In this study, a stochastic diagnosis method based on the changing information of not only a linear correlation but also a higher-order nonlinear correlation is proposed in a form suitable for online signal processing in time domain by using a personal computer, especially in order to find minutely the mutual relationship between sound and vibration emitted from rotational machines. More specifically, a conditional probability hierarchically reflecting various types of correlation information is theoretically derived by introducing an expression on the multidimensional probability distribution in orthogonal expansion series form. The effectiveness of the proposed theory is experimentally confirmed by applying it to the observed data emitted from a rotational machine driven by an electric motor.
1. Introduction
Some studies on the diagnoses for many machines, for example, the estimation and prediction of presence of crack
in rotational machines without stopping the machines, have become important
recently because of increased use of various complex industrial systems [1–9]. Most of these
investigations have been carried out especially in a frequency domain by the
use of signal information on sound or vibration wave emitted from the machines.
These standard methods in a frequency domain are useful for the purpose of
analyzing the failure of individual machine based on the mechanism from a
bottom-up viewpoint. However, these methods require a lot of procedures and times
on signal processing for measured sound or vibration data.
In the actual machine systems, simultaneous
time series data on sound and vibration signals can be easily observed by the
use of multichannel measurement instruments. The correlation statistics between
the sound and vibration emitted from an identical machine contain important
information on the driving condition of the machine. For the diagnosis of the
fault without stopping the machine, it is necessary to consider various kinds
of latent correlation information of not only the lower order but also the higher
orders in order to investigate and evaluate precisely the mutual relationship
between sound and vibration emitted from the machine.
The
standard diagnoses in frequency domain seem to be inadequate for evaluating
total effects on the compound of the mutual relationship between sound and
vibration waves in complicated circumstances, such as the actual driving
condition. In order to evaluate universally the mutual correlation
characteristics and detect the failure of machines by the use of a simple
procedure in the actual complex working environment, it is necessary to
introduce some signal processing methods, especially in a time domain.
From
the above viewpoints, in this theory, a diagnosis method based on a conditional
probability reflecting the linear and nonlinear correlation information with
lower and higher orders is proposed in order to grasp minutely and universally
the mutual relationships between sound and vibration emitted from machines.
More specifically, a conditional probability expression for multivariables is
theoretically derived by introducing a multidimensional joint probability
function in a series-type expression. First, in Section 2, based on the observation
data of sound and vibration emitted from machines, an expansion series
expression of the conditional probability distribution to evaluate the
correlation relationship between the sound and vibration signals is
theoretically derived. Next, in Section 3, by the use of the expansion
expression of the conditional probability distribution derived in Section 2, a
prediction method of the probability for a failure of machines is theoretically
derived on the basis of the observation data in a sequential time series form
of sound and vibration. The fundamental principle of the proposed diagnosis
method is based on the detection of the changing information on various kinds
of correlation characteristics between sound and vibration as much as possible.
Finally, in Section 4, by applying the proposed methodology to the measurement
data from a rotational machine driven by an electric motor, the effectiveness
of theory is confirmed experimentally.
2. Theory2.1. Conditional Probability Expression of Fault Occurrence by Considering Correlation Relationship between Sound and Vibration
Since there is a certain correlation
relationship between sound and vibration emitted from an identical machine
according to the driving condition of the machine, the diagnoses of the machine
are possible by detecting the changing information of the correlation
characteristics. First, let us introduce a random variable y with two exclusive values of 0 and 1
corresponding to normal situation without fault of machines and a failure
situation with the fault. In order to evaluate quantitatively the complicated
relationship between sound and vibration emitted from a machine, let us
introduce an expansion expression of conditional probability considering not
only the linear correlation but also the nonlinear correlation information
among these variables. In order to derive a mathematical expression in a
general form, two kinds of variables (i.e., sound and vibration) are expressed
as x1 and x2.
In the case when paying attention to the variables x1, x2,
and y,
all the information on mutual correlations among x1, x2,
and y is included in the conditional probability
distribution P(y|x1,x2).
First,
the joint probability distribution P(x1,x2,y) is expanded into an orthonormal polynomial
series [10] on the basis of the fundamental probability distributions P0(x1), P0(x2),
and P0(y),
which can be artificially chosen as the probability distributions describing
approximately the dominant parts of the actual fluctuation pattern as follows:
P(x1,x2,y)=P0(x1)P0(x2)P0(y)∑m1=0∞∑m2=0∞∑n=0∞Am1m2n×φm1(1)(x1)φm2(2)(x2)φn(3)(y),Am1m2n=〈φm1(1)(x1)φm2(2)(x2)φn(3)(y)〉,
where (2)
〈⋅〉 denotes the averaging
operation with respect to the random variables. Here, (2)
φm1(1)(x1), (2)
φm2(2)(x2),
and (2)
φn(3)(y) are orthonormal polynomials with three
weighting functions (2)
P0(x1), (2)
P0(x2),
and (2)
P0(y),
and must satisfy, respectively, the following orthonormal relationships:
∫φm1(1)(x1)φm1′(1)(x1)P0(x1)dx1=δm1m1′,∫φm2(2)(x2)φm2′(2)(x2)P0(x2)dx2=δm2m2′,∑y=01φn(3)(y)φn′(3)(y)P0(y)=δnn′.
Thus, the information on the various types of linear and nonlinear correlations among x1, x2,
and y is reflected hierarchically in each expansion coefficient Am1m2n.
Next, from (1)), the following expression can be obtained:
P(x1,x2)=∑y=01P(x1,x2,y)=P0(x1)P0(x2)∑m1=0∞∑m2=0∞Am1m20φm1(1)(x1)φm2(2)(x2).
Thus, by using (1)) and (4), the conditional probability distribution function
containing all the information on the regression relationship can be derived
under the employment of the well-known Bayes’ theorem [11] as follows:
P(y|x1,x2)=P(x1,x2,y)P(x1,x2)=P0(y)∑m1=0∞∑m2=0∞∑n=0∞Am1m2nφm1(1)(x1)φm2(2)(x2)φn(3)(y)∑m1=0∞∑m2=0∞Am1m20φm1(1)(x1)φm2(2)(x2).
As the fundamental probability density
functions P0(x1) and P0(x2) in the first terms of the expansion series expression of (1), the
well-known Gaussian distribution is adopted because this probability density
function is the most standard one:
P0(x1)=12πσx12exp{−(x1−μx1)2σx12},P0(x2)=12πσx22exp{−(x2−μx2)2σx22}
with
μx1=〈x1〉,σx12=〈(x1−μx1)2〉,μx2=〈x2〉,σx22=〈(x2−μx2)2〉.
Furthermore, as the fundamental probability function P0(y) on the level-quantized random variable y,
the binomial distribution can be chosen [12]
P0(y)=((N−M)/h)!((y−M)/h)!((N−y)/h)!p(y−M)/h(1−p)(N−y)/h
with
N=1;the maximum value ofy,M=0;the minimum value ofy,h=1;the level difference interval ofy.
Thus, from (3), the
orthonormal functions can be determined as
φm1(1)(x1)=1m1!Hm1(x1−μx1σx1),φm2(2)(x2)=1m2!Hm2(x2−μx2σx2),φn(3)(y)=1((N−M)/h)(n)n!(1−pp)n/21hn×∑j=0n(nj)(−1)n−j(p1−p)n−j(N−y)(n−j)(y−M)(j)
with
x(n)=x(x−h)(x−2h)⋯(x−(n−1)h),x(0)=1,
where Hm(•) is the Hermite polynomial with nth order [13].
By substituting (8)–(10) into (2) and
(5), the conditional probability distribution function P(y|x1,x2) can be obtained in the concrete form.
It may be natural to impose a condition A001=0 upon the expansion coefficient in (2) in order
to get an appropriate explanation for the dominant part of the probability
distribution only by the first term. Thus, the unknown parameter p can be explicitly determined by the use of
mean value of the variable y as
p=μ−MN−M,μ=〈y〉.
Furthermore,
four parameters μx1, μx2, σx12, and σx22 in (6) can be determined as (7), by considering the following conditions on
expansion coefficients
A100=A010=A200=A020=0.
In the probability distribution expression of infinite series type, by adopting
the probability distribution function describing the dominant parts of the
phenomena as the fundamental probability distribution, it is enough to consider
only a few expansion terms in order to express the whole fluctuation form. In
order to roughly express the whole shape of the probability distribution
function, it is fundamentally important to first catch the statistical
information on two lower-order moments like mean and variance.
2.2. Prediction of Fault Probability Based on The Observation of Sound and Vibration
Based on the conditional probability
distribution P(y|x1,x2),
the probability P(y) expressing the occurrence or nonoccurrence of
a fault can be predicted on the basis of the observation data on sound x1 and vibration x2 as follows:
P(y)=〈P(y|x1,x2)〉x1,x2.
Thus, by the use of (5), P(y) can be expressed concretely as
P(y)=P0(y)∑n=0∞Bnφn(3)(y),Bn=〈∑m1=0∞∑m2=0∞Am1m2nφm1(1)(x1)φm2(2)(x2)∑m1=0∞∑m2=0∞Am1m20φm1(1)(x1)φm2(2)(x2)〉x1,x2.
3. Experiment
The proposed method is applied to detect the
fault of a rotational machine by observing simultaneously the sound and
vibration waves emitted from the machine. The scatted diagram between the sound
and vibration in two cases before and after occurrence of the fault in the
machine is shown in Figures 1 and 2, respectively. It is obvious that the correlation relationship between the sound and vibration in the case of failure situation occurring of the fault changes from the correlation characteristic before the occurrence of the fault. Therefore, by detecting the changing information of the correlation
characteristic between the sound and vibration, it is possible to predict the
fault of the machine in principle. (The sensitivity for the fault detection of
the proposed method is discussed in the appendix.)
Scatter diagram between the observed data of sound and vibration in normal situation without the fault.
Scatter diagram between the observed data of sound and vibration in failure situation with the fault.
The signal processing for the observed
data by the use of the proposed diagnosis method in Sections 2 and 3 is
summarized as follows.
(i) Time series data of sound and
vibration waves including two cases of occurrence and nonoccurrence of a fault are
simultaneously measured.
(ii) The expansion coefficients Am1m2n in (2) are evaluated by averaging the observed
data, concretely, by the use of the relationship
Am1m2n=1S∑i=1Sφm1(1)(x1,i)φm2(2)(x2,i)φn(3)(yi),
where the expansion coefficients are experimentally estimated. In (), (17)
S is a large number of the observed total data, the
subscript (17)
i of (17)
x1 and (17)
x2 denotes the (17)
ith sampled data in time series, and (17)
yi is 0 or 1 corresponding to the normal
situation without the fault or a failure situation with the fault. Parameters (17)
μx1, (17)
σx12, (17)
μx2, (17)
σx22 in () and (17)
p in () are estimated by averaging the
observed data as
μx1=1S∑i=1Sx1,i,σx12=1S∑i=1S(x1,i−μx1)2,μx2=1S∑i=1Sx2,i,σx22=1S∑i=1S(x2,i−μx2)2,μ=1S∑i=1Syi.
(iii) The conditional probability
containing the whole information on the regression relationship is concretely
evaluated by the use of (5) and the expansion coefficients Am1m2n estimated in (ii).
(iv) In order to predict the probability
of the fault occurrence based on the two variables x1 and x2 (i.e., sound and vibration waves), different
kinds of data on x1 and x2 from those employed in the evaluation of Am1m2n are observed.
(v) By using (16) and using newly observed
these data of x1 and x2 for the prediction of P(y),
the expansion coefficients Bn can be
evaluated as
Bn=1T∑i=1T∑m1=0∞∑m2=0∞Am1m2nφm1(1)(x1,i)φm2(2)(x2,i)∑m1=0∞∑m2=0∞Am1m20φm1(1)(x1,i)φm2(2)(x2,i),
where T is the number of data observed for the
prediction and the subscript i denotes the ith sampled data in time series.
(vi) Thus, by the use of (15), the
probability P(y) of the fault occurrence can be predicted.
The above procedure from (iv) to (vi) based on
time series data can be carried out easily in online signal processing form by using a personal computer.
The RMS values of the sound pressure level (dB)
and the acceleration amplitude (m/s2) emitted from a rotational
machine driven by an electric motor are simultaneously measured. The RMS values of the acceleration amplitude
are obtained by composing three axis values.
More specifically, two kinds of time series datasets (i.e., Dataset 1
and Dataset 2) of sound and vibration in two different time intervals are
measured with a sampling interval of 0.1 second, in two cases of a fault
occurrence and nonoccurrence, successively. As an example of faults, the
existence of a scratch in a cogwheel is considered as a trial. By adopting the
2000 data points of Dataset 1 which contain both cases of the fault occurrence
and nonoccurrence as the learning data, the expansion coefficients are first
evaluated by the use of (17) with m1≤3, m2≤3, n≤2,
and S=2000.
Next, after dividing each dataset into 20 subdatasets which consist of 500 data
points with 400 overlapping data points, the probability of the fault
occurrence P(y) is predicted by the use of (15) and (19) with m1≤3, m2≤3,
and T=500 based on the 500-sampled data in each subdataset. The relationship between each dataset and subdatasets is illustrated in Figure 3. The experimental setup and the signal processing procedures for Datasets 1 and 2 are illustrated in Figures 4 and 5.
Relationship between Datasets and subdatasets.
Experimental setup for the measurement of sound and vibration.
Signal processing procedures for Datasets 1 and 2.
The predicted results of the probabilities
of y=0 corresponding to the normal situation without
the fault and y=1 corresponding to the failure situation with
the fault are shown in Table 1 for Dataset 1 and in Table 2 for Dataset 2,
respectively. From these results, it is obvious that the fault probability
shows clearly large values after the subdatasets in the failure situation are
used in order to predict the probability P(y) because the correlation characteristics between
sound and vibration are changed after the occurrence of the failure.
Probabilities predicted by using sequentially subdatasets of sound and vibration (for Dataset 1).
Data points
P(y=0)
P(y=1)
1–500
0.5333
0.4667
101–600
0.5745
0.4255
201–700
0.5444
0.4556
301–800
0.5169
0.4831
401–900
0.5176
0.4824
501–1000
0.4956
0.5044
601–1100
0.4784
0.5216
701–1200
0.4599
0.5401
801–1300
0.4480
0.5520
901–1400
0.4359
0.5641
1001–1500
0.4266
0.5734
1101–1600
0.4231
0.5769
1201–1700
0.4168
0.5832
1301–1800
0.4087
0.5913
1401–1900
0.4120
0.5880
1501–2000
0.4078
0.5922
Probabilities predicted by using sequentially subdatasets of sound and vibration (for Dataset 2).
Data points
P(y=0)
P(y=1)
1–500
0.6008
0.3992
101–600
0.5933
0.4067
201–700
0.5728
0.4272
301–800
0.5470
0.4530
401–900
0.5245
0.4755
501–1000
0.5033
0.4967
601–1100
0.4776
0.5224
701–1200
0.4613
0.5387
801–1300
0.4481
0.5519
901–1400
0.4381
0.5619
1001–1500
0.4299
0.5701
1101–1600
0.4216
0.5784
1201–1700
0.4135
0.5865
1301–1800
0.4071
0.5929
1401–1900
0.4021
0.5979
1501–2000
0.3966
0.6034
4. Conclusion
In this paper, in order
to predict minutely and universally the fault probability based on the mutual
relationship between sound and vibration emitted from rotational machine, a
method to estimate the correlation information especially in the time domain
has been proposed in the form suitable for online signal processing by employing
a personal computer. Our proposed method has utilized not only the linear
correlation of lower order but also the nonlinear correlation information of
higher order between variables. The validity and effectiveness of the proposed
method have been confirmed experimentally by applying it to the observation
data emitted from a rotational machine.
The proposed method can
evaluate numerically the occurrence of a fault as probability. If the fault probability predicted by using (15) and (16) shows larger value, it
can be considered that the possibility
of the occurrence of fault becomes higher. In this situation, it is necessary
to check carefully whether the fault actually occurs or not by stopping the
machine. The proposed method has an advantage to evaluate numerically the
occurrence of fault in online signal processing form. Furthermore, the fault
detection with high reliability can be achieved because the proposed method
utilizes total information on sound and vibration.
The proposed approach is obviously quite
different from the ordinary approach, and it is still at the early stage of
study. Thus, there are a number of problems to be addressed in the future,
building on the results of the basic study in the paper. Some of the problems
are given in the following.
(1) In order to estimate more accurately the occurrence
of failures, it is essential to more accurately estimate the conditional
probability distribution P(y|x1,x2) given by the expansion expression in (5). From
the theoretical viewpoint, the conditional probability distribution can be
expressed with higher precision, by employing many expansion coefficients Am1m2n of higher order. From the practical viewpoint,
however, the reliability of the conditional probability distribution in the
proposed expression tends to be lacking especially for the higher-order
correlation information because only a finite number of data can be observed in
practice. It is thus a problem as how
the optimal number of terms in the above expansion expression should be
determined according to the statistical property of the phenomenon and the
available number of data.
(2) The proposed method should be applied to
actual diagnoses for other kinds of failures of machines, and the proposed
theory should be extended to more realistic situation with many kinds of faults
in the machine.
(3) The proposed theory should be further
improved to fit the actual situations in the presence of external and internal
noises.
AppendixSensitivity for The Fault Detection
From (1), the joint probability density functions of x1 and x2 is obtained as follows:
P(x1,x2)=∑yP(x1,x2,y)=P0(x1)P0(x2)∑m1=0∞∑m2=0∞Am1m20φm1(1)(x1)φm2(2)(x2).
The expansion coefficients Am1m2n are defined by (2). Then, the conditional
probability density function can be given by
P(x1|x2)=P(x1,x2)P(x2)=P0(x1)∑m1=0∞∑m2=0∞Am1m20φm1(1)(x1)φm2(2)(x2)∑m2=0∞A0m20φm2(2)(x2).
The probability density function of x1 can be predicted on the basis of the observed
data of x2 as follows:
P(x1)=〈P(x1|x2)〉x2=P0(x1)∑m1=0∞〈∑m2=0∞Am1m20φm2(2)(x2)∑m2=0∞A0m20φm2(2)(x2)〉x2φm1(1)(x1).
The probability density function in
orthogonal expansion series has theoretically convergent property by
considering expansion terms with higher orders. Therefore, after evaluating in
advance the expansion coefficients Am1m20 by the use of the measured data on sound and
vibration in normal situation without fault, by using (A.3),
the prediction of P(x1) can be obtained form the measured data of x2 in the normal situation as shown in Figure 6. On the other hand, by using the measured data x2 in a failure situation with the fault, the
predicted probability density function P(x1) shows the tendency to diverge, as shown in Figure 7. Therefore, by introducing an indicator expressing the difference between
two probability density functions shown in Figures 6 and 7, it is possible to
evaluate quantitatively the occurrence of the fault. It remains to continue as one of future works.
Prediction of the probability distribution for sound pressure level based on the measurement data of vibration in normal situation.
Prediction of the probability distribution for sound pressure level based on the measurement data of vibration in failure situation.
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