Fault Index Statistical Study for Gear Fault Detection Using Stator Current Space Vector Analysis Shahin Hedayati Kia, Humberto Henao, Senior Member, IEEE, and Gérard-André Capolino, Fellow, IEEE
factor, it is possible to define a frequency bandwidth where main fault-related frequencies are located. This has led to propose a simple algorithm for an efficient GTSDF detection. The real-time data acquisition, processing and analysis for online condition monitoring of gears promise immediate fault detection and hence a fast maintenance action. This can be achieved with digital signal processors (DSPs) and field programmable arrays (FPGAs) which have been widely used in electrical drives for torque, speed and position controls [7]-[12]. For instance, a complete DSPbased fault diagnosis algorithm which covers both fault detection and decision making process stages has been proposed. This last procedure is an important part of the fault diagnosis process which needs statistical analysis on extracted fault features. This leads to define a threshold beyond of which a reliable decision can be made and the state of gear teeth can be evaluated [10]. FPGAs are attractive alternatives which can outperform the fully serialized DSP treatment in order to achieve the algorithm parallelism which yields to drastic minimization of the execution time [11]. In this regard, the stator current entropy information analysis with fuzzy logic detections is realized on a FPGA-based platform in order to diagnose multiple combined faults, namely broken rotor bars, bearings and load unbalance [12]. This paper aims to perform some basic statistical analysis on a normalized fault index which has been proposed for GTSDF study with the goal to define a reliable threshold for fault detection. It has been previously observed that the fault index can be directly influenced by the induction machine load level [13]. Particularly, at light load the defective tooth is not fully in a mesh with the drive gear and the computed fault index in healthy gear becomes adjacent to the faulty one. The proposed method is relied on the SCSVIF analysis which has been implemented on a rugged reconfigurable real-time embedded system. Both pinion and wheel GTSDF have been studied through experiments performed on a setup based on a 250W three-phase squirrel-cage induction machine connected to a single-stage spur or helical gears.
Abstract — This paper presents a statistical analysis on a fault index computed based on the stator current space vector instantaneous frequency on a real-time platform for online non-invasive gear tooth surface damage fault detection in a single-stage gearbox. This analysis is crucial for defining a threshold beyond of which the condition of gear teeth can be considered defective with a significance level less than the critical value. Besides, the efficacy of fault detection algorithm realized on a real-time platform will be evaluated for both pinion and wheel tooth surface damage faults using a set-up based on a 250W three-phase squirrel-cage induction machine shaft-connected to a single-stage spur or helical gearboxes. Index Terms — AC motor protection, Fault diagnosis, FPGA, Gears, Induction motors, Monitoring, Real-time systems, Signal processing.
G
I.
INTRODUCTION
EARS are important parts in the mechanical power transmission for the majority of industrial applications and their online condition monitoring can avoid catastrophic failures and can reduce maintenance costs [1]. The vibration measurement is a well-known tool and has proved its effectiveness for fault detection, diagnosis and prognosis of gears up to now [2], [3]. The main drawbacks of vibration measurements are their sensitivity to sensors positions and the background noise due to external mechanical excitations [4]. Besides, the installation of vibration sensors faces difficulties in those applications in which rotating parts of the system are not accessible or where there are implementation constraints due to limited space and high temperature [5]. By contrast, the machine electrical signature analysis (MESA) uses noninvasive sensors and leads to the fault diagnosis of both electrical and mechanical parts with minimum costs. The stator current space vector instantaneous frequency (SCSVIF) has been recently proposed for gear tooth surface damage fault (GTSDF) detection which is known as a gear tooth localized fault [6]. It has been shown that, with the knowledge of some basic data of eletromechanical system in addition to the torsional natural frequency and the damping
II. GTSDF DETECTION METHOD
Φ Shahin Hedayati Kia, Humberto Henao and Gérard-André Capolino are with the Department of Electrical Engineering, University of Picardie “Jules Verne,” 80039 Amiens Cedex 1, France (e-mails:
[email protected];
[email protected];
[email protected]).
978-1-4799-7743-7/15/$31.00 ©2015 IEEE
It has been shown that localized faults in gears produce large mechanical impacts at the rotation frequency corresponding with the fault location (pinion or wheel side
481
Gearbox
Torque sensor
Induction motor
Torque Real-time display system
Brake control unit
Host PC
fr2 fr1
Digitally-controlled brake
Current sensors Fig. 1. Proposed experimental set-up.
for a one-stage gear) in the mechanical torque [6]. This produces fault-related frequency components ffaulty in the stator current which can be written as: f faulty = f s ± hf fp (1)
The results are used to compute the SCSVIF and subsequently its spectrum by using the DFT.
where h=1,2,3,… and ffp is the fault profile frequency. It should be noted that the most sensitive frequency components in (1) are related to the closest ones of the damped natural frequency for the electromechanical system as it was mentioned [6]. For the set-up shown in Fig. 1, the damped natural frequency fd can be estimated as:
f d = 1 − ς 2 f0
f0 =
1 2π
Kc
(J
m
2
I D [ n] =
3
I Q [ n] =
I sA [ n ] −
1
1 6
1 6
I sC [ n ]
(4)
1
I sC [ n] 2 2 At the next stage, the SCSVIF can be determined if the first order derivative is used [13]. The realization of SCSVIF[n] on a FPGA is affordable since it uses only basic mathematical operations: 1 SCSVIF [ n ] = × 2π
(2)
+ Jp )
Jm J p
Fs
where f0 is the torsional natural frequency, ζ is damping factor, Jm is the driven induction motor rotor inertia, Jp is the load inertia seen from the gearbox pinion side (gearbox, pulley-belt transmission and brake) and Kc is the torsional stiffness of the coupling between them related to the implemented torque sensor (Fig. 1). It was also analytically proved that the stator current space vector instantaneous frequency (SCSVIF) and mechanical torque have similar information concerning the GTSDF for which the fault frequencies fsv-faulty can be obtained as: f sv − faulty = hf fp (3) By knowing the torsional natural frequency of the mechanical system, it is possible to define a frequency bandwidth for some values of h for which the fault-related frequencies are located.
(( I
Q
I sB [ n] −
I sB [ n ] −
[ n] − IQ [ n − 1]) I D [ n] − ( I D [ n] − I D [ n − 1]) IQ [ n]) I D2 [ n ] + I Q2 [ n ]
(5) Since successive fault impacts on the mechanical torque excite the mechanical system torsional vibration, sensitive fault frequencies are located around the equivalent damped natural frequency fd. By knowing the rotor mechanical speed, it is possible to determine a normalized fault index in the frequency domain in order to evaluate the gear fault severity. The fault index FIg gives exactly the energy related to the reconstructed fault profile as it has been shown [6]:
FI g =
2 Nw
L
∑ max ⎛⎜⎝ X ⎡⎣Q ( l ) − M , Q ( l ) + M ⎤⎦
l = L0
fr
2
⎞ ⎟ ⎠
(6)
with ⎛ l × f fp ⎞ 1 Q ( l ) = Round ⎜ ⎟ and Δf = Tacq ⎝ Δf ⎠ where Δf is the spectrum frequency resolution, Tacq is the acquisition time, M is an integer number which defines a
A. Fault Detection Algorithm The fault detection algorithm is depicted in Fig. 2. Initially, three phase currents IsA(t), IsB(t) and IsC(t) are sampled at Fs frequency for N points which can be estimated as a function of the acquisition time in order to obtain enough frequency resolution to detect ffp harmonics.
482
IsA(t) IsB(t) IsC(t)
IsA[n] IsB[n]
SCSVIF [n] SCSVIF computation IsC[n] using (5)
Sampling with Fs for N points
IsA[n]
DFT
Rotor speed estimation
Computation of fault index (6)
Tracking of kffp frequencies in the bandwidth including fd frequency
Decision making process
Fig. 2. GTSDF detection algorithm using SCSVIF analysis.
bandwidth for the spectrum peak tracking, fr is the estimated rotor mechanical speed and L0 and L define the minimum and maximum harmonics of fault-related frequencies given by the formula (3).
The first one means the probability for deciding faulty gear hypothesis if it is being the true one whereas the latter stands for the probability for deciding faulty gear hypothesis if it is being the false one. This can be described by two formulas:
B. Rotor speed estimation It is essential to estimate the rotor mechanical speed with adequate precision in parallel with SCSVIF computation in order to track hffp frequencies in the spectrum which are function of the rotor speed. The rotor space harmonics, which are good candidates for the rotor speed estimation, have been used for hffp frequencies localization in the SCSVIF spectrum [6]. A simplified form for computation of principal slot harmonic is used for rotor speed estimation which can be represented by [14], [15]:
{
+∞
} ∫
PD , FI g (ξ ) = Pr FI g > ξ ; H1 =
{
ξ
+∞
} ∫
PFA, FI g (ξ ) = Pr FI g > ξ ; H 0 =
1 2π σ 1 1
⎛ x − μ1 ⎞ − ⎜⎜ ⎟⎟ ⎝ 2σ 1 ⎠
2
⎛ x − μ0 −⎜⎜ ⎝ 2σ 0
2
e
⎞ ⎟⎟ ⎠
dx (11)
e dx (12) 2π σ 0 where H1 is the faulty gear hypothesis, H0 is the healthy gear hypothesis, ξ is a gear fault detection threshold, μ1, μ0, σ1 and σ0 are mean and standard deviation values deduced from faulty and healthy gears data respectively. Also, the cumulative distribution function (CDF) of the probability density function f(x) can be defined as:
⎛η ⎞ (7) f PSH = ⎜ (1 − s ) + 1⎟ f s p ⎝ ⎠ where fs is the supply frequency, η is the number of rotor slots, s is the slip value and p is the number of induction machine pole pairs. Knowing the η value and the rated slip value of the induction machine sr, it is possible to localize the fPSH in the stator current spectrum in the following frequency bandwidth:
CDF (ξ ) =
ξ
ξ
∫
f ( x)dx
(13)
−∞
The expression (13) can be used to obtain the CDF of detection (CDFD,FIg) and false alarm (CDFFA,FIg): CDFD , FI g (ξ ) = 1 − PD , FI g (ξ ) (14)
CDFFA, FI g (ξ ) = 1 − PFA, FI g (ξ )
⎛η ⎞ ⎛η ⎞ [ ⎜ (1 − sn ) + 1⎟ f s ⎜ + 1⎟ f s ] (8) ⎝p ⎠ ⎝p ⎠ After localization of fPSH in the spectrum, the mechanical rotor speed in Hz can be determined as: 1 f r = ( f PSH − f s ) (9)
(15)
The optimal threshold value ξOPT can be obtained with the criterion which has been proposed [16]: ξOPT = arg max( PD , FI g (ξ ) − PFA, FI g (ξ )) (16) ξ
The above expression can be reformulated by using (14) and (15) as: ξ OPT = arg max(CDFFA, FI g (ξ ) − CDFD , FI g (ξ )) (17)
η
For a known value of supply frequency fs, the speed estimation error based on the spectrum frequency resolution Δf can be determined as: 1 f err = ± Δf (10)
ξ
The computation of the optimal threshold needs data collection for the healthy and worst case faulty conditions at different load levels.
η
It can be noticed that this error is much less than Δf and thus this method is implemented in the algorithm for rotor speed estimation.
III. EXPERIMENTAL RESULTS A. Electromechanical Set-Up A 250W, 50Hz, 400V, star-connected, 0.77A, 4-pole, 1380rpm, 24 rotor bars, three-phase squirrel-cage induction motor with electrical parameters shown in Table I is connected to a digital controllable brake through a one-stage spur or helical gearboxes with a number of teeth at the input (pinion) Nr1=25 and at the output (wheel) Nr2=75 (Fig. 1). fr1
C.
Decision making process Once the fault index FIg is computed a reliable decision need to be made in order to avoid any false alarm and fault missing. This needs the definition of detection and false alarm probability functions PD,FIg and PFA,FIg respectively.
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and fr2 represent the rotation frequencies at input and output stages of the gearbox respectively. A digital controllable brake system can simulate the load by keeping the rotation speed constant at the output stage of the gearbox through a pulley-belt transmission system. The system instrumentation is composed of three current sensors with the same 0.1V/A sensitivity with a frequency bandwidth of [1Hz, 20MHz]. A torque sensor with 5kHz frequency bandwidth is implemented between the induction machine shaft and the input stage of the gearbox for a torsional vibration measurement. The pinion and wheel tooth surface damage faults for spur and helical gears are illustrated in Fig. 3. The on-line condition monitoring system is based on a particular architecture which relies on a real-time processor linked to a reconfigurable FPGA Xilinx Spartan-6 LX45 through a PCI bus for bilateral data transfer. Four analog inputs (AI0 to AI3) are connected to the FPGA through a signal conditioning unit and 24-bit resolution analog-to-digital converters. The real-time processor operates with a realtime operating system which allows executing algorithms with both accurate timing and high reliability. It is principally dedicated to the rotor speed estimation, the localization of fault-related frequencies in the SCSVIF spectrum and the computation of fault index. The reconfigurable FPGA can be used as a co-processor to generate complex signals such as the SCSVIF while the real-time processor performs other signal analysis. The LabVIEW programming environment installed on a host PC has been used for the algorithm development on the realtime processor and the reconfigurable FPGA. The host PC is connected to the real-time system through an Ethernet (a)
(b)
network and it has been mainly used for the implementation of the generated code on the real-time processor and handling user commands. B. Measurements The GTSDF algorithm as it is shown in Fig. 1 needs the knowledge of mechanical equivalent damped natural frequency and the damping factor of the proposed set-up. This can be determined according to the physical properties of concerned elements as shown in Table I. The obtained mechanical resonance is f0 =158.3Hz. Assuming a negligible damping factor for this experimental set-up, the damped natural frequency is fd =158.3Hz. This last data in addition to the induction motor rated-speed and the gearbox ratio are considered as inputs of the proposed algorithm which automatically determines the corresponding fault frequency bandwidth in the SCSVIF spectrum. The faulty condition corresponds with the pinion and wheel tooth surface damage faults on both spur and helical gears. It should be noted that the fault fundamental frequency ffp is related to the pinion tooth surface damage with ffp=fr1 and to the wheel tooth surface damage with ffp=fr2 respectively. Knowing the induction motor rated-speed, the nearest kffp frequency component to the mechanical natural frequency f0 is 161Hz (kffp=7fr1=21fr2). Since the damping factor is assumed small, the frequency bandwidth [18fr2 24fr2] for the fault index (FIg) computation using (6) with L0=18 and L=24 is selected. This last frequency bandwidth covers both pinion and wheel tooth damage faults while the gear ratio is 3. The computation of the fault index (FIg) for both spur and helical gears in real-time have been illustrated (Figs. 4 and 5). In these last experiments, the sampling frequency and the acquisition time were fixed at Fs=2kHz and Tacq=4s respectively in order to achieve in one hand an acceptable error for the first order derivative and in the other hand the sufficient frequency resolution for fault-related frequency localization in the SCSVIF spectrum. The error related to the rotor speed estimation is ferr ≈0.01Hz which is much less than the spectrum frequency resolution Δf =0.25Hz. Also, the effect of load on FIg is studied by varying the load level from the minimum load (20% of rated load) to the rated load
(c)
(d)
Percentage of load
Fig.3. Gear tooth localized faults: a) Helical gear wheel tooth surface damage fault. b) Helical gear pinion tooth surface damage fault. c) Spur gear wheel tooth surface damage fault. d) Spur gear pinion tooth surface damage fault.
0.3
20
40
60
80
100 (a)
0.2 0.1 0 0 0.3
TABLE I FIg
ELECTRICAL AND MECHANICAL PARAMETERS OF THE EXPERIMENTAL SET-UP Induction Machine and Gearbox Rs 40.98Ω Lm 1.025H Lls 0.105H R’r 31.81Ω L’lr 0.105H Jm 0.0006kg.m2 Kc 424Nm/rad Jp 0.0015kg.m2
50
100 Sample 150
200
250
(b)
0.2 0.1 0 0 0.3
50
100 Sample 150
200
250
(c)
0.2 0.1 0 0
50
100
Sample 150
200
250
Fig. 4. Fault index computation for spur gear: a) Pinion tooth surface damage fault. b) Wheel tooth surface damage fault. c) Healthy gear.
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TABLE II STATISTICAL ANALYSIS ON COMPUTED FAULT INDEX (μ: mean value and σ: standard deviation). Gear type Spur Helical Load level (% of rated load) 20 40 60 80 100 20 40 60 0.0950 0.0641 0.0713 0.0581 0.0579 0.0850 0.0733 0.0690 μ Healthy gear 0.0150 0.0113 0.0131 0.0073 0.0062 0.0133 0.0131 0.0135 σ 0.1408 0.2144 0.1729 0.1384 0.1341 0.1581 0.2277 0.1992 μ Pinion tooth surface damage fault 0.0145 0.0202 0.0123 0.0137 0.0167 0.0146 0.0234 0.0154 σ 0.1094 0.1216 0.1600 0.1664 0.1469 0.0806 0.1047 0.1235 μ Wheel tooth surface damage fault 0.0129 0.0127 0.0145 0.0094 0.0101 0.0117 0.0108 0.0143 σ
0.3
40
60
80
100 (a)
0.2 0.1
FIg
0 0 0.3
50
100 Sample 150
200
250
(b)
0.2 0.1 0 0 0.3
50
100 Sample 150
200
250
(c)
0.2
1
(d)
0.1 0 0
100 0.0564 0.0081 0.1318 0.0077 0.1003 0.0093
helical gears are illustrated respectively in Figs. 6.a-6.f and 7.a-7.f. This allows verifying a normal distribution model for collected fault indexes. It should be noted that in both spur and helical gear cases the CDF of wheel tooth surface damage fault is adjacent to the CDF of healthy gear (Figs. 6 and 7) and hence can be considered for threshold definition as it was presented in (17). There are determined with ξOPTSpur=0.101 and ξOPT-Helical=0.086 for spur and helical gears respectively (Figs. 6.g and 7.g). The probability of fault detection when the fault index is greater than ξOPT is higher
Percentage of load
20
80 0.0580 0.0068 0.1568 0.0164 0.1147 0.0099
50
100 Sample 150
200
(f)
(c)
250
0.8
Fig. 5. Fault index computation for helical gear: a) Pinion tooth surface damage fault. b) Wheel tooth surface damage fault. c) Healthy gear. CDF
0.6
for each of which 50 samples are collected. All tests have been performed in the steady-state condition. An initial statistical analysis is performed on the collected data and the results are listed in Table II. It can be observed that the worst case is related to the wheel tooth surface damage fault at minimum load level condition where the wheel faulty tooth is not totally in contact with drive gear for both spur and helical gears. For the load levels greater than minimum load, the mean values of fault index (μ) in faulty conditions are higher than the healthy ones for both spur and helical gear configurations. The highest fault index mean value is related to 40% of rated load as it can be seen in Table II. Besides, this last value decreases once the load level increases which highlight evidently the damping effect of load on the fault index. Moreover, knowing that 99.7% of collected data are within μ±3σ interval, it can be deduced that there is a clear overlap between the fault index probability density functions of healthy and faulty gears. This is the reasons for which it is necessary to define a threshold for the fault index beyond of which a reliable decision needs to be made and the state of gear tooth can be considered defective. To define this last parameter, all previously collected fault indexes at different load levels (250 samples for each gear working condition, i.e. healthy gear, pinion surface damage fault and wheel surface damage fault conditions) are used for computation of cumulative distribution functions (CDFs) which are defined based on probability functions in (14) and (15) and shown for spur and helical gears in Figs. 6 and 7. Both empirical CDF and CDF based on normal distribution (CDF-ND) for healthy, wheel and pinion tooth surface damage faults in spur and
(e)
(g) 0.4
(b)
0.2
(a) 0 0
0.05
0.1
0.15
0.2
0.25
0.3
Fault index [pu] Fig. 6. CDF of fault index for spur gear: a) CDF-ND for healthy gear. b) Empirical CDF for healthy gear. c) CDF-ND for wheel tooth surface damage fault. d) Empirical CDF for wheel tooth surface damage fault. e) CDF-ND for pinion tooth surface damage fault. f) Empirical CDF for pinion tooth surface damage fault. g) ξOPT-Spur estimation using (17). 1
(f)
0.8
(c)
CDF
0.6
(g) 0.4
(b)
(e)
0.2
(a) 0 0
(d) 0.05
0.1
0.15
0.2
0.25
0.3
Fault index [pu] Fig. 7. CDF of fault index for helical gear: a) CDF-ND for healthy gear. b) Empirical CDF for healthy gear. c) CDF-ND for wheel tooth surface damage fault. d) Empirical CDF for wheel tooth surface damage fault. e) CDF-ND for pinion tooth surface damage fault. f) Empirical CDF for pinion tooth surface damage fault. g) ξOPT-Helical estimation using (17).
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for spur gear in comparison with the helical gear since in this last case the CDF of wheel tooth surface damage fault is very close to the healthy CDF. If the maximum value within ξOPT-Spur=0.101 and ξOPT-Helical=0.086 is selected, the false alarm hypothesis H0 will be rejected, i.e. the GTSDF is considered defective, with the 0.0359 level of significance in spur gear (the false alarm probability PFA,FIg(ξ) computed based on healthy spur gear fault index data with ξ=0.101) and the 0.0167 level of significance in helical gear (the detection probability PFA,FIg(ξ) computed based on healthy helical gear fault index data with ξ=0.101) respectively for both pinion and wheel tooth surface damage fault conditions. These last values are less than the classical significance level 0.05 which is commonly used for null hypothesis rejection in standard hypothesis tests [17].
[9]
IV. CONCLUSION
[14]
[10]
[11]
[12]
[13]
In this paper, a statistical analysis is performed on the fault index computed on a real-time platform in order to detect GTSDF with highest sensitivity and reliability. The proposed fault index is based on the energy evaluation of fault-related frequencies which are localized around the main torsional natural frequency of the electromechanical system in the SCSVIF spectrum. In this paper, both pinion and wheel tooth surface damage faults are studied in singlestage spur and helical gears. The FPGA resources were used to compute the SCSVIF whereas the real-time system resources were dedicated to the rest of algorithm computation. The fault index is collected at different load levels in both healthy and GTSDF conditions in order to determine the CDF for spur and helical gears. The results of these last operations are used for estimation of a threshold beyond of which the state of gear tooth is considered defective.
[15]
[16]
[17]
Shahin Hedayati Kia received the M.Sc. in electrical engineering from the Iran University of Science and Technology (IUST), Tehran, Iran, in 1998 and the M.Sc. and the Ph.D. in power electrical engineering from the University of Picardie “Jules Verne”, Amiens, France, in 2005 and 2009, respectively. From 2008 to 2009, he was a lecturer at INSSET de SaintQuentin, France. From September 2009 to September 2011 he was a postdoctoral associate at the School of Electronic and Electrical Engineering of Amiens (ESIEE Amiens). Since September 2011, he is an Assistant Professor at the University of Picardie “Jules Verne,” Amiens, France in the Department of Electrical Engineering.
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Humberto Henao (M’95–SM’05) received the M.Sc. degree in electrical engineering in 1983, the M.Sc. degree in power system planning in 1986, and the Ph.D. degree in electrical engineering in 1990. In 1994, he joined the Ecole Supérieure d’Ingénieurs en Electrotechnique et Electronique, Amiens, France, as an Associate Professor. In 1995, he joined the Department of Electrical Engineering, University of Picardie “Jules Verne,” Amiens, as an Associate Professor, where he has been a Full Professor since 2010. He is currently the Department Representative for international programs and exchanges. He also leads the research activities in the field of condition monitoring and diagnosis for power electrical engineering. His main research interests are modeling, simulation, monitoring, and diagnosis of electrical machines and drives. Gérard-André Capolino (A’77–M’82–SM’89–F’02) was born in Marseille, France. He received the B.Sc. degree in electrical engineering from the Ecole Centrale de Marseille (ECM), Marseille in 1974, the M.Sc. degree from the Ecole Supérieure d’Electricité (Supelec), Paris, France, in 1975, the Ph.D. degree from the Aix-Marseille University (AUM), Marseille, in 1978, and the D.Sc. degree from the Institut Polytechnique de Grenoble (Grenoble INP), Grenoble, France, in 1987. He had several faculty positions in Yaoundé, Cameroun, Le Creusot, France and Marseille, France. In 1994, he joined the University of Picardie “Jules Verne,” Amiens, France, as a Full Professor and was appointed Chair Professor in 2013.
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