POLISH ACADEMY OF SCIENCES COMMITTEE OF MACHINE ENGINEERING
SCIENTIFIC PROBLEMS
OF MACHINES OPERATION
AND MAINTENANCE
ZAGADNIENIA EKSPLOATACJI MASZYN
TRIBOLOGY • RELIABILITY • TEROTECHNOLOGY
DIAGNOSTICS • SAFETY • ECO-ENGINEERING
TRIBOLOGIA • NIEZAWODNOŚĆ • EKSPLOATYKA
DIAGNOSTYKA • BEZPIECZEŃSTWO • EKOINŻYNIERIA
3 (167) Vol. 46
2011
Institute for Sustainable Technologies – National Research Institute, Radom
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SCIENTIFIC BOARD
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and
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SCIENTIFIC PROBLEMS
OF MACHINES OPERATION
AND MAINTENANCE
3 (167) 2011
CONTENTS
W. Batko, L. Majkut: Classification of phase trajectory portraits
in the process of recognition in the changes in the technical
condition of monitored machines and constructions ................. 7
J.M. Czaplicki, A.M. Kulczycka: Steady-state availability
of a multi-element symmetric pair .............................................. 15
W. Grzegorzek, S. Ścieszka: Prediction on friction characteristics
of mine hoist disc brakes using artificial neural networks .......... 27
A. Katunin: The construction of high-order b-spline wavelets
and their decomposition relations for fault detection
and localisation in composite beams ......................................... 43
A. Sowa: Problem of computer-aided technical state evaluation
of rail-vehicle wheel sets ............................................................ 61
M. Styp-Rekowski, L. Knopik, E. Mańka: Probabilistic formulation
of steel cables durability problem............................................... 69
M. Ważny: An outline of a method for determining the density
function of the time of exceeding the limit state with the use
of the Weibull distribution ......................................................... 77
SPIS TREŚCI
W. Batko, L. Majkut: Klasyfikacja obrazów trajektorii fazowych
w procesie rozpoznawania zmian stanu monitorowanych maszyn
i konstrukcji ................................................................................... 7
J.M. Czaplicki, A.M. Kulczycka: Graniczny współczynnik gotowości
wieloelementowej pary symetrycznej ............................................ 15
W. Grzegorzek, S. Ścieszka: Prognozowanie charakterystyk ciernych
hamulców maszyn wyciągowych z zastosowaniem sztucznych
sieci neuronowych ......................................................................... 27
A. Katunin: Konstrukcja falek b-splinowych wyższych rzędów
i ich zależności dekompozycji dla detekcji i lokalizacji
uszkodzeń w belkach kompozytowych .......................................... 43
A. Sowa: Problemy wspomaganej komputerowo oceny
stanu technicznego zestawów kołowych pojazdów szynowych..... 61
M. Styp-Rekowski, L. Knopik, E. Mańka: Probabilistyczne ujęcie
zagadnienia trwałości lin stalowych............................................... 69
M. Ważny: Zarys metody określenia funkcji gęstości czasu
przekroczenia stanu dopuszczalnego z wykorzystaniem
rozkładu Weibulla.......................................................................... 77
Classification of phase trajectory portraits in the process of recognition...
7
SCIENTIFIC PROBLEMS
OF MACHINES OPERATION
AND MAINTENANCE
3 (167) 2011
WOJCIECH BATKO*, LESZEK MAJKUT
*
Classification of phase trajectory portraits in the process
of recognition in the changes in the technical condition
of monitored machines and constructions
K e y w o r d s
Phase trajectory, attractor, recurrence, diagnostics.
S ł o w a k l u c z o w e
Trajektoria fazowa, atraktor, rekurencja, diagnostyka.
S u m m a r y
A methodology of the functioning correctness control of machines and structural components
is described in the article. The proposed new approach to the construction
of a vibration-based monitoring system is presented. A methodology based on the quantitative
analysis of attractor, phase trajectory and recurrence quantification analysis (RQA) is presented in
detail.
* AGH, University of Science and Technology, Mickiewicza Avenue 30, 30-059 Kraków, Poland.
W. Batko, L. Majkut
8
Introduction
Performing the review of systems – functioning in the industrial practice –
monitoring changes in a machine and structure technical conditions, it is
possible to generate several synthesising statements and conclusions of a
general nature being the assessment of their solutions.
• Monitoring systems, which trace changes of the special numerical estimates
(e.g. effective, peak and average values of the measuring signal or their
mutual combinations) as well as the determined functional patterns
(meaning: shaft neck motion trajectory in a bearing sleeve, spectrum density
function, correlation, coherence, cepstrum, envelopes etc.), ensure control of
the state of the machine.
• Criteria values for the monitored diagnostic symptoms are determined by the
appropriate standards, regulations, and findings resulting either from
maintenance experiences or from the assumption of acceptable projections of
object damages.
• Structural and exploitation features of the monitored object are not taken into
consideration to a satisfying degree in the process of building monitoring
systems.
These assessments are not pretending to list all problems occurring in the
construction of monitoring systems. However, they can constitute an inspiration
in searching for new methodological guidelines for the monitoring systems that
without limitations as shown in the presented synthesis.
The aim of this paper is to indicate some possibilities in this scope. It
seems that the quantitative analysis of certain geometrical features, which are
graphical signal representations, can be a good tool for the realisation of such
tasks. Such an analysis can help in finding new diagnostic symptoms related to
the analysis of the monitored object dynamics.
Description of the system dynamics in the phase space
The phase space of a dynamic system is a mathematical space of
orthogonal coordinates representing all variables necessary for the
determination of the instantaneous state of the system. The total description of
the system dynamics in the phase space can also be obtained when the system
attractors are known. An attractor is a certain set in the phase space towards
which the trajectories initiated in various domains of the phase space (i.e.
trajectories for various initial conditions) are heading as the time progressed.
An attractor can be a point, a closed curve, or a fractal. The possibilities of
using the attractor in a form of a boundary cycle are described in Section 3. The
possibilities of utilising a quantitative analysis of trajectory in the case when the
Classification of phase trajectory portraits in the process of recognition...
9
point is the attractor are given in Section 4, and the quantitative analysis of the
trajectory recurrence are given in Section 5.
Both the attractor and phase trajectory are multidimensional curves. The
trajectory projection on a certain plane, formed by two perpendicular axes of the
phase space, can be analysed without loosing the generalities of considerations [1].
The most obvious coordinates of such plane used in the topological
analysis of vibrations are velocity and displacement. Instead of analysing
velocity as a function of displacement, it is possible to analyse displacement as a
function of velocity. Apart from advantages due to fewer time series
integrations (time and cost of calculations), another benefit is the fact that they
can be determined directly on the object being under diagnostics. The authors,
in their investigations, were using a speedometer VS80 produced by Brüel &
Kjær and accelerometer type PCB 356A16 of the PCB Pizotronics Company.
Another way to construct the projection plane is to use the delay method as
known from chaos theory. It is enough in this method to determine one time
series (e.g. vibration acceleration) and on its basis determine the whole phase
space.
The trajectory reconstruction from the individual time series requires the
creation of additional variables. In searching for new variables, the Takens
Theorem can be helpful, since it states that each point in the phase space a(n) is
represented by a sequence of time series values.
)])1((,),(),([)( ττ −++= mnynynyna K (1)
where: m is a phase space dimension, τ – delay time.
The most often applied procedure of selecting the phase space dimension m
is the method of the False Nearest Neighbours (FNN).
The criterion of the time delay selection, which utilises non-linear
dependencies between observations, is the method of Mutual Information. The
number of mutual information I(xi,T) is determined from the following
dependence [11]:
[ ]))((log))(((log))(),(((log1
),( 222 TnxpnxpTnxnxpN
TxI ii
n
iii +−−+= ∑ (2)
where: p(xi(n)) – the probability density function of the analysed series, T – the
delay time, p(xi(n),xi(n+T)) – combined probability density, N – number of
samples in the time series.
According to this method for the time delay τ, the smallest time value T in
Equation (2) for which the mutual information function obtains the local
minimum should be assumed.
W. Batko, L. Majkut
10
Qualitative analysis of the attractor in a form of the boundary cycle
The proposed diagnostics method is based on the determination of
displacement and velocity for the arbitrary selected point of the investigated
system, where vibrations originated as a result of excitation. The mono-
harmonic excitation of a frequency lower than the first natural frequency of
vibrations of the element undergoing diagnostics was assumed in the study
model excitation. Attractors determined for the loaded beam of various axial
cracking length (delamination) d are shown in Fig. 1a [8], and Fig. 1b shows the
changes of the attractor determined for the beam with a transverse cracking of a
depth – a. The area of allowable solutions Ω was selected in such a way as to
have the crack propagation rate being equal to the determined value. Such a
selection of the allowable solution area enables one to assess the time remaining
to the damage of the analysed beam [2–6].
-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
d=0
d=0.05 m
d=0.10 m
d=0.15 m
d=0.20 m
x
xo
(a)
(b)
Fig. 1. Attractors of the cracked beam
Rys. 1. Atraktory pękniętej belki
The diagnosed system in which trajectory exceeds the allowable solutions
area Ω is not suitable, because of the condition for which the Ω area was
determined.
Quantitative analysis of the phase trajectory
The first damage index proposed by the authors is related to the distance
change of the point in the trajectory from the point that is the attractor of this
trajectory, which is the scalar damage index (the authors are using the sum of
the relative vectors difference r) for each time instant (of each sample n).
Classification of phase trajectory portraits in the process of recognition...
11
∑−
=
n z
zu
nr
nrnr
N )(
)()(1WU r (3)
where: ru – vector of the distance of points in the trajectory from the attractor
determined for the damaged element, rz – vector for the not damaged element.
The second damage index is related to the Poincare map. It is constructed
by the stroboscopic ‘viewing’ of the trajectory phase pattern at constant time
intervals. If ‘pictures’ are taken at time intervals corresponding to the period
of the first frequency of natural vibrations, the map is a straight line. When the
same time intervals are applied for the formation of the Poincare map of the
trajectory of the system with different inertial-elastic parameters (e.g. of a
damaged object), the map will not be a straight line. The proposed damage
index WUφ is determined from the following equation:
∑−
=
n z
zu
n
nn
N )(
)()(1WU
φ
φφ
φ (4)
where: φ u – vector of polar coordinates of the Poincare map for the damaged
element, φ z – vector for the not damaged element.
The simulation of the impulse response of the beam determined the
example of the phase trajectory with a transverse cracking of various depths [9].
Both damage indexes as a function of a crack depth are shown in Fig. 2.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
-3
wska
zn
ik u
szko
dze
nia
a/h
WUr
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
0.005
0.01
0.015
0.02
0.025
wska
zn
ik u
szko
dzen
ia
a/h
WUφ
Fig. 2. Damage Indexes based on the phase trajectory analysis
Rys. 2. Wskaźniki uszkodzenia oparte o analizę trajektorii fazowej
Other examples of this diagnostic method application can be found in the
authors’ papers [6, 9].
W. Batko, L. Majkut
12
Application of recurrence diagrams
A recurrence diagram is a diagram presenting the repeatability (recurrence)
of processes, effects, or system states. An important advantage of the diagram is
the possibility of its application both for large and small data sets, including
non-stationary ones. The diagram presents the following dependence [10]:
( ) MjiHR ji ,,1,,, K=−−= ji XXη (5)
where: Xi, Xj – states in the space mR , M – number of states, H – Heaviside's
function, X – standard of vector X in the space mR (the most often it is the
Euclidean or maximum standard), η – non-negative real number, the so-called:
cut-off parameter.
As can be seen, the basis of the diagram determined by Equation (5) is the
zero-one square matrix RMM. Value Ri,j = 1 is marked by a black point in the
diagram, while Ri,j = 0 is marked by a white point (no point).
0 2000 40000
0.02
0.04
0.06
0.08
0.1recurrence rate
0 2000 40000
0.02
0.04
0.06
0.08
0.1recurrence rate
0 2000 40000
0.02
0.04
0.06
0.08
0.1recurrence rate
0 2000 40000
0.02
0.04
0.06
0.08
0.1recurrence rate
0 2000 40000
0.02
0.04
0.06
0.08
0.1recurrence rate
0 2000 40000
0.02
0.04
0.06
0.08
0.1recurrence rate
a=0.15 h a=0.2 h a=0.25 h
a=0.1 ha=0.05 ha=0 h
Fig. 3. Changes in recurrence rate as a function of the crack depth
Rys. 3. Przebiegi wybranych funkcji uszkodzenia w funkcji uszkodzenia
In the dependence on the nature and properties of the considered problem,
the black points in the diagram form various structures. These can be individual
points, points collected along curves of various lengths, or straight lines
arranged horizontally, perpendicularly, or skewed. The most important values
allowing one to perform the quantitative diagram analysis are Recurrence Rate,
Determinism, lmax, Trend, Entropy, Laminarity, and Trapping time [10]. These
and several other values characterising (describing quantitatively) the recurrent
diagram can be determined for the whole or part of the recorded signal. The
Classification of phase trajectory portraits in the process of recognition...
13
analysis of the waveform part is based only on the determination of the sought
values in the observation window. When shifting the window by one or more
samples, it is possible to determine certain functions that were assumed in this
work as damage functions. In the waveforms and their changes, the symptoms
related to the cracking of the beam are sought.
The waveforms of the recurrence rate that are dependent on the crack
depth are shown in Fig. 3. The view in the upper left window is related to the
artificial noise added to the signal.
The possibilities of using this and other functions of damages together with
the analysis of measuring error influences are described in [7].
Conclusions
An utilisation of the proposed diagnostics method based on phase trajectory
analysis allows for the fast and effective diagnostics of damages.
Analysis of damage indexes as a function of damage indicates a high
sensitivity of the proposed method (the possibility of detecting damages in the
early stage of their formation). The early detection of damages of structural
elements allows for the optimisation of repairs (their necessity and scope),
avoiding losses related to forced shutdowns, and decreasing costs of not needed
spare parts storage and costs related to unexpected breakdowns.
All proposed indexes are also characterised by a high sensitivity in a
damage function. This high sensitivity means large changes of the damage index
in a function of a damage degree, which allows for the detection and analysis of
the damage progressing. In other words, a comparison of the current trajectory
with the trajectory from previous diagnostics allows one to check whether the
crack opening is propagating or remains stationary.
The method does not filter non-linear effects or the changes of the
frequency structure of the monitored diagnostics signals related to the
development of damages, which can be its special advantage.
A practical application of the trajectory changes is useful as a control
method for the beginning and development of damage. This can be the most
distinctive feature of the method, which is easily adaptable for practical
applications.
References
[1] Abarbanel H.D.I.: Analysis of observed chaotic data, Springer, 1996.
[2] Batko W.: Technical stability – a new modeling perspective for building solutions
of monitoring systems for machine state, Zagadnienia Eksploatacji Maszyn, 151, 2007,
147–157.
W. Batko, L. Majkut
14
[3] Batko W., Majkut L.: The phase trajectories as the new diagnostic discriminates of foundry
machines and devices usability. Archives of Metallurgy and Materials, 52, 2007, 389–394.
[4] Batko W., Majkut L.: Classification of phase trajectory portraits in the process of recognition
the changes in technical condition of monitored machines and constructions, Archives of
Metallurgy and Materials 55, 2010, pp. 757–762.
[5] Batko W., Majkut L.: Damage identification in prestressed structures using phase
trajectories, Diagnostyka 44, 2007, 63–68.
[6] Batko W., Majkut L.: Wykorzystanie trajektorii fazowej jako informacji o stanie
technicznym obiektu. Biuletyn WAT, 2010.
[7] Batko W., Majkut L.: Zastosowanie diagramów rekurencyjnych do oceny stanu technicznego
obiektu, Pomiary, Automatyka, Kontrola, vol. 57, nr 07/2011, s. 794–800.
[8] Majkut L.: Diagnostyka wibroakustyczna belek z pęknięciami wzdłużnymi, Biuletyn
Wojskowej Akademii Technicznej, 59 (2010), p. 181–196.
[9] Majkut L.: Diagnostyka wibroakustyczna uszkodzeń elementów konstrukcyjnych,
Wydawnictwo ITeE, Radom 2010.
[10] Marwan N., Romano M., Thiel M., Kurths J.: Recurrence plots for the analysis of complex
systems, Physics Reports 438, 2002, pp. 237–329.
[11] Nichols J.M., Seaver M., Trickey S.T.: A method for detecting damage-induced
nonlinearities in structures using information theory, Journal of Sound and Vibration,
297:1–16, 2006.
Klasyfikacja obrazów trajektorii fazowych w procesie rozpoznawania zmian stanu
monitorowanych maszyn i konstrukcji*
S t r e s z c z e n i e
W artykule omówiono metodykę nadzoru poprawności funkcjonowania maszyn i konstrukcji
wsporczych, bazującą na systemach monitoringu drganiowego. W szczególności zaprezentowano
metodykę opartą na ilościowej analizie graficznych reprezentacji monitorowanego sygnału
w postaci atraktora, trajektorii fazowej oraz analizy ilościowej rekurencji.
* The work was performed within the realization of the statute studies No.11.11.130.885.
Steady-state availability of a multi-element symmetric pair
15
SCIENTIFIC PROBLEMS
OF MACHINES OPERATION
AND MAINTENANCE
3 (167) 2011
JACEK M. CZAPLICKI*, ANNA M. KULCZYCKA
*
Steady-state availability of a multi-element symmetric pair
K e y w o r d s
Semi-Markov process, multi-element symmetric pair, steady-state availability calculation.
S ł o w a k l u c z o w e
Proces semi-Markowa, wieloelementowa para symetryczna, graniczny współczynnik gotowości.
S u m m a r y
This article is a continuation of the consideration in the study "Semi-Markov process for a pair
of elements,” published in 2011. However, this paper discusses the issues concerning a multi-element
symmetric pair and not just a pair of elements alone. The problems with the operation of this type
of system are identified and discussed. An analysis of this multi-element system was done with
a proposed modelling method of its operation. The presented methods take into account different
cases, depending on the result of an empirical study, exactly the type of probability distributions
of times of elements states which the system are composed. The modelling method provides solutions
that allow the basic reliability parameters of the multi element system to be obtained.
Introduction
There are a number of technical systems consisting of one basic element
and a second one held in reserve. There are also a number of such pairs in
* Silesian University of Technology, Faculty of Mining and Geology, Mining Mechanisation Institute,
Akademicka 2, 44-100 Gliwice, Poland, e-mail: [email protected], [email protected].
J.M. Czaplicki, A.M. Kulczycka
16
mining engineering, e.g. a one-belt conveyor carrying a valuable mineral to its
destination and a second conveyor serving as a spare. Such a solution makes
sense if the stream of the mineral is high. Generally, there are five problems
associated with the operation of a system of this kind.
At the very beginning is the problem:
(a) Whether it is an economically rational decision to add a spare unit to
operating element?
Further problems are typically operational ones, namely:
(b) Which method of system utilisation should be selected?
(c) What is the intensity of the failures of spare element?
(d) Which mathematical model should be applied to adequately describe the
operation of the system and what is the best way to assess the basic system
parameters?
The last concern that can be associated with some generalisations is as
follows:
(e) Each unit of the system consists of a certain number of elements connected
in series.
To obtain an answer to the first question – applying the necessary
economic considerations – it is necessary to get answers to questions (b) to (d),
keeping in mind that the last point should be taken into account if the system is
of such a structure.
Consider problem (b). Analysing the reliability of pairs of elements
operating in several different technical fields, around fifty years ago, engineers
discovered that the problem of the manner of system utilisation is important. As
a rule, three different methods of system operation were taken into account, that
is a “symmetric pair,” a “pair in order” and a “pair half-loaded” (e.g. Czaplicki
2010, Chapter 7).
The operation order for a symmetric pair is as follows. One element
executes its duties, and the second one is in a cold-type reserve. When a failure
occurs in the working element, the second element commences its duties
without delay. The first element is then in a repair state. When the repair is
finished – a renewal occurs – the first element becomes the reserve. This
situation exists until the moment when a failure occurs in the second element.
The situation is then reversed. A failure of the system occurs when a failure
occurs in the working element during the repair of the other.
Mining practice has shown that this method of operation of the system is
most convenient or its equivalent – a pair in which elements are switched into
operation deterministically from time to time, not waiting for a failure to occur.
This is important for mechanical elements; but for electronics items, it may be
invalid. In mechanical devices, some unwanted processes may occur during their
long standstill, e.g., fluids sediments gathering at bottom, extensive belt sag, etc.
Presume here that the system of interest is operating under the “symmetric
pair” regime, i.e. both elements are used uniformly in a long run in a stochastic
Steady-state availability of a multi-element symmetric pair
17
sense. Thus, for a discussion of reliability, it does not matter whether the system
can be treated as a symmetric pair or a pair deterministically switched on-and-
off on a time schedule.
This problem (c) was discussed at the very beginning of the problem
formation (for example: Gnyedenko 1964, 1969, Gnyedenko et al. 1965,
Kopociński 1973). Each unit can be in its two own states, namely, a work and
repair, and one state being a result of the system construction – standstill/reserve.
Now, the problem is whether an element can or cannot fail when it is in a standstill
state. The possibility of a failure in a spare part was one of the preliminary
assumptions, making the reserve a “warm type.” Some results were presented in
the cited papers in connection with different types of reserves; however, they were
mainly in the shape of Laplace transforms. Although, it looks like the most
important result, especially from a practical point of view, it is based on the
assumption of a “cold type” reserve. Usually, the intensity of the failures of an
element in reserve is none or very small and, for this reason, can be neglected.
Method of system modelling
A first step in an analysis of system operation is the choice of the method
of modelling its operation. If an empirical investigation shows that the
probability distributions of the times of element states can be described by
exponential distributions, the whole system operates according to the Markov
process. This is the simplest case and the corresponding process of changes of
states for the system is well known and was recently recalled by Czaplicki (2010
Chapter 7). However, in many practical cases, this assumption does not hold.
The times of states are usually independent of each other but their probability
distributions are not exponential, or only one is exponential. If so, the process of
the changes of the states of the system can be described by a semi-Markov
process. Let us discuss such an option in detail.
The method of analysis of such a system developed in the Markov process
can be restated as follows: We start from an analysis of a series system, and
when the appropriate characteristic functions are obtained, we consider a pair of
elements.
Analysis of a series system
Consider an exploitation repertoire1 si; i=1,2, …,n+1 for states of the
system. It consists of n+1 states; n states of repair because the system has n
1 An exploitation repertoire is a defined set of the possible states of a given object, i.e. states in
which object can be.
J.M. Czaplicki, A.M. Kulczycka
18
elements and 1 state of work. Denote the set of the repair states of the system by
(0) and the set of the work states of the system by (1).
If so, an exploitation graph for the process of changes of states can be
illustrated as is shown in Fig. 1a. In this figure, information on the
corresponding probabilities of a transition between states is given. Figure 1b, in
turn, shows the principle of passages between the states with information on the
probability distributions ( )tQij concerning the transition from state i to state j.
Notice:
(i) Qij(t); i ,j=1,2, …,n+1denotes the probability distribution of time that
process stay in state i and will jump to state j.
(ii) In all cases, one subscript is 1 because system analyzed is a series one.
(iii) The probability distributions ( )tQij are determined by the equations
( ) ( )tQptQ ijijij = ; 1p 1ii
≡∧ (1)
(iv) Obviously, 1p1n
2j
j1=∑
+
=
.
System
work state
s1
Repair e1
s2
sn+1Repair en
(0)
(1)1
12p
1
( )1n1p+
.
.
.
.
.
.
Fig. 1. An exploitation graph of the process of the changes of states for a series system with:
a) probabilities of transition between states, b) probability distributions of passages between states
Rys. 1. Graf eksploatacyjny dla procesu zmiany stanów systemu szeregowego:
a) prawdopodobieństwa przejść pomiędzy stanami, b) rozkłady prawdopodobieństw przejść
pomiędzy stanami
System work state
s1
Repair e1
s2
sn+1Repair en
(0)
(1)( )tQ21
( )tQ12
( )( )tQ 11n+
( )( )tQ 1n1 +
.
.
.
.
.
.
a) b)
Steady-state availability of a multi-element symmetric pair
19
Now we can construct the embedded Markov chain for the semi-Markov
process of the system. We have the following:
P = = (2)
Based on the total probability principle, the following equations hold:
( )( )( )
( )( ) ( )tdQtQ1...tQ1p 12
0
1n11312 ∫∞
+−−=
. . . (3)
( )( )( ) ( )( )
( )( )tdQtQ1...tQ1p 1n1
0
n1121n1 +
∞
+ ∫ −−=
If so, the semi-Markov kernel is given by equation:
O(t) = (4)
It is a characteristic feature of series systems that non-zero elements
besides the first element are only in the first row and in the first column.
The ergodic probability distribution for the Markov chain is determined by
the following matrix equation:
ΠΠΠΠ P = ΠΠΠΠ (5)
The probability distribution ΠΠΠΠ consists of n+1 elements, because this is the
number of states of the process. Therefore, ΠΠΠΠ = (Π1 ... Πn+1).
( )
( )( )
( )
( )( )
+
+
000tQ
............
000tQ
tQ...tQ0
11n
21
1n112
p12
1
1
0
0
…0
…
… p1(n+1)
… 0
… …
0
…
P00 P10
P01 P11
J.M. Czaplicki, A.M. Kulczycka
20
By changing Equation (5) into the coordinate form, we have the following:
1
1n
2i
i Π=Π∑+
=
ii11 p Π=Π ; i ≠ 1
(6)
11n
1i
i =Π∑+
=
Solving this set of equations we get
Π1 = ½; Πi = (½)p1i ; i = 2, …, n+1 (7)
We can now determine the ergodic probability distribution for the semi-
Markov process. Elements of it are defined by the following elements:
M
miii
Π=ρ , ∑
+
=
Π=
1n
1i
iimM (8)
where mi is the average time of a given state.
These parameters can be obtained from the following relationships:
( )dxxxdQm0
i1i ∫∞
= ; i = 2,3,…,n+1; ( )∑ ∫∑+
=
∞+
=
==
1n
2i 0
i1i1
1n
2i
i1i11 dxxxdQpmpm
(9)
The steady-state availability Αs of the series system is
As = ρ1 (10)
The patterns that are derived concern a general case when all of the
elements of the system are different. However, as a rule, all items in a series
system are identical; and, for this reason,
Q21(t) = Q31(t) = ... = Q(n+1)1(t) = G(t)
(11)
Q12(t) = Q13(t) = ... = Q1(n+1)(t) = F(t)
Steady-state availability of a multi-element symmetric pair
21
If G(t) denotes the probability distribution of element repair time and F(t)
denotes the probability distribution of element work time. In further analysis
this assumption will hold.
Now, we try to replace the whole series system by one stipulated element
of reliability characteristics adequate for the system. Two probability
distributions are needed: the probability distribution Fs(t) of the work time of
the system, and the probability distribution Gs(t) of its repair time.
Due to the assumption that all elements connected in series are identical,
the probability distribution of repair time of the system is identical to the
probability distribution repair time of an element of the system, i.e.
G(t) ≡ Gs(t). (12)
Unfortunately, with the second probability distribution is not so simple.
Having the steady-state availability As of the system, we can calculate the
average time of work E(Tws) for the system using well-known formula
)(1
)( r
s
sws TE
A
ATE
−
= (13)
where E(Tr) is the average time of repair.
But that is all. In a general case, we have no possibility of getting further
information on the random variable that is of interest. Nonetheless, there is an
exception to this rule.
If the probability distribution of the work time of the system element is
exponential, then the probability distribution of the work time of the system is
also exponential and the intensity of the failures of the system is the sum of all
intensities of the elements of the system. In such a case, we have complete
information on the probability distribution of the work time.
If such regularity is not observed there are two possibilities. We can:
(i) use information gained from practice or
(ii) apply a simulation technique.
Neglect solution (ii). Consider the first one.
It has been observed in mining engineering that, for many pieces of
equipment, the mean work time and the corresponding standard deviation
remain in a certain stochastic proportion, i.e. this ratio stays approximately
constant. If so, it can be presumed that the unknown standard deviation of work
time is kE(Tws) and k is a certain constant; usually k < 1. Thus, having
information on two basic parameters of the random variable, we can presume a
certain probability distribution, say the Weibull one, which will represent the
probability distribution of the work time of the system. Such a Weibull
distribution should have an expected value that equals E(Tws) and the standard
deviation kE(Tws).
J.M. Czaplicki, A.M. Kulczycka
22
Hence, the following equations must hold:
E(Tws) = ( )α−−
λα+Γ/111
( )( ) ( ) α−
λ
α+Γ−
α
+Γ=
2
122
ws /12
1TkE
for the probability distribution of work time of a series system given by:
fs(t) = αλ tα-1
α
λ− te , t > 0, α > 0, λ > 0 (15)
Consider the reliability of a system of pair of elements.
Analysis of a symmetric pair of elements2
We may study an exploitation repertoire for the process of changes of
states of symmetric pair. Each element can be in three states: work (W), repair
(R), and standstill in reserve (S). Therefore, the set of theoretically possible
states consists of 23 = 8 elements; however, technically, the system can be in
five states. They are as follows:
S1, …, S5 = WS, WR, SW, RW, RR.
An exploitation graph is shown in Fig. 2.
Fig. 2. Exploitation graph for a symmetric pair; possible transitions between states and
corresponding probabilities
Rys. 2. Graf eksploatacyjny pary symetrycznej; możliwe przejścia pomiędzy stanami
i odpowiadające im prawdopodobieństwa
2 A lecture on a pair of elements system that has the process of changes of states following the
semi-Markov scheme was given during the XL Winter Reliability School on January 2012 [6].
However, considerations were orientated on only two elements.
WR
S1
S2
S3
S5S4
1
p25
p45
1
p21
p52
p54
p43
WS
RW RR
SW
(0)
(1)
14)
Steady-state availability of a multi-element symmetric pair
23
The passage probabilities can be calculated from following patterns:
( )[ ] ( ) ( )[ ] ( )dttgtF1dttqtQ1p1p0
ss
0
43454543 ∫∫∞∞
−=−=−=
( )[ ] ( ) ( )[ ] ( )dttgtF1dttqtQ1p1p0
ss
0
21252521 ∫∫∞∞
−=−=−= (16)
( )[ ] ( ) ( )[ ] ( )dttgtG1dttqtQ1p1p0
ss
0
52545452 ∫∫∞∞
−=−=−=
The semi-Markov kernel O(t) – the matrix of transition between states is
determined as
O(t) = (17)
The embedded Markov chain for the semi-Markov process of the
symmetric pair system is
P = = (18)
The ergodic probability distribution for the Markov chain can be obtained
by solving Equation (5), keeping in mind that the matrix ΠΠΠΠ = (Π1 ... Π5) and
that the sum of all probabilities obviously equals zero.
10
01 0
P 11 P
P P
( )
( ) ( )
( )
( ) ( )
( ) ( )
0tGp0tGp0
tFp0tGp00
000tF0
tFp000tGp
0tF000
s54s52
s45s43
s
s25s21
s
5452 p0p0
45
25
p
0
p
0
0p00
0010
000p
1000
43
21
0
J.M. Czaplicki, A.M. Kulczycka
24
Now we can create the formula for the expected values for all five states:
( ) ( )∫ ∫∞ ∞
===
0 0
s14141 dtttfdtttdQmm
( ) ( ) =+=+= ∫∫∞∞
dtttdQpdtttdQpmpmpm0
2525
0
2121252521212
( ) ( )∫∫∞∞
+=
0
s25
0
s21 dtttfpdtttgp
( ) ( )∫∫∞∞
===
0
s
0
32323 dtttfdtttdQmm
( ) ( ) =+=+= ∫∫∞∞
0
4545
0
4343454543434 dtttdQpdtttdQpmpmpm
( ) ( )∫∫∞∞
+=
0
s45
0
s43 dtttfpdtttgp
( ) ( ) =+=+= ∫∫∞∞
0
5454
0
5252545452525 dtttdQpdtttdQpmpmpm
( ) ( ) ( ) ( )∫∫∫∞∞∞
+=+=
0
s5452
0
s54
0
s52 dtttgppdtttgpdtttgp
Thus, the ergodic probability distribution for the semi-Markov process
consists of the following five elements:
M
miii
Π=ρ i=1,2,…,5 ∑
=
Π=
5
1i
iimM (20)
The steady-state availability of the multi-element symmetric pair is given
by the following formula:
∑ ∑= =
Π=ρ=
4
1i
4
1i
iii mM
1A (21)
(19)
Steady-state availability of a multi-element symmetric pair
25
Final remarks
The presented considerations and modelling approach allows the most
important parameter of the system reliability to be obtained, that is, the steady-
-state availability. At the same time, the problems and questions related to the
functioning and operation from a multi-element system reliability point of view
are raised in response to both, the first work of 2011, as well as in the present
work.
References
[1] Czaplicki J.M.: Mining equipment and systems. Theory and practice of exploitation and
reliability. CRC Press, Taylor & Francis Group. Balkema. 2010.
[2] Гнeдeнкo Б.В.: O дублировании с восставлением. АН СССР. Техническая кибернетика.
4, 1964.
[3] Гнeдeнкo Б.В.: Резервирование с восставлением и суммирование случайново
числа слагаемых. Colloquium on Reliability Theory. Supplement to preprint volume.
pp. 1–9, 1969.
[4] Гнeдeнкo Б.В., Бeляeв Ю.K., Coлoвьeв A.Д.: Математические методы в теории
надёжности. Изд. Наука, Mocквa, 1965.
[5] Kopociński B: An outline of renewal and reliability theory. PWN, Warsaw, Poland, 1973.
[6] Czaplicki J.M., Kulczycka A.M.: Semi-Markov process for a pair of elements. Lecture given
on XL Reliability Winter School, Szczyrk, Poland, 8–14 Jan., 2012 Scientific Problems
of Machines Operation and Maintenance, 2 (166), vol. 46, 2011/2012, pp. 7–16.
Graniczny współczynnik gotowości wieloelementowej pary symetrycznej
S t r e s z c z e n i e
Artykuł jest kontynuacją rozważań zawartych w pracy pod tytułem: „Proces semi-Markowa
dla pary elementów” opublikowanej w 2011 roku. W tym artykule są omawiane zagadnienia
dotyczące wieloelementowej pary symetrycznej, a nie tylko samej pary elementów. Zostały
zidentyfikowane i omówione problemy związane z funkcjonowaniem tego typu systemu.
Dokonano analizy systemu wieloelementowego wraz z propozycją metody modelowania jego
działania. Przedstawione metody uwzględniają różne przypadki w zależności od wyniku badań
empirycznych; od rodzaju rozkładu prawdopodobieństwa czasów stanów elementów, z których
składa się system. Zaprezentowana metoda modelowania dostarcza rozwiązań, które pozwalają na
uzyskanie podstawowych parametrów niezawodnościowych rozważanego systemu wieloelemento-
wego.
W. Grzegorzek, S. Ścieszka
26
Prediction on friction characteristics of mine hoist disc brakes using...
27
SCIENTIFIC PROBLEMS
OF MACHINES OPERATION
AND MAINTENANCE
3 (167) 2011
WOJCIECH GRZEGORZEK*, STANISŁAW ŚCIESZKA
*
Prediction on friction characteristics of mine hoist disc
brakes using artificial neural networks
K e y w o r d s
Mine hoist disc brakes, prediction on friction characteristics, coefficient of friction, neural
networks.
S ł o w a k l u c z o w e
Hamulce maszyn wyciągowych, prognozowanie charakterystyk ciernych, współczynnik tarcia,
sieci neuronowe.
S u m m a r y
Safety and reliability are the main requirements for brake devices in the mining winding
installations. Trouble-free performance under changing braking parameters is mandatory.
Therefore, selection of the right materials for the friction brake elements (pads and discs) is the
most challenging task for brake system designers. The coefficient of friction for the friction couple
should be relatively high (≈ 0.4); but, above all, it should be stable. In order to achieve the desired
brake friction couple performance, a new approach to the prediction of the tribological processes
versus friction materials formulation is needed. The paper shows that the application of the
artificial neural network (ANN) can be productive in modelling complex, multi-dimensional
functional relationships directly from experimental data. The ANN can learn to produce an
input/output relationship, and the model of friction brake behaviour can be established.
* Silesian University of Technology, Faculty of Mining and Geology, Institute of Mining
Mechanisation, Akademicka 2A Street, 44-100 Gliwice, Poland; [email protected],
W. Grzegorzek, S. Ścieszka
28
Introduction
Mine hoist brake systems have several distinctive design features and
specific operational requirements, which make them differ from an automotive
brakes and even other industrial brake systems. Winders installed in mines are
designed to raise and lower, in fully controlled manner, the mass in excess of 40
Mg in a mine shaft over one kilometre deep. Instead of the ϕ 300 mm size
typical for automotive disc brake, there might be, e.g., 16 brake callipers acting
on two approximately 6 metre diameter discs connected to the drum (Fig. 1)
able to stop the payload moving at a speed of up to 20 m/s (Table 1) [1, 2].
Fig. 1. Multi-rope friction sheaf hoist with hydraulic disc brake system, where: 1 – multi-rope
drum, 2 – journal bearing, 3 – brake disc connected to the drum, 4 – hydraulic brake calliper,
5 – callipers stand
Rys. 1. Wielolinowa maszyna wyciągowa z hydraulicznym hamulcem tarczowym,
gdzie: 1 – bęben wielolinowy, 2 – łożysko, 3 – tarcza hamulca, 4 – hydrauliczne szczękowe
zespoły robocze, 5 – stojak
Table 1. Mine host brake’s design and operational parameters [2]
Tabela 1. Parametry konstrukcyjne i użytkowe hamulców maszyn wyciągowych [2]
No Parameter Operating range Dimension
1
2
3
4
5
6
7
8
Initial sliding speed, v
Normal pressure, p
Maximal friction energy density, ρt
Brake disc’s surface temperature, T
Duration of braking, th
Friction surface, At
Radius of friction, Rt
Friction torque, Mt
10 – 20
0.9 – 1.5
55 – 240
60 – 380
6 – 17
0.12 – 0.48
2.20 – 3.11
0.9 – 2.8
m/s
MPa
kW/m2
˚C
s
m2
m
MNm
Prediction on friction characteristics of mine hoist disc brakes using...
29
To stop the dram (together with ropes, skips and payload), friction
materials in the form of brake pads mounted in the brake callipers (Fig. 1) are
forced hydraulically against both sides of the disc. Friction causes the discs and
attached moving parts of the winder to stop. The friction energy is converted
into heat during retardation, which, in consequence, means temperature
elevation on the friction surfaces of brake pads and discs. The extreme
tribological loading on the friction brake elements take place during emergency
braking, which can be initiated at full speed by the power lost, a control
malfunction, or faulty operation.
The emergency braking must be done with control giving a constant
predetermined retardation independent of the braking condition. The above
design features and operational requirements are particularly challenging for the
friction pads material, because it should maintain a stable coefficient of friction
within the range of predefined tribological conditions.
This paper describes the application of a neural network method for
modelling tribological processes in the winding-gear disc brakes and
subsequently might be used for the pad material optimisation.
Brake friction material consists of dozen or more different constituents,
combining organic, metallic, and ceramic phases. Their performance
characteristics include the coefficient of friction, resistance to wear, stiffness,
thermal conductivity, and environmental impact. The brake performance is
influenced by tribological conditions between a brake disc and brake pad,
characterised by sliding speed, pressure, and temperature distribution. Tribological
tests were carried out with particular emphasis on accurate measurement of the
friction and wear properties of the brake pair. The size of the brake disc (Fig. 1),
work safety, and cost considerations concerned with any winding gear operation
caused the testing to be carried out on a small scale tribotester. These tests were
set up to provide a comparison between various friction materials tested against
the same disc in both criteria, i. e. friction and wear [2].
In this paper, only the friction criterion (coefficient of friction) was taken
into consideration, because the stability of the emergency braking is the
operational priority in winding installations. The modelling and prediction of
the tribological processes within disc brake by the application of the artificial
neural network (ANN) method FFBP (Feed Forward Back Propagation) type
consisted of relating the friction process (coefficient of friction) versus friction
material formulation and testing conditions. The neural computation ability to
model complex non-linear, multi-dimensional functional relationship directly
from experimental data, without any prior assumption about input/output
relationship, has been used in this paper.
In the course of this work, it was found that the neural network method is a
powerful approach to the analysis of the experimental results and that the accuracy
of prediction of tribological processes obtained by the method was significantly
better than the results achieved by the multiple regression analysis [2].
W. Grzegorzek, S. Ścieszka
30
Experimental method and results
Brake friction materials tested
In these tribological experiments, specially prepared samples of the brake
pad materials were used. The materials are intended to work in dry conditions
most of the time, even though they may be unintentionally lubricated by the
rainwater or even oil. There are four main components of brake pad materials,
namely, the binder, reinforcing fibres, organic and inorganic fillers, metal
powders and composite premix master batch which complements mixture ratio
to 100% (Table 2).
Table 2. Range of the components volume fraction [2]
Tabela 2. Objętościowy udział składników [2]
No Component Volume fraction,
%
1
2
3
4
5
Binder (phenolic resin)
Reinforcing fibres
Fillers (inorganic and organic)
Additives (metals powders)
Composite premix master batch
4.00 – 6.50
16.30 – 17.80
38.35 – 41.88
26.85 – 29.32
7.00 – 12.00
The purpose of the binder is to maintain the brake pads structural integrity
under mechanical and thermal stresses. It has to hold the components of the
brake pad together [3, 4]. The choice of binders for brake pads is an important
issue, because if it does not remain structurally stable at high temperature other
components such as the fibres and powders will disintegrate. The purpose
of reinforcing fibres is to provide mechanical strength to the brake pad. Friction
materials typically use a mixture of different types of reinforcing fibres
(ceramic, aramid, metallic) with complementing properties. The fillers in
a brake pad are present for the purpose of improving its manufacturability.
Fillers play an important role in modifying certain characteristics of brake pads,
namely, noise suppression, heat stability or the friction coefficient stability.
Metal powders (brass, copper and steel) have very high heat conductivities;
therefore, they are able to remove heat from the friction surfaces very quickly.
Premix master batch is used as a complement of a composite to 100%. All
testing samples of friction material, no matter what composition (material
formulation), were produced by the same manufacturing procedure defined by
conditions of dry mixing and hot moulding (170°C and 90 MPa).
Prediction on friction characteristics of mine hoist disc brakes using...
31
Range of experimental testing
The proper friction brake design requires complete information about
tribological characteristics of the friction pair (pad material versus disc material)
in the full range of the operational parameters, namely, pressure, and sliding
speed and temperature on the friction surfaces.
Carrying out experimental tests on the industrial installations is not always
possible. In some cases this, inability arises from the fact that the installation
has not been built or come from the safety codes, or simply from cost
consideration. The investigations on friction characteristics of the brake
materials for mine hoist brake systems were made on the scaled-down inertia
dynamometer (Fig. 2) [2] due to the above reasons.
The experiments were conducted in accordance with the principles of the
tribological similarity developed by Sanders et. al. [5].
Fig. 2. Schematic diagram of scaled-down inertia dynamometer, where: 1 – electric motor,
2 – flexible coupling, 3 – flywheel, 4 – brake disc, 5 – specimens holder, 6 – torquemeter
Rys. 2. Schemat stanowiska bezwładnościowego do badań tarcia, gdzie: 1 – silnik elektryczny,
2 – sprzęgło, 3 – masa bezwładnościowa, 4 – tarcza hamulca, 5 – głowica do mocowania próbek,
6 – dźwignia pomiarowa momentu tarcia
The numerical values of input parameters for operational variables and
friction materials formulations were established making use of factorial and
simplex designs. Experimental sample formulations are shown in Table 3.
Friction tests were performed on the inertia disc brake dynamometer
(Fig. 2) and the following set of conditions were used during testing:
1. Running – in of samples friction surface:
– sliding speed v = 6.5 m/s
– pressure p = 1.2 MPa
– initial temperature T = 60°C
2. Pressure test:
– pressure p = 1.2; 2.4; 3.6; 4.8 MPa
W. Grzegorzek, S. Ścieszka
32
– sliding speed v = 9.0 m/s
– initial temperature T = 60°C
3. Sliding speed test:
– sliding speed v = 4; 6; 8; 10; 12 m/s
– pressure p = 3.0 MPa
4. Temperature test:
– initial temperature T = 60; 100; 140; 180; 240; 280; 320; 360°C
– sliding speed v = 9 m/s
– pressure p = 3.0 MPa
Complete set of arithmetic mean values of the coefficient of friction is
presented in Tables 4, 5 and 6.
Table 3. Samples composition
Tabela 3. Składy próbek
Binder,
Phenolic resin
Composite premix
masterbatch
Reinforcing
fibres Fillers
Additives,
Metal powders Ingredient
X1 X2 X3
No sample %
1 6.50 7.00 17.30 40.71 28.49
2 6.50 12.00 16.30 38.35 26.85
3 4.00 7.00 17.80 41.88 29.32
4 6.50 9.50 16.80 39.53 27.67
5 5.25 12.00 16.55 38.94 27.26
6 4.00 9.50 17.30 40.71 28.49
7 5.25 9.50 17.05 40.12 28.08
8 5.67 10.33 16.80 39.53 27.67
Friction process modelling in brakes by means of neural computation
Application of Feed Forward Back Propagation (FFBP) type of ANN method
In the first step on application of FFBP type ANN analysis friction model
was designed. In the model design, the experimental data presented in Tables 4
to 6 were used. The set of arithmetic mean values of the coefficient of friction
was obtained for 17 combinations of the test parameters (pressure, sliding speed
and temperature) and for 8 combinations of the friction materials compositions.
Friction tests were conducted three times for every pair of the combinations (test
parameters and materials compositions) leading to completion of the set of input
data for ANN which was composed of 468 vectors.
The set of input data was divided on three equal parts: the training, the
validation, and the testing data sets using suitable for the purpose software [2].
Before the neural computation, the data scaling was performed in order to reach
variability range of the characteristic suitable to the neuron activation function
Prediction on friction characteristics of mine hoist disc brakes using...
33
W. Grzegorzek, S. Ścieszka
34
Prediction on friction characteristics of mine hoist disc brakes using...
35
W. Grzegorzek, S. Ścieszka
36
(mini-max function). In the case of input data xin, the scaling range was from 0.1
to 1.0 (Equation 1) and for output data the scaling range was from 0.1 to 0.9
(Equation 2).
1.09.0x-x
x-xx
minmax
minin +⋅= (1)
1.08.0x-x
x-xx
minmax
minout +⋅= (2)
ANN design process includes value evaluation of coefficient of training η,
which is inherent to the sorted out problem. The design process also covers the
test stage recognition in which training can be considered as completed. The
completion of the ANN’s training can be determined by analysis of the root-
mean-square value of error, E and the maximum error Emax (Equations 3 and 4).
( )∑=
−=
p
j
)L(jj yz
1
2
p
1E (3)
( ) )(
max maxE Ljjyzabs −= (4)
where:
p – number of vectors,
L – output layer index, j
z – expected value in the network output,
z – mean value from jz ,
j(L)y – neuron answer in output layer.
In addition, the efficiency of the ANN training action can be represented by
the coefficient of determination, B (Equation 5).
( )
( )∑
∑
=
=
−
−
=p
j
)L(j
p
j
)L(jj
zy
yz
1
2
1
2
B (5)
The coefficient of determination B reflects the degree of accuracy in
representations between the investigated process and the model. The closer the
Prediction on friction characteristics of mine hoist disc brakes using...
37
coefficient of determination to unity (B→1), the better is the representation of
the expected values by the values generated by the model.
The architecture of the network FFBP type consists of three layers (Fig. 3),
namely, the input layer, the hidden layer, and the output layer.
Fig. 3. Structure of the ANN type FFBP used for friction modelling process
Rys. 3. Budowa sieci neuronowej typu FFBP wykorzystanej do modelowania procesów tarcia
During the training process, the values of the coefficient of friction were
observed from output as the result of input parameter insertion into the network
(Table 7).
Table 7. Input and output vectors in network FFBP type for friction model
Tabela 7. Wektory wejściowe i wyjście sieci typu FFBP dla modelu tarcia
Input vectors Output vector
Materials parameters Friction parameters Friction
coefficient
1 2 3 4 5 6 7
X1 X2 X3 v p T µ
Preliminary analysis shows that optimal value of the coefficient of training
is η = 0.06, which was achieved as a compromise between the accuracy and
swiftness of the training process.
Fixing a number of neurons in the hidden layer was undertaken in the next
step of building up the architecture of a network.
FFBP type ANNs are able to represent any functional relationship between
input and output, if there are enough neurons in the hidden layer [6]. However,
W. Grzegorzek, S. Ścieszka
38
too many neurons in the hidden layer may cause “over fitting”. The optimal is to
use a network that is just large enough to provide an adequate fit. In order to
reach the optimal solution, training was performed with arbitrarily selected
number of 20 neurons in the hidden layer, and subsequently the training
procedure was repeated several times with one neuron less in the hidden layer.
Results from these tests, including calculated values of the root-mean-square
error, E and the maximum error, Emax, are presented on Figure 4.
0.002
0.006
0.010
0.014
0.018
0.022
0.026
0.030
4567891011121314151617181920
training validation testing
Neurons of hidden layer
Erro
r E
a)
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
4567891011121314151617181920
training validation testing
Neurons of hidden layer
Erro
r E
max
b)
Fig. 4. Number of neurons in the hidden layer effect on error: a) E, b) Emax
Rys. 4. Wpływ kolejno usuwanych neuronów warstwy ukrytej na wartość błędu: a) E, b) Emax
An increase in both errors E and Emax was noticed below 12 neurons in
the hidden layer; therefore, the structure of network 6-12-1 was recommended
for modelling friction process in the disc brake.
Determination of the friction conditions and the composition of friction materials
effect on the coefficient of friction
An attempt was made to determine magnitude of the effect by friction
parameters (v, p, T) and materials parameters (X1, X2, X3) on the coefficient of
friction. The magnitude of effect determination is based on the selecting method
of the input feature for ANN [7]. The method is called “weight pruning”.
In the method assumption is that the significance of the inputs to the
network is equivalent to the magnitude of the effect made by the parameter on
the analysed process.
The results of the weight pruning process in the form of the significance
level for all input parameters are presented in Figure 5.
All input parameters to the network (Table 7) have a significance level
above 70%. The results indicate that friction parameters (v, p, T) and material
parameters (X1, X2, X3) taken into consideration in the experimental part have a
high and equivalent effect on coefficient of friction. The results confirm the
friction model correctness.
Prediction on friction characteristics of mine hoist disc brakes using...
39
76.92 82.05 78.21
89.10 83.97
80.13
0
10
20
30
40
50
60
70
80
90
100
X1 X2 X3 v p T
Sig
nif
ican
ce level,
%
Input parameter
Fig. 5. Significance level for input parameters to the network
Rys. 5. Istotność parametrów wejściowych sieci
In the next step of ANN modelling evaluation, the comparative analysis
was made between the quality of several models, namely, the model developed
by neural network with structure 6-12-1 (Figure 3) and models obtained by
multiply regression analysis (MRA) (Table 8). Errors E and Emax for neural
network friction model are 2 to 5 times lower than MRA models, which clearly
indicates the higher quality of ANN type FFBP modelling of friction process.
The same conclusion can be made analysing values of the coefficient of
determination B (Figure 6).
0.4
0.5
0.6
0.7
0.8
0.4 0.5 0.6 0.7 0.8
Co
mp
ute
d f
ricti
on
co
eff
icie
nt
Expected friction coefficient
Training results B = 0.985
a)
0.4
0.5
0.6
0.7
0.8
0.4 0.5 0.6 0.7 0.8
Co
mp
ute
d f
ric
tio
n c
oe
ffic
ien
t
Expected friction coefficient
Testing results B = 0.949
b)
Fig. 6. Comparison between expected and computed results, for:
a) training data set, b) testing data set
Rys. 6. Porównanie wyników oczekiwanych i obliczonych dla: a) zbioru trenującego,
b) zbioru testującego
W. Grzegorzek, S. Ścieszka
40
Table 8. Results of the comparison between several friction models
Tabela 8. Wyniki dla modeli procesu tarcia
Training results Testing results No Model name
E Emax B E Emax B
1 2 3 4 5 6 7
1 MRA-Linear 0.024 0.097 0.799 0.025 0.116 0.781
2 MRA-Logarithmic 0.024 0.104 0.785 0.026 0.123 0.759
3 MRA-Polynomial 0.021 0.096 0.845 0.023 0.094 0.813
4 MRA-Piecewise 0.020 0.058 0.860 0.021 0.070 0.848
5 Neural Network 0.007 0.020 0.985 0.013 0.034 0.949
High conformity between values of the coefficient of friction obtained
(Figures 7 to 9) from the model and expected values confirm the purposefulness
of the ANN type FFBP application for modelling friction and other tribological
processes in friction brakes.
v = 9 m/s; p = 3MPa
Fig. 7. Values of coefficient of friction as a function of temperature, obtained from:
a) experimental data, b) network responses
Rys. 7. Wartości współczynnika tarcia w funkcji temperatury, uzyskane z:
a) wyników badań, b) odpowiedzi sieci
Prediction on friction characteristics of mine hoist disc brakes using...
41
T = 60˚C; p = 3MPa
Fig. 8. Values of coefficient of friction as a function of sliding velocity, obtained from:
a) experimental data, b) network responses
Rys. 8. Wartości współczynnika tarcia w funkcji prędkości poślizgu, uzyskane z:
a) wyników badań, b) odpowiedzi sieci
T = 60˚C; v = 9m/s
Fig. 9. Values of coefficient of friction as a function of pressure, obtained from: a) experimental
data, b) network responses
Rys. 9. Wartości współczynnika tarcia w funkcji nacisku, uzyskane z: a) wyników badań,
b) odpowiedzi sieci
Concluding remarks
In the course of the work, it was found that the neural network method is
a powerful approach to the analysis of the experimental results and that the
accuracy of the prediction of friction processes in disc brakes obtained by the
ANN type FFBP method were significantly better than the results achieved by
the multiply regression analysis, (MRA).
W. Grzegorzek, S. Ścieszka
42
References
[1] Ścieszka S.F.: Friction Brakes. Gliwice – Radom: ITeE; 1998.
[2] Grzegorzek W.: Modelling tribological processes in winding gear disc brakes by mean
of a neural network method. PhD thesis (in polish). Gliwice: SUT; 2003.
[3] Chan D., Stachowiak G.W.: Review of automotive brake friction materials. J. Automobile
Engineering. 2004; 218: 953–966.
[4] Blau P.: Compositions, functions and testing of friction brake materials and their additives.
Technical Report Oak Ridge National Laboratory/TM – 2001.
[5] Sanders P.G., Dalka T.M., Bash R.H.: A reduced – scale brake dynamometer for friction
characterization. Tribology International. 2001; 34: 609–615.
[6] Aleksendric D., Duboka C.: Automotive friction material development by means of neural
computation, Conference Proceedings “Braking 2006”, York: 2006; 167–176.
[7] Sokołowski A.: Neural network application for tool point condition monitoring. PhD thesis
(in polish). Gliwice: SUT; 1994.
Prognozowanie charakterystyk ciernych hamulców maszyn wyciągowych
z zastosowaniem sztucznych sieci neuronowych
S t r e s z c z e n i e
Bezpieczeństwo i niezawodność działania to główne wymagania stawiane hamulcom maszyn
wyciągowych. Niezawodna, bezproblemowa praca hamulców w zmieniających się warunkach
otoczenia i obciążenia jest wymagana i egzekwowana przez dozór górniczy. Dlatego wybór
materiałów na elementy pary hamulcowej (okładzina cierna, tarcza hamulca) jest dużym
wyzwaniem dla konstruktorów. Współczynnik tarcia dla tej pary ciernej powinien być względnie
wysoki (około 0,4), ale przede wszystkim wymaga się, aby był stabilny. Dla osiągnięcia
pożądanego efektu pracy hamulca zastosowano nowe narzędzie dla predykcji i kontroli procesów
tribologicznych w funkcji parametrów tarcia i składu chemicznego materiału okładziny
hamulcowej. Zastosowanie sztucznych sieci neuronowych jest przydatne w modelowaniu
złożonych, wieloczynnikowych zależności w oparciu o dane pochodzące z eksperymentów
laboratoryjnych. Sztuczne sieci neuronowe mogą być wytrenowane do wytworzenia relacji
wejście/wyjście i do modelowania oraz przewidywania charakterystyk użytkowych w hamulcach
ciernych.
The construction of high-order b-spline wavelets and their decomposition...
43
SCIENTIFIC PROBLEMS
OF MACHINES OPERATION
AND MAINTENANCE
3 (167) 2011
ANDRZEJ KATUNIN*
The construction of high-order b-spline wavelets
and their decomposition relations for fault detection
and localisation in composite beams
K e y - w o r d s
B-spline wavelets, composite beams, faults detection.
S ł o w a k l u c z o w e
Falki B-splajnowe, belki kompozytowe, detekcja uszkodzeń.
S u m m a r y
B-spline scaling functions and wavelets have found wide applicability in many scientific and
practical problems thanks to their unique properties. They show considerably better results in
comparison to other wavelets, and they are used as well in mathematical approximations, signal
processing, image compression, etc. But only the first four wavelets from this family were
mathematically formulated. In this work, the author formulates the quartic, quintic and sextic
B-spline wavelets and their decomposition relations in explicit form. This allows for the
improvement of the sensitivity of fault detection and localisation in composite beams using
discrete wavelet transform with decomposition.
* Department of Fundamentals of Machinery Design, Faculty of Mechanical Engineering, Silesian
University of Technology, Konarskiego 18A, 44-100 Gliwice, Poland, e-mail: [email protected].
A. Katunin
44
Introduction
There is a great interest in the investigation of compactly supported
wavelets. This interest is due to the computational capabilities of such wavelets
and the wide range of their applications. Forerunners in the development of
compactly supported wavelets are Daubechies [1, 2], Cohen, and Feauveau [3].
Spline and B-spline wavelets were also introduced in [4, 5]. The compactly
supported orthonormal B-spline wavelets have been found to be a powerful tool
in many scientific and practical applications, including mathematical
approximations, the finite element method, image processing, and compression
and computer-aided geometric design. Thanks to some of their exceptional
properties and mathematical simplicity, they are also applied and give very good
results in various areas of applied sciences in comparison to other known
wavelets.
The last decade demonstrates an augmentation of interest of B-spline
wavelets. The scientific group of Lakestani presents several works on solving
integral and integro-differential equations using linear B-spline [6], quadratic
B-spline [7] and cubic B-spline scaling functions [8]. This wavelet family also
found and application in image compression, and the standard of the image
compression – JPEG2000 – is based on the B-spline wavelet transform and
B-spline factorisation [9]. In problems of the signal processing, B-spline
wavelets were used for the development digital filters [10], which show
excellent results. Analysing the above-cited works, one can notice a tendency
towards the reduction of errors when increasing the order of the B-spline
wavelet. In [11] the author presented the possibility of the application B-spline
wavelets for diagnostic signal processing. The effectiveness of these wavelets
in comparison to other chosen wavelets and the above-mentioned tendency was
presented. Therefore, it is necessary to investigate the effectiveness of higher-
-order B-spline wavelets to improve results, which can be applied in various
scientific and technical problems.
There are many methods and techniques for fault detection in problems of
technical diagnostics. A large group of these methods are based on signal
processing using several transforms (e.g. DFT, STFT) and other techniques
(e.g. cepstrum analysis, signal demodulation, etc.), but not all of these
techniques can be used for problems of fault detection in the early damage phase
[12]. The classic modal analysis may be used only for damage detection, but
conclusions about presence of the fault can be made based on the shift of
frequency spectrum only, which is a very poor feature in the light of the damage
identification problem. Signal processing based on Wavelet Transform (WT)
makes it possible to analyse modal shapes in the space domain. Such an analysis
is very sensitive to the singularities in the modal shapes, which makes possible
the accurate location of a fault, even when the fault is very minimal. It is
possible due to the wavelet decomposition algorithm, while one cannot obtain
The construction of high-order b-spline wavelets and their decomposition...
45
such information using modal analysis. WT found an application in fault
detection in mechanical systems like gearboxes, rolling bearings, rotors, etc., but
it can also be applied to structural health monitoring [13, 14]. In [12], the
authors show that wavelet analysis makes it possible to detect the type of
damage using Continuous WT based on scalogram evaluation. For lightly
damaged structures, the authors proposed a method based on Discrete WT,
which allows the use of decomposition analysis. In the above-cited work,
Daubechies (db8) and Morlet wavelets were used for the approximation.
The problem of fault detection and localisation in beams has been studied
in several works. Many of them have been based on simulation results or
theoretical models, e.g. [15]. However, fault detection and localisation in
experimental research is a more difficult problem, because of the limitation of
the number of measuring points and the presence of noise. The authors of [16]
presented both the model-based and the experiment-based approach and
confirmed the difficulty of fault detection and localisation in real tasks. The
authors used the ‘symlet4’ wavelet for the analysis.
Based on the obtained results in [11] of the comparison of DSD parameters
of different wavelets, the author decided to construct higher-order (quartic,
quintic and sextic) B-spline wavelets and scaling functions and their
decomposition relations. Due to this, using Discrete B-spline WT for fault
detection and localisation in composite beams is possible. Pre-notched
composite specimens were excited by a random noise signal and displacement
was measured using a laser scanning vibrometer. For the signal processing,
Discrete B-spline WT was used and fault detection and localisation was
evaluated based on the analysis of detailed coefficients by means of the
decomposition of the signal. The efficiency of the approximation using B-spline
wavelets was compared with other families of orthogonal wavelets. The
obtained results indicate the effectiveness of high-order B-spline in fault
detection and localisation. Several examples are presented.
Construction of B-spline wavelets
General order B-spline wavelets
The B-spline wavelet can be defined recursively by the convolution [17]:
( ) ( )∫∞
∞−
−= dttx mm 11ϕϕϕ (1)
where
( )
<≤
=
otherwise 1
10for 01
xxϕ (2)
A. Katunin
46
The construction of the scaling function of m-th order B-spline wavelet is
based on the two-scale relation:
( ) ( )∑=
−=
m
k
mkm kxpx
0
2ϕϕ (3)
where kp is the two-scale sequence and can be expressed as a combination:
,21
=
−
k
mp
m
k for mk <≤0 (4)
The two-scale relation for m-th order B-spline wavelets is given by:
( ) ( )∑−
=
−=
23
0
2
m
k
mkm kxqx ϕψ (5)
where
( )∑=
−
+−
−=
m
l
mmk
k lkl
m)(q
0
21 121 ϕ (6)
The decomposition relation for m-th order B-spline wavelet is given by:
( )( )
( ),2
2
2∑
−
+−
=−
−
−
k kl
kl
mkxb
kxalx
ψ
ϕ
ϕ Zl ∈ (7)
where decomposition sequences ka and kb are as follows:
∑ +−+−
+
−=
l
m,llmk
k
k cq)(
a 2122
1
2
1 (8)
∑ +−+−
+
−−=
l
m,llmk
k
k cp)(
b 2122
1
2
1 (9)
In (8) and (9) the coefficients sequence mkc , is presented by m-th order
Fundamental Cardinal Spline (FCS) function [18]:
( ) ∑∞
−∞=
−+=
k
mm,km kxm
cxL2
ϕ (10)
The construction of high-order b-spline wavelets and their decomposition...
47
To obtain the coefficient sequences, the authors of [17] used an analytical
relation for B-spline wavelets with order m < 3. For higher values of m,
obtaining the analytical solutions became very difficult, and for values of m
greater than 5, it is impossible in the light of Abel-Ruffini theorem. Therefore,
the analytical formula was omitted here. Another way of obtaining the
coefficient sequences is to form the bi-infinite system of equations [18] as
follows:
,2
0,,∑∞
−∞=
=
−+
k
jmmk kjm
c δϕ Zj ∈ (11)
The explicit form of (11) for m = 2 can be written as (cf. [17]):
=
−
−
M
M
M
M
O
O
0
1
0
0
321
321
0
2
1
0
1
2
444
444
c
c
c
c
c
)()()(
)()()(
ϕϕϕ
ϕϕϕ (12)
The coefficients sequence ck,m is infinite for ≥m 3, so that (10) does not
vanish identically outside any compact set. However, these coefficients decay to
zero exponentially fast as ,k ∞→ which implies decaying to zero of (10) as
.x ±∞→
Quartic B-spline wavelet (m = 5)
Quartic B-spline φ5(x) scaling function is given by the next recurrence
relation:
+−+−
−+−+−
+−+−
−+−+−
=
24
625
6
125
4
25
6
5
24
24
655
2
65
4
55
2
5
6
24
155
2
25
4
35
2
5
4
24
5
6
5
4
5
6
5
6
24
234
234
234
234
4
5
xxxx
xxxx
xxxx
xxxx
x
)x(ϕ (13)
A. Katunin
48
where the support changes in the range [0,m] with step 1 referring to the
property of B-spline scaling functions. Two-scale sequences 5p and 5q are
presented in (14) and (15). Based on them, two-scale relations for φ5(x) and
ψ5(x) can be constructed using (3) and (5), respectively.
Fig. 1. Scaling and basic functions of quartic B-spline wavelet
Rys. 1. Funkcja skalująca i bazowa falki B-splajnowej rzędu 5
Decomposition sequences were calculated using (8), (9) and (12). For
quartic B-spline wavelet some of them are presented in Tab. 1.
=
16
1
16
5
8
5
8
5
16
5
16
15 ;;;;;p (14)
−−
−
−−
−
=
5806080
1
1935360
169
725760
2141
181440
5197
1161216
149693
165888
54289
145152
74339
145152
74339
165888
54289
1161216
149693
181440
5197
725760
2141
1935360
169
5806080
1
5
;
;;;
;;;
;;;
;;;
q (15)
The construction of high-order b-spline wavelets and their decomposition...
49
Table 1. Quartic B-spline decomposition sequences
Tabela 1. Współczynniki dekompozycji dla falki B-splajnowej rzędu 5
k 2−ka 2−kb
0 0.27944 0.26081
1 -0.03765 -0.1238
2 -0.48157 -0.44529
3 -0.05206 0.08078
4 0.87419 0.73019
5 -0.20474
6 -1.15096
7 0.54171
8 1.59650
M
M
M
Quintic B-spline wavelet (m = 6)
Let us go to the next example φ6(x) given by
+−+−+−
−+−+−
+−+−+−
−+−+−
+−+−+−
=
5
32454183
4120
20
1289
4
409
2
89
2
19
24
4
231
4
231
2
71
2
21
2
3
12
20
79
4
39
2
19
2
9
12
20
1
422424
120
2345
234
5
2345
234
5
2345
5
6
xxxxx
xxxx
x
xxxxx
xxxx
x
xxxxx
x
)x(ϕ (16)
and shown in Fig. 2. Two-scale sequences 6p and 6q are given by (17) and
(18), respectively. Because of the symmetry of 6q , only the half sequence was
presented.
=
32
1
16
3
32
15
8
5
32
15
16
3
32
16 ;;;;;;p (17)
A. Katunin
50
−
−
−
−
=
...
q
;42577920
21112517
;25546752
10504567;
319334400
74131711
;70963200
6127141;
319334400
6322333
;127733760
314487;
9676800
1249
;638668800
1021;
1277337600
1
6 (18)
Decomposition sequences for quintic B-spline are given in Tab. 2.
Fig. 2. Scaling and basic functions of quintic B-spline wavelet
Rys. 2. Funkcja skalująca i bazowa falki B-splajnowej rzędu 6
Table 2. Quintic B-spline decomposition sequences
Tabela 2. Współczynniki dekompozycji dla falki B-splajnowej rzędu 6
k 2−ka 2−kb
0 0.29214 -0.24695
1 0.29398 -0.35522
2 -0.41198 0.46725
3 -0.55802 0.48370
4 0.48972 -0.77160
5 1.24143 -0.65901
6 1.22365
7 0.82492
8 -1.91952
9 -0.69623
10 2.84275
M
M
M
The construction of high-order b-spline wavelets and their decomposition...
51
Sextic B-spline wavelet
The last presented wavelet scaling function φ7(x) is given by the following:
+−+−+−
−−−+−+−
−−+−+−
++−+−+−
−++−+−
−+−+−+−
=
720
117649
120
16807
48
2401
36
343
48
49
720
7
720
720
208943
24
7525
48
6671
36
1169
48
203
24
7
120
360
59591
3
700
24
3227
9
364
24
161
12
7
48
360
12089
3
196
24
1253
9
196
24
119
12
7
36
720
1337
24
133
48
329
36
161
48
77
24
7
48
120
7
120
7
48
7
36
7
48
7
120
7
120
720
23456
23456
23456
23456
23456
23456
6
7
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
xxxxxx
x
)x(ϕ (19)
and two-scale sequences are given by (20) and (21). Because of the anti-
symmetry of 7q , only the half sequence was presented. Graphical
interpretation of φ7(x) and ψ7(x) is presented in Fig. 3. Decomposition sequences
are tabulated in Tab. 3.
Fig. 3. Scaling and basic functions of sextic B-spline wavelet
Rys. 3. Funkcja skalująca i bazowa falki B-splajnowej rzędu 7
=
64
1
64
7
64
21
64
35
64
35
64
21
64
7
64
17 ;;;;;;;p (20)
A. Katunin
52
−
−
−
−
−
=
...
q
;02846638080
11258310392;
1897758720
605806759
;01423319040
2338911899;
569327616
33355811
;03321077760
456405947;
09963233280
193344049
;07970586624
11212661;
04428103680
170777
;07970586624
1637;
003985293312
1
7 (21)
Table 3. Sextic B-spline decomposition sequences
Tabela 3. Współczynniki dekompozycji dla falki B-splajnowej rzędu 7
k 2−ka 2−kb
0 0.05808 -0.09463
1 0.57369 0.73914
2 -0.00377 -0.02005
3 -0.85444 -1.08620
4 -0.23318 0.16139
5 1.22519 1.54667
6 -0.37510
7 -2.14425
8 0.77637
9 2.82425
10 -1.62267
11 -3.18547
M
M
M
Notice that, in Tables 1 through 3, decomposition sequences were limited
to unique values. In the case of ka , there is symmetry of the sequence;
therefore, only half of it is presented. In the case of kb , the sequence is
symmetrical for even m and anti-symmetrical for odd m.
Comparative analysis of approximation effectiveness of some (semi)-orthogonal
wavelet families
The evaluation of the approximation effectiveness can be executed using
the degree of scalogram density (DSD) parameter. In technical diagnostics, DSD
was used by A. Timofiejczuk [19]. DSD is a statistical scalar parameter, which
The construction of high-order b-spline wavelets and their decomposition...
53
is based on the normalisation of the set of wavelet coefficients from the
scalogram, their filtering for some non-zero threshold and determination using
the following dependence:
L
NDSD −= 1 (22)
where N denotes the number of wavelet coefficients greater than threshold value
and L is the number of all wavelet coefficients.
In this section, we will compare DSD for various wavelet families for three
types of signals, which most frequently occur in diagnostic signal processing: the
harmonic one, the harmonic with variable frequency (chirp), and the triangular
pulse. One may consider only orthogonal or semi-orthogonal (e.g. B-spline)
wavelets, because Discrete WT is possible only using such wavelets.
Wavelets and their decomposition relations from Section 2 were
implemented into MATLAB®. Then, the DSD test was performed with
a threshold value of 0.01, a scale parameter of 1–256 and time of 2 s with
sampling rate 0.0001 s. The obtained results are presented in Figs. 4 through 6
for investigated types of signals. Note, that first-order Daubechies and B-spline
wavelets are identical to the Haar wavelet.
Fig. 4. DSD parameter for the harmonic component
Rys. 4. Stopień zagęszczenia skalogramu dla składowej harmonicznej
As it can be observed, the B-spline wavelet gives the best DSD parameter
for each considered type of signal. In cases of harmonic components, DSD
reveals asymptotic convergence to unity with the increase in the order of the
wavelet. For harmonic components, the growth of DSD is stabilised after the
fifth order; therefore, the construction of B-spline wavelets with an order higher
than seventh is not profitable. Analysing DSD values in Fig. 6, one can conclude
A. Katunin
54
that DSD has a decreasing tendency with the increase in the order of the
wavelet. However, values of DSD are very good for all considered orders of B-
spline wavelets and the changes of DSD are minor, i.e. B-spline wavelets can be
used in diagnostic signal processing for pulse components as well.
Fig. 5. DSD parameter for the harmonic component with variable frequency
Rys. 5. Stopień zagęszczenia skalogramu dla składowej harmonicznej ze zmienną częstotliwością
Fig. 6. DSD parameter for the pulse component
Rys. 6. Stopień zagęszczenia skalogramu dla składowej impulsowej
Fault detection in composite beams
Specimens preparation and experimental setup
The specimens were manufactured from 24-layered glass fiber-reinforced
epoxy laminate in the form of unidirectional impregnated fibers. The
configuration of the specimens was selected in order to achieve transversal
The construction of high-order b-spline wavelets and their decomposition...
55
isotropic properties. The structural formula and material properties of the
specimens can be found in [20]. The dimensions of the specimens were defined
as follows: length L = 250 mm, width W = 25 mm and thickness H = 5.28 mm.
Three specimens (one sound and two pre-notched) were considered. Notches,
whose depth h is 1 mm, were located at the distance l of 0.28L and 0.6L,
respectively. Fig. 7 shows the scheme of the investigated specimens.
Fig. 7. Dimensions of the specimens
Rys. 7. Wymiary próbek
The specimens were clamped on one side at length of 0.08L. For the
excitation, the random noise signal was generated and amplified by a power
amplifier and exerted to the beam through the TIRA TV-51120 modal shaker.
For measurements, Laser Doppler Vibrometers (LDV) were used, which
provided highly precise values. The scanning LDV (Polytec PSV-400) was used
for sensing the response signal of the beam, and a second LDV (Polytec PDV-
100) was used for achieving the reference signal. Measurements were provided
in the bandwidth of 1 to 3200 Hz with a sample frequency of 8192 Hz. On the
effective measurement length Leff of 215 mm (from 0.1L to 0.96L), a line with
44 measurement points was defined. The interval between points was 5 mm.
The experimental setup is presented in Fig. 8.
Fig. 8. Experimental setup
Rys. 8. Stanowisko badawcze
A. Katunin
56
Then, the testing of above-mentioned specimens was carried out.
Frequency response functions obtained during the modal analysis were stored
and, based on them, the natural frequencies and modal shapes were determined.
The displacement data for selected modal shapes of resonant vibrations were
acquired and exported to MATLAB®.
Analysis and experimental results
In obtained frequency spectra of the first four natural modes of vibration
were selected and considered in the analysis. Then, the discrete wavelet
transform with high-order B-spline wavelets was performed. Preliminary
analysis indicates that symmetric B-spline wavelets give better results in the
decomposition process; therefore, the quintic B-spline wavelet was used in the
next analysis. After decomposition, detail coefficients of signals for each case
were investigated. Additionally, the soft threshold filtering was conducted for
de-noising detail coefficients. Exemplary detail coefficients before and after
de-noising are shown in Fig. 9. After these operations, zero-value detail
coefficients for healthy specimen were obtained. Therefore, the graphical
presentation of detail coefficients was omitted for this specimen. The
approximation and detail coefficients of pre-notched specimens are depicted in
Fig. 10. The first column contains approximations (Y-axis – approximation
coefficient). The second column contains detail coefficients for the specimen
with the notch at 0.6L (Y-axis – details coefficient). The third column contains
detail coefficients for the specimen with the notch at 0.28L (Y-axis – details
coefficient). For all, the X-axis is the distance, L [mm].
Fig. 9. De-noising of the detail for the specimen with the notch at 0.6L for the 1st mode shape
Rys. 9. Odszumienie współczynników detalu dla próbki z pęknięciem w 0,6L dla pierwszej postaci
własnej
As seen in Fig. 9, the de-noising operation allows one to remove the
measurement noise and to present changes in the detail coefficients. However,
because of the high-order of the applied wavelet and the consequently larger
number of vanishing moments of this wavelet, fault localisation became more
The construction of high-order b-spline wavelets and their decomposition...
57
difficult and detail coefficients could not be used to visualise the exact location
of the fault (see Figs. 10e–g and Figs. 10k–l). On the other hand, the small
number of vanishing moments of the wavelet could influence the accuracy
of the decomposition process. By analysing details from the decomposition by
means of the geometry of the wavelet, one can notice that the fault localisation
can be provided by the evaluation of the highest value of the de-noised detail
coefficients.
Fig. 10. Decomposition of measured signals for four mode shapes of beams
Rys. 10. Dekompozycja sygnałów pomiarowych dla czterech postaci własnych belek
Discussion
In the present work, high-order B-spline wavelets were proposed. The
analytical formulation of quartic, quintic, and sextic B-spline wavelets and their
decomposition relations were presented. The comparative analysis of wavelets,
which could be used for Discrete WT, indicates the highest effectiveness
of B-spline wavelets, especially for higher orders. The construction of wavelets
with an order higher than 7 is not well grounded. The analytical formulation of
these wavelets and their decomposition relations could be difficult, but the
practical application of them will also be limited because of the increasing the
number of vanishing moments and the effective support.
A. Katunin
58
One of presented wavelets, the quintic B-spline wavelet, was applied for
fault detection and localisation in composite beams. The selection of the
appropriate wavelet to such analysis is crucial. However, in analysis, one can
noticed the effectiveness of the above-mentioned wavelet. As previously shown,
fault localisation using a decomposition procedure with B-spline wavelets is
possible after detail coefficient de-noising and gives precise results.
The accuracy of the damage localisation is directly dependent on the number
of measurement points. With a higher number of measurement points,
the displacements of a given modal shape can be determined more accurately.
An application of high-order B-spline wavelets is not only limited to
the problem presented above. They can also be used for numerical solving
of differential equations, where the wavelet scaling function is a differential
operator. Moreover, they could find an application in structural health
monitoring of complex problems, pattern recognition problems, signal
processing in biomedical applications, etc.
In further works, the use of the presented wavelets for detection and
localisation of faults in multi-damaged structures will be investigated. An
additional task will be the evaluation of structural life assessment based on
detail coefficients.
References
[1] Daubechies I.: Orthonormal bases of compactly supported wavelets, Communications on
Pure and Applied Mathematics 41, 1988, pp. 909–996.
[2] Daubechies I.: Ten lectures on wavelets, Society of Industrial and Applied Mechanics
(SIAM), Philadelphia, PA, 1992.
[3] Cohen A., Daubechies I., Feauveau J.-C.: Biorthogonal bases of compactly supported
wavelets, Communications on Pure and Applied Mathematics 45, 1992, pp. 485–560.
[4] Chui C.K.: An introduction to wavelets, Academic Press, 1992.
[5] Chui C.K., Wang J.: A general framework of compactly supported splines and wavelets,
Journal of Approximation Theory 71, 1992, pp. 54–68.
[6] Lakestani M., Razzaghi M., Dehghan M.: Solution of nonlinear Fredholm-Hammerstein
integral equations by using semiorthogonal spline wavelets, Mathematical Problems
in Engineering 1, 2005, pp. 113–121.
[7] Malenejad K., Aghazadeh N.: Solving nonlinear Hammerstein integral equations by using
B-spline scaling functions, Proc. of the World Congress of Engineering, London, 2009.
[8] Dehghan M., Lakestani M.: Numerical solution of Ricatti equation using the cubic B-spline
scaling functions and Chebyshev cardinal functions, Computer Physics Communications
181(5), 2010, pp. 957–966.
[9] Taubman D.S., Marcellin M.W.: JPEG2000: image compression fundamentals, standards
and practice, Kluwer Academic Publishers, 2002.
[10] Samadi S., Achmad M.O., Swamy M.N.S.: Characterization of B-spline digital filters, IEEE
Transactions on Circuits and Systems 51(4), 2004, pp. 808–816.
[11] Katunin A., Korczak A.: The possibility of application of B-spline family wavelets
in diagnostic signal processing, Acta Mechanica et Automatica 3(4), 2009, pp. 43–48.
[12] Katunin A., Moczulski W.: Faults detection in composite layered structures using wavelet
transform, Diagnostyka 1(53), 2010, pp. 27–32.
The construction of high-order b-spline wavelets and their decomposition...
59
[13] Hou Z., Noori M., Amand R.: Wavelet-based approach for structural damage detection,
J. Eng. Mech., 126(7), 2000, pp. 677–683.
[14] Moyo P., Brownjohn J.M.W.: Detection of anomalous structural behaviour using wavelet
analysis, Mechanical Systems and Signal Processing 16(2-3), 2002, pp. 429–445.
[15] Douka E., Loutridis S., Trochidis A.: Crack identification in plates using wavelet analysis,
Journal of Sound and Vibration 270, 2004, pp. 279–295.
[16] Zhong S., Oyadiji S.O.: Crack detection in simply supported beams using stationary wavelet
transform of modal data, Structural Control and Health Monitoring 18(2), 2010,
pp. 169–190.
[17] Ueda M., Lodha S.: Wavelets: an elementary introduction and examples, University
of California, Santa Cruz, 1995.
[18] Chui C.K.: On cardinal spline wavelets, Wavelets and their applications, Jones and Bartlett,
Boston, 1992, pp. 419–438.
[19] Timofiejczuk A.: Methods of analysis of non-stationary signals, Silesian University
of Technology Publishing House, Gliwice, 2004 [in Polish].
[20] Katunin A., Moczulski W.: The conception of a methodology of degradation degree
evaluation in laminates, Eksploatacja i Niezawodnosc – Maintenance and Reliability 41,
2009, pp. 33–38.
Konstrukcja falek b-splinowych wyższych rzędów i ich zależności dekompozycji
dla detekcji i lokalizacji uszkodzeń w belkach kompozytowych
S t e s z c z e n i e
B-splajnowe funkcje skalujące i falki znajdują szerokie zastosowanie w wielu zagadnieniach
naukowych i praktycznych dzięki ich wyjątkowym właściwościom. Pokazują one znacznie lepsze
wyniki w porównaniu z innymi falkami i są z powodzeniem stosowane w matematycznych
aproksymacjach, przetwarzaniu sygnałów, kompresji obrazów itd. Ale tylko pierwsze cztery falki
z tej rodziny zostały sformułowane matematycznie. W niniejszej pracy autor sformułował falki
B-splajnowe wyższych rzędów i ich zależności dekompozycji w postaci jawnej. Pozwalają one na
zwiększenie dokładności przy detekcji i lokalizacji uszkodzeń w belkach kompozytowych
z zastosowaniem dyskretnej transformacji falkowej z dekompozycją.
A. Sowa
60
Problems of computer-aided technical state evaluation of rail-vehicle wheel sets
61
SCIENTIFIC PROBLEMS
OF MACHINES OPERATION
AND MAINTENANCE
3 (167) 2011
ANDRZEJ SOWA∗
Problems of computer-aided technical state evaluation
of rail-vehicle wheel sets
K e y w o r d s
Technical diagnosis, technical state evaluation, wheel sets, rail-vehicles.
S ł o w a k l u c z o w e
Diagnostyka techniczna, ocena stanu technicznego, zestawy kołowe, pojazdy szynowe.
S u m m a r y
This article presents issues related to the construction of a computer-aided system
evaluation of the technical condition of rail-vehicle wheel sets. Physical features which
make a diagnostic feature vector are discussed and classified in the paper. This vector
may be used for the identification of a vehicle condition state. Examples of formulas for
distinguishing the classes of the technical states of rail-vehicles based on the evaluation
of the thickness of a rim and the thickness and height of a wheel flange have been
specified. The article also presents the guidelines for building a database for the
computer-aided evaluation system of wheel sets.
∗ Institute of Rail Vehicles, Cracow University of Technology, Al. Jana Pawła II nr 37, 31-864
Cracow, Poland, email: [email protected].
A. Sowa
62
Introduction
Wheel sets constitute a set of rail vehicles. Their technical state change
during operation due to the wear processes and damage (examples shown in
[4]), and it determines whether the vehicle is certified fit for use. It requires
carrying out measurements of certain features in the units of the technical
support and comparing their values with the boundary values in order to identify
the current technical state of the wheel sets. It enables one to make a decision on
certifying the vehicle fit for use or directing it to the servicing system.
Measurements and evaluation results are recorded in a traditional “paper”
form. The introduction of the computer-aided evaluation of the wheel set
technical state is a much better solution. This system enables the registration of
the research results, and it automatically generates the decision. It also actively
controls the registration process through the analysis of the archive data
conducted in a real time. The issues concerning the development of such
a system is the subject of this paper.
Diagnostic features of rail vehicle wheel sets and their classification
Wheel sets can be described by a set of a whole range of physical features,
i.e. the features of the internal structure [2]. The technical state of the wheel set
can be defined as a characteristic determined by the vector of these features in
the following form [5]:
( ) ( )[ ] ( )[ ] ( )[ ][ ]tfx,....,tfx,tfxt,a n,an,a,a 2211=X (1)
where:
( )at,X – the vector of the technical state after the period of operation t in
the conditions a,
nxx ,...,1 – the values of the internal structure features,
ana ff ,,1 ,..., – the functions describing the changes in the values of these
features in operation,
n – the number of the components of the technical state vector.
These characteristics are the operation safety features and their set contains
29 items [5]. They can be divided into groups of measurable features and
non-measurable features. The measurable features are divided into the primary
ones (individual and collective) and the secondary ones (internal and external).
The majority of measurable features are the primary features which can be
measured in a direct way (for instance: wheel flange thickness, wheel diameter)
Problems of computer-aided technical state evaluation of rail-vehicle wheel sets
63
or an indirect way (tyre thickness) with the use of proper physical value
converters. The measurable features of an individual type are evaluated on the
basis of one measurement result (an unbalance moment or the resistance
of a wheel set). For the collective features, the checked value is an average of
some results (e.g. the thickness of a tyre measured in three planes every 120°).
The secondary features apply to the additional bonds among the primary
features, and they have the appropriate boundary values of these bonds. The
internal secondary features apply to one wheel set under evaluation, and the
external secondary features apply to the bonds among the primary features of
the wheel set of a given bogie or rail vehicle.
The non-measurable features also undergo evaluation, and this evaluation is
essential while deciding whether the whole rail vehicle can be certified fit for
operation. The examples of such features are, e.g., the purity of a wheel sound
or the aligning of control signs on a tyre and a wheel centre. In order to take the
non-measurable features into account in a computer-aided process of decision-
making, a certain binary-type value, for instance [1,0] or [true, false], should be
assigned to the result of the control of these features.
Having the measured or assigned values of individual diagnostic features and
neglecting their origin, one can identify the current technical condition
of a wheel set, in a certain operation point t, on the basis of the diagnostic
feature vector ( )at,Y , in the following form [5]:
( ) ( )[ ] ( )[ ] ( )[ ][ ]tytytyat aaa ,2929,22,11 ,...,,, ϕϕϕ=Y (2)
where:
291,..., yy – a wheel set diagnostic features,
ana ,,1 ,...,ϕϕ – the forms of the functions describing the changes in the
values of these diagnostic features during operation.
The evaluation procedure of the individual features which are the vector
( )at,Y components is not uniform and requires the use of appropriate relation
operators for the comparison of the measured and boundary values of a given
feature or for the identification of non-measurable unfitness of a wheel set.
Additionally, for certain features, a uniform trend of value changes in the
course of operation cannot be defined. Such trends may apply to individual
operational periods determined by the machining of wheels. It is illustrated in
Figure 1, which shows the example of the changes in the thickness of a tyre and
the changes of the thickness and height of the wheel flange.
Every reconditioning of the wheel profile significantly reduces the thickness
of a tyre, whereas the reduction of this thickness in the rail vehicle operational
A. Sowa
64
period is systematic but relatively insignificant. It is different in the case of the
wheel flange thickness, which decreases significantly, and the reconditioning
of the wheel profile causes this dimension to be restored close to the nominal
value. The individual reconditioning of the profile must be recorded in the
register of the research results. The wheel flange height is practically
unchanging in the presented case. It may result from the errors in the recording.
In order to avoid this problem, one has to properly design the database structure
as well as the structure of the evaluation of measurement results correctness,
which should be used by a computer system for the evaluation of the technical
condition during the data handling.
Y
Ow
Og
O = D - D1
26 27
28 29
30 26
28 30
32 34
40
50
60
70
80
26 28
30 32
34
Fig. 1. An example of the technical condition state vector of the wheel set (Y) for the chosen
diagnostic features y1, y2, y3, i.e. the thickness of a rim and the thickness and height of the wheel
flange
Rys. 1. Przykładowy wektor stanu technicznego zestawu kołowego (Y) dla wybranych cech
diagnostycznych y1, y2, y3, tj. grubości obręczy oraz grubości i wysokości obrzeża koła
Problems of computer-aided technical state evaluation of rail-vehicle wheel sets
65
Formal classification of the technical condition of rail vehicles taking into
account the wheel set
Defining the classification forms of technical condition states for a simple
case of the evaluation of one feature of a given element requires taking into
account at least one boundary value of this feature’s changes. A completely
different situation refers to the features of a wheel set that is evaluated. These
features constitute a certain set. With more than two boundary values of some
features, each measured feature value should be compared with a few boundary
values. With the existing number of features, there is the possibility to initiate
incorrect actions, because the final operational decision referring to a rail
vehicle also depends on the differences of feature values between the individual
wheel sets of the given rail vehicle.
The variety of occurring limitations also requires using more than three
classes of technical condition which are usually mentioned in professional
literature, e.g. in [1, 2, 3].
The formulae which allow for the automatic generation of the evaluation of
the technical condition of the wheel set and consequently enable one to make an
operational decision with the reference to a given tested rail vehicle may be
formed on the basis of the boundary values of the features included in [5]. For
the features whose changes are illustrated in Fig. 1, the ranges of boundary
values are shown in Table 1. On the basis of Table 1, one can build examples of
formulae that determine classes of the technical condition states of the tested
vehicle and the appropriate decisions from a set of four operational decisions
and two maintenance decisions. These formulae are as follows:
– for the usability state zS and decision 0U ,
≥∧≤≤
∧≤≤∧≥
∈⇔⇔
∧
)53()3325(
)3225()40(:4,3,2,1
43
21
0yy
yyyiSU iz (3)
– for the conditional usability state 1zwS and decision 1U ,
<≤∧
<≤∧≤≤
∈⇔⇔
∧
)5348(
)2522()3632(:4,3,2
4
32
11y
yyyiSU izw (4)
– for the conditional usability state 2zwS and decision 2U ,
[ ] 4030:1 122 <≤∈⇔⇔
∧
yyiSU izw (5)
A. Sowa
66
– for the conditional usability state 3zwS and decision 3U ,
[ ] 48:4 433 <∈⇔⇔
∧
yyiSU izw (6)
– for the non-usability state 1nzS and decision 1O ,
>∨<
∨>∨<
∈⇔⇔
∧
)33()22(
)36()25(:3,2
33
22
11yy
yyyiSO inz (7)
– for the non-usability state 2nzS and decision 2O ,
[ ] 30:1 122 <∈⇔⇔
∧
yyiSO inz (8)
Table 1. The boundary values of the selected safety features for the tyre wheel set for an electric
locomotive
Tabela 1. Wartości graniczne wybranych cech bezpieczeństwa dla zestawu kołowego z obręczą
do lokomotywy elektrycznej
Item
number
Denotation
and name
of the feature
Relation to boundary
values
in [mm]
Decision
type Meaning of decision
1 y1 ≥ 40 U0
operation without
restrictions
2 30 ≤ y1 < 40 U2
freight or passenger
traffic v < 70 km/h
3
y1 – the thickness
of the tyre O
y1 < 30 O2 change of a tyre
4 25 ≤ y2 ≤ 32 U0
operation without
restrictions
5 32 < y2 ≤ 36 U1 v < 140 km/h
6
y2 – the height
of the flange Ow
y2 <25 ∨ y2 > 36 O1
reconstruction of
a profile
7 25 ≤ y3 ≤ 33 U0
operation without
restrictions
8 22 ≤ y3<25 U1 v < 140 km/h
9
y3 – the thickness
of the flange Og
y3<22 ∨ y3 > 33 O1
reconstruction of
a profile
10 y4 ≥ 53 U0
operation without
restrictions
11 y4 <53 U1 v < 140 km/h
12
Y4 – the sum
of the flange
thicknesses in the
set Ogl + Ogp y4 <48 U3 freight traffic
In the formulae (3 to 8), various operational operators and a various
number of boundary values of the evaluated features are present. It requires
Problems of computer-aided technical state evaluation of rail-vehicle wheel sets
67
constructing appropriate variants of the procedures of the computer evaluation
that will take into account all possible situations. It is not too difficult a task
because, regardless the programming language, one can easily construct
appropriate simple or complex conditional instructions of the type: if ... then
...else.
The guidelines for constructing the database for the computer-aided
evaluation of the technical condition state of wheel sets
The realisation of the system of the computer-aided evaluation of the
technical condition of wheel sets also requires, apart from building the
operational use application, the appropriate designing of the database [6], which
enables storing a wide variety of essential information. This data refers to the
following:
– the values of measured or evaluated features,
– the boundary values of measured or evaluated features,
– the trends of feature value changes and the relation operators of their
evaluation,
– the repertory of operation decisions,
– the means of the wheel set identification and their location in rail vehicles,
and
– the persons conducting diagnostic tests.
The properly designed structure of the database allows for the recording of
the information obtained from the mentioned groups in the properly related
database tables. It creates both the possibility of registering the research results
with the use of the feature values describing the wear and damage of wheel sets
and taking appropriate operational decisions as well as making and analysing
the trends of these unfavourable changes. With the growth of the database, it
makes it possible to use archive recordings for the active control of the
introduction of the data of the current measurement results. Proper warning
functions may be then automatically started in each attempt of recording values
that are unjustified by the observed trend of changes. Moreover, introducing
into the database the information about location of the wheel sets in the rail
vehicle enables the current analysis of the secondary feature values. Other data
also has its significance, e.g. the identification of the persons conducting
research favours an increase in the reliability of the obtained results.
Conclusions
The issues connected to the computer-aided evaluation of the technical state
of a wheel set presented in this article have considerable significance for the
A. Sowa
68
process of the control of rail vehicle operation. The appropriate application
which realises the data registration functions referring to wheel sets in vehicles
and the results of the tests run on these wheel sets and which generates
the appropriate operational decisions is currently being constructed. It is also
expected that there will be the possibility of conducting the analysis of the
research results. It can be, however, effectively applied after an experimental
period of using this application and after collecting enough data in
the databases.
The presented research results obtained within the M8/15/DS/2012 project
were financed from the subsidies granted by the Ministry of Science and Higher
Education.
References
[1] Będkowski L.: Elements of technical diagnostics, WAT, Warszawa 1991 (in Polish).
[2] Hebda M., Niziński S., Pelc H.: Fundamentals of motor vehicle diagnostics, WKŁ, Warszawa
1980 (in Polish).
[3] Niziński S.: Maintenance elements of technical objects, Publishing House of University
of Warmia and Mazury in Olsztyn, (Wydawnictwo Uniwersytetu Warmińsko-Mazurskiego),
Olsztyn 2000 (in Polish).
[4] Sowa A.: Wear curves used in construction of technical condition vector of diagnostic object.
Maintenance Problems vol.2/2007. ITeE Radom 2007, p. 65–76 (in Polish).
[5] Sowa A.: Diagnostic feature vector used in evaluation of vehicle wheel set technical state.
Maintenance Problems vol.2/2009. ITeE Radom 2009, p. 61–72 (in Polish).
[6] Sowa A.: The database for the evaluation system of the technical condition of rail-vehicle
wheel sets. Monograph „Problems of maintenance of sustainable technological systems”.
The Polish Maintenance Society, Warsaw 2010. Volume I, p. 88–100.
Problemy wspomaganej komputerowo oceny stanu technicznego zestawów kołowych
pojazdów szynowych
S t r e s z c z e n i e
W artykule przedstawiono zagadnienia związane z budową systemu wspomaganej
komputerowo oceny stanu technicznego zestawów kołowych pojazdów szynowych. Cechy
fizykalne tworzące wektor cech diagnostycznych. można wykorzystać do identyfikacji stanu
technicznego pojazdu. Na podstawie ocen przykładowych cech określono formuły pozwalające na
wyodrębnienie klas stanów technicznych pojazdów szynowych. Przedstawiono również wytyczne
do budowy bazy danych dla wspomaganego komputerowo systemu oceny stanu technicznego
zestawów kołowych.
Probabilistic formulation of steel cables durability problem
69
SCIENTIFIC PROBLEMS
OF MACHINES OPERATION
AND MAINTENANCE
3 (167) 2011
MICHAŁ STYP-REKOWSKI*, LESZEK KNOPIK*, EUGENIUSZ MAŃKA*
Probabilistic formulation of steel cables durability problem
K e y w o r d s
Steel cable, durability, probabilistic method.
S ł o w a k l u c z o w e
Lina stalowa, trwałość, metoda probabilistyczna.
S u m m a r y
This paper introduces a procedure for defining the probability of achievement of quantity
value received as a state symptom. The results of magnetic investigations of steel cables confirmed
the usefulness of the presented procedure in monitoring of the object state, in this case, of a cable
mechanism.
Introduction
The durability of steel cables is determined by many factors [1, 2] creating
a multielement set. The significance of individual factors that exist in the set
depends on the kind of mechanism in which the cable is used. It is different for
* University of Technology and Life Sciences in Bydgoszcz, Prof. S. Kaliski Avenue 7, 85-796 Bydgoszcz, Poland.
M. Styp-Rekowski, L. Knopik, E. Mańka
70
stay-cables of some constructions (mast, chimney), and different for the cable
of hoists and lifts (shaft, ski). However, in each case, extortion influence on the
cable has a random character. Therefore, the cable durability is considered
a random problem, in which probabilistic methods are used. This paper presents
a method that makes it possible to estimate the probability of the cable reaching
the acceptable border state.
Verifying the usefulness of the proposed method is the aim of this work. In
practical conditions, it may to contribute to an increase in the time of the
exploitation of the cables and the whole mechanism with the assurance
of indispensable safety.
Statistical analysis of magnetic investigations results
The presented analytic research results were conducted on the basis of
results received in magnetic investigations of cables [3]. Because these signals
are of a random character, statistical analysis is indispensable.
The first step was the choice of hypothetical distributions of empirical
investigation results. The programs STATGRAPHICS and STATISTICA were
used to study and introduce research results, both analytic and empirical, and
they contain a spacious collection of distributions. As a result of their review
and analysis, the following distributions were accepted:
• Gamma, for which the frequency function of variable x can be expressed by
formula:
b/x1p
pex
)p(Γb
1)x(f −−
= (1)
for p, b, x > 0; and,
• Weibull, with the frequency function of variable x expressed with equation:
a
b
x
1aa exba)x(f
−−
⋅⋅⋅= (2)
In both cases, variables belong to the range x ∈ (0,+ ∞).
The preliminary, estimated opinion of the usefulness of the chosen
distribution indicated agreement with empirical results.
To check whether the studied population has the defined type of
distribution, the tests of goodness of fit are used. In practice, the two most often
used tests are chi-square (χ2) and Kolmogorov (λ)[4].
The chi-square test is used for both continuous and step distributions. The
populations are divided by class of value and for each class from the
hypothetical distribution of theoretical sizes and compared the empirical values
Probabilistic formulation of steel cables durability problem
71
by means of the suitable statistics (χ2). The sample size of population is limited
applying this test. It has to be large because its elements are divided by class of
value, which should be sufficiently numerous. It is assumed that each class of
value should contain at least 8 test results.
In second test, λ Kolmogorov, empirical and hypothetical distribution
functions are compared. If the general population has a distribution concordant
with the hypothesis, then the value of empirical and hypothetical distribution
functions in all studied points should be close to each other. The continuity of
hypothetical distribution function is the condition that essentially limits the
applicability of this test.
In the results of preliminary analysis of the obtain results of empirical
investigations, chi-square (χ2) and Kolmogorov (λ) [5] were proven to be
indispensable for the analytic research of cables, for the following reasons:
• The sample size of the population from over 300 tests is sufficient to apply
chi-square test.
• The distribution functions of received hypothetical distributions (the gamma
and Weibull) are continuous.
The statistical analysis used n = 326 values of measurements of the
amplitudes of magnetic recorder plotter inclinations, which were recorded for
transmission of tape recorder movement pR = 20 mm/m. A comparison of results
for transmission values 10 and 20 mm/m indicated that higher value had greater
accuracy. The number of signal values resulted from the fact that twenty
measuring sections were tested, which permitted to register mentioned to be
above the number of peaks. Average value of all registered amplitudes is
02.4x = , and the standard deviation s = 2,04. The value of the coefficient of
changeability, expressed by the quotient v = s/ x = 0.51, shows that the analysed
statistical data have the comparatively large dispersion of values.
The preliminary analysis indicates that the distribution of probability of the
studied statistical feature is asymmetrical. The attempt of adjustment of
theoretical distribution to empirical data distribution indicates that, among the
considered ones, the gamma distribution is better; therefore, it received the
following analysis. The gamma distribution has a frequency function according
to Formula (1), in which expression Γ (p) is gamma function is described by the
following equation:
∫∞
−−
=
0
x1p dxex)p(Γ (3)
Average value of the random variable X in the gamma distribution is
obtained from the following formula:
EX = p.b, (4)
M. Styp-Rekowski, L. Knopik, E. Mańka
72
And variance is determined by
D2X = p.b2 (5)
Formulae (4) and (5) are the basis to determining the initial values of
parameters p and b using empirical data. X and S2 were marked as the
estimators of average the values and variances, which were obtained from
following formulae:
∑=
=
n
1i
ixn
1X (6)
∑=
−=
n
11
2i
2 )xx(n
1S (7)
If by ^
bp and ^
b there are marked the estimators of parameters respectively
p as well as b, then the equation (4) and (5) for moments method obtain form:
^^
bpX = (8)
S2 = ^
p^
b 2 (9)
From Equations (8) and (9), the estimators ^
p and ^
b of parameters p as
well as b of distribution (1) were determined, according to formulae:
^
b = S2/ X (10)
22^
)X/(Sp = (11)
Parameter p is the parameter of form (shape), and it does not depend on the
unit in which random variable X is measured; however, b is the parameter of
scale. Defined with formulae (10) and (11) notes are treated as preliminary
notes (initial) of the values of parameters p and b, in process of the exact values
estimation. To get the more exact notes of parameters p and b, in this work the
method of the largest credibility was applied. In this case statistical pack of
program STATISTICA was used.
Probabilistic formulation of steel cables durability problem
73
Results of investigations and their analysis
The procedure was applied to the results of experimental investigations of
steel cable. The results are recorded in graphic form on a recorder tape. An
exemplary fragment of the defectogram is shown in Fig. 1. Obtained results,
with the division of individual classes of value, are shown in Table 1.
Table 1. Statement of measurements results – power of a set in classes
Tabela 1. Zestawienie wyników pomiarów – liczebności w klasach
⟨xi; xi+1) ⟨0; 1) ⟨1; 2) ⟨2; 3) ⟨3; 4) ⟨4; 5) ⟨5; 6) ⟨6; 7) ⟨7; 8) ⟨8; 9) ⟨9;10)
ni 19 43 61 71 56 32 18 13 8 5
The analysis of data in Table 1 presents that the studied empirical
distribution expansion is not symmetrical. Class ⟨ 3; 4) has the largest size, so
one should accept that the modal value of distribution belongs to this class.
Moreover, one can notice that the median and modal values do not coincided to
the average value of the analysed distribution.
L2-3
Fig. 1. Fragment of recorded magnetic investigations
Rys. 1. Fragment zapisu wyników badań magnetycznych
The highest probabilities of data from measurements gave the following
values for the distribution parameters:
p = 3.2455 (12a)
b = 1.1960 (12b)
For gamma distribution with frequency Function (1), the skew is described
by the following formula:
pb
11 = (13)
M. Styp-Rekowski, L. Knopik, E. Mańka
74
For analysed data, b1 = 0,555. This result confirms fact that the skew of the
studied distribution is visible (comparatively large).
Therefore, for examining goodness of fit of empirical distribution and
received the gamma distribution with parameters (12), the tests χ2, as well as test
λ-Kolmogorov, were applied. For test χ2, the calculated value is
83,42=oblχ (14a)
However, from statistical tables, for significance level α = 0.05, the value of
this statistic is
07,142=tablχ (14b)
The value of parameter p calculated for (14a), is 0.68. The obtained results
confirm the goodness of fit of the gamma distribution with the empirical one.
For test λ-Kolmogorov, the λobl value is equal to 0.40, and value λtab received
from statistical tables, for significance level α = 0.05, is equal to 1.36. These
values testify to the very good goodness of fit of the gamma distribution with
the empirical distribution.
In images below present the results of statistical analyses in graphic form.
Graphs of the empirical and theoretical distribution functions are shown in
Fig. 2. They are very close, which testifies to the good adjustment of the results
of experiments to the assumed hypothetical distribution.
0,00
0,20
0,40
0,60
0,80
1,00
1 2 3 4 5 6 7 8 9 10
Fe
Ft
Fig. 2. Graphs of distribution functions: Fe – empirical, Ft – theoretical
Rys. 2.Wykresy dystrybuant Fe – empirycznej i Ft – teoretycznej
In Fig. 3 the graphs of probability frequency functions were introduced for
both distributions. The compatibility of the probability frequency functions,
empirical fe and theoretical ft, of the analysed distributions is visible.
Tested distributions differ at points of maximum probability density. This
means that modes of distribution, theoretical and empirical, are different, but
only slightly.
Probabilistic formulation of steel cables durability problem
75
Table 2 indicates that the probabilities do not cross the value of the random
variable, which was accepted as the admissible (boundary) value.
Table 2. Probability of boundary values
Tabela 2. Prawdopodobieństwo nieprzekroczenia wartości progowej
Boundary
value Probability
Boundary
value Probability
8 0.95101 14 0.99901
9 0.97335 15 0.99951
10 0.98580 16 0.99976
11 0.99256 17 0.99988
12 0.99616 18 0.99994
13 0.99804
0
10
20
30
40
50
60
70
80
1 2 3 4 5 6 7 8 9 10
fe
ft
Fig. 3. Graphs of frequency functions: fe – empirical, ft – theoretical
Rys. 3. Wykresy gęstości: fe – empirycznej i ft – teoretycznej
Fig. 4 presents the graphs of the dependence of probability on the value of
the threshold, e.g. value accepted as boundary operational safety. It is a
visualisation of data contained in Table 2.
0,95
0,96
0,97
0,98
0,99
1,00
8 10 12 14 16 18
Fig. 4. Relation between probability and assumed boundary value
Rys. 4. Zależność prawdopodobieństwa od przyjętej wartości progowej
M. Styp-Rekowski, L. Knopik, E. Mańka
76
Presented in Table 2 and in Fig. 4 results of calculations of probability of
not crossing of boundary value indicate, that for boundary values greater than 11
probabilities have crossed 0,99 and they are not differ more than several pro miles.
On this basis, one can formulate a practical conclusion that observing
recorded magnetic investigations makes it possible to estimate the probability of
the occurrence of cable weakness limits in regard to safety limits of cables
operating in a shaft hoist.
Closure
These investigations have cognitive and practical aspects. Cognitive element
is that, in statistical categories, the results of the magnetic investigations of cables
have a distribution very closed to the gamma distribution.
The confirmation of the possibility of application of described procedure to
continuous diagnostic investigations is the practical element. Monitoring the
change of the signal which has direct relationship with cable operational
features – its weakness, one can react in suitable moment, e.g. to reduce
working load of devices or to stop its, what will permit to avoid breakdown
generated extensive damages.
References
[1] Mańka E., Skrok T., Styp-Rekowski M.: Eksploatacyjne czynniki determinujące proces
zużywania lin górniczych wyciągów szybowych. Materiały XVI Międzynarodowej
Konferencji N-T TEMAG’08. Politechnika Śląska, Gliwice – Ustroń 2008, s. 227–237.
[2] Hansel J.: Podstawy teorii i inżynierii bezpieczeństwa systemów maszynowych transportu
pionowego. Materiały VI Międzynarodowej Konferencji „Bezpieczeństwo pracy urządzeń
transportowych w górnictwie”. Centrum Badań i Dozoru Górnictwa Podziemnego, sp. z o.o.
Lędziny – Ustroń 2010, s. 20–28.
[3] Kwaśniewski J.: Badania magnetyczne lin stalowych. Wydawnictwo AGH, Kraków 2010.
[4] Bendat J.S., Piersol A.G.: Metody analizy i pomiaru sygnałów losowych. PWN, Warszawa
1976.
[5] Benjamin J.R., Cornel C.A.: Rachunek prawdopodobieństwa, statystyka matematyczna
i teoria decyzji dla inżynierów. WNT, Warszawa 1977.
Probabilistyczne ujęcie zagadnienia trwałości lin stalowych
S t r e s z c z e n i e
W artykule przedstawiono procedurę określania prawdopodobieństwa osiągnięcia określonej
wartości wielkości przyjętej jako symptom stanu. Wykorzystując wyniki magnetycznych badań lin,
potwierdzono jej przydatność w monitorowaniu stanu obiektu, w tym przypadku mechanizmu
linowego.
An outline of a method for determining the density function of the time...
77
SCIENTIFIC PROBLEMS
OF MACHINES OPERATION
AND MAINTENANCE
3 (167) 2011
MARIUSZ WAŻNY*
An outline of a method for determining the density
function of the time of exceeding the limit state with the use
of the weibull distribution
K e y w o r d s
A diagnostic parameter, the Weibull distribution, destructive processes, reliability, probability.
S l o w a k l u c z o w e
Parametr diagnostyczny, rozkład Weibulla, procesy destrukcyjne, niezawodność, prawdopodo-
bieństwo.
S u m m a r y
This article presents an attempt at an analytical description of technical state changes within
a selected group of technical objects. The occurring changes of the technical state of these objects
are identified by diagnostic parameter values. The changes are identified by diagnostic parameter
values. The technical state of a device deteriorates with the time of its maintenance due to the
effect of numerous destructive factors. The conducted studies are based on the assumption that the
intensity of changes of the deviation of diagnostic parameter values adopts the Weibull constants.
The dynamics of changes of diagnostic parameter values is described by the difference equation
that was transformed into a differential equation. Its solution in the form of a density function
enables one to determine the reliability of a device in terms of an examined diagnostic parameter.
The density function of the time of exceeding the limit state by a diagnostic parameter was
determined using material from the literature [9], whose continuation is this article.
* Military University of Technology, General Sylwester Kaliski 2 Street, 00-908 Warsaw 49,
Poland, phone (22) 683-76-19.
M. Ważny
78
Introduction
Aeronautical engineering obliges design engineers, manufacturers and
users to meet requirements connected with maintaining high values of safety
and reliability parameters. Examining the safety and reliability of an aircraft in
the maintenance process involves the prediction of the technical state of its
particular devices and systems and the aircraft itself as a platform combining all
the above-mentioned elements. Analysing an aircraft as an object whose task,
for example, is to ensure the transportation of passengers and cargo, we can
assume that the maintenance conditions are of special importance compared
with other popular means of transport [4]. The influence of a series of factors
causes that the values of the parameters describing the technical state of an
aircraft change over time. Destructive processes manifesting themselves in the
form of overload, friction, vibrations, wear, etc. have a crucial effect on
technical state changes in aircraft devices.
The technical state of aircraft devices is mainly evaluated through a set of
diagnostic parameters. The effect of destructive processes manifests itself in the
change of diagnostic parameter values causing a rise in the deviation from the
nominal values of these parameters. The values of deviations from the nominal
values are used to estimate the reliability of a device.
The classifications of the correlation between the effect of destructive
processes and the change of diagnostic parameter values are presented in the
paper [9]. In this article, the density function of changes of diagnostic parameter
deviations is determined based on the following assumptions:
− The technical state of a device is determined by one dominant diagnostic
parameter. Its current value is denoted by “x”.
− The changes of a diagnostic parameter value due to the destructive effect
of ageing processes occurs with the passing of calendar time.
− The deviation of a diagnostic parameter from the nominal value is
np xxz −= (1)
where:
px – the measured value of a diagnostic parameter,
nx – the nominal value of a diagnostic parameter.
− If z∈[0,zd], then an element of a device is regarded as operable;
otherwise, an element of a device is regarded as inoperable,
− An increase of a diagnostic parameter deviation in the function of calendar
time satisfies the following relationship:
cdt
dz= ,
An outline of a method for determining the density function of the time...
79
where:
c – the mean value, a variable velocity depending on ageing processes,
t – the calendar time.
This determines the density function of changes in values of diagnostic
parameter deviations.
Determining the density function of changes in values of diagnostic
parameter deviations
It is assumed that the intensity of the increase in deviations has the
following form:
( )1−
=α
θ
αλ tt (2)
where:
θα i – the constants in the Weibull distribution with the following
denotations:
α – the shape factor,
θ – the scale factor.
The random dynamics of changes of diagnostic parameter values including
the deviation is described by the difference equation. Let tzU , denote the
probability that at the time t, the value of a diagnostic parameter deviation
adopts the value “z”.
The differentiated equation has the following form:
t,zzt,ztt,z UttUttU∆
αα
∆∆
θ
α∆
θ
α
−
−−
++
−=
111 (3)
where:
z∆ – the increase in deviation of a diagnostic parameter over the time
interval t∆ .
Equation (3) has the following form in function notation (4):
( ) ( ) ( )tzzutttzuttttzu ,,1, 11∆−∆+
∆−=∆+
−− αα
θ
α
θ
α (4)
M. Ważny
80
where:
( )t,zu – the density function of a diagnostic parameter deviation;
−
−
tt ∆
θ
αα 11 – the probability that over the time interval t∆ there is no
parameter deviation;
tt ∆−1α
θ
α – the probability that over the time interval t∆ there is the
increase in the parameter deviation “ z∆ ”;
and the following condition is met .tt 11≤
−
∆
θ
αα
We transform Equation (4) into a partial differential equation. We assume
the following approximation:
( ) ( )( )
( ) ( )( ) ( )
( )2
2
2
2
1z
z
t,zuz
z
t,zut,zut,zzu
,tt
t,zut,zutt,zu
∆∆∆
∆∆
∂
∂+
∂
∂−=−
∂
∂+=+
(5)
We substitute the relationships expressed in (5) into Equation (4) and
obtain equation (6).
( ) ( )( )
( )
2
2211
2
1
z
t,zuzt
z
t,zuzt
z
t,zu
∂
∂+
∂
∂−=
∂
∂−−
∆
θ
α∆
θ
ααα (6)
We examine the increase of a parameter deviation per unit of time (when
∆t = 1), so
ct
z=
∆
∆, tcz ∆∆ =⇒ , c
t 1=
⇒∆
,
where: c denotes the deviation increase per a unit of time.
The final form of Equation (6) is as follows:
( )
( )
( )
( )
( )
2
21
21
2
1
z
t,zut
c
z
t,zut
c
z
t,zu
tt
∂
∂+
∂
∂−=
∂
∂−−
43421321β
α
γ
α
θ
α
θ
α (7)
An outline of a method for determining the density function of the time...
81
As it can be seen in Equation (7), the form of the coefficients depends on
the parameter values α. For α = 1, the coefficients have the following form:
( )θ
γc
t = ; θ
β
2c
= .
For α = 2, the coefficients have the following form:
( ) tc
tθ
γ2
= ; ( ) tc
tθ
β
22
= .
The solution of Equation (7) has the following form:
( )( )
( )( )
( )tA
tBz
etA
t,zu 2
2
2
1−
−
=
π
(8)
where:
B(t) – the average value of a parameter deviation for the time of the
service life t,
( ) ( )∫=
t
dtttB
0
γ (9)
A(t) – the value of the variance of a diagnostic parameter deviation for the
time of the service life t.
( ) ( )∫=
t
dtttA
0
β (10)
We calculate integrals (9) and (10) and obtain the following:
( )ααααα
θθαθ
α
θ
α
θ
αt
ct
ct
cdtt
cdtt
ctB
ttt
=−==== ∫∫ −− 01
0
0
1
0
1 (11)
( )ααα
θαθ
α
θ
αt
ct
ct
ctA
tt
2
0
2
0
12 1
=== ∫ − (12)
M. Ważny
82
Hence, the relationship depicted in Equation (8) has the following form:
( )
α
α
θ
θ
α
θ
π
tc
tc
z
e
tc
t,zu
2
2
2
2
2
1
−
−
= (13)
The relationship depicted in Equation (13) presents the density function of
a diagnostic parameter deviation from the nominal value.
Let
bc
=
θ
and ac
=
θ
2
.
Hence, Equation (13) has the following form:
( )
( )α
α
α
π
at
btz
eat
t,zu 2
2
2
1−
−
= (14)
By using the density function (14), we can determine the relationship for
the reliability of a device in terms of an examined diagnostic parameter. This
relationship has the form of Equation (15) as follows:
( ) ( )∫∞−
=
dz
dzt,zutR (15)
where:
zd – the permissible deviation value of the diagnostic parameter a u(z, t)
is determined by Equation (14).
Determining the distribution of time when a diagnostic parameter exceeds
the permissible state
Using the deviation density function, we can write down the probability of
exceeding the deviation value of a diagnostic parameter in the following form:
( )
( )
dzeat
z,tQ at
btz
z
d
d
α
α
α
π
2
2
2
1−
−
∞
∫= (16)
An outline of a method for determining the density function of the time...
83
(21)
The density function for the distribution of time of exceeding the
permissible value of the diagnostic parameter zd equals
( ) ( )dz,tQt
tf∂
∂= (17)
If we consider (16), this equation takes the form
( )
( )
dzeatt
tf
dz
at
btz
∫∞ −
−
∂
∂=
α
α
α
π
2
2
2
1
Hence,
( )
( )
dzeatt
tf
dz
at
btz
∫∞ −
−
∂
∂=
α
α
α
π
2
2
2
1 (18)
We search for the time derivative of the integrand of this relationship (18)
and obtain the following:
( ) ( ) ( )′
⋅+
′
=
∂
∂−
−−
−−
−α
α
α
α
α
α
ααα
πππ
at
btz
at
btz
at
btz
eat
eat
eatt
222
222
2
1
2
1
2
1 (19)
We can then calculate the component derivatives of Equation (19) as
follows:
( )
( )α
αα
α
α
αα
α
π
α
ππ
απ
π
αππ
π attatat
ta
at
taat
at 22222
222
10
2
1
2
1
112
1
−=−=
⋅−
=
′
−
−−
(20)
( ) ( )( )( ) ( )
( )
( )( ) ( )
( )
( )
( ) ( ) ( )α
α
α
α
α
α
α
α
α
ααα
α
ααααα
α
ααααα
αα
αα
αα
at
btz
at
btz
at
btz
at
btz
eat
tbtztbbtz
eat
tabtzattbbtz
at
tabtzattbbtzee
2
21
2
2
121
2
121
2
2
2
2
22
2
12
2
222
2
222
−
−
−
−−
−−
−−−
−−
−
⋅−+−
=
=−−−−
−=
=
−−−−
−=
′
M. Ważny
84
(23)
We then substitute the above-defined formulas into Equation (19):
( )[ ]
( ) ( ) ( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )( )t,zu
at
btztbbtz
t
t,zuat
tbtztbbtz
t
t,zuat
tbtztbbtz
t,zut
eat
tbtztbbtz
ate
attt,zu
tat
btz
at
btz
−+−
+−=
=
−+−
+−=
=
−+−
+−=
=
−+−
⋅+−=
∂
∂
+
−
−
−−
−−
−
1
2
21
21
2
21
2
2
2
2
2
2
2
2
2
2
2
2
2
1
22
22
α
ααα
α
ααα
α
ααα
α
ααα
αα
ααα
αα
α
αα
α
αα
ππ
α α
α
α
α
From this relationship (18), we obtain the following:
( )
( )
dzeatt
tf
dz
at
btz
∫∞ −
−
=
∂
∂
=
2
12
2
πα
α
α
( ) ( )( )dzt,zu
tat
btztbbtz
dz
∫∞
+
−
−+−=
22
21
2ααα
α
ααα
In order to calculate the integral (23), we need to determine an
antiderivative. We assume the following form of the antiderivative of the
integrand in the relationship presented in (23):
( ) ( ) ( )t,zt,zut,zw θ= (24)
The derivative of the indefinite integral with respect to the variable “ z ” is
equal to the integrand of the relationship depicted in (23).
Hence,
( )( ) ( )
( ) ( ) ( )( )t,zu
tat
btztbbtz
z
t,zt,zut,z
z
t,zu
−
−+−=
∂
∂+
∂
∂
+ 22
21
2αααθ
θα
ααα
(25)
(22)
An outline of a method for determining the density function of the time...
85
We calculate the derivative ( )
z
t,zu
∂
∂ as follows:
( )( )
( )( )
( )
−−=
−−=
∂
∂−
−
α
α
α
α
α
α
α
π at
btzt,zu
at
btze
atz
t,zuat
btz
2
2
2
12
2
(26)
By substituting (26) into (25), we obtain the following:
( )( )
( ) ( )( )
z
t,zt,zut,z
at
btzt,zu
=
∂
∂+
−−
θθ
α
α
( ) ( )( )t,zu
tat
btztbbtz
−
−+−=
+ 22
21
2ααα
α
ααα
( )( )
( )( )
( )( ) ( )
−−+−
=
∂
∂+
−−
−−
+
−− PIIPILIILI
tat
btztbbtzt,zu
z
t,zt,z
at
btzt,zu
22
21
2αααθ
θα
ααα
α
α
44444 344444 214342144 344 21
(28)
By using the relationship (28), we can determine the function ( )tz,θ in
such a way that the left side of the relationship (28) equals the right side. So
PILI −=− →→→→ ( )( )( )
t
btztbt,z
2
2αα
ααθ
−+−= (29)
PIILII −=− →→→→ ( )
tz
t,z
2
αθ−=
∂
∂ (30)
After reducing this expression, we obtain the following:
( )( ) ( )
t
btz
t
btzbtt,z
22
2ααα
ααθ
+−=
−+−= (31)
The antiderivative has the following form:
( ) ( )( )
+−=
t
btzt,zut,zw
2
α
α (32)
(27)
M. Ważny
86
where:
( )
( )α
α
α
π
at
btz
eat
t,zu 2
2
2
1−
−
=
We can then calculate the integral as follows:
( ) ( )( )
( )( )
+=
+−=
∞
t
btzt,zu
t
btzt,zutf d
z
z
d
d 22
αα
αα (33)
where:
( )
( )α
α
α
π
at
btz
d
d
eat
t,zu 2
2
2
1−
−
= (34)
Hence, the density function of the time of exceeding the permissible value
of the diagnostic parameter “ dz ” has the following form:
( )( )
( )α
α
α
α
π
αat
btz
dz
d
de
att
btztf 2
2
2
1
2
−
−
⋅+
= (35)
We need to check whether the integral (36) is equal to 1.
( )( )
12
1
20
2
2
?at
btz
d dteatt
btzd
=+
∫∞ −
−α
α
α
α
π
α (36)
The above relationship can be written down in the form of Equation (37)
( ) ( )
12
1
22
1
20
2
0
2
22
?
B
at
btz
A
at
btz
d dteatt
btdte
att
zdd
=+ ∫∫∞ −
−
∞ −
−
44444 344444 214444 34444 21
α
α
α
α
α
α
α
π
α
π
α (37)
We then calculate the integral A as follows:
( )
dtettat
zdte
att
zA at
tbbtzz
dat
btz
d
ddd
∫∫∞ +−
−
∞ −−
==
0
2
2
0
2
2222
1
2
1
22
1
2
α
αα
α
α
αα π
α
π
α (38)
An outline of a method for determining the density function of the time...
87
In order to determine the above-mentioned integral, we perform the
following substitution:
α
tu =
dttdu1−
=α
α ⇒ 1−
=α
αt
dudt
Hence,
t
due
uta
zA
dd
au
ubbuzz
d
222
1
2
2
0
1
2
1
2 −
+−
−
∞
∫ =⋅=
απ
α
α
( )
dueutta
zdd
au
ubbuzz
u
d
222
0
2
2
1
11
2
1
2
∞ +−
−
−∫ =⋅
⋅
=
απ
α
α
321
(39)
dueuua
zd
dd
d
z
au
z
ub
zbuz
d2
2
22
2
2
121
0
1
2
1
2
+−
−
∞
∫=
π
We then perform one more substitution:
ωω
b
zuu
z
b d
d
=⇒=
duz
bd
d
=ω ⇒ ωdb
zdu d
=
As a result, we get the following relationship:
444 3444 21D
qd
dd
a
bz
d
dbz
a
dd
z
a
zbz
d
deb
z
b
z
b
ze
a
z
db
ze
b
z
b
ze
a
zA
d
dd
d
d
∫
∫
∞ +−
∞
+−
⋅
⋅=
==
0
2
1
0
2
1
2
12
2
2
2
2
11
2
1
2
1
2
1
2
ω
ωωπ
ω
ωωπ
ω
ω
ω
ω
(40)
M. Ważny
88
where:
bz
aq
d
= .
Based on the patterns posted on the table of integrals [3], we can write:
qe
qD
π2= .
After further transformations, the solution of Equation (40) has the
following form:
2
11
2
1
2
11
2
2
1
2
1
2
2
1
2
1
2
21
2
1
2 1
=⋅=⋅=⋅=⋅⋅=
=⋅=⋅=
d
d
dd
d
dd
d
a
bz
d
d
a
bz
d
bz
a
d
d
b
bz
d
q
d
a
bz
d
z
z
bzb
z
z
bz
b
z
z
e
bz
a
b
ze
a
z
e
bz
a
b
ze
a
z
e
q
b
ze
a
zA
d
d
d
dd
π
π
π
π
π
π
(41)
Before calculating integral B, we write it down as Equation (42):
( ) ( )
dtett
t
a
bdte
atzt
btB at
btz
at
btz dd
2
0
2
2
2
0
1
222
1
∫∫∞ −
−−
−
∞
⋅=⋅=α
α
α
α
α
α
α
α
π
α
π
α (42)
We then make the following substitution:
α
tu =
dttdu1−
=α
α ⇒ 1−
=α
αt
dudt
( ) ( )
dueua
bdu
te
ut
a
bB au
buz
au
buz dd
∫∫∞
−−
−
−−
∞
−
=⋅=
0
2
2
1
2
2
0
1 1
22
11
22 παπ
α
α
α (43)
An outline of a method for determining the density function of the time...
89
∫∫
∫∫
∞
+
−
∞
+
−
∞
+−
−
∞+−
−
==
=⋅=⋅=
0
2
1
0
2
1
2
2
0
2
21
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
222
1
22
1
22
1
22
1
22
dueu
ea
bdue
ue
a
b
dueua
bdue
ua
bB
uz
a
uz
b
a
bzuz
a
uz
b
z
a
z
bz
uz
a
uz
bu
z
bz
au
ubbuzz
d
d
d
d
d
d
d
d
d
dd
d
dd
ππ
ππ
(44)
We then substitute again as follows:
uz
b
d
=ω ⇒ ω
b
zu d
=
duz
bd
d
=ω ⇒ ωdb
zdu d
=
As a result, we obtain the following:
44 344 21E
qda
bz
db
z
z
a
d
a
bz
deb
ze
a
bd
b
ze
b
ze
a
bB
dd
d
d
∫∫∞
⋅
+−
∞
+−
⋅=⋅=
0
2
1
0
2
12
2
2
1
22
1
22ω
ωπ
ω
ωπ
ω
ωω
ω
(45)
where:
bz
aq
d
=
Similar to the dependency expressed in Equation (40), using patterns posed
in [3], we can write
qe
qE
π2=
As a result of the above transformations, the solution of integral B has the
form of the following relationship (46):
2
12
22
2
221
=⋅⋅⋅=⋅⋅=
b
z
bz
a
e
e
a
b
e
bz
a
b
ze
a
bB d
da
bz
a
bz
bz
a
dda
bz
d
d
d
dπ
π
π
π
(46)
M. Ważny
90
The above results indicate that the relationship depicted in (36) is true,
i.e. ( )dztf is the density function for the distribution of the time of exceeding the
permissible state.
Summary
Determining the density function for the distribution of the time of
exceeding the limit state is an extremely significant issue. Based on the above-
mentioned function, we can determine the residual durability of a device, which
will constitute the subject of further analyses. A significant element of this
paper involves the utilisation of the Weibull distribution to determine both the
density function of changes in diagnostic parameter deviations and the density
function of the time of exceeding the limit state by a diagnostic parameter.
Contemporary aircraft are equipped with various electronic devices
supporting both its functions and flight. These devices undergo periodic
inspections during which the values of diagnostic parameters are recorded. The
monitoring of diagnostic parameter values is contingent upon the destructive
effect of factors deteriorating the technical state of devices and systems. Such
changes are often described by means of the Weibull distribution. Therefore, the
utilisation of the Weibull distribution seems to be justified.
The description presented in this paper may be used not only in the field of
aeronautical engineering but also in all other fields where the technical state of
devices is determined on the basis of analysing the changes of diagnostic
parameters.
Scientific work funded by the National Centre for Researches and
Development in 2011–2013 as a research project.
Bibliography
[1] Abezgauz G.: Rachunek probabilistyczny. Poradnik. Wydawnictwo Ministra Obrony
Narodowej, Warszawa 1973.
[2] Gniedenko B.W., Bielajew J.K., Sołowiew A.D.: Metody matematyczne w teorii
niezawodności. Wydawnictwo Naukowo-Techniczne, Warszawa 1968.
[3] Gradsztejn I.S., Ryżyk I.M.: Tablice całek, sum, szeregów i iloczynów. Państwowe
Wydawnictwo Naukowe, Warszawa 1964.
[4] Tomaszek H., Wróblewski M.: Podstawy oceny efektywności eksploatacji systemów
uzbrojenia lotniczego. Dom Wydawniczy Bellona, Warszawa 2001.
[5] Tomaszek H., Żurek J., Loroch L.: Zarys metody oceny niezawodności i trwałości
elementów konstrukcji lotniczych na podstawie opisu procesów destrukcyjnych.
Zagadnienia Eksploatacji Maszyn, Zeszyt 3 (139), Radom 2004.
An outline of a method for determining the density function of the time...
91
[6] Tomaszek H., Szczepanik R.: Metoda określania rozkładu czasu do przekroczenia stanu
granicznego. Zagadnienia Eksploatacji Maszyn, Zeszyt 4 (144), Radom 2005.
[7] Tomaszek H., Żurek J., Stępień S.: Eksploatacja statku powietrznego z odnową
i ryzykiem jego utraty. Zagadnienia Eksploatacji Maszyn, Zeszyt 4 (156), Radom 2008.
[8] Pamuła W.: Niezawodność i bezpieczeństwo. Wydawnictwo Politechniki Śląskiej, Gliwice
2011.
[9] Ważny M.: The outline of the method for determining the density function of changes in
diagnostic parameter deviations with the use of the Weibull distribution. Scientific Problems
of Machines Operation and Maintenance.
Zarys metody określenia funkcji gęstości czasu przekroczenia stanu dopuszczalnego
z wykorzystaniem rozkładu Weibulla
S t r e s z c z e n i e
W artykule podjęto próbę analitycznego opisu zmiany stanu technicznego wybranej grupy
obiektów technicznych. Zachodzące zmiany stanu technicznego tychże obiektów identyfikowane
są za pomocą wartości parametrów diagnostycznych. W wyniku oddziaływania licznej grupy
czynników destrukcyjnych stan techniczny urządzeń wraz z upływem czasu ich eksploatacji ulega
pogorszeniu. Podstawą przeprowadzonych rozważań było przyjęcie założenia, że intensywność
zmian odchyłki wartości parametrów diagnostycznych przyjmuje stałe o rozkładzie Weibulla.
Dynamikę zmian wartości parametrów diagnostycznych opisano za pomocą równania
różnicowego, dla którego, po przekształceniu do postaci równania różniczkowego, wyznaczono
rozwiązanie w postaci funkcji gęstości umożliwiającej określenie niezawodności urządzenia ze
względu na rozpatrywany parametr diagnostyczny. Posiłkując się materiałem zamieszczonym
w [9], której niniejszy artykuł jest kontynuacją, wyznaczono funkcję gęstości czasu przekroczenia
stanu dopuszczalnego przez parametr diagnostyczny.