METHOD OF THE EXCESSION
FOR RESOURCE
RESEARCHES OF RISK
TECHNICAL SYSTEMS
Nikolay
Ivanov Petrov
Technical University – Sofia
Abstract: - Irrespective
of rising of the producing responsibility of the functional elements of the risk technical systems (automobiles,
railway and marine transport, aircrafts, chemical installations and munitions,
informational society tortured by terrorism) is doing rising of their volume
and complexity. That is provoked by the necessity for uninterrupted
modernization of the management and communication systems. At the same time the
tiredness of the material and the growing old are the reason for the appearance
of casual and sudden failures, provoking
accidents and crashes. That is way the rising of the exploitation reliability
of the risk technical systems (RTS) is connected to the minimization of
the control time of their working capacity, fast diagnosis of the failure at
them and doing of resource examinations (defining of the lasting technical
resource till the consecutive basic, middle or flowing repair). At the statue
is suggested a method of the excess ion for doing a resource examinations. It
is based on the equality of the initial reactions of the examined non-linear
system (RTS) and the received by analytical iterations linear systems, at the
entry of which activates another reaction, only for the holomorfs systems. That
reaction is determined by the analytical connection between the two systems.
Key words: resource
examinations; method of the excession; risk technical systems
Introduction: In the middle and
the last period of a technical usage, the change of the intensity flow of
failures (IFF) marked with 
is characterized with expressed
non-linearity and determinates by formulas [4, 5, 6, 7, 12, 14]:
(1) 
,
where: 
is basic number
RTS by respective tape objects, observe in a interval 
; 
is summarized
number renewable and un-renewable failures in a interval ; 
failure
time of 
th object of RTS for the time 
; 
- summarized failure time of a group
objects off relevant tape RTS, observed in interval time .
The non-linear
evaluation of the graph of 
determinated by
usage conditions is
shown by fig. 1.
Fig. 1. Perfectly graff of the evaluation of
the
intensity flow of failures of RTS
It is necessary to
formulate an analytical mathematic model for description of the alteration
dynamic of the intensity of failure’s stream intensity at the middle and limit
interval of the technical usage (TU).
Problem Solution: The intensity of
the flower failures 
could be presented at the
process of TE using the uninterrupted and uniquely determined function 
at the observation interval [5, 12, 14]. In
the fig. 1 it is shown technical recourse (TR) until the final of
TU 
, the remained TR 
after 
moment and the
permissible value of the IFF 
for the whole period
of TU. Let the value of the at the moment 
(fig. 1) be known, i.e. 
.
Then the function
could be expanded in a
(2) 
,
with limited
quantity of non-linearity terms for
every current moment 
.
Equation (2) is presented in the following way:
The expression in middle brackets we mark
with 
, i.e.:
With a non-linearity use process, mutuality correlation
of the every next value of 
(fig. 1), and that
derivates in current moment 
, must be admissible accuracy will by evaluation with
equation (3):
(3)
,
where: is a parameter,
which is determinate by the asymmetric of the trajectory recourse in comparison
with the case of the regular expense of the recourse and has dimension [
].
The
current value of the parameter , could be determinated by two measurements in
sufficient large interval
(fig. 1 ) [with expressive evaluation of the IFF], with
equations:
(4) 
,
(5) 
[ t/fit ].
The product 
, where 
, characteri-zes the current (for moment 
) excess (asymmetric),
of the ratio 
. It has dimension of the derivate in point 
.
The current excess for
random moment in the time , is made for
a-priory value of by equation (2.1.4) and is determinated by the result:
(6) 
, 
Corresponding to equation (3), it follows:
(7) 
,
where 
is determinated from:
, 
.
In equation (7), 
is
remainder resource of
the final of TU
(fig. 1); is total resource specifying the
works producer RTS, of the final TU by CR;
is an interval of the
value of IFF 
in moment 
(fig. 1),
when 
. The ratio (
) is shown in equation (6) of the present
paragraph with purpose to normalize the next equation.
In an open mode functional , in case of irregular expense of recourse (3)
and functional 
, in case of the regular expense of recourse, are done
correlations with equation (8). This is on the base of the
axiom for regular and invariant a
priory – a posteriori determination of a
(8) 
,
where: 
are the permissible values of
the IFF , determinated by norms of flay serviceability [21, 22,
23].
These parameters
are determinated on the affirmative in contemporary aviation standards staging
for relatively constancy of , in the whole period of service and exploitation. They
determine the regular expenses of resource in this period. After the transfer of equations for and in equation (8), it is presented a posteriori evaluation
of , with the
moment 
(factual a posteriori
with moment 
) in mode of the non-linear
differential equation:
(9)
,
where parameters and 
are determinated with
functional of a time 
, according to
equation (7):
(10) 
.
(11) 
; 
.
The parameter
in equation (10), determinates the coefficient
of excess, and the asymmetric of the resource trajectory in actual case of TU (fig. 1) in comparison with case of the regular expense of resource in
perfect case. He has dimension 
.
The parameter is determinated while index
of convergence of the recourse trajectory in actual case of TU, in
comparison with case of the regular expense of resource in perfect case with 
. In the non-linear differential equation (9) parameters and , respecting coefficient of
excess 
, are determinated by current value of and his hodograph is
shown in fig. 1. It this way, with service and use of the modes RTS
by current technical state, it is secured an adequacy of the value of non-
linearity of the use process.
The determination
of the equation (9), with sufficient analytical approximation, must by get by method,
developed in [22, 27]. It is based on the identicality of output reaction of
research non-linear system, equation (9) and getting with analytical iterations linear system (or order
with analytical inerrability equation). The input of the linear system has
influence, unity for holomorphic systems and determination of the analytical
value between two systems.
The solution of
the equation (9)
1. The
first analytical iteration of the solution of the equation:
The member of the left side of equation (9)
we replace, with condition 
. Then we get:
,
(12)
.
The
determination, of the equation with given first condition (first analytical
iteration) is:
(13) 
,
where: 
.
2.
The solution of the complement of the influence
(right side of the equation (9) by
liberalization), with purpose getting of the second analytical iteration
(14) 
While it is supplemented 
by (14) in right side of equation (9) and doing
substitution in first member of the left
part of (9) by convention 
, it is obtained the next equation, with general mode:
,
and general mode:
(15) 
.
3. The solution of the equation (14), represents second analytical iteration
(16) 
,
.
Until the getting
of the second determination (16), functional 
is expansed in a Mac Loren series with accuracy of the fifth
member 
[22].
4. The third
analytical iteration - determination of the addition of the influence
for the third analytical iteration
(17)
Supplement and 
in right side of
equation (2.1.8), but first member in left side of (2.1.8) is changed with 
. It is obtained the equation:
that in distribution mode has the form:
(18)
.
5. Solution of the
equation (2.1.17), with third analytical iteration:
(19) 
,
where: 
- is determinated as a non-dimensional quantity.
The equation (19) could be represented in the following way:
(20) 
,
where: 
After replacing 
on the left side of (20) it forms the equation:
(21) 
,
by which at first iteration
we get the value of the remainder TR, with TU by technical station:
(22) 
,
where: 
.
Whit getting the second equation (22), 
is determinated with 
, and expression:
,
is agent with 
. This transfer is actual if 
.
6. The
analyze of the accuracy of third analytical iteration
With analytical
iteration from kind (16) and (19) in the solution (19) there are shown members from produces of 
and 
with grow degree
(higher than forth degree). This
member is little with realization of condition:
(23) .
According to equation (20), repair is depended from the
time . His iteration value
could be determinated from the equation (20), after determination of the time 
by the equation:
(24) 
and replacing in (22). In the realization of 
, the complement iterations practical aren’t necessary.
In non-execution
of that condition, the repairing of the remainder recourse, determinates iteration by the
(25) 
,
where:
,
,
.
In the realizing of the
condition 
, complement iterations aren’t necessary and it is
realizable 
. In contrary a case it is repaid procedures (25) at replacing
the time with 
.
7. According the
equation (22 ), determination of the
remainder resourse of the RTS, consist in determination of the current value of
parameters and .
The general
resource 
in regulare service
and use, is shown in passports of the RTS. The permissible IFF , respective 
is determinated
according to paragraph §2.2.
8. According to
equation (22 ), in the values of the multiplier
, the remainder resourse of the RTS is equal to the regulare
remainder resourse 
by the producer.
Respectively in:
·
Realizing of condition 
it is possible an
increasing of the TR with the time interval 
;
·
Realizing of condition 
it is necessary
decreasing of the TR with in the time interval 
, because of his quickleness to spend in comparison
with forecast by the producer.
The
presented analytical algorithm, could be used for control of the between
repair, regulate and control intervals
of different kinds technical systems and used
mixed systems for TS
[22].
In that case, in
the equation (22) follows to be
substituted with the remainder regulated between repair,
regulates and controls intervals, and must be substituted by
the admissible average of IFF at the end
of the relevant time interval.
At an applied
plan the method demands the little time intervals , and actualizing the current value of the coefficient of the
excess of the curve of intensity of the failures’ stream. The
defining of the coefficient of the excess is doing according
formula (11) by the current evaluation of IFF.
In comparison
with the popular algebra methods from [2; 12; 13; 14; 17; 18] for a valuation
of current technical condition and
prognostication of the remainder resource and
between repair intervals, developed method for differential stochastic
prognostication (method of excess) is characterized by the following:
Conclusions
The suggested at the paper
method of the excess ion for resource investigations of RTS, suggests the
following priorities:
1. Make a possibility
for determination of the asymmetric of resourse trajectory in comparison with case of a regular spending of
TR.
2. Remove is comparative complicated calculations (solving of systems by algebraic equation), as the analysis is made by current determination of the same parameter (coefficient of excess).
3. The algorithm
for determination of the remained resource or between repair intervals is shown at Appendix 1 and Appendix
2 at literature [13] and
is characterized with engineering simplicity, which permits its realization at conditions of TU, without
complicate calculating configurations.
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