<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Publishing DTD v1.3 20210610//EN" "JATS-journalpublishing1-3.dtd">
<article article-type="research-article" dtd-version="1.3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xml:lang="ru"><front><journal-meta><journal-id journal-id-type="publisher-id">vrgup</journal-id><journal-title-group><journal-title xml:lang="ru">Вестник Ростовского государственного университета путей сообщения</journal-title><trans-title-group xml:lang="en"><trans-title>Vestnik Rostovskogo gosudarstvennogo universiteta putej soobŝeniâ</trans-title></trans-title-group></journal-title-group><issn pub-type="ppub">0201-727X</issn><publisher><publisher-name>Ростовский государственный университет путей сообщения</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="doi">10.46973/0201-727X_2026_2_194</article-id><article-id custom-type="elpub" pub-id-type="custom">vrgup-344</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="ru"><subject>МАШИНОСТРОЕНИЕ</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="en"><subject>MECHANICAL ENGINEERING</subject></subj-group></article-categories><title-group><article-title>Оценка остаточного ресурса конструкций при случайном нагружении на основе одномерной свёрточной нейронной сети</article-title><trans-title-group xml:lang="en"><trans-title>Residual life assessment of structures under random loading based on a one-dimensional convolutional neural network</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Ерпалов</surname><given-names>А. В.</given-names></name><name name-style="western" xml:lang="en"><surname>Erpalov</surname><given-names>A. V.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Ерпалов Алексей Викторович, старший научный сотрудник, кандидат технических наук</p></bio><bio xml:lang="en"><p>Erpalov Aleksey Victorovich, Senior Researcher, Candidate of Engineering Sciences</p></bio><email xlink:type="simple">erpalovav@susu.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Хорошевский</surname><given-names>К. А.</given-names></name><name name-style="western" xml:lang="en"><surname>Khoroshevsky</surname><given-names>K. A.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Хорошевский Кирилл Антонович, инженер</p></bio><bio xml:lang="en"><p>Khoroshevsky Kirill Antonovich, Engineer</p></bio><email xlink:type="simple">khoroshevskiika@susu.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Гадолина</surname><given-names>И. В.</given-names></name><name name-style="western" xml:lang="en"><surname>Gadolina</surname><given-names>I. V.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Гадолина Ирина Викторовна, старший научный сотрудник, кандидат технических наук, доцент</p></bio><bio xml:lang="en"><p>Gadolina Irina Viсtorovna, Senior Researcher, Candidate of Engineering Sciences, Associate Professor</p></bio><email xlink:type="simple">gadolina@mail.ru</email><xref ref-type="aff" rid="aff-2"/></contrib></contrib-group><aff-alternatives id="aff-1"><aff xml:lang="ru"><institution>Южно-Уральский государственный университет (национальный исследовательский университет) (ФГАОУ ВО «ЮУрГУ (НИУ)»)</institution><country>Россия</country></aff><aff xml:lang="en"><institution>South Ural State University (National Research University) (FSAEIHE SUSU (NRU))</institution><country>Russian Federation</country></aff></aff-alternatives><aff-alternatives id="aff-2"><aff xml:lang="ru"><institution>Институт машиноведения им. А. А. Благонравова Российской академии наук</institution><country>Россия</country></aff><aff xml:lang="en"><institution>Institute of Machines Sciense named after A. A. Blagonravov of the Russian Academy of Sciences (IMASH RAS)</institution><country>Russian Federation</country></aff></aff-alternatives><pub-date pub-type="collection"><year>2026</year></pub-date><pub-date pub-type="epub"><day>29</day><month>07</month><year>2026</year></pub-date><volume>0</volume><issue>2</issue><elocation-id>194–209</elocation-id><permissions><copyright-statement>Copyright &amp;#x00A9; Ерпалов А.В., Хорошевский К.А., Гадолина И.В., 2026</copyright-statement><copyright-year>2026</copyright-year><copyright-holder xml:lang="ru">Ерпалов А.В., Хорошевский К.А., Гадолина И.В.</copyright-holder><copyright-holder xml:lang="en">Erpalov A.V., Khoroshevsky K.A., Gadolina I.V.</copyright-holder><license xml:lang="ru" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>Данная работа распространяется под лицензией Creative Commons Attribution 4.0.</license-p></license><license xml:lang="en" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>This work is licensed under a Creative Commons Attribution 4.0 License.</license-p></license></permissions><self-uri xlink:href="https://vestnik.rgups.ru/jour/article/view/344">https://vestnik.rgups.ru/jour/article/view/344</self-uri><abstract><p>Предложена модель оценки остаточного ресурса элементов конструкций при случайном вибрационном нагружении на основе одномерной свёрточной нейронной сети (1D-CNN), на вход которой подаётся распределение максимумов процесса напряжений, а на выходе формируется относительная долговечность. Для обучения сети сформирован набор из 12 000 реализаций стационарных и нестационарных случайных процессов, включая процессы с ударными воздействиями и реальные эксплуатационные записи; эталонные значения долговечности получены методом «падающего дождя» с линейным суммированием повреждений, а гиперпараметры оптимизированы байесовским методом (минимальная ошибка RMSE = 0,0403). Экспериментальная верификация на образцах из лёгкого сплава МА-15, испытанных на электродинамическом вибростенде при трёх уровнях нагружения, показала погрешность прогноза от 6 до 25 %, возрастающую при высоких уровнях вследствие малоцикловой усталости. С учётом разброса экспериментальных данных модель признана верифицированной.</p></abstract><trans-abstract xml:lang="en"><p>A model is proposed for assessing the residual life of structural elements under random vibration loading, based on a one-dimensional convolutional neural network (1D-CNN) that takes the stress-process maxima distribution as input and outputs the relative durability. A training set of 12,000 realizations of stationary and non-stationary random processes – including processes with shock impacts and real operational records – was generated; the reference durability values were obtained by the rainflow method with linear damage summation, and the hyperparameters were tuned by Bayesian optimization (minimum error RMSE = 0.0403). Experimental verification on MA-15 light-alloy specimens tested on an electrodynamic shaker at three loading levels yielded a prediction error of 6 to 25 %, increasing at high levels owing to low-cycle fatigue. Taking the scatter of the experimental data into account, the model is considered verified.</p></trans-abstract><kwd-group xml:lang="ru"><kwd>остаточный ресурс</kwd><kwd>усталостная долговечность</kwd><kwd>случайное нагружение</kwd><kwd>одномерная свёрточная нейронная сеть</kwd><kwd>метод «падающего дождя»</kwd></kwd-group><kwd-group xml:lang="en"><kwd>residual life</kwd><kwd>fatigue life</kwd><kwd>random loading</kwd><kwd>one-dimensional convolutional neural network</kwd><kwd>rainflow method</kwd></kwd-group></article-meta></front><back><ref-list><title>References</title><ref id="cit1"><label>1</label><citation-alternatives><mixed-citation xml:lang="ru">A comparative review of time- and frequency-domain methods for fatigue damage assessment / M. Muñiz-Calvente [et al.] // International Journal of Fatigue. – 2022. – Vol. 163. – Art. 107069. – DOI 10.1016/j.ijfatigue.2022.107069.</mixed-citation><mixed-citation xml:lang="en">A comparative review of time- and frequency-domain methods for fatigue damage assessment / M. Muñiz-Calvente [et al.] // International Journal of Fatigue. – 2022. – Vol. 163. – Art. 107069. – DOI 10.1016/j.ijfatigue.2022.107069.</mixed-citation></citation-alternatives></ref><ref id="cit2"><label>2</label><citation-alternatives><mixed-citation xml:lang="ru">Dirlik, T. Dirlik and Tovo-Benasciutti spectral methods in vibration fatigue: a review with a historical perspective / T. Dirlik, D. Benasciutti // Metals. – 2021. – Vol. 11, No. 9. – Art. 1333. – DOI 10.3390/met11091333.</mixed-citation><mixed-citation xml:lang="en">Dirlik, T. Dirlik and Tovo-Benasciutti spectral methods in vibration fatigue: a review with a historical perspective / T. Dirlik, D. Benasciutti // Metals. – 2021. – Vol. 11, No. 9. – Art. 1333. – DOI 10.3390/met11091333.</mixed-citation></citation-alternatives></ref><ref id="cit3"><label>3</label><citation-alternatives><mixed-citation xml:lang="ru">Miner, M. A. Cumulative damage in fatigue / M. A. Miner // Journal of Applied Mechanics. – 1945. – Vol. 12, No. 3. – P. A159–A164. – DOI 10.1115/1.4009458.</mixed-citation><mixed-citation xml:lang="en">Miner, M. A. Cumulative damage in fatigue / M. A. Miner // Journal of Applied Mechanics. – 1945. – Vol. 12, No. 3. – P. A159–A164. – DOI 10.1115/1.4009458.</mixed-citation></citation-alternatives></ref><ref id="cit4"><label>4</label><citation-alternatives><mixed-citation xml:lang="ru">Vibration fatigue by spectral methods: from structural dynamics to fatigue damage – theory and experiments / J. Slavic, M. Boltezar, M. Mrsnik [et al.]. – Amsterdam : Elsevier, 2021. – 300 p. – DOI 10.1016/C2019-0-04580-3.</mixed-citation><mixed-citation xml:lang="en">Vibration fatigue by spectral methods: from structural dynamics to fatigue damage – theory and experiments / J. Slavic, M. Boltezar, M. Mrsnik [et al.]. – Amsterdam : Elsevier, 2021. – 300 p. – DOI 10.1016/C2019-0-04580-3.</mixed-citation></citation-alternatives></ref><ref id="cit5"><label>5</label><citation-alternatives><mixed-citation xml:lang="ru">Park, J. B. Fatigue damage model comparison with formulated tri-modal spectrum loadings under stationary Gaussian random processes / J. B. Park, C. Y. Song // Ocean Engineering. – 2015. – Vol. 105. – P. 72–82. – DOI 10.1016/j.oceaneng.2015.05.039.</mixed-citation><mixed-citation xml:lang="en">Park, J. B. Fatigue damage model comparison with formulated tri-modal spectrum loadings under stationary Gaussian random processes / J. B. Park, C. Y. Song // Ocean Engineering. – 2015. – Vol. 105. – P. 72–82. – DOI 10.1016/j.oceaneng.2015.05.039.</mixed-citation></citation-alternatives></ref><ref id="cit6"><label>6</label><citation-alternatives><mixed-citation xml:lang="ru">Braccesi, C. Random fatigue. A new frequency domain criterion for the damage evaluation of mechanical components / C. Braccesi, F. Cianetti, L. Tomassini // International Journal of Fatigue. – 2015. – Vol. 70. – P. 417–427. – DOI 10.1016/j.ijfatigue.2014.07.005.</mixed-citation><mixed-citation xml:lang="en">Braccesi, C. Random fatigue. A new frequency domain criterion for the damage evaluation of mechanical components / C. Braccesi, F. Cianetti, L. Tomassini // International Journal of Fatigue. – 2015. – Vol. 70. – P. 417–427. – DOI 10.1016/j.ijfatigue.2014.07.005.</mixed-citation></citation-alternatives></ref><ref id="cit7"><label>7</label><citation-alternatives><mixed-citation xml:lang="ru">Trapp, A. Fatigue assessment of non-stationary random loading in the frequency domain by a quasi-stationary Gaussian approximation / A. Trapp, P. Wolfsteiner // International Journal of Fatigue. – 2021. – Vol. 148. – Art. 106214. – DOI 10.1016/j.ijfatigue.2021.106214.</mixed-citation><mixed-citation xml:lang="en">Trapp, A. Fatigue assessment of non-stationary random loading in the frequency domain by a quasi-stationary Gaussian approximation / A. Trapp, P. Wolfsteiner // International Journal of Fatigue. – 2021. – Vol. 148. – Art. 106214. – DOI 10.1016/j.ijfatigue.2021.106214.</mixed-citation></citation-alternatives></ref><ref id="cit8"><label>8</label><citation-alternatives><mixed-citation xml:lang="ru">Zorman, A. Short-time fatigue-life estimation for non-stationary processes considering structural dynamics / A. Zorman, J. Slavič, M. Boltežar // International Journal of Fatigue. – 2021. – Vol. 147. – Art. 106178. – DOI 10.1016/j.ijfatigue.2021.106178.</mixed-citation><mixed-citation xml:lang="en">Zorman, A. Short-time fatigue-life estimation for non-stationary processes considering structural dynamics / A. Zorman, J. Slavič, M. Boltežar // International Journal of Fatigue. – 2021. – Vol. 147. – Art. 106178. – DOI 10.1016/j.ijfatigue.2021.106178.</mixed-citation></citation-alternatives></ref><ref id="cit9"><label>9</label><citation-alternatives><mixed-citation xml:lang="ru">Wolfsteiner, P. Fatigue assessment of non-stationary random vibrations by using decomposition in Gaussian portions / P. Wolfsteiner // International Journal of Mechanical Sciences. – 2017. – Vol. 127. – P. 10–22. – DOI 10.1016/j.ijmecsci.2016.05.024.</mixed-citation><mixed-citation xml:lang="en">Wolfsteiner, P. Fatigue assessment of non-stationary random vibrations by using decomposition in Gaussian portions / P. Wolfsteiner // International Journal of Mechanical Sciences. – 2017. – Vol. 127. – P. 10–22. – DOI 10.1016/j.ijmecsci.2016.05.024.</mixed-citation></citation-alternatives></ref><ref id="cit10"><label>10</label><citation-alternatives><mixed-citation xml:lang="ru">Wolfsteiner, P. Fatigue life due to non-Gaussian excitation – an analysis of the fatigue damage spectrum using higher order spectra / P. Wolfsteiner, A. Trapp // International Journal of Fatigue. – 2019. – Vol. 127. – P. 203–216. – DOI 10.1016/j.ijfatigue.2019.06.005.</mixed-citation><mixed-citation xml:lang="en">Wolfsteiner, P. Fatigue life due to non-Gaussian excitation – an analysis of the fatigue damage spectrum using higher order spectra / P. Wolfsteiner, A. Trapp // International Journal of Fatigue. – 2019. – Vol. 127. – P. 203–216. – DOI 10.1016/j.ijfatigue.2019.06.005.</mixed-citation></citation-alternatives></ref><ref id="cit11"><label>11</label><citation-alternatives><mixed-citation xml:lang="ru">Non-stationarity and non-Gaussianity in vibration fatigue / J. Slavič [et al.] // Conference Proceedings of the Society for Experimental Mechanics Series. – 2020. – Vol. 97. – P. 73–76. – DOI 10.1007/978-3-030-12676-6_7.</mixed-citation><mixed-citation xml:lang="en">Non-stationarity and non-Gaussianity in vibration fatigue / J. Slavič [et al.] // Conference Proceedings of the Society for Experimental Mechanics Series. – 2020. – Vol. 97. – P. 73–76. – DOI 10.1007/978-3-030-12676-6_7.</mixed-citation></citation-alternatives></ref><ref id="cit12"><label>12</label><citation-alternatives><mixed-citation xml:lang="ru">Assessment of fatigue damage induced by non-Gaussian bimodal processes with emphasis on spectral methods / S. Gao [et al.] // Ocean Engineering. – 2021. – Vol. 220. – Art. 108489. – DOI 10.1016/j.oceaneng.2020.108489.</mixed-citation><mixed-citation xml:lang="en">Assessment of fatigue damage induced by non-Gaussian bimodal processes with emphasis on spectral methods / S. Gao [et al.] // Ocean Engineering. – 2021. – Vol. 220. – Art. 108489. – DOI 10.1016/j.oceaneng.2020.108489.</mixed-citation></citation-alternatives></ref><ref id="cit13"><label>13</label><citation-alternatives><mixed-citation xml:lang="ru">Marques, J. M. E. Variance of the fatigue damage in non-Gaussian stochastic processes with narrow-band power spectrum / J. M. E. Marques, D. Benasciutti // Structural Safety. – 2021. – Vol. 93. – Art. 102131. – DOI 10.1016/j.strusafe.2021.102131.</mixed-citation><mixed-citation xml:lang="en">Marques, J. M. E. Variance of the fatigue damage in non-Gaussian stochastic processes with narrow-band power spectrum / J. M. E. Marques, D. Benasciutti // Structural Safety. – 2021. – Vol. 93. – Art. 102131. – DOI 10.1016/j.strusafe.2021.102131.</mixed-citation></citation-alternatives></ref><ref id="cit14"><label>14</label><citation-alternatives><mixed-citation xml:lang="ru">Marques, J. M. E. More on variance of fatigue damage in non-Gaussian random loadings – effect of skewness and kurtosis / J. M. E. Marques, D. Benasciutti // Procedia Structural Integrity. – 2020. – Vol. 25. – P. 101–111. – DOI 10.1016/j.prostr.2020.04.014.</mixed-citation><mixed-citation xml:lang="en">Marques, J. M. E. More on variance of fatigue damage in non-Gaussian random loadings – effect of skewness and kurtosis / J. M. E. Marques, D. Benasciutti // Procedia Structural Integrity. – 2020. – Vol. 25. – P. 101–111. – DOI 10.1016/j.prostr.2020.04.014.</mixed-citation></citation-alternatives></ref><ref id="cit15"><label>15</label><citation-alternatives><mixed-citation xml:lang="ru">Marques, J. M. E. Variability of the fatigue damage due to the randomness of a stationary vibration load / J. M. E. Marques, D. Benasciutti, R. Tovo // International Journal of Fatigue. – 2020. – Vol. 141. – Art. 105891. – DOI 10.1016/j.ijfatigue.2020.105891.</mixed-citation><mixed-citation xml:lang="en">Marques, J. M. E. Variability of the fatigue damage due to the randomness of a stationary vibration load / J. M. E. Marques, D. Benasciutti, R. Tovo // International Journal of Fatigue. – 2020. – Vol. 141. – Art. 105891. – DOI 10.1016/j.ijfatigue.2020.105891.</mixed-citation></citation-alternatives></ref><ref id="cit16"><label>16</label><citation-alternatives><mixed-citation xml:lang="ru">Benasciutti, D. The use of fractional order statistics for estimating nonparametric confidence intervals for quantiles of the fatigue damage computed in service random loadings / D. Benasciutti, J. M. E. Marques // IOP Conference Series: Materials Science and Engineering. – 2023. – Vol. 1275, No. 1. – Art. 012020. – DOI 10.1088/1757-899X/1275/1/012020.</mixed-citation><mixed-citation xml:lang="en">Benasciutti, D. The use of fractional order statistics for estimating nonparametric confidence intervals for quantiles of the fatigue damage computed in service random loadings / D. Benasciutti, J. M. E. Marques // IOP Conference Series: Materials Science and Engineering. – 2023. – Vol. 1275, No. 1. – Art. 012020. – DOI 10.1088/1757-899X/1275/1/012020.</mixed-citation></citation-alternatives></ref><ref id="cit17"><label>17</label><citation-alternatives><mixed-citation xml:lang="ru">Orlando, A. Structural response and fatigue assessment of a small vertical axis wind turbine under stationary and non-stationary excitation / A. Orlando, L. Pagnini, M. P. Repetto // Renewable Energy. – 2021. – Vol. 170. – P. 251–266. – DOI 10.1016/j.renene.2021.01.123.</mixed-citation><mixed-citation xml:lang="en">Orlando, A. Structural response and fatigue assessment of a small vertical axis wind turbine under stationary and non-stationary excitation / A. Orlando, L. Pagnini, M. P. Repetto // Renewable Energy. – 2021. – Vol. 170. – P. 251–266. – DOI 10.1016/j.renene.2021.01.123.</mixed-citation></citation-alternatives></ref><ref id="cit18"><label>18</label><citation-alternatives><mixed-citation xml:lang="ru">Niu, Q. An empirical mode decomposition-based frequency-domain approach for the fatigue analysis of nonstationary processes / Q. Niu, S. Yang, X. Li // Fatigue &amp; Fracture of Engineering Materials &amp; Structures. – 2018. – Vol. 41, No. 9. – P. 1980–1996. – DOI 10.1111/ffe.12836.</mixed-citation><mixed-citation xml:lang="en">Niu, Q. An empirical mode decomposition-based frequency-domain approach for the fatigue analysis of nonstationary processes / Q. Niu, S. Yang, X. Li // Fatigue &amp; Fracture of Engineering Materials &amp; Structures. – 2018. – Vol. 41, No. 9. – P. 1980–1996. – DOI 10.1111/ffe.12836.</mixed-citation></citation-alternatives></ref><ref id="cit19"><label>19</label><citation-alternatives><mixed-citation xml:lang="ru">A novel approach for stress cycle analysis based on empirical mode decomposition / R. Li [et al.] // MFPT 2018: Intelligent Technologies and Equipment for Human Performance Monitoring. – 2018. – P. 4–12.</mixed-citation><mixed-citation xml:lang="en">A novel approach for stress cycle analysis based on empirical mode decomposition / R. Li [et al.] // MFPT 2018: Intelligent Technologies and Equipment for Human Performance Monitoring. – 2018. – P. 4–12.</mixed-citation></citation-alternatives></ref><ref id="cit20"><label>20</label><citation-alternatives><mixed-citation xml:lang="ru">An improved VMD-based denoising method for time domain load signal combining wavelet with singular spectrum analysis / J. Fu [et al.] // Mathematical Problems in Engineering. – 2020. – Vol. 2020. – Art. 1485937. – DOI 10.1155/2020/1485937.</mixed-citation><mixed-citation xml:lang="en">An improved VMD-based denoising method for time domain load signal combining wavelet with singular spectrum analysis / J. Fu [et al.] // Mathematical Problems in Engineering. – 2020. – Vol. 2020. – Art. 1485937. – DOI 10.1155/2020/1485937.</mixed-citation></citation-alternatives></ref><ref id="cit21"><label>21</label><citation-alternatives><mixed-citation xml:lang="ru">Soman, R. Semi-automated methodology for damage assessment of a scaled wind turbine tripod using enhanced empirical mode decomposition and statistical analysis / R. Soman // International Journal of Fatigue. – 2020. – Vol. 134. – Art. 105475. – DOI 10.1016/j.ijfatigue.2020.105475.</mixed-citation><mixed-citation xml:lang="en">Soman, R. Semi-automated methodology for damage assessment of a scaled wind turbine tripod using enhanced empirical mode decomposition and statistical analysis / R. Soman // International Journal of Fatigue. – 2020. – Vol. 134. – Art. 105475. – DOI 10.1016/j.ijfatigue.2020.105475.</mixed-citation></citation-alternatives></ref><ref id="cit22"><label>22</label><citation-alternatives><mixed-citation xml:lang="ru">Median ensemble empirical mode decomposition / X. Lang [et al.] // Signal Processing. – 2020. – Vol. 176. – Art. 107686. – DOI 10.1016/j.sigpro.2020.107686.</mixed-citation><mixed-citation xml:lang="en">Median ensemble empirical mode decomposition / X. Lang [et al.] // Signal Processing. – 2020. – Vol. 176. – Art. 107686. – DOI 10.1016/j.sigpro.2020.107686.</mixed-citation></citation-alternatives></ref><ref id="cit23"><label>23</label><citation-alternatives><mixed-citation xml:lang="ru">Метод оценки долговечности конструкций при стационарном и нестационарном случайных нагружениях с применением вариационной модовой декомпозиции / А. В. Ерпалов, К. А. Хорошевский, Е. А. Румянцева, И. В. Гадолина // Заводская лаборатория. Диагностика материалов. – 2024. – Т. 90, № 9. – С. 63–74. – DOI 10.26896/1028-6861-2024-90-9-63-74.</mixed-citation><mixed-citation xml:lang="en">Method for assessing the durability of structures under stationary and non-stationary random loading using variational mode decomposition (VMD) / A. V. Erpalov, K. A. Khoroshevskii, E. A. Rumyanceva, I. V. Gadolina // Industrial Laboratory. Diagnostics of Materials. – 2024. – Vol. 90, No. 9. – P. 63–74. – DOI 10.26896/1028-6861-2024-90-9-63-74.</mixed-citation></citation-alternatives></ref><ref id="cit24"><label>24</label><citation-alternatives><mixed-citation xml:lang="ru">Niesłony, A. Mean stress effect correction in frequency-domain methods for fatigue life assessment / A. Niesłony, M. Böhm // Procedia Engineering. – 2015. – Vol. 101. – P. 347–354. – DOI 10.1016/j.proeng.2015.02.042.</mixed-citation><mixed-citation xml:lang="en">Niesłony, A. Mean stress effect correction in frequency-domain methods for fatigue life assessment / A. Niesłony, M. Böhm // Procedia Engineering. – 2015. – Vol. 101. – P. 347–354. – DOI 10.1016/j.proeng.2015.02.042.</mixed-citation></citation-alternatives></ref><ref id="cit25"><label>25</label><citation-alternatives><mixed-citation xml:lang="ru">Niesłony, A. Frequency-domain fatigue life estimation with mean stress correction / A. Niesłony, M. Böhm // International Journal of Fatigue. – 2016. – Vol. 91. – P. 373–381. – DOI 10.1016/j.ijfatigue.2016.02.031.</mixed-citation><mixed-citation xml:lang="en">Niesłony, A. Frequency-domain fatigue life estimation with mean stress correction / A. Niesłony, M. Böhm // International Journal of Fatigue. – 2016. – Vol. 91. – P. 373–381. – DOI 10.1016/j.ijfatigue.2016.02.031.</mixed-citation></citation-alternatives></ref><ref id="cit26"><label>26</label><citation-alternatives><mixed-citation xml:lang="ru">A unified mean stress correction model for fatigue thresholds prediction of metals / Y. Liu [et al.] // Engineering Fracture Mechanics. – 2020. – Vol. 223. – Art. 106787. – DOI 10.1016/j.engfracmech.2019.106787.</mixed-citation><mixed-citation xml:lang="en">A unified mean stress correction model for fatigue thresholds prediction of metals / Y. Liu [et al.] // Engineering Fracture Mechanics. – 2020. – Vol. 223. – Art. 106787. – DOI 10.1016/j.engfracmech.2019.106787.</mixed-citation></citation-alternatives></ref><ref id="cit27"><label>27</label><citation-alternatives><mixed-citation xml:lang="ru">Oh, G. Effective stress and fatigue life prediction with mean stress correction models on a ferritic stainless steel sheet / G. Oh // International Journal of Fatigue. – 2022. – Vol. 157. – Art. 106707. – DOI 10.1016/j.ijfatigue.2021.106707.</mixed-citation><mixed-citation xml:lang="en">Oh, G. Effective stress and fatigue life prediction with mean stress correction models on a ferritic stainless steel sheet / G. Oh // International Journal of Fatigue. – 2022. – Vol. 157. – Art. 106707. – DOI 10.1016/j.ijfatigue.2021.106707.</mixed-citation></citation-alternatives></ref><ref id="cit28"><label>28</label><citation-alternatives><mixed-citation xml:lang="ru">Выбор частоты дискретизации и оптимального метода цифровой обработки сигнала в задачах, учитывающих случайный процесс нагружения, для оценки долговечности / И. В. Гадолина, Н. Г. Лисаченко, Ю. А. Свирский, Д. А. Дубин // Заводская лаборатория. Диагностика материалов. – 2019. – Т. 85, № 7. – С. 64–72. – DOI 10.26896/1028-6861-2019-85-7-64-72.</mixed-citation><mixed-citation xml:lang="en">The choice of the sampling frequency and optimal method of signal digital processing in the problems considering a random loading process for assessing durability / I. V. Gadolina, N. G. Lisachenko, Yu. A. Svirskiy, D. A. Dubin // Industrial Laboratory. Diagnostics of Materials. – 2019. – Vol. 85, No. 7. – P. 64–72. – DOI 10.26896/1028-6861-2019-85-7-64-72.</mixed-citation></citation-alternatives></ref><ref id="cit29"><label>29</label><citation-alternatives><mixed-citation xml:lang="ru">Gadolina, I. V. Varied approaches to loading assessment in fatigue studies / I. V. Gadolina, N. A. Makhutov, A. V. Erpalov // International Journal of Fatigue. – 2021. – Vol. 144. – Art. 106035. – DOI 10.1016/j.ijfatigue.2020.106035.</mixed-citation><mixed-citation xml:lang="en">Gadolina, I. V. Varied approaches to loading assessment in fatigue studies / I. V. Gadolina, N. A. Makhutov, A. V. Erpalov // International Journal of Fatigue. – 2021. – Vol. 144. – Art. 106035. – DOI 10.1016/j.ijfatigue.2020.106035.</mixed-citation></citation-alternatives></ref><ref id="cit30"><label>30</label><citation-alternatives><mixed-citation xml:lang="ru">Хорошевский, К. А. Расчётно-экспериментальное исследование влияния демпфирования конструкции на её долговечность при случайном нагружении / К. А. Хорошевский, А. В. Ерпалов, И. В. Гадолина // Вестник ЮУрГУ. Серия «Математика. Механика. Физика». – 2023. – Т. 15, № 4. – С. 47–57. – DOI 10.14529/mmph230406.</mixed-citation><mixed-citation xml:lang="en">Khoroshevsky, K. A. A computational and experimental study of the influence of structural damping on a structure's durability under random loading / K. A. Khoroshevsky, A. V. Erpalov, I. V. Gadolina // Bulletin of the South Ural State University. Series "Mathematics, Mechanics, Physics". – 2023. – Vol. 15, No. 4. – P. 47–57. – DOI 10.14529/mmph230406.</mixed-citation></citation-alternatives></ref><ref id="cit31"><label>31</label><citation-alternatives><mixed-citation xml:lang="ru">1D convolutional neural networks and applications: a survey / S. Kiranyaz, O. Avci, O. Abdeljaber [et al.] // Mechanical Systems and Signal Processing. – 2021. – Vol. 151. – Art. 107398. – DOI 10.1016/j.ymssp.2020.107398.</mixed-citation><mixed-citation xml:lang="en">1D convolutional neural networks and applications: a survey / S. Kiranyaz, O. Avci, O. Abdeljaber [et al.] // Mechanical Systems and Signal Processing. – 2021. – Vol. 151. – Art. 107398. – DOI 10.1016/j.ymssp.2020.107398.</mixed-citation></citation-alternatives></ref><ref id="cit32"><label>32</label><citation-alternatives><mixed-citation xml:lang="ru">Chen, J. Fatigue modeling using neural networks: a comprehensive review / J. Chen, Y. Liu // Fatigue &amp; Fracture of Engineering Materials &amp; Structures. – 2022. – Vol. 45, No. 4. – P. 945–979. – DOI 10.1111/ffe.13640.</mixed-citation><mixed-citation xml:lang="en">Chen, J. Fatigue modeling using neural networks: a comprehensive review / J. Chen, Y. Liu // Fatigue &amp; Fracture of Engineering Materials &amp; Structures. – 2022. – Vol. 45, No. 4. – P. 945–979. – DOI 10.1111/ffe.13640.</mixed-citation></citation-alternatives></ref><ref id="cit33"><label>33</label><citation-alternatives><mixed-citation xml:lang="ru">Li, X. Remaining useful life estimation in prognostics using deep convolution neural networks / X. Li, Q. Ding, J.-Q. Sun // Reliability Engineering &amp; System Safety. – 2018. – Vol. 172. – P. 1–11. – DOI 10.1016/j.ress.2017.11.021.</mixed-citation><mixed-citation xml:lang="en">Li, X. Remaining useful life estimation in prognostics using deep convolution neural networks / X. Li, Q. Ding, J.-Q. Sun // Reliability Engineering &amp; System Safety. – 2018. – Vol. 172. – P. 1–11. – DOI 10.1016/j.ress.2017.11.021.</mixed-citation></citation-alternatives></ref><ref id="cit34"><label>34</label><citation-alternatives><mixed-citation xml:lang="ru">A novel method of multiaxial fatigue life prediction based on deep learning / J. Yang, G. Kang, Y. Liu, Q. Kan // International Journal of Fatigue. – 2021. – Vol. 151. – Art. 106356. – DOI 10.1016/j.ijfatigue.2021.106356.</mixed-citation><mixed-citation xml:lang="en">A novel method of multiaxial fatigue life prediction based on deep learning / J. Yang, G. Kang, Y. Liu, Q. Kan // International Journal of Fatigue. – 2021. – Vol. 151. – Art. 106356. – DOI 10.1016/j.ijfatigue.2021.106356.</mixed-citation></citation-alternatives></ref><ref id="cit35"><label>35</label><citation-alternatives><mixed-citation xml:lang="ru">Machine learning-based predictions of fatigue life and fatigue limit for steels / L. He, Z. Wang, H. Akebono, A. Sugeta // Journal of Materials Science &amp; Technology. – 2021. – Vol. 90. – P. 9–19.</mixed-citation><mixed-citation xml:lang="en">Machine learning-based predictions of fatigue life and fatigue limit for steels / L. He, Z. Wang, H. Akebono, A. Sugeta // Journal of Materials Science &amp; Technology. – 2021. – Vol. 90. – P. 9–19.</mixed-citation></citation-alternatives></ref><ref id="cit36"><label>36</label><citation-alternatives><mixed-citation xml:lang="ru">Jimenez-Martinez, M. Fatigue damage effect approach by artificial neural network / M. Jimenez-Martinez, M. Alfaro-Ponce // International Journal of Fatigue. – 2019. – Vol. 124. – P. 42–47.</mixed-citation><mixed-citation xml:lang="en">Jimenez-Martinez, M. Fatigue damage effect approach by artificial neural network / M. Jimenez-Martinez, M. Alfaro-Ponce // International Journal of Fatigue. – 2019. – Vol. 124. – P. 42–47.</mixed-citation></citation-alternatives></ref><ref id="cit37"><label>37</label><citation-alternatives><mixed-citation xml:lang="ru">Zhang, X.-C. A deep learning based life prediction method for components under creep, fatigue and creep-fatigue conditions / X.-C. Zhang, J.-G. Gong, F.-Z. Xuan // International Journal of Fatigue. – 2021. – Vol. 148. – Art. 106236. – DOI 10.1016/j.ijfatigue.2021.106236.</mixed-citation><mixed-citation xml:lang="en">Zhang, X.-C. A deep learning based life prediction method for components under creep, fatigue and creep-fatigue conditions / X.-C. Zhang, J.-G. Gong, F.-Z. Xuan // International Journal of Fatigue. – 2021. – Vol. 148. – Art. 106236. – DOI 10.1016/j.ijfatigue.2021.106236.</mixed-citation></citation-alternatives></ref><ref id="cit38"><label>38</label><citation-alternatives><mixed-citation xml:lang="ru">Modelling fatigue life prediction of additively manufactured Ti-6Al-4V samples using machine learning approach / J. Horňas, J. Běhal, P. Homola [et al.] // International Journal of Fatigue. – 2023. – Vol. 169. – Art. 107483. – DOI 10.1016/j.ijfatigue.2022.107483.</mixed-citation><mixed-citation xml:lang="en">Modelling fatigue life prediction of additively manufactured Ti-6Al-4V samples using machine learning approach / J. Horňas, J. Běhal, P. Homola [et al.] // International Journal of Fatigue. – 2023. – Vol. 169. – Art. 107483. – DOI 10.1016/j.ijfatigue.2022.107483.</mixed-citation></citation-alternatives></ref><ref id="cit39"><label>39</label><citation-alternatives><mixed-citation xml:lang="ru">Graphical feature construction-based deep learning model for fatigue life prediction of AM alloys / H. Wu, A. Wang, Z. Gan, L. Gan // Materials. – 2024. – Vol. 18, No. 1. – Art. 11. – DOI 10.3390/ma18010011.</mixed-citation><mixed-citation xml:lang="en">Graphical feature construction-based deep learning model for fatigue life prediction of AM alloys / H. Wu, A. Wang, Z. Gan, L. Gan // Materials. – 2024. – Vol. 18, No. 1. – Art. 11. – DOI 10.3390/ma18010011.</mixed-citation></citation-alternatives></ref><ref id="cit40"><label>40</label><citation-alternatives><mixed-citation xml:lang="ru">Multiaxial fatigue life prediction for various metallic materials based on the hybrid CNN-LSTM neural network / F. Heng, J. Gao, R. Xu [et al.] // Fatigue &amp; Fracture of Engineering Materials &amp; Structures. – 2023. – Vol. 46, No. 5. – P. 1979–1996. – DOI 10.1111/ffe.13977.</mixed-citation><mixed-citation xml:lang="en">Multiaxial fatigue life prediction for various metallic materials based on the hybrid CNN-LSTM neural network / F. Heng, J. Gao, R. Xu [et al.] // Fatigue &amp; Fracture of Engineering Materials &amp; Structures. – 2023. – Vol. 46, No. 5. – P. 1979–1996. – DOI 10.1111/ffe.13977.</mixed-citation></citation-alternatives></ref><ref id="cit41"><label>41</label><citation-alternatives><mixed-citation xml:lang="ru">A physics-informed neural network approach to fatigue life prediction using small quantity of samples / D. Chen, Y. Li, K. Liu, Y. Li // International Journal of Fatigue. – 2023. – Vol. 166. – Art. 107270. – DOI 10.1016/j.ijfatigue.2022.107270.</mixed-citation><mixed-citation xml:lang="en">A physics-informed neural network approach to fatigue life prediction using small quantity of samples / D. Chen, Y. Li, K. Liu, Y. Li // International Journal of Fatigue. – 2023. – Vol. 166. – Art. 107270. – DOI 10.1016/j.ijfatigue.2022.107270.</mixed-citation></citation-alternatives></ref><ref id="cit42"><label>42</label><citation-alternatives><mixed-citation xml:lang="ru">An image recognition based multiaxial low-cycle fatigue life prediction method with CNN model / X. Sun, T. Zhou, K. Song, X. Chen // International Journal of Fatigue. – 2023. – Vol. 167. – Art. 107324. – DOI 10.1016/j.ijfatigue.2022.107324.</mixed-citation><mixed-citation xml:lang="en">An image recognition based multiaxial low-cycle fatigue life prediction method with CNN model / X. Sun, T. Zhou, K. Song, X. Chen // International Journal of Fatigue. – 2023. – Vol. 167. – Art. 107324. – DOI 10.1016/j.ijfatigue.2022.107324.</mixed-citation></citation-alternatives></ref><ref id="cit43"><label>43</label><citation-alternatives><mixed-citation xml:lang="ru">Remaining useful life prediction via a deep adaptive transformer framework enhanced by graph attention network / P. Liang, Y. Li, B. Wang, X. Yuan, L. Zhang // International Journal of Fatigue. – 2023. – Vol. 174. – Art. 107722.</mixed-citation><mixed-citation xml:lang="en">Remaining useful life prediction via a deep adaptive transformer framework enhanced by graph attention network / P. Liang, Y. Li, B. Wang, X. Yuan, L. Zhang // International Journal of Fatigue. – 2023. – Vol. 174. – Art. 107722.</mixed-citation></citation-alternatives></ref><ref id="cit44"><label>44</label><citation-alternatives><mixed-citation xml:lang="ru">Remaining useful life prediction of aero-engine enabled by fusing knowledge and deep learning models / Y. Li, Y. Chen, Z. Hu, H. Zhang // Reliability Engineering &amp; System Safety. – 2023. – Vol. 229. – Art. 108869.</mixed-citation><mixed-citation xml:lang="en">Remaining useful life prediction of aero-engine enabled by fusing knowledge and deep learning models / Y. Li, Y. Chen, Z. Hu, H. Zhang // Reliability Engineering &amp; System Safety. – 2023. – Vol. 229. – Art. 108869.</mixed-citation></citation-alternatives></ref><ref id="cit45"><label>45</label><citation-alternatives><mixed-citation xml:lang="ru">Sun, H. A novel artificial neural network model for wide-band random fatigue life prediction / H. Sun, Y. Qiu, J. Li // International Journal of Fatigue. – 2022. – Vol. 157. – Art. 106701. – DOI 10.1016/j.ijfatigue.2021.106701.</mixed-citation><mixed-citation xml:lang="en">Sun, H. A novel artificial neural network model for wide-band random fatigue life prediction / H. Sun, Y. Qiu, J. Li // International Journal of Fatigue. – 2022. – Vol. 157. – Art. 106701. – DOI 10.1016/j.ijfatigue.2021.106701.</mixed-citation></citation-alternatives></ref><ref id="cit46"><label>46</label><citation-alternatives><mixed-citation xml:lang="ru">Novel models for fatigue life prediction under wideband random loads based on machine learning / H. Sun, Y. Qiu, J. Li [et al.] // Fatigue &amp; Fracture of Engineering Materials &amp; Structures. – 2024. – Vol. 47, No. 9. – P. 3342–3360. – DOI 10.1111/ffe.14371.</mixed-citation><mixed-citation xml:lang="en">Novel models for fatigue life prediction under wideband random loads based on machine learning / H. Sun, Y. Qiu, J. Li [et al.] // Fatigue &amp; Fracture of Engineering Materials &amp; Structures. – 2024. – Vol. 47, No. 9. – P. 3342–3360. – DOI 10.1111/ffe.14371.</mixed-citation></citation-alternatives></ref><ref id="cit47"><label>47</label><citation-alternatives><mixed-citation xml:lang="ru">Fatigue life prediction based on modified narrow-band method under broadband random vibration loading / S.-D. Wu, D.-G. Shang, P.-C. Liu [et al.] // International Journal of Fatigue. – 2022. – Vol. 159. – Art. 106832. – DOI 10.1016/j.ijfatigue.2022.106832.</mixed-citation><mixed-citation xml:lang="en">Fatigue life prediction based on modified narrow-band method under broadband random vibration loading / S.-D. Wu, D.-G. Shang, P.-C. Liu [et al.] // International Journal of Fatigue. – 2022. – Vol. 159. – Art. 106832. – DOI 10.1016/j.ijfatigue.2022.106832.</mixed-citation></citation-alternatives></ref><ref id="cit48"><label>48</label><citation-alternatives><mixed-citation xml:lang="ru">A new frequency-domain method for fatigue prediction of offshore structures under wideband random loadings / B. Wu, W. Lu, X. Li [at al.] // Ocean Engineering. – 2023. – Vol. 281. – Art. 114812. – DOI 10.1016/j.oceaneng.2023.114812.</mixed-citation><mixed-citation xml:lang="en">A new frequency-domain method for fatigue prediction of offshore structures under wideband random loadings / B. Wu, W. Lu, X. Li [at al.] // Ocean Engineering. – 2023. – Vol. 281. – Art. 114812. – DOI 10.1016/j.oceaneng.2023.114812.</mixed-citation></citation-alternatives></ref><ref id="cit49"><label>49</label><citation-alternatives><mixed-citation xml:lang="ru">Random vibration fatigue behavior of directionally solidified superalloy: experiments and evaluation of life prediction methods / H. Lu, J. Wang, Y. Lian [et al.] // International Journal of Fatigue. – 2023. – Vol. 175. – Art. 107746. – DOI 10.1016/j.ijfatigue.2023.107746.</mixed-citation><mixed-citation xml:lang="en">Random vibration fatigue behavior of directionally solidified superalloy: experiments and evaluation of life prediction methods / H. Lu, J. Wang, Y. Lian [et al.] // International Journal of Fatigue. – 2023. – Vol. 175. – Art. 107746. – DOI 10.1016/j.ijfatigue.2023.107746.</mixed-citation></citation-alternatives></ref><ref id="cit50"><label>50</label><citation-alternatives><mixed-citation xml:lang="ru">Zhang, H. Data-based deep learning for random vibration fatigue life prediction of car seat frame / H. Zhang, S. Wang // Nonlinear Dynamics. – 2025. – Vol. 113, No. 5. – P. 4121–4145. – DOI 10.1007/s11071-024-09972-3.</mixed-citation><mixed-citation xml:lang="en">Zhang, H. Data-based deep learning for random vibration fatigue life prediction of car seat frame / H. Zhang, S. Wang // Nonlinear Dynamics. – 2025. – Vol. 113, No. 5. – P. 4121–4145. – DOI 10.1007/s11071-024-09972-3.</mixed-citation></citation-alternatives></ref><ref id="cit51"><label>51</label><citation-alternatives><mixed-citation xml:lang="ru">Алюнов, Д. Ю. Рекуррентная нейронная сеть для контроля ширины спектра нестационарного случайного сигнала / Д. Ю. Алюнов // Вестник Чувашского университета. – 2023. – № 2. – С. 5–17. – DOI 10.47026/1810-1909-2023-2-5-17.</mixed-citation><mixed-citation xml:lang="en">Alyunov, D. Yu. Recurrent neural network for controlling the spectrum width of a non-stationary random signal / D. Yu. Alyunov // Bulletin of the Chuvash University. – 2023. – No. 2. – P. 5–17. – DOI 10.47026/1810-1909-2023-2-5-17.</mixed-citation></citation-alternatives></ref><ref id="cit52"><label>52</label><citation-alternatives><mixed-citation xml:lang="ru">Федотов, М. В. Предиктивная аналитика технического состояния систем тепловозов с использованием нейросетевых прогнозных моделей / М. В. Федотов, В. В. Грачев // Бюллетень результатов научных исследований. – 2021. – № 3. – С. 102–114. – DOI 10.20295/2223-9987-2021-3-102-114.</mixed-citation><mixed-citation xml:lang="en">Fedotov, M. V. Predictive analytics of the tech-nical condition of diesel locomotive systems using neural network predictive models / M. V. Fedotov, V. V. Grachev // Bulletin of Scientific Research Results. – 2021. – No. 3. – P. 102–114. – DOI 10.20295/2223-9987-2021-3-102-114.</mixed-citation></citation-alternatives></ref><ref id="cit53"><label>53</label><citation-alternatives><mixed-citation xml:lang="ru">Ломакина, Л. С. Диагностирование и прогнозирование состояний технических и технологических объектов на основе ансамблевых технологий машинного обучения / Л. С. Ломакина, А. Н. Двитовская, К. А. Корелин // Информационные и математические технологии в науке и управлении. – 2025. – № 4 (40). – С. 26–37. – DOI 10.25729/ESI.2025.40.4.003.</mixed-citation><mixed-citation xml:lang="en">Lomakina, L. S. Diagnostics and forecasting of technical and technological object states based on ensemble machine learning technologies / L. S. Lomakina, A. N. Dvitovskaya, K. A. Korelin // Information and Mathematical Technologies in Sci-ence and Management. – 2025. – No. 4 (40). – P. 26–37. – DOI 10.25729/ESI.2025.40.4.003.</mixed-citation></citation-alternatives></ref></ref-list><fn-group><fn fn-type="conflict"><p>The authors declare that there are no conflicts of interest present.</p></fn></fn-group></back></article>
