PSI - Issue 50

E.A. Afanaseva et al. / Procedia Structural Integrity 50 (2023) 1–5 Afanaseva E.A and Afanaseva O.S. / Structural Integrity Procedia 00 (2023) 000–000

3 3

3. Approbation of the method

Let’s demonstrate that the algorithm of individual prediction by the leader-item described in 2 can be applied, for example, to the process of friction of the coupling nodes of the landing gear of the aircraft. The size of the gap in the interfaces ∆ (microns) measures in ”time”, the role of which is played by take-o ff and landing cycles. Information about the nature of the dependence ∆=∆ ( t ),where t is the number of take-o ff and landing cycles, and the magnitude of its spread is provided by experimental data on the wear of the front landing gear axle boxes for ten di ff erent items, which are given in Gromakovskii et al. (1988) and shown in Fig. 1. The analysis of experimental data showed that the values of the coe ffi cients of the correlation matrix have an order of 0 . 95 ÷ 0 . 98 which is the justification for the use of the leader prediction method. Sample No.2 was used as a leader-item as a result of random selection. The take-o ff and landing cycles of t ∈ [0; 1200] was used as an observation base at the initial stage of operation. According to the above methodol ogy, all parameters of the model (2)-(6) were determined. In Fig. 2, solid lines show experimental data, dashed-dotted lines — the results of the forecast based on (4), and dashed — 95% confidence interval for mathematical expectation.

Fig. 1. Wear curves of the front landing gear of the aircraft depending on take-o ff and landing cycles

As can be seen from Fig. 2, using the methodology for constructing the confidence interval (5), (6) which uses the classical definition of variance, wide confidence intervals were obtained for samples No. 3, No. 6 and No. 9. The above ”classical” method of individual prediction of generalized displacement based on (1)–(6) has one draw back: the forecast is not tied to the final experimental value of the movement at the end of the base time interval t = t n and, accordingly, the first point of the prediction interval ( t > t n ). And if the experimental error and other random fac tors are not taken into account, i.e. the measured experimental value of the generalized displacement at the point t = t ∗ n is considered accurate, then the confidence interval at this point should be zero. Therefore, we modify the (1)–(6) as follows: since the value of the generalized displacement at the point t = t n is known, it is necessary to estimate the standard deviation taking into account the conditional probability Venttsel’ and Ovcharov (1969). Then instead of the value of the standard deviation s 0 in (5) you need to use the value s ∗ = s 1 · 1 − ( r ∗ ) 2 , r ∗ = n i = 1 ( p i − ˆ p )( p 0 i − ˆ p 0 ) n i = 1 ( p i − ˆ p ) 2 · n i = 1 ( p 0 i − ˆ p 0 ) 2 , (7)