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IssuesArchive of Issues2018-2pp.177-183

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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
References
1.  A. I. Vesnitsky and A. I. Potapov, "Transverse Vibrations of Ropes in Mine Hoists," in Dynamics of Systems, No. 7 (Gor'k. Univ., Gor'kiy, 1975), pp. 84-89.
2.  L. B. Kolosov and T. I. Zhigula, Longitudinal-Transverse Vibrations of a String of a Rope of a Lifting Installation," Izv. Vyssh. Uchebn. Zaved. Gorn. Zh. No. 3, 83-86 (1981).
3.  V. N. Anisimov and V. L. Litvinov, "Longitudinal Oscillations of a Viscoelastic Rope of Variable Length," in Proc. 4th All-Russia Sci. Conf. "Mathematical Modeling and Boundary-Value Problems." Mathematical Models of Mechanics, Strength, and Reliability of Structural Elements. Part 1 (SamGTU, Samara, 2007), pp. 25-27 [in Russian].
4.  W. D. Zhu and Y. Chen, "Theoretical and Experimental Investigation of Elevator Cable Dynamics and Control," J. Vibr. Acoust., No. 1, 66-78 (2006).
5.  Y. Shi, L. Wu, and Y. Wang, "Nonlinear Analysis of Natural Frequencies of Cable System," J. Vibr. Engng, No. 2, 173-178 (2006).
6.  L. Wang and Y. Zhao, "Multiple Internal Resonances and Non-Planar Dynamics of Shallow Suspended Cables to the Harmonic Excitations," J. Sound Vibr. 319 (1-2), 1-14 (2009).
7.  Y. Zhao and L. Wang, "On the Symmetric Modal Interaction of the Suspended Cable: Three-to-One Internal Resonance," J. Sound Vibr. 294 (4-5), 1073-1093 (2006).
8.  Yu. P. Samarin and V. N. Anisimov, "Forced Transverse Vibrations of a Flexible Link during Acceleration," Izv. Vyssh. Uchebn. Zaved. Mashinostr., No. 12, 17-21 (1986).
9.  A. I. Vesnitsky, Waves in Systems with Moving Boundaries and Loads (Fizmatlit, Moscow, 2001) [in Russian].
10.  O. A. Goroshko and G. N. Savin, Introduction to Mechanics of Deformable One-dimensional Bodies of Variable Length (Naukova Dumka, Kiev, 1971) [in Russian].
11.  V. N. Anisimov, V. L. Litvinov, and I. V. Korpen, "On a Method of Analytical Solution of Wave Equation Describing the Oscillations System with Moving Boundaries," Vestnik Samarsk. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, No. 3(28), 145-151 (2012).
12.  Yu. P. Samarin, "On Wave Phenomena in Regions with Moving Boundaries," in Volga Mathematical Collection, Issue 5 (KPI, Kuibyshev, 1967), pp. 337-340 [in Russian].
13.  E. L. Nikolai, "On Transverse Vibrations of a Section of a String, the Length of which Changes Uniformly," in Works on Mechanics (Gostekhizdat, Moscow, 1955), pp. 328-331 [in Russian].
14.  W. D. Zhu and N. A. Zheng, "Exact Response of a Translating String with Arbitrarily Varying Length under General Excitation," J. Appl. Mech. 75 (3), 031003 (2008).
15.  Z. W. Engel and J. Nizioł, "Problems of Mechanical Vibrations in the Works by Cracow's Scientists," Czasopismo Techniczne. M, No. 1, 77-85 (2008).
16.  B. H. Anisimov and V. L. Litvinov, "Investigation of Resonance Characteristics of Mechanical Objects with Moving Borders by Application of the Kantorovich-Galyorkin Method," Vestnik Samarsk. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, No. 1(18), 149-158 (2009).
17.  V. N. Anisimov and V. L. Litvinov, Resonant Properties of Mechanical Objects with Moving Boundaries (SamGTU, Samara, 2009) [in Russian].
18.  H. Ding and L. Chen, "Galerkin Methods for Natural Frequencies of High-Speed Axially Moving Beams," J. Sound Vibr. 329 (17), 3484-3494 (2010).
19.  A. A. Lezhneva, "Flexural Oscillations of a Beam of Variable Length," Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 1, 159-161 (1970) [Mech. Solids (Engl. Transl.)],
20.  A. A. Lezhneva, "Free Flexural Oscillations of a Beam of Variable Length," in Schilarship Notes, No. 156 (Perm. Univ., Perm', 1966), pp. 143-150 [in Russian].
21.  G. N. Savin and O. A. Goroshko, Dynamics of a Thread of Variable Length (Naukova Dumka, Kiev, 1962) [in Russian].
22.  V. L. Litvinov, "Investigation of Free Vibrations of Mechanical Objects with Moving Boundaries by Means of an Asymptotic Method," Zh. Srednevolzhsk. Mat. Obshch. 16 (1), 83-88 (2014).
23.  V. L. Litvinov, "Transverse Vibrations of a Viscoelastic Rope of Variable Length, Having Flexural Rigidity, Taking into Account the Influence of the Medium's Resistance Forces," in Modern problems in the theory of functions and their applications (Izd-vo Nauchn. Kniga, Voronezh, 2014), pp. 153-157 [in Russian].
24.  V. N. Anisimov and V. L. Litvinov, "Analysis of the Influence of the Motion of Boundaries with the Investigation of the Resonance Properties of Systems with the Damping," Vestnik Samarsk. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, No. 2(19), 147-152 (2009).
25.  V. N. Anisimov and V. L. Litvinov, "Calculation of the Natural Frequencies of Transverse Vibrations of a Viscoelastic Rope Moving in the Longitudinal Direction and Having a Flexural Rigidity," in Proc. 5th All-Russia Sci. Conf. "Mathematical Modeling and Boundary Value Problems." Mathematical Models of Mechanics, Strength, and Reliability of Structural Elements. Part 1 (SamGTU, Samara, 2008), pp. 38-42 [in Russian].
26.  V. N. Anisimov, I. V. Korpen, and V. L. Litvinov, "Resonant Properties of a Rope of Variable Length, Having Flexural Stiffness with Allowance for the Action of Damping Forces," in Proc. 8th All-Russia Sci. Conf. "Mathematical Modeling and Boundary Value Problems." Mathematical Models of Mechanics, Strength, and Reliability of Structural Elements (SamGTU, Samara, 2011), pp. 15-20 [in Russian].
27.  Z. Liu and G. Chen, "Analysis of Flat Non-Linear Free Oscillations of the Supporting Rope Taking into Account the Effect of Flexural Rigidity," J. Vibr. Engng, No. 1, 57-60 (2007).
Received 22 January 2015
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