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POLISH ACADEMY OF SCIENCES COMMITTEE OF MACHINE ENGINEERING

SCIENTIFIC PROBLEMS

OF MACHINES OPERATION

AND MAINTENANCE

ZAGADNIENIA EKSPLOATACJI MASZYN

TRIBOLOGY • RELIABILITY • TEROTECHNOLOGY

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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


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


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


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).


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


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


AND MAINTENANCE

3 (167) 2011

JACEK M. CZAPLICKI*, ANNA M. KULCZYCKA

*


K e y w o r d s

Semi-Markov process, multi-element symmetric pair, steady-state availability calculation.


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


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.


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)


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


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)


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).


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)


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


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)


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


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


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.


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],

[email protected].


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


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].


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).


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


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


4. Temperature test:

– initial temperature T = 60; 100; 140; 180; 240; 280; 320; 360°C

– sliding speed v = 9 m/s


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


33


34


35


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


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,


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.


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


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


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).


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


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


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.


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


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)


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)


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


Table 2. Quintic B-spline decomposition sequences


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


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


=

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


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


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


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


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.


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


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.


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)


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


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)


[ ] 4030:1 122 <≤∈⇔⇔

∧

yyiSU izw (5)

A. Sowa

66


[ ] 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


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


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


AND MAINTENANCE

3 (167) 2011

MICHAŁ STYP-REKOWSKI*, LESZEK KNOPIK*, EUGENIUSZ MAŃKA*


K e y w o r d s

Steel cable, durability, probabilistic method.


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


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)


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.


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)


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.


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


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


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


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= ,


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)


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)


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)


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)


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)


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.


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


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.

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