METHOD FOR DETECTION OF FAULTY CONDITIONS USING THE
THEOREM OF HYPOTHESES
(FOR INSTANCE THE ENGINE TV3-117 TO MI-8)
M.Sc. Zagorski N.
Technical University of Sofia, Branch Plovdiv
Abstract: Refusals and faults of the functional systems of the aircraft affect the reliable operation of these systems and the safety of
conducting flight operations. To detect the causes of failures require additional inspections, testing, time, material costs. Problems
associated with determining the causes of failures and malfunctions can be resolved with the use of mathematical methods for identification.
The state analysis is performed in terms of operating mode, which is extremely difficult to obtain comprehensive information. An important
part of recognition is the choice of parameters describing the state of the object. They must contain sufficient information to allow
implementation of the recognition process for the selected number of diagnoses.
Keywords: aircraft, mathematical methods for identification, causes of failures and malfunctions
where: P(Si) is a probability to perform the state Si, determined by
statistical data (a priori probability of occurrence of the condition);
P(kj/Si) – probability of occurrence of attributes in objects with kj
status Si; P(ki) – probability of occurrence of the trait kj in all
facilities regardless of the status of the object.
1. Introduction
Diagnosis of aviation gas turbine engines (GTE) is performed in
the production and use, and storage need. At each stage of GTE,
there are certain technical requirements (technical terms - TT). An
obligatory condition for normal operation is the correlation
parameters of GTE with TT. But the systems and units of engine
failures can occur that violate this line.
Summary of the Bayesian formula applies to cases where the
examination is done on a set of K-signs, including signs k1, k2, ...,
kν). Any signs of τ there kj bits (kj1, kj2, ..., kjs, ..., kjτ). As a result of
the survey can be found the realization of the trait Kj* = kjs and the
whole complex of signs K*1.
The collection, compilation and analysis of recorded failures
and malfunctions of various functional elements of the aviation
GTE is the basis for an extensive study of the causes of these
events, many of which have a high potential risk for the operation
of air transport system [1,4].
Bayesian formula for the complex of signs has the form
(2)
A GTE is a system of many interacting structural units,
functional systems, making it difficult to assess the technical
condition without the use of special methods and equipment.
2. How to solve the problem of detection of faulty
conditions
There are two basic approaches to solve the problem of
recognition: deterministic and probabilistic.
,
This study used data for 122 pcs. TV3-117 engines of various
modifications operated helicopters Mi-8 and Mi-17 in the Air Force
and airline operators in Kazakhstan, Kyrgyzstan, Armenia, Belarus,
Slovakia and Bulgaria in the period from 1976 to 2010, with total
engine working time during the period of 136,800 hours. For this
period, 50 units were recorded failures of functional elements of the
engine oil system.
If probabilistic methods are required to recognize the
construction of crucial rule with which the available sets of signs,
the object may be referred to one of the possible states Si. When it is
necessary to assess the reliability of the decision and the degree of
risk of wrong decision.
Failures and malfunctions of the functional elements and
components of the engine oil system are listed as follows:
From probability theory, as a result of the multiplication
theorem and the formula for the total probability theorem is known
of the hypotheses (Bayesian formula). For diagnostic purposes, are
attempts to use the hypotheses of the theorem and its application to
technical objects, but they are not in widespread use.
- Failure of the reducing valve of the engine oil MA-78 – 13
pcs;
- Failure of the fuel oil cooler – 9 pcs;
Bayesian formula has the form:
P S
∗
3. Indications of failure of components and
products of oil system the engine TV3-117
If deterministic methods to identify the task of forming a
geometric language is to partition the space of signs in areas of
states (diagnoses).
P S /k
∗/
If a symptom occurs, then obviously it indicates a faulty item,
unit or product. Much more complicated is the situation when the
defect is manifested by several signs. In such a case, even a highly
qualified specialist is not always able to determine the cause of the
malfunction. Further inspection, control, time and material costs.
Problems associated with determining the causes of failures and
malfunctions can be resolved with the use of detection methods.
This allows to shorten the time to search for reasons and to reduce
material costs.
An important part of recognition is the choice of parameters
describing the state of the object. They must contain sufficient
information to allow implementation of the recognition process for
the selected number of diagnoses [2,4].
(1)
∗
for (i = 1, 2, ..., n), where: P(Si/K*) is the probability of diagnosis Si
after becoming the results of research on the complex of signs K,
P(Si) is the probability of occurrence of state Si, determined by
statistical data (a priori probability of occurrence of the condition).
* Indicates a specific index value (realization) of the trait.
One of the main tasks of technical diagnostics is a recognition
of the technical condition of the site in terms of limited information.
The state analysis is performed in terms of operating mode, which is
extremely difficult to obtain comprehensive information.
/
/
- Failure of the magnetic chip detector CC-78 – 6 pcs;
- Failure of induction sensor ИД-8 oil pressure – 5 pcs;
,
59
І – Occurrence of a sign in a state of disrepair;
- Refusal of the receiver П-2Тр oil temperature – 5 pcs;
II – Occurrence of a sign in two faulty conditions;
- Refusal of the directory УИЗ-3 oil pressure and temperature –
3 pcs;
III – Occurrence of two signs in a state of disrepair;
- Cut valve malfunction – 3 pcs;
IV – Occurrence of two signs in two faulty conditions;
- Destruction of gears in the box of the drive motor – 2 pcs;
V – Occurrence of two signs in three faulty conditions;
- Failure of the thermostatic valve – 2 pcs;
VI – Occurrence of three signs in two faulty conditions.
- Failure of returning pumps – 2 pcs.
In each option is necessary to consider the alleged defective
condition and signs with regard to their possible manifestation. It is
therefore appropriate to define the conditions under which signs
will be considered cases- I, and failed states - such as variations- II.
With the emergence of the above failures of functional elements
and components of the engine oil system crew observed and
registered the following signs of failure:
The conditions are determined by the need to perform the
following operations:
- Oil pressure is lower than normal – 18 pcs;
- Oil temperature is higher than normal – 12 pcs;
I. The implementation of the estimate on cases can be classified
- Dashboard lights “Chips in the engine” – 7 pcs;
as:
- Presence of smoke and oil leak from the output device for
stopping the engine – 4 pcs;
І-а) The co-occurrence of all signs;
І-b) Implicit in occurrence (no occurrence) at the first sign.
Implicit occurrence of the trait means that this feature is less
pronounced;
- Oil pressure at УИЗ-3 dropped to “zero” – 3 pcs;
- Increased oil consumption – 2 pcs;
- Emergence of oil vapor and odor of smoke in cockpit – 2 pcs;
І-c) Implicit in occurrence (no occurrence) on the second sign;
- Occurrence of metal chips in the oil – 1 pc;
І-d) Implicit in occurrence (no occurrence) of both signs.
II. Consider in each case variations (II) to the following
conditions:
- Oil pressure is higher than normal – 1 pc.
If a symptom occurs, then obviously it indicates a faulty item,
unit or product. Much more complicated is the situation when the
defect is manifested by several signs. In such a case, even a highly
qualified specialist is not always able to determine the cause of the
malfunction. Further efforts are needed inspection, control, time and
material costs. Problems associated with determining the causes of
failures and malfunctions can be resolved with the use of detection
methods. This allows to shorten the time to search for reasons and
to reduce material costs [5].
ІІ-а) considered the first state of disrepair (S1);
ІІ-b) the second considered disrepair (S2);
ІІ-c) considered the third state of disrepair (S3).
These conditions are only applicable to this option, which
simultaneously exhibit two or more attributes in two or more faulty
conditions.
Any disrepair should be considered in all four cases: I-a), I-b),
I-c) and I-d) – the developer or occurrence of symptoms.
4. Conditions for the realization of a mathematical
model to determine the faulty conditions
For convenience in handling diagnostic matrix is constructed
from the general form presented in the Table. 1, which consists of
columns, giving the specific signs and in the last column – the
probability P(kj/Si), and lines are placed in the values of the
probabilities of faulty conditions.
With the emergence of the above failures probability of
occurrence of the complex K* of signs could be determined by the
expression
(3)
∗
∗
∑
.
/
Table 1: Diagnostic matrix of general form
Probability of occurrence of the sign kj
by k1
by k2
by k3
P(k1/Si)
P(k2/Si)
P(k3/Si)
For the complex of symptoms of generalized Bayesian formula
can be written as:
(4)
/
∗
∗/
∑
∗/
Faulty
conditions
Si
,
S1
S2
S3
while
(5) ∑
/
∗
1,
P(k1/S1)
P(k1/S2)
P(k1/S3)
P(k2/S1)
P(k2/S2)
P(k2/S3)
P(k3/S1)
P(k3/S2)
P(k3/S3)
P(Si)
P(S1)
P(S2)
P(S3)
Table 1 is used for obtaining the necessary dependencies
summary account of the Bayesian formula.
i.e. one of two states necessarily be realized while simultaneously
implementing the two conditions is impossible.
To examine the expression of the most complex Vth option –
occurrence of two signs (k1 and k2) in three non-compliant states
(S1, S2 and S3).
Basic can be used to apply the theorem of hypotheses, but its
direct use would not give the desired result. To achieve this result it
is necessary to make several changes, transformations and
generalizations of the Bayesian formula to determine the conditions
of application necessary to construct mathematical models estimate
and to establish an algorithm for search problems [1,2] .
The Vth option should be considered in four cases in occurrence
or not occurrence the two signs in three variations, i.e. three faulty
conditions.
In practical tasks, especially when many signs of failures and
malfunctions of individual functional systems, components and
assemblies applicable condition of independence between the signs,
even in the presence of correlation between them [3,5].
Variation II-a) - the first state of disrepair (S1): for case I-a) - in
occurrence simultaneously on two grounds (k1 and k2) in disrepair
(S1). Using the generalized Bayesian formula [2,4,8] can be
obtained the following expression for P(S1/k1k2):
By revealing the theorem's hypotheses can identify possible
options and combinations of signs of failed states:
60
(6) P(S1/k1k2) = P(S1)P(k1/S1)P(k2/S1) /
[P(S1)P(k1/S1)P(k2/S1) + P(S2)P(k1/S2)P(k2/S2) +
+ P(S3)P(k1/S3)P(k2/S3)]
5. Conclusion
Received expressions of type (6) represent the mathematical
model for determining the state of disrepair. This model can be
applied not only to each functional system of aircraft or aviation
gas-turbine engines, but also to any technical system in general.
In the numerator of expression (6) is the product of significance
P(Si) - probability of occurrence of failed it i-state (in this case-S1) P(S1), with significance P(K*/Si) - probability of occurrence of the
complex of signs (in the case - both signs of occurrence-k1 and k2)
in disrepair (the case-S1) - P(k1k2/S1) or P(k1/S1)P(k2/S1). Thus, the
numerator will be a phrase P(S1)P(k1k2/S1) or P(S1)P(k1/S1)P(k2/S1).
6. Acnowledgement Researches, the results of which are presented in this
publication are funded by internal competition TU-2012.
In the denominator is the sum of the product of the values of
P(Si) - probability of occurrence of a combination of failed states (in
this case - S1, S2 and S3 – determine the number of addends) - P(S1),
P(S2) and P(S3), the importance of P(K*/Si) – probability of
occurrence of the complex of signs (in the case - both signs of
developing- k1 and k2), the combination of faulty conditions (in this
case – S1, S2 and S3): P(k1/S1)P(k2/S1)+P(k1/S2)P(k2/S2)+
P(k1/S3)P(k2/S3). As a result, the denominator is obtained expression
P(S1)P(k1/S1)P(k2/S1)+P(S2)P(k1/S2)P(k2/S2)+P(S3)P(k1/S3)P(k2/S3).
7. Bibliography
1. Асенов С. Избор на основни и частни показатели на
експлоатационната и ремонтна технологичност. Юбилейна
научна сесия „40 години от първия полет на човек в космоса”,
ВВВУ „Г.Бенковски”, 2001 г.
2. Венцель Е. Теория вероятностей, 10-е издание,
стереотипное, Издательский центр „Академия”, Москва, 2005,
572 стр.
After replacing the considered expressions obtained dependence
presented in (6).
3. Гмурман В., Руководство к решению задач по теории
вероятностей и математической статистике, 11-е издание,
переработанное, ООО „Высшее образование”, Москва, 2008,
480 стр.
With a similar action and reflection can be obtained the
following expressions: for case I-b) - implicit in developing the first
/ ; for case I-c) – implicit in
sign in disrepair S1 –
developing the second sign in disrepair S1 –
/
; for if I-d)
- implicit in the development of both signs in disrepair
/ .
4. Гмурман В. Теория вероятностей и математическая
статистика, 12-е издание, переработанное, ООО „Высшее
образование”, Москва, 2008, 480 стр.
Having performed similar computational operations Variation
II-b) will obtain the expression for the second fault condition (S2)
and variation II-c) – the third state of disrepair (S3).
5. S. Brandt. Data Analysis Statistical and Computational
Methods for Scientists and Engineers, Third Edition, SpringerVerlag, New York, 2003, 686 p.
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