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 [45, 67, 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 RTSobserved 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 knowni.e. .  

Then the function  could be expanded in a Taylor’s series [2]:  


(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 Taylor series [19, 22]:     


(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:






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






          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 




          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:  


(19)            ,   


where:  - is determinated as a non-dimensional quantity.


          The equation (19) could be represented in the following way:        



(20)                         ,                   




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:    




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 Newton’s formula [27]:



(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:




          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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