Time-frequency space vector modulus analysis of motor current for planetary gearbox fault diagnosis under variable speed conditions

Mechanical Systems and Signal Processing 121 (2019) 636–654

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Time-frequency space vector modulus analysis of motor current for planetary gearbox fault diagnosis under variable speed conditions Xiaowang Chen, Zhipeng Feng ⇑ School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China

a r t i c l e

i n f o

Article history: Received 13 September 2017 Received in revised form 5 August 2018 Accepted 28 November 2018

Keywords: Fault diagnosis Induction motors Variable speed Planetary gearbox Space vector analysis

a b s t r a c t Motor current signature analysis is a promising technique for electromechanical system fault detection, and has been studied under steady states. In this paper, a planetary gearbox fault diagnosis method under variable speed conditions using stator current signal is proposed. In order to thoroughly understand the frequency characteristics of current signals, an analytical amplitude modulation and phase modulation (AM-PM) current model considering gear fault modulation effects is presented. Two more aspects of endeavor are made to highlight gear fault signatures in stator current signals in context of inconspicuousness and time variability. Firstly, to address the sideband complexity and power supply dominance issues inherent with stator current signals, squared space vector modulus (SVM) together with its time-varying spectral characteristics in planetary gearbox fault cases under variable speed conditions are mathematically derived. Secondly, to reveal time-varying fault features in details, polynomial chirplet transform (PCT) is improved by iterative algorithm, and merits of fine time-frequency resolution and cross term free nature are achieved. The effectiveness of the proposed method is illustrated by numerical simulation, and is further validated by lab experiments on a real world 4 kW induction motor driven planetary gearbox test rig. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Planetary gearboxes are widely integrated in power transmissions of various industrial electromechanical systems [1–3], owing to their merits of high transmission ratio and large load capacity in a compact package. Since the failure of gear train is among the reasons which cause longest shut down and largest economic loss, condition monitoring and fault diagnosis of planetary gearboxes are quite essential. Vibration signal analysis is commonly used in machinery fault diagnosis. However, planetary gearboxes have complicated structure and involute kinematics, particularly the planet gear revolution around the sun gear and the resultant time-varying gear mesh location relative to a fixed vibration sensor. This leads to great complexity of vibration signals, and high difficulty in planetary gearbox fault diagnosis via vibration signal analysis [1–4]. For planetary gearbox fault diagnosis, motor current signature analysis (MCSA) has advantages over vibration analysis, because: (1) stator current signals are free from the extra amplitude modulation (AM) effect due to time-varying vibration ⇑ Corresponding author. E-mail address: [email protected] (Z. Feng). https://doi.org/10.1016/j.ymssp.2018.11.049 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

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Nomenclature Zs, Zp, Zr, number of gear teeth number of planet gears and gear meshing frequency N, fm(t) fs-c(t), fp-c(t), fr-c(t) gear rotating frequency relative to planet carrier fs(t), fp(t), fr(t) gear fault frequency ðr Þ ðr Þ sun gear and planet carrier rotating frequencies f s ðt Þ; f c ðt Þ ffault(t), xfault(t) fault frequency fim(t) induction motor speed fsupply(t), xsupply(t) power supply frequency Tmotor(t), TL(t), T0(t), Tr0(t), TF motor and load torque xrotor(t), xr0(t), hr(t) rotor speed and rotor angle F(h,t), Fs(h,t), Fr(h,t) magnetomotive force U(t), Us(t), Ur(t) air gar flux s(t) motor slip p, J pole pair number and rotary inertia V(t), Rs, Is voltage, resistance and current Isa(t), Isb(t), three-phase stator currents current amplitude Im, Ir a, g(t), u coefficients and initial phase Jn(), d(t) Bessel function and Dirac function space vector and its two parts SV(t), Id(t), Iq(t) A(t), /(t) amplitude and phase of squared SVM b1,. . .,bn polynomial parameters

propagation paths [4]; and (2) stator current signals can be measured from electric wire or circuit board and have easier accessibility than vibration signals [5,6]. These appealing advantages have attracted increasing investigations on current based planetary gearbox fault detection. Motor-planetary gearbox models were established in [7,8] to study the gear fault effect on the electrical signature of induction machine, and gear transmission errors were detected by inspecting the presence of spectral sidebands around power supply frequency. A resonance residual technique was applied to stator current signal, seeking for more evident spectral presentation of planetary gearbox fault feature [9]. Gear tooth profile error in a planetary gearbox was also revealed as waveform transients of stator current signal in both mathematical modeling and experimental validation [10]. Planetary gearboxes often work under unsteady states (variable speeds for example), and the motor current is therefore nonstationary. However, most of the reported works on planetary gearbox fault diagnosis focus on steady states, and rely on either time domain or frequency domain analysis which is insufficient for nonstationary signal processing. Time-frequency analysis offers an approach to reveal the frequency contents and their variability from nonstationary current signals. Wigner-Ville distribution was utilized to analyze phase modulation (PM) feature of current signals under steady states for mechanical load fault detection [11]. Wavelet transform was used to analyze current signals for detecting transients due to gear fault or load fluctuation of a fixed shaft gearbox [12]. These works inspire us to study planetary gearbox fault diagnosis under unsteady states via time-frequency analysis of current signals. However, to the best of our knowledge, the effect of planetary gearbox fault on stator current signals under unsteady states and the time-varying fault characteristics have not been thoroughly studied, and still need to be investigated indepth. In addition, time-frequency analysis method with satisfactory readability is required to reveal the frequency contents of nonstationary current signals and their time variability. In order to address the above issues, three major contributions are made in this paper. (1) To thoroughly understand the spectral characteristics of current signals under variable speed conditions, a time-varying AM-PM current model is derived, considering gear fault modulation effects on stator current signals. Stator currents were modeled as amplitude modulation (AM) processes in induction motor and motor bearing fault cases [13,14], and were simplified as phase modulation (PM) in cases of mechanical load and gear fault [11,15]. However, studies in [10] exhibited AM feature of stator current signals when gear defect exists. In this paper, the AM-PM effect of gear fault on stator current is derived through magnetomotive force and permeance wave approach. (2) To highlight gear fault signatures against dominant power supply frequency harmonics, space vector modulus (SVM) is exploited for its capability to extract modulation features. It is difficult to extract gear fault frequency directly from single phase current signals, because their spectra involve intricate sidebands. More importantly, power supply frequency harmonics are strongly dominant, masking the gear fault frequency of interest. To effectively reveal gear fault information, the squared SVM of stator currents under variable speed conditions is mathematically derived, its explicit time-varying Fourier spectrum is deduced, and gear fault symptoms are summarized. (3) To effectively reveal the time-varying gear fault features, polynomial chirplet transform (PCT) is improved and applied on squared SVM, by exploiting

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its adaptability in handling various nonstationary cases. PCT extends chirplet transform to more general cases when the instantaneous frequency (IF) of a signal can be expressed by polynomial functions [16,17]. It outperforms conventional time-frequency analysis methods [18] with better time-frequency resolution and cross term free nature. Integrated with iterative algorithm [19], iterative polynomial chirplet transform (IPCT) is implemented with designed basis functions specific for frequency components of interest, and fine time-frequency readability is achieved. The remainders of this paper are structured as follows. In Section 2, the AM-PM current model with gear fault under variable speed conditions, the time-varying space vector modulus analysis, and the principle of IPCT for multi-component signal analysis, are elaborated in details. In Section 3, numerical simulation analysis is performed to illustrate the effectiveness of proposed method. Experimental validations for sun, planet and ring gear fault detection are carried out in Section 4. Finally, conclusions are drawn in Section 5. 2. Time-frequency space vector analysis of motor current 2.1. Planetary gearbox fault frequencies Planetary gearboxes have complicated structure and involute kinematics which are far different from those of fixed shaft ones. Fig. 1 shows a brief schematic of a single stage planetary gearbox with a fixed ring gear and three equally spaced planet gears. In planetary gearboxes, planet gears mesh simultaneously with both sun gear and ring gear, and meanwhile revolve with the planet carrier around sun gear, as they spin around their own centers. This unique kinematic makes gear fault frequencies not equal to gear rotating frequencies as they do in the fixed shaft gearbox cases. Instead, they depend on the gear rotating frequencies relative to the planet carrier [4], as

f s ðtÞ ¼ N  f s - c ðtÞ ¼ N  ½f m ðtÞ=Z s 

ð1Þ

f p ðtÞ ¼ f p - c ðt Þ ¼ f m ðt Þ=Z p

ð2Þ

f r ðtÞ ¼ N  f r - c ðtÞ ¼ N  ½f m ðtÞ=Z r 

ð3Þ

where f s ðt Þ, f p ðt Þ, f r ðtÞ stands for the sun, planet and ring gear fault frequency respectively. Their rotating frequencies relative to the planet carrier f s - c ðtÞ; f p - c ðtÞ; f r - c ðtÞ can be calculated given the sun, planet and ring gear tooth number Z s ; Zp ; Zr , the number of planet gears N, and the gear meshing frequency

h i ðrÞ ðrÞ ðrÞ f m ðt Þ ¼ Z r  f c ðt Þ ¼ Z s  f s ðt Þ  f c ðtÞ ðrÞ

ð4Þ

ðrÞ

where f s ðtÞ and f c ðtÞ are the sun gear and planet carrier rotating frequencies respectively. In real-world electromechanical drivetrains, either sun gear, ring gear or planet carrier is connected to the induction motor for certain speed outputs. Therefore, sun, planet and ring gear fault frequencies are all proportional to the induction motor speed f im ðt Þ. 2.2. Time-varying amplitude modulation and phase modulation effect The role of planetary gearbox in an induction motor driving system is illustrated in Fig. 2. In Refs. [11,15], researchers have studied the phase modulation effect of load fluctuation due to fixed-shaft gear defect on motor stator current. However, the amplitude modulation effect is neglected and all characteristic frequencies are assumed to be constant, which fails to represent general situations and nonstationary features. A more general amplitude modulation and phase modulation (AM-PM) model of motor current under variable speed conditions is proposed in this paper.

Sun gear

Planet gears

Ring gear Fig. 1. Schematic of a single stage planetary gear set (Gears are simplified as solid circles for better view).

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I sa I sb I sc Induction motor

Planetary gearbox

Load

Fig. 2. Role of a planetary gearbox in drive train (Dotted lines stand for the coupling and other parts in between).

For the sake of clarity and simplicity, during derivation we consider only the gear fault effect on current signals, and the fundamental frequencies. The load torque of induction motor T L ðtÞ can be generally formulated as

Z t  T L ðt Þ ¼ T 0 ðtÞ þ T F sin xfault ðsÞds þ u

ð5Þ

0

where T 0 ðt Þ stands for the instantaneous average load torque, and the second term indicates gear fault induced torque oscillation with amplitude T F and initial phase u. xfault ðtÞ ¼ 2pf fault ðtÞ denotes the time-varying gear fault frequency calculated according to Eqs. (1) through (3). The rotor speed xrotor ðt Þ is affected by the oscillation of load torque, and the rotor angle can be calculated as

Z

t

hr ðt Þ ¼

xrotor ðsÞds ¼ 0

Z

t

¼ 0

Z

1 J

Z

t

0

½T L ðtÞ  T motor ðtÞdtds ¼

0

TF xr0 ðsÞds  _ fault ðsÞ J xfault ðsÞx

t

¼

Z s

xr0 ðsÞds  0

TF _ fault ðtÞ J x2fault ðt Þx 2

Z

t

Z

1 J

Z Z s T r0 ðtÞ þ T F sin

t

0

0

Z s  cos xfault ðmÞdm þ u ds

0

Z

0

t



xfault ðmÞdm þ u dtds

0



t

ð6Þ

xfault ðsÞds þ u

sin 0

where T motor ðt Þ means the motor torque, T r0 ðtÞ is the retained torque under variable speed conditions, xr0 ðtÞ indicates the instantaneous average motor speed, and J is the total rotary inertia of rotating parts. The total magnetomotive force F ðh; tÞ of both stator and rotor can then be cast in stator reference frame as [11]

F ðh; tÞ ¼ Fs ðh; tÞ þ Fr ðh; tÞ  Z t    Z t Z t ¼ Fs sin ph  xsupply ðsÞds þ p=2 þ Fr sin ph  xsupply ðsÞds  gðtÞsin xfault ðsÞds þ u 0

0

ð7Þ

0

_ 2fault ðtÞ is the phase modulation coefficient with pole pair number p. where xsupply ðtÞ ¼ 2pf supply ðt Þ, and gðtÞ ¼ pT F =½J x2fault ðtÞx Under the assumption of constant air gap permeance, the air gap flux UðtÞ has the same frequency structure as magnetomotive force, i.e.,

Z

Uðt Þ ¼ Us ðt Þ þ Ur ðt Þ ¼ Us sin



t

Z

xsupply ðsÞds  p=2 þ Ur sin 0

t

Z

xsupply ðsÞds þ gðtÞsin 0



t

xfault ðsÞds þ u

ð8Þ

0

The stator voltage V ðtÞ, stator current Is ðt Þ in an arbitrary phase and flux UðtÞ satisfy the relationship of _ ðt Þ where Rs is the stator resistance. Since the stator voltage is imposed by voltage source, the stator current V ðt Þ ¼ Rs Is ðtÞ þ U is in a linear relation with the time derivative of flux, and can be modeled by

h i _ ðt Þ =Rs ¼ Is0 ðtÞ  U _ ðt Þ=Rs Is ðt Þ ¼ V ðtÞ  U

8 9 > > > >Z t > Z t > Z t  < =   p ¼ Im 1  axsupply ðtÞ cos xsupply ðsÞds   Ir cos xsupply ðsÞds þ gðtÞsin xfault ðsÞds þ u > > 2 0 0 > 0 > > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} : ; phase modulation



Z t  Z t  xfault ðsÞds þ u þ gðtÞxfault ðtÞcos xfault ðsÞds þ u  xsupply ðtÞ þ g_ ðtÞsin 0 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}

ð9Þ

amplitude modulation

where Im >> Ir are respectively the magnitude of stator resistive current and stator active current [20], and

a ¼ U_ s = ðRs Im Þ << 1 is the amplitude ratio.

Eq. (9) indicates that gear fault has both PM and AM effect on stator current signal, with the carrier frequency equal to the power supply frequency and the PM and AM modulating frequency equal to the gear fault frequency. According to the

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convolution property of Fourier transform, the Fourier spectrum of (9) at time instant t amounts to convolution of the AM term and the carrier signal Fourier spectra. Meanwhile, the Fourier spectrum of a PM signal involves a Bessel series expansion of infinite orders. These frequency components are time-varying under variable speed conditions. Therefore, the stator current signal has a very intricate sideband structure around the power supply frequency. Moreover, power supply frequency component has much higher magnitude than sidebands due to gear fault modulation since Im >> Ir , leaving the latter inconspicuous. These factors lead to difficulty in gear fault frequency identification. 2.3. Time-varying space vector modulus analysis Because of the dominance of power supply frequency component f supply ðt Þ, as well as the spectral complexity in current signals, fault related frequency components can hardly be intuitively recognized [21,22]. Space vector analysis is explored to address this issue. In this section, the space vector modulus (SVM) of motor current is deduced based on the current model Eq. (9). Its time-varying frequency structure is further derived with explicit expressions. The space vector SVðt Þ and its constituents Id ðt Þ, Iq ðt Þ are obtained as a combination of the three phase stator currents Isa ðt Þ, Isb ðtÞ and Isc ðtÞ, which are modeled by (9) and are shifted from each other by 2p=3, i.e.,

SVðtÞ ¼ Id ðtÞ þ jIq ðt Þ Id ðt Þ ¼

ð10Þ

rffiffiffi 2 1 1 Isa ðtÞ  pffiffiffi Isb ðt Þ  pffiffiffi Isc ðtÞ 3 6 6

ð11Þ

1 1 Iq ðt Þ ¼ pffiffiffi Isb ðt Þ  pffiffiffi Isc ðtÞ: 2 2

ð12Þ

SVM is defined as the modulus of space vector SVðt Þ. According to the AM-PM current model in (9), the squared SVM in planetary gearbox fault cases under variable speed conditions can be derived as 2

jSVMðt Þj2 ¼ jId ðtÞ þ jIq ðt Þj

 Z t  Z t 2 2 3 3  ¼ I2m 1  axsupply ðt Þ þ I2r xsupply ðt Þ þ g_ ðt Þsin xfault ðsÞds þ u þ gðtÞxfault ðtÞcos xfault ðsÞds þ u 2 2 0 0  Z t  Z t     3Im Ir 1  axsupply ðt Þ xsupply ðt Þ þ g_ ðtÞsin xfault ðsÞds þ u þ gðtÞxfault ðtÞcos xfault ðsÞds þ u 0 0 Z t    p xfault ðsÞds þ u þ  cos gðtÞsin ð13Þ 2 0

Through Bessel series expansion [23], it can be further rewritten as

Z t  Z t  jSVMðt Þj2 ¼ Aðt Þ þ A1 ðt Þsin xfault ðsÞds þ /1 ðtÞ þ A2 ðtÞsin 2xfault ðsÞds þ /2 ðt Þ 0

þ

1 X k¼1

þ

1 X n¼1

0

Z t  X Z t  1 A3k ðtÞsin kxfault ðsÞds þ /3k ðtÞ þ A4l ðtÞcos ð1 þ lÞxfault ðsÞds þ /4l ðtÞ 0

l¼1

Z t  A5n ðtÞcos ð1  nÞxfault ðsÞds þ /5n ðtÞ

0

ð14Þ

0

where AðtÞ is the instantaneous average amplitude, /ðtÞ are time dependent initial phases, AðtÞ are the time-varying amplitude of associated frequency components. They are derived as

h i  2 2 Aðt Þ ¼ 32 I2m 1  axsupply ðt Þ þ 32 I2r x2supply ðtÞ þ 34 I2r g2 ðt Þx2fault ðtÞ þ g_ ðt Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i 2 2 A1 ðtÞ ¼ 3I2r xsupply ðt Þ g_ ðtÞ þ g2 ðt Þx2fault ðtÞ; A2 ðtÞ ¼ 34 I2r g2 ðtÞx2fault ðtÞ þ g_ ðtÞ ;   A3k ðt Þ ¼ 3J k ½gðtÞIm Ir 1  axsupply ðt Þ xsupply ðt Þ;  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 A4l ðt Þ ¼ 32 J l ½gðt ÞIm Ir 1  axsupply ðt Þ g_ ðt Þ þ g2 ðtÞx2fault ðt Þ; q   ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 A5n ðt Þ ¼  32 J n ½gðtÞIm Ir 1  axsupply ðtÞ g_ ðt Þ þ g2 ðt Þx2fault ðtÞ;   h i g2 ðtÞx2fault ðtÞg_ 2 ðtÞ ðt Þ þ 2u; /3k ðtÞ ¼ ku; u ; / ð t Þ ¼ arctan /1 ðtÞ ¼ arctan gðtÞxg_ ðfault þ 2 tÞ 2gðtÞg_ ðtÞxfault ðtÞ h i h i ðt Þ ðtÞ þ ð1 þ lÞu; /5n ðtÞ ¼ arctan gðtÞxg_ ðfault þ ð1  nÞu: /4l ðtÞ ¼ arctan gðtÞxg_ ðfault tÞ tÞ

ð15Þ

and J n ðÞ is the Bessel function of the first kind with an integer order n. More detailed derivation can be found in Appendix A.

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At each time instant t, applying Fourier transform to (14), yields the time-varying Fourier spectrum

TFRðt; f Þ ¼ Aðt Þdð f Þ þ A1 ðt Þd½ f  f fault ðtÞexp½ j/1 ðt Þ þ A2 ðt Þd½ f  2f fault ðt Þexp½ j/2 ðt Þ 1 1 X X A3k ðtÞd½f  kf fault ðtÞexp½j/3k ðtÞ þ A4l ðtÞd½f  ðl  1Þf fault ðtÞexp½j/4l ðtÞ þ þ

k¼1 1 X

l¼1

A5n ðtÞd½f  ðn þ 1Þf fault ðtÞexp½j/5n ðtÞ

ð16Þ

n¼1

where dðÞ stands for Dirac delta function. Eq. (16) implies that only the gear fault frequency and its harmonics remain in the time-varying spectrum of squared SVM, while the power supply frequency components are rejected. Gear fault frequencies are directly exhibited without complex sidebands or masking by other components, such feature facilitates gear fault identification. 2.4. Iterative polynomial chirplet transform According to Eqs. (1) through (4), gear fault characteristic frequencies are proportional to each other. Under variable speed conditions, their trends follow the motor speed changing profile, thus being time-varying. Since time or frequency domain analysis alone is insufficient to reveal the time-varying frequency structure of nonstationary signals, timefrequency analysis is used. Polynomial chirplet transform is essentially a linear time-frequency analysis method. It generalizes chirplet transform with a quite flexible kernel function which can be adaptively designed to match instantaneous frequency in various polynomial forms [16–17], thus achieving fine time-frequency resolution and cross-terms free nature for complex nonstationary signal analysis. For a signal xðt Þ 2 L2 ðRÞ, its PCT is defined as

Z

PCTðt0 ; f; b1 ; :::; bn ; rÞ ¼

WRb1 ;:::;bn ðtÞ ¼ exp j

þ1

1

^xðt ÞWRb ;:::;b ðt ÞWM b1 ;:::;bn ðt; t 0 ÞfðrÞ ðt  t 0 Þexpðjxt Þdt; 1 n

! nþ1 X 1 bk1 t k ; k k¼2

WM b1 ;:::;bn ðt; t 0 Þ ¼ exp j

nþ1 X

ð17Þ

! ðk1Þ

bk1 t 0

t ;

ð18Þ

k¼2

^ where WRb1 ;:::;bn ðt Þ and WM b1 ;:::;bn ðt; t 0 Þ are respectively the nonlinear operators for frequency rotating and shifting, xðt Þ is the analytical signal of xðt Þ, and characteristic parameters of the polynomial kernel function are given by ðb1 ; :::; bn Þ, and fðrÞ ðt  t0 Þ denotes a nonnegative, symmetric and normalized real Gaussian window function with a window length r at a time span centered around t 0 . The time-frequency readability obtained by PCT method depends on the suitability of kernel function. To reduce energy leakage and achieve flexibly high time-frequency concentration, kernel functions which best match the target time-varying frequencies need to be designed with appropriate polynomial parameters. Since the motor current signal is multicomponent, and each characteristic frequency component has different IF trajectories, PCT with only one settled kernel function cannot best match all time-varying features. To solve such problem, the improved iterative polynomial chirplet transform (IPCT) is proposed. In IPCT, such parameters are obtained iteratively with measured motor speed f im ðt Þas well as planetary gearbox configurations. Firstly, the coarse time-frequency representation of the analyzed signal is obtained via Morlet CWT, from which the frequency compositions can be estimated. Then, specific for each frequency components, its best matched polynomial parameters are structured by polynomial fitting. Each characteristic frequency component is iteratively extracted and its accurate time-frequency information is independently generated by PCT. The superposition of generated time-frequency details constitutes the TFR of original analyzed signal. The advantages of IPCT method over commonly used Morlet continuous wavelet transform (CWT), Wigner-Ville Distribution (WVD), and the original PCT method, can be demonstrated with a simple two-component synthetic signal  R

   R

  s1 ðt Þ ¼ cos 2p 160t 2  160t þ 50 dt þ cos 4p 160t2  160t þ 50 dt . Set the sampling frequency 400 Hz, t 2 ð0; 1 and signal to noise ratio 5 dB. TFRs of s1 ðt Þ generated by CWT, WVD, PCT and IPCT are respectively displayed in Fig. 3(a–d). Clearly, the traditional CWT method suffers from poor time-frequency concentration, resulting in smearing of instantaneous frequency trajectories in Fig. 3(a). Much interferences exist in Fig. 3(b) because of the inevitable cross term/inner terms caused by WVD. Despite the ability of PCT to generate best matched window functions for high time-frequency resolution, only one time-varying frequency component is clearly revealed in Fig. 3(c), whereas other components with different instantaneous frequency trends are still blurred. In comparison, the IPCT method generate optimal window functions iteratively for each characteristic frequency component, thus exhibiting complete time-frequency details of s1 ðtÞin Fig. 3(d). Besides, the TFR generated by IPCT achieves both high time–frequency resolution and outer/inner interference free nature. 2.5. Analysis procedure Procedures of the proposed time-frequency space vector modulus analysis for planetary gearbox fault diagnosis under variable speed conditions is illustrated with block diagram in Fig. 4, and detailed steps are summarized as follows:

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(a)

(c)

(b)

(d)

Fig. 3. TFRs of synthetic signal s1(t) by: (a) CWT, (b) WVD, (c) PCT, and (d) IPCT.

(1) Given the three-phase stator currents Isa ðtÞ, Isb ðtÞ and Isc ðtÞ, calculate the squared space vector modulus jSVMðt Þj2 based on Eqs. (10) through (13). (2) Generate wavelet scalograms of squared SVM using conventional continuous wavelet transform (CWT), and estimate its frequency structure. (3) Calculate the polynomial coefficients of selected characteristic frequency components via polynomial fitting. (4) Perform IPCT on squared SVM, with the optimal kernel functions obtained in step (3). (5) Identify gear fault frequencies from the time-frequency representation achieved in step (4). 3. Numerical simulation In this section, the proposed planetary gearbox fault detection method is illustrated through numerical simulated signal analysis. In addition to gear tooth fault, gear manufacturing and gearbox assembling errors are inevitable in practices. As

Data preparation Isa(t) Isb(t) Isc(t)

Squared space vector modulus |SVM(t)|2

Preprocessing

Analysis

Continuous wavelet scalogram

Perform IPCT on |SVM(t)|2

Frequency structure estimation Polynomial parameters fitting

Time-varying spectrum of |SVM(t)|2

Gear fault feature extraction Fig. 4. Block diagram of time-frequency space vector modulus analysis for gear fault detection.

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643

such, more gear fault characteristic frequencies contribute to modulation effect on motor current, even under healthy condition of planetary gearboxes. Suppose more than one characteristic frequency components modulate the motor current, Eq. (9) is rewritten as

Z t    p Is ðt Þ ¼ Im 1  axsupply ðtÞ cos xsupply ðsÞds  2 0 8 9 > > > >Z t > Z t > < = XZ xsupply ðsÞds þ z¼1 gz ðtÞsin xz ðsÞds þ uz  Ir cos > > 0 0 > > > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} : ; 

phase modulation

XZ 

Z t  Z t  _ z ðtÞsin g x x  xsupply ðtÞ þ z ðsÞds þ uz þ gðtÞxz ðtÞcos z ðsÞds þ uz z¼1 0 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}

ð19Þ

amplitude modulation

where Z is the number of modulation frequency components. Considering instrumentation error and background noise interferences, the three-phase stator currents are finally generated by

hR i 1 hR i1 0 t t 0 1 cos 0 xsupply ðsÞds  p2 cos 0 xsupply ðsÞds þ PMðt Þ na ðtÞ Isa ðt Þ B B hR iC hR iC C C B B   t t B C B C p C 2p C B @ Isb ðt Þ A ¼ Im 1  axsupply ðtÞ B B cos 0 xsupply ðsÞds þ 6 C  Ir AMðtÞB cos 0 xsupply ðsÞds þ PM ðt Þ þ 3 C þ @ nb ðtÞ A A A @ @ h i h i Rt Rt Isc ðt Þ nc ðtÞ cos 0 xsupply ðsÞds  76p cos 0 xsupply ðsÞds þ PM ðt Þ  23p 0

0

1

ð20Þ where AMðt Þ and PMðt Þ respectively stand for amplitude modulation and phase modulation terms in (19), and na ðtÞ; nb ðtÞ; nc ðtÞ are added white Gaussian noise. Without loss of generality, we set Z ¼ 2 while assuming both localized sun gear fault and sun gear shaft misalignment of a planetary gearbox which has the same gear configuration as in the lab experiments in Section 4 (see Table 1). The power supply frequency is set f supply ðtÞ ¼ 0:23t 3  3:41t2 þ 10:28t þ 40 in a 10 s speed fluctuation process. Then the modulation f 2 ðt Þ ¼

ðrÞ f s ðt Þ

frequencies

in

(19)

are

calculated

as

ðrÞ

f 1 ðt Þ ¼ f s ðt Þ ¼ 2:63f s ðtÞ ¼ 2:63f im ðt Þ ¼ 1:315f supply ðt Þ

and

¼ f im ðtÞ ¼ 0:5f supply ðt Þ, if we consider pole pair number p ¼ 2 and motor slip almost zero. Set the amplitude

ratio a ¼ 2  104 , the amplitude of sun gear fault induced torque oscillation T 1 ¼ 10 N  m, and sun gear shaft misalignment induced torque oscillation T 2 ¼ 1 N  m. Let Im ¼ 8, Ir ¼ 0:01, J ¼ 0:2 kg  m2 , u1 ¼ u2 ¼ 0, and signal-to-noise ratio of added noises 40 dB, the three phase stator currents Isa ðtÞ, Isb ðtÞ and Isc ðtÞ are generated according to Eq. (20). Fig. 5(a) shows a close-up view of the simulated three-phase stator currents Isa ðt Þ, Isb ðtÞ and Isc ðt Þ, and Fig. 5(b) displays the corresponding motor speed f im ðtÞ. CWT is firstly applied to the phase current Isa ðt Þ for presenting time-varying characteristics, and the obtained wavelet scalogram is shown in Fig. 5(c). The time-varying modulation sidebands jf supply ðt Þ  f s ðt Þj ðrÞ

and jf supply ðtÞ  f s ðtÞj, which are expected to appear at locations where red dash curves and blue dash-dotted curves indicate in Fig. 5(c), can hardly be observed because of the strong dominance of power supply frequency component. Besides, evident smearing phenomenon occurs especially in the frequency region of 30–60 Hz, due to the inherent property of CWT in achieving high time resolution but low frequency resolution in high frequency region, and vice versa. To solve these problems and pursue a simple yet clear distribution of fault characteristics, time-varying space vector modulus analysis is employed. The detrended squared SVM has mean value of nearly zero (1.18e13), variance of 0.06, and peak to peak value of 1.88. Its waveform is plotted in Fig. 6(a). CWT is then applied to squared SVM for estimating its frequency structure, and the resultant scalogram shown in Fig. 6(b) implies that both the sun gear fault frequency ðrÞ

f s ðtÞ and sun gear rotating frequency f s ðt Þ are directly revealed. Such observation validating the effectiveness of SVM analysis in planetary gearbox fault detection. However, time-frequency smearing still exists in the wavelet scalogram in Fig. 6(b), due to the poor adaptability of wavelet basis to changes in signal. Such smearing phenomenon under noise background may hinder the accurate fault identification, as shown in Fig. 6(b) around 50–70 Hz frequency band from 0 to 4.5 s. Based on the given motor speed, the polynomial parameters of sun gear fault frequency and sun gear rotating frequency are calculated for IPCT. The achieved TFR of squared space vector modulus by IPCT method is exhibited in Fig. 6(c), which shows much higher

Table 1 Planetary Gearbox configuration. Gear

Sun

Planet

Ring

Number of gear teeth

13

38 (3)

92

Note: The number of planet gears is indicated in the parenthesis.

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(a)

(b)

(c) Fig. 5. Direct analysis of simulated phase current. (a) waveform of simulated phase current, (b) motor speed, (c) continuous wavelet scalogram of simulated phase current.

time-frequency concentration than that in wavelet scalogram. Comparing Fig. 6(c) with Fig. 5(c), it is obvious that the fault feature is better highlighted and more clearly presented through time-frequency space vector analysis with IPCT, than by conventional analysis of phase current with CWT.

4. Experimental evaluation 4.1. Test settings Experiments are carried out to evaluate the proposed method in real-world planetary gearbox fault diagnosis under variable speed conditions. Fig. 7 shows the test rig. A NGW21 planetary gearbox is driven by a 4 kW SNMA YP-50-4-4 threephase squirrel-cage induction motor with delta-connection and two pole pairs (connected to the sun gear shaft). A CZ-5 magnetic powder brake is applied to the output shaft (connected to the planet carrier). The induction motor is controlled by an ABB ACS550 variable frequency drive, and three phase stator current signals are measured by Fluke 80i-110s AC/DC current clamps mounted between the variable speed drive and induction motor. The stator current signals are collected with LMS SCADAS DAQ system and LMS Test. Xpress 8A software at a sampling rate of 20 k Hz. Four groups of tests are carried out: baseline case when all gears are healthy, faulty sun, planet and ring gear cases when one tooth of the sun gear, one of the planet gears, and the ring gear alone is locally chipped. The faulty gears are respectively shown in Fig. 8(a–c). In all experiments, a load of 30 lb-in is applied to the gearbox output shaft by the magnetic powder brake. The power supply frequency of induction motor is set as a linear function f supply ðtÞ ¼ 2t þ 30 Hz to simulate an accelerating process within 10 s. Gear configurations of the used planetary gearbox are listed in Table 1, basing on which the characteristic frequencies are calculated by measured motor speed f im ðt Þ, and are further given in Table 2. These characteristic frequencies will be used for fault identification.

X. Chen, Z. Feng / Mechanical Systems and Signal Processing 121 (2019) 636–654

(a)

645

(b)

(c) Fig. 6. Time-frequency space vector modulus analysis of simulated stator currents. (a) waveform of detrended squared SVM, (b) continuous wavelet scalogram of squared SVM, (c) TFR of squared SVM by IPCT.

Induction Motor

Planetary gearbox

Fig. 7. Experimental system.

Magnetic powder brake

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(a)

(b)

(c)

Fig. 8. Local defects on: (a) sun gear, (b) planet gear and (c) ring gear.

4.2. Baseline signal analysis Firstly, the condition when all gears are theoretically healthy is set as baseline. Under a speed up process of 10 s, three phase stator currents are measured and plotted in Fig. 9(a). The wavelet scalogram of Isa ðtÞ is shown in Fig. 9(b), on which only the power supply frequency component f supply ðtÞ is dominant while other components have much lower amplitude. This is because Im >> Ir , as structured in (9). The motor speed multiplied with pole pair number pf im ðt Þ is also plotted in Fig. 9(b) as a black dash curve, which has almost the same frequency varying trend with f supply ðtÞ since the motor slip sðtÞ is quite small. To reveal potential fault frequency features and facilitate fault identification, we follow the proposals in Section 2.3 and utilize space vector modulus analysis. The squared space vector modulus is calculated according to (10)–(13), and its waveforms is plotted in Fig. 9(c). As explained in (14)–(16), potential fault frequency components are expected to be directly presented in TFR of squared SVM. But in its wavelet scalogram shown in Fig. 9(d), only first 4 orders of sun gear rotating frequencies are identified despite poor time-frequency resolution. Basing on results of continuous wavelet scalogram, IPCT is further exploited to achieve clear time-frequency distribution of squared SVM, and results are exhibited in Fig. 9(e). Obviously, only first 4 orders of sun gear ðr Þ

rotating frequencies (1–4) f s ðt Þ are clearly identified in TFR of squared SVM. Basing on former analysis in Section 3.2, such phenomenon is probably due to sun gear shaft misalignment, and does not imply gear defect.

4.3. Sun gear fault detection In this case, local chipping of the sun gear is brought in while other gears remain to be healthy. Other experimental settings are same with those in baseline case. The three phase stator currents are measured and plotted in Fig. 10(a), and the wavelet scalogram of Isa ðtÞ is shown in Fig. 10(b). Similarly, only the power supply frequency component f supply ðt Þ is visible on the TFR obtained by direct analysis of phase current, while potential fault features are concealed. With help of motor speed as plotted in Fig. 10(b), and PCT based time-frequency space vector modulus analysis, we further attempt to extract the fault characteristics. The squared SVM is calculated according to (10)–(13), while its waveform, continuous wavelet scalogram and PCT based time-frequency distribution are respectively presented in Fig. 10(c–e). From Fig. 10(d) we can still pinpoint the harmonics of ðr Þ

sun gear rotating frequency f s ðtÞ, which are also visible in baseline case. Such phenomenon again validates the existence of manufacturing/installing errors. Besides, another frequency component which is invisible in baseline case is also pinpointed. Owing to the high time-frequency resolution through IPCT, this component is identified as sun gear fault frequency f s ðt Þ, and such appearance indicates defect on sun gear of the planetary gearbox.

Table 2 Planetary gearbox characteristic frequencies. Sun

Planet

ðrÞ

Rotating frequency

f s ðtÞ ¼ f im ðtÞ

Local fault frequency

2.63f s ðt Þ

ðr Þ

Ring

Carrier

ðrÞ

–

ðrÞ

0.37f s ðtÞ

0.12f s ðtÞ –

0.17f s ðtÞ 0.29f s ðtÞ

ðr Þ

ðr Þ

X. Chen, Z. Feng / Mechanical Systems and Signal Processing 121 (2019) 636–654

(a)

647

(b)

(d)

(c)

(e) Fig. 9. Baseline signal analysis. (a) Waveform of three phase currents, (b) continuous wavelet scalogram of phase current a, (c) waveform of squared SVM, (d) continuous wavelet scalogram of squared SVM, (e) TFR of squared SVM by IPCT.

4.4. Planet gear fault detection Experiment is further carried out under same process as in above two cases, but only the planet gear is locally chipped while other gears are theoretically in perfect condition. We follow the former analysis procedure and compare the results with those in other cases for fault feature extraction. The waveform as well as the continuous wavelet scalogram of phase current Isa ðtÞ are shown in Fig. 11(a and b), from which little fault related information can be conveniently detected. To better reveal fault characteristics, squared SVM analysis is exploited. The waveform, continuous wavelet scalogram and further achieved IPCT based TFR of squared SVM are respectively exhibited in Fig. 11(c–e). Similar to the baseline case, the 1, 2 and 4 times sun gear rotating frequencies are still visible in TFR of squared SVM. In addition, the six times planet gear

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X. Chen, Z. Feng / Mechanical Systems and Signal Processing 121 (2019) 636–654

(a)

(b)

(c)

(d)

(e) Fig. 10. Sun gear fault signal analysis. (a) Waveform of three phase currents, (b) continuous wavelet scalogram of phase current a, (c) waveform of squared SVM, (d) continuous wavelet scalogram of squared SVM, (e) TFR of squared SVM by IPCT.

fault frequency 6f p ðtÞ is also discovered as shown in Fig. 11(e), while it is not found in baseline or sun gear fault case. Such phenomenon shows the high probability of planet gear defect, which is in accordance with the experimental setting. 4.5. Ring gear fault detection In this final case we apply a condition when only the ring gear is faulty and other gears do not have any defects. Still, under a speed up process of 10 s, the three phase motor currents are measured as shown in Fig. 12(a). The motor speed as well as continuous wavelet scalogram of phase current Isa ðtÞare plotted in Fig. 12(b), on which we can only observe

X. Chen, Z. Feng / Mechanical Systems and Signal Processing 121 (2019) 636–654

(a)

(b)

(c)

(d)

649

(e) Fig. 11. Planet gear fault signal analysis. (a) Waveform of three phase currents, (b) continuous wavelet scalogram of phase current a, (c) waveform of squared SVM, (d) continuous wavelet scalogram of squared SVM, (e) TFR of squared SVM by IPCT.

the power supply frequency f supply ðtÞ. To detect potential gear fault, the proposed time-frequency space vector modulus analysis is employed. Following the procedure in Section 2.5, the squared SVM is calculated and its waveform is plotted in Fig. 12(c). To estimate its frequency structure for further polynomial chirplet transform, its continuous wavelet scalogram is generated as shown in Fig. 12(d). Finally, IPCT is applied on squared SVM for clearly exhibiting its time-frequency constitution. As revealed on Fig. 12(e), 4 times the ring gear fault frequency 4f r ðt Þ is pinpointed in addition to harmonics of the sun gear rotating frequency. Considering the difference between such result and results in baseline case, we can effectively diagnose the ring gear fault. In above lab experiments, the fault frequencies of sun, planet and ring gear are successfully detected on TFR of squared SVM, and corresponding gear faults are correctly diagnosed. Such results validate the former derivation of time-varying space vector modulus, and demonstrate the effectiveness of IPCT based time-frequency space vector modulus analysis in planetary gearbox fault diagnosis under variable speed conditions.

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(a)

(b)

(c)

(d)

(e) Fig. 12. Ring gear fault signal analysis. (a) Waveform of three phase currents, (b) continuous wavelet scalogram of phase current a, (c) waveform of squared SVM, (d) continuous wavelet scalogram of squared SVM, (e) TFR of squared SVM by IPCT.

5. Conclusions Gear faults generate load torque oscillations, thus resulting in both PM and AM effects on induction motor current signals. An AM-PM signal model is proposed to better describe induction motor stator currents in planetary gearbox fault cases under variable speed conditions. SVM can highlight gear fault frequencies against power supply frequency dominant in arbitrary single-phase current. The explicit time-varying Fourier spectrum of squared SVM under variable speed conditions is derived, wherein dominant frequencies correspond to the gear fault frequency harmonics. IPCT has good time-frequency readability, and is suited for modulation signal analysis. It is applied to squared SVM to extract time-varying gear fault

X. Chen, Z. Feng / Mechanical Systems and Signal Processing 121 (2019) 636–654

651

features. Both numerical simulation and lab experiments on an induction motor driven planetary gearbox test rig are performed to illustrate and validate the proposed method. Analysis results demonstrate the high time-frequency readability and the capability to highlight gear fault features of IPCT based SVM time-frequency analysis method. Localized fault on the sun, planet and ring gears are successfully diagnosed. Acknowledgement This work is supported by National Natural Science Foundation of China (51475038, 51875034). Appendix A Based on (9)–(12), the squared space vector modulus is structured as

2 3  2 jSVMðtÞj2 ¼ jId ðtÞ þ jIq ðt Þj ¼ I2m 1  axsupply ðtÞ 2  Z t  Z t 2 3 þ I2r xsupply ðt Þ þ g_ ðt Þsin xfault ðsÞds þ u þ gðtÞxfault ðtÞcos xfault ðsÞds þ u 2 0 0  Z t     3Im Ir 1  axsupply ðtÞ xsupply ðt Þ þ g_ ðt Þsin xfault ðsÞds þ u 0   Z t  Z t  p xfault ðsÞds þ u  cos gðtÞsin xfault ðsÞds þ u þ þgðt Þxfault ðtÞcos 2 0 0

ð21Þ

Eq. (21) can further be expanded into 10 terms

jSVMðtÞj2 ¼

Z t  2 3 2 3 3 2 2 xfault ðsÞds þ u Im 1  axsupply ðt Þ þ I2r x2supply ðt Þ þ I2r g_ ðt Þsin 2 2 2 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 1

2

3

Z t  Z t  3 þ I2r g2 ðt Þx2fault ðtÞcos2 xfault ðsÞds þ u þ3I2r xsupply ðtÞg_ ðtÞsin xfault ðsÞds þ u 2 0 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} 4

Z t  þ3I2r xsupply ðtÞgðtÞxfault ðt Þcos xfault ðsÞds þ u 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}

5

6

Z t  Z t  þ3I2r g_ ðtÞgðtÞxfault ðt Þsin xfault ðsÞds þ u cos xfault ðsÞds þ u 0 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} 7

  Z t    p 3Im Ir 1  axsupply ðtÞ xsupply ðt Þcos gðtÞsin xfault ðsÞds þ u þ 2 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 8

Z t   Z t     p 3Im Ir 1  axsupply ðtÞ g_ ðtÞsin xfault ðsÞds þ u cos gðtÞsin xfault ðsÞds þ u þ 2 0 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 9

Z t   Z t     p 3Im Ir 1  axsupply ðtÞ gðtÞxfault ðt Þcos xfault ðsÞds þ u cos gðtÞsin xfault ðsÞds þ u þ 2 0 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} 10

According to the identities of trigonometric functions, the first 7 terms of (22) can be simplified as

ð22Þ

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i 2 3 2 3 3 h 2 Im 1  axsupply ðt Þ þ I2r x2supply ðt Þ þ I2r g2 ðtÞx2fault ðt Þ þ g_ ðtÞ 2 2 4  i  Z t 3 2h 2 2 þ Ir g ðtÞx2fault ðtÞ  g_ ðtÞ cos 2 xfault ðsÞds þ 2u þ 3I2r xsupply ðtÞ 4 0    qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z t gðtÞxfault ðtÞ 2  g_ ðtÞ þ g2 ðtÞx2fault ðtÞsin xfault ðsÞds þ arctan þu g_ ðtÞ 0  Z t  3 þ I2r g_ ðtÞgðtÞxfault ðtÞsin 2 xfault ðsÞds þ 2u 2 0 i 2 3  3 3 h 2 ¼ I2m 1  axsupply ðtÞ þ I2r x2supply ðtÞ þ I2r g2 ðtÞx2fault ðtÞ þ g_ ðtÞ þ 3I2r xsupply ðtÞ 2 2 4    qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z t gðtÞxfault ðtÞ 2 xfault ðsÞds þ arctan þu  g_ ðtÞ þ g2 ðtÞx2fault ðtÞsin g_ ðtÞ 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h



2 i 2 3 2 2 3 2 2 Ir g ðtÞx2fault ðtÞ  g_ ðtÞ Ir g_ ðtÞgðtÞxfault ðtÞ þ þ 4 2 i33 2 2h Z t g2 ðtÞx2fault ðtÞ  g_ 2 ðtÞ 55 xfault ðsÞds þ 2u þ arctan4  sin42 2g_ ðtÞgðtÞxfault ðtÞ 0 Z t  Z t  ¼ AðtÞ þ A1 ðtÞsin xfault ðsÞds þ /1 ðtÞ þ A2 ðtÞsin 2xfault ðsÞds þ /2 ðtÞ

jSVMðt Þjð217Þ ¼

0

ð23Þ

0

where Aðt Þ is the instantaneous average amplitude, /1;2 ðtÞ are time dependent initial phases, A1,2(t) are the time-varying amplitudes of associated frequency components, as

h i  2 2 Aðt Þ ¼ 32 I2m 1  axsupply ðtÞ þ 32 I2r x2supply ðt Þ þ 34 I2r g2 ðt Þx2fault ðtÞ þ g_ ðt Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i 2 2 A1 ðtÞ ¼ 3I2r xsupply ðt Þ g_ ðtÞ þ g2 ðt Þx2fault ðtÞ; A2 ðtÞ ¼ 34 I2r g2 ðtÞx2fault ðtÞ þ g_ ðtÞ ;   h i g2 ðtÞx2 ðtÞg_ 2 ðtÞ ðt Þ þ 2u: þ u; /2 ðtÞ ¼ arctan 2gðtÞg_fault /1 ðtÞ ¼ arctan gðtÞxg_ ðfault tÞ ð tÞ x ðtÞ

ð24Þ

fault

According to the identity

exp½ jzsinðhÞ ¼

1 X

Jm ðzÞexpð jmhÞ

ð25Þ

m¼1

where J m ðzÞ is the first class of Bessel function with order m and argument z, the last 3 terms of (22) can be expanded as a sum of infinite harmonic terms

Z t      p jSVMðt Þjð2810Þ ¼ 3Im Ir 1  axsupply ðtÞ xsupply ðt Þcos gðt Þsin xfault ðsÞds þ u þ 2 0 Z t   Z t     p xfault ðsÞds þ u cos gðtÞsin xfault ðsÞds þ u þ  3Im Ir 1  axsupply ðtÞ g_ ðtÞsin 2 0 0 Z t   Z t     p xfault ðsÞds þ u cos gðtÞsin xfault ðsÞds þ u þ  3Im Ir 1  axsupply ðtÞ gðtÞxfault ðtÞcos 2 0 0 Z t  1 X   p Jk ½gðtÞcos kxfault ðsÞds þ ku þ ¼  3Im Ir 1  axsupply ðtÞ xsupply ðtÞ 2 0 k¼1 Z t      p xfault ðsÞds þ u þ  3Im Ir 1  axsupply ðtÞ cos gðtÞsin 2 0  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z t g ð t Þ x ðtÞ 2 fault 2 xfault ðsÞds þ arctan þu  g_ ðtÞ þ g2 ðtÞxfault ðtÞsin g_ ðtÞ 0 Z t  1 X   Jk ½gðtÞ sin kxfault ðsÞds þ ku ¼ 3Im Ir 1  axsupply ðtÞ xsupply ðtÞ k¼1

ffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 þ Im Ir 1  axsupply ðtÞ g_ ðtÞ þ g2 ðtÞx2fault ðtÞ 2

0

X. Chen, Z. Feng / Mechanical Systems and Signal Processing 121 (2019) 636–654

653

Z t Z t  gðtÞxfault ðtÞ  cos xfault ðsÞds þ arctan x þ u þ gðtÞsin fault ðsÞds þ u g_ ðtÞ 0 0 ffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 2 2  Im Ir 1  axsupply ðtÞ g_ ðtÞ þ g ðtÞxfault ðtÞ 2 Z t Z t  gðtÞxfault ðtÞ  cos xfault ðsÞds þ arctan x ð s Þd s þ u þ u  gðtÞsin fault g_ ðtÞ 0 0 Z t  1 X   Jk ½gðtÞ sin kxfault ðsÞds þ ku ¼ 3Im Ir 1  axsupply ðtÞ xsupply ðtÞ 0

k¼1

ffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 þ Im Ir 1  axsupply ðtÞ g_ ðtÞ þ g2 ðtÞx2fault ðtÞ 2 Z t  Z t 1 X gðtÞxfault ðtÞ  Jl ½gðtÞcos xfault ðsÞds þ lxfault ðsÞds þ lu þ u þ arctan g_ ðtÞ 0 0 l¼1 ffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2  Im Ir 1  axsupply ðtÞ g_ ðtÞ þ g2 ðtÞx2fault ðtÞ 2   Z Z t 1 t X gðtÞxfault ðtÞ  Jn ½gðtÞcos xfault ðsÞds  nxfault ðsÞds  nu þ u þ arctan g_ ðtÞ 0 0 n¼1     Z Z 1 1 t t X X ¼ A3k ðtÞsin kxfault ðsÞds þ /3k ðtÞ þ A4l ðtÞcos ð1 þ lÞxfault ðsÞds þ /4l ðtÞ k¼1 1 X

þ

0

0

l¼1

Z t  A5n ðtÞcos ð1  nÞxfault ðsÞds þ /5n ðtÞ

ð26Þ

0

n¼1

whereA3k ðtÞ,A4l ðt Þ,A5n ðtÞare the time-varying amplitudes of associated frequency harmonics, and /3k ðtÞ,/4l ðtÞ,/5n ðtÞ are corresponding phases, as

  A3k ðt Þ ¼ 3J k ½gðt ÞIm Ir 1  axsupply ðt Þ xsupply ðt Þ; q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 A4l ðt Þ ¼ 32 J l ½gðt ÞIm Ir 1  axsupply ðt Þ g_ ðt Þ þ g2 ðtÞx2fault ðt Þ; q   ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 A5n ðt Þ ¼  32 J n ½gðtÞIm Ir 1  axsupply ðtÞ g_ ðt Þ þ g2 ðt Þx2fault ðtÞ; h i h i ð tÞ ðtÞ þ ð1 þ lÞu; /5n ðtÞ ¼ arctan gðtÞxg_ ðfault þ ð1  nÞu: /3k ðtÞ ¼ ku; /4l ðtÞ ¼ arctan gðtÞxg_ ðfault tÞ tÞ

ð27Þ

Therefore, combining (23)–(24) and (26)–(27), the squared space vector modulus in (21) is summarized in form of

Z t  Z t  jSVMðtÞj2 ¼ AðtÞ þ A1 ðt Þsin xfault ðsÞds þ /1 ðtÞ þ A2 ðtÞsin 2xfault ðsÞds þ /2 ðt Þ 0

þ

1 X k¼1

þ

1 X n¼1

0

Z t  X Z t  1 ð1 þ lÞxfault ðsÞds þ /4l ðtÞ A3k ðtÞsin kxfault ðsÞds þ /3k ðtÞ þ A4l ðtÞcos 0

l¼1

Z t  A5n ðtÞcos ð1  nÞxfault ðsÞds þ /5n ðtÞ

0

ð28Þ

0

References [1] P.D. Samuel, D.J. Pines, A review of vibration-based techniques for helicopter transmission diagnostics, J. Sound Vib. 282 (2005) 475–508. [2] T.Y. Wang, F.L. Chu, Q.K. Han, Fault diagnosis for wind turbine planetary ring gear via a meshing resonance based filtering algorithm, ISA Trans. 67 (2017) 173–182. [3] Y.G. Lei, J. Lin, M.J. Zuo, Z.J. He, Condition monitoring and fault diagnosis of planetary gearboxes: a review, Measurement 48 (2014) 292–305. [4] Z.P. Feng, M.J. Zuo, Vibration signal models for fault diagnosis of planetary gearboxes, J. Sound Vib. 331 (2012) 4919–4939. [5] F.Z. Cheng, Y.Y. Peng, L.Y. Qu, W. Qiao, Current-based fault detection and identification for wind turbine drivetrain gearboxes, IEEE Trans. Ind. Appl. 53 (2017) 878–887. [6] S.H. Kia, H. Henao, G.-A. Capolino, Torsional vibration assessment using induction machine electromagnetic torque estimation, IEEE Trans. Ind. Electron. 57 (2010) 209–219. [7] M.H. Marzebali, S.H. Kia, H. Henao, G.-A. Capolino, J. Faiz, Planetary gearbox torsional vibration effects on wound-rotor induction generator electrical signatures, IEEE Trans. Ind. Appl. 52 (2016) 4770–4780. [8] Z. Daneshi-Far, H. Henao, G.-A. Capolino, Planetary gearbox effects on induction machine in wind turbine: modeling and analysis, in: Proc. 20th ICEM, Marseille, France, Sep. 2012, pp. 1790–1796. [9] J.D. Zhang, J.S. Dhupia, C.J. Gajanayake, Stator current analysis from electrical machines using resonance residual technique to detect faults in planetary gearboxes, IEEE Trans. Ind. Electron. 62 (2015) 5709–5721. [10] J.R. Ottewill, A. Ruszczyk, D. Broda, Monitoring tooth profile faults in epicyclic gearboxes using synchronously averaged motor currents: mathematical modeling and experimental validation, Mech. Syst. Sig. Process. 84 (2017) 78–99. [11] M. Blödt, M. Chabert, J. Regnier, J. Faucher, Mechanical load fault detection in induction motors by stator current time-frequency analysis, IEEE Trans. Ind. Appl. 42 (2006) 1454–1463.

654

X. Chen, Z. Feng / Mechanical Systems and Signal Processing 121 (2019) 636–654

[12] C. Kar, A.R. Mohanty, Monitoring gear vibrations through motor current signature analysis and wavelet transform, Mech. Syst. Sig. Process. 20 (2006) 158–187. [13] S.M.A. Cruz, A.J.M. Cardoso, Stator winding fault diagnosis in three-phase synchronous and asynchronous motors, by the Extended Park’s Vector Approach, IEEE Trans. Ind. Appl. 37 (2001) 1227–1233. [14] J. Zarei, J. Poshtan, An advanced Park’s vectors approach for bearing fault detection, Tribol. Int. 42 (2009) 213–219. [15] S.H. Kia, H. Henao, G.-A. Capolino, Gear tooth surface damage fault detection using induction machine stator current space vector analysis, IEEE Trans. Ind. Electron. 62 (2015) 1866–1878. [16] Z.K. Peng, G. Meng, F.L. Chu, Z.Q. Lang, W.M. Zhang, Y. Yang, Polynomial chirplet transform with application to instantaneous frequency estimation, IEEE Trans. Instrum. Meas. 60 (2011) 3222–3229. [17] Y. Yang, W.M. Zhang, Z.K. Peng, G. Meng, Multicomponent signal analysis based on polynomial Chirplet transform, IEEE Trans. Ind. Electron. 60 (2013) 3948–3956. [18] Z.P. Feng, M. Liang, F.L. Chu, Recent advances in time–frequency analysis methods for machinery fault diagnosis: a review with application examples, Mech. Syst. Sig. Process. 38 (2013) 165–205. [19] X.W. Chen, Z.P. Feng, Iterative generalized time-frequency reassignment for planetary gearbox fault diagnosis under nonstationary conditions, Mech. Syst. Sig. Process. 80 (2016) 429–444. [20] M. Barnes, in: Practical Variable Speed Drives and Power Electronics, Newnes, Oxford, 2003, pp. 41–48. [21] A.R. Mohanty, C. Kar, Fault detection in a multistage gearbox by demodulation of motor current waveform, IEEE Trans. Ind. Electron. 53 (2006) 1285– 1297. [22] S. Kia, H. Henao, G.-A. Capolino, Analytical and experimental study of gearbox mechanical effect on the induction machine stator current signature, IEEE Trans. Ind. Appl. 45 (2009) 1405–1415. [23] M. Abramowitz, I.A. Stegun, in: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972, pp. 359–361.