Research in Area of Longevity of Sylphon Scraies

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1 IOP Conference Series: Earth and Environental Science PAPER OPEN ACCESS Research in Area of Longevity of Sylphon Scraies To cite this article: Natalia Y Golovina and Svetlana Y Krivosheeva 2018 IOP Conf. Ser.: Earth Environ. Sci View the article online for updates and enhanceents. This content was downloaded fro IP address on 11/07/2018 at 16:39

2 IOP Conf. Series: Earth and Environental Science 115 (2018) Research in Area of Longevity of Sylphon Scraies Natalia Y Golovinа, Svetlana Y Krivosheeva Industrial University of Tyuen, the Branch of IUT in Surgut, Entuziastov Street, 38, Surgut, Russia E-ail: ntgolovina@rabler.ru Abstract. The proposed ethod of deterining the sensitivity threshold for bellows expansion copensator by fitting a power polynoial of the epirical curves of distribution of durability for a large nuber of saples, followed by statistical analysis. For a reliable justification of resource products with regard to dispersion of their durability is of great iportance the evaluation of the sensitivity threshold in cycles. Currently, the appointent of a guaranteed resource is ade with high strength requireents. The proposed ethod allows to define sensitivity threshold value for the cycles between the curves of the distribution of longevity as having a distinct threshold, and not having it. Method is in good agreeent with other known ethods. Thus, by statistical processing of results of fatigue tests of joints obtained values of the sensitivity threshold in cycles and confidence interval for atheatical expectation of the agnitude of the threshold. Introduction Setting the reliable indicators is deterined largely by a distribution function of operating tie to failure. The value of operating tie to failure in the increasing order reflects the epirical distribution function of endurance. For reliable study of ites resource with a view of the dispersion of its endurance it is iportant to estiate the threshold of sensitivity to N 0 cycles. Currently, the appointent of the safe life is ade without taking into account the values of N 0. Consequently a large endurance argin n N is necessary in order to account the dispersion of endurance. Safe life based on the average rated life is deterined fro the expression: N safe life = N aver rated n N (1) As a rule such setting of safe life leads to increased strength properties of ites and to increasing of laboriousness, etal consuption and excessive ass. Deterination N 0 values allows us to justify the safe life of ites. Researchers have found that the endurance has lower and upper boundaries under constant axiu voltage [8]. The lower boundary of endurance N 0 denotes the threshold of sensitivity to cycles, i.e. the largest possible nuber of cycles, Ϭ ax scarcely causes daage at a given voltage. The value N 0 separates insensitive zone of operating tie to changing load. It is difficult to estiate the N 0 epirically, as a considerable nuber of tests is required. Even ore difficult it is to deterine the threshold of sensitivity at relatively low voltages due to the large test duration. Materials and ethods for solving the proble There are currently several ethods for deterining the threshold of sensitivity: graphic, syetrical for bringing, least squares, axiu likelihood. However, the effectiveness of these ethods is not resolved until now. Application of these ethods to the results of fatigue tests of syphons has not given results: Content fro this work ay be used under the ters of the Creative Coons Attribution 3.0 licence. Any further distribution of this work ust aintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd 1

3 IOP Conf. Series: Earth and Environental Science 115 (2018) The ethod of reducing to the syetrical for is applicable only to a coparatively large volues of saples (ore than 60), so it was not considered. 2. The ethod of least squares, as well as its graphic expression a graphical ethod gives a solution only for the curve distribution with clearly defined lower boundary of endurance and not clearly upper liit. If endurance curves do not coply at least with one condition it eans that the values N 0, calculated on these ethods becoe unrealistic and even negative. Figure 1 shows the distribution curves of endurance [9]. The ethod of least squares does not give solutions for the curves 1, 2, 3, the solution for the curve 4 atches the value N 0 = cycles. 3. The axiu likelihood ethod does not give convergence solutions for saples less than 36, including curves (Figure 1). The need for solving this proble is recognized by the authors [1]. 4. An approxiate ethod [7] does not take into account the conditions of existence of the threshold of sensitivity; therefore it gives lower results (straight lines in Figure 1) P, % Figure 1. Distribution curves of endurance In this research, the sensitivity threshold deterination for copensator ade of steel 12X18H10T, is perfored by approxiation of exponential polynoial of epirical curves distribution of endurance for a large nuber of saples, followed by statistical processing. When the approxiation is used under condition of sensitivity threshold display on the lower and upper boundaries of the epirical distribution curves. Endurance test results of the products at a level of voltage Ϭ ax were located in the variation series in ascending order of endurance and were grouped by voltages with a deviation fro one another not ore than MPa. The value of the probability of destruction P, estiated at the accuulated frequency, was calculated by the forula [10] 3 N 4 P i = i 0.5 n i ite nuber in the variational series, n the total nuber of tested ites. (2) 2

4 IOP Conf. Series: Earth and Environental Science 115 (2018) As noted in [2-4], sign of the sensitivity threshold N 0 is the curvature of the curve P lgn, built on probability paper, bup up. Analysis of the data presented in [6], shows that the distribution curves of endurance in the liit at probabilities of failure P 0 and P 1 for a right angle with lgn axis. Thus, the proble reduces to a dot approxiating of the epirical distribution curves with zero derivative at the boundary points that are sought. To increase the accuracy, the approxiation was ade with the transition to a unifor probability scale. And the perpendicularity of distribution curve to the axis lgn in the liit is persists. The solution of the proble can be represented in the for of approxiating diensional polynoial. y(x) = a k x k x represents the values of P. We consider at P = 0, x = 0, N a = a 0. Naturally, on the segent N 1 N 2... N 0 epirical distribution function has a range of values, that eans that the values of the desired distribution curve were set with an error. To solve the proble it is necessary to construct a function that gradually took place nearby the predeterined values. Polynoial is used in this function, at which the iniu of the functional is reached [5], n+1 x n+1 S[f(x k ) = α [ f (x k )] 2 dx + P k [f(x k ) y k ] 2 x 0 α soothing paraeter; P k ass coefficient (P k > 0); y k corresponds to the values N k. (3) (4) Boundary condition for sought function will be: f (x 0 ) = j a j x j 1 0 = 0, f (x n+1 ) = j a j x j 1 n+1 = 0 (5) After differentiating the general expression for a syste of equations has the following for: 2i(i 1)j(j 1) S[f(x a k )] == α i i + j 3 i=0 n+1 a j + 2 P k ( a j x j k y k ) x i k = 0 k (6) Thus, for deterining + 3 the unknowns a 0, a 1,... a, y 0, y n + 1, we have + 3, equations. Taking into consideration that x 0 = 0, x n + 1 = 1, the final syste of equations for deterining the coefficients in a general for can be written as: α i=0 ij(i 1)(j 1) i + j 3 n+1 a j + P k a j x k i+1 y 0 x i i 0 y n+1 x n+1 n = y k x k i k=1 (7) a 1 = 0 ja j = 0 (8) 3

5 IOP Conf. Series: Earth and Environental Science 115 (2018) Accuracy of approxiation of a function to set values depends on the paraeter α and ass coefficient P k. The higher the ass coefficient P k, the greater contribution the interpolation conditions ake into the functional and the closer the soothing function passes to the set values. Since the tests were conducted on identical equipent, with the sae error, it is natural to take the values of the coefficients P k equal at all points of the epirical distribution curve. Selecting α soothing paraeter is a proble. At a low α value the soothing will be insignificant, at inflated α values the function will be extreely sooth. In this case, the choice of α paraeter deterines by the behavior of the first derivative of the approxiating polynoial, which should not be less than zero. As a result of the approxiation of the epirical distribution curves of the fifth degree polynoial the corresponding values of sensitivity thresholds N 0 were obtained. Figure 2 shows the results of calculations N 0 values, depending on the voltage σ ax. N 0 values can be viewed as a rando variable having a range of possible values. In order to get the averaged curve lgn 0 = f (σ ax), having a probability of P = 0.5, these presented results are approxiated by a polynoial in the eighth power (dash-dot line). σax, МPа P=99.9% 10 3 P=50% 10 4 N Figure 2. The epirical regression line for the sensitivity threshold N 0 and its bounder of the confidence region It is obvious that the resulting dependency without a large error can be replaced by a linear dependence, that is, the connection between the norally distributed rando variable y = lgn 0 and the value of x = Ϭ ax can be established by eans of linear regression analysis. The epirical regression line has the for Y = a + b (x-x ) (9) where Y - the evaluation of the conditional expectation value of y = lgn 0 for a set value x; x - saple average value of x. Paraeter estiation of the regression line was calculated fro the forula: x = 1 x i i=1 (10) 4

6 IOP Conf. Series: Earth and Environental Science 115 (2018) the nuber of voltage levels. Finally, for = 29, we have: or: a = y = 1 y i i=1 b = i=1 (x i x )y i (x i x ) 2 i=1 Y = (x ) lgn 0 = Ϭ ax (11) The regression line corresponding to the resulting equation is shown in Figure 2 (solid line). At probabilistic calculations of ite strength to enhance their reliability it is not appropriate to use the average sapled values of the rando variable, but the values of the boundaries of the confidence intervals, in particular, we should take the lower liit of the confidence interval instead of the saple average values of N 0. The boundaries of the confidence interval Δ for the expectation N 0 are deterined by the expression: Y- t α,k S y < < Y+t α,k S y (12) t α,k - student criterion for the significance level α and the nuber of degrees of freedo K; S y the variance of expected value estiation. On Figure 2 the lower liit 99.9% of confidence region with a significance level α = is arked by the dashed line. The upper liit of endurance was not viewed as it had no practical interest. Results The proposed statistical processing ethod allows to deterine the N 0 value for the distribution curves of endurance with a distinct sensitivity threshold, and without it. The ethod is consistent with other known ethods, there is a pronounced tendency of the distribution curve to N e. For exaple, for distribution curve 4 (Figure 1) N 0 = cycles (least squares N 0 = cycles). Conclusion Thus, by eans of statistical processing of the copensators fatigue test results we obtained the values of sensitivity threshold on cycles and the confidence interval for the expectation values of N 0. Disclosure stateent No potential conflict of interest was reported by the authors. References [1] Bezenov SA (2011) Methodical aspects of assessent of the fatigue resistance of etallic aterials characteristics. New Materials and Technologies in Metallurgy and Machinebuilding 1: [2] Golovina NY (2014), Sustainability issues forced transverse vibrations of flexible etal pipes. Nauchnoe obozrenie 10: [3] Golovina NY (2005), New approach in the study of forced transverse vibrations of flexible etal pipes for stability. Proceedings of the higher educational institutions. Oil and gas. 1: [4] Golovina NY, Krivosheeva SY (2015), The ipact of energy dissipation in the durability of flexible etal pipes. Nauchnoe obozrenie 12: [5] Marchuk GI (1980) Methods of Coputational Matheatics, 456. [6] Roessle, M.L., Fatei, A.: Strain-controlled fatigue properties of steels and soe siple approxiations. Int J Fatigue 2000;22:

7 IOP Conf. Series: Earth and Environental Science 115 (2018) [7] Serensen SV (1975) Bearing capacity calculations and achine parts for durability, 488. [8] Stepanov MN (2005), Statistical ethods for processing the results of echanical tests (directory), 399. [9] Stepanov M.N., Giatsintov EV (1973) Fatigue light structural alloys, 320. [10] Weibull W (1964), Fatigue testing and analysis of results,