Email this article Initial-boundary value problem for the equation of forced vibrations of a cantilever beam
- Authors: Sabitov K.B.1, Fadeeva O.V.1
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Affiliations:
- Samara State Technical University
- Issue: Vol 25, No 1 (2021)
- Pages: 51-66
- Section: Differential Equations and Mathematical Physics
- URL: https://journal-vniispk.ru/1991-8615/article/view/60601
- DOI: https://doi.org/10.14498/vsgtu1845
- ID: 60601
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Abstract
In this paper, an initial-boundary value problem for the equation of forced vibrations of a cantilever beam is studied. Such a linear differential equation of the fourth order describes bending transverse vibrations of a homogeneous beam under the action of an external force in the absence of rotational motion during bending.
The system of eigenfunctions of the one-dimensional spectral problem, which is orthogonal and complete in the space of square-summable functions, is constructed by the method of separation of variables. The uniqueness of the solution to the initial-boundary value problem is proved in two ways: (i) using the energy integral; (ii) relying on the completeness property of the system of eigenfunctions.
The solution to the problem was first found in the absence of an external force and homogeneous boundary conditions, and then the general case was considered in the presence of an external force and inhomogeneous boundary conditions. In both cases, the solution of the problem is constructed as the sum of the Fourier series.
Estimates of the coefficients of these series and the system of eigenfunctions are obtained. On the basis of the established estimates, sufficient conditions were found for the initial functions, the fulfillment of which ensures the uniform convergence of the constructed series in the class of regular solutions of the beam vibration equation, i.e. existence theorems for the solution of the stated initial-boundary value problem are proved. Based on the solutions obtained, the stability of the solutions of the initial-boundary value problem is established depending on the initial data and the right-hand side of the equation under consideration in the classes of square-summable and continuous functions.
About the authors
Kamil B. Sabitov
Samara State Technical University
Email: sabitov_fmf@mail.ru
ORCID iD: 0000-0001-9516-2704
SPIN-code: 3011-3873
Scopus Author ID: 6603447719
http://www.mathnet.ru/rus/person11101
Dr. Phys. & Math. Sci.; Professor; Dept. of Higher Mathematics
244, Molodogvardeyskaya st., Samara, 443100, Russian FederationOksana V. Fadeeva
Samara State Technical University
Author for correspondence.
Email: faoks@yandex.ru
ORCID iD: 0000-0003-1704-9524
SPIN-code: 9266-7262
Scopus Author ID: 57223162055
http://www.mathnet.ru/rus/person41418
Cand. Phys. & Math. Sci.; Associate Professor; Dept. of Higher Mathematics
244, Molodogvardeyskaya st., Samara, 443100, Russian FederationReferences
- Tikhonov A. N., Samarskii A. A. Uravneniia matematicheskoi fiziki [Equations of Mathematical Physics]. Moscow, Nauka, 1966, 724 pp. (In Russian)
- Rayleigh L. Teoriia zvuka [The Theory of Sound]. Moscow, Gostehizdat, 1955, 503 pp. (In Russian)
- Krylov A. N. Vibratsiia sudov [The Ship Vibration]. Leningrad, Moscow, 1936, 442 pp. (In Russian)
- Collatts L. Zadachi na sobstvennye znacheniia s tekhnicheskimi prilozheniiami [The Eigenvalue Problem with Technical Applications]. Moscow, Nauka, 1968, 503 pp. (In Russian)
- Biderman V. L. Teoriia mekhanicheskikh kolebanii [Theory of Mechanical Vibrations]. Moscow, Vyssh. shk., 1980, 408 pp. (In Russian)
- Timoshenko S. P. Kolebaniia v inzhenernom dele [Fluctuations in Engineering]. Moscow, Fizmatlit, 1967, 444 pp. (In Russian)
- Rudakov I. A. Periodic solutions of the quasilinear beam vibration equation with homogeneous boundary conditions, Differ. Equ., 2012, vol. 48, no. 6, pp. 820–831. https://doi.org/10.1134/S0012266112060067.
- Li S., Reynders E., Maes K., De Roeck G. Vibration-based estimation of axial force for abeam member with uncertain boundary conditions, J. Sound Vibrat., 2013, vol. 332, no. 4, pp. 795–806. https://doi.org/10.1016/j.jsv.2012.10.019.
- Rudakov I. A. Periodic solutions of the quasilinear equation of forced beam vibrations with homogeneous boundary conditions, Izv. Math., 2015, vol. 79, no. 5, pp. 1064–1086. https://doi.org/10.1070/IM2015v079n05ABEH002772.
- Wang Y.-R., Fang Z.-W. Vibrations in an elastic beam with nonlinear supports at both ends, J. Appl. Mech. Tech. Phys., 2015, vol. 56, no. 2. https://doi.org/10.1134/S0021894415020200.
- Sabitov K. B. Fluctuations of a beam with clamped ends, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2015, vol. 19, no. 2, pp. 311–324 (In Russian). https://doi.org/10.14498/vsgtu1406.
- Sabitov K. B. A remark on the theory of initial-boundary value problems for the equation of rods and beams, Differ. Equ., 2017, vol. 53, no. 1, pp. 86–98. https://doi.org/10.1134/S0012266117010086.
- Sabitov K. B. Cauchy problem for the beam vibration equation, Differ. Equ., 2017, vol. 53, no. 5, pp. 658–664. https://doi.org/10.1134/S0012266117050093.
- Kasimov S. G., Madrakhimov U. S. Initial-boundary value problem for the beam vibration equation in the multidimensional case, Differ. Equ., 2019, vol. 55, no. 10, pp. 1336–1348. https://doi.org/10.1134/S0012266119100094.
- Sabitov K. B., Akimov A. A. Initial-boundary value problem for a nonlinear beam vibration equation, Differ. Equ., 2020, vol. 56, no. 5, pp. 621–634. https://doi.org/10.1134/S0012266120050079.
- Naimark M. A. Lineinye differentsial’nye operatory [Linear Differential Operators]. Moscow, Nauka, 1969, 528 pp. (In Russian)
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