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A Model for Single-Point Bearings Defects in Electric Motors
8th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes (SAFEPROCESS) August 29-31, 2012. Mexico City, Mexico
A Model for Single-Point Bearings Defects in Electric Motors A. Castellino, C. De Angelo, G. Bossio Grupo de Electrónica Aplicada, Universidad Nacional de Río Cuarto X5804BYA, Argentina. ([email protected], [email protected], [email protected]) Abstract—A mathematical model of an induction motor including single-point bearing defects is proposed in this work. This model allows studying the behavior of different current-based fault diagnosis strategies in front of bearing faults. For this to be achieved, an eccentricity at bearing fault frequency is added to the model of a wound rotor induction motor. Through simulating this model, the effects of an outer race single-point defect on the motor variables are analyzed. Particularly, the behavior of the electromagnetic torque under fault condition is analyzed in this effort. Keywords—Mathematical Model, Bearing faults, Single-point defects, Induction Motor 1. INTRODUCTION Induction motors (IM) can undergo different types of faults resulting from operation. Among the faults that can affect IMs, those that occur in bearings are the most frequent and they constituting over 40 % of total faults in IMs (Bellini, et al. 2008; Cabanas, et al. 1998; Drif and Cardoso, 2008; Silva and Cardoso, 2005; Zhou, et al. 2007). The main objective of studying the types of faults that affect IMs is the development of strategies for early detection of them (Bellini, et al. 2008). In this sense, the use of mathematical models constitutes an alternative tool to analyze the effects of faults on the machines (Ong, 1998; Nelles, 2002). A statistic model that describes the motor response to an excitation due to a single-point defect in the motor bearing is presented by Stack, et al. (2006), which uses information from the mechanical vibration of the motor. The model shows similar results to those obtained by Mc Fadden and Smith (1984), in which the effects of a single-point bearing defect is modeled by means of a pulse train combined with a static eccentricity model. Finally, an analytical model is presented by Blodt, et al. (2008) to obtain the expressions for the motor currents in presence of bearing faults. In that work, the authors consider bearing faults according to Mc Fadden and Smith (1984). In this work, a simulation model for the study of single-point bearing faults in IMs is presented and developed. Such model specifically considers the particular case of an outer race single-point defect. However, it can be easily modified to extend the analysis to defects in the inner race and in the cage. The results obtained in this work are mainly meant to be used in the development of strategies for an early detection and diagnosis of bearing faults.
Bearing faults can be classified in different ways. One of these ways consists of considering the size of the area affected by fault. According to such classification criteria, generalized faults are those that affect a significant area of the damaged bearing components. On the other hand, singlepoint damages are those confined to a specific area of some components of the bearing, while the rest of the area remains undamaged (Stack, et al. 2004). From the perspective of diagnosis strategies, single-point defects present the particularity of generates periodic alterations on the motor variables at a specific and predictable frequency. As it is described by Eren, et al. (2004), the frequency associated to each component of the bearing system can be obtained by considering the bearing geometry as well as the relative speed between the fault and the bearing rotating elements translation. Therefore, the equations that describe the frequency associated to each bearing component can be expressed as: n D cos ( β ) Outer race f fpe = f r 1 − B (1) fault frequency 2 DC Inner race fault frequency
f fpi =
n DB cos ( β ) f r 1 + DC 2
(2)
Cage fault frequency
f fc =
1 DB cos ( β ) f r 1 − 2 DC
(3)
Figure 1 shows a ball-bearing with angular contact, indicating also the terms of (1) to (3).
DB : Diameter of ball or roller DC : Mean diameter of the cage
β : Contact angle f r : Rotor speed [Hz] Figure 1: Ball-Bearing with angular contact
2. BEARING FAULTS 978-3-902823-09-0/12/$20.00 © 2012 IFAC
n : Number of balls or rollers
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SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
3. BEARING FAULT MODEL This work proposes a model to evaluate the behavior of an IM with single-point outer race bearing defect. However, the model can be modified to analyze single-point on the inner race as well. In both cases, the effects of fault on the loading area of the bearing for each cycle of the rotor must be considered and to be included in the model (Harris and Kotzalas, 2006).
axial variation due to eccentricity that may be produced in an IM (Bossio, et al. 2004). However, in this work, it is assumed that the system remains unchanged in the axial direction. Through the previous consideration it is possible to eliminate a variable in the calculation of inductances, reducing in this way its complexity. Then, to obtain L nsms , L nrmr and L nsmr , the following expression can be used (Forchetti, et al. 2004): 2π
3. 1. IM Model whit bearing fault
(stator, s, and rotor, r); I i and i represent current and flux vectors in the stator and rotor circuits; and Vs represents stator voltage vector. On the other hand, rotor voltage is considered null due to the rotor short circuit. Fluxes s and r are calculated as follows: s L ss L sr I s = L r rs L rr I r Each sub-matrix in (6) can be defined as Lasbs Lascs Lasas + Lls Lbsbs + Lls Lbscs L ss = Lbsas Lcsas Lcsbs Lcscs + Lls
Lasar L sr = Lbsar Lcsar
Larbr Lbrbr + Llr Lcrbr Lasbr Lbsbr Lcsbr
(10)
0
In general, the behavior of an IM can be described by a mathematical model that combines both, electric and mechanical systems of the machine. The equations that compose each system are described as follows, considering a three-phase IM with a short-circuited winding rotor (Krause, et al. 1996). The electrical system is defined by the following equations: d s Vs = R s I s + (4) dt d R r Ir = − r (5) dt where R i represents the resistance matrix for each circuit
Larar + Llr L rr = Lbrar Lcrar
−1
Lmn (θ r ) = µ 0 r L nm (φ ,θ r ) N n (φ , θ r ) ( g (φ , θ r ) ) dφ
+ Llr
Larcr Lbrcr Lcrcr
Lascr Lbscr = LTrs Lcscr
(6)
where, Lmn (θ r ) is the inductance of the phase n respect to phase m , being both of them of stator, of rotor, or one of them of the rotor and the other of the stator, indistinctly, µ 0 is vacuum permeability; r is the mean radius between the rotor and the stator nm (φ ,θr ) is the windings distribution function
N n (φ ,θ r ) is the modified windings distribution function g −1 (φ , θ r ) is the inverse air-gap function
φ is the angular position measured in stationary reference frame; θr is the time dependent rotor angular position. Once defined these parameters, bearing faults are included into the motor model. The effect of the fault consists in an eccentricity phenomena produced every time any rotating element of bearing contact with the fault; this phenomena is considered according to Mc Fadden and Smith (1984) and Blodt, et al. (2008). To satisfy the above requirement, an air-gap function is proposed in this work. This function takes the form, g e (φ , θ r , t ) = g 0 1 − e0 cos (φ + ψ ( t ) ) (θ r ) (11)
(7)
where, g0 is the mean air-gap length for a motor without fault;
e0 is the relative eccentricity; (8)
(9)
(θ r ) is a function that enables eccentricity, which will be described ahead. ψ ( t ) is a function that determines angular position of the fault respect to a stationary reference frame. By assuming that ω d is the angular frequency of fault respect to a stationary reference frame, then ψ ( t ) can be defined as
In (7) each element L nsms corresponds to inductance between
follows:
ψ ( t ) = ωd t
phases n and m of the stator, being n, m = a, b, c . Similarly, in (8) each element L nrmr corresponds to inductance between phases n and m of the rotor, being L ls and L lr leakage inductances for each stator and rotor phase, respectively, both assumed to be constant. In addition, each element of matrix (9), defined as L nsmr , corresponds to mutual inductance between stator phase n and rotor phase m . For the calculation of the elements of matrixes (7) and (9), it is possible to use a general expression which includes the
(12)
From equation (12), it is possible to distinguish three different cases as described below. Fault in the outer race
In this particular case and due to the fact that the outer race of the bearing is coupled with the machine housing, the displacement speed is null, ωd = 0 , and therefore ψ ( t ) = 0 .
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Under these conditions, (11) can be rewritten as: g e (φ , θ r ) = g 0 1 − e0 cos (φ ) (θ r )
(13)
Figure 2.a illustrates a fault in the outer race of the bearing, assuming such fault arbitrarily located at φ = 0 .
In order for this objective to be achieved, a trapezoidal function that is described by the following expression (infinite series) is proposed, (Hsu, 1973): A TF A ( t ) = ( d1 + d 2 ) + 2 Sg (t ) (19) TF π ( d1 + d 2 ) where ∞
1 cos ( nωF d1 ) − cos ( nωF d2 ) cos ( nωF t ) 2 n n=1
Sg (t ) =
(20)
2π 1 = ,is the fault period obtained from ωF f F equations (1) to (3); A is the amplitude of the function; d1 , and
TF =
a) b) Figure 2: Single-point bearing fault a) Fault in the outer race b) Fault in the inner race.
half the width of the upper base of the trapeze; and d 2 , half the lower base of the trapeze. In addition, (20) can be rewritten using (14), which yields,
Fault in the inner race
S g (θ r ) =
∞
n =1
In standard electrical motors, the inner race of the bearing is integrated with the motor axis. In consequence, a fault in the inner race rotates at the rotor speed, as it is represented in Fig. 2.b. Therefore, if ω d = ω r , then ψ ( t ) = ωr t . Also, it is possible to rewrite ψ ( t ) for a change of variables as follow,
θ r = ωr t
(14)
ψ ( t ) = θr
(15)
and therefore
By replacing (15) in (11), the following air-gap function for fault in the inner race of the bearing can be obtained, g e (φ , θ r ) = g 0 1 − e0 cos (φ + θ r ) (θ r ) . (16) Fault in the cage For faults in the cage ω d = 0.5 ω r . Then, using (15) the air gap function results in, θ g e (φ ,θr ) = g0 1 − e0 cos φ + r (θr ) . (17)
2
Once the expression for the eccentricity produced by the fault has been defined, it becomes necessary to find (θ r ) As mentioned previously, in order for the system model to mimic the behavior of the real system, the static eccentricity must be produced every time a rotating element crosses the single-point defect under consideration. During the remaining time, the air gap function must take the mean value of the air gap function for the motor without fault, that is g0 . This behavior is achieved by using a periodic function (θ r ) whose amplitude in t is defined by (18). k t= ∀ k → integer (18) fF where f F is the fault frequency associated to each element of the bearing, as it is stated in equations (1) to (3). On the other hand, this function must be zero at the remaining time, so that eccentricity effect is eliminated.
ω 1 cos ( nω F d1 ) − cos ( nω F d 2 ) cos n F θ r n2 ω r
(21)
In this way, (19) can be written as a function of the rotor position, A TF A (θ r ) = ( d1 + d 2 ) + 2 S g (θ r ) (22) TF π ( d1 + d 2 ) Under these conditions, (21) and (22) describe a trapezoidal function with the required characteristics previously established. Once the air gap function ge (φ ,θ r ) , was defined, the remaining elements of (10) are described so as to complete the equation. First, the winding distribution function is defined. This function is assumed sinusoidal as for the rotor as for the stator. Therefore, the winding distribution function for phase m of the stator is obtained as follows, N (23) n ms (φ ,θ r ) = s (1 − cos( Pφ + βm ) ) 2 Likewise, the winding distribution function for phase n of the rotor is obtained as follows, n nr (φ , θ r ) =
Nr 1 − cos ( Pφ − θ r ) + β n 2
{
}
(24)
where N s and N r are the number of equivalent windings of the stator and rotor, respectively, whereas P is the number of pairs of poles of the motor. Similarly, β m and β n represents the phase angles corresponding to phases m (stator) and n (rotor), respectively. Then, for n, m = a, b, c , it can be π 4π obtained β a = 0 , β b = and β c = . 3 3 The modified winding distribution function of phase m can be calculated for both the rotor and the stator as, 1 N m (φ , θ r ) = nm (φ , θ r ) − U (φ , θ r ) (25) −1 2 π g (φ , θ r ) where, 2π
U (φ , θ r ) =
n (φ ,θ ) g (φ ,θ ) dφ −1
m
0
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r
r
(26)
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
The expression
g −1 ( φ , θ r )
Table 1: Parameters of the Air-Gap Function
can be defined as the mean
value of the air gap inverse function and it can be determined using the following equation: 2π 1 g −1 (φ ,θ r ) = g −1 (φ ,θ r ) dφ (27) 2 π 0 The next step in the process of developing the mathematical model consists in the modeling of the mechanical system. Regarding the motor mechanical system, for the equilibrium conditions to be satisfied, the following equation must be fulfilled, d ωr J = Te − TL + Bωr (28) dt where, J is the system inertia, Te is the electromagnetic torque, TL is the load torque and B is the friction coefficient. The rotor speed is related to its angular position as follows, dθ r = ωr (29) dt
d2
n
A 8000
80
d1
A
5 × 10
1
−4
From the geometric characteristics of the bearing as well as from the characteristics of motor speed at no-load and full load states defined in Table A1, the fault frequencies of (1) are obtained Outer race fault frequency at no-load state f fpe _ v = 229 [ Hz ] Outer race fault frequency at full load state f fpe _ L = 226 [ Hz ] On the other hand, the eccentricity produced by fault is obtained considering a 1.2 mm-diameter fault, which in turn produces a relative eccentricity of e0 = 0.1015 . To obtain the air gap function ge (φ ,θ r ) that fulfills the
On the other hand, the electromagnetic torque can be defined as a function of the motor co-energy, Wco as
requirements previously established, the values indicated in Table 1 are adopted. Under these conditions, ge (φ ,θ r ) takes
∂W Te = co (30) dθ r Is , Ir =cte where, the derivative of co-energy respect to the rotor angular position is defined as:
the form presented in Fig. 4 for a complete turn of the rotor, as a function of the rotor position for an arbitrary angle φ = 0 . -4
5 g(φ,θr)
∂Wco 1 T ∂L ss ∂L sr 1 ∂L rr = Is I s + ITs I r + ITr Ir ∂θ r ∂θ r ∂θ r ∂θ r 2 2
x 10
(31)
4.8 4.6 0
With the above equations the mathematical model for an IM single-point bearing fault conditions is completed.
1
2
3 4 θr [rad]
5
6
Figure 4: Air gap function for an arbitrary angle φ = 0 , as a
4. MODEL IMPLEMENTATION
function of the rotor position θr .
The mathematical model described in the previous section was simulated using Simulink® from Matlab®. In order for the model implementation to be carried out, the parameters corresponding to a wound rotor three-phase induction motor are used, which were obtained through noload and blocked rotor tests at the lab. Such parameters are presented in Table 1 of the Appendix. Once these parameters are defined, then the motor is simulated at no-fault conditions first. For this simulation, (4) to (10) are used to simulate the electrical system, and (28) to (31) are used to simulate the mechanical system. For the motor without fault, it is assumed that the air-gap length is g e (φ , θ r ) = g 0 (32) In these conditions the motor is simulated with and without fault, considering successively the no-load and full load states. For the fault condition a single-point defect in the outer race is incorporated to the model using in this case parameters of a bearing 6007 of SKF, corresponding to the load side motor bearing.
As it can be observed in Fig. 4, the air gap function takes its maximum value at
θr = k ωr TF
→ ∀k integer
(33)
For the remaining values of θr the function takes the value of the mean air gap g0 . Once defined this function it is used to calculate the rotor and stator inductances, both own and mutual, as well as the mutual inductances between rotor and stator, as shown in (7) to (9). In addition, the derivatives of the above functions necessary to implement (31) are also calculated. Once the model for the motor with fault is defined, simulation is carried out considering the same load conditions than those for the motor without fault. 5. SIMULATION Results obtained from the implementation of the model for the motor with and without fault are presented in this section, considering the motor at no-load and full load conditions in both cases.
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Figure 5 illustrates the self inductance of phase a of stator for the motor healthy and with fault. It is the element [1,1] of (7). 0.125
In this figure can be observed that the mean value of the torque remains unchanged respect to healthy condition, but its dispersion increases considerably.
L [H]
healthy
An analysis through Fourier Transform allows evaluating the frequency content of the variable under analysis. Figures 8 and 9 present results from this frequency analysis of torque under the conditions stated in previous paragraphs.
0.12 0.115
0
1
2
3
4
5
6
θr [rad] 0.125
Particularly, Fig. 8 illustrates the different cases analyzed. The upper sub-figure shows the frequency spectrum for a motor without fault and for the two load conditions considered so far. The intermediate sub-figure illustrates the a frequency spectrum for a motor with fault and under no-load conditions whereas the sub-figure in the bottom analyzes the motor with fault and under full load condition.
L [H]
fault 0.12 0.115
0
1
2
3
4
5
6
θr [rad]
Figure 5: Self-Inductance of stator phase a. From Fig. 5 can be concluded that the considered fault affects inductances of the IM. That is, it produces a periodic perturbation at the fault frequency given by (1) and expressed as a function of the rotor position through (33). On the other hand, Fig. 6 illustrates the derivative of the mutual inductances between phase a of the rotor and a of the stator, for the motor healthy and with fault, represented by element [1,1] of matrix (9).
0.2
0.1 0 healthy
-0.1 1
2
3 θr [rad]
4
5
0
0
200
300 f [Hz]
400
500
600
Fault & no-load
0
2
100
0.1
fault
-0.1 1
0
0.2
0.1
0
Healthy & no-load Healthy & full load
0.1
6
τ [Nm]
0
δL/δθr [H/rad]
From the analysis of both of the spectra of the motors with fault, it can be observed and concluded that an increase in load originates an increase in the amplitude of the fault components. Moreover, due to the speed change produced by the increase in the motor load, a displacement in the fault components frequency is observed. This behavior clearly illustrates the interdependence between fault frequency and the motor speed, as defined in (1). τ [Nm]
δL/δθr [H/rad]
As it can be observed in Fig. 6, the presence of fault produces a periodic perturbation on the derivatives of inductances. These perturbations are present at outer race fault frequency and they occur for those values of θr given by (33).
From the same figure, it can be observed that the presence of fault originates new components in the motor electromagnetic torque spectrum. Such components are not present in motors without fault for any load state.
3
4
5
0
100
200
300 f [Hz]
400
500
600
0.2
6
τ [Nm]
θr [rad]
Figure 6: Derivative of mutual inductance between phase a of the rotor and phase a of the stator. Once the inductances and their derivatives are calculated, a simulation of the whole model is carried out. As a first step, a simulation of the model of the motor at no-load state is carried out for the motor healthy and with fault, successively. The effects of fault over the electromagnetic torque are evaluated and described as follows:
Faulth & full load 0.1 0
0
100
200
300 f [Hz]
400
500
600
Figure 8: Electromagnetic torque spectrum for all considered conditions. Figure 9 shows the spectrum of the motor for fault and noload conditions, indicating in this particular case the frequency for each spectrum component. 0.15
Figure 7 presents the temporal evolution of the electromagnetic torque at full-load condition for a motor healthy and with fault.
k*ffpe
τ [Nm]
τ [Nm]
60 Fault
2*fs
0.1
Healthy
40
ff pe ff pe- 2*fs
k*ff pe- 2*fs
k*ffpe+ 2*fs
ffpe+ 2*fs
0.05 20 0 0
1
2
3
4
5
6
7
0
t [seg]
Figure 7: Electromagnetic torque at full load condition.
0
100
200
300 f [Hz]
400
500
600
Figure 9: Torque frequency spectrum for the motor under fault and no-load condition. 1374
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
The spectrum in the Fig. 9 shows a 100-Hz component; that is, the double supply frequency 2 × fs . In addition, components at frequencies multiple of characteristic fault frequency can be observed in the spectrum; that is k × f fpe . Finally, sidebands associated to each fault frequency component and separated 2 × f s of these can also be observed in the graphic. This particular distribution respond to ( k × f fpe ) ± ( 2 × fs ) , where k is a natural number. 6. CONCLUSIONS The results obtained from the proposed model allow concluding that the presence of fault produces a perturbation on the motor self and mutual inductances as well as on their derivatives. Such perturbation is produced at the characteristic fault frequency of the bearing fault. Likewise, the frequency analysis of torque demonstrates that due to fault, components at multiples of the fault frequency are originated. Around every one this fault components sidebands at double the motor supply frequency are originated. Such behavior is validated by the results presented by Stack, et al. (2006) and Mc Fadden and Smith (1984). On the other hand, an increase in load produces an increase in the amplitude of these components as well a displacement at frequency given by the variation in the motor speed depending on the load state. With the objective of generalizing the model developed in this paper, it is proposed as future work, extending the analysis to other types of faults like single-point faults in the inner race and faults in the cage. ACKNOWLEDGEMENTS
Eren, L. Karahoca, A., Devaney, M.J. (2004). Neural network based motor bearing fault detection. Instrumentation and Measurement Technology Conference 2004. Proceedings of the 21st IEEE. Forchetti, D., Bossio, G., García, G., Valla, M.I. (2004). Modelado de la Máquina de Inducción con Excentricidad del Entrehierro Incluyendo el Efecto de la Ondulación de Par. AADECA 2004. ID #72. (ISBN Nº 950-99994-3-1). Harris, T. A., Kotzalas, M. N. (2006). Essential Concepts of Bearing Technology. Rolling Bearing Analysis. Fifth Edition. Taylor & Francis Group, LLC.USA. Hsu, H.P. (1973). Análisis de Fourier. Fondo Educativo Interamericano. Prentice Hall. Colombia. Krause, P.C., Wasynczuk, O., Sudhoff, S. (1996). Analysis of Electric Machinery. IEEE PRESS. Mc Fadden, P., Smith, J. (1984). Model for the vibration produced by a single point defect in a rolling element bearing. Sound Vibration Journal. Vol. 96, no. 4, pp. 69–82. Nelles, O. (2002). Nonlinear System Identification, 1st edition, Springer Verlag. Ong, C. M. (1998). Dynamic Simulation of Electric Machinery. New Jersey: Prentice Hall. Silva, J.A. Marquez Cardoso, A.J. (2005). Bearing Diagnosis Failure in Three-Phase Induction Motors by Extended Park’s Vector Approach. 31st Annual Conference of IEEE Industrial Electronics Society. IECON 2005. Stack, J. Habetler, T.G., Harley, R.G. (2006). Fault-signature modelling and detection of inner-race bearing faults. Industry Applications, IEEE Transactions on. 42(1): 61-68. Stack, J. Habetler, T.G., Harley, R.G. (2004). Fault classification and fault signature production for rolling element bearings in electric machines. Industry Applications, IEEE Transactions on. 40(3): 735-739. Wei Zhou, Habetler, T.G., Harley, R.G. (2007). Bearing Condition Monitoring Methods for Electric Machines: A General Review. Diagnostics for Electric Machines, Power Electronics and Drives. SDEMPED 2007. IEEE, vol., no., pp.3-6, 6-8 Sept. 2007.
This work was supported by UNRC, CONICET, MinCyTCba, and ANPCyT.
Appendix: PARAMETERS OF THE MODELED MOTOR Table A1: Parameters of the Modeled Motor
REFERENCES
Rate Power
Pn
5.5 [kW]
Bellini A., Filippetti, F., Tassoni, C., Capolino, G.A. (2008). Advances in Diagnostic Techniques for Induction Machines. Industrial Electronics, IEEE Transactions on. 55(12): 41094126. Bossio, G. De Angelo, C. Solsona, J. Garcia, G. Valla, M.I. (2004). A 2-D model of the induction machine: an extension of the modified winding function approach. IEEE Trans. Energy Conversion. vol. 19, no. 1, pp. 144–150. Blodt, M. Granjon, P., Raison, B. Rostaing, G. (2008). Models for Bearing Damage Detection in Induction Motors Using Stator Current Monitoring. Industrial Electronics, IEEE Transactions on. 55(4): 1813-1822. Fernández Cabanas, M., García Melero, M., Alonso Orcajo M. Cano Rodríguez, J. Solares Sariego, J. (1998). Técnicas para el mantenimiento y diagnóstico de máquinas rotativas. ABB service S.A. Marcombo. Barcelona. Drif, M. and Marques Cardoso, A.J. (2008). Air-gap Eccentricity Fault Diagnosis, in Three-Phase Induction Motors by the Complex Apparent Power Signature Analysis. Industrial Electronics, IEEE Transactions on, vol.55, no.3, pp.14041410.
Rate Current
In
11.1 [A]
Rate Phase Voltage
V fn
220 [V]
Speed no-load
2996.4 [rpm]
Speed Full Load
ωv ωL
Supply Frequency
fe
50 [Hz]
Pair of Poles
P
1
Stator Resistance
Rst
0.67 []
Rotor Resistance
Rrot
0.43 []
Magnetization Reactance
Xm
38.17 []
Stator Leakage Reactance
X ls
0.57 []
Rotor Leakage Reactance
X lr
0.57 []
Windings per Phase
N st = N rot
80
Effective axial length Mean Radius Rotor-Stator
L
0.115 [m] 0.075 [m] 0.45 x 10−3 [m]
1375
Mean Air Gap length Inertia
r g0 J
2951.1 [rpm]
0.08 [Kg m2]
A Model for Single-Point Bearings Defects in Electric Motors A. Castellino, C. De Angelo, G. Bossio Grupo de Electrónica Aplicada, Universidad Nacional de Río Cuarto X5804BYA, Argentina. ([email protected], [email protected], [email protected]) Abstract—A mathematical model of an induction motor including single-point bearing defects is proposed in this work. This model allows studying the behavior of different current-based fault diagnosis strategies in front of bearing faults. For this to be achieved, an eccentricity at bearing fault frequency is added to the model of a wound rotor induction motor. Through simulating this model, the effects of an outer race single-point defect on the motor variables are analyzed. Particularly, the behavior of the electromagnetic torque under fault condition is analyzed in this effort. Keywords—Mathematical Model, Bearing faults, Single-point defects, Induction Motor 1. INTRODUCTION Induction motors (IM) can undergo different types of faults resulting from operation. Among the faults that can affect IMs, those that occur in bearings are the most frequent and they constituting over 40 % of total faults in IMs (Bellini, et al. 2008; Cabanas, et al. 1998; Drif and Cardoso, 2008; Silva and Cardoso, 2005; Zhou, et al. 2007). The main objective of studying the types of faults that affect IMs is the development of strategies for early detection of them (Bellini, et al. 2008). In this sense, the use of mathematical models constitutes an alternative tool to analyze the effects of faults on the machines (Ong, 1998; Nelles, 2002). A statistic model that describes the motor response to an excitation due to a single-point defect in the motor bearing is presented by Stack, et al. (2006), which uses information from the mechanical vibration of the motor. The model shows similar results to those obtained by Mc Fadden and Smith (1984), in which the effects of a single-point bearing defect is modeled by means of a pulse train combined with a static eccentricity model. Finally, an analytical model is presented by Blodt, et al. (2008) to obtain the expressions for the motor currents in presence of bearing faults. In that work, the authors consider bearing faults according to Mc Fadden and Smith (1984). In this work, a simulation model for the study of single-point bearing faults in IMs is presented and developed. Such model specifically considers the particular case of an outer race single-point defect. However, it can be easily modified to extend the analysis to defects in the inner race and in the cage. The results obtained in this work are mainly meant to be used in the development of strategies for an early detection and diagnosis of bearing faults.
Bearing faults can be classified in different ways. One of these ways consists of considering the size of the area affected by fault. According to such classification criteria, generalized faults are those that affect a significant area of the damaged bearing components. On the other hand, singlepoint damages are those confined to a specific area of some components of the bearing, while the rest of the area remains undamaged (Stack, et al. 2004). From the perspective of diagnosis strategies, single-point defects present the particularity of generates periodic alterations on the motor variables at a specific and predictable frequency. As it is described by Eren, et al. (2004), the frequency associated to each component of the bearing system can be obtained by considering the bearing geometry as well as the relative speed between the fault and the bearing rotating elements translation. Therefore, the equations that describe the frequency associated to each bearing component can be expressed as: n D cos ( β ) Outer race f fpe = f r 1 − B (1) fault frequency 2 DC Inner race fault frequency
f fpi =
n DB cos ( β ) f r 1 + DC 2
(2)
Cage fault frequency
f fc =
1 DB cos ( β ) f r 1 − 2 DC
(3)
Figure 1 shows a ball-bearing with angular contact, indicating also the terms of (1) to (3).
DB : Diameter of ball or roller DC : Mean diameter of the cage
β : Contact angle f r : Rotor speed [Hz] Figure 1: Ball-Bearing with angular contact
2. BEARING FAULTS 978-3-902823-09-0/12/$20.00 © 2012 IFAC
n : Number of balls or rollers
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3. BEARING FAULT MODEL This work proposes a model to evaluate the behavior of an IM with single-point outer race bearing defect. However, the model can be modified to analyze single-point on the inner race as well. In both cases, the effects of fault on the loading area of the bearing for each cycle of the rotor must be considered and to be included in the model (Harris and Kotzalas, 2006).
axial variation due to eccentricity that may be produced in an IM (Bossio, et al. 2004). However, in this work, it is assumed that the system remains unchanged in the axial direction. Through the previous consideration it is possible to eliminate a variable in the calculation of inductances, reducing in this way its complexity. Then, to obtain L nsms , L nrmr and L nsmr , the following expression can be used (Forchetti, et al. 2004): 2π
3. 1. IM Model whit bearing fault
(stator, s, and rotor, r); I i and i represent current and flux vectors in the stator and rotor circuits; and Vs represents stator voltage vector. On the other hand, rotor voltage is considered null due to the rotor short circuit. Fluxes s and r are calculated as follows: s L ss L sr I s = L r rs L rr I r Each sub-matrix in (6) can be defined as Lasbs Lascs Lasas + Lls Lbsbs + Lls Lbscs L ss = Lbsas Lcsas Lcsbs Lcscs + Lls
Lasar L sr = Lbsar Lcsar
Larbr Lbrbr + Llr Lcrbr Lasbr Lbsbr Lcsbr
(10)
0
In general, the behavior of an IM can be described by a mathematical model that combines both, electric and mechanical systems of the machine. The equations that compose each system are described as follows, considering a three-phase IM with a short-circuited winding rotor (Krause, et al. 1996). The electrical system is defined by the following equations: d s Vs = R s I s + (4) dt d R r Ir = − r (5) dt where R i represents the resistance matrix for each circuit
Larar + Llr L rr = Lbrar Lcrar
−1
Lmn (θ r ) = µ 0 r L nm (φ ,θ r ) N n (φ , θ r ) ( g (φ , θ r ) ) dφ
+ Llr
Larcr Lbrcr Lcrcr
Lascr Lbscr = LTrs Lcscr
(6)
where, Lmn (θ r ) is the inductance of the phase n respect to phase m , being both of them of stator, of rotor, or one of them of the rotor and the other of the stator, indistinctly, µ 0 is vacuum permeability; r is the mean radius between the rotor and the stator nm (φ ,θr ) is the windings distribution function
N n (φ ,θ r ) is the modified windings distribution function g −1 (φ , θ r ) is the inverse air-gap function
φ is the angular position measured in stationary reference frame; θr is the time dependent rotor angular position. Once defined these parameters, bearing faults are included into the motor model. The effect of the fault consists in an eccentricity phenomena produced every time any rotating element of bearing contact with the fault; this phenomena is considered according to Mc Fadden and Smith (1984) and Blodt, et al. (2008). To satisfy the above requirement, an air-gap function is proposed in this work. This function takes the form, g e (φ , θ r , t ) = g 0 1 − e0 cos (φ + ψ ( t ) ) (θ r ) (11)
(7)
where, g0 is the mean air-gap length for a motor without fault;
e0 is the relative eccentricity; (8)
(9)
(θ r ) is a function that enables eccentricity, which will be described ahead. ψ ( t ) is a function that determines angular position of the fault respect to a stationary reference frame. By assuming that ω d is the angular frequency of fault respect to a stationary reference frame, then ψ ( t ) can be defined as
In (7) each element L nsms corresponds to inductance between
follows:
ψ ( t ) = ωd t
phases n and m of the stator, being n, m = a, b, c . Similarly, in (8) each element L nrmr corresponds to inductance between phases n and m of the rotor, being L ls and L lr leakage inductances for each stator and rotor phase, respectively, both assumed to be constant. In addition, each element of matrix (9), defined as L nsmr , corresponds to mutual inductance between stator phase n and rotor phase m . For the calculation of the elements of matrixes (7) and (9), it is possible to use a general expression which includes the
(12)
From equation (12), it is possible to distinguish three different cases as described below. Fault in the outer race
In this particular case and due to the fact that the outer race of the bearing is coupled with the machine housing, the displacement speed is null, ωd = 0 , and therefore ψ ( t ) = 0 .
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Under these conditions, (11) can be rewritten as: g e (φ , θ r ) = g 0 1 − e0 cos (φ ) (θ r )
(13)
Figure 2.a illustrates a fault in the outer race of the bearing, assuming such fault arbitrarily located at φ = 0 .
In order for this objective to be achieved, a trapezoidal function that is described by the following expression (infinite series) is proposed, (Hsu, 1973): A TF A ( t ) = ( d1 + d 2 ) + 2 Sg (t ) (19) TF π ( d1 + d 2 ) where ∞
1 cos ( nωF d1 ) − cos ( nωF d2 ) cos ( nωF t ) 2 n n=1
Sg (t ) =
(20)
2π 1 = ,is the fault period obtained from ωF f F equations (1) to (3); A is the amplitude of the function; d1 , and
TF =
a) b) Figure 2: Single-point bearing fault a) Fault in the outer race b) Fault in the inner race.
half the width of the upper base of the trapeze; and d 2 , half the lower base of the trapeze. In addition, (20) can be rewritten using (14), which yields,
Fault in the inner race
S g (θ r ) =
∞
n =1
In standard electrical motors, the inner race of the bearing is integrated with the motor axis. In consequence, a fault in the inner race rotates at the rotor speed, as it is represented in Fig. 2.b. Therefore, if ω d = ω r , then ψ ( t ) = ωr t . Also, it is possible to rewrite ψ ( t ) for a change of variables as follow,
θ r = ωr t
(14)
ψ ( t ) = θr
(15)
and therefore
By replacing (15) in (11), the following air-gap function for fault in the inner race of the bearing can be obtained, g e (φ , θ r ) = g 0 1 − e0 cos (φ + θ r ) (θ r ) . (16) Fault in the cage For faults in the cage ω d = 0.5 ω r . Then, using (15) the air gap function results in, θ g e (φ ,θr ) = g0 1 − e0 cos φ + r (θr ) . (17)
2
Once the expression for the eccentricity produced by the fault has been defined, it becomes necessary to find (θ r ) As mentioned previously, in order for the system model to mimic the behavior of the real system, the static eccentricity must be produced every time a rotating element crosses the single-point defect under consideration. During the remaining time, the air gap function must take the mean value of the air gap function for the motor without fault, that is g0 . This behavior is achieved by using a periodic function (θ r ) whose amplitude in t is defined by (18). k t= ∀ k → integer (18) fF where f F is the fault frequency associated to each element of the bearing, as it is stated in equations (1) to (3). On the other hand, this function must be zero at the remaining time, so that eccentricity effect is eliminated.
ω 1 cos ( nω F d1 ) − cos ( nω F d 2 ) cos n F θ r n2 ω r
(21)
In this way, (19) can be written as a function of the rotor position, A TF A (θ r ) = ( d1 + d 2 ) + 2 S g (θ r ) (22) TF π ( d1 + d 2 ) Under these conditions, (21) and (22) describe a trapezoidal function with the required characteristics previously established. Once the air gap function ge (φ ,θ r ) , was defined, the remaining elements of (10) are described so as to complete the equation. First, the winding distribution function is defined. This function is assumed sinusoidal as for the rotor as for the stator. Therefore, the winding distribution function for phase m of the stator is obtained as follows, N (23) n ms (φ ,θ r ) = s (1 − cos( Pφ + βm ) ) 2 Likewise, the winding distribution function for phase n of the rotor is obtained as follows, n nr (φ , θ r ) =
Nr 1 − cos ( Pφ − θ r ) + β n 2
{
}
(24)
where N s and N r are the number of equivalent windings of the stator and rotor, respectively, whereas P is the number of pairs of poles of the motor. Similarly, β m and β n represents the phase angles corresponding to phases m (stator) and n (rotor), respectively. Then, for n, m = a, b, c , it can be π 4π obtained β a = 0 , β b = and β c = . 3 3 The modified winding distribution function of phase m can be calculated for both the rotor and the stator as, 1 N m (φ , θ r ) = nm (φ , θ r ) − U (φ , θ r ) (25) −1 2 π g (φ , θ r ) where, 2π
U (φ , θ r ) =
n (φ ,θ ) g (φ ,θ ) dφ −1
m
0
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(26)
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The expression
g −1 ( φ , θ r )
Table 1: Parameters of the Air-Gap Function
can be defined as the mean
value of the air gap inverse function and it can be determined using the following equation: 2π 1 g −1 (φ ,θ r ) = g −1 (φ ,θ r ) dφ (27) 2 π 0 The next step in the process of developing the mathematical model consists in the modeling of the mechanical system. Regarding the motor mechanical system, for the equilibrium conditions to be satisfied, the following equation must be fulfilled, d ωr J = Te − TL + Bωr (28) dt where, J is the system inertia, Te is the electromagnetic torque, TL is the load torque and B is the friction coefficient. The rotor speed is related to its angular position as follows, dθ r = ωr (29) dt
d2
n
A 8000
80
d1
A
5 × 10
1
−4
From the geometric characteristics of the bearing as well as from the characteristics of motor speed at no-load and full load states defined in Table A1, the fault frequencies of (1) are obtained Outer race fault frequency at no-load state f fpe _ v = 229 [ Hz ] Outer race fault frequency at full load state f fpe _ L = 226 [ Hz ] On the other hand, the eccentricity produced by fault is obtained considering a 1.2 mm-diameter fault, which in turn produces a relative eccentricity of e0 = 0.1015 . To obtain the air gap function ge (φ ,θ r ) that fulfills the
On the other hand, the electromagnetic torque can be defined as a function of the motor co-energy, Wco as
requirements previously established, the values indicated in Table 1 are adopted. Under these conditions, ge (φ ,θ r ) takes
∂W Te = co (30) dθ r Is , Ir =cte where, the derivative of co-energy respect to the rotor angular position is defined as:
the form presented in Fig. 4 for a complete turn of the rotor, as a function of the rotor position for an arbitrary angle φ = 0 . -4
5 g(φ,θr)
∂Wco 1 T ∂L ss ∂L sr 1 ∂L rr = Is I s + ITs I r + ITr Ir ∂θ r ∂θ r ∂θ r ∂θ r 2 2
x 10
(31)
4.8 4.6 0
With the above equations the mathematical model for an IM single-point bearing fault conditions is completed.
1
2
3 4 θr [rad]
5
6
Figure 4: Air gap function for an arbitrary angle φ = 0 , as a
4. MODEL IMPLEMENTATION
function of the rotor position θr .
The mathematical model described in the previous section was simulated using Simulink® from Matlab®. In order for the model implementation to be carried out, the parameters corresponding to a wound rotor three-phase induction motor are used, which were obtained through noload and blocked rotor tests at the lab. Such parameters are presented in Table 1 of the Appendix. Once these parameters are defined, then the motor is simulated at no-fault conditions first. For this simulation, (4) to (10) are used to simulate the electrical system, and (28) to (31) are used to simulate the mechanical system. For the motor without fault, it is assumed that the air-gap length is g e (φ , θ r ) = g 0 (32) In these conditions the motor is simulated with and without fault, considering successively the no-load and full load states. For the fault condition a single-point defect in the outer race is incorporated to the model using in this case parameters of a bearing 6007 of SKF, corresponding to the load side motor bearing.
As it can be observed in Fig. 4, the air gap function takes its maximum value at
θr = k ωr TF
→ ∀k integer
(33)
For the remaining values of θr the function takes the value of the mean air gap g0 . Once defined this function it is used to calculate the rotor and stator inductances, both own and mutual, as well as the mutual inductances between rotor and stator, as shown in (7) to (9). In addition, the derivatives of the above functions necessary to implement (31) are also calculated. Once the model for the motor with fault is defined, simulation is carried out considering the same load conditions than those for the motor without fault. 5. SIMULATION Results obtained from the implementation of the model for the motor with and without fault are presented in this section, considering the motor at no-load and full load conditions in both cases.
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Figure 5 illustrates the self inductance of phase a of stator for the motor healthy and with fault. It is the element [1,1] of (7). 0.125
In this figure can be observed that the mean value of the torque remains unchanged respect to healthy condition, but its dispersion increases considerably.
L [H]
healthy
An analysis through Fourier Transform allows evaluating the frequency content of the variable under analysis. Figures 8 and 9 present results from this frequency analysis of torque under the conditions stated in previous paragraphs.
0.12 0.115
0
1
2
3
4
5
6
θr [rad] 0.125
Particularly, Fig. 8 illustrates the different cases analyzed. The upper sub-figure shows the frequency spectrum for a motor without fault and for the two load conditions considered so far. The intermediate sub-figure illustrates the a frequency spectrum for a motor with fault and under no-load conditions whereas the sub-figure in the bottom analyzes the motor with fault and under full load condition.
L [H]
fault 0.12 0.115
0
1
2
3
4
5
6
θr [rad]
Figure 5: Self-Inductance of stator phase a. From Fig. 5 can be concluded that the considered fault affects inductances of the IM. That is, it produces a periodic perturbation at the fault frequency given by (1) and expressed as a function of the rotor position through (33). On the other hand, Fig. 6 illustrates the derivative of the mutual inductances between phase a of the rotor and a of the stator, for the motor healthy and with fault, represented by element [1,1] of matrix (9).
0.2
0.1 0 healthy
-0.1 1
2
3 θr [rad]
4
5
0
0
200
300 f [Hz]
400
500
600
Fault & no-load
0
2
100
0.1
fault
-0.1 1
0
0.2
0.1
0
Healthy & no-load Healthy & full load
0.1
6
τ [Nm]
0
δL/δθr [H/rad]
From the analysis of both of the spectra of the motors with fault, it can be observed and concluded that an increase in load originates an increase in the amplitude of the fault components. Moreover, due to the speed change produced by the increase in the motor load, a displacement in the fault components frequency is observed. This behavior clearly illustrates the interdependence between fault frequency and the motor speed, as defined in (1). τ [Nm]
δL/δθr [H/rad]
As it can be observed in Fig. 6, the presence of fault produces a periodic perturbation on the derivatives of inductances. These perturbations are present at outer race fault frequency and they occur for those values of θr given by (33).
From the same figure, it can be observed that the presence of fault originates new components in the motor electromagnetic torque spectrum. Such components are not present in motors without fault for any load state.
3
4
5
0
100
200
300 f [Hz]
400
500
600
0.2
6
τ [Nm]
θr [rad]
Figure 6: Derivative of mutual inductance between phase a of the rotor and phase a of the stator. Once the inductances and their derivatives are calculated, a simulation of the whole model is carried out. As a first step, a simulation of the model of the motor at no-load state is carried out for the motor healthy and with fault, successively. The effects of fault over the electromagnetic torque are evaluated and described as follows:
Faulth & full load 0.1 0
0
100
200
300 f [Hz]
400
500
600
Figure 8: Electromagnetic torque spectrum for all considered conditions. Figure 9 shows the spectrum of the motor for fault and noload conditions, indicating in this particular case the frequency for each spectrum component. 0.15
Figure 7 presents the temporal evolution of the electromagnetic torque at full-load condition for a motor healthy and with fault.
k*ffpe
τ [Nm]
τ [Nm]
60 Fault
2*fs
0.1
Healthy
40
ff pe ff pe- 2*fs
k*ff pe- 2*fs
k*ffpe+ 2*fs
ffpe+ 2*fs
0.05 20 0 0
1
2
3
4
5
6
7
0
t [seg]
Figure 7: Electromagnetic torque at full load condition.
0
100
200
300 f [Hz]
400
500
600
Figure 9: Torque frequency spectrum for the motor under fault and no-load condition. 1374
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The spectrum in the Fig. 9 shows a 100-Hz component; that is, the double supply frequency 2 × fs . In addition, components at frequencies multiple of characteristic fault frequency can be observed in the spectrum; that is k × f fpe . Finally, sidebands associated to each fault frequency component and separated 2 × f s of these can also be observed in the graphic. This particular distribution respond to ( k × f fpe ) ± ( 2 × fs ) , where k is a natural number. 6. CONCLUSIONS The results obtained from the proposed model allow concluding that the presence of fault produces a perturbation on the motor self and mutual inductances as well as on their derivatives. Such perturbation is produced at the characteristic fault frequency of the bearing fault. Likewise, the frequency analysis of torque demonstrates that due to fault, components at multiples of the fault frequency are originated. Around every one this fault components sidebands at double the motor supply frequency are originated. Such behavior is validated by the results presented by Stack, et al. (2006) and Mc Fadden and Smith (1984). On the other hand, an increase in load produces an increase in the amplitude of these components as well a displacement at frequency given by the variation in the motor speed depending on the load state. With the objective of generalizing the model developed in this paper, it is proposed as future work, extending the analysis to other types of faults like single-point faults in the inner race and faults in the cage. ACKNOWLEDGEMENTS
Eren, L. Karahoca, A., Devaney, M.J. (2004). Neural network based motor bearing fault detection. Instrumentation and Measurement Technology Conference 2004. Proceedings of the 21st IEEE. Forchetti, D., Bossio, G., García, G., Valla, M.I. (2004). Modelado de la Máquina de Inducción con Excentricidad del Entrehierro Incluyendo el Efecto de la Ondulación de Par. AADECA 2004. ID #72. (ISBN Nº 950-99994-3-1). Harris, T. A., Kotzalas, M. N. (2006). Essential Concepts of Bearing Technology. Rolling Bearing Analysis. Fifth Edition. Taylor & Francis Group, LLC.USA. Hsu, H.P. (1973). Análisis de Fourier. Fondo Educativo Interamericano. Prentice Hall. Colombia. Krause, P.C., Wasynczuk, O., Sudhoff, S. (1996). Analysis of Electric Machinery. IEEE PRESS. Mc Fadden, P., Smith, J. (1984). Model for the vibration produced by a single point defect in a rolling element bearing. Sound Vibration Journal. Vol. 96, no. 4, pp. 69–82. Nelles, O. (2002). Nonlinear System Identification, 1st edition, Springer Verlag. Ong, C. M. (1998). Dynamic Simulation of Electric Machinery. New Jersey: Prentice Hall. Silva, J.A. Marquez Cardoso, A.J. (2005). Bearing Diagnosis Failure in Three-Phase Induction Motors by Extended Park’s Vector Approach. 31st Annual Conference of IEEE Industrial Electronics Society. IECON 2005. Stack, J. Habetler, T.G., Harley, R.G. (2006). Fault-signature modelling and detection of inner-race bearing faults. Industry Applications, IEEE Transactions on. 42(1): 61-68. Stack, J. Habetler, T.G., Harley, R.G. (2004). Fault classification and fault signature production for rolling element bearings in electric machines. Industry Applications, IEEE Transactions on. 40(3): 735-739. Wei Zhou, Habetler, T.G., Harley, R.G. (2007). Bearing Condition Monitoring Methods for Electric Machines: A General Review. Diagnostics for Electric Machines, Power Electronics and Drives. SDEMPED 2007. IEEE, vol., no., pp.3-6, 6-8 Sept. 2007.
This work was supported by UNRC, CONICET, MinCyTCba, and ANPCyT.
Appendix: PARAMETERS OF THE MODELED MOTOR Table A1: Parameters of the Modeled Motor
REFERENCES
Rate Power
Pn
5.5 [kW]
Bellini A., Filippetti, F., Tassoni, C., Capolino, G.A. (2008). Advances in Diagnostic Techniques for Induction Machines. Industrial Electronics, IEEE Transactions on. 55(12): 41094126. Bossio, G. De Angelo, C. Solsona, J. Garcia, G. Valla, M.I. (2004). A 2-D model of the induction machine: an extension of the modified winding function approach. IEEE Trans. Energy Conversion. vol. 19, no. 1, pp. 144–150. Blodt, M. Granjon, P., Raison, B. Rostaing, G. (2008). Models for Bearing Damage Detection in Induction Motors Using Stator Current Monitoring. Industrial Electronics, IEEE Transactions on. 55(4): 1813-1822. Fernández Cabanas, M., García Melero, M., Alonso Orcajo M. Cano Rodríguez, J. Solares Sariego, J. (1998). Técnicas para el mantenimiento y diagnóstico de máquinas rotativas. ABB service S.A. Marcombo. Barcelona. Drif, M. and Marques Cardoso, A.J. (2008). Air-gap Eccentricity Fault Diagnosis, in Three-Phase Induction Motors by the Complex Apparent Power Signature Analysis. Industrial Electronics, IEEE Transactions on, vol.55, no.3, pp.14041410.
Rate Current
In
11.1 [A]
Rate Phase Voltage
V fn
220 [V]
Speed no-load
2996.4 [rpm]
Speed Full Load
ωv ωL
Supply Frequency
fe
50 [Hz]
Pair of Poles
P
1
Stator Resistance
Rst
0.67 []
Rotor Resistance
Rrot
0.43 []
Magnetization Reactance
Xm
38.17 []
Stator Leakage Reactance
X ls
0.57 []
Rotor Leakage Reactance
X lr
0.57 []
Windings per Phase
N st = N rot
80
Effective axial length Mean Radius Rotor-Stator
L
0.115 [m] 0.075 [m] 0.45 x 10−3 [m]
1375
Mean Air Gap length Inertia
r g0 J
2951.1 [rpm]
0.08 [Kg m2]