University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies Legacy Theses 2001 Bearing condition monitoring and fault diagnosis Chen, Ping Chen, P. (2001). Bearing condition monitoring and fault diagnosis (Unpublished master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/23398 http://hdl.handle.net/1880/40657 master thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca
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University of Calgary
PRISM: University of Calgary's Digital Repository
Graduate Studies Legacy Theses
2001
Bearing condition monitoring and fault diagnosis
Chen, Ping
Chen, P. (2001). Bearing condition monitoring and fault diagnosis (Unpublished master's thesis).
University of Calgary, Calgary, AB. doi:10.11575/PRISM/23398
http://hdl.handle.net/1880/40657
master thesis
University of Calgary graduate students retain copyright ownership and moral rights for their
thesis. You may use this material in any way that is permitted by the Copyright Act or through
licensing that has been assigned to the document. For uses that are not allowable under
copyright legislation or licensing, you are required to seek permission.
Downloaded from PRISM: https://prism.ucalgary.ca
NOTE TO USERS
This reproduction is the best copy available.
THE UNIVERSITY OF CALGARY
Bearing Condition Monitoring and Fault Diagnosis
by
Ping Chen
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
DEPARTMENT OF MECHANICAL AND MANUFACT'UMNG ENGINEERING
CALGARY, ALBERTA
DECEMBER, 2000
0 Ping Chen 2000
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ABSTRACT
Bearing condition monitoring and fault diagnosis have been studied for many years.
Popular techniques include those through advanced signal processing and pattern
recognition technologies. Recently, some interesting results were published using pattern
recognition for bear& diagnosis by means of feahms extracted from vibration signals
through time domain and kquency domain analyses [Sun, et al, 19981. In this work,
segmentation parameters are proposed to f i d e r improve the sensitivity and reliability of
the technique. Parameters extracted from the segmentation analysis reflect the variation
of vibration signals associated with the bearing dynamics. A three-layered artificial
neural network is applied to accomplish the non-linear mapping fkom the feature space to
the two dimensional classification space. The mapping is conducted to create the best
cluster effect for training samples belonging to the same class. Successll non-linear
mapping through the neural network eliminates intra-class transformations as used in
[Sun, et al, 19981. Numerical experiments are performed to illustrate the effectiveness of
the method.
I am deeply indebted to Dr. Q. Sun, my supervisor, who has been a strong source of
inspiration throughout my project work. I have benefited greatly &om her invaluable
guidance and motivation. Her guidance has been very supportive in helping me complete
this project.
1 would also like to give a special thanks to the National Research Council of
Canada for its tinancia1 support and the Association of American Railroad for providing
the bearing testing &a
TABLE OF CONTENTS
. . .......................................................................................................... APPROVAL PAGE u
ACKNOWLEDGEMENTS .Om.mO.mOm..O.....mO. ~ ~ ~ ~ m o ~ ~ w w ~ m ~ H ~ ~ ~ ~ ~ m ~ ~ ~ o ~ ~ m ~ e m ~ m ....mm...~w..~.....m~~wm~.m.~m~m~...m iv
TABLE OF CONTENTS ................................................................................................. v
. . LIST OF TABLES .......................................................................................................... vu ... LIST OF FIGURES ....................................................................................................... vrrr
CHAPTER ONE : INTRODUCTIONoooooooooooeoooooooooooooooooooooooooooooooooo.ooooo.o~oo 1
1.6.1 Time Domain Techniques ............................................................................. 10 1.6.2 Frequency Domain Techniques ...................................................................... 1 1 1.6.3 Time-Frequency Analysis ............................................................................ IS
........................................................................................... 1.7 Objective of the Thesis 16 ...................................................................................... 1.8 Organization of the Thesis 18
....................................................................... 3.2.1 Probability Density Function 4 1 ................................................................ 3.2.2 Root Mean Square and Peak Value 43
CHAPTER SIX: CONCLUSION AND FUTURE WORK ......m..........m 1 1'8
.............................................................................. 6.1 S u w of Results Obtained 118 .................... 6.2 Limitations of the Present Method and Directions for Future Work 121
Multilayer feedforward artificial neural networks have been widely adopted for
many ANN applications. They have been applied successfidly to solve complicated
problems by training them in a supervised manner with the popular algorithm known as
the error back-propagation algorithm maykin, 19941. The basic idea of back-propagation
was first described by Werbos in his Ph.D. thesis [Werbos, 19741, in the context of
general networks with neural networks representing a special case. The development of
the back-propagation algorithm represents a "landmark" in neural networks in that it
74
provides a computationally efficient method for the training of multiplayer perceptrom.
The trained network based on the error back-propagation algorithm often produces
surprising results in applications where explicit derivation of input-output relationship is
almost impossible.
Training of feedforward neural networks takes place in an iterative fashion. Each
iteration cycle involves a forward-propagation pass followed by an error backward-
propagation pass to update the connection weights. Figure 4.4 depicts a portion of the
multiplayer neural network with the two passes.
The forward-propagation pass starts when the input nodes receive their activation
levels in the form of an input pattern. Then, forward-propagation proceeds through the
hidden layers up to the output layer by computing the activation levels of the nodes in
those layers. Finally, a set of outputs is produced as the actual response of the network.
During the forward pass the synaptic weights of the network are all fixed.
Weight adjustment is accomplished by propagating the error function of the output
back through the net and modifying all the weights. The iterative method propagates error
hc t ion required to adapt weights back &om nodes in the output layer to nodes in the
hidden layers in accordance with the training rule. The weights are adjusted so as to make
the actual response of the network move closer to the desired response.
Training sets are repeatedly presented and weights modified until the error between
the predicted and actual output is less than a specified value (error criterion). Once the
neural network has been trained in this way, it should be possible to relate input patterns
with the appropriate output patterns [Chiou, et al., 1992). To use the trained ANN, a new
input set is simply presented to the network and the network calculates an output solution.
Properly-trained ANNs are able to give reasonable answers when presented with inputs
that they have never seen. Typically, a new input will lead to an output with similar
features to the comct output for input vectors with similar features used in training.
75
Therefore, it is possible to train a neural network on a representative set of input/target
pairs and get good results without training it on alf possible input/output pairs.
4.2.2 Error Back-Propagation Training Algorithm
Before the network could be used for the non-linear transforming purposes in this
work, we decide to apply the supervised learning technique to train the neural network
using a set of known inputs and corresponding outputs.
Inputs are the features extracted from bearing vibration signals with known bearing
conditions. Desired outputs are the cluster centers arbitrarily chosen for each class.
Assume there are total K classes, K cluster centers u,, k = 1, . . ., K, are chosen in the first
quadrant of a 2D coordinate h e in order to locate the desired output associated with
each of the K classes. Although their arrangement is somewhat arbitrary, we placed them
evenly on a unit circle in the first quadrant of a 2D space. Non-linear mapping is applied
to cluster the entire samples belonged to the same category in the feature space around
their own specified cluster center ar in the classification space.
Consider a three-layered ANN with only one hidden layer as shown in Figure 4.5. In
the figure, index i refers to nodes at the input layer, index j ~ f e r s to nodes at the hidden
layer, and index k refers to nodes at the output layer. widenotes the weight of the
connection between node i in the input layer and j in the hidden layer, while v~denotes
the weight of the connection between node j in the hidden layer and k in the output layer.
Assume xi, i = 1, . . ., I, are input signals, a neuron j in the hidden layer can be described by
writing the following pair of equations:
and
where 6, represents the internal activity level of the neuron j, y, is the output of the neuron
j and f () is the activation hmction of the hidden layer.
Similarly, a neuron k in the output layer can be described by writing the following
pair of equations:
and
where c, represents the internal activity level of the neuron k, or is the output of the
neuron k and fc) is the activation hc t ion of the output layer which is assumed to be the
same as the hidden layer.
Let p be the index representing the training set and P the total number of samples
involved in training the network. At any iteration, the sum of squared errors for the pth
training sample between the target and actual output is defined as:
Where:
K is the number of output nodes of the network.
O k d represents the target output at node k.
0, is the actual output at node k.
The average squared error Em among all the training sets can be calculated as
Obviously, the value of the error fiulction depends on the weights of the network.
For a giving training set, Em represents the performance function as the measure of
training set learning performance. The objective of the learning process is to minimize the
performance function Em through adjusting the weights at every neuron. We consider a
simple method of tmhing in which the weights are updated on a sample-by-sample basis.
The adjustments to the weights are made in accordance with the respective errors
computed for each sample presented to the network. The arithmetic average of these
individual weight changes over the training set is therefore an estimate of the true change
that would result from modifying the weights based on minimizing the performance
function E, over the entire training set. The gradients calculated at each training pattern
are added together to determine the change in the weights. It can be seen that the
performance function depicts the accuracy of the neural network mapping after a number
of training cycles have been implemented.
The gradients of the error surface with respect to the weights between the output and
hidden layers d E / h h is calculated as follows:
Substituting Eq. (4.10) into the above equation leads to:
In Equation (4.13), argument p is omitted from E for brevity. The gradient aE/&k,
determines the direction of search in weight space for the weight v,. Change in weights
between the output and hidden layers Avk is proportional to the gradient aE/&& :
where 7 is the Leaming rate of the back-propagation algorithm. At the nth iteration of the
training process, weights at every neurons in the output layer are updated using the
increment calculated in eq. (4.16):
Now we consider the weight adjustment from input Layer to the hidden layer, the
gradients of the error surface with respect to the weights between the hidden and input
layers dE/aWji is calculated as follows:
Combine Eq. (4.9) and (4.10) with the above equation., we can obtain:
The weight adjustment between the hidden and input layers Aw,~ is proportional to
the gradient dE/&vji :
At the nth iteration of the training process, weights at every neurons in the hidden layer
are updated using the increment calculated in Eq. (4.20):
Backpropagation networks often have one or more hidden layers with sigmoid
activation fimction followed by an output layer of linear or sigmoid bction. Multiple
layers of neurons with nonlinear activation fbctions allow the network to learn nonlinear
and linear relationships between input and output vectors. An example of a continuously
differentiable nonlinear activation hct ion commonly used in multilayer neural networks
is the sigrnoidal function which is smooth (i.e., differentiable everywhere). Based on
numerical experiments conducted in this work and what is available in the literature,
sigmoidal functions work best for supervised neural nets; i.e., the inputs and the
corresponding outputs are known a priori [Karkoub, et al., 19911. Moreover, the use of
80
the logistic hc t ion is biologically motivated, since it attempts to account for the
refkctory phase of real neurons pineda, 1988 3. The same sigrnoidal nonlinearity in the
form of a Logistic hc t ion is chosen for all the hidden and output neurons of the ANN
used in this work.
With the application of the sigmoid activation hc t ion as shown in Eq. (4.7),
derivatives of f (v) with respect to v can be derived. Combine with Eq. (4.1 I), we have
the following expression:
And combining with Eq. (4.9) results in:
Combine Eqs. (4.16), (4.1 7) and (4.22), weight adjustments at neurons in the output
layer can be expressed as follows
Weight adjustments at neurons in the hidden layer can be expressed after combining
Eqs. (4.20), (4.21), (4.22) and (4.23) as follows:
It is to be noted that if the network has more than one hidden layer, the same
procedure is extended to adjust the weights at all the additional hidden layers.
If we define the weight space to be
then the weight adjustments in the hidden and output layers can be expressed as:
where the change in the weight space A is defined as:
4 2 3 Convergence
The back-propagation algorithm is implemented by the method of gradient descent.
Typically, the effectiveness and convergence of the error back-propagation learning
algorithm depend significantly on the value of the learning rate constant q . In general,
however, the optimum value of 7 depends on the problem being solved, and there is no
single learning constant value that would be suitable for all cases. This problem seems to
be common for all gradient-based optbization schemes. While gradient descent can be
an efficient method for obtaining the weight values that minimize an error, error surfaces
frpquently possess properties that make the procedure slow to converge. The smaller we
make the learning rate parameter q , the slower but smoother the procedure will be
leading to optimal point in the weight space.
Although one can speed up the rate of learning by setting q to a large value, the
resulting large changes in the weights may result in unstable behavior (i.e., oscillatory).
Also, a smaller value of .q may be desirable when close to the target to avoid
82
overshooting the optimal point. A simple method of increasing the rate of learning and
yet avoiding instability is to modify the updating rule as shown in Eq. (4.27) by including
a momentum tenn.
Momentum can be added to error back-propagation learning algorithm by
including the search direction in the weight space at the previous iteration AW(~-'). This
is usually done according to:
where 0 < a < 1 is referred to as the momentum constant and the first term in Eq. (4.29)
is called the momentum term. When a = 0, search direction is in the gradient descent
direction and Eq. (4.29) is identical to Eq. (4.27). When a = 1, search direction is pamllel
to that of the previous iteration and the gradient is simply ignored. The weight
adjustments of the output layer according to the generalized updating rule is
For the hidden layer, it is
The incorporation of momentum in the back-propagation algorithm has highly
beneficial effects on learning behavior of the algorithm. The momentum term typically
helps to speed up convergence, and to achieve an efficient and more reliable learning
profile. Momentum allows a network to respond not only to the local gradient, but also to
the shape of the error d a c e . Acting like a low pass filter, momentum allows the
83
network learning to ignore small sudden changes on the error surface. It also helps to
avoid being trapped by the local minima.
4.2.4 Stopping Criteria
There are some typical termination criteria, each with its own practical merit, which
may be used to terminate the weight adjustments. Two commonly used criteria are
introduced in the following:
i) The maximum value of the average squared error Em is equaled to or less than a
sufficiently small threshold which is chosen as the criterion for convergence as stated
here:
E - w ' ) ~ E (4.32)
where W' is the weight vector which denotes a minimum, E is a sufficiently small
error threshold. The back-propagation computation iterates by presenting new epochs of
training samples to the network until the parameters of the network stabilize their values
and the average squared error Em computed over the entire training set is at the small
threshold. The drawback of this convergence criterion is that, when the shape of the error
space is flat the criterion can be reached without finding the minimum W' . ii) The absolute rate of change in the weight vectors per iteration is sufficiently small
as follows:
IwC) - ,,,,b-') 1 6 I (4.33)
The network learning has converged when the consecutive weight adjustments reaches
the small threshold. This convergence criterion has the drawback of the network being
trapped in the quasi minimum if the network converges to a value that is diverse with the
ideal minimum.
The first convergence criterion is utilized for the network learning in this work. The
experimental results show that it has been a very simple and effective stopping criterion.
4.2.5 Initial Weights and Cumulative Weight Adjustment
The weights of the network to be trained are typically initialized with random
values. If all weights start out with equal values, and if the solution requires unequal
weights, the network may not be trained properly. Also, the network may fail to learn
with the error increasing as the leaning continues. In fact, many empirical studies of the
algorithm point out that continuing training beyond a certain low-error plateau may result
in the undesirable drift of the weights. This causes the error to increase again afker being
converged previously. To counteract the drift problem, network learning should be
restarted with other random weights.
There are two schemes of updating the weights in the error back-propagation
learning. Scheme 1 (Figure 4.6) is called incremental updating which is based on the
single training sample error reduction and makes a small adjustment of weights which
follows each presentation of the training sample. Scheme 2 (Figure 4.7) is called batch
updating which implements the minhbt ion of the error fhction computed over the
complete cycle of P samples with gradient descent searching, provided the learning
constant q is sufficiently small.
The advantage of scheme 1 is that the searching for optimal solution is along the
gradient descent direction on the error surface. Moreover, during the computer
simulation, the weight adjustments determined by the algorithm do not need to be stored
and compounded over the learning cycle consisting of P joint error signal. However, the
network trained this way may be skewed toward the most recent training sample in the
cycle. To counteract such a problem, either a small learning constant 7 should be used or
cumulative weight changes be imposed as follows:
85
for both output and hidden layers, where A W ( ~ ) represents the change in the weight
space for the pth training pattern. The weight adjustment in this scheme is implemented at
the conclusion of the complete learning cycle. It takes the average effects of all the
training cycles. Provided that the learning rate is small enough, the cumulative weight
adjustment can still implement the algorithm close to the w e n t descent minhkation.
Although both scheme 1 and scheme 2 can bring satisfactory solutions, attention
should be paid to the fact that the training works the best under random conditions. It
would thus seem advisable to use the incremental weight updating after each pattern
presentation, but choose patterns in a random sequence. This introduces much-needed
noise into the training and alleviates the problems of averaging and skewed weights
which would tend to favor the most recent training patterns.
4.3 Experimental Determination of Optimal Neural Network
4 e 3 e 1 e Network Architecturm with Optimal Hidden Layer
The multilayered ANN trained with the back-propagation algorithm is applied to
perform the nonlinear input-output mapping. One of the most important attributes of a
multilayered neural network design is choosing the architecture. The number of input
nodes is simply determined by the dimension of the input vector.
In this thesis, the input-output relationship of the network defines a mapping from a
19-dimensional feature space to a Zaimensional classification space. Thus, the number of
input nodes is chosen to be nineteen and the number of neurons in the output layer is two.
This inputoutput mapping is assumed to be infinitely continuously differentiable. In
assessing the capability of the neural network, two fundamental questions arise:
I ) Determine the number of hidden layers:
It was Cybenko who demonstrated rigorously for the first time that a single hidden
layer is suilicient to uniformly approximate any continuous h c t i o n with support in a
86
unit hypercube [Cybenko, 19881. He introduced the universal approximation theorem
which states that a single hidden layer is d c i e n t for a multilayered neural network to
compute a uniform approximation to a giving training set represented by the set of inputs
and a desired (target) output. A single hidden layer is chosen for the ANN used in this
work.
2) Determine the size of the hiden layer:
Size of each hidden layer is mostly determined through trial and error process
depending on individual problems. The exact analysis of the issue is rather difFcult
because of the complexity of the network mapping and due to the non-deterministic
nature of the training procedures. If there are too few nodes the neural network will fail to
memorize the training process and lead to underfitting. Too many neurons can contribute
to overfitting, in which all training points are well fit, but the fitting curve takes wild
oscillations between these points. Based on trial and error, the size of 24 is found to be
the optimal compromise between underfitting and overfitting with faster convergence
compared to 20 and 26 hidden nodes (see Table 4.1). Therefore, the finally obtained
optimal architecture of the network is 19-24-2 as shown in Figure 4.8. It is shown that the
input layer has 19 nodes, each of which represents a parameter in the feature space
denotedas as [h fk Cf Kv Clf If Fi ~ ~ v - e a r k - c CRY-= m - c b t : l j - c mf-= ORV - s
m - . gty -I bkv - .* mu -.T my - S lr. The output layer has two neurons, each of which
contains a coordinate value in the 2D space denoted as 1% y.lT.
4.3.2. Accelerated Convergence through Learning-Rate Adaptation
Situation arises when a constant learning rate q does not produce satisfactory
performance. For example, on a flat e m r surface, too many steps may be required to
compensate for the small gradient value. In this work we use a heuristic technique to
87
determine the variable learning rate in order to accelerate the convergence of bck-
propagation learning. Four heuristics are considered as guidelines maykin, 19941 :
Heuristic 1. Every adjustable network parameter should have its own adjustable
learning-rate parameter. The back-propagation algorithm may be slow to converge
because of a fixed learning rate that may not suit a l l portions of the error suxface. In other
words, a learninggrate parameter appropriate for the adjustment of one weight is not
necessarily appropriate for the adjustment of other weights in the network. This method
recognizes this fact by assigning a different learning-rate parameter to each adjustable
weight @ m e t e r ) in the network.
Heuristic 2. Every learning-rate parameter should be allowed to vary fmm one
iteration to the next. Typically, the error surface behaves differently in different regions
and different dimensions. In order to match this variation, heuristic 2 allows the learning
rate to vary from iteration to iteration.
Heuristic 3. When the derivative of the performance fimction with respect to a
weight has the same algebraic sign for several consecutive iterations, the learning-rate
parameter for that particular weight should be increased. The current operating point in
the weight space may lie on a relatively flat portion of the error surface along a particular
weight dimension. This results in the derivative of the performance function with respect
to that weight with the same algebraic sign, that is, the same gradient direction, for
several consecutive iterations. Heuristic 3 states that in such a situation the number of
iterations required to move across the flat portion of the error surface may be reduced by
increasing the learning-rate parameter appropriately.
Heuristic 4. When a E / h n alternates for several consecutive iterations of the
algorithm, the learning-rate parameter for that weight should be decreased. This is the
opposite situation to the above. When the current operating point in weight space lies on a
portion of the error surface along a weight dimension of interest that exhibits peaks and
88
valleys (i.e., the surface is highly curved), then it is possible for the derivative of the
performance h c t i o n with respect to that weight to change its algebraic sign from one
iteration to the next. In order to prevent the weight adjustment from oscillating, the
learning-rate parameter for that particular weight should be decreased appropriately.
It should be noted that the use of a non-uniform and time-varying learning rate
modities the back-propagation algorithm significantly. Specifically, the modified
algorithm no longer performs a gradient descent search. Rather, the adjustments applied
to the weights are based on (1) the partial derivatives of the error surface with respect to
the weights, and (2) estimates of the curvatures of the error surface at the current
operating point in weight space along the various weight dimensions.
Let t l (n) denote the learning rate assigned to the weight at iteration n for both
hidden and output layers. The learning-rate update rule is defined as follows:
where 0 < y < 1 is a positive constant called the control step-size parameter for the
leaming rate adaptation procedure. The partial derivatives d ~ ' " ' / d r . r ( " ) and
E ~ ' / . " ' refer to the derivative of the error surface with respect to the weight
w./" at iterations n and n-1 respectively. It can be observed that when the partial
derivative has the same algebraic sign on two consecutive iterations, the adaptation
procedure increases the learning rate for the weight W.V. Correspondingly, the learning
along that direction will be fast. When the derivative alternates on two consecutive
iterations, the adaptation procedure decreases the leaming rate for the weight W.V.
Consequently, the learning along that direction will be slow.
Many parameters of the network can be adjusted during training to provide optimal
performance. Unfortunately, a systematic method for selection of the most appropriate
89
parameters does not exist. Thus construction of neural networks typically requires a trial
and error approach. Based on many trials, we determined the optimal settings of the
control parameters to be:
When the initial learning rate p has the value of 0.01, the learning takes twice as
much time; while when rp is 0.1, the convergence becomes unstable in some regions.
The momentum a of 0.9 accelerates the learning rate most, but only to the extent that the
network can learn without the increase of the error function. If the control step-size
parameter y has the value of 0.1, the learning rate will grow too fast to ensure stable
convergence. On the other hand, the learning rate reduces too slowly when y is 0.02.
Thus the above optimal settings of the control parameters results in a near optimal
learning rate for the local terrain.
Figure 4.1 Architecture graph of a multiplayer neural network with two hidden layers
signals
\ Activation
Synaptic Weights
Figure 4.2 A neuron model
Figure 4 3 Sigmoid activation kction
Figure 4.4 Illustration of the directions of two passes:
Forward propagation pass and Back-propagation pass
Figure 4.8 The neural network used for non-linear mapping
Note: The network has 19 input layer nodes, 24
hidden layer nodes, and 2 output layer nodes.
Table 4.1 Performance comparison of hidden layer with different size
- j Number of hidden nodes 20 24 26
CHAPTER FIVE:
BEARING DEFECT DIAGNOSIS
5.1 Experimental Studies
The developed method is applied to diagnose the defects of the tapered roller
bearings used in railroad fkight cars. To train the neural network for the non-linear
mapping, we used a total number of 1 15 samples with known defect information operated
under various conditions such as different loads and speeds. Severity of the defects is also
reflected by single vs. multiple spas. Since the present work intends to focus on bearing
failure due to fatigue spalls, we decide to use samples representing the following
conditions:
Table 5.1. Bearing conditions represented with class numbers
These data were provided through NRC by the Association of American Railroads
(AAR). A bearing test rig has been set up in the Transportation Technology Center (TTC)
of the AAR. Figure 5.4 illustrates the laboratory roller bearing test rig. The roller bearing
mounted in the test rig is clearly shown in Figure 5.5. The test m g s used in the
laboratory tests include both AP class E (6 x 11) 70-ton capacity bearings, and AP Class
F (6 1R x 12) 100-ton capacity bearings. The component dimensions of these two types
of bearings are described in Table 5.2.
f Clm Numbor 1- 2
a
3 4
f 5 6
Boaring Conditions Good Bearing
Single Cup Spall Multiple Cup Spalls (Figure 5.1)
Single Cone Spall Multiple Cone Spalls (Figure 5.2)
Broken Roller Figure 5.3)
97
Each AP class bearing (EBtF) are embedded with defects of different types as listed
in Table 5.1. Experiments are performed with two separate radial loads representing
empty and Mly loaded fieight car:
Type E bearings: 8,000 lb. and 27,500 lb.
Type F bearings: 8,000 ib. and 33,000 lb.
Each test is conducted at different train speeds ranging fkom 30 to 80 miles per hour
(MPH) at increment of 10 miles per hour.
Test data are collected from acoustic sensors and accelerometers in parallel for all
bearings under test. Analog signals are digitized with a sampling rate of 270 kHz. The
digital signals are stored in files, each of which contains 540,000 points representing 2
seconds of signal collection time. Tachometers are also used to measure the exact shaft
rotation speed to provide a reference for synchronization.
5.2 Feature Selection
The obtained vibration signals are processed and analyzed through time domain,
fiequency domain and segmentation analyses. A total number of nineteen feature
parameters are calculated for measured signals. Time domain parameters include Root
Mean Square value (Rv), Peak value (Pk), Crest factor (Cf), Kurtosis value (Kv),
Clearance factor (Clf) and Impulse factor (If). They can be used to indicate either the
severity of the bearing defects or the spikiness of the vibration amplitude associated with
the defect-induced impulses.
Frequency index (Fi) is the parameter extracted h m fiequency domain proposed to
highlight significant fiequency contents that may be associated with the bearing defect
characteristics hquencies [Sun, et al., 19981. Although the defect characteristic
fkequencies could be used to help determine the location of the defect, automatic
98
detection of impulses at these fkequencies is not a simple task. This is because frequency
spectra often show much stronger peaks at much higher fiequencies representing high
order structural resonance compared to those at the characteristic fiequencies. Vibrational
energy of the bearing spreads across a wide fkquency band and can be easily buried in
the noise. Figure 5.6 shows the fhquency spectrum of a bearing with outer race defect.
The dominant fkequency can be seen to be around 4300Hz which is far beyond the range
of the roller passing outer race fkquency as shown in Table 2.1. No explicit relations
between the spectrum and the defect characteristic kquencies can be constructed in the
case. Therefore, it is not advised to depend solely on the fiequency spectrum, which
necessitates the pattern recognition analysis for more reliable diagnosis.
Segmentation analysis is applied to characterize non-stationary signals through
segmenting the signal into quasi-stationary components based on the understanding of
bearing dynamics. Impulses can only be generated from the passage of defects when they
are inside the load zone. Defects located on different bearing components will generate
impacts with different fkquencies and modulation patterns when passing through the
load zone. Correlation exists between the location of the defects and the impulse patterns
observed in the vibration signal. We decide to divide the signal in one shaft or cage
revolution into six segments, so that at least one segment will be completely inside the
load zone and one completely outside of the load zone. Segmentation parameters are
determined based on the calculation of standard deviation of the time domain parameters
in various segments using cage hquency and shaft fkquency respectively. A segmented
vibration signal obtained fbm bearing with h e r race defect and the spectra of each
segment are illustrated in Figure 5.7. It is obvious that the peaks in the spectra of the last
two segments are much more dominant than the those in other four segments, which
corresponds to the impulse generating region in the time domain waveform.
99
5.3 Results of the Artificial Neural Network
Nineteen parameters are first calculated for each measured signal of the 1 15 samples
to form the feature space, as listed in Table 5.3. These parameters are then normalized
and used as input to train the neural network to perform the non-linear mapping as
discussed in chapter 4. Before training, it is often usefhl to scale the inputs and targets so
that they always fall within a specified range. This preprocessing is helpful for efficient
and stable behavior of the training process. We choose to scale all numbers such that they
fall into the range of a sigmoid function, i.e., between 0 and 1. The minimum and
maximum values of each feature parameter for the total 115 samples, that is, the
minimum and maximum of each column in Table 5.3 are used to normalize the column
into the sigrnoid range. These values are also exploited in normalizing the test data for
diagnosis as detailed later. Each training data set consists of the normalized nineteen
input parameters and the specified cluster centers as target outputs of the network as
shown in Table 5.4.
An error criterion of 0.01 is achieved through a trial and error approach. The
network training was pursued for 9,000 iterations when the error criterion was reached.
The actual outputs at the end of training are compared with the target outputs and listed in
Table 5.5. The averaged error fhction Em of the trained network is calculated to be 0.009
also shown in the table, which fkther co- that the learning has converged to the
expected criterion. If the error criterion is chosen to be 0.005, the network converges after
16,000 iterations and only leads to 5% reduction of the error hct ion Em. An error
criterion of 0.02 was also tested, the network learning converged after 6,000 iterations.
However, with the value of Em being 0.019 the samples belonging to different classes
were not well clustered in the classification space and had some overlapping.
Feature extracttion without segmentation parameters fiom the same bearing was
performed to compare with the developed method. The same experimental data were
100
used for the pattern recognition. The learning process took the same network more than
40,000 iterations to converge to the same error criterion.
After the network training was complete, the actual network outputs were plotted on
a 2D space and the mapping result is shown in Figure 5.8, where the Arabic numbers
represent different bearing conditions as listed in Table 5.1. The black dots in the
classification space represent the designated cluster centers. Although their arrangement
was somewhat arbitrary, we placed them evenly on a unit circle in the first quadrant of a
2D space as shown in Figure 5.9. There were three reasons for this configuration. Firstly,
a unit circle was chosen so that the outputs will fall into the range of a sigmoid function,
that is, between 0 and 1. Secondly, a larger circle does not necessarily lead to a better
clustering effect. In fact, although the between-class distance may increase with the
diameter, the within-class distance may also increase. Consequently, class separability
will not be improved. The third reason states that if cluster centers were arrayed on a unit
square in the first quadrant of a 2D space as shown in Figure 5.10, the convergence of the
network took longer. Also, some regions are left unexplored because the mapped samples
could not distribute evenly in the first quadrant. Finally, the coordinates of the cluster
centers, that is, the desired [xc are chosen to be:
It can be observed from Figure 5.8 that samples belonging to different classes are
separated in different regions and clustered around their own pre-defined cluster centers
in the classification space. The neural network has successfully performed the high
fidelity dimension reducing non-linear mapping. The intra-class transformation [Sun, et
101
al., '19981 is eliminated thereby. Simple piecewise linear classifications can then be
applied to partition the classification space.
5.4 Classification
Once the sample data have been transformed h m the feature space to the
classification space with high fidelity, they are ready to be classified. For the present
study, we used a distribution h e classification method due to the deficient knowledge of
the bearing defect distribution. Discriminant hc t ions are used to partition the
classification space.
Consider K classes: S,, . . . , Sk,. . ., SK with defining prototypes y,,,(k' for each class rn =
1, . . ., ktk. The discriminant bc t ion is defined such that for any point z belonging to Sk,
there exists a function gdz) such that
g k ( ~ ) > ~ ( ~ ) VZ €St and tlk* j (5-1)
In other words, within the region Sk, the kth discriminant function will have the
largest value. For linearly separable patterns, it is convenient to use piecewise linear
discriminant hctions. If we d e h e the distance of a point z to a class Sk to be the
distance fiom the closest prototype point in Sk, that is,
We could define the above to be the discriminant function. Therefore, the decision will be
made based on the smallest distance between a point in the classification space to any
class.
Mathematically, this can be written as:
Accordingly, the discriminant hct ion can be defined as:
Appareny in a 2D space, boundaries are defined when two functions @(z) and
g j ( ~ ) both become maximum and
= d=) (5.5)
Figure 5.1 1 shows patterns of the 123 samples in the classification space. Boundaries
are generated as described above. Once classification space is constmcted, it can be used
for diagnosis.
5.5 Diagnosis
After completing the pattern classifier, a total number of 31 test data (not used in
training) from bearings with defects of different types as listed in Table 5.1 were used to
test the effectiveness of the developed method. They were taken under different loads and
at different speeds.
By calculating the time domain, fnquency domain and segmentation parameters, a
point can be located in the feature space for each measured signal. Table 5.6 shows the
calculated feature parameters for the 31 test data. The minimum and maximum values
used to normalize the feature parameters of training samples are also adopted to
normalize those of the test data. The normalized parameters are listed in Table 5.7 and
fed through the trained neural network. The network outputs corresponding to each
103
measured signal were plotted in the classification space denoted by different symbols as
shown in Figure 5.11. The results show that all the testing data were correctly recognized.
The developed methdd is very effective in bearing defect diagnosis.
S/N 54900 *. 3 UULTIRECUPSPUW TEST BRG #9 b
Figure 5.1 Multiple cup spalls
Fi gum 5 2 Multiple cone spalls
Figure 5.4 Roller bearing test rig at TI%
Figure 5.3. Broken roller
F w In #
Figure 5.6. Frequency spectrum versus bearing defect characteristic fraluencies
Figure 5.5 Roller bearing mounted in test rig
Time domain waveform in one rewolution divided into six segments
Fmquency spectrum of the waveform in one revolution
Figure 5.7a. An time/fiequency display of the signal in one revolution
Figure 5.7b. Frequency spectra of each segment
Figure 5.8 Result of nonlinear mapping using neural networks " 1 " - Good Bearing "2" - Single Cup Spall "3 " - Multiple Cup Spalls "4" - Single Cone Spall "5" - Multiple Cone S p a s "6" - Broken RoUer
Figure 5.9 Cluster centers evenly spaced on a unit circle
1
Figure 5.10 Cluster centers arrayed on a unit square
Figure 5.11 Classification and diagnosis results
Table 5.2 Bearing component dimensions
bem Description
Sae Designation (k hes)
Wpical Carload pns) Number of RPlem
Cbler lenglh lbler D&mebr
lblerPRh DiameBr Cone BOR (DiaMBOer)
Cup OD @--D)/2 Bearing HMlh
1/2 -hided Cup Angle (deg) Cos(Angle)
hches
'Ibm Num
hches hches hches hches hches hches hches
Deg Beta
E
6 x l l
;FD a4 m am470 7.- S68;F#)
7.U3725 6-
10 QM
F
61/2xl2
100 23
lS530 .
7.- -6-
AS3750 80622s 7.00000 3.0
Table 53. Calculated 19 parameters for 1 15 samples
Table 5.4 Normalized training sets
'able 5.5. Network outputs compared with target outputs
Table 5.6. Calculated parameters of 3 1 test data
Table 5.7. N o r m M parameten of 3 1 test data
CHAPTER SIX:
CONCLUSION AND FUTURE WORK
6.1 Summary of Results Obtained
The signal processing and pattern recognition techniques described in chapters 3,4,
and 5, were applied to vibration signals obtained from rolling element bearings used in
railway hight cars.
Vibration signals obtained from bearings were processed and analyzed through time
domain, fiequency domain and segmentation analyses. T i e domain parameters include
Root Mean Square value (Rv), Peak value (Pk), Crest factor (Cf), Kurtosis value (Kv),
Clearance factor (Clf) and Impulse factor (If). They provide information such as the
spikiness and the energy level of the vibration signals. Frequency index (Fi) is the
parameter extracted from kquency domain proposed to highlight significant frequency
contents that may be associated with the bearing defect characteristics fkquencies.
Earlier work on pattern recognition for bearing defect diagnosis using these parameters
showed promising results and was simple to implement [Sun et al., 19981.
The sensitivity and reliability of the pattem recognition analysis is fUrther improved
by including the newly developed segmentation parameters. Since the vibration signal of
a bearing with defects is generally non-stationary, segmentation d y s i s can be applied
to feature the description of such a signal through segmented quasi-stationary
components. Based on the observation that impuises can only be generated from the
passage of the defect contacting the mating surfaces under load, vibration signals present
certain patterns associated with def- inside or outside of the bearing load zone. Defects
located on different bearing components will generate impacts when passing through the
load zone with different fkquencies and modulation patterns. A correlation exists
119
between the variation pattern of signals and the location of defects on the bearing
components and impulses modulated with the shaft or cage frequency can be detected.
For the bearings studied in this work, the radial loads cause a stress distribution
over an angle range of about 120 degrees. A fixed-length segmentation scheme is used in
order to reduce the computational expense of the process. Signals in one shaft revolution
and cage revolution are evenly divided into six segments respectively so that at least one
segment will be completely inside the load zone and at least one segment can be
completely outside of the load zone. Descriptive features of these signal segments can be
calculated through time domain parameters. Segmentation parameters are thus
determined based on the calculation of standard deviation of the time domain parameters
among six segments. They can directly reflect the variation of vibration patterns
associated with the bearing load zone. The segmentation parameters referenced in both
shaft and cage rotations are employed to participate in constructing the feature space.
Segmentation parameters, together with the existing time and fkequency domain
parameters are used to construct the feature space. Thus the find feature space is
composed of 19 dimensions. A three-layered artificial neural network is used to
accomplish the nonlinear mapping fiom the 19-dimensional feature space to the 2-
dimensional classification space. Artificial neural networks allow us to construct
complicated non-linear relations between input and output when analytical description is
not available.
The ANN is chosen to have three layers since three-layered networks are
s a c i e n t for representing the non-linear relations between the input and output and they
have relatively simple architecture. The error back-propagation algorithm is used to train
the neural network. The same sigmoid activation h c t i o n in the form of a logistic
function is chosen for all the hidden and output neurons of the network as it allows the
network to learn non-linear relationships between input and output vectors. The nineteen
120
feature parameters extracted fiom vibration signals are fed as inputs to the network and
the output contains the corresponding coordinates in the 2D classification space. The
cluster centers are evenly spaced on a unit circle in the first quadrant of a 2D space in
order to locate the desired outputs associated with each of the classes. The neural network
is trained with known input sets each of which consists of 19 parameters and the
corresponding desired outputs that are the specified cluster centers. The finally obtained
optimal architecture of the network is 19-24-2 through a trial and error approach. A
heuristic technique, variable learning rate, is adopted in order to accelerate the
convergence of back propagation learning. A momentum is also incorporated to b l p
speed up convergence, and achieve a more efficient and reliable learning profile.
A total number of 115 samples with known defect information operated under
various conditions such as different laods and speeds are used to train the network.
Severity of the defects is also reflected by single vs. multiple spalls. The network training
was pursued for 9,000 iterations when an error criterion of 0.01 was reached. Feature
parameters without the segmentation parameters extracted fiom the same bearing was
also investigated. It took the same network more than 40,000 iterations to converge to the
same error criterion. The mapping result shows that the corresponding outputs in the
classification space are completely separated in different regions and clustered around the
prescribed centers associated with bearing in different conditions. The artificial neural
network has successfidly performed the high fidelity dimension reducing non-linear
mapping. The intraclass transformation is eliminated thereby. Successll mapping
allows application of the simple piecewise linear boundaries.
A total number of 3 1 test data (not used in training) fiom bearings with defects of
different types as listed in Table 5.1 were used to test the effectiveness of the developed
method. They are taken fiom bearings under different operating conditions. By
calculating the time domain, frequency domain and segmentation parameters of the
121
signals, these samples can be located in the feature space. Fed through the trained neural
network, each test sample is identified on the classification space. The classification
results show that they are all correctly recognized.
In summary, the developed method based on pattern recognition analysis has
improved the sensitivity and reliability in bearing fault diagnosis by including the
segmentation parameters. The successful non-linear mapping through the neural network
eliminates intra-class transformation process. The result shows the method is simple and
effective. It is suitable for the development of automatic monitoring and diagnostic
systems.
6.2 Limitations of the Present Method and Directions for Future Work
Although the present method accurately has recognized bearing conditions, the
results were obtained using experimental data. In actuality, we deal with bearings
mounted on moving trains. Vibration signals obtained fiom this environment are
expected to have diEerent characteristics than those obtained fiom a test rig in the lab.
Future work will be directed towards investigating the reliability of the existing method
diagnosis. Improvement, if any, needs to be made to further increase the sensitivity of the
method to non-defect related characteristics.
Also in this work, we focused on detecting and diagnosing bearing defects caused
by fatigue spalling as it is the predominant bearing failure mode. Further studies need to
be conducted to include other bearing failure modes.
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