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ADVANCED SIGNAL PROCESSING IN MECHANICAL FAULT
DIAGNOSIS
Sulo Lahdelma
Mechatronics and Machine Diagnostics Laboratory, Department of Mechanical
Engineering, P.O.Box 4200, FI-90014 University of Oulu, Finland
Phone: +358-8-5532083
E-mail: sulo.lahdelma@oulu.fi
Esko Juuso
Control Engineering Laboratory, Department of Process and Environmental Engineering,
P.O.Box 4300, FI-90014 University of Oulu, Finland
E-mail: esko.juuso@oulu.fi
ABSTRACT
Advanced signal processing methods combined with automatic fault detection enable
reliable condition monitoring for long periods of continuous operation. Root-mean-square
and peak values obtained from vibration signals are useful features for detecting various
faults. Unbalance, misalignment, bent shaft, mechanical looseness and some electrical
faults, for example, can be detected using features of displacement and velocity. Higher
order derivatives provide additional possibilities for detecting faults that introduce high-
frequency vibrations or impacts. Real order derivatives increase the number of signal
alternatives. New generalised moments and norms related to lp space with short sample
times and relatively small requirements for the upper cut-off frequency are feasible
approaches for on-line cavitation analysis and power control. In a lime kiln, features were
generated from the distribution of the signals )3(
x and )4(
x. Standard deviations of the
signal )4(
x in three frequency ranges were used in a very fast rotating centrifuge. The
nonlinear scaling used in the linguistic equation (LE) approach extends the idea of
dimensionless indices to nonlinear systems. In some cases, a single feature provides a good
solution, and several features can be combined by means of linear equations. Other
measurements, e.g. temperature and pressure, are scaled with similar nonlinear functions as
the vibration features. Case-based reasoning (CBR) is used when there are many fault
alternatives. Health indices can be formed from condition indices by means of the
weighted average. The health index and its inverse, called the measurement index, are
good indicators for the need of maintenance.
1. INTRODUCTION
Machine condition monitoring can be based on various techniques (Figure 1) which
contain numerous approaches, e.g. vibration condition monitoring can be classified into
time and frequency domain techniques (Figure 2). Various measurement technologies are
discussed in (1). Attempts to detect different types of machine faults reliably at an early
stage require the development of improved signal processing methods. Vibration
measurements provide a good basis for condition monitoring. The displacement x and
velocity x(1) react successfully to unbalance and misalignment, but they do not usually
allow the detection of impact-like faults, e.g. defective bearings and gears, at a sufficiently
early stage. The signals x and x(1) can be obtained from the acceleration x(2) through
analogue or numerical integration. Smith used the jerk, i.e. x(3) signal, when examining
slowly rotating bearings (2). The jerk had been used earlier for assessing the comfort of
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travelling. Higher, real and complex order derivatives bring additional methods to signal
processing (3,4,5). Different approaches have been reviewed in (6).
Figure 1. Some machine condition monitoring techniques.
Figure 2. Examples of vibration monitoring techniques.
Vibration indices based on several higher derivatives in different frequency ranges were
introduced by Lahdelma in 1992 (3). Fractional integrals and derivatives are discussed in (7).
Operating conditions can be detected with a Case-Based Reasoning (CBR) type application
with linguistic equation (LE) models and Fuzzy Logic. The basic idea of the LE
methodology, which was introduced by Juuso in 1991, is the nonlinear scaling that was
developed to extract the meanings of variables from measurement signals (8). Various fuzzy
models can be presented by means of LE models, and neural networks and evolutionary
computing can be used in tuning. The first LE application in condition monitoring was
presented in (9). The condition monitoring applications are similar to the applications
intended for detecting operating conditions in the process industry (10).
The combined approach has been summarised in (6). Features extracted from higher
derivatives x(3) and x(4) have been used in cavitation indicators. The index obtained from
x(4) is the best alternative but also the index obtained from x(3) provides good results
throughout the power range. The cavitation indicator also provides warnings of possible
risk during short periods of cavitation. (11) Cavitation can be detected with indicators based
on features of acceleration and higher derivatives in a fairly low frequency range (12,13). A
generalised central moment introduced in (14) operates well even with short sample times.
This paper deals with higher and real order derivatives in processing vibration
measurements, feature extraction and model-based fault detection, where condition indices
and fault models have special emphasis on nonlinear scaling and linguistic equations.
Examples are taken from experimental systems and real machines and process equipment.
2. SIGNAL PROCESSING
Some faults, such as unbalance, misalignment, bent shaft and mechanical looseness, can be
detected by means of displacement and velocity, i.e. signals )0(
xx = and )1(
xx =
& (Table 1).
On the other hand, bearing faults or faults in gears, as well as cavitation can be detected
more efficiently with the acceleration signal. Higher order derivatives provide more
sensitive solutions. The severity of faults can be assessed by comparing the features on a
different order of derivation. For example, a certain type of bearing fault may cause a
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seven times higher peak value for the signal )4(
x than in the non-faulty case. For the signal
)2(
x, the change is about a half of this. Therefore, we can assume that the fault has become
much more dangerous if the peak value of the signal )2(
xdoubles.
Additional flexibility can be achieved with the real order of derivation (4,15,16). For
sinusoidal signals tXtxx
ω
sin)( == we obtain
),sin()
2
sin(
)(
αα
αα
α
α
ϕω
π
αωω
+=+== tXtXx
d
t
xd .................................. (1)
where α is a real number, the amplitude ,XX
α
α
ω
= and the change of the phase angle
.
2
π
αϕ
α
=The velocity and acceleration are special cases of (1). Real order derivatives
may improve the sensitivity of the features (Figure 3). For an inner race fault in a roller
bearing, the maximum sensitivity 5.91 for kurtosis was achieved with 5.4=
α
as
compared with 1.99 obtained for the acceleration (16,17). On the other hand, the oil whirl is
not always detected in displacement or velocity signals as the increase of the whirl may be
hidden under the vibrations caused by the unbalance. The negative order of derivation, i.e.
integration, amplifies the amplitudes at a low frequency. The oil whirl can be detected
better in rms measurements if 0<
α
(4,6).
Requirements for the frequency range of the measurements depend on the faults under
consideration. Unbalance, misalignment, bent shaft, mechanical looseness and some
electrical faults, for example, cause vibrations on a relatively low frequency. On the other
hand, faults in roller bearings introduce high-frequency vibrations. Other similar cases are
fast rotating gears, cavitation, and loose stator coils in electrical motors. The high
sensitivity of the analysis methods may allow the use of lower frequency ranges. In a
Kaplan water turbine, for example, cavitation-free conditions were reliably detected with
features obtained in the frequency range 10-1000 Hz. The features of the higher derivatives
)3(
xand )4(
x have a good overall performance, especially in wider frequency ranges, 10-
3000 Hz and 10-4000 Hz (6).
3. FEATURE EXTRACTION
Feature extraction in vibration analysis is based on velocity )1(
x, acceleration )2(
x and
higher derivatives, )3(
x and )4(
x. The resulting features can be combined with other
measurements.
Table 1. Examples of signals and features in fault detection.
Nature of fault Signal Features
1. Unbalance x, )1(
x rms, peak
2. Misalignment x, )1(
x rms, peak
3. Bent shaft x, )1(
x rms, peak
4. Damaged rolling element
bearings
)2(
x, )3(
x,)4(
x peak, rms, crest factor,
kurtosis, lp norm
5. Mechanical looseness x, )1(
x rms, peak
6. Damaged or worn gears )2(
x, )3(
x,)4(
x peak, rms, lp norm
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7. Oil whirl )(
α
x, 0
<
α
, x, )1(
x rms, peak
8. Resonance x, )1(
x rms, peak
9. Poor lubrication )2(
x, )3(
x,)4(
x peak, rms, lp norm
10. Cavitation )2(
x, )3(
x,)4(
x peak, rms, lp norm
11. Electrical problems x, )1(
x rms, peak
12. Loose stator coils )2(
x, )3(
x,)4(
x rms, peak
Figure 3. The signals )2(
x and )5.4(
x from a faulty spherical double-row roller bearing of type
SKF 24124 CC/W33. The measurements were performed in the frequency range 3-2000 Hz. The
fault was on the bearing’s inner race and the rotation frequency was 2 Hz (16).
3.1 Statistical features
Root-mean-square (rms) and peak values are useful features for detecting various faults
(Table 1). For an electric motor, the feature )1(
rms
xprovides information on unbalance,
misalignment, mechanical looseness and some electrical faults. The feature )4(
p
xreacts to
bearing faults, lubrication problems, stator coil faults, and also for detecting cavitation.
Signals )(
α
x can be analysed for example with standard deviation,
()
,)
1
(2/1
1
2
)()(
∑
=
−= N
i
ixx
N
αα
α
σ
........................................................ (2)
and kurtosis,
()
,
)(
1
1
4
)()(
4
2∑
=
−= N
i
ixx
N
αα
α
α
σ
β
..................................................... (3)
where )(
α
x is the arithmetic mean of the signal values Nixi,...,1,
)( =
α
, and α is a real
number. Root-mean-square of )(
α
x, i.e. rms
x)(
α
is
α
σ
when .0
)( =
α
x
These features have been used for fault diagnosis in a test rig which consists of an
electric motor and a transmission between two axes with roller bearings (9). Independent
fault modes were rotor unbalance at two levels, three misalignment cases between the
motor and input shaft, bent shaft, and three bearings faults. The features were rms and
kurtosis of the acceleration )2(
x, the average of the highest three values of the jerk )3(
x,
and rms velocities rms
x)1( in two frequency ranges.
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The cavitation analysis for a Kaplan water turbine in (11) was based on two features: the
mean peak )(
α
mp
x and the fraction )(
α
h
F of the peaks exceeding the normal range
[]
αα
σ
σ
3,3− obtained from the signal )(
α
x, α = 1, 3 and 4. The fractions )(
α
h
F have low
values in the low power range where the spikes are less frequent. Signals )(
α
x, α = 2, 3 and
4, have been analysed in four frequency ranges by means of rms values, kurtosis and peak
values (12,13). The frequency ranges were 10-1000 Hz, 10-2000 Hz, 10-3000 Hz and 10-
4000 Hz. The kurtosis is a useful feature in the low power range but for the cavitation-free
area and the high power range, the kurtosis is close to value 3, which corresponds to a
Gaussian signal, i.e. the kurtosis does not give an indication of cavitation in the high power
range in the case. An alternative feature for kurtosis is peak value, which has fairly similar
changes in the low power range and small changes in the high power range.
Detecting bearing faults and unbalance in fast rotating bearings (18) was based on
standard deviations calculated for the signal )4(
xon three frequency ranges: 10-1000 Hz,
10-10000 Hz and 10-50000 Hz. Unbalance was clearly detected on the basis of the
standard deviations obtained from the lowest frequency range. Several signals had to be
combined for detecting the other faults. In this case the rotation frequency was in the range
65-525 Hz.
3.2 Signal distribution
The distributions of the signals )1(
x, )3(
x and )4(
x have been used in monitoring the
condition of the supporting rolls of a lime kiln (19,20). Fault situations were detected as a
large number of strong impacts. The bins )(
α
k
F of the histograms are based on the standard
deviation
α
σ
of the corresponding signal )(
α
x. The velocity signal only shows very small
differences between a serious surface problem and an excellent condition. For signals )3(
x
and )4(
x, large values for the features
α
σ
and the fractions )(
α
k
F, k=4 and 5 are related to
faulty situations, and large values for the fractions )(
α
k
F, k=1…3 are obtained in normal
conditions. Similar results can be obtained with bins defined by the absolute average of the
signals, and the resulting easier calculation is useful for developing intelligent sensors (21).
3.3 Moments and norms
New features are calculated by means of a generalised moment about the origin (17):
,
1
1
)(
∑
=
=N
i
p
i
px
N
M
α
α
τ
............................................. (4)
where the real number α is the order of derivation, the real number p is the order of the
moment, τ is the sample time, i.e. the moment is obtained from the absolute values of
signals )(
α
x. The number of signal values s
NN
τ
=
where Ns is the number of samples per
second. Alternatively, the signals values )(
α
i
x can be compared to the mean )(
α
x:
,
1)(
1
)( p
N
i
i
pxx
N
M
αα
α
τ
−= ∑
=
................................. (5)
The generalised central moment can be normalised by means of the standard deviation
α
σ
of the signal )(
α
x:
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()
.
1)(
1
)( p
N
i
i
p
pxx
N
M
αα
α
α
τ
σ
σ
−= ∑
=
...................... (6)
which was presented in (14). The order of derivation ranges from 1 corresponding to
velocity to 4, which corresponds to the signal x(4). The moment 1
2=
α
τ
σ
M, and the moment
4
α
τ
σ
M corresponds to the kurtosis of the signal. The standard deviation
α
σ
can be obtained
from (5) by taking the square root when 2
=
p.
There are many alternative ways of normalisation, Lahdelma and Juuso (17) introduce a
new norm
,)
1
()( /1
1
)(/1 p
N
i
p
i
pp
p
px
N
MM ∑
=
==
α
α
τ
α
τ
...... (7)
which is the lp norm
.
)(
pp
pxM
α
α
τ
≡.................................................. (8)
This norm has same dimensions as the corresponding signals )(
α
x. The lp norms are
defined in such away that .1 ∞<≤ p In (17) the order p is allowed to be less than one. The
absolute mean and the rms value are special cases of (8). Faults can also be detected with
other types of norm, e.g. maximum norm, or with a sum of the norms obtained for different
orders of derivatives. In a special case, integer orders can be used: .,...,1,0 10 n
n==
=
α
α
α
The rms values of displacement and velocity, which are special cases of (7), can be used
for detecting unbalance, misalignment, bent shaft and mechanical looseness (Table 1).
For bearing faults, displacement and velocity should be replaced by acceleration or
higher derivatives. The rms values can then be used to detect bearing faults, especially if
2≥
α
and the rotation frequency is not very low. The peaks of the signal have a strong
effect on the moments (4), (5) and (6). The moment (6) can be used in the same way as
kurtosis in the previous studies (13). The norm (7) combines two trends: a strong increase
caused by the power p and a decrease with the power 1/p. For the order p = 1, there is no
amplification. The significance of the highest peaks will decrease if p < 1. The moments
calculated for higher order derivatives )3(
xand )4(
xare more sensitive to impacts than the
ones calculated for velocity, and the sensitivity improves when the order p of the moment
increases.
The norm (7) gives the rms value when p = 2 and has good performance in the high
power range even with high p values, compared to kurtosis. In (11,12,13,14) these two features
were combined since kurtosis provides an indication of the strong cavitation in the low
power range and the rms values are needed in the high power ranges. The relative
)max( 75.2
4
3Mprovides a good indication for cavitation as explained in (21). The power
ranges classified into cases of short periods of cavitation are slightly wider than in (13) but
differences between the results of these approaches are rather small. The relative maximum
was obtained by by using 15 MW as a reference power. The cavitation-free area and the
strong cavitation cases are clearly detected with the signal )3(
x but a slightly higher order
of moment than for )4(
xis needed. A much higher order of moment is needed for the signal
)1(
x: on the strongest cavitation is detected with the relative )max( 8
1
3M. However, all
the other cavitation cases would be classified as cases of short periods of cavitation. The
signal can be divided into short samples and features calculated for these samples. The
sample time τ is an essential parameter in the calculation of moments and norms. The time
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s3=
τ
provided the most sensitive results, but sufficiently long signals are required to
produce reliable maximum moments and material for analysing short-term cavitation (14,21).
3.4 Other measurements
Temperature can increase considerably in faulty situations: the temperature of bearings
goes fast up if the lubrication fails, the excess load of an electric motor causes a
temperature increase, and a strong unbalance may also lead to a temperature rise when the
vibration energy is converted to heat. All of these are already severe situations which can
be avoided through early fault detection.
Pressure fluctuations also mean serious problems. Vibrations change the sound pressure
when the vibrating surfaces cause the motion of air molecules. For example, surface
unevenness of the rolls in soft-calenders causes a whining sound. At this stage, the roller
should be changed in order to avoid quality problems in the paper. Pressure fluctuations in
head box cause variations in the paper thickness. Strong pressure impacts in pipes can
cause dangerous situations.
Rotation speed fluctuations , which can be detected with a stroboscope, can reveal faults
in thyristor control. Operation is in these cases unstable. These faults can also be detected
with very accurate rotation speed measurements, also denoted as “modern stroboscope”. In
blowers, full rotation speed is not achieved when unbalance is strong. In unbalance cases,
the electric motors of the blower accelerate slowly in start-up and also the current intake is
higher than normal. Current probes can be used for detecting faults in rotor bars when the
probe is mounted on any phase lead.
The wearing of machines can be examined by means of an oil analysis, e.g. in worm
gears, increased metal contents are a sign of fault: increased copper content means a fault
in the wheel, and increased iron content a fault in the worm or bearings (22).
3.5 Nonlinear scaling
The analysis can be further improved by taking into account nonlinear effects (9,11,18,19). The
scaling function scales the real values of variables to the range of [-2, +2] with two
monotonously increasing functions: one for the values between -2 and 0, and one for the
values between 0 and 2. The membership definition f consists of two second-order
polynomials, and the scaled values, which are called linguistic levels j
X, are obtained by
means of the inverse function 1−
f. Nonlinear scaling has been used in previous studies for
statistical features (9,11), and features based on the signal distribution (19,20).
Both expertise and data can be used in developing the mapping functions (membership
definitions) (23). Usually, the functions are obtained with a data-driven approach from the
features. For each feature, the level 0 can be obtained as a median of the values in the
training set, and the levels -1 and 1 as medians of the lower and higher halves of the
values, respectively. For example, a cavitation index can be obtained by scaling the norm
(7) with the function 1−
α
f:
).max((
1p
CMrelativefI
α
τ
α
α
−
=......................................................... (9)
The index 4
C
I has the best classification result when the relative )max( 75.2
4
3M is the
original feature (21). The other measurements listed in section 3.4 can scaled with similar
nonlinear functions.
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4. MODEL-BASED FAULT DIAGNOSIS
Operating conditions can be detected by combining several features in case-specific
models. Model-based cavitation indices are needed for a detailed analysis (12,13) . Fault
models can be constructed with linguistic equations. The LE models are linear equations
0
1
=+
∑
=
ij
m
j
ji BXA ............................................................................... (10)
where j
X is a linguistic level for the variable j, j=1...m. Each equation i has its own set of
interaction coefficients ....1, mjA ji = The bias term i
B was introduced for fault diagnosis
systems. Reasoning is based on the error i
ε
, also called fuzziness, is calculated for each
LE model by means of
.
1
ij
m
j
jii BXA += ∑
=
ε
............................................................................. (11)
The sequence of the LE models is case-specific, and each equation has a weight factor
ki
w. The degree of membership of each equation, denoted as i
μ
, is based on the
distribution of error represented as a trapezoidal membership function developed on the
basis of the train case. The condition index k
C of each case k can be represented by means
of ,24 −= kk
C
μ
where the degree k
μ
is obtained as a weighted average of the degrees i
μ
.
The machine condition monitoring application presented in (9) was based on models
developed for normal operation and nine fault cases. In each case the model consists of
seven LE models developed for a sensor-specific variable group including the rotation
speed and five features obtained from the measurements of the sensor. For fast-rotating
bearings, the condition index Ind is a sum of the scaled standard deviations of the signal
)4(
x calculated for three frequency ranges (18): unbalance is detected in the lowest
frequency range )(41
1
41
σ
−
f and inner race fault with the features )(42
1
42
σ
−
f and )(43
1
43
σ
−
fbut
the complete index Ind is needed for detecting the outer race fault.
A case-based reasoning (CBR) approach was used for the test rig (9): the degree of
membership was calculated for all cases, and the case with the highest degree of
membership was chosen. The classification results were very good. There are some faulty
classified measurements but the mistakes are very logical, e.g. small unbalance and normal
state. A small misalignment and the normal state are also close to each other. In all the
cases, mistakes only occur between very similar classes. The system placed practically all
the bearing faults into the right classes. The fault models and the CBR system are
necessary since the five features obtained from the signals and the rotation speed need to
be combined.
5. CONDITION INDICES
The purpose with condition indices is to extract indirect measurements from the signal
samples. Possibility to use short samples is beneficial for automatic fault detection.
5.1 Cavitation index
The intelligent cavitation indicator developed in (11) for a Kaplan water turbine is based on
the nonlinear scaling of two features: peak height and the fraction of the peaks exceeding
the normal limit. The classification results obtained from the experimental cases involving
the water turbine were very good and logical. Features of velocity )1(
x, acceleration )2(
x
and higher derivatives, )3(
x and )4(
x were compared in (12,13). The indices obtained from
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)4(
x are the best alternatives. Generalised moments and norms can also be used in the
model-based cavitation indices. The generalised moment (4) indicates possible cavitation
but in detailed analysis (14) the moments should be combined with other features, e.g. rms
values used in (12). The norm (7) introduced in (17) can be used alone, see (9), if the order p
is chosen correctly. Combined indices, where several orders α are used, require tuning of
the scaling functions. The cavitation indicator also provides warnings of a possible risk on
short periods of cavitation.
5.2 Condition index
In the lime kiln application, the features were combined with a linguistic equation, i.e.
1=iin (10). The condition index IC of the supporting rolls is a number between -2 and 2,
and the interaction coefficients ,6...1,
=
jA ji are based on expertise (19,20):
A = [-2 1 1 1 -1 -1 -1]
includes the coefficients of the features and the cavitation index. The same coefficients are
used for both signals )3(
x and )4(
x. Compared to (10), the index IC corresponds to i
B−.
The index developed for the supporting rolls of a lime kiln provides an efficient indication
of faulty situations. Surface damage is clearly detected and friction increase is indicated at
an early stage. The features are generated directly from the higher order derivates of the
acceleration signals. All the supporting rolls can be analysed using the same system. The
index )4(
C
I is very good and logical for all the measurement points, which makes it already
suitable for practical applications. Condition indices for other measurements listed in
Section 3.4 can be formed by means of nonlinear scaling functions, see (9).
5.3 Health index
Both the cavitation and condition indices can be considered linearly related to the health
index SOL (21). The health index SOL can be calculated from the cavitation index by means
of
),1(
4
2
1
*
δ
−
+
−= C
I
SOL .................................................................... (12)
where
δ
is the value of SOL index when the cavitation index .2
*=
C
I For the lime kiln
application, the health index SOL and the measurement index MIT can be calculated from
the condition index, the only difference to the cavitation case is that the condition index is
a measure of good condition, i.e. value 2 corresponds to excellent condition.
The generalised health index (Figure 4) can combine several measurements:
SOL = SOL (vibration, pressure, temperature, electric current, rotation speed, …).
A feasible alternative is to calculate individual condition indices, transform them to SOL
indices, and finally calculate an overall health index by means of the weighted average.
Figure 4. Combined indices.
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5.4 Measurement index
Vibration signals can be utilised in process or machine operation by combining features
obtained from derivatives. The measurement indices presented in (17) are the weighted
sums of dimensionless vibration indices (6). The dimensionless features are obtained by
comparing to the feature values in good condition. The individual norms can be based on
rms values, peak values or kurtosis, corresponding to indices MIT1, MIT2 and MIT3,
respectively. The measurement index MITΣ can also combine several indices, e.g. an
index calculated from the rms and peak values provides good results (17). The frequency
ranges can be specific to each feature.
The health index SOL combines several measurement indices (21). The machine is in
good condition if 1.MITSOL == The measurement index MIT (17) is an inverse of the
index SOL. If the parameter 2.0=
δ
, the highest values of the index MIT were 5 in a
Kaplan turbine. Examples are explained in (21). The MIT index provides a good indication
of the need of maintenance.
6. CONCLUSIONS
Advanced signal processing and feature extraction methods of vibration measurements are
chosen in a specific way in order to detect different faults. Generalised moments and
norms obtained from higher or real order derivatives provide informative features for
diagnosing cavitation, and faults in bearings and gears. Short sample times and relatively
small requirements for frequency ranges make this approach feasible for on-line cavitation
analysis and power control. Useful features can be extracted from the signal distributions.
Several features from vibration signals and other measurements are combined in condition
and health indices. The scaling approach extends the condition indices to nonlinear
systems. All the necessary calculations can be performed in the intelligent sensors.
REFERENCES
1. B K N Rao, ‘Handbook of Condition Monitoring, 603 pp, Elsevier Advanced
Technology, Oxford, UK, 1996.
2. J D Smith, ‘Vibration monitoring of bearings at low speeds’, Tribology International
Vol 15, No 3, pp 139-144, 1982.
3. S Lahdelma, ‘New vibration severity evaluation criteria for condition monitoring’ (in
Finnish), Research report No. 85, University of Oulu, 1992.
4. S Lahdelma, ‘On the higher order derivatives in the laws of motion and their
application to an active force generator and to condition monitoring’, D.Sc.Tech.
thesis. Research report No. 101, University of Oulu, 1995.
5. S Lahdelma and V Kotila, ‘Complex Derivative – A New Signal Processing Method’,
Kunnossapito, Vol 19, No 4, pp 39-46, 2005.
6. S Lahdelma and E Juuso, ‘Advanced signal processing and fault diagnosis in condition
monitoring’, Insight, Vol 49, No 12, pp 719–725, 2007.
7. S G Samko, A A Kilbas, and O I Marichev, ‘Fractional Integrals and Derivatives.
Theory and Applications’, 976 pp, Gordon and Breach Science Publishers, 1993.
8. E K Juuso and K Leiviskä, ‘Adaptive expert systems for metallurgical processes, in
expert systems in mineral and metal processing’, Proceedings of the IFAC Workshop,
Espoo, Finland, IFAC Workshop Series, No 2, Pergamon, Oxford, UK, pp 119-124,
1992.
9. E K Juuso, M Kivistö and S Lahdelma, ‘Intelligent Condition Monitoring Using
Vibration Signals’, Proceedings of EUNITE 2004, Aachen, Germany, pp 381-390,
Verlag Mainz, Aachen, June 2004.
The Fifth International Conference on Condition Monitoring & Machinery Failure Prevention Technologies
888
10. E Juuso and K Leiviskä, Combining Monitoring and Process Data in Detecting
Operation Conditions in Process Industry, in Maintenance, Condition Monitoring and
Diagnostics, Proceedings of the 2nd International Seminar, Oulu, Finland, pp 145-156,
September 2005.
11. E Juuso and S Lahdelma, ‘Intelligent Cavitation Indicator for Kaplan Water Turbines’,
Proceedings of COMADEM 2006, Luleå, Sweden, 2006, pp 849-858, Luleå
University Press, June 2006.
12. E Juuso, S Lahdelma, and P Vähäoja, ’Feature extraction for vibration analysis of
cavitation in Kaplan water turbines, Proceedings of WCEAM-CM 2007, Harrogate,
UK, pp 943-952, Coxmoor Publishing, Oxford, UK, June 2007.
13. S Lahdelma, P Vähäoja, and E Juuso, ’Detection of Cavitation in Kaplan Water
Turbines’, Proceedings of 2007 Arctic Summer Conference on Dynamics, Vibrations
and Control, Ivalo, Finland, pp 80–87, Tampere University Press, Tampere, Finland,
August 2007.
14. S Lahdelma and E Juuso, ‘Vibration Analysis of Cavitation in Kaplan Water
Turbines’, IFAC World Congres, Seoul, Korea, 6 pp, July 2008.
15. S Lahdelma, ‘On the Derivative of Real Number Order and its Application to
Condition Monitoring’, Kunnossapito, Vol 11, No 4, pp 25-28, 1997.
16. S Lahdelma and V Kotila, ‘Real Order Derivatives – New Signal Processing Method’
(in Finnish), Kunnossapito, Vol 17, No 8, pp 39-42, 2003.
17. S Lahdelma and E Juuso, ’Signal Processing in Vibration Analysis’, Proceedings of
CM 2008 – MFPT 2008, Edinburgh, 12 pp, July 2008.
18. S Lahdelma, E Juuso and J Strackeljan, ‚Neue Entwicklungen auf dem Gebiet der
Wälzlagerüberwahung, in A Seeliger and P Burgwinkel (Ed.) Tagungsband zum
AKIDA 2006, Aachen, Germany, pp 447-460, ASRE, Band 63, 2006, RWTH Aachen,
November 2006.
19. S Lahdelma and E K Juuso, ’Intelligent condition monitoring for lime kilns’, in A
Seeliger and P Burgwinkel (Ed.) Tagungsband zum AKIDA 2006, Aachen, Germany,
pp 399-408, ASRE, Band 63, RWTH Aachen, November 2006.
20. E Juuso and S Lahdelma, ‘Advanced condition monitoring for lime kilns’.
Proceedings of WCEAM-CM 2007, Harrogate, UK, pp 931-942, Harrogate, UK.
Coxmoor, June 2007.
21. E Juuso and S Lahdelma, ’Intelligent Condition Indices in Fault Diagnosis’,
Proceedings of CM 2008 – MFPT 2008, Edinburgh, 12 pp, July 2008.
22. P Vähäoja, S Lahdelma and T Kuokkanen, ‘Condition Monitoring of Gearboxes Using
Laboratory-scale Oil Analysis’, Proceeding COMADEM 2004, Cambridge, UK, pp
104-114, Comadem, August 2004.
23. E K Juuso, ‘Integration of Intelligent Systems in Development of Smart Adaptive
Systems’, International Journal of Approximate Reasoning, Vol 35, No 3, pp 307-337,
2004.
The Fifth International Conference on Condition Monitoring & Machinery Failure Prevention Technologies
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