Research ArticleA Diagnostic System for Speed-Varying Motor Rotary Faults

Chwan-Lu Tseng, Shun-Yuan Wang, Shou-Chuang Lin, Jen-Hsiang Chou, and Ke-Fan Chen

Department of Electrical Engineering, National Taipei University of Technology, Taipei 10608, Taiwan

Correspondence should be addressed to Shun-Yuan Wang; sywang@ntut.edu.tw

Received 26 February 2014; Accepted 18 April 2014; Published 19 May 2014

Academic Editor: Her-Terng Yau

Copyright © 2014 Chwan-Lu Tseng et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

This study proposed an intelligent rotary fault diagnostic system for motors. A sensorless rotational speed detectionmethod and animproved dynamic structural neural network are used.Moreover, to increase the convergence speed of training, a terminal attractormethod and a hybrid discriminant analysis are also adopted.The proposedmethod can be employed to detect the rotary frequenciesof motors with varying speeds and can enhance the discrimination of motor faults. To conduct the experiments, this study usedwireless sensor nodes to transmit vibration data and employedMATLAB towrite codes for functionalmodules, including the signalprocessing, sensorless rotational speed estimation, neural network, and stochastic process control chart. Additionally, Visual Basicsoftware was used to create an integrated human-machine interface.The experimental results regarding the test of equipment faultsindicated that the proposed novel diagnostic system can effectively estimate rotational speeds and provide superior ability of motorfault discrimination with fast training convergence.

1. Introduction

The abnormal vibrations observed during motor rotaryfaults are typically caused by incorrect shaft connectionor friction between rotary components [1–4]. Traditionally,maintenance personnel had to shut down motors for exam-ination and maintenance. However, the vibration detectionmethod, which involves simply attaching an accelerometerto the motor [5, 6] for analysis and measurement, fulfillsthe demand for real-time, online, and minimally invasivediagnosis.

For experiments regarding vibration faults, time-domainsignals are typically converted into frequency-domain signalsbefore analysis. For example, Iorgulescu et al. [7] employeda fast Fourier transform (FFT) to achieve this objective, andGarcia-Perez et al. [8] adopted power spectral density (PSD)to complete a conversion to frequency-domain signals forfurther analysis.

Previous studies of motor faults have typically focused onmeasurements of single rotational speeds; however, changesin rotational speed alter fault characteristic frequencies,resulting in incorrect diagnosis. In recent years, scholars

have begun considering variations in rotational speeds whendiagnosing faults. For instance, Teotrakool et al. [9] usedwavelet packet decomposition to diagnose bearing faultsunder various rotational speeds. Arabaci and Bilgin [10]employed torque speed curves to diagnose rotor bar brokenin motors. Also, Blodt et al. [11] conducted a time-frequencyanalysis to detect torque vibrations under various rotationalspeeds. These studies further adopted a motor current sig-nature analysis to diagnose motor faults with various rotaryfrequencies.

However, these studies required the installation oftachometers and the a priori input of rotary frequencies orrotational speeds into diagnostic systems. Inspired by theresearch done by Ocak and Loparo [12], who employedthe physical characteristics of frequency spectra, this studyproposed a sensorless method that enables a system toautomatically detect the motor rotational speed, therebyreducing the tachometer installation cost.

The development of neural networks in modern intel-ligent motor fault diagnostic systems is quite mature [13,14]. Such networks possess substantial parallel computingabilities, adaptability, fault tolerance, self-learning abilities,

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 310626, 10 pageshttp://dx.doi.org/10.1155/2014/310626

2 Mathematical Problems in Engineering

and memory abilities and are, hence, suitable for diagnosingmotor faults. The disadvantages of traditional neural net-works are that the structure such as the appropriate numberof neurons is difficult to know and the training is timeconsuming. To solve the neuron choice problem, Chuanget al. [15] proposed a dynamic structural neural network(DSNN) that was applied to power quality classification. Still,the training of DSNN is time consuming.

In addition to the above mentioned considerations, thisstudy also incorporated hybrid discriminant analysis (HDA).HDA is a combination of principal component analysis(PCA) and discriminant analysis (DA). These two analysismethods are commonly used in statistics, medicine, andpattern recognition to categorize data and reduce data dimen-sionality. Researchers [16, 17], for example, used PCA in amotor fault analysis and Ayhan et al. [18] combined DA witha neural network for diagnosing the faults caused by rotor bardamage. HDA, which is based on PCA and DA, can be usedto identify optimal data classification results. Converted dataare then fed into a neural network to increase the speed oflearning and discrimination during training.

In this study, motor vibration signals are firstly measured.Subsequently, the vibrational time-domain signals are con-verted into frequency-domain signals using the fast Fouriertransform (FFT) algorithm. The sensorless rotational speedestimation method was used to obtain the current rotatoryfrequency and rotational speed from spectrograms as areference for extracting motor fault characteristics. Also, anormalization procedure was used to preprocess the spectralcharacteristics. To speed up the convergence of trainingdynamic structural neural network, a hybrid discriminantanalysis (HDA) was used to reduce the influence of insignif-icant characteristics. Different from the conventional motordiagnostic system, this study used the improved dynamicstructural network with the terminal attractor method [19]to update the weights (hereafter, called HTAIDSNN) for fastconvergence and proposed a novel diagnostic system whichis capable of detecting rotary faults in motors at differentmotor speeds. Furthermore, in practice, to eliminate theinfluence of noises or outliers, this study also established adual core diagnostic system that comprised the HTAIDSNNand statistical process control (SPC) techniques [20].

2. Vibration Characteristics of Motor RotaryFaults and the Normalization

Mechanical vibrations during the operation of machines areamong the main features of equipment damage or deterio-ration. When a motor experiences a fault, different types ofmotor faults will correspond with different fault character-istics on the spectrogram. Spectrograms are composed of aspecificmotor rotary frequency𝑓

𝑟and frequencymultipliers,

in which different fault types show different peak valuesfor each multiplier. For AC motors, the synchronous speed,rotational speed, and rotary frequencies can be obtainedusing

𝑁𝑠

=120 × 𝑓

𝑠

𝑃,

0 1 2

Am

plitu

de

(×fr)

(a)

Am

plitu

de

0 1 (×fr)

(b)

Am

plitu

de

0 1 2 3 (×fr)

(c)

Figure 1: Spectral characteristics: (a) bent shafts; (b) imbalancedrotors; and (c) loose screw components.

𝑁𝑟

= (1 − 𝑆) × 𝑁𝑠

𝑓𝑟

=𝑁𝑟

60,

(1)

where 𝑁𝑠represents the synchronous speed of the motors,

𝑓𝑠represents the supply frequency, P represents the number

of poles, S represents the slip rate, 𝑁𝑟represents the actual

speed, and 𝑓𝑟represents the rotary frequency.

After obtaining the rotary frequency of a motor, motorfault can be diagnosed using the rotary frequency and ampli-tudes related to frequency multipliers. Three most commontypes ofmotor rotary faults are bent shafts, imbalanced rotors,and loose screw components. The spectral characteristics ofthese types of faults are depicted in Figure 1. As indicatedin Figure 1, different types of faults correspond to differentcharacteristic frequency spectra; thus, the motor fault typecan be determined by examining the specific pattern of afrequency spectrum.

The corresponding spectral characteristics examined dur-ing motor fault diagnoses are called fault characteristicparameters. Equation (2) is the mathematical notion of thetypes of faults, fault characteristics, and fault characteristicparameters

𝐹 (𝑛) = 𝐶 (𝑎𝑛,1

, 𝑎𝑛,2

, 𝑎𝑛,3

, . . . , 𝑎𝑛,𝑘

, . . . , 𝑎𝑛,𝐾

) , (2)

where 𝐹(𝑛) denotes the fault type, 𝐶 represents the faultcharacteristics, 𝑎

𝑛,1, 𝑎𝑛,2

, . . . , 𝑎𝑛,𝑘

, . . ., and 𝑎𝑛,𝑘

represent thefault characteristic parameters of the 𝑛th-segment signal, 𝑘represents the 𝑘th characteristic frequency, and𝐾denotes thenumber of characteristic frequencies.

When faults occur, the amplitudes of specific fre-quency multipliers change accordingly. The fault character-istic parameters used in this study are defined as 0.5𝑓

𝑟,

Mathematical Problems in Engineering 3

1𝑓𝑟, 1.5𝑓

𝑟, 2𝑓𝑟, 2.5𝑓

𝑟, 3𝑓𝑟, and 3.5𝑓

𝑟. As shown in (3), a

vibration eigen-matrix 𝐻 of a healthy motor is compiledusing fault characteristics based on the 𝑛-segment vibrationaldata collected from healthy motors collected in this study:

𝐻 =[[[[

[

ℎ1,1

ℎ1,2

⋅ ⋅ ⋅ ℎ1,𝐾

ℎ2,1

ℎ2,2

⋅ ⋅ ⋅ ℎ2,𝐾

...... d

...ℎ𝑛,1

ℎ𝑛,2

⋅ ⋅ ⋅ ℎ𝑛,𝐾

]]]]

]

. (3)

Themeans and standard deviations of amplitudes of thesecharacteristic frequencies are then calculated by using

𝐻mean value [𝑘] =1

𝑛

𝑛

∑𝑞=1

ℎ𝑞,𝑘

, (4)

𝐻standard deviation [𝑘] = √1

𝑛

𝑛

∑𝑞=1

(ℎ𝑞,𝑘

− 𝐻mean value [𝑘])2

. (5)

The amplitude of each characteristic frequency of the 𝑛-segment signal is normalized by

ℎ𝑧𝑞,𝑘

=ℎ𝑞,𝑘

− 𝐻mean value [𝑘]

𝐻standard deviation [𝑘], (6)

𝐻𝑧

=[[[[

[

ℎ𝑧1,1

ℎ𝑧1,2

⋅ ⋅ ⋅ ℎ𝑧1,𝐾

ℎ𝑧2,1

ℎ𝑧2,2

⋅ ⋅ ⋅ ℎ𝑧2,𝐾

...... d

...ℎ𝑧𝑛,1

ℎ𝑧𝑛,2

⋅ ⋅ ⋅ ℎ𝑧𝑛,𝐾

]]]]

]

. (7)

When normalizing the vibration eigen-matrices of dif-ferent fault types, the amplitude means and standard devi-ations of the characteristic frequencies for healthy motorsare employed as the basis of normalization. Thus, all thenormalization processes are performed using the same basis.As shown in (8), (9), and (10), the normalized fault eigen-matrices 𝐵

𝑧, 𝐼𝑧, and 𝐿

𝑧are obtained by normalizing the

fault eigen-matrices of bent shafts, imbalanced rotors, andloose components, respectively. The compiled normalizedfault eigen-matrices are then used as training data for theneural networks. Fault eigen-matrices with a greater numberof segments provide more training data, thereby increasingthe discrimination ability of the networks. Consider thefollowing:

𝐵𝑧

=[[[[

[

𝑏𝑧1,1

𝑏𝑧1,2

⋅ ⋅ ⋅ 𝑏𝑧1,𝐾

𝑏𝑧2,1

𝑏𝑧2,2

⋅ ⋅ ⋅ 𝑏𝑧2,𝐾

...... d

...𝑏𝑧𝑛,1

𝑏𝑧𝑛,2

⋅ ⋅ ⋅ 𝑏𝑧𝑛,𝐾

]]]]

]

, (8)

𝐼𝑧

=[[[[

[

𝑖𝑧1,1

𝑖𝑧1,2

⋅ ⋅ ⋅ 𝑖𝑧1,𝐾

𝑖𝑧2,1

𝑖𝑧2,2

⋅ ⋅ ⋅ 𝑖𝑧2,𝐾

...... d

...𝑖𝑧𝑛,1

𝑖𝑧𝑛,2

⋅ ⋅ ⋅ 𝑖𝑧𝑛,𝐾

]]]]

]

, (9)

𝐿𝑧

=[[[[

[

𝑙𝑧1,1

𝑙𝑧1,2

⋅ ⋅ ⋅ 𝑙𝑧1,𝐾

𝑙𝑧2,1

𝑙𝑧2,2

⋅ ⋅ ⋅ 𝑙𝑧2,𝐾

...... d

...𝑙𝑧𝑛,1

𝑙𝑧𝑛,2

⋅ ⋅ ⋅ 𝑙𝑧𝑛,𝐾

]]]]

]

. (10)

3. Sensorless Rotational Speed Estimation

Traditionally, an additional tachometer or an encoder isemployed for measuring motor rotational speed. To reducethe measurement cost, this study proposes a sensorlessrotational speed estimation method that determined motorrotational speed by analyzing the vibrational frequency-domain signals of motors.

Previous literature [12] has indicated that the harmon-ics of rotary frequency exhibit greater amplitudes in thevibration frequency spectra of motors. Considering thisfinding, this study first identifies the local maximums inrelevant spectrograms and their corresponding frequencies.After identifying the frequencies that corresponded to thelocal maximums, an indicator 𝛼 with an initial value of0 is assigned to each frequency shown in Figure 2. If thecorresponding frequency for one of the local maximumshas a multiplicative relationship with a larger frequency, 1 isadded to 𝛼. If two relationships are determined, 2 is added tothe 𝛼 value, and so on. Because the actual frequency valuesdo not exhibit exact multiplicative relationships, errors lessthan 3Hz are allowed in this study. Finally, the frequencycorresponding to the greatest 𝛼 (𝛼 = 2) is selected as theestimated rotatory frequency. For example, a frequency of30Hz (the corresponding rotational speed is 1800 rpm) islower than the frequencies of 43.8, 60.7, and 120.2Hz, withthe frequencies of 60.7 and 120.2Hz being two and four times(second and fourth harmonic generation) higher than thatof 30Hz; thus, 𝛼 is set as 2. If the frequency is 60.7Hz,120.2Hz is a frequency that doubles the original value (secondharmonic); therefore, 𝛼 is set as 1. The frequency with thegreatest 𝛼 is selected as the desired rotatory frequency 𝑓

𝑟as

indicated in Figure 3.To verify the speed estimation method using the above

mentioned concept, a four-pole, one-hp, three-phase induc-tionmotormanufactured by the TECOElectric &MachineryCompany (Taiwan) is adopted for this experiment. Theexperiment performed ten times of measurement and esti-mation for rotational speeds from 900 rpm to 1800 rpm.Table 1 summarizes the statistics of the estimated motorrotational speeds. The standard deviations are smaller than4 rpm for each rotational speed. Moreover, the estimationerrors of rotational speeds are smaller than 2 rpm and thusthe accuracy of the proposed sensorless rotational speedestimation method is confirmed.

4 Mathematical Problems in Engineering

20 40 60 80 100 120 1400

10

20

30

40

50

(Hz)

(mV)

𝛼1 = 0

𝛼2 = 2

𝛼3 = 0 𝛼4 = 1𝛼5 = 0

Figure 2: Indicator 𝛼 for local maximums.

20 40 60 80 100 120 1400

10

20

30

40

50

(mV

)

(Hz)

fr = 30Hz

Figure 3: Rotary frequencies obtained using the estimationmethod.

Table 1: Statistics of the estimated motor rotational speeds.

Specifiedrotationalspeed(rpm)

Averagemeasuredspeed

(𝑓𝑟, rpm)

Averageestimatedspeed

(𝑓𝑟𝑒, rpm)

Standarddeviationof 𝑓𝑟𝑒

Averageestimation

error𝑓𝑟𝑒

− 𝑓𝑟

900 899.55 899.90 3.41 0.451000 1000.30 1001.90 3.52 1.601200 1199.56 1200.10 2.94 0.541350 1348.44 1348.80 2.87 0.361400 1398.20 1398.30 2.83 0.101600 1599.31 1600.40 3.55 1.091800 1799.52 1800.00 2.82 0.48

4. The Improved Dynamic StructuralNeural Network with the HybridDiscriminant Analysis

4.1. Improved Dynamic Structural Neural Network. In 2005,Chuang et al. proposed a DSNN capable of self-adjusting thenumber of neurons in the hidden layer [15]. Figure 4 shows

a schematic diagram of the DSNN framework on a pseudo-3D plane. The network is capable of adjusting the number ofneurons in the hidden layer, which can be calculated usingthe following equation:

𝑁ℎ

= 𝑥 ⋅ (𝑦 + 𝑝) (𝑧 + 𝑝) , (11)

where 𝑁ℎrepresents the number of neurons in the hidden

layer, 𝑥 represents the length of the hidden layer, 𝑦 representsthe height of the hidden layer, 𝑧 represents the width ofthe hidden layer, and 𝑝 represents the number to whichthe neurons can be expanded, which can prevent neuronsfrom increasing without limitation. Assume that there are 𝑀neurons in the input layer, 𝑁 neurons in the output layer,and 𝐻

𝑙neurons in the 𝑙th hidden layer, for 𝑙 = 1, 2, . . . , 𝑘.

The total connection in Figure 4 indicates that all neuronsin hidden layer are interconnected. Equation (12) shows theoutput vectors of the neurons in the first hidden layer. Theoutput vectors for neurons in the other hidden layers arepresented in (13), and the output vectors of the output layerare expressed in (14) as follows:

𝑜ℎ1

= 𝑓 [(𝑀

∑𝑚

𝑤ℎ1 ,𝑚

𝑥𝑚

) − 𝜃ℎ1

] , (12)

𝑜ℎ𝑗

= 𝑓 [

[

(

𝑗−1

∑𝑖=1

𝐻𝑖

∑ℎ𝑖

𝑤ℎ𝑗 ,ℎ𝑖

𝑜ℎ𝑖

) − 𝜃ℎ𝑗

]

]

, 𝑗 = 2, 3, . . . , 𝑘. (13)

𝑜𝑛

= 𝑓 [

[

(𝐻𝑘

∑ℎ𝑘

𝑤𝑛,ℎ𝑘

𝑜ℎ𝑘

) − 𝜃𝑛

]

]

, (14)

where 𝑜ℎ1, 𝑜ℎ𝑗, and 𝑜

𝑛are the outputs of the neurons in the first

and 𝑗th hidden layer and the output layer, respectively. 𝑓[⋅] isthe activation function; 𝑤

ℎ1 ,𝑚is the weight value connecting

the input layer and the first hidden layer neurons; 𝑤ℎ𝑗 ,ℎ𝑖

isthe weight value connecting the 𝑖th and 𝑗th hidden layerneurons; 𝑤

𝑛,ℎ𝑘is the weight value connecting the 𝑘th hidden

layer and the output layer neurons; and 𝜃ℎ1, 𝜃ℎ𝑗, and 𝜃

𝑛are

the threshold or bias values of the neurons in the first and 𝑗thhidden layer and the output layer. In (12), (13), and (14), thearguments of all variables are expressed as 𝑚 = 1, 2, 3, . . . , 𝑀,ℎ1

= 1, 2, 3, . . . , 𝐻1, ℎ𝑖

= 1, 2, 3, . . . , 𝐻𝑖, ℎ𝑗

= 1, 2, 3, . . . , 𝐻𝑗,

and 𝑛 = 1, 2, 3, . . . , 𝑁, representing the numbers of neuronsin the respective layers.

4.2. The Terminal Attractor Method. The error function isdefined as (15), where 𝑑

𝑛is the expected output and 𝑜

𝑛is the

actual output vector. The number of neurons in the outputlayer is represented by 𝑛 = 1, 2, . . . , 𝑁. Therefore,

𝐸 =1

2

𝑁

∑𝑛=1

(𝑑𝑛

− 𝑜𝑛)2

. (15)

During the learning process for a neural network, thesteepest descent method is the option most commonlyemployed for reducing the error function in (15) to a mini-mum. However, this method tends to constrain the function

Mathematical Problems in Engineering 5

Hidden layer

Input layer Output layerTotal connection

x

Total connection

Total connection

(y + p) × (z + p)

x1

x2

xM

· · ·

......

......

......

−𝜃h1−𝜃h2 −𝜃h𝑘−1 −𝜃h𝑘 −𝜃n

yN

y2

y1

wh1,mwh2,h1 wh𝑘,h𝑘−1 wn,h𝑘

h1 h2 hk−1 hk

Figure 4: Pseudo-3D schematic of the HTAIDSNN hidden layers.

in a local optimal solution.This leads to the fact that the errorcannot converge to zero within a limited time. To resolvethis issue, Wang and Hsu incorporated heuristic terminalattractors into back-propagation neural networks in 1991 [19].Assume that 𝑤 is the aggregated weight vector and let thederivative of 𝑤 satisfy

𝑑𝑤

𝑑𝑡= −𝛾Δ𝑤, Δ𝑤 =

𝜕𝐸

𝜕𝑤, 𝛾 =

𝐸𝑘

‖Δ𝑤‖2, (16)

where 0 < 𝑘 < 1. Consequently, the derivative of errorfunction becomes

𝑑𝐸

𝑑𝑡= −(Δ𝑤)

𝑇(𝐸𝑘

‖Δ𝑤‖2⋅ Δ𝑤) = −𝐸𝑘, (17)

which determines a terminal attractor. Using the concept, theupdate law of the weight vector in this paper is given as

𝑤 (𝑛 + 1) = 𝑤 (𝑛) + 𝜇𝛾 (𝑛) Δ𝑤 (𝑛) , if 𝛾 > 1,

𝑤 (𝑛 + 1) = 𝑤 (𝑛) + 𝜇Δ𝑤 (𝑛) , if 𝛾 < 1,(18)

where𝜇 is the learning rate.Hereafter, the proposed improveddynamic structural neural network is abbreviated as HTAID-SNN (heuristic terminal attractor improved dynamic systemneural networks) and the learning rate 𝜇 is set to 0.9.

4.3. Applying Hybrid Determinant Analysis to HTAIDSNN.To identify the eigen-matrix of principal components, 𝑋(𝑗),which possesses various classes of data, is considered as

𝑋(𝑗) =

[[[[[[

[

𝑥(𝑗)

11𝑥(𝑗)

12⋅ ⋅ ⋅ 𝑥(𝑗)

1𝑁𝑗

𝑥(𝑗)

21𝑥(𝑗)

22⋅ ⋅ ⋅ 𝑥(𝑗)

2𝑁𝑗

...... d

...𝑥(𝑗)

𝑘1𝑥(𝑗)

𝑘2⋅ ⋅ ⋅ 𝑥(𝑗)

𝑘𝑁𝑗

]]]]]]

]

, (19)

where 𝑗 denotes the class number in sequence, 𝑁𝑗denotes

the times of collection in the 𝑗th class, 𝑘 denotes the numberof factors and characteristics considered, and 𝑥(𝑗)

𝑘𝑁𝑗

denotesthe value of the 𝑘th characteristic at the 𝑁

𝑗th time under the

𝑗th class. Let 𝑥(𝑗)𝑖

= [𝑥(𝑗)

1𝑖, 𝑥(𝑗)

2𝑖, . . . , 𝑥

(𝑗)

𝑘𝑖]𝑇

, which representsthe eigenvector collected in the 𝑖th time under the 𝑗th class,and let 𝑖 = 1, 2, . . . , 𝑁

𝑗. The average of the characteristics in

each class is described using (20), where 𝑚𝑗

∈ R𝑘 and 𝐶represents the number of classes considered. The average ofall class characteristics is shown in (21), where 𝑚 ∈ R𝑘:

𝑚𝑗

=∑𝑁𝑗

𝑖=1𝑥(𝑗)

𝑖

𝑁𝑗

, (20)

𝑚 =∑𝐶

𝑗=1𝑁𝑗

⋅ 𝑚𝑗

∑𝐶

𝑗=1𝑁𝑗

. (21)

The covariance matrix for the data is presented using (22),where 𝑇 represents the transpose or transposition symbol.

6 Mathematical Problems in Engineering

Finally, through (23), the transformation matrix 𝑊PCA isobtained following PCA:

𝑆 =1

𝐶

𝐶

∑𝑗=1

1

𝑁𝑗

𝑁𝑗

∑𝑖=1

(𝑥(𝑗)

𝑖− 𝑚) (𝑥

(𝑗)

𝑖− 𝑚)𝑇

, (22)

𝑊PCA = argmax𝑊

det (𝑊𝑇𝑆𝑊) . (23)

Using (19), (20), and (21), this study independently estab-lishes matrices for within-class variance (24) and between-class variance (25). The ratio of 𝑆

𝐵to 𝑆𝑊

is larger, and thediscrimination is better. Finally, using the Fisher DAmethod,the transformation matrix 𝑊FDA is obtained by using (26)

𝑆𝑊

=𝐶

∑𝑗=1

𝑁𝑗

∑𝑖=1

(𝑥(𝑗)

𝑖− 𝑚𝑗) (𝑥(𝑗)

𝑖− 𝑚𝑗)𝑇

, (24)

𝑆𝐵

=𝐶

∑𝑗=1

𝑁𝑗

⋅ (𝑚𝑗

− 𝑚) (𝑚𝑗

− 𝑚)𝑇

, (25)

𝑊FDA = argmax𝑊

det (𝑊𝑇𝑆𝐵𝑊)

det (𝑊𝑇𝑆𝑊

𝑊). (26)

Although between-class variances can increase during per-forming PCA, within-class variances also increase.Therefore,DA appears to be a more appropriate method for relevantcomputations.However, for data classeswith a limited samplesize or uneven sample distribution, PCA is superior toDA. Toenhance the discrimination of data analysis, HDA [20] is usedto obtain the optimal transformation matrix for given data.Equation (27) shows the transformation matrix of HDA:

(𝑊HDA, 𝜆∗, 𝜂∗) = argmax

𝑊,𝜆,𝜂

det (𝑊𝑇 [(1 − 𝜆) ⋅ 𝑆𝐵

+ 𝜆 ⋅ 𝑆] 𝑊)

det (𝑊𝑇 [(1 − 𝜂) ⋅ 𝑆𝑊

+ 𝜂 ⋅ 𝐼] 𝑊),

(27)

where 𝐼 represents the identity matrix and 𝜆 and 𝜂 representparameters that are adjustable between 0 and 1. Beginningat 𝜆 = 0 and 𝜂 = 0, the parameters can increase by 0.1to obtain the optimal HDA transformation matrix 𝑊HDA.Equation (28) shows the newmatrixHDA 𝑋(𝑗) obtained afterthe transformation:

HDA𝑋

(𝑗) = 𝑊𝑇HDA ⋅ 𝑋(𝑗). (28)

To perform the HDA, firstly the seven frequency-multiplier amplitudes considered in Section 2 are measured20 times for variousmotor faults.The normalized fault eigen-matrices are then compiled using (7), (8), (9), and (10).Thesematrices are substituted into (19), where the total numberof classes 𝑗 is 4 and the times or frequency for every class𝑁𝑗is equal to 20. The covariance matrix 𝑆, within-class

variance matrix 𝑆𝑊, and between-class variance matrix 𝑆

𝐵

are obtained by substituting them into HDA. The adjustableparameters 𝜆 and 𝜂 are set to 0.4 and 0.7, respectively. Finally,a new fault eigen-matrix is obtained with (28). Generally,when using hybrid determinants or principal components,

0 50 100 150 200 250

0

20

40

60

80

100

HealthShaft bending

Rotor imbalanceMechanical looseness

−20−50

2nd

hyb

rid d

iscrim

inan

t com

pone

nt

1st hybrid discriminant component

Figure 5: Distribution of the first two hybrid discriminant compo-nents.

the distribution of the first two components is analyzed[20], because the components are the most suitable forrepresenting data traits. Figure 5 shows the distribution of thefirst two hybrid discriminant components.

Figure 5 shows that HDA provides comprehensive dataclassification and small within-class distance or variance andenhanced data discrimination. When clustering or groupingfor fault classification is used, the relationship between onlytwo component values is typically considered. This studycombined theHTAIDSNNwith the fault eigen-matrices con-verted by HDA. Based on the compilation of the fault eigen-matrices, this study established a neural network trainingmatrix, as shown in Figure 6, where hd

𝑖denotes the 𝑖th hybrid

discriminant or discrimination and 𝑖 = 1, 2, . . . , 7. hd𝑖is used

as the input neuron value. Regarding the parameters of neuralnetworks, the HTAIDSNN possessed seven neurons in theinput layer and four neurons in the output layer. Equation (11)shows the framework parameters of the hidden layer, where𝑥 = 4, 𝑦 = 5, and 𝑧 = 2. Furthermore, the number towhich the neurons could be expanded is 𝑝 = 2; therefore,the initial number of neurons in each hidden layer is 10. Theinitial weight and bias or threshold values are both randomnumbers ranging between −1 and 1. The terminal criterionfor concluding training is 5000 iterations or a minimumerror smaller than 10−6. These criteria prevented excessiveconvergence times when the system is constrained in localoptimal solutions.

Figure 7 represents a comparison of the error conver-gence curves for the data processed by the HTAIDSNN andHTAIDSNN with HDA and PCA. The curves show that theHTAIDSNN method requires 22.73 seconds to complete 195iterations of network training. The HDA method requires14.97 seconds to complete 131 iterations of network training,whereas the PCA requires more time and training iterations,that is, 18.51 seconds and 163 iterations. It is clear that the

Mathematical Problems in Engineering 7

HDA healthy motor

HDA shaft bending

HDA rotor imbalance

HDA mechanical loosenessMatrix 1

Matrix 20

h1,1 h1,2 h1,7

· · ·

· · ·

· · ·

· · ·

· · ·

b1,1 b1,2 b1,7

i1,1 i1,2 i1,7

l1,1 l1,2 l1,7

hd1 hd2 hd7

h20,1 h20,2 h20,7

b20,7

i20,7

l20,7

⋰ ⋰ ⋰ ⋰

⋰

⋰

⋰

⋰

Figure 6: Neural network training matrix.

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Epochs

Mea

n sq

uare

erro

r (M

SE)

Error function

HTAIDSNN with HDA input

HTAIDSNN with PCA input

HTAIDSNN

HTAIDSNN with HDA inputHTAIDSNN with PCA inputHTAIDSNN

Minimal error = 0.000001

Minimal error = 0.000001Time of total learning = 14.974833Epoch of total learning = 131

Minimal error = 0.000001Time of total learning = 18.514932Epoch of total learning = 163

Time of total llearning = 22.725933Epoch of total learning = 195

Figure 7: Convergence errors of HTAIDSNN with/without theaddition of PCA and HDA.

Accelerometerwith wirelesssensor node

Flying wheel

Figure 8: Equipment employed in the motor fault experiment.

Table 2: Classification statistics of the HTAIDSNN-basedmethods.

Motorcondition HTAIDSNN

HTAIDSNNwith PCA

HTAIDSNNwith HDA

Healthymotor 95.9% 98.4% 99.95%

Shaft bending 94.0% 96.3% 98.12%Rotorimbalance 92.2% 98.1% 98.92%

Mechanicallooseness 98.2% 99.5% 99.89%

HTAIDSNN with HDA has superior convergence perfor-mance. The improvement of the convergence rate is mainlydue to the specific classification information provided byHDA.

Table 2 shows the statistics of fault classification usingthe HTAIDSNN-based methods. From the table, the perfor-mance of the HTAIDSNN with HDA is superior to the othertwo methods.

5. Implementation of a FaultDiagnostic System

5.1. Description of Experimental Equipment. The equipmentemployed in this study to conduct the motor vibration exper-iment included a four-pole, one-hp, three-phase inductionmotor manufactured by the TECO Electric & MachineryCompany (Taiwan). An A7200-MA VFD (TECO Electric &Machinery Company) was also installed to change the rota-tional speed, and the source frequency was 60Hz. Figure 8shows the wireless sensor nodes that are attached to themotor in the motor vibration experiment. An MMA7260three-axis accelerometer was installed on a wireless sensornode (employed for transmitting vibration data) to measurevibration displacement.

During the experiment, screw nuts were installed on themotor flying wheel as unbalanced weights to generate thevibration fault signals that had frequency characteristics sim-ilar to that of an imbalanced rotor or bent shaft. The screwson the motor equipment stand were adjusted to emulate

8 Mathematical Problems in Engineering

Signal processing

WSN Motor vibrationmeasurementEstimate rotation

frequency and rpm

Data normalizationand HDA

Motor vibration data

Signal processing andfeature extraction

HTAIDSNN training

Hybrid discriminantanalysis

Non-real-time (training)

HTAIDSNNclassification

SPC control chartmonitoring

Whether there

Yes Yes

LV1(no fault)

LV2(maybe fault)

LV3(fault)

No No

LV2(maybe fault)

Real-time (diagnosis)

Healthy motorvibration measurement

Mean and standarddeviation calculation

Real-time (measurement) Non-real-time (initialization)

is a fault?Whether there

is a fault?

Figure 9: Flowchart of the intelligent speeding-varying rotary fault diagnostic system for motors.

Figure 10: Status of an imbalanced rotor at a rotary frequency of30Hz.

vibration fault signals resulting from loose components asshown in Figure 8.

5.2. Description of System Procedures. As depicted inFigure 9, this study initially retrieves the motor vibrationsignals using wireless sensor nodes and obtains the currentrotary frequency through the sensorless rotational speedestimationmethod. Subsequently, the fault characteristics areextracted from the frequency-multiplier amplitudes of therotary frequency, and the characteristic data are normalized.Finally, after HDA conversion, the neural network feeds the

Figure 11: Diagnosis information for an imbalanced rotor at a rotaryfrequency of 30Hz.

compiled hybrid discriminant fault eigen-matrices into theHTAIDSNN for training of learning and memorizing forthe relationships between the fault characteristics and faultclasses.

In addition to the HTAIDSNN mentioned previously,control charts for stochastic process control (SPC) areincluded for fault diagnosis.The result of this combination is adual core diagnostic system that employed parallel processing[21].

Mathematical Problems in Engineering 9

Figure 12: Status of a bent-shaft rotor at a rotary frequency of22.5Hz.

Figure 13: Diagnosis information for a bent-shaft rotor fault at arotary frequency of 22.5Hz.

The SPC technique distinguishes between random out-liers and abnormal statuses or states regarding the quality ofmonitored products to provide warnings concerning abnor-mal trends in the production process. Since the workplace isin general a noisy environment, the SPC charting used in thiswork is to monitor the shift and drift of the characteristicfrequencies and rule out the abruptly environmental noiseinfluence.The rule adopted in this paper is as follows: if thereis only one point out of control limit, a warning is assigned.An abnormality in this study is defined as the situationsin which two of three continuous or consecutive points arelarger than control limit.

In the dual core diagnostic system, if the neural networkand control chart are free from abnormalities, the systemgenerates an LV1 signal to indicate the normal state. Ifeither the neural network or the control chart generates anabnormal signal, an LV2 signal is generated to indicate thatthe motor may be faulty and require reconfirmation or stay

alerted. If both are abnormal, an LV3 signal is generated toindicate that the motor is faulty and should be immediatelyrepaired by technicians.

5.3. Description of the Human-Machine Interface. Thehuman-machine interface designed in this study primarilyincludes a motor diagnostic system. Figure 10 shows themotor monitoring interface designed for the imbalancedrotor experiment conducted at a rotary frequency of30Hz. This interface comprises a time-domain graph, aspectrogram, and a rotational speed estimation graph forthe vibration signal. The frequency spectrum displays thepeaks of the recent vibration signals at 0.5𝑓

𝑟, 1𝑓𝑟, 1.5𝑓

𝑟, 2𝑓𝑟,

2.5𝑓𝑟, 3𝑓𝑟, and 3.5𝑓

𝑟. Finally, the speed estimation graph

displays the five most recent rotational speeds. The tolerablerotational speed change is 50 rpm. If the change of speedis larger than 50 rpm, the rotational frequency and speedare recalculated accordingly. To exclude the influences ofnoise or disturbance, the changes on the rotational speed areonly displayed after the occurrence of the changes had beenverified.

Figure 11 shows the interface designed for the diagnosticsystem used in the imbalanced rotor experiment conductedat a rotary frequency of 30Hz. This interface includes aneural network detection and Shewhart control chart. Boththe neural network and the control chart in the figure detectan abnormality, and the overall diagnosis is marked in red(LV3). Figure 12 shows vibration signal, frequency spectrum,and the estimated rotational speed of a bent-shaft motor.Figure 13 indicates the diagnosis information of the motor.In the figure, the neural network detects an abnormality, andthe Shewhart chart points out that there is a point out ofcontrol limit. By the rule defined in this work, this situation isacceptable unless the consecutive point is also out of controllimit. Consequently, the overall diagnosis is classified as LV2marked in yellow.

6. Conclusion

This study proposes a novel intelligent diagnostic systemfor speed-varying motors, which improves the convergencerate of traditional dynamic structural neural network train-ing by introducing the terminal attractor and the hybriddiscriminant analysis. The implemented system integratesthe required functions: sensorless speed estimation, neuralnetwork-based diagnosis, and SPC charting, with a human-machine interface to perform rotary fault diagnosis. Thesimulation results indicate that the proposed system isequipped with quick convergence rate for neural networktraining. Also, motor rotary faults are detected effectivelyby the developed diagnostic function and statistics processcontrol chart.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

10 Mathematical Problems in Engineering

Acknowledgment

This work was supported financially in part by the NationalScience Council, Taiwan, under Grant no. NSC102-2221-E-027-082.

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