Simulation of continuous random process according to the specified sequence of extremes

A method has been developed for converting a discrete sequence of extrema into a continuous process. The relevancy of the problem is attributed to the necessity of an approximate estimation of spectral density in in testing materials and structures under random (irregular) loading. A great bulk of available experimental data thus can be used in development and validation of calculation methods for assessing durability in the multi-cycle region. Postulating the continuity of random stress processes and their first derivative we propose to connect piecewise the available starting points (namely, the extrema of the random process) with half-cosine functions under the condition of compatibility at the points of extrema. A distinctive feature of the method is the provision of 100% coincidence of the values and sequences of extrema in the initial discrete and simulated continuous processes. The issue of choosing the magnitude of half-periods for these half-cosine functions is addressed on the basis of information obtained from the analysis of real stress records in the form of a regression equation linking half-periods and half-ranges for some realizations of the random process for transport vehicles. The regression dependences of the half-periods and semi-ranges of bending stresses (part of a railway train) and torsion (torsion shaft of a tracked vehicle) are shown as an example. An analysis of the correlation of two random variables (half-periods and half-ranges) according to empirical data has shown that the correlation exists and is significant for the observed number of points thus providing the basis for using the regression formula for an approximate choice of the frequency composition of the process. Moreover, the lower restrictions are imposed on the number of points (at least 5) in the half-period. Since the extrema of the initial and simulated processes coincide in accordance with the principle of the proposed simulation, the distribution of the amplitudes of complete cycles, as well as the results of schematization by other known methods are identical, therefore, the estimate of the durability by hypotheses based on a linear one is also identical. The validation of the method consists in consideration of the chain: 1) the initial continuous process; 2) the discrete process of extrema; 3) simulated continuous process according to the proposed method. Auxiliary distributions, such as distributions of maximum, minimum and average cycle values also coincide in accordance with the principle of modeling. The method is proposed to be used in analysis of the comparability of two competing approaches in assessing the loading in the problems of assessing durability, namely: those that use cycle-counting methods and methods based on the spectral density of processes. Since the spectral densities of the processes can differ due to an approximate choice of the frequencies on the basis of a regression formula, methods on their base can give estimates of the durability that differ from those obtained by schematization methods. To study this phenomenon, further computational experiments are required. The developed method can be very useful for the experiment design.

1. Tucker L., Bussa S. The SAE Cumulative Fatigue Damage Test Programm / In book: Fatigue under Complex Loading. Ed. R. M. Wetzel. — Society of Automotive Engineering, 1977. P. 1 – 44.

2. De Jonge J. B., Schütz D., Lowak H., et al. A standardized load sequence for flight simulation tests on transport aircraft wing structures. — The Netherlands: NLR TR 73029U, National Aerospace Laboratory (NLR) — Report TR 73, 1973 (TWIST).

3. Fischer R., Haibach E. Modeling functions loading in experiments on the evaluation of materials / In: Behavior of steel under cyclic loads / Dahl V., ed. — Moscow: Metallurgiya. 1983. P. 368 – 405 [Russian translation].

4. Lebedinkii S. G. Design modeling of propagation of the fatigue cracks in the steel of molded parts of the railway structures / Journal of machinery manufacture and reliability. 2018. Vol. 47. N 1. P. 62 – 66. DOI: 10.3103/S1052618818010107.

5. Kogaev V. P. Strength calculations at stresses variable in time. — Moscow: Mashinostroenie, 1993. — 364 p. [in Russian].

6. Yuanpei Hu at al. Effects of loading frequency and loading type on high-cycle and very-high-cycle fatigue of a high-strength steel / Materials. 2018. Vol. 11. P. 1456. DOI: 10.3390/ ma11081456.

7. Marques J., Benasciutti D., Tovo R. Variance of fatigue damage in stationary random loadings: comparison between time- and frequency-domain results / Proc. Struct. Integr. 2019. Vol. 24, P. 398 – 407. DOI: 10.1016/j.prostr.2020.02.037.

8. Mršnik M., Slavič J., Boltežar M. Vibration fatigue using modal decomposition / Mech. Syst. Signal Proc. 2018. Vol. 98. P. 548 – 556. DOI: 10.1016/j.ymssp. 2017.03.052.

9. Cetrini A., Cianetti F., Letizia M., et al. On-line fatigue alleviation for wind turbines by a robust control approach / Int. J. Electr. Power Energy Syst. 2019. Vol. 109. P. 384 – 394. DOI: 10.1016/j.ijepes.2019.02.011

10. Böhm M. Fatigue life assessment with the use of spectral method for materials subjected tostandardized wind loading spectrums / AIP Conf. Proc. 2018. Vol. 2029. #020005. DOI: 10.1063/1.5066467.

11. Gadolina I. V., Lisachenko N. G., Svirskiy Yu. A., et al. The choice of the sampling frequency and optimal method of signal digital processing in the problems considering random loading process for assessing durability / Zavod. Lab. Diagn. Mater. 2019. Vol. 85. N 10. P. 64 – 72. DOI: 10.26896/1028-6861-2019-85-7-64-72 [in Russian].

12. Gadolina I. V., Dubin D. A., Serebriakova I. L., et al. Stabilization of the Design Loading Characteristics in the problem of durability estimation of tracked vehicle parts/ Reliability, strength, and wear resistance of Machines and Structures. 2020. N 1. P. 31 – 37. DOI: 10.3103/S1052618820010069.

13. Interstate Standard GOST 25.101–83. Strength calculation and testing. Representation of random loading of machine elements and structures and statistical evaluation of results. — Moscow: Standartinform, 2005. — 25 p. [in Russian].

14. Gadolina I. V., Gryzlova T. P., Dubin D. A., et al. Studies of the loading of transport vehicles in time and frequency domain / Proceedings of the 4th international scientific-technical conference dedicated to the 80th anniversary of IMASh. Survivability and structural materials, 2018. — Moscow: IMASh RAN. P. 84 – 86 [in Russian].

15. R Core Team (2020). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org.