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 intensity flow of failures (IFF) marked with is characterized with expressed non-linearity and determinates by formulas [4, 5, 6, 7, 12, 14]:
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 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
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):
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:
(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:
Corresponding to equation (3), it follows:
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
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:
where parameters and are determinated with functional of a time , according to equation (7):
(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:
The determination, of the equation with given first condition (first analytical iteration) is:
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
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:
3. The solution of the equation (14), represents second analytical iteration
Until the getting of the second determination (16), functional is expansed in a Mac Loren series with accuracy of the fifth member .
4. The third analytical iteration - determination of the addition of the influence for the third analytical iteration
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:
5. Solution of the equation (2.1.17), with third analytical iteration:
where: - is determinated as a non-dimensional quantity.
The equation (19) could be represented in the following way:
After replacing on the left side of (20) it forms the equation:
by which at first iteration we get the value of the remainder TR, with TU by technical station:
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:
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:
and replacing in (22). In the realization of , the complement iterations practical aren’t necessary.
of that condition, the repairing of the remainder recourse, determinates iteration by the
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.
· 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 .
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:
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  and is characterized with engineering simplicity, which permits its realization at conditions of TU, without complicate calculating configurations.
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