Analytical Framework of Gearbox Monitoring Based on the Electro-mechanical Coupling Mechanism

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ScienceDirect Energy Procedia 105 (2017) 3138 – 3145

The 8th International Conference on Applied Energy – ICAE2016

Analytical framework of gearbox monitoring based on the electro-mechanical coupling mechanism Kai Chena, Jibin Hua, Zengxiong Penga a

Beijing Institute of Technology,5 South Zhongguancun Street, Haidian District, Beijing postcode:100081, China

Abstract Along with the development of the Hybrid Electric Vehicles (HEVs), the permanent magnetic synchronous motor (PMSM) and the planetary gearbox is widely used because of its high power density. There is much significant electro-mechanical coupling phenomenon in the mechanical installations. The aim of this paper is the analytical study of a planetary gearbox by using the stator current signature analysis in the PMSM based on the electro-mechanical coupling mechanism. This paper proposes a mathematical framework based on the electro-mechanical coupling model of the PMSM-planetary gearbox system, considering the load torque oscillation and the time-varying mesh stiffness. The model simulation result shows that from the mathematical framework, the load torque oscillation frequency and the time-varying mesh stiffness frequency can be predicted through the current frequency. The proposed mathematical model is verified by the model simulation. And a test-bed based on a 60kW three-phase PMSM connected to a planetary gearbox has been used. Fourier transform is applied to the demodulated torque signal and current signal for denoising and removing the intervening neighbouring features. © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 8th International Conference on Applied Energy. Keywords: PMSM, motor current signal analysis, fault diagnosis, planetary gearbox, Fourier transform

1. Introduction Planetary gearboxes and permanent magnet synchronous machines (PMSM) are preferred in many drive-trains applications, such as wind turbines, maritime and automobiles, as they allow for a larger transmitted torque to weight ratio, easily control, high reliability and efficiency, and reduced maintenance costs. However, failures in these components can often result in catastrophic accidents, large down-times and expensive maintenance. Therefore, detecting such failures at early stage can significantly reduce the associated capital losses and down-time. The study of the gearboxes fault detection in electromechanical systems has been mainly performed using vibration and acoustic based methods [1-5]. Vibration signals have numerous disadvantages like signal background noise due to external perturbations, sensitivity to the censor location, and their invasive installation environment [6]. Motor current signal analysis(MCSA) can present an alternative for mechanical analysis based on the electrical signatures of electric motors. The potential of stator current analysis is mentioned in [7-9]. It has been shown that the mechanical faults can be reflected on the motor current. Some cases are studied where load torque oscillations cause

1876-6102 © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 8th International Conference on Applied Energy. doi:10.1016/j.egypro.2017.03.677

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changes in the stator current spectrum, showing the feasibility of the MCSA monitoring of mechanica l perturbations. Mechanical faults such as static and dynamic air-gap eccentricities and the bearing faults have been successfully detected through AC motor stator current [10-12]. Publications related fixed-axis gearbox monitoring using MCSA in induction machine-based electromechanical systems are presented in [13-17]. Primarily, a multistage gearbox for a mechanical fault case is studied in [14] and [15]. It is shown that the rotating as well as the mesh frequency components can be detected in the stator current spectrum. However, the monitoring of the planetary gearbox using MCSA in PMSM-based electromechanical systems has been challenging so far because of the difficulties in modeling the planetary gear-set such as a large number of degrees of freedom and nonlinearity in the PMSM model. The electro-mechanical drive-train consisting of a PMSM connected to a load through a planetary gearbox is modeled by Jaspreet [18]. Then the PMSM stator current response under healthy and faulty gear tooth conditions can be evaluated. Nevertheless, Jaspreet show the result without any proposed explicit theoretical development for the analyzed frequency components. Fourier transform(FT) is used to decompose a signal into many levels with different frequency bandwidth. This will demodulate the signal for denoising and removing the intervening neighbouring features. In this paper, the electro-mechanical drive-train consisting of a PMSM connected to a load through a planetary gearbox is modeled. The PMSM is modeled by Park’s equation [19]. The planetary gearbox is modeled by a lumped multi-body dynamic model [20]. If a fault exists in one of the gears, such as spalled tooth, it results in a reduced gear meshing stiffness as the faulty tooth passes through the gear meshing [6]. Then a theoretical framework based on the observation of the torque spectrum is developed for a PMSM connected to a planetary gearbox without any faults for the purpose of mechanical analysis. It is shown that due to the torsional vibration induced by the load oscillation in the output wheels and the stiffness variation of the gear teeth contact, the gearbox adds the rotation and mesh frequency components into the torque signature. This effect makes the stator current multicomponent phase modulated. Then t he Fourier transform(FT) is used to decompose the stator current signal in the test-bed to verify the theoretical framework. 2. Dynamic model of the electro-mechanical drive-train 2.1. PMSM model A PMSM model and the electromagnetic torque can be represented in a representation as [21]

Λ

A Zr ˜ Λ  V  R ˜ L1 ˜ G ˜ O cf

Te

 3 / 2  P / 2  ΛT  GT ˜ O c  L1

T

(1)

˜W˜Λ

Where, Λ is the magnetic flux vector and V is the terminal voltage vector, I is the current vector, vector G is a constant matrix, R is the resistance matrix, P is the number of the poles and L is the inductance matrix, A(ωr) is a function of electrical rotor speed ωr, and are described as T

Λ

ª¬Oqs

Ods º¼ ˈV

G

>0 1@

T

ˈA Zr

ª¬vqs

T

vds º¼ ˈI

L1 ˜  Λ  G ˜ O cf ˈR

  R ˜ L1  Zr ˜ W ˈL

ª Lq « ¬0

0º ˈW Ld ¼»

diag > rs

rs @

ª 0 1º « » ¬ 1 0 ¼

(2)

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it is obvious that the stator current can be influenced by the rotor speed variation that may arise due to torsional vibration excited by the gearbox failures. 2.2. Planetary Gearbox model In subsequent analysis, a four-planet equally spaced planetary gear set is assumed in the drive-train. For purpose of simulations, a lumped-parameter multi-body dynamic model of the planetary gears is developed, details of which can be found in [22], which can be expressed in matrix form as MQ  t   C  Cb Q  t   K  K b Q  t

F t

(3)

where M is the mass matrix, C is the damping matrix, K(t) is the gear meshing stiffness matrix, Cb and Kb are the bearing damping and stiffness matrices, F(t) is the external applied torques vector, and Q(t) is the degrees of freedom vector. Q(t) for the two coordinates of translational vibration which is described as Q=[uc ur us up1, … , upN]T (4) 2.3. The Overall Model In the proposed electro-mechanical drive-train, the PMSM is connected to the sun gear of the planetary gearbox. The ring gear is fixed and the shafts are assumed to be rigid. The carrier of the planetary gear box is connected to a load whose required torque is proportional to the square of its angular velocity. Such load characteristic is often encountered in pumps, fans, propellers, etc. Equation (1) and (3) can be combined in a single state space representation describing the states of the proposed electro-mechanical drive-train as

ª/ º « » «Q » «Q » ¬ ¼

0 0 ª A  Zr º ª / º ªV  R ˜ L1 ˜ G ˜ O cf º » « »« » « 0 I 0 » « 0 » «Q »  « » «¬ S  M 1  K  K b  M 1  C  Cb »¼ «¬Q »¼ «¬ M 1 F0 ¼

(5)

Where, F0 represents the external force and torque vector applied on the planetary gearbox from the load. The effect of the electromagnetic torque acting on planetary gearbox is captured in the matrix S. 3. Influence of torque oscillations on stator current The method used to study the influence of the periodic load torque variation on the stator current is based on the MMF [24]. First, the rotor and stator MMFs are calculated, which are directly related to the current flowing in the windings. The second important quantity is the air gap permeance Λ, which is directly proportional to the inverse of the air gap length g. The magnetic field in the airgap can then be determined by multiplying the permeance by the sum of rotor and stator MMFs. The equivalent magnetic flux in one phase is obtained by integration of the magnetic field in each turn of the phase winding. The induced phase voltage, related to the current by the stator voltage equation, is then deduced from the magnetic flux. 3.1. Effect on rotor MMF Under a mechanical fault, the load torque as a function of time is modelled by a constant component Tconst and an additional component varying at the characteristic frequency fc (which can be for example

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the rotational frequency fr). The first term of the variable component Fourier series is a cosine with frequency fc. For the sake of clarity, higher order terms are neglected in the following, and only the fundamental term is considered. The load torque can therefore be described by Tload  t Tconst  Tc cos Zc t  Tmesh cos Zmesh t

(6)

Where Tc is the amplitude of the load torque oscillation and ωc=2πfc, Tmesh is the amplitude of the torque oscillation due to the periodic changes in tooth stiffness, and ωmesh=2πfmesh. We can deduce from the torque and the rotor speed that the mechanical rotor position θr is:

Tr t

³

t

t0

Zr W dW

Tc T cos Zc t  mesh cos Zmesh t  Zr 0t 2 2 J Zc J Zmesh

(7)

Then θ′ is the mechanical angle in the rotor reference frame (R) and can be obtained as

T c T  Zr 0 t 

Tc T cos Zc t  mesh cos Zmesh t 2 2 J Zc J Zmesh

(8)

So the rotor MMF and stator MMF is calculated as:

Fr T , t Fs T , t

ª º T T cos Zmesh t » Fr cos « pT  pZr 0t  p c 2 cos Zc t  p mesh 2 J Zc J Zmesh ¬ ¼ Fr cos  pT  pZr 0t  M s

(9)

3.2. Effect on Flux density and Stator Current The airgap flux density B(θ,t) is the product of total MMF and airgap permeance Λ. The relation between the flux and the stator current in a considered phase is given by the stator voltage equation. So the resulting stator current will be in a linear relation to the time derivative of the phase flux Φ(t) and will have an equivalent frequency content.

I  t ist  t  irt  t

I s sin  pZr 0t  Ms  I r sin pZr 0t

 I rc ª¬cos  pZr 0  Zc t  cos  pZr 0  Zc t º¼  I rmesh ª¬cos  pZr 0  Zmesh t  cos  pZr 0  Zmesh t º¼

(10)

Therefore, the stator current I(t) can be considered as the sum of two components. The term ist(t) results from the stator MMF. The term irt(t), which is a direct consequence of the rotor MMF, shows the phase modulation due to the considered torque oscillations. 3.3. Stator current frequency analysis

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The FT of the equation (10) is :

I rt  f

Ir

S

ªG  f  f r 0  G  f  f r 0 º¼ j¬

 I rcS ª¬G  f   f r 0  f c  G  f   f r 0  f c º¼

 I rcS ª¬G  f   f r 0  f c  G  f   f r 0  f c º¼

 I rmeshS ª¬G  f   f r 0  f mesh  G  f   f r 0  f mesh º¼

 I rmeshS ª¬G  f   f r 0  f mesh  G  f   f r 0  f mesh º¼

(11)

Then it is very clear that the fault leads to side band components of the fundamental at |fs±fc| and |fs±fmesh|. 4. test verifying The proposed theoretical framework is experimentally verified using an electromechanical drivetrain as shown is figure 6.. The electricity generated by the load generator is dissipated at a resistive load bank. Three current sensors measure the current of the PMSM. And the rotor speed of the PMSM is measured by a brushless resolver. The torque of the load generator is measured by a torque measurement system. All of the above data is collected by a data acquisition system. The drivetrain was operated under different operating conditions as shown in table I to show both the effectiveness of the proposed method over the existing ones and its robustness to detect faults under different operating conditions. The test data is disposed by the FT algorithm. PMSM

Torque exciter

Transmission

load generator

Figure 6 electrical drivetrain test rig Table 1. parameters for experimental operating conditions Case 1 2 3 4

Speed(rpm) 1000 1000 1000 500

Load torque(N·m) 20 10 5 10

Load torque frequencyf load (Hz) 10 30 10 5

Gear fault frequency f GF(Hz) 16.67 16.67 16.67 8.33

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The stator current spectrum of the healthy gearbox and unhealthy gearbox under operation condition 1 is shown as an example. For the existing load, the input and output gearbox rotating frequencies are particularly sensitive in the bandwidth. Moreover, this effect is also present in the bandwidth related to the 1st time harmonics. No matter what the load is in the Table I, the frequency components are not changed. It is supposed that the load frequency components fload and the fault frequency f GF are located at 10Hz and 16.67Hz. These observations are collected in Table II. f s/Hz

Fault frequency

f L =10Hz

f L=30Hz

f L=5Hz

50

16.67

40

20

-

60

80

-

33.33

33.33

-

66.67

66.67

-

-

-

20

-

-

30

-

-

16.67

-

-

33.33

25

8.33

In order to examine the influence of the load frequency on the stator current, the test is performed for three main frequencies 10Hzǃ30Hz with the line frequency 50Hz and 5Hz with the line frequency 25Hz. From the table II, the results show that the stator frequencies accord with the equation (27), namely is |fs±fc| and |fs±fmesh|. 5. Conclusions In this paper, a planetary gearbox study method has been proposed. It is based on the electromechanical model. The previous works on this subject are limited to observe the planetary gearbox characteristics without any theoretical framework. In this paper, a new formulation for the load torque oscillation and the fault gear mesh stiffness is presented, using the FT method to analyse the planetary gearbox. Then the following conclusions are drawn in this paper. x A new formulation for the load torque oscillation and the time-varying mesh stiffness in electromechanical coupling system is presented. It is very clear that the fault leads to side band components of the fundamental at fs±fc and fs±fGF, where fc is the load oscillation frequency and fGF is the gear fault frequency. x The electro-mechanical model has been created. And using the FT method, the load torque oscillation and gear fault are identified. In Fig. 5(a), the frequency 40Hz and 50Hz shows the load oscillation influence on the stator current. Fig. 5(b) shows the influence of gear fault on the stator current, which has the amplitude on the frequency 33.3Hz and 66.67Hz. x Application of FT to decompose the demodulated current signal of a test-bed to various levels and verifying the frequency domain signature at the theoretical level. Figure 8~Figure 11 show that stator current frequency reflect the various load oscillations and the gear fault. The electro-mechanical coupling mechanism is widely used in mechanical engineering. This paper extends it to the fault diagnosis in the planetary gearboxes. There are also other faults in the planetary gearboxes such as transmission error, wheel eccentricity, and et.al. These faults may be taken into account in the further studies.

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Acknowledgements This work is supported by the National Science Foundation of China (NSFC) ‘51305032’. References [1] Randall. R. B, “New method of modeling gear faults,” American Society of Mechanical Engineers, 1981. [2] B. Li, M. Chow, Y. Tipsuwan, and J. Huang, “Neural- network based motor rolling bearing fault diagnosis,” IEEE Trans. Ind. Electron., vol. 47, no. 5, pp. 1060-1069, Oct.2000. [3]Samuel. P. D, and Pines. D. J, “A review of vibration-based techniques for helicopter transmission diagnostics,” Journal of Sound and Vibration, vol. 282, no. 1-2, pp.475-508, 2005. [4]Y. Zhan, and V. Makis, “A robust diagnostic model for gearboxes subject to vibration monitoring,” Journal of Sound and Vibration, vol. 290, no. 3-5, pp. 928-955, Mar. 2006. [5]Zuo. M. J, and Xianfeng. F, “Gearbox fault detection using Hilbert and wavelet packet transform,” Mechanical Systems and Signal Processing, vol. 20, no. 4, pp. 966-982, 2006. [6]Feki. N, Clerc. G, and Velex. P, “An integrated electro- mechanical model of motor- gear units-Applications to tooth fault detection by electric measurements,” Mechanical Systems and Signal Processing, vol. 29, pp. 377-390, 2011. [7]W. T. Thomson, “On- line current monitoring to detect electrical and mechanical faults in three-phase induction motor drives,” in Proc. Int. Conf.Life Manage. Power Plants, pp. 66-73, Dec. 1994. [8]J. R. Stack, T. G. Habetler, and R. G. Harley, “Fualt classification and fault signature production for rolling element be arings in electric machines,” IEEE Transaction. Ind, Appl., vol.40, no.3, pp.735-739, May/Jun. 2004. [9]M. Blodt, D. Chabert, J. Regnier, and J. Faucher, “Mechanical load fault detection in induction motors by stator current t imefrequency analysis,” IEEE Trans. Ind. Appl., vol. 42, no. 6, pp. 1454-1463, Nov./Dec. 2006. [10]Ebrahimi. B. M, Faiz. J, and Roshtkhari. M. J, “Static-, dynamic-, and mixed-eccentricity fault diagnoses in permanentmagnet synchronous motors,” IEEE transactions on industrial electronics, pp. 4727-4739, 2009. [11]S. Nandi, R. M. Bharadwaj, and H. A. Toliyat, “Performance analysis of a three-phase induction motor under mixed eccentricity condition,” IEEE Transactions of Energy Conversions, vol. 17, no. 3, pp. 392-399, Sep. 2002. [12]R. Schoen, T. Habetler, F. Kamran, and R. Bartheld, “Motor bearing damage detec tion using stator current monitoring,” IEEE Transactions on Industrial Application, vol. 31, no. 6, pp. 1274-1279, Nov./Dec. 1995. [13]S. Rajagopalan, T. G. Habetler, R. G. Harley, T. Sebastian, B. Lequesne, “Current/voltage-based detection of faults in gears coupled to electric motors,” IEEE Transactions on industry applications, vol. 42, no. 6, pp. 1412-1420, 2006. [14]A. R. Mohanty, and C. Kar, “Fault detection in a multistage gearbox by demodulation of motor current waveform,” IEEE Transactions on industrial electronics, vol. 53, no. 4, pp. 1285-1297, 2006. [15]A. R. Mohanty, and C. Kar, “monitoring gear vibration through motor current signature analysis and wavelet transform,” Mechanical Systems and Signal Processing, vol. 20, no. 1, pp. 158-187, Jan. 2006. [16]C. Kar, and A. R. Mohanty, “Vibration and current transient monitoring for gearbox fault detection using multiresolution Fourier transform,” Journal of sound and vibration, vol. 311, no. 1-2, pp. 109-132,Mar. 2008. [17]H. K. Shahin, H. Humberto, and G. A. Capolino, “Analytical and experimental study of gearbox mechanical effect on the induction machine stator current signature,” IEEE Transactions on industry applications, vol. 45, no. 4, pp. 1405-1415, July/August. 2009. [18]Jidong. Zhang, Liu. Hong and J. S. Dhupia, “Gear fault detection in planetary gearbox using stator current measurement of AC motors,” ASME 2012 5th Annual Dynamic Systems and Control Conference, Oct. 2012. [19]P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, “Analysis of electric machinery and drive systems,” IEEE Press, New York, 2002. [20]A. Kahraman, “Natural modes of planetary gear trains,” Journal of Sound and Vibration, vol. 173, no. 1, pp. 125-130, 1994. [21]P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, “Analysis of electric machinery and drive system,” IEEE Press, New York, 2002.

Kai Chen et al. / Energy Procedia 105 (2017) 3138 – 3145 [22]A. Kahraman, “Natural modes of planetary gear train,” Journal of sound and vibration, vol. 173, no. 1, pp. 125-130, 1994. [23]R. Yacamini, K. S. Smith, and L. Ran, “Monitoring torsional vibrations of electro- mechanical systems using stator currents,” Trans. ASME, J. Vib. Acoust. Stress Reliab. Des., vol. 120, no. 1, pp. 72-79, Jan. 1998. [24]P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, “Analysis of electric machinery and drive system,” IEEE Press, New York, 2002

Nomenclature Λ

the magnetic flux vector

V

the terminal voltage vector

I

the current vector

R

the resistance matrix

P

the number of the poles

L

the inductance matrix

M

the mass matrix

C

the damping matrix

K(t)

the gear meshing stiffness matrix

Tc

the amplitude of the load torque oscillation

Tmesh

the amplitude of the torque oscillation due to the periodic changes in tooth stiffness

θr

the mechanical rotor position

θ′

the mechanical angle in the rotor reference frame (R)

Biography Kai Chen was born in hu’nan, China, in August 1988. He received the M.S.degree in college of mechanical vehicle engineering, from Hunan university, and will received Ph.D in school of mechanical engineering from Beijing institute of Technology. He is currently studying the unbalanced magnetic force.

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