THE INVERSE PROBLEM OF RECOVERING AN UNSTEADY LINEAR LOAD FOR AN ELASTIC ROD OF FINITE LENGTH

Yana A. Vahterova; Gregory V.  Fedotenkov

doi:10.5937/jaes0-28073

Yana A. Vahterova Moscow Aviation Institute (National Research University), Department of Resistance of Materials Dynamics and Strength of Machines, Moscow, Russian Federation
Gregory V. Fedotenkov 1Moscow Aviation Institute (National Research University), Department of Resistance of Materials Dynamics and Strength of Machines, Russian Federation 2Lomonosov Moscow State University, Institute of Mechanics, Moscow, Russian Federation

DOI: https://doi.org/10.5937/jaes0-28073

Keywords: inverse problem, elastic rod, influence function, Fourier series, integral transformations, integral equations, Tikhonov regularisation, quadrature formulas

Abstract

The main purpose of the paper is to obtain solutions for new non-stationary inverse problems for elastic rods. The objective of this study is to develop and implement new methods, approaches and algorithms for solving non-stationary inverse problems of rod mechanics. The direct non-stationary problem for an elastic rod consists in determining elastic displacements, which satisfies a given equation of non-stationary oscillations in partial derivatives and some given initial and boundary conditions. The solution of inverse retrospective problems with a completely unknown space-time law of load distribution is based on the method of influence functions. With its application, the inverse retrospective problem is reduced to solving a system of integral equations of the Volterra type of the first kind in time with respect to the sought external axial load of the elastic rod. To solve it, the method of mechanical quadratures is used in combination with the Tikhonov regularisation method.

References

Kozel, A.G. (2018). Influence of shear stiffness of the base on the stress state of a sandwich plate. Fundamental and Applied Problems of Engineering and Technology, vol. 6, no. 332, 25-35.

Starovoitov, E.I., Kozel, A.G. (2019). Influence of the stiffness of the Pasternak base on the deformation of a circular sandwich plate. Problems of Mechanical Engineering and Automation, vol. 2, 25-35.

Kozel, A.G. (2019). Areas of physical nonlinearity in a three-layer plate based on parsnips. Sreda Publishing House, Cheboksary.

Kozel, A.G. (2019). Deformation of a physically nonlinear three-layer plate based on Pasternak. Mechanics. Research and Innovation, vol. 12, 105-112.

Okonechnikov, A.S., Tarlakovsky, D.V., Fedotenkov, G.V. (2019). Transient interaction of rigid indenter with elastic half-plane with adhesive force. Lobachevskii Journal of Mathematics, vol. 40, no. 4, 489-498.

Lokteva, N.A., Paimushin, V.N., Serdyuk, D.O., Tarlakovsky, D.V. (2017). Interaction of a plane harmonic wave with a plate limited in height. Scientists Notes of Kazan University. Series: Physics and Mathematics, vol. 159, no. 1, 64-74.

Lokteva, N.A., Tarlakovskii, D.V. (2019). Analysis of vibration insulation properties of a plate in an elastic medium under the influence of different types of waves. Springer Nature Switzerland, Cham.

Lokteva, N.A., Tarlakovskii, D.V. (2019). Interaction of a spherical wave with a rectangular plate in a ground. Springer Nature Switzerland, Cham.

Astapov, A.N., Pogodin, V.A., Rabinskiy, L.N. (2020). CCCM specific surface estimation in process of low-temperature oxidation. Periodico Tche Quimica, vol. 17, no. 34, 793-802.

Egorova, O.V., Rabinskiy, L.N., Zhavoronok, S.I. (2020). Use of the higher-order plate theory of I.N. Vekua type in problems of dynamics of heterogeneous plane waveguides. Archives of Mechanics, vol. 72, no. 1, 3-25.

Babaytsev, A.V., Kuznetsova, E.L., Rabinskiy, L.N., Tushavina, O.V. (2020). Investigation of permanent strains in nanomodified composites after molding at elevated temperatures. Periodico Tche Quimica, vol. 17, no. 34, 1055-1067.

Kurbatov, A.S., Orekhov, A.A., Rabinskiy, L.N., Tushavina, O.V., Kuznetsova, E.L. (2020). Research of the problem of loss of stability of cylindrical thinwalled structures under intense local temperature exposure. Periodico Tche Quimica, vol. 17, no. 34, 884-891.

Bulychev, N.A, Rabinskiy, L.N., Tushavina, O.V. (2020). Effect of intense mechanical vibration of ultrasonic frequency on thermal unstable low-temperature plasma. Nanoscience and Technology, vol. 11, no. 1, 15-21.

Okonechnikov, A.S., Tarlakovski, D.V., Ul'yashina, A.N., Fedotenkov, G.V. (2016). Transient reaction of an elastic half-plane on a source of a concentrated boundary disturbance. IOP Conference Series: Materials Science and Engineering, vol. 158, no. 1, 012073.

Rabinskiy, L.N., Tushavina, O.V. (2019). Investigation of an elastic curvilinear cylindrical shell in the shape of a parabolic cylinder, taking into account thermal effects during laser sintering. Asia Life Sciences, vol. 2, 977-991.

Dobryanskiy, V.N., Rabinskiy, L.N., Tushavina, O.V. (2019). Validation of methodology for modeling effects of loss of stability in thin-walled parts manufactured using SLM technology. Periodico Tche Quimica, vol. 16, no. 33, 650-656.

Dobryanskiy, V.N., Rabinskiy, L.N., Tushavina, O.V. (2019). Experimental finding of fracture toughness characteristics and theoretical modeling of crack propagation processes in carbon fiber samples under conditions of additive production. Periodico Tche Quimica, vol. 16, no. 33, 325-336.

Antufev, B.A., Egorova, O.V., Medvedskii, A.L., Rabinskiy, L.N. (2019). Dynamics of shell with destructive heat-protective coating under running load. INCAS Bulletin, vol. 11, 7-16.

Antufev, B.A., Egorova, O.V., Rabinskiy, L.N. (2019). Dynamics of a cylindrical shell with a collapsing elastic base under the action of a pressure wave. INCAS Bulletin, vol. 11, 17-24.

Gorshkov, A.G., Medvedsky, A.L., Rabinsky, L.N., Tarlakovsky, D.V. (2004). Waves in continuous media. Fizmatlit, Moscow.

Porutchikov, V.B. (1986). Methods of the dynamic theory of elasticity. Nauka, Moscow.

Rabotnov, Yu.N. (1988). Mechanics of a deformable solid. Nauka, Moscow.

Gelfand, I.M., Shilov, G.E. (1959). Generalized functions and actions on them. Fizmatlit, Moscow.

Skvortsov, A.A., Zuev, S.M., Koryachko, M.V. (2018). Contact melting of aluminum-silicon structures under conditions of thermal shock. Key Engineering Materials, vol. 771, 118-123.

Blinov, D.G., Prokopov, V.G., Sherenkovskii, Yu.V., Fialko, N.M., Yurchuk, V.L. (2002). Simulation of natural convection problems based on low-dimensional model. International Communications in Heat and Mass Transfer, vol. 29, no. 6, 741-747.

Vakhterova, Ya.A., Serpicheva, E.V., Fedotenkov, G.V. (2017). The inverse problem of identifying a non-stationary load for a Timoshenko beam. Bulletin of the Tula State University. Engineering Sciences, vol. 4, 82-92.

Dinzhos, R.V., Lysenkov, E.A., Fialko, N.M. (2015). Features of thermal conductivity of composites based on thermoplastic polymers and aluminum particles. Journal of Nano- and Electronic Physics, vol. 7, no. 3, 03022.

Ryndin, V.V. (2019). Statement of the second law of thermodynamics on the basis of the postulate of nonequilibrium. Periodico Tche Quimica, vol. 16, no. 32, 698-712.

Hadamard, J. (1932). Cauchy's problem and hyperbolic linear particle derivative equations. Hermann, Paris.

Skvortsov, A.A., Pshonkin, D.E., Luk'yanov, M.N., Rybakova, M.R. (2018). Deformations of aluminum alloys under the influence of an additional load. Periodico Tche Quimica, vol. 15, no. 30, 421-427.

Tikhonov, A.N., Arsenin, V.Ya. (1979). Methods for solving ill-posed problems. Nauka, Moscow.

Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V., Yagola, A.G. (1990). Numerical methods for solving ill-posed problems. Nauka, Moscow.

Markova, E.V. (1999). Numerical methods for solving non-classical linear Volterra equations of the first kind and their applications. Irkutsk State Pedagogical University, Irkutsk.

Vatulyan, A.O. (2007). Inverse problems in solid mechanics. Fizmatlit, Moscow.

Fedotenkov, G.V., Tarlakovsky, D.V., Vahterova, Y.A. (2019). Identification of non-stationary load upon timoshenko beam. Lobachevskii Journal of Mathematics, vol. 40, no. 4, 439-447.