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Reliability and Lifetime of Mechanical Units in Operation and Test
Vladimir B. Algin1,a, Hyoung-Eui Kim2,b
1BELAVTOTRAKTOROSTROENIE Scientific-Engineering Enterprise and the Institute of
Mechanics and Reliability of Machines of the National Academy of Sciences of Belarus
12, Akademicheskaya Str., 220072 Minsk, Belarus
2Reliability Assessment Center, Korea Institute of Machinery and Materials,
Yuseong P.O Box 101, 171 Jang Dong, Yu Seong Gu, Daejeon, Korea, 305-343
aalgin@ncpmm.bas-net.by, bkhe660@kimm.re.kr
Keywords: Mechanical unit, Reliability, Load, Lifetime, Operation conditions, Limiting state, Test
conditions.
Abstract. The paper discusses the problem of reliability calculation for mechanical units under
load. Because the rated reliability of the unit as a system with many independent elements can tend
to zero, the methods of reliability calculation of the units as real objects with the common factors
(loads, operation conditions) are proposed. The choice of the life test modes and test time depends
essentially on unit-related information like their operation conditions. The paper describes one of
approaches, passing to this problem. Within its framework, the accumulations of damages are made
at several typical points of the loaded system.
Introduction
A shaft with N elements (Fig. 1), assuming that reliability (probability of no-failure) Pi for each
element is known, can be considered as a case in point. To result in the total reliability of the shaft,
the single element's probabilities should be multiplied. Thus, the total reliability depends on a
number of the elements allocated in the shaft. It is generally accepted in many fundamental works
that the more elements are considered, the closer this reliability tends to zero. Hence the reliability
of the loaded system can not be correctly calculated having only reliabilities of single elements. To
analyze such a system correctly, its work conditions and common acting factors should be
considered.
Fig. 1 Fig. 2
This approach can be examined for a chain of the same elements with the random load Q
(Fig. 2), where strength of elements is a random variable; the load is the common factor. A method
of reliability calculation for the loaded chain is known [1]. But in mechanical units the elements and
parameters of their loads and load-carrying abilities are different. Therefore the first section of the
paper is dedicated to methods for calculating the reliability of units as loaded mechanical systems.
In the second section the problem of test modes for the loaded units is examined.
1 Reliability calculation for systems with various loaded elements
Calculation of system with the elements having the dependent damages. It can be accepted
for the further discussion that the operating time has fixed value L= Lj and the elements failure is
caused by accumulation of damages. Then probability of no-failure PLj for every element is rated
with distributions of its extent of damage QLj and the load-carrying ability R (see Fig. 3). The unit
has elements with various pairs of distributions QLj and R. Therefore it is impossible to use the
approach suitable for calculation of the chain reliability. If the damage-related extents of various
P1 P2 PN
Q
P1 P2 PN
Q
Key Engineering Materials Vols. 326-328 (2006) pp. 549-552
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elements are strongly dependent, this feature can be take into account offering an approach
described below.
Now a concept of “equivalent
pairs of distributions” for random
variables can be introduced. A pair
of distributions {fRI(RI), fQI(QI)} is
equivalent to a pair {fRII(RII),
fQII(QII)} if distribution fYI(Y) of
random variable YI = RI − QI
coincides with distribution fYII(Y) of
random variable YII = RII − QI, i.e.
fYI(Y)=fYII(Y). Using this concept, the
equivalent pair distributions with the same distribution fQa(Q) for each system element (x=a, b, c,
…) can be selected: {fQa(Q), fRxa(R)}, {fQa(Q), fRxb(R)}, {fQa(Q), fRxc(R)} and so on. It allows finding
the function of distribution of load-carrying ability of the system as an integrated function of
minimal values for the equivalent distributions fRx(R). This question is described in detail in [2]. As
a result, the system is reduced to one equivalent element with known distributions of the load-
carrying ability FE(R) and the extent of damage FQa(Q). Now it is not a problem to determine the
probability of no-failure P(Lj). Repeating described procedure for other values L, the reliability
function of system P(L) is derived.
As an example, a planetary reducer (Fig. 4) and the results of its calculation (Fig. 5) are
presented: 1 is the reliability function P(L) and 2 is the density of distribution of lifetime f (L).
Densities of distributions of lifetime for reducer elements are presented as well: the bearing of
satellite with contact fatigue (a); sun gear teeth with contact fatigue (b); satellite teeth with contact
fatigue (c) and bending fatigue (d).
Fig. 4 Fig. 5
Reliability calculation with taking into account the operation conditions as common factor.
Concepts of loading are various for different machine parts. But all loads are determined by the
operation conditions. To rate the loaded systems in the general case, the calculation can be
performed under the scheme “operation conditions — lifetime”. For this purpose lifetime-strength
curves of an element are used (Fig. 6). The operation conditions are described by probabilistic
manner in a form of the relative durations for the commonly accepted typical conditions (see
Fig. 7).
The relative durations of operation conditions αk and the load-carrying ability of elements are
random variables. Using a method of statistical modelling (Monte Carlo), in each cycle of
modelling the durations of the operation conditions αk (k=1, 2, …, K) and load-carrying abilities of
QLj, R
f(QLj),
f(R) R
QLj
Fig. 3
a
c, d
b
Experimental Mechanics in Nano and Biotechnology550
elements are reproduced. Then using lifetime-strength curves the lifetimes of elements can be found
at these conditions. This results in lifetimes of elements and the total system under all conditions at
the end of the cycle. To obtain the condition ∑αk =1 in each cycle, the special correction is used.
When the statistical modelling is carried out, the characteristics of the initial and corrected (related
to the total sum) values are different. Therefore closeness of parameters of distributions αk to the
given values is provided by means of multi-step optimization procedure.
Fig. 6 Fig. 7
Probabilistic lifetime calculation with taking into account a complicated logic of limiting
states of the machine, its units and parts. The previous part of the paper considered that the
limiting state of a system occurs with the first limiting state of an element. Actually the lifetime of a
complex object is usually determined by a complicated way. For example, the machine lifetime is
exhausted if the limiting states of the frame, the engine and one of transmission units are reached.
To describe such states, the scheme of limiting states is introduced.
As an example, lifetime calculation of the machine with the scheme of limiting states (1, 1, 1) is
presented in Table 1. The record (1, 1, 1) means that the limiting state of the machine occurs if the
limiting states are reached with one part of the first type (the number standing in the first position),
one part of the second type (the number standing in the second position) and one of parts of the
third type (the number standing in the third position). Elements lifetimes are lognormally distributed
with the following coefficients of variation: 1) 0.50; 2) 0.45; 3) 0.55; 4) 0.45 and 5) 0.45.
Eighty percentage lifetimes of machine parts and the machine as the whole: A) machine lifetime
(first failure); B) machine lifetime with taking into account the scheme of limiting state (1, 1, 1)
1)
Frame
(type 1)
2) Engine
(type 2)
3)
Gearbox
(type 3)
4) Forward
driving axle
(type 3)
5) Rear
driving axle
(type 3)
A) Machine
(first
failure)
B) Machine:
scheme of
limiting state
(1, 1, 1)
1,0 1,0 1,0 1,0 1,0 0,687 1,423
Table 1
2 Life accelerated test
Object representation. Three following cases described below can be marked out.
The first case. Full information about object and its load mode in operation are known. The
weakest element is known. The test mode is directed on testing this element.
-5
0
5
10
15
20
25
0 0,2 0,4 0,6 0,8 1
Relative duration of operation conditions
Density of distribution
Condition 1
Condition 2
Condition 3
Condition 4
0
1
2
3
4
5
6
7
8
9
0 10 20 30 40
Load-carrying ability (fatigue endurance)
Logarithm of lifetime
Condition 1
Condition 2
Condition 3
Condition 4
Key Engineering Materials Vols. 326-328 551
The second case. Elements data is unknown. The load mode of the object in operation is known
as well as typical processes of object damage.
The third case. Assignment and general device of object (for example, the kinematic scheme of
transmission) is known.
The choice of the accelerated test mode essentially depends on information on object.
Example. In Fig. 8 the
choice of test time for
transmission is shown
taking into account the
loading and accumulation
of damages at its typical
points. The load mode of
transmission in operation
is predicted. Calculation
is carried out in relative
units. Only the ratios and
efficiencies of
transmission units are
used.
Horizontal lines
represent extents of
damage at typical points
of transmission for the
accepted operation
conditions. Inclined lines
(bevels) represent the
change of damage-related
extents depending on test
time. The inclination is determined by a test torque level and parameters of S-N curves of elements.
Also it is necessary to consider differences in dynamic loadings under operation and test. The points
of crossing the horizontal and inclined lines give an expected time for obtaining the limiting states
of the most loaded parts which are located in the units of transmission mentioned above.
Conclusion
The dependent behaviors of elements are the basic problem in calculating the complex systems. The
developed techniques allow to reproduce the real connections with elements loadings and to avoid
tending to zero for reliability in calculating systems with many loaded elements. Next prominent
aspect in calculating units is the description and taking into account a complicated logic of their
limiting states. The choice of lifetime test mode depends on the initial information on object and its
operation conditions. The method based on calculating extents of damages at several typical points
of the loaded system has been presented.
References
[1] K.C. Kapur and L.R. Lamberson: Reliability in Engineering Design (John Willey & Sons, New
York 1977).
[2] V.B. Algin: Dynamics, reliability and lifetime designing for transmissions of mobile machines
(Navuka & Teknika, Minsk 1995, in Russian).
Fig. 8
Experimental Mechanics in Nano and Biotechnology552
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