Nikolay
Ivanov Petrov

*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

(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

** Problem Solution: **The intensity of
the flower failures

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

(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

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

In that case, in
the equation (22) _{} follows to be
substituted with the remainder regulated** **between repair,
regulates and controls intervals, and

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.

[1] *Avizienis, A. and L. Lapr. *Dependability
Basic Concept and Terminology. Springer Verlag,

[2]
*Bartsch**,** H. J.* Mathematische formeln. VEB
Fachbuchverlag, Leipzig, 1984.

[3] *Davidov**, **P**.* Technical
diagnosis of radio-electronic devices and systems. Publ. House “Sov. Radio”, Ìîscow,
1988.

[4] *Dimitrov**, **J**. *Reliability of the
real way transport.

[5] *Dimitrov**, Ê**.**, **D**. **Danchev**. * Reliability of the Constructor Machines
and System. Publ. House “Technique”,

[6] *Goble,
W.M., Tucker T.W.* Control system quality is free.

[7]
*Gindev, E.G. ** *Reliability of the
aviation electronic technique.

[8] *Gindev, E.G. *Introduction* *in Theory and Practice of the Reliability. Part 1. Basic of the Applied Reliability. Academic
Publishing “Prof. Marin Drinov”,

[9] *Gnedenko**, **B**.,** U. Beliaev,
**À. **Solaviev. *Mathematical
Methods in Theory of the Reliability. Publ. House “Nauka”, Ìîscow, 1965.

[10] *ISO 3534-1 Statistics*. Vocabulary and
Symbols. Part 1: Probability and General Statistical Terms, First edition 1993.* *

[11] *Kaplan, E., P.** **Meier.* Nonparametric
estimation from incomplete observations, *Journal of the American Statistical Associations*, Wiley,

[12] *Petrov, N.I. *Optimization
and Management of the Technical Usage of the Military Aviation Systems.
Dissertation for getting a scientific degree “Doctor Scientist”, Military Academy “G.St. Rakovski”, Sofia, 2001.

[13] *Petrov, N.I. *Use
Reliability of the Risk Technical Systems. “Asen Zlatarov” University,* *Publ. House
“Uchkov”,

[14] *Petrov, N.I. *Analytical
Model* *for Examination
of the Technical Condition of Aviation Systems. MCP’99, MCME’99,

[15] *Petrov, N.I.*
Method for Resolution of Nonlinear Differential Equations Describing the
Process of Exhaustion of the Technical Resource of Aircrafts. EDS-2002,

[16] *RTO lectures**.* Aging Aircrafts
Fleets: Structural and Other Subsystem
Aspects.