation frequency (1x) and its second harmonic (2x). However, this highly depends on the coupling type [7].
II.
C. Determination of the speed and torque relations
MODELING
For misalignment effects on the motor currents to be analyzed, a model in d-q d-q variables [8] for the induction motor was used. The system (motor-load) was simulated using five different misalignment levels, with 80 % load torque. t orque. The mechanical dynamic for the system motor-load was modeled with the objective of including angular misalignment to the system as follows.
With the objective of including flexible coupling misalignment effects in the dynamic simulation of the motor-load system, the corresponding mathematical expressions are obtained as follows. From the relation between angles obtained in (4), and reorganizing it, tan θk =
A. Model for the aligned system If misalignment phenomenon is not included, the mechanic dynamic of the motor-load system with flexible coupling can be expressed as,
m = Tm − Bm ωm + K a ( θl − θm ) J m ω l = Tl − Bl ωl − K a ( θl − θm ) J l ω
ωk =
cos α
( cos α ) ( cos θl )
(4)
⎛ 1
1− ( sin α )
2
( cos θl )
2
(5)
where, α is the misalignment angle and θk is an auxiliary Universal Joint θm
K a
θk
θl
J l
Fig. 2. Flexible coupling modeled as a universal joint.
2
ω l
(8)
cos α 1− ( sin α )
2
( cos θl )
2
ω l
(9)
(10)
cos α 1− ( sin α )
2
( cos θl )
2
T k
(11)
-
(12)
Therefore, this component at 2 f r r will be observed in the motor torque, which characterizes angular misalignment. III.
angle.
J m
+ ( sin θl )
⎞⎞ sin 2 α ⎛ sin 2 α cos2θ ⎟ T l l ⎝⎜ cosα 2cosα ⎝⎜ 2cosα ⎠⎟ ⎠⎟
T =⎜ k
cos α
2
Tk ωk = T l ω l
T b is determined, as described in sub-section II.C, as follows,
(7)
It allows including flexible coupling misalignment effects in the dynamic simulation of the motor-load system. By reorganizing the previous equation, it can be observed that an oscillating component at twice the rotor frequency f r r appears in the torque T k .
and, by the relation between angles,
Tb = T a
ω l
Therefore, replacing (9) into (10) and solving it, yields
(3)
⎛ 1 ⎞ tan θl ⎟ ⎝ cos α ⎠
2
2
Tl =
tan −1 ⎜
2
Then, as power at each shaft is the same,
with,
θk =
( cos θk ) = cos α ( cos θl )
ωk =
(2)
Ta = K a ( θk − θm )
Through applying the cosine properties, (8) can be ex pressed as follows, follows,
as shown in Fig. 2 [9][10]. For angular misalignment, the mechanic dynamic (1) changes due to the inclusion of the joint, resulting res ulting in,
l = Tl − Bl ωl − T b J l ω
(6)
Replacing the value of θk obtained in (6) into (7), yields
For the angular misalignment phenomenon to be included in the flexible coupling, it was modeled as a universal joint,
m = Tm − Bm ωm + T a J m ω
tan θl
1
ωk
B. Model for the misaligned system
cos α
Deriving it with respect to time (t) and reorganizing it again,
(1)
where, J represents where, J represents the inertia, B inertia, B the the dynamic friction coefficient, T the torque, ω the speed and θ the angular position. All the terms with sub-index m represent the motor variables whereas the terms with sub-index l represent load. K a represents the flexible coupling elasticity constant, which can be obtained from the manufacturing data (see Appendix). Appendix).
1
A NALYSIS THROUGH SIMULATION SIMULATION
A. Misalignment Effects on Torque Torque As it can be observed in the expressions presented in the previous section, angular misalignment phenomenon produces oscillations on the torque and speed. A motor working under misalignment conditions undergoes a perturbation frequency that doubles that of rotation. Such perturbations due to misalignment are produced by the previously mentioned oscillations.
0.2 0.02
] m0.15 N [ e u 0.1 q r o T
] 0.015 W k [ r e 0.01 w o P 0.005
0.05 0
0 0
20
40
60
80
0
Frequency [Hz]
20
40
60
80
Frequency [Hz]
Fig. 3. Electromagnetic Torque Spectrum for the aligned system.
Fig. 6. Instantaneous Active Power Spectrum for the aligned system.
0.2 0.02
] m0.15 N [ e u 0.1 q r o T
] 0.015 W k [ r e 0.01 w o P 0.005
0.05
0
0 0
20
40
60
0
80
Frequency [Hz]
20
40
60
80
Frequency [Hz]
Fig. 7. Instantaneous Active Power Spectrum for a five-degree misalignment situation.
Fig. 4. Electromagnetic Torque Spectrum for a five-degree misalignment situation.
x 10
−3
] w10 K [ e d u t i l p 5 m A
] m 0.1 N [ e d u t i l 0.05 p m A
0 1
0 1
2
3
4
Angle Alpha [degrees]
2
3
4
Angle Alpha [degrees]
5
5
Fig. 8. Evolution of the amplitude of the 2 f 2 f rr IAP IAP component, as a function of the misalignment angle.
Fig. 5. Electromagnetic torque component at misalignment frequency 2 f , , as r r as a function of the misalignment misalignment angle.
Figures 3 and 4 show the motor electromagnetic torque spectrum, where the component at perturbation frequency due to misalignment can be clearly observed. Figure 3 shows the torque spectrum for the aligned system. On the contrary, Figure 4 shows the torque spectrum for a five-degree misalignment situation. The evolution of the amplitude of the electromagnetic torque component at misalignment frequency 2 f r r , as a function of the misalignment degree, can be observed in Fig. 5.
B. Misalignment Effects on power power In a similar way, oscillation effects can be observed in the Instantaneous Active Power (IAP) consumed consumed by the motor. For a three-phase motor at constant load, the instantaneous active power consumed by the motor is constant. On the contrary, for perturbations, as for instance those due to an increase of the misalignment angle, the instantaneous active power undergoes such perturbations at the misalignment frequency 2 f rr . Figure 6 shows misalignment effects on the instantaneous active power spectrum for the aligned system whereas Fig. 7 shows it for a five-degree misalignment situation. The evolution of the amplitude of the 2 f r r IAP component, as a function of the misalignment degree, can be observed in Fig. 8.
C. Misalignment Effects Effects on Currents The effects of angular misalignment in the coupling between the motor and load shafts can be studied through analyzing the stator current spectrum obtained from simulation. The current spectrum shows two sidebands around the fundamental component, given by,
f cd = f ± nf r
(13)
where f is where f is the supply frequency, f frequency, f r the motor rotational speed in Hz, and n a constant. From the obtained results, the amplitudes of the components at f at f + 2 f r r , y f y f − 2 2 f r is proportional to the misalignment degree. Figure 9 shows the simulation results for the aligned system. As it can be appreciated, only the component at the supply frequency appears in the spectrum. Figure 10 shows the stator current spectrum for a five-degree misalignment angle. The figure also shows the sidebands at frequencies f + 2 + 2 f r r and and f f - 2 f r r. Figure 11 shows the amplitude evolution of the components at frequencies f ± 2 ± 2 f r r , as functions of misalignment degree.
] 0.02 W k [ r e w0.01 o P
] 0.4 A [ t n e r r 0.2 u C
2f
r
0
0 0
20
40
60
80
100
0
120
Frequency [Hz]
10
20
30
40
50
Frequency [Hz]
Fig. 9. Stator Current Spectrum for the aligned system.
Fig. 12. IAP Spectrum for t wo-degree misalignme misalignment nt and with 80 % load. −3
x 10
] 0.4 A [ t n e r r 0.2 u C 20
40
60
80
100
Load 40 % Load 60 % Load 80 %
0
120
Frequency [Hz]
49
fl−2fr
0.4
50.5
−3
x 10
fl+2fr
] 10 W 8 K [ e d 6 u t i l p 4 m A
0 1
50
Fig. 13. IAP Fault Frequency for five different load levels.
] A [ 0.3 e d u t i l 0.2 p m A0.1
49.5
Frequency [Hz]
Fig. 10. Stator Current Spectrum for five-degree misalignment angle.
2
3
4
Angle Alpha [degrees]
1º 2º 3º
20
40
60
80
Load [%]
Fig. 14. Amplitude of the 2 f rr IAP IAP component with respect to the misalignment angle.
A NALYSIS THROUGH EXPERIMENTAL EXPERIMENTAL RE RESULTS SULTS
Experimental tests were carried out at a laboratory. A 5,5kW induction motor, coupled to and identical motor were used for the tests. The additional motor performs as load and is coupled to the IM by means of a flexible Gummi-type coupling (see Appendix). By lifting the rear part of the motor, through adding sup plementary devices, it was possible to produce different levels of shaft misalignment. misalignment. Such misalignment levels go from zero degree (motor-load aligned), to five degree. It is important to notice that flexible couplings absorb more misalignment stress than rigid couplings before reaching high vibration levels.
The previous figure shows a decrease in the amplitude of the 2 f r r frequency component for a three-degree misalignment. This reduction is a consequence of the raise in the system rigidness due to an increase of the misalignment degree. B. Misalignment Effects Effects on Currents The following figure shows the results for the stator current spectrum, under two-degree misalignment and 80% load. From the experimental analysis of the stator current spectrum, it can be deduced that the angular misalignment effects between the motor and load cannot be clearly distinguished. However, a variation of the f + 2 f r r component component amplitude for misalignment can be observed in the spectrum.
A. Misalignment Effects on the Instantaneous Instantaneous Active Powe Powerr
0.3
Angular misalignment effects on the instantaneous active power show the same frequency components than that on the motor torque. Figure 12 shows the IAP component at frequency 2 f rr for a two-degree misalignment and with 80 % load. The IAP amplitude at fault frequency as a function of load can be seen in Fig. 13. The amplitude evolution of the IAP component at 2 f rr frequency as a function of load for different misalignment degrees can be observed in Fig. 14. This figure also shows that under no load conditions the component at frequency f + 2 f rr matches the supply frequency f , which difficults distinguishing the misalignment effect.
0º
2
5
Fig. 11. Amplitude of the current components at frequencies f frequencies f ± ± 2 2 f r r, with respect to the misalignment misalignment angle.
IV.
Load 20 %
] 6 W k [ r e 4 w o P 2
0 0
Load 0 %
8
−f
] A [ 0.2 e d u t i l p m 0.1 A
−2f
r
f
r
2f r
r
0 0
20
40
60
Frequency [Hz]
80
100
Fig. 15. Stator Current Spectrum for two-degree misalignment and 80 % load.
0.02
] c e 2 s / m1.5 m [ e d 1 u t i l p 0.5 m A 0
0º
] A [ 0.015 t n e r r u 0.01 C
1º 2º 3º
0.005 0 0
20
40
60
2f
r
0
80
20
40
60
80
100
120
Frequency [Hz]
Load [%]
Fig. 16. Stator current f current f + + 2 2 f r r component component under misalignment as a function of load.
Fig. 18. Horizontal vibration velocity spectrum – Components 1x, 2x and 3x for two-degree misalignment and 80 % load.
0.4
1.5
] c e s / m 1 m [ e d u t i l 0.5 p m A
0º
] A [ 0.3 t n e r r 0.2 u C
1º 2º 3º
0.1 0 0
20
40
60
Load [%]
0
80
The trend of the stator current f current f + 2 + 2 f r r component as a function of load, for different levels of misalignment can be seen in Fig. 16. Such component increases under misalignment besides its effects for the motor at no load conditions, previously mentioned for power. It must be noted that the presence of two sidebands in the stator current spectrum corresponding to the components at frequencies f frequencies f + f rr and and f f − f f rr can also be seen in Fig. 15. These components depend on the motor eccentricity level, which in part is a consequence of misalignment. The eccentricity frequencies are generally present even for no anomaly in the motor. This is due to the fact that it is almost impossible to assemble a motor with no eccentricity, generally coming from the motor own characteristics as well as from abrasion of the rolling parts. In addition, the process of getting the motor and load aligned makes it difficult to obtain a zero-degree misalignment. The combination of all the effects mentioned above will result in static and dynamic eccentricity of the motor even for those completely healthy. That means that the com ponents at frequencies f frequencies f + f + f rr and f and f − f f r r will be always present in the current spectrum. Figure 17 shows the variation of the stator current components at f ± f ± f r r frequency as a function of load for different misalignment degrees. VIBRATION ANALYSIS
One of the techniques more widely used in the diagnosis of angular misalignment is the analysis of the motor radial and axial vibrations. This technique allows distinguishing the effects due to the motor unbalance from those due to misalignment while evaluating if such misalignment is either angular or parallel. However, on one side, for this analysis, it is necessary to install sensors on the motor, which may difficult its implementation. On the other side, the different components of the vibration signals as well as their relation highly depend on the flexible coupling used, like with the motor currents [7].
2º 3º
20
40
60
Load [%]
Fig. 17. Trend of the f the f ± f r r stator current component as a function of load, for different misalignment degrees.
V.
1º
80
Fig. 19. Evolution of the horizontal vibration velocity 2x component.
With the objective of validating the analysis through currents, the analysis through vibrations to detect misalignment was used. The vibration data were obtained from two acceleration transducers mounted horizontally and vertically over the motoraccording case. Thetospeed values (1995). for suchIt vibrations were obtained ISO 10816 is important to notice that the highest vibration level corresponds to the horizontal axis. This is due to the height of test bench used for the analysis. Figure 18 shows the fundamental components of the horizontal vibration velocity spectrum for angular misalignment. The figure also shows the vibrations at rotational frequency f frequency f r r (1x) due to eccentricity and those at frequencies 2 f 2 f r r (2x) and 3 f r r (3x) mainly due to angular and parallel misalignment but also to the residual eccentricity of the motor. The amplitude evolution of the 2x component in the horizontal velocity spectrum, as a function of the misalignment degree, can be observed in Fig. 19. The same figure also shows the increase of the 2x component depending on the motor load for different misalignment degrees. VI.
THERMOGRAP HERMOGRAPHIC HIC ANALYSIS
To quantify the severity of the angular misalignment, their effects over the flexible coupling were analyzed by infra-red thermography. In Fig. 20 shows the temperature distribution on the coupling rubber for the aligned shafts. The thermographic picture for the two-degree misalignment case is shown in Fig. 21, where the great increase in the rubber tem perature can be appreciated.
dustry Applications Conference, 2003. 38th IAS Annual Meeting. Conference Vol. 2, 12-16 Oct. 2003 P Page(s):1347 age(s):1347 - 1351 1351 vol.2 [5] Obaid, R.R.; Habetler, T.G.; “Effect of load on detecting mechanical faults in small induction motors”. Diagnostics for Electric Machines, Power Electronics and Drives, 2003. SDEMPED 2003. 4th IEEE International Symposium on 24-26 Aug. 2003 Page(s):307 – 311. [6] Obaid, R.R.; Habetler, T.G.; Tallam, R.M.; “Detecting load unbalance and shaft misalignment using stator current in inverter-driven induction motors”.. Electric Machines and Drives Conference, 2003. IEMDC'03. motors” IEEE Internacional Volume Volume 3, 1-4 June 2003 Page(s):1454 Page(s):1454 - 1458 vol.3. [7] Piotrowski, “Shaft “Shaft Alignment Handbook”, Handbook”, Third Edition-1995. [8] Krause P., Wasynczuk O. and Sudhoff S., “ Analysis “ Analysis of electric machinery”,, IEEE Press, New york, 1986. ery” [9] Xu M. and Maranconi R.D., “Vibration “ Vibration analysis of a motor-flexible cou pling-rotor system subjet to misalignment and unbalance, Part I: theoretical model and analysis. Journal of Sound and Vibration”. Vibration”. 176(5), pp. 663-679. 1994. [10] Hamzaoui N., Boisson C. and Lesueur C. , “Vibro-acoustic “Vibro-acoustic analysis and identification of defects in rotating machinery. Part I: Theoretical model. Journal of Sound Sound and Vibr Vibration” ation”.. 216(4), pp. 553-570. 1998.
Fig. 20. Thermographic picture of the coupling without misalignment.
APPENDIX Table 1: Coupling characteristics and misalignment admissible load values
Coupling A35
Fig. 21. Thermographic picture of the coupling with two-degree misalignment.
VII.
CONCLUSIONS
Effects of misalignment on an induction motor input currents were analyzed in the present effort. The results obtained from simulation show that misalignment produces sidebands in the stator currents at frequencies f frequencies f + 2 f r r , and f and f − 2 2 f r r . It was demonstrated through experimental results that the rotor curvature or deformation produces f ± f r r components. This is mainly due to a combination of the air-gap static and dynamic eccentricity produced by misalignment but also due to the residual eccentricity of the motor. With the objective of validating the analysis of currents to detect misalignment, the analysis of the motor through vibrations was carried out in the present work. Both analysis, through currents and vibrations, demonstrate that it is difficult to determine the fault severity as a function of the characteristic component. However, even when the misalignment degree cannot be determined, the presence of sidebands at 2 f rr in the current spectrum (or 2 f r r components in the power spectrum), indicates the existence of a possible shaft misalignment. R EFERENCES EFERENCES [1] Manés F. Cabanas, Manuel G. Melero, Javier G. Aleixandre, J. Solares, “Shaft misalignment diagnosis of induction motors using current spectral analysis: a theoretical approach” approach” International Conference on Electric Machines, ICEM 96, Vigo 10-12 Septiembre 1996. [2] Manés F. Cabanas, et al., al., “Effects of shaft misalignment on the current and axial flux spectra of induction motors”. motors”. ELECTRIMACS 96, Nantes, Francia, 17-19 Septiembre 1996. [3] Manés F Cabanas, Carlos H. Rojas, Manuel García Maleno, Gonzalo A. Orcajo, Manuel P. Donsión. “Relación entre los modos de vibración y la combinación de excentricidad estática y dinámica en el entrehierro de los motores de inducción”. EDUNIV, Revista Científica-Ingeniería Energética, año 2000. [4] Obaid, R.R.; Habetler, T.G.; “Current-based algorithm for mechanical fault detection in induction motors with arbitrary load conditions” conditions”.. In-
Tn
Torsion
K=Tn/torsion
9 Kpm = 90 Nm
4º (0.06981 rad)
1289.155 Nm/rad
Misalignment maximum tolerance
angular
1 [º]
parallel
0.4 [mm]
Table 2. Induction Motor Characteristics
Induction Motor Power Voltage Frequency Nominal Current Current Nominal Speed Power Factor Rs Rr Lls=Llr Lm
5.5 kW 380 V 50 Hz 11.1 A 1470 rpm 0.85 0.95 Ω 0.36 Ω 4.7 mH 122 mH
