Gear Tooth Surface Damage Fault Detection Using ... - IEEE Xplore

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 3, MARCH 2015

Gear Tooth Surface Damage Fault Detection Using Induction Machine Stator Current Space Vector Analysis Shahin Hedayati Kia, Humberto Henao, Senior Member, IEEE, and Gérard-André Capolino, Fellow, IEEE Abstract—A noninvasive technique for the diagnosis of gear tooth surface damage faults based upon the stator current space vector analysis is presented. The torque oscillation profile produced by the gear tooth surface damage fault in the mechanical torque experimented by the driven electrical machine is primarily investigated. This profile consists of a mechanical impact generated by the fault followed by a damped oscillation that can be identified through the mechanical system torsional natural frequency and damping factor. Through theoretical developments, it is shown that the periodic behavior of this particular shape produces faultrelated frequencies in the stator current and harmonics integer multiple of the rotation frequency in the stator current space vector instantaneous frequency. The fault signature related to the gear tooth surface damage fault is predicted through the numerical simulation. The simulation results are validated through experimental tests, illustrating a possible noninvasive gear tooth surface damage fault detection with a fault sensitivity comparable to invasive methods. A dedicated experimental setup, which is based on a 250-W squirrel-cage three-phase induction machine that is shaft connected to a single-stage gear, has been used for this purpose. Index Terms—AC motor protection, fault diagnosis, gearbox, induction machine, monitoring, motor current signature analysis, signal processing, space vector, stator current analysis, vibration measurement.

N OMENCLATURE TLH (t), TLF (t) Tfp (t) Te (t), TL (t)

TKc (t) χLH (t), χfp (t) ffp

Load torque in healthy and faulty conditions. Periodic torque component with zero mean produced by gear tooth localized fault. Induction machine electromagnetic torque and load torque seen from the gearbox pinion side. Load torque measured by torque sensor. Phase modulation components related to healthy and faulty conditions. Fundamental frequency of fault profile related to pinion or wheel rotation frequencies for a one-stage gearbox.

Manuscript received February 14, 2014; revised June 30, 2014; accepted July 31, 2014. Date of publication September 24, 2014; date of current version February 6, 2015. The authors are with the Department of Electrical Engineering, University of Picardie Jules Verne, 80039 Amiens, France (e-mail: Shahin. [email protected]; [email protected]; Gerard. [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2014.2360068

f0 , fd ς fr1 , fr2 fr ωs Ωm (t), Ω(t) p Nw M L0 , L F Ig

Torsional natural and damped natural frequencies. Damping factor. Input and output rotation frequencies. Rotor angular frequency. Stator fundamental angular frequency. Induction machine and pinion side mechanical rotation speeds. Number of pole pairs. Length of data used for gear fault index computation. Integer number that defines a bandwidth for spectrum peak tacking. Minimum and maximum harmonics of sensible fault frequencies given by fd and ζ. Gear fault index. I. I NTRODUCTION

O

NLINE condition monitoring of electromechanical systems is considered as a key task for many industrial applications. This leads to early detection of incipient faults before occurrence of serious failures. The vibration signal is representative of the behavior of periodic events in the mechanical system. This behavior will be changed in case of any kind of mechanical abnormality [1]. This is why the vibration signal analysis is the most popular up to this time, which is known as an efficient method for mechanical fault detection, and has taken an important place in the preventive maintenance planning of electromechanical systems. Gears, which are known as key elements in the mechanical power transmission, have received significant attention in the field of condition monitoring for years [2]. The common gear faults are related to gear tooth irregularities, i.e., the chipped tooth, tooth breakage, root crack, spalling, wear, pitting, and surface damage, which are typical localized faults [3]. When such faults occur, extra mechanical impacts are generated at the rotational frequency in the vibration signal. The shape of mechanical impacts is related to the mechanical structure resonance excited by the tooth localized fault when the damaged tooth is engaged. This results in a wide frequency distribution in the vibration spectrum [4]. Despite the broad usage of vibration measurements for condition monitoring and fault detection of gears, they have numerous drawbacks, including the high cost, the inaccessibility in mounting the vibration transducers, and the sensitivity to the installation location [5]. Furthermore, the transducers such as accelerometers measure the mechanical transverse vibration,

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KIA et al.: GEAR TOOTH SURFACE DAMAGE FAULT DETECTION BY STATOR CURRENT SPACE VECTOR ANALYSIS

whereas the gear tooth localized faults have a direct influence on the shaft torsional vibration [6]. The gear fault detection based upon the machine electrical signature analysis offers significant advantages over the vibration analysis due to its effective cost and the requirement of minimum changes in the system installation. Early works on this specific topic have reported only the influence of mechanical anomalies as the load fluctuation on the stator current spectrum [7]–[9]. The input and output rotation frequencies of a one-stage gearbox are initially detected around the supply frequency in the stator current spectrum [8]. Afterward, an effort has been performed for tooth breakage fault diagnosis in a multistage gearbox through the amplitude evaluation of the gear mechanical characteristic frequency components, explicitly the rotations, mesh, and mesh-related frequencies, which are localized across the supply frequency in the stator current spectrum [5], [10], [11]. The rotation frequency components are easily identified using the classical Fourier transform, whereas the discrete wavelet transform was suggested to remove the noise, hence resulting in better identification of the mesh and mesh-related frequency components, which are commonly localized at the higher frequency band of the stator current spectrum. It was shown that the amplitudes of some of them are sensitive to the gear tooth breakage fault in the stator current spectrum. Nevertheless, the reasons for such increases or decreases in amplitude are still ambiguous. It is also mentioned that the severity of vibration is higher in the case of one-tooth breakage fault than in a tooth breakage fault with two or more teeth, inducing to an important conclusion that the gear tooth fault detection may be much easier at an early stage [5]. A rigorous and thoughtful attempt was carried out with enough theoretical backgrounds highlighting sufficient reasons for which the gear mechanical characteristic frequency components can be detected in the stator current spectrum [12]. It relies on the observation of the mechanical torque spectrum experimented by the induction machine and the concept of the stator current phase modulation due to load torque oscillations [13]. These former theoretical developments have been successfully validated through a modeling approach by using a simple dynamic model that considers a realistic behavior of gears, i.e., including in the model the gear transmission errors and pinion and wheel eccentricities, which requires only a minimum number of gear parameters [14]. Moreover, the induction machine is used as a torque sensor through the estimation of electromagnetic torque for condition monitoring and fault diagnosis of gears in electromechanical systems [14], [15]. The effectiveness of this last technique has been demonstrated in the case of loss of lubrication fault in a gear-based motor drive [16], gear surface wear fault in a reduced-scale railway traction system [17], and tooth breakage fault in a high-ratio gear in cement kiln drives [18]. A 3-D dynamic model of spur and helical gears, including both shafts and bearings, have been rigidly coupled to the induction machine model investigating a feasible noninvasive detection of gear tooth faults, specifically tooth spalls and pits [19], [20]. The simulation results depict the presence of a periodic signature in the d-axis current component. However, the proposed technique has not yet been experimentally verified.

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This work aims at introducing the new idea of the fault signature analysis for noninvasive gear fault detection. The principal motivation is to study the effect of gear tooth localized fault at an early stage, in the stator current, according to its signature in the torque experimented by the driven electrical machine. Indeed, the gear localized fault may not necessary influence the amplitude of mesh and mesh-related frequency components in the stator current, which was the main assumption for noninvasive gear tooth fault detection in the previous works [5], [10], [11], [21]. It will be illustrated that, in such a case, a fault profile appears in the mechanical torque due to the occurrence of any tooth localized faults in gears. This profile consists of an initial mechanical impact generated by the gear tooth surface damage fault followed by a damped oscillation that can be identified through the mechanical system torsional natural frequency and damping factor, which is essentially related to the impulse response of the mechanical system to the fault mechanical impact [22]. This shape is periodic with a specific time interval related to the rotation frequency corresponding to the fault location in a gear (pinion or wheel side for a onestage gear). Since any periodic function can be represented by its Fourier series, it is possible to show that each particular fault signature produces fault-related frequency components with amplitudes corresponding to the Fourier series coefficients in the stator current spectrum and hence harmonics, explicitly integer multiple of rotation frequencies in the stator current space vector instantaneous frequency (SCSVIF) spectrum. The stator current space vector has been widely used for mechanical and electrical fault detection in induction machines [23]. In this regard, the synchronous signal averaging method was applied to the instantaneous amplitude of the stator current space vector for gear tooth fault detection [24]. The main disadvantage of this method is the requirement of an extra sensor for the measurement of the shaft angular position. In [25], it has been shown that the analytical decomposition of the stator current can be achieved through the stator current space vector. This technique has been used only for classical load torque oscillation fault diagnosis, mainly by incorporating the synthetic and experimental signal analysis. The SCSVIF has been recently employed for fault profile extraction in case of gear tooth localized fault [26], [27]. The proposed method uses three stator phase currents, instead of torque or accelerometer sensors, for gear fault diagnosis. It requires only three ac current sensors for measuring three stator phase currents for the computation of the SCSVIF. This property leads to early-stage noninvasive gear fault detection, which has been rarely investigated. In this paper, the original analysis in [26] and [27] are consolidated, in order to better understand the phenomenon, with aims at offering a concrete diagnosis technique for noninvasive gear tooth surface damage fault detection. Hence, it will be shown through necessary mathematical developments that the SCSVIF contains information similar to the mechanical torque experimented by the driven electrical machine. Furthermore, the fault profile in SCSVIF is predicted through the numerical simulation, by applying the extracted fault signature from the measured torque to the classical d−q model of the induction machine. The simulation results are validated through

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experiments, showing possible noninvasive gear tooth surface damage fault detection. The effect of load and inertia is analyzed as well. Knowing the mechanical system torsional natural frequency and damping factor, it is possible to determine the frequency bandwidth, where the main fault-related harmonics are located. This leads to propose a simple algorithm for gear tooth surface damage fault detection, which makes its implementation on a real-time platform possible, for online condition monitoring of gear-based electromechanical systems. A prototype test setup based on a 250-W three-phase squirrelcage induction machine connected to a single-stage helical gearbox has been used for experiments. The new approach can be applied to several industrial applications, mainly to railway traction systems and wind turbines, which have mechanical transmission models that are very similar to what is used as prototype in this work [15], [28].

A. Machine Current Signature Analysis

TLH (t) = T0 + Tosc (t)

(1)

where T0 is the average load torque, and Tosc (t) is the load torque oscillation related to both healthy gear pinion and wheel rotations plus mesh and mesh-related frequency components, which has been previously detailed [12]. The fault mechanical impact generates an extra periodic component in addition to the healthy condition. Therefore, the load torque in faulty condition TLF (t) can be formulated as TLF (t) = TLH (t) + Tfp (t)

(2)

where Tfp (t) is a periodic component with a zero mean value produced by the gear tooth localized fault. Since any periodical signal can be represented by its Fourier series, the Tfp (t) term can be given by Tk cos(kωfp t + ϕk ),

+∞

χk cos(kωfp t + αk ),

χk ∝

k=1

pTk (5) Jeq (kωfp )2

where αk is the phase related to χfp (t), Jeq is the induction machine side equivalent inertia, and p is the number of induction machine pole pairs. Assuming that sin(x) ≈ x and cos(x) ≈ 1 for x 1 since χLH (t) and χfp (t) have small amplitudes, the stator phase current in (4) can be simplified as IA (t) ≈ Ihealthy (t) + Ir cos(ωs t + ηr )χfp (t)

(6)

with

The stator current can be rewritten using (5) as

The tooth localized faults, in contrast to tooth distributed faults, in gears produce large mechanical impacts at the rotation frequency periodicity corresponding with the fault location in the vibration signal, which is mainly due to the transmission of the mechanical torque [17], [18]. Based on previously experimental studies, the load torque observed by an induction machine coupled to a healthy gear (TLH (t)) consists of a mean value and a time-varying part, which can be written as follows [12]:

+∞

χfp (t) =

Ihealthy (t) ≈ Is sin(ωs t) + Ir sin(ωs t + ηr ) + Ir cos(ωs t + ηr )χLH (t).

II. T HEORETICAL B ACKGROUND

Tfp (t) =

phase modulation components related to the normal and fault profiles, respectively. This last phase modulation part can be defined as [12]

ωfp = 2πffp

(3)

k=1

where ffp is the fundamental frequency of the fault profile (related to pinion or wheel rotation frequencies for a one-stage gear), k is an integer, and Tk and ϕk can be obtained from the fault profile Fourier series coefficients. Therefore, the stator phase current can be written as IA (t) ≈ Is sin(ωs t) + Ir sin (ωs t + χLH (t) + χfp (t) + ηr) (4) where ωs = 2πfs is the supply frequency; ηr is the phase related to the phase modulation term; χLH (t) and χfp (t) are the

IA (t) ≈ Ihealthy (t) + Ir + Ir

+∞ χk

k=1 +∞ k=1

2

cos ((ωs − kωfp )t + ϕl,k )

χk cos ((ωs + kωfp )t + ϕr,k ) 2

(7)

with ϕl,k = ηr − αk

ϕr,k = ηr + αk .

Thus, fault frequency components in the stator current can be formulated as ffaulty = |fs ± mffp |

(8)

with m = 1, 2, 3, . . .. B. Simplified Mechanical Model In order to understand the way to localize the mechanical impact generated by tooth surface damage fault in the spectrum, a simplified representation of the mechanical test rig that will be used for experimental tests is shown in Fig. 1 [26], [27]. This mechanical model essentially represents the driven induction motor rotor inertia Jm , the load inertia seen from the gearbox pinion side (gearbox, pulley–belt transmission, and brake) Jp , and the torsional stiffness Kc of the coupling between them related to the torque sensor (see Fig. 4). In Fig. 1, Te (t) is the induction machine electromagnetic torque; TKc (t) is the load torque applied to the induction machine shaft and measured by the implemented torque sensor; TL (t) is the load torque seen from the gearbox pinion side; Ωm (t) and Ω(t) are the induction machine and pinion side mechanical rotation speeds, respectively. The differential equation for this mechanical system can be written as follows. • For the induction machine, Te (t) − TKc (t) = Jm Ω˙ m (t).

(9)


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a mechanical system with small damping factor, the resulting spectrum of this phenomenon will be given by a fundamental component at a multiple frequency of ffp nearest to fd , with some sideband harmonics of ffp . Then, the integer k defined in (3) will be finally restricted to some values, which depend on the equivalent damped natural frequency and damping factor. Fig. 1.

Simplified representation of the mechanical system.

C. SCSVIF

• For the gearbox pinion side ˙ TKc (t) − TL (t) = Jp Ω(t).

(10)

• For the coupling between induction machine and gearbox, T˙Kc (t) = Kc (Ωm (t) − Ω(t)) .

(11)

Expressions (9)–(11) can be reformulated as 1 1 Kc Te (t) Kc TL (t) T¨Kc (t)+TKc (t)Kc + + . (12) = Jm Jp Jm Jp In case of gear tooth surface damage fault and assuming a constant electromagnetic torque Te (t) [12], the mechanical impact will be generated in the load torque TL (t), and hence, a fault profile will be produced in TKc (t) due to the mechanical resonance at the torsional natural frequency of the mechanical system ω0 , which can be determined as (Jm + Jp ) ω0 = 2πf0 = Kc . (13) Jm Jp The inertias and torsional stiffness element described in Fig. 1 can be determined based upon the information concerning its material type and dimensions. It should be noted that, in (9)–(11), the damping factor is neglected since this value is small for a mechanical system with steel couplings and rigid shafts [22]. In practice, any mechanical structure holds a certain amount of damping, which can limit the magnitude of torsional vibration [29]. Nevertheless, there is no straightforward procedure for the determination of the mechanical system damping factor since it can be associated to variety of sources, such as internal friction in the rotating elements (e.g., bearings), fluid frictions (e.g., lubrication in gear systems), hysteresis in shafting and between fitted parts, etc. [29], [30]. This last factor can be derived empirically from field measurements based upon the applied forcing torques to the mechanical system as it is mentioned [22], [29]. Considering the damping factor of the mechanical system, the fault profile can be represented as Tfp (t) = TKc e−ςω0 t sin(2πfd t) 2πfd = 1 − ς 2 ω0

(14)

where TKc is the initial torque amplitude related to the fault profile impact, fd is the damped natural frequency, and ζ is the damping factor. If this impact is repetitive at the fault profile frequency ffp defined in (3), it can be demonstrated that, for

The space vector analysis has been widely used for detection of faults such as the opened wound rotor windings, the airgap eccentricity, the interturn winding short-circuits, the rotor cage faults, the unbalance in the supply voltage, mechanical load misalignments, and the driven converter power switch faults in electromechanical systems [23]. The current space-vector components ID (t) and IQ (t) can be determined as a function of stator current phase variables isA (t), isB (t), and isC (t) as follows: √ 2 1 1 ID (t) = √ isA (t) − √ isB (t) − √ isC (t) 3 6 6 1 1 IQ (t) = √ isB (t) − √ isC (t) 2 2 I¯sv (t) = ID (t) + jIQ (t). (15) Using (4) as the phase A and ±120◦ phase shift for IB (t) and IC (t), expression (15) can be rewritten as 3 {Is sin(ωs t) ID (t) = 2 + Ir sin (ωs t + χLH (t) + χfp (t) + ηr )} 3 {Is cos(ωs t) IQ (t) = − 2 + Ir cos (ωs t + χLH (t) + χfp (t) + ηr )} . (16) The instantaneous phase of the stator current space vector can be written as IQ (t) φsv (t) = arctan . (17) ID (t) The SCSVIF is then given by IF¯isv (t) =

I˙Q ID − I˙D IQ 1 ˙ 1 × . φsv (t) = 2 + I2 2π 2π IQ D

(18)

Using (16), the aforementioned relation can be simplified and rewritten as 1 2π I 2 + Is Ir cos (χfp (t) + χLH (t) + ηr ) × 2 r 2 Is + Ir + 2Is Ir cos (χfp (t) + χLH (t) + ηr ) d [χfp (t) + χLH (t)] . (19) × dt

IF¯isv (t) = fs +

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Fig. 2.


Reconstruction of fault profile in the time domain.

Assuming that cos(χfp (t) + χLH (t) + ηr ) ≈ 1 since χfp (t) and χLH (t) are commonly small, the SCSVIF can be approximated as C d · [χLH (t)] 2π dt +∞ C − χk sin(kωfp t + αk ) 2π

IF¯isv (t) ≈ fs +

k=1

pTk χk ∝ Jeq kωfp

C=

Ir . Is + Ir

(20)

The fault-related frequencies in the stator current are scattered in a wide frequency range, which makes their tracking difficult, as it is shown in (8). The expression (20) shows that frequency components of both SCSVIF and mechanical torque are similar with the same restriction imposed by the equivalent damped natural frequency and damping factor described before. Moreover, the SCSVIF expression and the instantaneous frequency of the stator phase current using the Hilbert transform are exactly the same, but the first one can be achieved without considering the Bedrosian theorem, as it was stated in [25]. Thus, SCSVIF can be used to simplify the fault diagnosis process.

Fig. 3.

Algorithm of fault detection using SCSVIF analysis. TABLE I E LECTRICAL AND M ECHANICAL PARAMETERS OF THE E XPERIMENTAL S ETUP

D. Fault Profile Reconstruction It is possible to reconstruct the periodic fault profile due to a faulty gear in the time domain through a simple analysis of the spectrum of the mechanical torque or of the SCSVIF (see Fig. 2). This technique initially computes the discrete Fourier transform (DFT) of the used signal (measured torque or SCSVIF). Then, the obtained spectrum is multiplied by a vector with zeros at all elements, except for kffp frequencies (elements equal to 1) which are of interest. The computation of the inverse Fourier transform of this multiplication reconstructs the fault profile in the time domain. In practice, some of the kffp frequency components exist even in healthy condition. Therefore, the reconstruction process can be realized with kffp frequency components, which have great sensitivity to the fault. It is also obvious that the input should be a stationary signal, in order to obtain correct results. This technique is used to investigate the degree of similarity between the fault profiles extracted from mechanical torque and SCSVIF. The energy of the reconstructed fault profile could be a good indicator for fault detection, as it was mentioned in [26] and [27]. The Parseval theorem states the energy conservation property between time and frequency domains established by [31] E=

Nw n=1

|x[n]|2 =

Nw 1 |X[n]|2 Nw n=1

(21)

where Nw is the number of data points in a window, x[·] is the reconstructed fault profile, and X[·] is the DFT coefficients selected for the reconstruction process x[·]. The energy of the reconstructed signal can thus be determined from the measured load torque or the SCSVIF in the frequency domain, according to their DFTs. E. Fault Detection Algorithm The fault detection algorithm is depicted in Fig. 3. Initially, the 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. The results are used to compute the SCSVIF using (15) and, subsequently, its spectrum using DFT. It is also essential to estimate the rotor speed in parallel, in order to localize kffp frequencies in the spectrum as a function of the rotor speed. The fault sensitive frequency component selection is the key part of this algorithm since kffp components may exist for some initial values of k in healthy condition due to the inherent rotor eccentricity of the induction machine as well as the gear pinion and wheel eccentricities [Tosc (t) term in the expression (1)]. The proposed method relies


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Fig. 4. (a) Schematic of the proposed setup. (b) Pinion–wheel contact at tooth surface damage point. (c) Wheel tooth surface damage fault. (d) Pinion tooth surface damage fault.

on the selection of those frequency components, which appear after occurrence of fault. Since the successive fault impacts on the mechanical torque excite the mechanical system torsional vibration, the sensitive fault harmonics are located around the equivalent damped natural frequency and damping factor, as suggested by the response of the simplified mechanical model. At the final stage, the normalized squared energy is computed as a fault index F Ig in the frequency domain using expression (22), allowing evaluation of the gear fault severity. Hence

L 1 max |X [Q(l)−M, Q(l)+M ]|2 Nw F Ig =

l=L0

(22)

fr

with

Q(l) = Round

l × ffp Δf

and

Δf =

1 Tacq

where Nw is the data length; Δf is the spectrum frequency resolution; Tacq is the acquisition time; M is an integer number, which defines a bandwidth for spectrum peak tracking; fr is the estimated rotor speed; and L0 and L define the minimum and maximum harmonics of sensible fault frequencies given by the equivalent damped natural frequency and damping factor. III. S IMULATION AND E XPERIMENTAL R ESULTS A. Electromechanical Setup A 250-W, 50-Hz, 400-V, star-connected, 0.77-A, four-pole, 1380-r/min, 24-rotor-bar, and three-phase squirrel-cage induction motor with electrical parameters shown in Table I is connected to a digital controllable brake through a one-stage helical

gear with a number of teeth at the input Nr1 = 25 and at the output Nr2 = 75 (see Fig. 4). fr1 and fr2 represent the rotation frequencies at input and output stages of the gear, respectively. The digital controllable brake system can simulate the load by keeping the rotation speed constant at the output stage of the gear through a pulley–belt transmission system. The system instrumentation consists of three commercial wideband current transformers with the same sensitivity of 0.1 V/A and frequency bandwidth of [1 Hz, 20 MHz] and three voltage sensors with the same 1/200 transformation ratio and 40-MHz frequency bandwidth. In addition, an accelerometer with 500-mV/g sensitivity and 22-kHz frequency bandwidth is installed near the input stage, in order to include only one sensor for the mechanical transversal vibration measurement. In addition, a torque sensor with 5-kHz frequency bandwidth is implemented between the induction machine shaft and the input stage of the gear for torsional vibration measurement. For data collection, a 24-bit resolution modular data acquisition system with builtin adjustable signal conditioning filters has been used. The acquisition is performed with sampling frequency Fs = 5 kHz, and the acquisition time is Tacq = 10 s for all signals, in order to observe only remarkable frequency components for both electrical and mechanical signals and to achieve a frequency resolution of Δf = 0.1 Hz to localize fault sensitive rotational harmonics at different load levels. The acquisition is performed in a stationary condition, in order to obtain a valid fault profile reconstruction. Several tests have been carried out on the setup in both healthy and faulty conditions at five levels of load and two levels of inertia, respectively, in order to study their effects on the energy criterion. Each test is repeated at least ten times, in order to evaluate the reliability and repeatability of the proposed algorithm. For the faulty condition, four groups of tests have been done, including the pinion, the wheel, and

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the simultaneous synchronous and asynchronous pinion–wheel gear tooth surface damage faults, in order to study the effect of each particular fault on the vibration, the measured torque, and the SCSVIF. The position of the pinion damaged tooth is synchronized to the wheel damaged tooth for the simultaneous pinion–wheel gear tooth surface damage fault condition. A delay of eight teeth between the wheel damaged tooth and the pinion damaged tooth is considered for the simultaneous asynchronous pinion–wheel gear tooth surface damage fault. In these previous tests, each faulty pinion and wheel includes only one tooth with entirely damaged surface, with depths of about 0.3 mm, in comparison with the healthy gear (see Fig. 4). The last test is related to the healthy condition. The P-faulty, W-faulty, SSPW-faulty, and SAPW-faulty gears mentioned later are related to the pinion, the wheel, and the simultaneous synchronous and asynchronous pinion–wheel tooth surface damage faults, respectively. The proposed fault detection algorithm is based on the knowledge of the mechanical equivalent damped natural frequency and damping factor of the used setup. Then, the mechanical parameters used in expression (13) were computed, according to physical properties of concerned elements, and are shown in Table I. The obtained mechanical resonance is then f0 = 158.3 Hz. Assuming a negligible damping factor for this experimental setup, the damped natural frequency is fd = 158.3 Hz. B. Vibration Versus Mechanical Torque Initially, the correlation between vibration and torque measurements is investigated, and then, the results of experimental tests for the induction machine working close to the rated load are illustrated here. The rotation frequencies in this last case are fr1 = 23.1 Hz and fr2 = 7.7 Hz. The fault profile frequencies are related to ffp−p = fr1 for the pinion; ffp−w = fr2 for the wheel; and ffp−sspw = fr2 and ffp−sapw = fr2 for the simultaneous synchronous and asynchronous gear tooth surface damage faults, respectively. The fundamental frequency of the simultaneous pinion–wheel gear tooth damage faults is the same compared with the wheel gear tooth fault frequency since the gear speed ratio (fr1 /fr2 ) is an integer number equal to 3. The fault signatures are clearly identified in analyzed signals (see Figs. 5 and 6) with 1/fr1 , 1/fr2 , and 1/fr1 time periods corresponding to P-faulty, W-faulty, SSPWfaulty conditions. In the SAPW-faulty case, the pinion-related signatures with 1/fr1 time period and the wheel-related fault signatures with 1/fr2 time period are identified as well. The measured mechanical impacts of fault in P-faulty, SSPW-faulty, and SAPW-faulty cases [see Fig. 5(b), (d), and (e)] are greater than that of the W-faulty condition since the accelerometer is close to the input stage of the gear [see Fig. 5(c)]. These last signatures can be also observed in the ac part of measured torque (see Fig. 6). The measured torque of the healthy gear includes a mean value plus several oscillations, as it is shown by the expression (1). It is mostly dominated by the torsional vibration induced by the gear characteristic and the mechanical system structure frequencies [12], [14]. In this particular case, an additional oscillation is observed at 100 Hz in the measured torque of the healthy gear. It is verified that this frequency

Fig. 5. Vibration in steady-state working condition at rated load: (a) healthy gear; (b) P-faulty gear; (c) W-faulty gear; (d) SSPW-faulty gear; (e) SAPW-faulty gear.

component is related to pulley–belt transmission element by removing the belt from the test rig [see Fig. 6(a)]. In addition to these last oscillations, a periodic profile Tfp (t) appears in the measured torque, in the case of tooth surface damage fault, as it is shown in Fig. 6(b)–(e). This signature can be well identified by applying the reconstruction algorithm to the torque signal, which will be used in further section. Moreover, it is observed that the mechanical impacts occur at the same instant in the vibration and the measured torque. C. Simulation Results This section deals with the study of gear tooth localized fault influence on the SCSVIF through the numerical simulation. The profile generated by the gear tooth fault Tfp (t), in addition to a mean value T0 , is applied to a classical induction machine d−q model supplied by a pure three-phase sinusoidal voltage,


Fig. 7.

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Scheme of simulation analysis for SCSVIF extraction.

Fig. 8. Spectra of the measured torque ac part at rated load: (a) healthy gear; (b) P-faulty gear.

Fig. 6. AC part of measured torque in steady-state working condition at rated load: (a) healthy gear; (b) P-faulty gear; (c) W-faulty gear; (d) SSPW-faulty gear; (e) SAPW-faulty gear.

in order to compute the result of SCSVIF without any post reconstruction procedure (see Fig. 7). This can facilitate the prediction of the generated fault signature in SCSVIF, for each particular faulty case. In other words, the reconstruction is an efficient procedure to remove all unrelated harmonics from the SCSVIF using the proposed spectrum filter since the measured stator current is commonly dominated by time and space harmonics. Nevertheless, the SCSVIF in the numerical simulation case is noiseless, and hence, a better interpretation can be achieved. Therefore, at the initial stage, Tfp (t) is extracted from the measured torque using the fault profile reconstruction algorithm (see Fig. 2) for normal and all faulty conditions. The simulated amplitude of the sinusoidal three-phase voltage supply is determined based upon the fundamental frequency component of the three-phase voltage measurement. Torque measurement shows that, in normal condition and at rated load, the spectrum of

Fig. 9. Spectra of the simulated SCSVIF ac part at rated load: (a) healthy gear; (b) P-faulty gear.

load torque is principally integrated by gearbox rotational and mesh harmonic frequencies (fr1 = 23.1 Hz, fr2 = 7.7 Hz, and fmesh = 577.5 Hz) with an additional oscillation observed at 100 Hz [see Fig. 8(a)]. In the P-faulty condition, in addition to these last frequency components, the spectrum includes k1 fr1 frequency components with significant amplitudes, for k1 = 5, . . . , 34 [see Fig. 8(b)]. The more remarkable amplitude is localized at 7fr1 = 161.7 Hz, which is near to the damped natural frequency fd = 158.3 Hz. By applying the fault profile reconstruction to both normal and P-faulty conditions and by using only sensitive harmonics, the obtained load torques were applied to the induction machine model (see Fig. 7) in order to verify the proposed approach. It can be observed in Fig. 9(a)

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Fig. 10. Spectra of the measured SCSVIF ac part at rated load: (a) healthy gear; (b) P-faulty gear; (c) W-faulty gear; (d) SSPW-faulty gear; (e) SAPW-faulty gear.

and (b) that the simulated SCSVIF shows clearly the same position as in the load torque of fault rotational harmonics around 7fr1 = 161.7 Hz, for k = 5, . . . , 10. The obtained harmonics for k = 11, . . . , 34 are not presented because their amplitudes are less than −40 dB of the noise level. The proposed fault detection approach was applied to the experimental setup for healthy and all faulty conditions [see Fig. 10(a)–(e)]. Similar results can be observed for the P-faulty case, as it is depicted in Fig. 10(a) and (b). For W-faulty, SSPW-faulty, and SAPWfaulty conditions, the most sensitive frequency components in the measured torque are related to k2 fr2 with k2 = 15, . . . , 66; k2 fr2 with k2 = 15, . . . , 84; and k2 fr2 with k2 = 15, . . . , 120, respectively. In all these last cases, the most significant amplitude is obtained at 21fr2 = 161.7 Hz. The indicated harmonics in the measured SCSVIF spectrum for W-faulty, SSPW-faulty, and SAPW-faulty conditions are k fr2 with k = 15, . . . , 30 [see Fig. 10(c)–(e)]. These harmonics cover kfr1 frequency components with k = 5, . . . , 10 since the gear ratio is 3 (i.e., fr1 = 3fr2 ). For the W-faulty case, the most sensitive harmonics are related to 19fr2 , 20fr2 , 21fr2 , and 22fr2 . It can be observed that similar to the P-faulty case, in the SSPW-faulty condition, the sensitive harmonics are related to 6fr1 = 18fr2 , 7fr1 = 21fr2 , 8fr1 = 24fr2 , and 9fr1 = 27fr2 since the position of pinion and wheel tooth surface damage faults are synchronized. In the SAPW-faulty condition, 18fr2 −22fr2 , 24fr2 , and 25fr2 frequency components are the most sensitive harmonics to the gear tooth surface damage fault [see Fig. 10(e)]. After reconstruction of fault profile from the measured torque and from numerical SCSVIF estimation, the comparison between healthy and faulty cases [see Fig. 11(a)–(j)] shows the presence of the fault profile Tfp (t) in mechanical torque due to the tooth surface damage fault, as predicted in (14), with periodicity related to the fault profile frequency (ffp−p = fr1 ,

ffp−w = fr2 , ffp−sspw = fr2 , and ffp−sapw = fr2 ). In the case of the SSPW-faulty condition, the simultaneous contact of the pinion damaged tooth with the wheel damaged tooth generates a higher torque variation at the simultaneous contact point than the single P-faulty condition, as it is depicted in Fig. 11(b) and (d). Moreover, the fault profiles observed in mechanical torques match with the fault profiles reproduced by numerical SCSVIF estimation, without any filtering and computation of fault profile from the SCSVIF through measurement, as it is depicted in Fig. 11(k)–(o). It is also fundamental to investigate the effect of voltage supply distortion and inverter-fed conditions on the SCSVIF and consequently on the fault profile harmonics. The voltage supply distortion gives rise mainly to the magnitude increase of odd harmonics in the stator current. These harmonics appear also in the SCSVIF spectrum particularly at 100, 200, and 300 Hz, which are translated versions of 150-, 250-, and 350Hz frequency components (see Fig. 10). The most sensitive fault harmonic (i.e., 7fr1 for P-faulty and 21fr2 for W-faulty, SSPW-faulty, and SAPW-faulty cases at rated load) in the SCSVIF spectrum is the nearest frequency component to the torsional natural frequency of the mechanical system, which is not affected by distortion harmonics of the load interval from the minimum load (20% of rated load) to the rated load. Nevertheless, some sideband harmonics may coincide with distortion harmonics at a specific load working condition. This is the case of 26fr2 harmonic for W-faulty, SSPW-faulty and SAPW-faulty cases at rated load. Therefore, these last harmonics are removed from the fault index computation. For inverter-fed induction machines, well-known supply voltage harmonics are related to 6h ± 1 (i.e., 5, 7, 11, 13,. . .), which produce the same effect in the SCSVIF as line-fed voltage supply distortion. The harmonics related to the pulsewidth


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Fig. 11. Reconstruction of fault profile from the measured torque at rated load: (a) healthy gear, (b) P-faulty gear, (c) W-faulty gear, (d) SSPW-faulty gear, (e) SAPW-faulty gear. Computation of fault profile from the SCSVIF through numerical simulation at rated load: (f) healthy gear, (g) P-faulty gear, (h) W-faulty gear, (i) SSPW-faulty gear, and (j) SAPW-faulty gear. Computation of fault profile from the SCSVIF through measurement at rated load: (k) healthy gear, (l) P-faulty gear, (m) W-faulty gear, (n) SSPW-faulty gear, (o) SAPW-faulty gear.

modulation carrier frequency are all located beyond the frequency bandwidth of interest and do not influence the fault detection criterion. To verify this fact, an experiment is performed in an open-loop inverter-fed condition at 50 Hz, at rated load. The identical phenomenon is observed, concerning the presence of fault profile harmonics kffp in the SCSVIF in comparison with the line-fed working condition. D. Experimental Results Initially, it was assumed that the experimental setup mechanical damping factor is small in practice. Then, based on the reconstructed fault profile from the mechanical torque for the P-faulty case, this assumption was verified by using the curve fitting approach. The obtained damping factor is ζ = 0.046, which verifies the proposed mechanical model. The damping factor gives the frequency bandwidth related to fault sensitive harmonics as a function of fault profile frequency. Considering that significant amplification of fault profile harmonics is situated around the damped natural frequency (fd = 158.3 Hz), obtained results in SCSVIF estimation [see Fig. 10(a)–(e)] are effectively justified. Then, when mechanical damping is small, fault sensitive frequency harmonics to be considered for fault detection are the multiple frequency of ffp nearest to fd and some sideband harmonics of ffp . For instance, in the P-faulty case, the main 7fr1 = 161.7 Hz and two sideband

(5fr1 , 6fr1 , 8fr1 , 9fr1 ) harmonics are the most representative harmonics. In the case of W-faulty, these harmonics are related to 21fr2 = 161.7 Hz and (Nr2 /Nr1 ) × 2 sideband harmonics (15fr2 −20fr2 , 22fr1 −27fr2 ) as it can be observed in Fig. 10(c). The application of the profile reconstruction method to the SCSVIF has been performed by localizing fault profile harmonics previously described for pinion and wheel tooth surface damage faults. The obtained results [see Fig. 11(k)–(o)] validate the observation in the mechanical torque predicted initially by expression (20) and numerical simulations. Furthermore, the effect of mechanical load on the fault profile is shown for both healthy and faulty conditions (see Fig. 12). It can be noticed that the generated fault profile is amplified at 40% of the rated load, in comparison with 60% and 80% of the rated load for P-faulty and SAPW-faulty conditions. Moreover, in all faulty conditions, a periodic fault profile appears in the reconstructed SCSVIF with fr1 or fr2 frequencies according to the fault location, i.e., pinion or wheel side. The effect of load inertia on the extracted fault profile is studied in the numerical simulation by multiplying the equivalent system inertia Jeq (Jeq = Jp + Jm = 0.0021 kg · m2 ) by two. The SCSVIF for SAPW-faulty condition, for which the fault signature has large fault profile amplitude, is shown in Fig. 13. The result shows that the magnitude of SCSVIF is proportional to the inverse of inertia value, as it can be predicted using the expression (5).

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Fig. 12. Computation of fault profile from the SCSVIF through measurement at three load levels. 80% of rated load: (a) healthy gear; (b) P-faulty gear; (c) W-faulty gear; (d) SSPW-faulty gear; (e) SAPW-faulty gear. 60% of rated load: (f) healthy gear; (g) P-faulty gear; (h) W-faulty gear; (i) SSPW-faulty gear; (j) SAPW-faulty gear. 40% of rated load: (k) healthy gear; (l) P-faulty gear; (m) W-faulty gear; (n) SSPW-faulty gear; (o) SAPW-faulty gear.

Fig. 13. Effect of inertia on the SCSVIF obtained from numerical simulation for SAPW-faulty gear: (a) with J = Jeq (system equivalent inertia) and (b) with J = 2 × Jeq .

The fault index is determined using (22) with Nw = 50 000, Δf = 0.1 Hz (Taq = 10 s), L0 = 15, L1 = 27 (to cover pinion and wheel tooth surface damage fault using only fr2 ) and M = 3 for each faulty condition, at five load levels, for 10 tests in each case. The statistical behavior of fault index is evaluated by utilization of the cumulative distribution function (CDF), mean (μ), and standard deviation (σ) of healthy and faulty conditions, according to 50 experiments acquired for each studied case and for both measured torque and SCSVIF (see Fig. 14).

With regard to the measured torque, the CDFs of P-faulty, SSPW-faulty, and SAPW-faulty conditions are sufficiently distant from the CDF of the healthy one [see Fig. 14(c)–(e)]. The largest F Ig mean value is related to the SAPW-faulty case with μSAPW = 0.295 [see Fig. 14(e)]. The CDFs of P-faulty and SSPW-faulty cases are similar because the P-faulty mean and standard deviation values (μP = 0.224, σP = 0.041) are very close to those values linked to the SSPW-faulty condition (μSSPW = 0.211, σSSPW = 0.039) [see Fig. 14(c) and (d)]. The W-faulty gear is the worst case since its CDF is adjacent to the CDF of the healthy gear [see Fig. 14(a) and (b)]. The analysis of SCSVIF fault index based on their CDFs, mean, and standard deviation is also illustrated in Fig. 14(f)–(j). It can be seen that similar to the CDFs of measured torque fault indexes, the CDF of SAWP-faulty gear is far from the CDF of healthy gear, with the largest mean value μSAPW = 0.533 [see Fig. 14(j)]. As it can be predicted, the CDF of P-faulty and SSPW-faulty conditions are close to each other, to some extent, since the mean value of P-faulty (μP = 0.339) is close to the mean value of SAWP-faulty (μSSPW = 0.367), as it is depicted in Fig. 14(h) and (i). Moreover, as it was shown for measured torque, the worst case is related to W-faulty gear [see Fig. 14(g)]. A threshold needs to be defined to attain high sensitivity and reliability for gear tooth surface damage fault detection. This can be determined where the difference between CDFs of healthy and W-faulty gears reaches to its upper limit. The computed threshold value in this case is 0.2, which is illustrated in Fig. 14. It can be concluded that the fault index


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R EFERENCES

Fig. 14. CDF of measured torque fault index (F Ig ): (a) healthy gear; (b) W-faulty gear; (c) P-faulty gear; (d) SSPW-faulty gear; (e) SAPWfaulty gear. CDF of measured SCSVIF fault index (F Ig ): (f) healthy gear; (g) W-faulty gear; (h) SSPW-faulty gear; (i) P-faulty gear; (j) SAPW-faulty gear.

criterion based on measured torque and SCSVIF gives similar results, and it is efficient, reliable, and sensitive enough for both pinion and wheel gear tooth surface damage fault detection. Moreover, the fault detection algorithm (see Fig. 3) is simple, since it requires only some basic data concerning the electromechanical system, i.e., the torsional resonance frequency and damping factor, and independent to the type of gear, which makes its implementation on real-time systems possible. IV. C ONCLUSION In this paper, a noninvasive technique based on the SCSVIF for a fault diagnosis of gear tooth surface damage fault has been presented. The principal idea relies on the fact that, in gear localized fault condition, an extra component appears in the mechanical torque experimented by the driven electrical machine due to the presence of a torque impact, which induces a periodic fault signature amplified by torsional natural resonance. This periodic fault signature can be observed in SCSVIF as a frequency modulation, which can be estimated by using a simple algorithm. The proposed fault index is based on energy introduced by this modulation, and it proves its effectiveness for a noninvasive fault diagnosis of gear tooth surface damage fault detection in a one-stage gear-based electromechanical system at different load levels.

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Humberto Henao (M’95–SM’05) received the M.Sc. degree in electrical engineering from the Technical University of Pereira, Pereira, Colombia, in 1983, the M.Sc. degree in power system planning from the Universidad de los Andes, Bogota, Colombia, in 1986, and the Ph.D. degree in electrical engineering from the Institut National Polytechnique de Grenoble, Grenoble, France, in 1990. From 1987 to 1994, he was a Consultant for companies such as Schneider Industries and GEC Alstom, with the Modeling and Control Systems Laboratory, Mediterranean Institute of Technology, Marseille, France. In 1994, he joined the Ecole Supérieure d’Ingénieurs en Electrotechnique et Electronique, Amiens, France, where he was an Associate Professor. In 1995, he joined the Department of Electrical Engineering, University of Picardie Jules Verne, Amiens, where he was initially an Associate Professor and has been a Full Professor since 2010. He is currently the Department Representative for international programs and exchanges. He also leads research activities in the field of condition monitoring and diagnosis for power electrical engineering. His main research interests include modeling, simulation, monitoring, and diagnosis of electrical machines and drives.

Shahin Hedayati Kia received the M.Sc. degree in electrical engineering from the Iran University of Science and Technology, Tehran, Iran, in 1998 and the M.Sc. and Ph.D. degrees 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 the Institut Supérieur des Sciences et Techniques (INSSET) de Saint-Quentin, 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 has been an Assistant Professor with the Department of Electrical Engineering, University of Picardie Jules Verne.

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 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 has held several faculty positions in Yaoundé, Cameroon, Le Creusot, France, and Marseille. In 1994, he joined the University of Picardie Jules Verne, Amiens, France, as a Full Professor. He was appointed Chair Professor in 2013.