PSI - Issue 20

Gusev E.L. et al. / Procedia Structural Integrity 20 (2019) 294–299

297

Gusev E.L. et.al. / Structural Integrity Procedia 00 (2019) 000–000

4

*

*

* N N R u t R t  * ; )

* ( ) min max ( ; ) N N 

* R u t R t 

max (

( ) ,

N  

1

min T t T  

min T t T  

max

max

* ( ; ) R u t R t ( ) N N

*  

min T t T   max

,



(3)

max

max

N   

1

.

In these designations: N* - the optimal number of parameters of the prediction model of optimal complexity * N R ; * N u - the optimal vector of parameters of the optimal forecasting model of the parametric family corresponding to the parameter N, : γ max - the maximum allowable required precision of the solution of the problem of forecast; R*(t) - the real dependence on the time determining the properties of the composite (residual life, durability, etc.) 5. Comparative calculations on the basis of predictive models of optimal complexity Comparative computational experiments have been carried out within the framework of refined variational formulations of inverse prediction problems on the basis of the principle of multiplicity of prediction models and the introduced concept of prediction models of optimal complexity, by Gusev (2016, 2018), Gusev and Babenko (2015), Gusev and Bakulin (2017, 2018). Based on comparative computational experiments the study was conducted forecasting of a residual resource polivalente composites (PVC) are susceptible to aging processes, by Bulmanis and Startsev (1988). The strength of the composite (R) was considered as a residual resource. On the basis of the developed multivariable prediction models (1), (2) the estimation errors of prediction made when the use of models of longevity with a small number of parameters, such as model by Bulmanis and Startcev (1998), where the number of parameters does not exceed four, and models of durability with the optimum complexity of the optimal number of parameters, by Gusev (2016, 2018), Gusev and Babenko (2015), Gusev and Bakulin (2017, 2018), based on the solution of extremal problems of the form (3). It is shown that the use of durability models with a small number of parameters, such as the model Bulmanis V.N. may not be adequate to the complexity of the tasks of forecasting and lead to significant errors in the forecast. Choosing as system support functions in the presentation of descriptions, impact-sponding polymer, composite materials factors Fj as a series system of exponential functions, describing the nature of the aging processes occurring in the composite , in accordance with the equations of Arrhenius, receive the following specific prediction model of the form (1) ,(2)   0 2 1 2 1 (u ; ) exp( ) 1 . n k k k R t R u u t       (4) For Fig. 1 the results of the comparative analysis of the residual resource prediction results are presented on the basis of the optimal prediction model of the optimal complexity corresponding to the optimal number of parameters (curve 2), and on the basis of the prediction model corresponding to a fixed number of parameters n=4 . Curve 1 is the real time dependence of the residual resource. A vertical line separates the time interval on the left that precedes the prediction and the time interval of the spring on which the prediction is made. Experimental measurements of the residual life of the composite R1, R2,..., Rm were carried out at the time interval of retrospection [0, Tmin], where Tmin=15, in years t=0,1,2,...,Tmin=15. On the basis of the results of the experiments, a model of optimal complexity was constructed, which is the solution of an extreme problem of the form (3). Forecasting based on the forecasting model of optimal complexity and optimal number of parameters (the optimal number of parameters n*=7) was carried out on the temporary prediction interval [Tmin, Tmax] following