 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| V.N. Anisimov, I.V. Korpen, and V.L. Litvinov, "Application of the Kantorovich-Galerkin Method for Solving Boundary Value Problems with Conditions on Moving Borders," Mech. Solids. 53 (2), 177-183 (2018) |
| Year |
2018 |
Volume |
53 |
Number |
2 |
Pages |
177-183 |
| DOI |
10.3103/S0025654418020085 |
| Title |
Application of the Kantorovich-Galerkin Method for Solving Boundary Value Problems with Conditions on Moving Borders |
| Author(s) |
V.N. Anisimov (Syzran' Branch of the Samara State Technical University, ul. Sovetskaya 45, Syzran', 446001 Russia, anisimov170159@mail.ru)
I.V. Korpen (Syzran' Branch of the Samara State Technical University, ul. Sovetskaya 45, Syzran', 446001 Russia)
V.L. Litvinov (Syzran' Branch of the Samara State Technical University, ul. Sovetskaya 45, Syzran', 446001 Russia, vladlitvinov@rambler.ru) |
| Abstract |
The approximate Kantorovich-Galerkin method is considered for solving problems describing the vibrations of viscoelastic objects with conditions on moving boundaries and analyzing the resonance properties of these objects. The method makes it possible to take into account the effect of forces of environmental resistance on the system, flexural rigidity, and also boundary conditions with weak nonstationarity. The mathematical formulation of the problem involves a partial differential equation with respect to the desired displacement function and inhomogeneous boundary conditions. The Kantorovich-Galerkin method makes it possible to take into account the initial conditions, but they do not affect the resonance properties of linear systems, so in this case they are not taken into account. By introducing a new function into the problem, the boundary conditions are reduced to homogeneous ones. The solution is carried out in dimensionless variables to within a second order of smallness with respect to small parameters characterizing the velocity of the boundary motion and viscoelasticity. Using the Kantorovich-Galerkin method, an approximate solution of high accuracy of the problem of forced longitudinal vibrations of a viscoelastic rope of variable length, one end of which is wound on a drum, and the second is rigidly fixed, is found. The results obtained for the amplitude of oscillations corresponding to the $n$th dynamical mode are presented. The phenomenon of steady resonance and passage through resonance is investigated using numerical methods. A graphical dependence of the maximum amplitude of the rope oscillations as it passes through the resonance, depending on the coefficient characterizing the viscoelasticity of the object based on the Voigt model, is presented. The accuracy of the Kantorovich-Galerkin method is estimated. |
| Keywords |
resonance properties, oscillations of systems with moving boundaries, boundary motion laws, viscoelastic properties, amplitude of oscillations |
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|
| Received |
22 January 2015 |
| Link to Fulltext |
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