AbstractAbout AuthorsReferences
Based on the results of the tests carried out on the bent beams made of various viscoelastic composite materials, the optimal values of the α-orders of the fractional derivative included in the differential equation of the rheological viscoelastic model were selected. The selection was carried out using three deviation characteristics. A new method is proposed for finding the optimal values of the parameters of an equation with a fractional derivative c and λ, which allow using this equation as a mathematical model of the experiment. All calculations were carried out using a specially written program in Python. It is shown that the obtained analytical solutions of the impact of the applied force and the response of a structure made of various composite materials give satisfactory convergence.
T.A. MATSEEVICH, Doctor of Sciences (Physico-mathematical) (This email address is being protected from spambots. You need JavaScript enabled to view it.),
L.V. KIRYANOVA, Candidate of Sciences (Physico-mathematical) (This email address is being protected from spambots. You need JavaScript enabled to view it.),
V.A. SMIRNOV, Candidate of Sciences (Engineering) (This email address is being protected from spambots. You need JavaScript enabled to view it.),
P.S. IVANOV, Senior Lecturer, Department of Higher Mathematics (This email address is being protected from spambots. You need JavaScript enabled to view it.)
L.V. KIRYANOVA, Candidate of Sciences (Physico-mathematical) (This email address is being protected from spambots. You need JavaScript enabled to view it.),
V.A. SMIRNOV, Candidate of Sciences (Engineering) (This email address is being protected from spambots. You need JavaScript enabled to view it.),
P.S. IVANOV, Senior Lecturer, Department of Higher Mathematics (This email address is being protected from spambots. You need JavaScript enabled to view it.)
National Research Moscow State University of Civil Engineering (26, Yaroslavskoye Highway, Moscow, 129337, Russian Federation)
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14. Bagley R.L., Torvi, P.J. On the fractional calculus model of viscoelastic behavior. J. Rheol. 1986. Vol. 30, pp. 133–155. DOI: 10.1122/1.549887
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16. Ingman D., Suzdalnitsky J. Control of damping oscillations by fractional differential operator with time-dependent order. Computer Methods in Applied Mechanics and Engineering. 2004. No. 193. Vol. 52, pp. 5585–5595. DOI: 10.1016/j.cma.2004.06.029
17. Naber M. Linear fractionally damped oscillator. International Journal of Differential Equations. 2010. Vol. 2010. 197020. DOI: 10.1155/2010/197020
18. Kiryanova L.V., Matseevich T. Sturm-Liouville problem with mixed boundary conditions for a differential equation with a fractional derivative and its application in viscoelasticity models. Axioms. 2023. Vol. 12. No. 8. P. 779. DOI: 10.3390/axioms12080779
19. Podlubny I. Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. London: Academic Press. 1999. 340 p.
20. Кирьянова Л.В. Разностная схема для неоднородной модели Бегли-Торвика // Вестник Академии наук Чеченской Республики. 2020. № 1. Т. 48. С. 14–18. DOI: 10.25744/vestnik.2020.48.1.002
20. Kiryanova L.V. Difference scheme for the inhomogeneous Begley-Torvik model. Vestnik of the Academy of Sciences of the Chechen Republic. 2020. No. 1. Vol. 48, pp. 14–18. (In Russian). DOI: 10.25744/vestnik.2020.48.1.002
1. Tamrazyan A.G., Khetagurov A.T., Esayan S.G. Calculation of aging viscoelastic bodies by modeling their rheological characteristics. Seismostoikoe stroitel’stvo. Bezopasnost’ sooruzhenii. 2002. No. 2, pp. 34–37. (In Russian).
2. Тамразян А.Г. Динамическая устойчивость сжатого железобетонного элемента как вязкоупругого стержня // Вестник МГСУ. 2011. № 1–2. С. 193–196.
2. Tamrazyan A.G. Dynamik stability of the compressed reinforced concrete element as viscoelastic bar. Vestnik MGSU. 2011. No. 1–2, pp. 193–196. (In Russian).
3. Handbook of Fractional Calculus with Applications. De Gruyter. Series ed. Tenreiro Machado J.A. 2019. Vol. 1–8.
4. Тарасов В.Е. Модели теоретической физики с интегродифференцированием дробного порядка. М.; Ижевск: Ижевский институт компьютерных исследований, 2011. 568 с.
4. Tarasov V.E. Modeli teoreticheskoi fiziki s integro-differentsirovaniem drobnogo poryadka [Models of theoretical physics with fractional order integro-differentiation]. M. Izhevsk: Institute of Computer Science. 2010. 568 p.
5. Atanacković T.M., Pilipović S., Stanković B., Zorica D. Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles. London: Wiley. 2014. 406 p.
6. Sandev T., Tomovshi Z., Fractional Equations and Models: Theory and Applications. Berlin: Springer. 2019. 363 p.
7. Shitikova M.V., Krusser A.I. Models of viscoelastic materials: A review on historical development and formulation. Advanced Structured Materials. 2022. Vol. 175. pp. 285–326. DOI: 10.1007/978-3-031-04548-6_14
8. Shabani M., Jahani K., Di Paola M., Sadeghi M.H. Frequency domain identification of the fractional Kelvin-Voigt’s parameters for viscoelastic materials. Mechanics of Materials. 2019. Vol. 137. 103099.DOI: 10.1016/j.mechmat.2019.103099
9. Lagos-Varas M., Movilla-Quesada D., Arenas J.P., Raposeiras A.C., Castro-Fresno D., Calzada-Pérez M.A.,Vega-Zamanillo A., Maturana J. Study of the mechanical behavior of asphalt mixtures using fractional rheology to model their viscoelasticity. Const. Build. Mat. 2019. Vol. 200, pp. 124–134. DOI: 10.1016/j.conbuildmat.2018.12.073
10. Wang Y., Harris J.M. Seismic attenuation models: multiple and fractional generalizations. SEG Technical Program Expanded Abstracts. 2020, pp. 2754–2758. DOI: 10.1190/segam2020-3421172.1
11. Popov I.I., Shitikova M.V., Levchenko A.V., Zhukov A.D. Experimental identification of the fractional parameter of the fractional derivative standard linear solid model for fiber-reinforced rubber concrete. Mechanics of Advanced Materials and Structures. 2023. DOI: 10.1080/15376494.2023.2191600
12. Ерохин С.В., Алероев Т.С. Параметрическая идентификация порядка дробной производной в модели Бегли–Торвика // Математическое моделирование. 2018. Т. 30. № 7. С. 93–102.DOI: 10.1134/S2070048219020030
12. Erokhin S.V., Aleroev T.S. Parametric identification of the order of a fractional derivative in the Begley–Torvik model. Math. Model. 2018. Vol. 30. No. 7, pp. 93–102. (In Russian). DOI: 10.1134/S2070048219020030
13. Xu Y., Guo Y.-Q., Huang X.-H., Dong Y.-R., Hu Z.-W.,Kim J. Experimental and theoretical investigation of viscoelastic damper by applying fractional derivative method and internal variable theory. Buildings. 2023. Vol. 13 (1). 239. DOI: 10.3390/buildings13010239
14. Bagley R.L., Torvi, P.J. On the fractional calculus model of viscoelastic behavior. J. Rheol. 1986. Vol. 30, pp. 133–155. DOI: 10.1122/1.549887
15. Caputo M. Vibrations of an infinite viscoelastic layer with a dissipative memory. Journal of the Acoustical Society. 1974. Vol. 56, pp. 897–904. DOI: 10.1121/1.1903344
16. Ingman D., Suzdalnitsky J. Control of damping oscillations by fractional differential operator with time-dependent order. Computer Methods in Applied Mechanics and Engineering. 2004. No. 193. Vol. 52, pp. 5585–5595. DOI: 10.1016/j.cma.2004.06.029
17. Naber M. Linear fractionally damped oscillator. International Journal of Differential Equations. 2010. Vol. 2010. 197020. DOI: 10.1155/2010/197020
18. Kiryanova L.V., Matseevich T. Sturm-Liouville problem with mixed boundary conditions for a differential equation with a fractional derivative and its application in viscoelasticity models. Axioms. 2023. Vol. 12. No. 8. P. 779. DOI: 10.3390/axioms12080779
19. Podlubny I. Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. London: Academic Press. 1999. 340 p.
20. Кирьянова Л.В. Разностная схема для неоднородной модели Бегли-Торвика // Вестник Академии наук Чеченской Республики. 2020. № 1. Т. 48. С. 14–18. DOI: 10.25744/vestnik.2020.48.1.002
20. Kiryanova L.V. Difference scheme for the inhomogeneous Begley-Torvik model. Vestnik of the Academy of Sciences of the Chechen Republic. 2020. No. 1. Vol. 48, pp. 14–18. (In Russian). DOI: 10.25744/vestnik.2020.48.1.002
For citation: Matseevich T.A., Kiryanova L.V., Smirnov V.A., Ivanov P.S. Optimization of parameters of a viscoelastic model of elements of structures made from composite materials based on experimental data. Zhilishchnoe Stroitel’stvo [Housing Construction]. 2023. No. 11, pp. 32–36. (In Russian).DOI: https://doi.org/10.31659/0044-4472-2023-11-32-36