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.
Citation:
K. B. Sabitov, O. V. Fadeeva, “Initial-boundary value problem for the equation of forced vibrations of a cantilever beam”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 25:1 (2021), 51–66
\Bibitem{SabFad21}
\by K.~B.~Sabitov, O.~V.~Fadeeva
\paper Initial-boundary value problem for the equation of forced vibrations of~a~cantilever beam
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2021
\vol 25
\issue 1
\pages 51--66
\mathnet{http://mi.mathnet.ru/vsgtu1845}
\crossref{https://doi.org/10.14498/vsgtu1845}
\zmath{https://zbmath.org/?q=an:1474.35223}
\elib{https://elibrary.ru/item.asp?id=45604170}
Linking options:
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https://www.mathnet.ru/eng/vsgtu/v225/i1/p51
This publication is cited in the following 16 articles:
A. K. Urinov, M. S. Azizov, “Initial-Boundary Value Problem for a Degenerate High Even-Order Partial Differential Equation with the Bessel Operator”, Lobachevskii J Math, 45:2 (2024), 864
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U. D. Durdiev, “Obratnaya zadacha ob istochnike dlya uravneniya vynuzhdennykh kolebanii balki”, Izv. vuzov. Matem., 2023, no. 8, 10–22
U. D. Durdiev, Z. R. Bozorov, “Nonlocal inverse problem for determining the unknown coefficient in the beam vibration equation”, J. Appl. Industr. Math., 17:2 (2023), 281–290
A. K. Urinov, M. S. Azizov, “About an initial boundary problem for a degenerate higher even order partial differential equation”, J. Appl. Industr. Math., 17:2 (2023), 414–426
A. K. Urinov, D. D. Oripov, “O razreshimosti odnoi nachalno-granichnoi zadachi dlya vyrozhdayuschegosya uravneniya vysokogo chetnogo poryadka”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 27:4 (2023), 621–644
Yu. P. Apakov, S. M. Mamazhonov, “Boundary Value Problem for an Inhomogeneous Fourth-Order Equation with Lower-Order Terms”, Diff Equat, 59:2 (2023), 188
U. D. Durdiev, “Inverse Source Problem for the Equation of Forced Vibrations of a Beam”, Russ Math., 67:8 (2023), 7
U. D Durdiev, “Obratnaya zadacha po opredeleniyu neizvestnogo koeffitsienta uravneniya kolebaniya balki v beskonechnoy oblasti”, Differentsialnye uravneniya, 59:4 (2023), 456
O. V. Fadeeva, “Inverse Problems for the Equation of Vibrations of a Canister Beam to Find the Source”, Prikladnaya matematika i mekhanika, 87:4 (2023), 661
U. D. Durdiev, “Inverse Problem of Determining the Unknown Coefficient in the Beam Vibration Equation in an Infinite Domain”, Diff Equat, 59:4 (2023), 462
Yu. P Apakov, S. M Mamazhonov, “Kraevaya zadacha dlya neodnorodnogo uravneniya chetvertogo poryadka s mladshimi chlenami”, Differentsialnye uravneniya, 59:2 (2023), 183
A. Grigorenko, T. Duyun, “SIMULATION OF NATURAL FREQUENCIES AND FORCED OSCILLATION MAGNITUDES OF A VERTICAL MILLING MACHINE”, Bulletin of Belgorod State Technological University named after. V. G. Shukhov, 8:6 (2023), 76
A. K. Urinov, M. S. Azizov, “Nachalno-granichnaya zadacha dlya uravneniya v chastnykh proizvodnykh vysshego chetnogo poryadka s operatorom Besselya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 26:2 (2022), 273–292
U. D. Durdiev, “Inverse Problem of Determining an Unknown Coefficient in the Beam Vibration Equation”, Diff Equat, 58:1 (2022), 36
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