Jacek Rysiński,
Robert Drobina,
Jerzy Tomaszewski
Probability of the Critical Length of
a Fatigue Crack Occurring at the Tooth Foot
of Cylindrical Geared Wheels of the Drive
System of a Fiomax 2000 Ring Spinner
DOI: 10.5604/12303666.1227895
Faculty of Mechanical Engineering
and Computer Science
University of Bielsko Biala,
ul. Willowa 2, 43-309 Bielsko-Biała, Poland
E-mail: [email protected]
[email protected]
[email protected]
Abstract
In the present paper, we describe a method of determination of the probability of reaching
the critical crack length at the tooth root of the cylindrical geared wheels of the drive
system of the Fiomax 2000 ring spinner. The Paris-Erdogan formula was utilised for calculations of the fatigue crack length depending on the number of load cycles. Experimental
investigations were performed on cylindrical geared wheels. The wheel specimens were
manufactured from 1.6523 steel (UE) according to a technical specification relevant to
the drive system of the ring spinner. The experiments were performed using a professional
pulsator (pulsating test machine). Based upon the experiments (series of 12 tests), material constants and were calculated. These parameters were utilised in the Paris law of
crack propagation for further calculations. Moreover it was also ascertained that these
unknowns are related via the deterministic relationship. Therefore a function allowing for
approximation of constant in dependence on exponent m was derived. In the next step, for
the values of parameter chosen – belonging to the variability interval, established from
experimental data – we determined the times of reaching the critical length of the fatigue
crack. It was stated that the best approximation distribution describing the simulated
random values of times of reaching the critical length of the tooth crack for the drive system of the ring spinner is the asymptotic Gumbel’s distribution. Knowing the distribution
and number of cycles until reaching the critical crack length at the tooth root, one can
evaluate the fatigue life of the damaged wheel in the ring spinner (Fiomax) drive system
for the assumed probability. The goal of the present paper is evaluation of the working
time of the elements of the drive system of a ring spinner until the occurrence of damage.
The highest fatigue life of geared wheels was achieved within the interval (4.3 – 4.5)x 105
cycles. However, it is recommended to change of the geared wheel in case of the spotting
of early symptoms of defect. For the stretching apparatus, the authors of the present paper
suggest the exchange of the idler geared wheels at least once per year.
Key words: tooth foot crack, Paris-Erdogan formula, probability distribution, cycle
number, crack critical length, evaluation, fatigue life, cracked tooth, ring spinner, ring
spinning frame.
Nomenclature
a0, a1, a2constants of an equation determined based upon numerical
simulation,
A1, A2, A3, A4, A5 constants related to the
equation for calculations of
the intensity factor coefficient,
determined for known technical specification of the geared
cylindrical wheel and known
loading conditions,
134
b
width of the geared cylindrical
wheel,
BEX idler gear wheel,
CEX idler gear wheel,
C
the Paris formula material constant,
dl increase in crack length,
EEX idler gear wheel,
FEX idler gear wheel,
Gconstant utilized for determination
of tooth shape coefficient,
Gv total stretch of spinning stream,
Hvmain draft of fiber assembly,
HzEX idler gear wheel,
hfp addendum height of tooth profile,
l
crack length,
lkr critical length of crack,
lmin introductory crack length,
m exponent of Paris curve (law),
mn normal module,
mymean value of random value distribution,
N
number of fatigue (loading) cycles,
NwEXidler gear wheel,
u
expected value of random value
distribution,
Rreliability,
VEX idler gear wheel,
Vvintroductory stretch of spinning flow,
xnominal value of coefficient of displacement of tooth profile (correction coefficient),
ziteeth number for standard cylindrical wheel,
znteeth number related to the transverse pitch diameter,
αn normal pressure angle,
αprobability estimated based upon
random value distribution,
β
helix angle,
ΔK coefficient of stress intensity,
σ
maximal stress in the crack tip,
σ 2y variation of random value y,
ϑauxiliary angle used for determination of tooth shape coefficient,
θ
working time of gear until damage,
ρhfradius of the transition curve (filet radius) for the tangential tooth
shape.
Rysiński J, Drobina R, Tomaszewski J. Probability of the Critical Length of a Fatigue Crack Occurring at the Tooth Foot of Cylindrical Geared Wheels of the Drive System of a Fiomax 2000 Ring Spinner.
FIBRES & TEXTILES in Eastern Europe 2017; 25, 1(121): 134-144. DOI: 10.5604/12303666.1227895
Introduction
cal ring spinning still has the dominant
role in the spinning sector of the textile industry due to the quality of yarn
achieved as well as its universality.
During the spinning process, one of the
essential factors influencing the production output is rupture of the spinning
flow. This phenomenon causes the most
essential problems within the whole
technology of yarn production because
it influences the production output, quality and number of defective products.
The breakage or rapture resistance of
yarn can be considered as one of the
most important indicators which characterise the technological and organisation
level of a spinning mill, and it comprises the basis for objective evaluation of
its work. The guarantee of breakage resistance is not an easy task due to the
complex mechanism of yarn breakage.
The number of ruptures is influenced by
the following factors [31-33]:
n yarn/thread tension and its variability,
n mass distribution of feeding and output product,
n strength of yarn,
n winding angle,
n mechanical defects of the stretching
apparatus and spindles,
n damage to driving system, especially
geared wheels,
n disruptions of operation of turning-winding system: shuttle-ring.
The propagation of a fatigue crack at the
tooth root of geared wheels in the ring
spinner drive system (i.e. in its basic elements) can generate damage to the whole
system or can decrease the quality of the
end-product. The propagation of cracks
reaching the critical length at the tooth
root of the cylindrical geared wheel is
connected with the phenomenon of exceeding the limit, and in consequence
instant fatigue breakage occurs, precluding further work of the machine, which
diminishes the output of the manufacturing process. In industrial practice, such
damage has to be eliminated immediately. Knowledge of phenomena occurring
during machine operation diminishes
the risk of a breakdown of the whole
machine [40]. Introductory initiation of
tooth cracks in geared wheels in the normal service conditions of a particular machine can remain unidentified. Damage at
the tooth foot – in rare cases – can happen
very rapidly, damaging other working elements of a particular machine.
In the textile industry, there are several
methods for yarn manufacture, depending on the structure of yarn required for
a special end-product. During the spinning process, the manufacturer aims for
achievement of a product of the highest
physical properties. In the case of ring
spinning the maximal ultimate rotational
velocity of a spindle is equal to approx.
25 000 rev/min, whereas in the case of
rotor spinning this velocity is equal to
150 000 rev/min. This essential increase
in the rotational velocity of machine
working elements has a negative influence on the reliability of the machine.
Moreover in the case of ring spinning,
the upper ultimate rotational velocity of
spindles is limited by the dimensions of
the machine working elements. Classi-
a)
The number of ruptures can be evaluated only when the working conditions of
yarn manufacturing are known as well
as when we can assume that rupture occurs if its tension exceeds its strength.
Unfortunately, although in industrial
conditions the average yarn tension does
not exceed the strength, yarn is subject
to rupture. Due to this fact, for evaluating the number of ruptures expected, it
is necessary to determine the variability
of yarn tension and strength. However,
b)
c)
the phenomenon of rapture itself should
be analysed by means of probabilistic
methods [34] i.e. based upon probability
theory [35]. Yarn ruptures are also influenced by the uneven work of the machine
drive system and changes in the velocity
of machine working elements caused especially by the structure of the cop. Other
reasons for wearing out machine working
elements in ring spinning are the rotation
of the cop in order to obtain yarn twist
and strains on construction elements of
the machine. Failure of the driving element of the spinning machine impacts the
quality of the yarn, leading to destruction
of the driving system in extreme cases. In
ring spinners, the stretching apparatus is
an essential element of the drive system.
Frequent changes in rotational velocity
have a direct influence on the dynamics
of loading changes. The stretching apparatus consists of a series co-operating
gears which are utilized for the setting of
stretches. Geared teeth are subject mainly
to high-cyclic fatigue phenomena.
Due to the cyclic loading of gear teeth
(entering in and losing contact), there is
a possibility of fatigue breakage of a particular tooth, which is usually connected
to exceeding (at the tooth) the bending
stresses of the foot – i.e. the so-called allowable/ultimate fatigue limit. Whereas
the cause of too high values of stresses
can be different e.g. a too low tooth module, over-loading, the notch effect as well
as defects of the transit surface (filet zone)
at the tooth foot. Tooth fatigue breakage
is usually initiated on the transit surface
and propagates along the arc placed along
the tooth foot. The described shape of the
crack forms invisibly on the outer surface
of the tooth surface perpendicular to its
main geometrical axis. Despite the propagation of the fatigue crack in its introductory phase, the operation of the gear
is still possible and allowable. However,
d)
Figure 1. Consecutive phases of propagation of a fatigue crack: a) initiation, b) propagation, c) critical/final crack which launches the process
of rapid growth of the crack, so- called full break of the tooth, d) fatigue crack at the tooth root after full break.
FIBRES & TEXTILES in Eastern Europe 2017, Vol. 25, 1(121)
135
Figure 2. Woehler
fatigue curve for
the bending tooth of
geared wheels.
Analysis of fatigue crack
propagation for the
sub-systems of the drive
system of a Fiomax 2000 ring
spinning machine
During design activities, the designers of
machines and devices, including textile
ones, pay special attention to immediate and fatigue strains and stresses [22].
However, nowadays, it is an insufficient
approach to the design task because the
user also considers such parameters of
a product as its fatigue life and reliability.
It is due to the fact that the last characteristics mentioned have a direct influence
on costs connected with operation and
maintenance procedures.
in the case of a continuously propagating
crack, after exceeding of the fatigue limit
or critical crack length, total breakage of
the tooth can happen. In the majority of
cases, the critical crack length of a crack
located at the tooth foot is approximately half of the tooth width (Figure 1,
see page 135). However, a detailed description of the situation depends on the
loading character and geometrical shape
of the toothing given via its technical
specification. Early performed diagnostics of the apparatus or whole drive system consisting in registration of damage
symptoms and their comparison with the
standard (benchmark) signals allows for
detection of incorrectness connected with
the machine operation routine. Such an
approach directly influences the fatigue
life of the machine.
Aiming for proper and continuous machine operation, it is very important to
operate it under recommended work conditions and according to the requirements
as well as regular performance of maintenance tasks (especially repairs). Frequent
checking of the drive and working systems of a spinner has a positive influence
on the reduction of costs connected with
the elimination of errors and defects occurring during the production process.
Evaluation of the technical state of spinning machines is usually performed
based on diagnostic investigations [1618], [23-26], [29]. These investigations
consists in different aspects e.g. analysis of vibrations [16, 23, 25], state of
surfaces of co-operating geared wheels
[17, 24], pitting [18, 24, 27] or scoring
[26-28]. Considering only the symptoms
of damage to machine elements is not
136
sufficient and in some cases can lead
to the opinion that the drive system is
evaluated as sufficient or working properly. However, non-identified damage
can appear suddenly, especially for such
elements as geared wheels, bearings and
shafts [19, 20, 22, 27, 28]. Therefore additional duties – besides possible damage identification via inspections – consist in the utilization of statistical and
probabilistic methods and modelling
routines [36, 37].
The characteristic feature of phenomena
taking place during the wear of working
elements of heavy textile machines (under variable loading conditions) is the
complete randomness of these processes.
In the case of the classic approach to evaluation of the fatigue life of working elements, mainly general statistical methods
are used. Advanced probabilistic models
are used very rarely due to really complicated mathematical modelling. Nevertheless in related references dedicated
to the maintenance of textile machinery
and devices, one can find probabilistic
models which are based on the following types of random value distributions:
exponential, Weibull, normal, Gumbel,
Ferecht, Reyleigh, Gamma and log-normal distributions [36]. The majority of
the distributions previously mentioned
imply the application of advanced numerical methods and techniques as well
as the conducting of investigations on
a large number of specimens. Therefore
during the identification of damage to
a particular system, at the beginning it
should be assumed that the probabilistic
model applied allows for description of
the phenomenon considered based on the
operational investigations conducted.
The fatigue life of the elements of textile machines and devices, especially
of geared wheels, is mainly determined
based on fatigue investigations. Based
on these experiments, one can derivate
a Woehler curve [30, 37, 38] which in
turn can be utilized for further calculations taking into account fatigue hypotheses e.g. Palmgren – Miners, Haibach,
Corten-Dolan or Szala [30, 37, 41]. In
the present paper, a method of fatigue life
determination based on fracture mechanics principles [37] has been described.
The application of the boundary element
method allowed to describe the propagation of a fatigue crack (a = f (N)). After inserting of this relationship into the
Paris – Erdogan formula (or other), the
fatigue life measured from the number of
loading cycles was obtained.
In the case of fatigue investigations of
geared wheels, we create fatigue charts
utilizing the bi-logarithmic co-ordinate
system lg σF – lg NF (Figure 2), where σF
is the bending stress (at the tooth foot) in
the case of pulsating loading (from zero
level), whereas NF is the relevant fatigue
life measured by means of the number of
cycles.
Based on this chart, one can determine the basic fatigue characteristics of
a geared wheel-specimen i.e. average
value of the unlimited fatigue strength of
the tooth foot related to bending stresses
σF lim (for a probability of damage equal
to 50%) according to the ISO 6336 standard. It is an approximate value because
it depends on the assumed basic number
of cycles. According to the ISO 6336
standard, for hardened steel, this value is
equal to NF lim = 3 · 106 cycles. The chart
discussed also allows determination
FIBRES & TEXTILES in Eastern Europe 2017, Vol. 25, 1(121)
Figure 3. Scheme of the drive system of the stretching apparatus of the Fiomax 2000 ring spinner [11].
of the average limited fatigue life, i.e.
σF > σF lim (for a probability of damage
equal to 50%) relevant to the fatigue life
assumed NF > NF lim. Moreover in the reverse approach it is also possible to determine the fatigue life NF for the assumed
level of stress σF.
(Figure 4.a), can exist in its internal volume e.g. defect of the crystal structure,
(Figure 4.b) or partly penetrate into the
material interior (Figure 4.c).
a)
In theoretical and practical considerations, the terms ‘crevice’, ‘rupture’ are
also considered. In so-called fracture mechanics, a ‘crack’ represents a real defect
b)
c)
Having determined the fatigue life (from
the beginning of crack initiation, throughthe reverse
approach
it ispossible
also possible
to determine
the fatigue
𝐹𝐹 <
𝑙𝑙𝑖𝑖𝑚𝑚 . Moreover
𝑁𝑁
<𝑁𝑁𝑁𝑁
.𝐹𝐹Moreover
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inthe
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the
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throughout
the
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until
damage),
itpossible
is possible
to determine
the reliability
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ispossible
todetermine
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ofthe
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until
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the
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drive
ing gear in the head of the ring spinner
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the Fiomax
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evenstretch:
stretch:
total
total
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total
even
stretch:
initial stretch, see Equation (2) and final𝐻𝐻𝑧𝑧
∙ 𝐵𝐵
𝐸𝐸𝑋𝑋
𝐻𝐻𝑧𝑧
∙ 𝐵𝐵
𝐵𝐵
(1) (1)
𝐸𝐸𝑋𝑋
𝐸𝐸𝑋𝑋 𝐸𝐸𝑋𝑋 = 22,199,
𝐻𝐻𝑧𝑧
ly, main stretch, see Equation (3).
𝐻𝐻𝑧𝑧
(1)
(1)
(1)
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𝐸𝐸𝑋𝑋
𝐻𝐻𝑧𝑧
𝐵𝐵
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𝐸𝐸𝑋𝑋
𝐺𝐺𝑣𝑣
𝐻𝐻𝑣𝑣
∙=0,3560
𝑉𝑉𝑣𝑣
= 0,3560
∙ ∙∙ ∙𝐵𝐵
(1)
𝐸𝐸𝑋𝑋
𝐸𝐸𝑋𝑋
𝐺𝐺𝑣𝑣=
=𝐻𝐻𝑣𝑣
𝐻𝐻𝑣𝑣=
∙ 𝑉𝑉𝑣𝑣
𝑉𝑉𝑣𝑣=
0,3560
=22,199,
22,199,
𝐺𝐺𝑣𝑣
==
𝐻𝐻𝑣𝑣
𝑉𝑉𝑣𝑣
==
=
22,199,
𝐺𝐺𝑣𝑣
𝐻𝐻𝑣𝑣
0,3560
=
22,199,
𝑁𝑁𝑤𝑤
𝐺𝐺𝑣𝑣
∙∙ ∙𝑉𝑉𝑣𝑣
0,3560
∙∙ ∙∙ 𝑁𝑁𝑤𝑤
=
𝑁𝑁𝑤𝑤
𝐸𝐸𝑋𝑋 𝐸𝐸𝑋𝑋
𝑁𝑁𝑤𝑤
𝐸𝐸𝑋𝑋
𝑁𝑁𝑤𝑤
𝐸𝐸𝑋𝑋
𝐸𝐸𝑋𝑋
During the operation of machines, the
stretch:
initial
stretch:
initialinitial
stretch:
initial
stretch:
initial
stretch:
initial
stretch:
geared wheels are subjected to variable
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼𝐸𝐸𝑋𝑋·∙ ∙B𝐵𝐵
∙ 𝐵𝐵
shaft
II
𝐸𝐸𝑋𝑋
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼∙·∙𝐹𝐹∙F𝐹𝐹
𝐹𝐹EX
∙𝐹𝐹EX
𝐵𝐵
(2) (2)
loading which can cause damage, espe𝐸𝐸𝑋𝑋 𝐸𝐸𝑋𝑋 ,
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
(2)
(2)
𝐸𝐸𝑋𝑋
𝐸𝐸𝑋𝑋
𝐸𝐸𝑋𝑋∙ ∙𝐵𝐵𝐵𝐵
𝐸𝐸𝑋𝑋
𝑉𝑉𝑣𝑣
(2)
𝐸𝐸𝑋𝑋
𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼ł𝑒𝑒𝑘𝑘
→𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼 ∙ 𝐹𝐹𝐸𝐸𝑋𝑋
𝑉𝑉𝑣𝑣
=𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
, 𝐶𝐶
Vυ
=
𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼
→𝑤𝑤𝑎𝑎
𝐼𝐼𝐼𝐼𝐼𝐼=
𝑉𝑉𝑣𝑣
=
,
𝑉𝑉𝑣𝑣
=
,
shaft
II
→
shaft
III
𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼
→𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼𝐼𝐼
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼𝐼𝐼
∙
𝐸𝐸
∙
𝑉𝑉𝑣𝑣
,
𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼
→𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼𝐼𝐼
cially fatigue damage. The phenomena 𝑤𝑤𝑎𝑎 ł𝑒𝑒𝑘𝑘 𝐼𝐼𝐼𝐼 →𝑤𝑤𝑎𝑎 ł𝑒𝑒𝑘𝑘 𝐼𝐼𝐼𝐼𝐼𝐼 𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼𝐼𝐼∙∙· ∙𝐸𝐸
∙E𝐸𝐸𝐸𝐸𝑋𝑋
𝐸𝐸EX𝐸𝐸𝑋𝑋
𝐶𝐶𝐸𝐸𝑋𝑋
shaft𝐼𝐼𝐼𝐼𝐼𝐼
III
·∙∙C∙𝐶𝐶
𝐸𝐸𝑋𝑋
𝐸𝐸𝑋𝑋 𝐸𝐸𝑋𝑋
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼𝐼𝐼
𝐶𝐶𝐸𝐸𝑋𝑋
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼𝐼𝐼
𝐸𝐸
𝐶𝐶∙EX
𝐸𝐸𝑋𝑋
𝐸𝐸𝑋𝑋
𝐸𝐸𝑋𝑋
(2)
which occur most frequently are defects
31,5
∙
27
∙
𝐵𝐵
31,5∙∙ ∙27
∙ 27
27∙∙ ∙𝐵𝐵
∙ 𝐵𝐵
𝐵𝐵
𝐸𝐸𝑋𝑋 𝐸𝐸𝑋𝑋 = 0,0287 ∙ 𝐵𝐵
31,5
31,5
𝐸𝐸𝑋𝑋
31,5
27
𝐵𝐵
𝐸𝐸𝑋𝑋
𝑉𝑉𝑣𝑣
=
=
1,0906,
of the drive system e.g. fatigue cracks 𝑉𝑉𝑣𝑣
𝐸𝐸𝑋𝑋
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼ł𝑒𝑒𝑘𝑘
→𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐸𝐸𝑋𝑋
=25
0,0287∙∙ ∙𝐵𝐵
∙ 𝐵𝐵
𝐵𝐵
𝑉𝑉𝑣𝑣
=𝐼𝐼𝐼𝐼𝐼𝐼
=1,0906,
1,0906,
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼
→𝑤𝑤𝑎𝑎
𝐼𝐼𝐼𝐼𝐼𝐼=
𝐸𝐸𝑋𝑋=
0,0287
Vυ
𝑉𝑉𝑣𝑣
=
0,0287
1,0906,
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼
→𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼𝐼𝐼
𝐸𝐸𝑋𝑋
37=
𝑉𝑉𝑣𝑣
=
=∙=
0,0287
𝐵𝐵
==
1,0906,
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
ł𝑒𝑒𝑘𝑘𝐼𝐼𝐼𝐼𝐼𝐼
𝐼𝐼𝐼𝐼𝐼𝐼
𝐸𝐸𝑋𝑋
shaft
II→𝑤𝑤𝑎𝑎
→ shaft
III
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼
→𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐸𝐸𝑋𝑋
32,0∙∙ 32,0
∙ 37
37∙∙ ∙25
∙∙25
25
32,0
37
25
32,0
∙37
near the tooth foot.
32,0
and finally,
main
stretch:
andfinally,
finally,
main
stretch:
and
finally,
main
stretch:
and
finally,
main
stretch:
and
finally,
main stretch:
and
main
stretch:
As a crack, one considers this imper𝑆𝑆𝐸𝐸𝑋𝑋∙ ∙𝐸𝐸
∙ 𝐸𝐸
∙ 𝐻𝐻𝑧𝑧𝐸𝐸𝑋𝑋
𝐸𝐸𝑋𝑋 𝐸𝐸𝑋𝑋
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
∙ 𝑉𝑉
𝑉𝑉𝐸𝐸𝑋𝑋
∙𝑉𝑉
𝐸𝐸𝐸𝐸𝑋𝑋
∙ 𝐻𝐻𝑧𝑧
𝐻𝐻𝑧𝑧
(3) (3)
𝐸𝐸𝑋𝑋∙∙ ∙𝐻𝐻𝑧𝑧
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝑆𝑆𝑆𝑆𝑆𝑆∙ ∙𝑉𝑉
shaft𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐸𝐸𝐸𝐸𝑋𝑋
(3)
(3)
𝐸𝐸𝑋𝑋
𝐸𝐸𝑋𝑋
𝐸𝐸𝑋𝑋
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
∙ ∙𝐸𝐸
𝐻𝐻𝑧𝑧
𝐸𝐸𝑋𝑋
𝐸𝐸𝑋𝑋 ,
𝑉𝑉𝑣𝑣
,
(3)
fection of the material structure to have 𝑉𝑉𝑣𝑣
𝐸𝐸𝑋𝑋
𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘 𝐼𝐼ł𝑒𝑒𝑘𝑘
→𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼 = 𝑆𝑆 ∙ 𝑉𝑉𝐸𝐸𝑋𝑋
𝑉𝑉𝑣𝑣
=
𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐼𝐼
→𝑤𝑤𝑎𝑎
𝐼𝐼𝐼𝐼
=
,
𝑉𝑉𝑣𝑣
=
,
Vυ
𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐼𝐼
→𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼𝐼𝐼
∙
𝐸𝐸
∙
𝐶𝐶
𝑉𝑉𝑣𝑣 𝑤𝑤𝑎𝑎 ł𝑒𝑒𝑘𝑘𝐼𝐼 𝐼𝐼→𝑤𝑤𝑎𝑎
,
→𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼II=
I →ł𝑒𝑒𝑘𝑘
shaft
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼𝐼𝐼∙∙ ∙𝐸𝐸
∙ 𝐸𝐸𝐸𝐸𝑋𝑋
𝐸𝐸𝐸𝐸𝑋𝑋
𝐶𝐶𝐸𝐸𝑋𝑋
𝐸𝐸𝑋𝑋∙∙ ∙𝐶𝐶
𝐸𝐸𝑋𝑋 𝐸𝐸𝑋𝑋
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼𝐼𝐼
𝐸𝐸
𝐶𝐶∙ 𝐶𝐶𝐸𝐸𝑋𝑋
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼𝐼𝐼
shaft𝐼𝐼𝐼𝐼𝐼𝐼
a particular size and shape. When the 𝑤𝑤𝑎𝑎 ł𝑒𝑒𝑘𝑘 shaft
𝑤𝑤𝑎𝑎ł𝑒𝑒𝑘𝑘
𝐸𝐸𝑋𝑋
(3)
𝐸𝐸𝑋𝑋
𝐸𝐸𝑋𝑋
32,0
∙
85
∙
43
∙
37
∙
𝐻𝐻𝑧𝑧𝐸𝐸𝑋𝑋
𝐻𝐻𝑧𝑧𝐸𝐸𝑋𝑋
material is loaded, the crack surfaces can
32,0
∙
85
∙
43
∙
37
∙
𝐻𝐻𝑧𝑧
𝐻𝐻𝑧𝑧
𝐸𝐸𝑋𝑋
𝐸𝐸𝑋𝑋
32,0
∙
85
∙
43
∙
37
∙
𝐻𝐻𝑧𝑧
𝐻𝐻𝑧𝑧
32,0
∙ 85 ∙ 43 ∙ 37 ∙ 𝐻𝐻𝑧𝑧𝐸𝐸𝑋𝑋𝐸𝐸𝑋𝑋 = 12,390
𝐻𝐻𝑧𝑧𝐸𝐸𝑋𝑋
𝐻𝐻𝑧𝑧
𝐸𝐸𝑋𝑋
𝑉𝑉𝑣𝑣
= 12,390
∙ ==20,33.
= 20,33.
𝐸𝐸𝑋𝑋
ł𝑒𝑒𝑘𝑘 𝐼𝐼ł𝑒𝑒𝑘𝑘
→𝑤𝑤𝑎𝑎
𝐼𝐼𝐼𝐼 =∙ 85 ∙ 43 ∙ 37 ∙ 𝐻𝐻𝑧𝑧𝐸𝐸𝑋𝑋
𝑉𝑉𝑣𝑣
=32,0
20,33.
𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘𝐼𝐼𝑤𝑤𝑎𝑎
𝐼𝐼 →𝑤𝑤𝑎𝑎
→𝑤𝑤𝑎𝑎
𝐼𝐼𝐼𝐼=
=
𝑉𝑉𝑣𝑣
=
12,390∙∙ ∙∙ 𝑁𝑁𝑤𝑤
20,33.
open or be displaced in relation to each 𝑉𝑉𝑣𝑣
𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
→𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐼𝐼𝐼𝐼II𝐼𝐼𝐼𝐼
21∙ 22
∙∙ 24
∙∙ 22
∙ 𝑁𝑁𝑤𝑤
𝑁𝑁𝑤𝑤
𝑉𝑉𝑣𝑣
=ł𝑒𝑒𝑘𝑘
==12,390
12,390
==20,33.
𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
ł𝑒𝑒𝑘𝑘𝐼𝐼𝐼𝐼
Vυ
𝐸𝐸𝑋𝑋
𝐸𝐸𝑊𝑊
𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
𝐼𝐼 𝐼𝐼→𝑤𝑤𝑎𝑎
ł𝑒𝑒𝑘𝑘
31,5∙∙ 31,5
∙ 21
21∙∙ ∙24
∙∙24
24
22∙∙ ∙𝑁𝑁𝑤𝑤
𝑁𝑁𝑤𝑤
shaft
I → shaft
𝐸𝐸𝑋𝑋
𝐸𝐸𝑊𝑊
31,5
21
24
𝑁𝑁𝑤𝑤
𝑁𝑁𝑤𝑤
31,5
∙21
∙22
22
𝑁𝑁𝑤𝑤
𝑁𝑁𝑤𝑤
𝐸𝐸𝑋𝑋
𝐸𝐸𝑊𝑊
31,5
∙
𝑁𝑁𝑤𝑤
𝐸𝐸𝑋𝑋
𝐸𝐸𝑊𝑊
𝐸𝐸𝑋𝑋
𝐸𝐸𝑊𝑊
other, whereas in the unloaded state these
surfaces can contact each other. Cracks
(1),
can penetrate throughout an element Equation
F
VBL
GBL (2) and (3).
VBL
GBL
F F
FF
GBL
GBL
GBL
FIBRES & TEXTILES in Eastern Europe 2017, Vol. 25, 1(121)
V
V
VV
Hz
Hz
HzHz
S S
S
B BB
B
Hz
B
V
E
E
EE
A
A
AA
C
C
C
Wałki z cholewką
C C Wałki
Wałki
z cholewką
cholewką
Wałki
z cholewką
Wałki
zz cholewką
VBL
VBL
VBL
E
E
EE
E
A
A AA
A
E
V
V
VV
A
137
V
Hz
Hz
HzHz
Hz
S S
S
a)
b)
Figure 5. a) Crack ρ ≠ 0, b) crevice ρ = 0.
inside the material. A crevice has a theoretically radius of curvature equal to zero
at its tip (Figure 5).
As a cracking process, we consider the
growth of a crack to consist in an increase in its characteristic dimension i.e.
length. Regarding the front (tip) of the
crack, we consider its edge to be inside
the material.
In general, three characteristic models of
crack propagation – proposed by Irwin
– are considered, i.e. a crack of mode
I (opening) excited by tension stress,
a crack of mode II (longitudinal shear)
excited by shear loading acting parallel
to the cracking plane and perpendicular
to the crack front, as well as a crack of
mode III (tangential shear) excited by
shear loading parallel to the crack front
(Figure 6).
Figure 6. Models of crack propagation: mode I – crack of opening, mode II – crack of
(in-plane) longitudinal shear, mode III – crack of (out-of-plane) tangent shear.
Figure 7. Point defects of crystals.
Figure 8. Linear defects in material (dislocations) [9].
138
The main factors which have an essential influence on the crack direction and
propagation are the loading and material
structure. The process of crack initiation
can take place in the following structure
forms [39]:
n slip bands,
n boundaries of grains,
n undersurface inclusions (very rare).
Other indicators like heat-chemical and
mechanical treatment of material surfaces, the work environment and the variability of loading also have an crucial
influence.
Depending on the material structure, the
causes of cracks arising/occurring can be
divided into following types:
n point defects (Figure 7) – in fact, one
can distinguish four types of defects:
1 – vacancies (empty place –Schotthy’s defect), 2 – inter-node atom,
3 – improper atom (inter-node), 4 –
improper atom in the exact structure
node,
n dislocations (Figure 8) – along an
edge (the plane of the additional plane
exists in a crystal volume), screw
dislocations (i.e. defect of the crystal
structure caused by the displacement
of part of the crystal volume around
the particular axis).
Cracks passing along the grain boundaries frequently exist in the case of high
stress amplitudes and at high temperatures. Sometimes even a relatively moderate force can cause that the inter-particle bonds are subject to damage. Weak
bonds caused that molecular crystals are
FIBRES & TEXTILES in Eastern Europe 2017, Vol. 25, 1(121)
susceptible to deformations. An exemplary crack propagation along the grain
boundaries is presented in Figure 9.
a)
Exemplary scenarios of crack propagations [10] are shown in Figure 10. Three
curves representing crack propagation
are drawn as a function depending on
the following ratio: the number loading
cycles N nin relation to the fatigue life
N (until breakage). The points on the abscissa axis are related to the percentage,
whereas the axis of ordinates represent
an increase in the crack length (semi-log
coordinate system).
As can be noticed, crack initiation is not
always connected with its propagation.
Similarly not every defect or initiated
crack could be identified or detected.
Exemplary models of crack propagation
are presented in Figure 11.
High stress concentration appears around
the crack tip. Crack propagation is caused
by an outbreak of slip planes, whereas
the propagation and length increase of
the crack occurs in the direction of the
presence of tangent stresses (Figure 11,
level 1 and 2). Under active loading, the
crack opens consecutively and an increase in its length occurs simultaneously
Δa (Figure 11, level 3). Strengthening of
the material and increasing stresses cause
blunting of the crack tip (Figure 11,
level 4). During one cycle of loading,
the crack propagates by Δa. Around the
elastic surrounding of the crack tip, small
plastic deformations occur. An increase
in plastic deformations takes place along
with an increase in loading. Assuming
that the acting load has a variable character, then due to changes in the directions
b)
Figure 9. Loss of cohesion of material: a)trans-crystal (crack crossing through grains); b) intercrystal (crack going along grain boundaries) [9].
Figure 10. Different scenarios of fatigue crack propagation [10].
Figure 11. Exemplary model of crack propagation [9].
FIBRES & TEXTILES in Eastern Europe 2017, Vol. 25, 1(121)
139
crack model considered (Fig. 6), one should distin
crack opening - 𝐾𝐾𝐼𝐼 , crack under longitudinal shea
shear - 𝐾𝐾𝐼𝐼𝐼𝐼𝐼𝐼 . In general, the stress intensity coe
formula:
Table 1. Equivalent materials for 20HNM steel – utilized for manufacturing of the geared
wheel specimens.
Poland
EU
USA
France
Japan
China
Russia
20HNM
1.6523
8617
20NCD2
SNCM220
20CrNiMo
20HGNM
Table 2. Parameters of the geared wheels investigated [11].
Normal module
mn
2.0
Pressure angle
αn
20 deg
Number of teeth
z1
39
Helix angle
β
0 deg
Correction coefficient
x
0
Radius of reference tooth shape
ρhf
0,5
Addendum height
hfp
2,5
(5)
where:
𝜍𝜍 external
the crack
where:
σ external
stressstress
at theatcrack
tip, tip, 𝛼𝛼 – p
α – parameter depending on the shape of
specimen and crack geometrical form.
the specimen and crack geometrical form.
𝐾𝐾 = 𝜍𝜍 ∙ 𝛼𝛼 ∙ 𝜋𝜋 ∙ 𝑙𝑙,
In the present paper, we utilise the formula for determination of the range of the
In the
present
paper,
we utilise
thea funcformula for deter
stress
intensity
coefficient
ΔK as
tion
of
the
range
of
the
relevant
stress,
intensity coefficient ∆𝐾𝐾 as a function of the range of
i.e.: Δσ = σF > ΔσF lim and the ultimate
∆𝜍𝜍𝐹𝐹length
ofthe
theultimate
crack l: length of the crack 𝑙𝑙:
𝑙𝑙𝑖𝑖𝑚𝑚 and
∆𝐾𝐾 = ∆𝜍𝜍 ∙ 𝜋𝜋 ∙ 𝑎𝑎 = 𝜍𝜍𝐹𝐹 ∙ 𝜋𝜋 ∙ 𝑎𝑎. (6)
Coefficients 𝐶𝐶 and 𝑚𝑚 – which appear in formula (4) – ar
Coefficients C and m – which appear in
determined
via
investigations.
formula
(4)experimental
–stresses
are material-based
conlevel 3). Strengthening of the material
and
increasing
cause
blunting
of the crack tip
Table 3. Constants C and m for the Paris model determined for the experimental data shown
stants determined via experimental in(Fig. 11, level 4). During one cycle of loading, the crack propagates by a . Around the elastic
in Figure 12.
Thevestigations.
parameters of the model were determined based on t
Test no.
1
2
3
4
5 crack tip,
6 small plastic deformations occur. An increase in plastic
surrounding
of the
wheel-specimens made of 1.6523 (20 HNM) steel. Simi
C × 1011
4.724
4.213
0.172
0.178
0.349
1.323
The parameters of the model were deterdeformations
takes
place
along
m
1.913
1.929
2.472
2.449
2.325
2.073
in different
countries
areresults
listed in
1 for comparison
mined based
on the
of Table
investigaTest no.
7
8
9
10 increase in
11 loading. Assuming
12
with an
that
the
acting
load
has
a
variable
character,
then due
tions of geared wheel-specimens made of
C × 1011
5.149
5.562
2.645
3.688
1.079
1.523
1.6523
(20
HNM)
steel.
Similar
(equivto changes in the directions of the acting force, the closing of edges near the crack tip takes
m
1.872
1.848
2.364
1.928
2.132
2.041
Table
1. Equivalent
materials for
alent)
materials considered
in 20HNM
different steel – utiliz
place as well as the arising of plastic deformations
(Fig.
11, in
level
5). 1 for comcountries
are
listed
Table
wheel specimens
parison.
Polska
UE
USA
France
Jap
Table 4. Constants of Paris law determined for the empirical
data
in Figure
13.of the stretching gear of the Fiomnax 2000 ring spinner
Prediction
of shown
the working
time
Technical specifications i.e. geometri20HNM 1.6523
8617
20NCD2 SNCM
Test no.
1
2
3
4
5
6
cal parameters
of the
geared
Usually
the
detection
of
a
fatigue
crack at the
tooth
foot wheels.
impliesare
the immediate
C × 1011
1.381
1.081
0.908
0.573
0.437
0.437
listed inspecifications
Table 2. Within
the framework
Technical
i.e.
geometrical
parameters
of isth
switching
the drive system.
The sudden suspension of the technological process
m
2.1
2.14
2.168
2.24 off of2.28
2.28
of the experiments, a pair of idler gear
2. Within
the
framework
ofspinner
the
pair of idl
Test no.
7
8
9
10
11
12
wheels
ofTherefore
a FIOMAX
wereforutifrequently
connected
with high
economic
losses.
there
is aexperiments,
need
theapreparation
C × 1011
0.282
0.240
0.240
0.221
0.211
0.172
lised. The adequate
tooth numbersnumbers
were were as f
were
Thea adequate
of methodology
for determination
of
the utilised.
time when
particular tooth
crack reaches the
critical
m
2.34
2.36
2.36
2.37
2.395
2.395
as follows: z1 = 39 and z2 = 64 [11], rerespectively.
The
wheels
were
subjected
to
heat
tre
length. The aims of these attempts are the
assuranceThe of continuous
technological
spectively.
wheels were
subjected processes as
heat (reaching
treatment,hardness:
carburization,
58 ± nitra2 HRC). Investig
well as minimisation of losses due tohardening
thetosudden
suspension
of
production.
tion and hardening (reaching hardness:
of the acting force, the closing of edges (for crack growing at tooth foot ) is de- pulsating
test reaching
machine,the
at acritical
maximal
pressure
equa
A by
model
which
prediction
of the58 time
crack
length force
(for crack
± 2 for
HRC). Investigations
were
pernear the crack tip takes place as well as rived
solving
theenables
modified
Paris law
loading
applied
aParis
constant-amplitude,
sinusoidal cour
formed
ona had
hydro-pulsating
test mathe arising of plastic deformations (Fig- [1, 12]:
growing at tooth foot ) is derived by solving
the
modified
law [1, 12]:
chine,
at
a maximal
pressure
force
equal
ure 11, level 5).
Table 2. Parameters of the geared wheels investigated [11
𝑑𝑑𝑙𝑙
(4)
𝑚𝑚
= 𝐶𝐶 ∙ ∆𝐾𝐾 𝑙𝑙
. (4) to 150 kN. The experimental tooth load𝑑𝑑𝑁𝑁
Normal
module
ing applied had a constant-amplitude, si Prediction of the working time
nusoidal
course of 30 Hz frequency [38].
Pressure
angle
of the stretching gear of the
Relationship (4) is a commonly apFiomnax 2000 ring spinner
(4) is a commonly
applied
equation
the velocitywere
of fatigue crack
Number
of teethdescribing
pliedRelationship
equation describing
the velocity
The following
measurements
crack Paris,
propagation.
in performed:
Usually the detection of a fatigue crack of fatigue
measurements
of fatigue
propagation.
in his Paris,
publication,
stated
of cracking
depends on stress
Helix
anglethat velocity
at the tooth foot implies the immedi- his publication, stated that velocity of crack propagation for 12 teeth of the
intensity coefficient 𝐾𝐾. According to this concept,
crack propagation is governed by the
coefficient
ate switching off of the drive system. cracking depends on stress intensity co- Correction
wheel-specimen,
And the loading actK. According
to this in
concept,
The sudden suspension of the technolog- efficient
variability
of local stresses
the crack
tip,
whereas
coefficient
𝐾𝐾 describes
the effect of load
ing
in
the
direction
to the pitch
Radius of reference toothnormal
shape
ical process is frequently connected with crack propagation is governed by the diameter, which caused a bending stress
action and the stress field in the area around (neighborhood of) the tip. Depending on the
high economic losses. Therefore there is variability of local stresses in the crack Addendum
equal to height
1500 MPa. Measurement of the
model
considered
(Fig. 6), one
distinguish
adequate
intensity
whereas
coefficient
K describes
the should
a need for the preparation of methodol- tip, crack
fatigue
crack length
were stress
performed
by coefficients:
ogy for determination of the time when effect of load action and the stress field means of a laboratory microscope, after
crack opening - 𝐾𝐾𝐼𝐼 , crack under longitudinal shear4 - 𝐾𝐾𝐼𝐼𝐼𝐼 and for a crack relevant to tangent
of) Theevery
a particular crack reaches the critical in the area around (neighborhood
5×10measurements
cycles from the
moment
of measurem
following
were
performed:
general,
the model
stress intensity
coefficient 𝐾𝐾 is expressed by the following
- 𝐾𝐾𝐼𝐼𝐼𝐼𝐼𝐼 . In on
tip. Depending
the crack
length. The aims of these attempts are the shear
crack detection.
12 teeth of the wheel-specimen, And the loading acting
the assurance of continuous technologi- considered
formula:(Figure 6), one should discal processes as well as minimisation of tinguish adequate stress intensity coeffi- In Figure 12, curves of growth of the fa(5)
losses due to the sudden suspension of cients: crack
𝐾𝐾 =opening
𝜍𝜍 ∙ 𝛼𝛼 ∙–𝜋𝜋K∙ I𝑙𝑙,, crack under tigue crack depending on the number of
longitudinal shear – KII and for a crack loading cycles are shown.
production.
where: 𝜍𝜍 external stress at the crack tip, 𝛼𝛼 – parameter depending on the shape of the
relevant to tangent shear – KIII. In genthe stressand
intensity
coefficient K
is Based on an analysis of the data shown in
A model which enables prediction of the eral,specimen
crack geometrical
form.
time for reaching the critical crack length expressed by Equation (5):
Figure 12, one can state that it is possible
140
FIBRES & TEXTILES in Eastern Europe 2017, Vol. 25, 1(121)
Strona 13 z 26
6
2.5
5
Coefficient C x10-11
Crack length, mm
2
1.5
1
0.5
4
3
2
1
0
200
300
400
500
600
700
800
Number of cycles x 10
900
3
1000
0
1.8
1100 1200
11
1.9
4.724
𝐶𝐶 × 10
2
2.1
2.2
Exponent, m
4.213
0.172
2.3
0.178
2.4
2.5
0.349
1.323
2.472
2.325
𝑚𝑚 cracks at the 1.913
Figure 12. Curves representing growth of fatigue
Figure 13.1.929
Approximation
function of2.449
C and m summarized
in 2.073
tooth foot for 12 geared wheels loaded at momentTest
M =no.
750 MPa, 7Table 3.
8
9
10
11
12
manufactured of 1.6523 (20HNM) steel.
11
5.149
5.562
2.645
3.688
1.079
1.523
𝐶𝐶 × 10
4.724 1.872 4.213 1.848 0.172 2.364 0.178 1.928 0.349 2.132 1.323 2.041
𝐶𝐶 × 1011
𝑚𝑚
1.913
1.929
2.472
2.449
2.325
2.073
𝑚𝑚
0.4
Test no.
7
8
9
10
11
12
0.35
In Fig. 13, a chart of the
relationship of constant 𝐶𝐶 against exponent 𝑚𝑚 is shown. It was made
5.149
5.562
2.645
3.688
1.079
1.523
𝐶𝐶 × 1011
0.3data gathered in Table 3.
utilizing 1.872
the experimental
1.848
2.364
1.928
2.132
2.041
𝑚𝑚
2
1.8
1.6
Relative frequency
Crack length, mm
1.4
1.2
0.2
In Fig. 13, a chart of the relationship
of constant 𝐶𝐶 against exponent 𝑚𝑚 is shown. It was made
1
0.8
utilizing the experimental data0.15
gathered in Table 3.
0.6
0.4
0.1
0.05
0.2
0
0.25
0
1
2
3
4
5
Number of cycles x 105
6
7
8
Figure 14. Comparison of fracture curves obtained based on the
solution of Equations (8) and (4) with empirical data relevant to
test no 3.
0
4
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
6
6.2
Number of cycles x 105
Figure 15. Bar chart (histogram) of times of reaching the critical value by
the fatigue crack for 1.6523 (20HNM) steel and stresses at the tooth foot
σF = 1500 MPa in the drive system of a Fiomax 2000 spinner.
Based on an analysis of the chart, one can Furthermore utilizing Equation (6) for
Fig.constants
13. Funkcja
aproksymująca
materiałowe
i 𝑚𝑚empirical
zestawione
w tabeli
approximation
of𝐶𝐶the
data,
we 3
conclude that
C and
m depend stałe
on one another and the adequate relation- obtained the constants of the model –
gathered
in Table
4.
ship
can be
by of
means
of a parBased
onexpressed
an analysis
the chart,
one can
conclude
that constants
𝐶𝐶 and 𝑚𝑚 depend on one
ticular function. For the data presented in
another
adequate relationship
canInbeFigure
expressed
by means
of a particular
function. For
are exemplary
Figure
13,and
thethe
approximation
relationFig. 13.
Funkcja
aproksymująca
stałe materiałowe
𝐶𝐶14,
i 𝑚𝑚there
zestawione
w tabelisolu3
tions
of
Equations
(8)
and
(4)
related
to
ship
between
C
and
m
is
proposed
in
the
the data presented in Fig. 13, the approximation relationship between 𝐶𝐶 and 𝑚𝑚 is proposed in
the data collected during test no 3, prefollowing form:
Determination of constants Based on
analysisform:
of the chart, one can conclude that constants 𝐶𝐶 and 𝑚𝑚 depend on one
thean
following
pared according to the items shown in
in Paris-Erdogan equation
Tables 3byand
4, respectively.
Based
on an For
(7)
another and the
adequate
be expressed
means
of a particular
function.
𝐶𝐶 𝑚𝑚
= 𝑎𝑎0 ∙relationship
𝑚𝑚𝑎𝑎 1 + 𝑎𝑎2. can
(7)
analysis of the charts, we can draw−12
the
In the second phase of investigations,
−8
the dataWhere,
presented
in Fig. 13,𝑎𝑎 the
between
and 𝑚𝑚 is
in been
coefficients
= approximation
7,49 × 10 , 𝑎𝑎relationship
&𝑎𝑎 =𝐶𝐶−1,675
× proposed
10
have
1 = −11,435,
that the2application of Equaconstants C and m for the Paris model (4) Where, coefficients a0 =0 7.49 × 10–8, conclusion
−12
form:&bya2means
tion (8) givesofsome
errors insquare
comparison
a1 =determined
−11.435,
= −1.675
× 10
haveof minimization
of the
method
the average
error between the
were established based on experimentalthe following
determined 𝑎𝑎by
means
of
the
method to empirical data describing the growth of
results. These constants were derived been
1
𝐶𝐶data
𝑚𝑚 given
= 𝑎𝑎0 ∙in
𝑚𝑚 Table
+ 𝑎𝑎2.
3 and those calculated utilizing the approximation function (7).(7)
Inserting
of the average
square er- the fatigue crack in time. If the points of
utilising the modified method of definite of minimization
−8
−12
coefficients
𝑎𝑎0 (4),
=given
7,49
×Table
10 the
,3𝑎𝑎and
= −11,435,
= −1,675
× approximation
10
have been
(7)
we in
obtain
equation:
chart
14 &𝑎𝑎
are2placed
on the
rorformula
between
theinto
data
integral [2]. The results obtained are pre-Where,
1following
curve,
then
we
obtain
the
compatibility
those
calculated
utilizing
the
approximasented in Table 3.
𝑑𝑑𝑙𝑙
𝑎𝑎 1 method of minimization
determined by means
the
the average square error between the (8)
= (𝑎𝑎 of∙ 𝑚𝑚
+ 𝑎𝑎2 ) ∙ (𝛥𝛥𝐾𝐾 (7)
𝑙𝑙 )𝑚𝑚 . of theofcrack
growth chart with the empirtion function
𝑑𝑑𝑁𝑁 (7). 0Inserting Equation
given(4),
in we
Table
3 andthethose
calculated
utilizing
function
(7). Inserting
ical the
data.approximation
A characteristic
phenomenon
is
obtain
following
EquaIn Figure 13, a chart of the relationshipdata into
the
incompatibility
of
data
determined
by
tion:
of constant C against exponent m isformula (7) into (4), we obtain the following equation:
Furthermore utilizing formula (6) for approximation
of the empirical
we obtained the
means of Equation
(8) withindata,
the midshown. It was made utilizing the exper𝑑𝑑𝑙𝑙
𝑎𝑎 1
𝑚𝑚
(8)
=
(𝑎𝑎
∙
𝑚𝑚
+
𝑎𝑎
)
∙
(𝛥𝛥𝐾𝐾
𝑙𝑙
)
.
dle
range
of
the
process
of
crack
growth,
(8)
imental data gathered in Table 3.
0 of the model
2
constants
- gathered in Table 4.
𝑑𝑑𝑁𝑁
to determine the working time of a wheel
(of the drive system) as a function of
crack growth (near tooth foot) depending
on the loading excited. For a crack length
greater than 2 mm, accelerated fatigue
crack growth was observed.
FIBRES & TEXTILES in Eastern Europe 2017, Vol. 25, 1(121)
141
Strona
Furthermore utilizing formula (6) for approximation of the empirical data, we obtained
the 16 z 26
constants of the model - gathered in Table 4.
most utilised geared wheels in the stretching ap
distribution density function given in the standard [3]:
years.
𝑓𝑓 𝑦𝑦 = 𝛼𝛼 ∙ 𝑒𝑒𝑥𝑥𝑝𝑝 −𝛼𝛼 ∙ 𝑦𝑦 − 𝑢𝑢 − 𝑒𝑒 −𝛼𝛼∙ 𝑦𝑦 −𝑢𝑢 .
Utilizing the commercial package Statistica, it
Central moments of the distribution (mean value and v
which approximates the bar chart (histogram) is
formulas:
distribution density function
given in the standa
the whereas
manufacture
relating
the assortment
and thickness
yarns, and
the minimal
change of the
in theprofile
final range
thetoempirical
cal crack
length lkr =of2,4 mm
𝛿𝛿
𝜋𝜋 2
2
(11)
𝑚𝑚𝑦𝑦 = 𝑢𝑢 + ; 𝜍𝜍𝑦𝑦 = 2 ,
model-based
overlap
other. apparatus
crack length
= 0,05
mm.
𝑓𝑓 𝑦𝑦𝛼𝛼 = 𝛼𝛼 ∙ 𝑒𝑒𝑥𝑥𝑝𝑝6∙𝛼𝛼
−𝛼𝛼 ∙ 𝑦𝑦 − 𝑢𝑢 − 𝑒𝑒 −𝛼𝛼∙ 𝑦𝑦 −𝑢𝑢
min be
mostand
utilised
geared data
wheels
in theeach
stretching
has lto
made
at least once per two
In industrial practice, an engineer is interwhere:
δ –δ Euler
constant,
to
𝛿𝛿 ≈ to
0,577.
where:Central
– moments
Euler
constant,
equal
ofequal
the distribution
(mean valu
years.
ested in the time when the crack reaches Based on an analysis of the data shown
δ
≈
0.577.
The distribution
the ultimate
size, after package
which final
tooth it in
thestated
chart, that
one the
can most
state suitable
that the longer
formulas: parameters (8) for 1.6523 (20 HNM)
Utilizing
the commercial
Statistica,
was
distribution
4
4
-5
breakage takes place. Thus one could the operating time of the geared wheel, the σyThe , u = 4.434·10
,α=
.
distribution
parameters
(8)3.605·10
𝛿𝛿
𝜋𝜋 2 for
which
approximates
asymptotic
Gumbel’s
distribution,
having the= 3.557·10
𝑚𝑚𝑦𝑦 = 𝑢𝑢 + ; 𝜍𝜍𝑦𝑦2 = 2 ,
utilize
Equation the
(8) bar
duechart
to the(histogram)
compat- islower
its fatigue
life. The highest fatigue
𝛼𝛼
6∙𝛼𝛼 fol1.6523
(20
HNM)
steel
were
as
i.e. the probability
that after the
passage of N
ibility ofdensity
resultsfunction
with empirical
in life[3]:
was achieved in the following interval Reliability
5
4
distribution
given in data
the standard
lows: where:
my = 4.594·10
σy = 3.557·10
, 𝛿𝛿 ≈ 0,577.
δ – Euler, constant,
equal
to
5
the upper range of the crack growth chart
(4.3 – 4.5) x 10 cycles. The scatter is con- critical
-5
length 4𝑙𝑙,𝑘𝑘𝑟𝑟α –= is3.605·10
expressed
u = 4.434·10
. by the following form
𝑓𝑓 𝑦𝑦 of=fracture).
𝛼𝛼 ∙ 𝑒𝑒𝑥𝑥𝑝𝑝 −𝛼𝛼 ∙ 𝑦𝑦 − 𝑢𝑢 − 𝑒𝑒 −𝛼𝛼∙ 𝑦𝑦 −𝑢𝑢 nected,
.
(curve
as usual, with the propagation of (10)
The distribution
parameters (8) for 1.6523 (20
𝑦𝑦 1
.
𝑅𝑅 = 0 𝑓𝑓 𝑦𝑦 𝑑𝑑𝑦𝑦
or a particular
damage mechanism.
4 afCentral moments of the distribution (mean valuedefects
and variance)
are expressed
by the following
Reliability
i.e. the 4,probability
that
σy = 3.557·10
u = 4.434·10
, α = 3.605·10-5.
ter
the
passage
of
N
cycles,
a particular
i
Prediction of the number
formulas:
the probability
In practice, it means that the recommend- crack Reliability
reaches thei.e.
critical
length lkr that
– isafter the pass
of fatigue
cycles
acting
2
𝛿𝛿
𝜋𝜋
ed change of the geared wheel should be Verification
of
the
numerical
model
2
expressed
by the
following
𝑚𝑚on the geared
= 2 of
,
(11)
critical
length
𝑙𝑙𝑘𝑘𝑟𝑟 – isformula:
expressed by the followin
𝑦𝑦 = 𝑢𝑢 + 𝛼𝛼 ; 𝜍𝜍𝑦𝑦wheel
6∙𝛼𝛼
made before the performance of 3 x 106
During
machine
operation,
diagnostics symptoms
𝑦𝑦
the spinner drive system until
.
𝑅𝑅 = 0 1 𝑓𝑓 𝑦𝑦 𝑑𝑑𝑦𝑦 (12)
where: δ – Euler constant, equal to 𝛿𝛿 ≈ 0,577. cycles (ISO 6336-3:2006, Calculation of
reaching the crack critical length load capacity of spur and helical gears initiation
of the damage process of the gear teeth in
5
The distribution parameters (8) for 1.6523 (20–HNM)
steel
were as follows:
mbending
y =4.594·10 ,
Part
3:
Calculation
of
tooth
Assuming that the probability distribuspinner.
Based on eye
inspection
of toothing, a crack n
Verification
of the
numerical
4
-5
σy =tion
3.557·10
= Equation
4.434·104 ,(8)
α =is 3.605·10
value ,mu in
uniform . strength).
Verification
of
the
numerical
model
model
approx.
0,5 mm was detected. The time of gear ope
(even) within
range mmin
≤m
,
i ≤ m
maxpassage
Reliability
i.e. thethe
probability
that
after
the
of Ni cycles, a particular crack reaches the During machine operation, diagnostics sym
Due to the
wear within
of the most
During
operation,
diagnostics
the time
(numbervalue
of cycles)
until the
𝑙𝑙𝑘𝑘𝑟𝑟machine
= 2,14 𝑚𝑚𝑚𝑚
was evaluated.
A fatigue crack
hat the probability
distribution
𝑚𝑚 in formula
(8) is uniform
(even)
the frequently co- length
critical
length
𝑙𝑙𝑘𝑘𝑟𝑟 –the
is expressed
by thecan
following
formula:
geared wheels, their continuous symptoms
wereofregistered
which
indi- of the gear t
initiation
the damage
process
crack
reaches
ultimate value
be operating
reducer,ofworking
at a process
constant rotational v
≤ 𝑚𝑚𝑖𝑖 ≤ 𝑚𝑚𝑚𝑚𝑎𝑎𝑥𝑥
, the time
(number of
cycles) until the crack
reaches(SHM)
the ultimate
monitoring
is recommended. Tak- stage
𝑦𝑦
catedofanthe
initiation
the damage
determined
. transformed rela- ing into account the quality of the prod- (12) spinner. Based on eye inspection of toothing, a
𝑅𝑅 = 0 1 𝑓𝑓using
𝑦𝑦 𝑑𝑑𝑦𝑦the
of the
gear
in theHNM)
drive system
of
was
made
of teeth
1.6523(20
steel, hardened
up to 6
(8):
be determinedtionship
using the
transformed relationship (8):
ucts as well as the manufacture profile the Fiomax
approx.
0,5
mm
was
detected.
The
time
of ge
200
spinner.
Based
on
eye
𝑙𝑙 𝑘𝑘𝑟𝑟
gear tooth are given in Table 2.
relating to the assortment and thickness the
inspection
of
toothing,
a crack
near
the
1
length 𝑙𝑙𝑘𝑘𝑟𝑟 = 2,14 𝑚𝑚𝑚𝑚 was evaluated. A fatigu
Verification
model 𝑑𝑑𝑙𝑙.
𝑁𝑁𝑖𝑖 = of the numerical
of yarns, the change(9)
of the most utilised The
probability
oflength
damage
occurrence
can be determin
tooth
foot of a equal
to approx.
(𝑎𝑎2 + 𝑎𝑎0∙ 𝑚𝑚1𝑖𝑖 ) ∙ ∆𝐾𝐾(𝑙𝑙)𝑚𝑚 𝑖𝑖
geared
wheels
in
the
stretching
apparatus
stage
of
the
reducer,
working
at
a constant rota
0,5
mm
was
detected.
The time
of
gear
𝑙𝑙
During machine
operation, diagnostics symptoms were registered which indicated that
an we know the density distribution function
𝑚𝑚𝑖𝑖𝑛𝑛
𝑓𝑓 𝑦𝑦 , w
(9) has to be made at least once per two years.
operation
until
reaching
the
critical
– value of
exponent
within of
the the
range
𝑚𝑚𝑚𝑚𝑎𝑎𝑥𝑥 system
, 𝑙𝑙𝑘𝑘𝑟𝑟 – of the Fiomax 200 was made of 1.6523(20 HNM) steel, hardened
initiation
of 𝑚𝑚theestablished
damage process
gear𝑚𝑚teeth
𝑚𝑚𝑖𝑖𝑛𝑛 ≤in𝑚𝑚the
𝑖𝑖 ≤ drive
2
utilizing
formula
if mm
moments
𝑚𝑚𝑦𝑦 and 𝜍𝜍𝑦𝑦 ,determ
crack
length
=(8),
2,14
wasofevaluatwhere: mi – value of exponent m estabthe manufacture profile relating to the assortment and
thickness
oflkryarns,
the
change
the
gear crack
tooth are
in Table
2.
package
ck length, spinner.
𝑙𝑙𝑚𝑚𝑖𝑖𝑛𝑛lished
– minimal
crack
length.
fatigue
wasgiven
spotted
on the
Based
eyerange
inspection
crack nearthethecommercial
tooth foot of
a lengthStaequal ed.
to A the
withinon
the
mmin ≤ of
mi toothing,
≤ mmax, a Utilizing
known.
The
of possible
most utilised
geared
wheels
in
thethe
stretching
apparatus
has to
be number
made at least
once percycles
two 𝑁𝑁𝑖𝑖 until reach
tistica,
it
was
stated
that
most
suitpinion of the first stage of the reducer,
lkr – critical
length,
lminThe
minimal
we presentapprox.
a histogram
of crack
the
of– time
Ni of
of gear
reaching
the critical
0,5 mm
wasdistribution
detected.
time
operation
until fatigue
reaching the critical crack The probability of damage occurrence can be d
formula
(4) after
some transforma
working atutilizing
a constant
rotational
velocity
years.able distribution which approximates determined
crack length.
thatobr/min.
we knowThe pinion
the densitywas
distribution
function 𝑓𝑓
h by the length
tooth foot
1.6523
(20HNM) Asteel/.
Athebar
prepared
barchart
chart
(histogram)
asymptotic
𝑙𝑙𝑘𝑘𝑟𝑟 =/material
2,14 𝑚𝑚𝑚𝑚
was evaluated.
fatigue
crack
waswas
spotted
on theispinion
of the first
n = 1460
made
Utilizing
the commercial
package
Statistica,
it was stated that the2,4
most suitable distribution 1
Gumbel’s
distribution,
having
the dis2
of
1.6523
(20
HNM)
steel,
hardened
up
In
Figure
15
(see
page
141),
we
presformula (8),
if −11,435
moments 𝑚𝑚𝑦𝑦 and 𝜍𝜍−1
2.14
≤ 𝑚𝑚 ≤𝑛𝑛2.395
for obr/min. The pinion 𝑁𝑁utilizing
account stage
200 values
of parameter
𝑚𝑚 atwithin
the interval
𝑖𝑖 = 𝑦𝑦
of the reducer,
working
a constant
rotational
velocity
=function
1460
∙ 𝑚𝑚
∙ 10−8having
density
given in is
theasymptotic
0,05 (7.49
whichtribution
approximates
the bar
chart (histogram)
distribution,
to 60 Gumbel’s
± 2 HRC.
Geometrical
parameters
𝑖𝑖 the − 1.675 ∙ 10
ent a histogram of the distribution of
ar the tooth
equal
𝜍𝜍𝐹𝐹 = 1374
,steel,
critical
crack length
𝑙𝑙𝑘𝑘𝑟𝑟[3,
= 5]:
2,4 𝑚𝑚𝑚𝑚
and
known. The number of possible cycles 𝑁𝑁𝑖𝑖 unt
standard
wasfoot
made
1.6523(20
HNM)
hardened
up to 60±2
HRC.
Geometrical
parameters
time
Niofofto
reaching
the𝑀𝑀𝑃𝑃𝑎𝑎
critical
fatigue
distribution density function given in the standard [3]:of the gear tooth are given in Table 2.
crack
length
the in
tooth
foot
= tooth
0,05
𝑚𝑚𝑚𝑚.
ack lengththe
𝑙𝑙𝑚𝑚𝑖𝑖𝑛𝑛gear
determined utilizing formula (4) after some tran
areby
given
Table
2. /material
The probability of damage occurrence
𝑓𝑓 𝑦𝑦 = 𝛼𝛼 ∙ 𝑒𝑒𝑥𝑥𝑝𝑝 −𝛼𝛼 ∙ 𝑦𝑦 − 𝑢𝑢 − 𝑒𝑒 −𝛼𝛼∙ 𝑦𝑦−𝑢𝑢 .
(10) can
1.6523 (20HNM) steel/. A bar chart was
2,4
be
determined
based
on
Equation
(9), as1
Theprepared
probability
of
damage
occurrence
can
be
determined
based
on
relationship
(9),
assuming
taking into account 200 val(10)and variance) are 𝑁𝑁
Central moments of the distribution (mean value
expressed
by the
following
𝑖𝑖 =
−11,435
suming
that
we
know
the
density
distribu−8
∙ 𝑚𝑚𝑖𝑖
− 1.675
moments
of 𝑢𝑢theanddistribution
of parameter
m within
the interval
0,05 (7.49 ∙ 10
thatues
we know
the density
distribution
function 𝑓𝑓Central
𝑦𝑦 , whose
parameters
𝛼𝛼 are calculated
formulas:
(mean value and variance) are expressed tion function f (y), whose parameters u and
2.14 ≤ m ≤ 2.395 for stresses near the
2
α are calculated utilizing Equation (8), if
and 𝜍𝜍𝑦𝑦 by
,determined
based on 2 adequate statistics, are
utilizing
formula
𝑚𝑚𝑦𝑦 critiEquation
tooth foot
equal(8),
to σifF =moments
1374 MPa,
𝛿𝛿 (11):
𝜋𝜋
𝑚𝑚𝑦𝑦 = 𝑢𝑢 + ; 𝜍𝜍𝑦𝑦2 = 2 ,
(11) on
moments my and σ y2, determined based
6∙𝛼𝛼fatigue crack length is
known. The number of possible cycles 𝑁𝑁𝑖𝑖 until reaching𝛼𝛼the critical
adequate
statistics,
are
known.
The numThewere
values
of stress
intensity
were
chosen based
values
stress intensity
coefficients
chosen
based
on rulescoefficients
given in [30],
where:
Fig. 16. ApproximationThe
of δthe
stressof
intensity
coefficient
– Euler
constant,
equal
to 𝛿𝛿 ≈ 0,577.
ber of possible cycles Ni until reaching the 2
determined utilizing formula (4) after somewhere:
transformations:
2
3
4
𝑙𝑙
𝑙𝑙
𝑙𝑙
5
𝑙𝑙
𝑙𝑙
𝑙𝑙crack length
𝑙𝑙is determined
determined by
means of the parameters
Beasy system for 1.6523 (20
critical
fatigue
n of the stress intensity 2,4
coefficient
𝑙𝑙 =
𝜍𝜍4∙ ∙as𝜋𝜋follows:
∙ 𝑙𝑙 ∙+𝐴𝐴𝐴𝐴15m+
𝐴𝐴2 ∙ , + 𝐴𝐴,3 ∙ (13)
+ 𝐴𝐴4 ∙
∆𝐾𝐾 𝑙𝑙 = 𝜍𝜍 ∙ 𝜋𝜋 (8)
∙ 𝑙𝑙 ∙ 𝐴𝐴1 + 𝐴𝐴2 ∙ +HNM)
𝐴𝐴3 ∆𝐾𝐾
∙ steel
+were
𝐴𝐴
∙y =4.594·10
1The distribution
(13)
utilizing
Equation
(4)
after
some
transfor𝑏𝑏
𝑏𝑏
𝑏𝑏
𝑏𝑏
𝑏𝑏
𝑏𝑏
𝑑𝑑𝑙𝑙.
𝑁𝑁𝑖𝑖 =
4
4
-5𝑏𝑏
−8 ∙ 𝑚𝑚 −11,435 − σ
𝑚𝑚 𝑖𝑖 , α = 3.605·10 .
4.434·10
y = 3.557·10
means of the Beasy system
1.675
∙ 10−11, )u ∙=∆𝐾𝐾(𝑙𝑙)
0,05 (7.49 ∙ 10
mations, see Equation (13).
𝑖𝑖
𝜌𝜌𝑓𝑓𝑃𝑃
𝜌𝜌𝑓𝑓𝑃𝑃
𝜋𝜋
𝐺𝐺
𝐺𝐺 4 , which,
Central moments for statistics 𝑁𝑁𝑖𝑖 are equal
to 𝑚𝑚𝑏𝑏i.e.
=the
5,221
∙ 10𝜋𝜋5 −
&
∙ 10
𝑦𝑦 =
𝑧𝑧𝑛𝑛 ∙ ∙𝑠𝑠𝑖𝑖𝑛𝑛
𝜗𝜗 crack
+ 3reaches
∙
Reliability
that
the
of𝑏𝑏−N=i cycles,
a particular
the − (14) ∙ 𝑚𝑚𝑛𝑛 =
𝑧𝑧𝑛𝑛probability
∙ 𝑠𝑠𝑖𝑖𝑛𝑛
𝜗𝜗𝜍𝜍𝑦𝑦after
+= 4,436
3 ∙ passage
𝑚𝑚
4.99.
𝑛𝑛 = −
Strona
19
z
26
3
𝑐𝑐𝑜𝑜𝑠𝑠⁡
(𝜗𝜗)
𝑚𝑚𝑛𝑛
3
𝑐𝑐𝑜𝑜𝑠𝑠⁡
(𝜗𝜗)
𝑚𝑚
The values
of
stress
intensity
coefficients
𝑛𝑛
The values of stress intensity coefficients were chosen based on rules given in [30], where:
in turn,
entered
into
relationship
(9) critical
allow length
for the𝑙𝑙𝑘𝑘𝑟𝑟determination
of
distribution
parameters
5
4
–
is
expressed
by
the
following
formula:
were
chosen
based
on
rules
given
in
[30],
al
to
𝑚𝑚
=
5,221
∙
10
&
𝜍𝜍
=
4,436
∙
10
,
which,
Bar chart
crack
𝑦𝑦 (histogram) of times
𝑦𝑦 of reaching the critical value2by the fatigue
𝑙𝑙
𝑙𝑙
𝑙𝑙 3
𝑙𝑙 4
5
−5
𝑦𝑦1
(14) (13) where Equations (14) and (15). (12)
&𝜋𝜋𝛼𝛼∙ 𝑙𝑙=∙ 2,898
∙ 10
. 𝐴𝐴3 ∙ 𝑅𝑅 =+ 𝐴𝐴
𝑢𝑢 ∆𝐾𝐾
=5,022∙
10
𝑙𝑙
=
𝜍𝜍
∙
𝐴𝐴
+
𝐴𝐴
∙
+
∙
+
𝐴𝐴
∙
,
.
𝑓𝑓
𝑦𝑦
𝑑𝑑𝑦𝑦
1
2
4
5
ow 1.6523
for the (20HNM)
determination
parameters
𝑀𝑀𝑃𝑃𝑎𝑎
for
steel of
anddistribution
stresses at the
tooth
0where
where additionally:
𝑏𝑏 foot 𝜍𝜍𝑏𝑏𝐹𝐹 = 1500
𝑏𝑏additionally:
𝑏𝑏
Where
The
distribution
density
function
ofspinner
time (i.e. number of cycles) until reaching the critical
in the
drive system
of
a
Fiomax
2000
𝜌𝜌
𝑕𝑕
𝜌𝜌
𝑕𝑕
𝑓𝑓𝑃𝑃
𝑓𝑓𝑃𝑃
𝜌𝜌𝑓𝑓𝑃𝑃
𝜋𝜋
𝐺𝐺
𝜌𝜌𝑓𝑓𝑃𝑃 = 0,25 𝑚𝑚𝑛𝑛 , 𝐺𝐺 = 𝑓𝑓𝑃𝑃 − 𝑓𝑓𝑃𝑃 + 𝑥𝑥,
𝑚𝑚𝑛𝑛(10),
, 𝐺𝐺 =has the
− following
+ 𝑥𝑥,
𝑓𝑓𝑃𝑃
𝑏𝑏 = by
𝑧𝑧𝑛𝑛 the
∙ 𝑠𝑠𝑖𝑖𝑛𝑛
𝜗𝜗 + 3determined
∙
−
𝑚𝑚𝑛𝑛==0,25
4.99.
(14)form:
length
crack−analyzed,
based on𝜌𝜌∙ relationship
𝑚𝑚 𝑛𝑛
𝑚𝑚 𝑛𝑛
𝑚𝑚 𝑛𝑛
𝑚𝑚 (15)
𝑛𝑛
e (i.e. number of cycles) until3 reaching the critical
𝑐𝑐𝑜𝑜𝑠𝑠⁡
(𝜗𝜗)
𝑚𝑚𝑛𝑛
cos
(ϑ)
Verification
of the numerical model
h
=
1,25
m
,
n 𝑛𝑛 ,
𝑕𝑕fP𝑓𝑓𝑃𝑃 = 1,25 𝑚𝑚
𝑕𝑕𝑓𝑓𝑃𝑃 = 1,25
𝑚𝑚𝑛𝑛 ,
nbased
analysis
of the data shown
in the
one form:
can state
that the
longer
the operating
on relationship
(10), has
the chart,
following
During
machine
operation,
diagnostics symptoms ϑwere
registered
whichbyindicated
an the
– angle,
calculated
means of
−5
5
2
2∙𝐺𝐺
−5
−5
5
−2,898∙10 ∙ 𝑦𝑦−5,022∙10
𝑦𝑦
=
2,898
∙
10
∙
𝑒𝑒𝑥𝑥𝑝𝑝
−2,898
∙
10
∙
𝑦𝑦
−
5,022
∙
10
.
−
𝑒𝑒
geared wheel, 𝑓𝑓the
lower
its
fatigue
life.
The
highest
fatigue
life
was
achieved
in
𝜗𝜗
–
angle,
calculated
by
means
of
the
equation: 𝜗𝜗 = 𝑧𝑧
𝜗𝜗
–
angle,
calculated
by
means
of
the
equation:
𝜗𝜗
=
∙
𝑡𝑡𝑔𝑔
𝜗𝜗
−
𝐻𝐻,
equation:
where additionally:
initiation of the damage process of the gear teeth in the drive system
𝑧𝑧𝑛𝑛 of the Fiomax 200
(16)
−5 ∙𝜌𝜌The
ng interval (4.3 – 4.5)
x 105 cycles.
scatter
connected,𝐴𝐴as=usual,
with
the−2,766; 𝐴𝐴 = 10,17;
(15)
𝑕𝑕 𝑓𝑓𝑃𝑃 5 is spinner.
𝑦𝑦−5,022∙10
𝑓𝑓𝑃𝑃
𝐴𝐴
0,625;
𝐴𝐴foot
−2,766;
𝐴𝐴4 = −16,
0,625;
𝐴𝐴
=
𝐴𝐴the
=tooth
−16,178;
𝐴𝐴a5length
= 10,293;
Based
on
eye
inspection
of
toothing,
near
to
0−5 ∙ 𝑦𝑦 − 5,022 𝜌𝜌
∙ 𝑓𝑓𝑃𝑃
105= 0,25
− 𝑒𝑒 −2,898∙10
.
1 =
2 =of
3 = 10,17;
1
2
3 a crack A
A𝐴𝐴equal
𝑚𝑚𝑛𝑛 , 𝐺𝐺 = 𝑚𝑚 − 𝑚𝑚 + 𝑥𝑥,
1 = 40.625; A2 = −2.766;
3 = 10.17;
𝑛𝑛
𝑛𝑛
n of defects or a particular damage mechanism.
A
=
−16.178;
A
=
10.293;
4
5
2,14
≤until
𝑚𝑚𝑖𝑖 ≤reaching
2,395
for
= 1 ⋯ crack
200.
mm ≤
was
the𝑖𝑖 critical
𝑚𝑚𝑖𝑖detected.
≤ 2,395 The
for time
𝑖𝑖 = 1of⋯gear
200.operation
(15)approx. 0,5 2,14
1,25
𝑚𝑚(14),
2.14 ≤ mi ≤ 2.395 for i = 1 ··· 200.
(13),
(15)
(16).geared
𝑓𝑓𝑃𝑃 =17,
𝑛𝑛 , change
it means thatEquation
the𝑕𝑕
recommended
ofcourse
the
wheel
shouldfunction
be madeagainst
before the number of working
In
Fig.
a chart
of
theand
of the
reliability
length 𝑙𝑙 = 2,14 𝑚𝑚𝑚𝑚 was evaluated. A fatigue crack was spotted on the pinion of the first
𝑘𝑘𝑟𝑟
2∙𝐺𝐺
6 𝜗𝜗 – angle, calculated by means of the equation: 𝜗𝜗 =
∙ 𝑡𝑡𝑔𝑔
𝜗𝜗 −
mance of 3 x10cycles
cycles
Calculation
of load capacity
of
spur
and
for(ISO
a gear6336-3:2006,
having a broken
tooth (breakage
near
tooth
foot)
isat𝐻𝐻,
presented.
𝑧𝑧𝑛𝑛are
FIBRES &are
in Eastern
Europe
2017,
Vol. 25,
The
based
on
the1(121)
approximation
constants
determined
basedrotational
on
theconstants
approximation
of
the
stress
intensity
coefficient
of the
reducer,
working
a constant
velocity
𝑛𝑛TEXTILES
= determined
1460
obr/min.
The
pinion
142
reliability function
against the number of workingstageThe
s - Part 3: Calculation
of
tooth
bending
strength).
𝐴𝐴1 foot)
= 0,625;
𝐴𝐴2 = −2,766; 𝐴𝐴3 = 10,17;
𝐴𝐴4 =
𝐴𝐴
10,293;
5 =simulation
(in HRC.
turn)ofGeometrical
via
of the
propagation
(in turn) HNM)
via
of obtained
the
thesimulation
fatigueparameters
crack
inofthe
BEASY of t
was obtained
made
of −16,178;
1.6523(20
steel, hardened
up propagation
to 60±2
reakage near tooth
is presented.
e wear of the 2,14
most≤ frequently
cooperating
geared
wheels,
their
continuous
𝑚𝑚𝑖𝑖 ≤ 2,395 for 𝑖𝑖 = 1 ⋯ 200.the gear
package
for1374
the stress
package
stress
near2.the tooth foot
equal to
MPa. near the tooth foot equal to 1374 MPa.
toothfor
are the
given
in Table
3500
1
0.9
3000
0.8
2500
Reliability R(N)
0.7
∆K
2000
1500
1000
0.5
0.4
0.3
0.2
500
0
0.6
0.1
0
1
2
3
4
Crack length, mm
5
6
7
Figure 16. Approximation of the stress intensity coefficient
determined by means of the Beasy system.
The constants are determined based on the
approximation of the stress intensity coefficient obtained (in turn) via simulation of
the propagation of the fatigue crack in the
BEASY package for the stress near the
tooth foot equal to 1374 MPa.
The BEASY package [21-22] is an engineering calculation system utilizing the
boundary element method for the solution of several different problems e.g. related to elastic theory, problems of elastic
contact as well as cracking. The modules
connected with fatigue and fracture calculations are based on the procedures
designed by the international research
team Flagro – NASA. Specialized files
i.e. NASMFM and NASMFC are added
to this package, which contain a considerable database of strength properties of
versatile materials.
In Figure 16, charts of changes in the
stress intensity coefficient and its approximation function are shown.
Central moments for statistics Ni are
equal to my = 5,221 ∙ 105 & σy = 4,436 ∙ 104,
which, in turn, entered into relationship
(9) allow for the determination of distribution parameters u = 5,022 ∙ 105 & α =
2,898 ∙ 10−5.
The distribution density function of time
(i.e. number of cycles) until reaching the
critical length by the crack analyzed, determined based on relationship (10), has
the following form, see Equation (16).
In Figure 17, a chart of the course of the
reliability function against the number of
working cycles for a gear having a broFIBRES & TEXTILES in Eastern Europe 2017, Vol. 25, 1(121)
0
4
4.2
4.4
4.6
4.8
5
5.2
5.4
Number of cycles x 105
5.6
5.8
6
Figure 17. Reliability function for toothing operation against number
of cycles, for stress equal to σ = 1316 MPa and value x = 0.005 mm.
ken tooth (breakage near tooth foot) is
presented.
Assuming an allowable value of the reliability function at a probability level
equal to 0.95, we can determine the time
of operation of a gear with a propagating
fatigue crack until reaching the critical
value (not exceeding θ = 1.45 hour), simultaneously assuming a level of loading not exceeding the allowable stresses σ = 1316 MPa (near tooth foot) for
1.6523 (20HNM) steel, according to the
standard [3].
In the case where nucleation of the prospective fatigue crack has been caused
by a sudden stopping of the production
process, where additionally the transient
stresses have exceeded the elasticity limit of the material and induced a fatigue
crack above the threshold level, then the
bending stress near the tooth foot can be
assumed as 1.3 times lower. The minimal
computational safety coefficient assumed
for calculation of the fatigue strength
near the tooth foot under bending conditions according to the standard [4] can
be assumed depending on the allowable
stresses. The operating time of a gear in
which the process of crack propagation
takes place elongates up to θ = 2.5 hours,
in the case of an assumption of reliability
not less than R = 0.95.
Summary
The modified probabilistic model of fatigue crack propagation at the tooth foot
of a geared wheel proposed allows prediction of the working time of a gear in which
an initiated fatigue crack propagates. One
of the essential factors permitting effective utilization of the model is knowledge
(proper evaluation) of the stresses at the
tooth foot (root) in the case of initiation
of a fatigue crack. Based on the guidelines
formulated in the standard [4], in cases
where the geared wheel has been subjected to more than the ultimate number of cycles Ng = 3 ∙ 106 (for carburized alloy and
hardened steel), the stresses at the tooth
foot can be considered as having values
under those allowable (ultimate) in the
standard [3]. In industrial practice, some
cases occur frequently i.e. a tooth crack
takes place when the particular geared
wheel was subjected to the ultimate number of cycles. Determination of the stress
value at the tooth foot for the moment of
crack initiation is the major problem in
the model proposed, since based on this
knowledge, one can predict the time until
the crack reaches the ultimate length.
The Uneveness of working of the spinner drive system has a direct influence
on yarn ends down. The stretching apparatus of the ring spinner consists of several mutually co-operating gears. Even
a trivial defect of a single element e.g.
the geared wheel, bearing or shaft disturb
or preclude the performance of the manufacturing process.
In industrial practice, the problem discussed is solved by installation of diagnostic gauges on a particular machine which
allow the monitoring of parameters of cooperating elements of the drive system. Additionally these solutions can be supported
by expert systems enabling an almost online performance of machine repairs. Due
to the high costs of adaptation of particular
143
procedures, it seems that the most useful is
the application of probabilistic methods for
evaluation of the fatigue lives of particular
elements of drive systems. These activities
could be planned and performed even at
the design phase. In the case of machines
which are currently operated, one can assure that during the repairs of particular
parts, the spare parts fulfil requirements
according to fatigue life.
The probabilistic models proposed enabled determination of the operating time
of a particular machine until the planned
repair. Knowledge of the fatigue life of
key elements of a drive system reduces
costs connected with the damage risk
of a particular machine as well as nonplanned break-downs. The method proposed can be also useful for other types
of ring spinners.
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Received 07.07.2016
Reviewed 19.10.2016
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