Abstract
The problem of vibrations of objects with moving boundaries formulated as a differential equation with boundary and initial conditions is a nonclassical generalization of a hyperbolic problem. To facilitate the construction of the solution to this problem and to justify the choice of the form of the solution, equivalent integro-differential equations with symmetric and time-dependent kernels and time-varying integration limits are constructed. The advantages of the method of integro-differential equations are revealed in the transition to more complex dynamic systems carrying concentrated masses vibrating under the action of moving loads. The method is extended to a wider class of model boundary value problems that take into account bending stiffness, the resistance of the external environment, and the stiffness of the base of a vibrating object. The solution is given in dimensionless variables and is accurate up to second-order values with respect to small parameters characterizing the velocity of the boundary. An approximate solution is found for the problem of transverse vibrations of a hoisting rope having bending stiffness, one end of which is wound on a drum and a load is fixed on the other.
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REFERENCES
L. B. Kolosov and T. I. Zhigula, “Longitudinal–transverse vibrations of an elevator rope,” Izv. Vyssh. Uch. Zaved. Gorn. Zh., No. 3, 83–86 (1981).
W. D. Zhu and Y. Chen, “Theoretical and experimental investigation of elevator cable dynamics and control,” J. Vibr. Acoust., No. 1, 66–78 (2006).
Y. Shi, L. Wu, and Y. Wang, “Nonlinear analysis of natural frequencies of a tether system,” J. Vib. Eng., No. 2, 173–178 (2006).
O. A. Goroshko and G. N. Savin, Introduction to the Mechanics of Deformable One-Dimensional Bodies of Variable Length (Naukova Dumka, Kiev, 1971) [in Russian].
V. L. Litvinov and V. N. Anisimov, “Transverse vibrations of a rope moving in longitudinal direction,” Vestn. Samar. Nauchn. Tsentra Ross. Akad. Nauk 19 (4), 161–165 (2017).
G. N. Savin and O. A. Goroshko, Dynamics of a Variable Length Thread (Naukova Dumka, Kiev, 1962) [in Russian].
Z. Liu and G. Chen, “Analysis of plane nonlinear free vibrations of a carrying rope taking into account the influence of flexural rigidity,” J. Vib. Eng., No. 1, 57–60 (2007).
J. Palm et al., “Simulation of mooring cable dynamics using a discontinuous Galerkin method,” The 5th International Conference on Computational Methods in Marine Engineering (2013).
V. L. Litvinov, “Investigation of free vibrations of mechanical objects with moving boundaries using the asymptotic method,” Zh. Middle Volga Mat. Society. 16 (1), 83–88 (2014).
V. L. Litvinov and V. N. Anisimov, Mathematical Modeling and Research of Oscillations of One-Dimensional Mechanical Systems with Moving Boundaries (Samar. Gos. Tekh. Univ., Samara, 2017) [in Russian].
A. A. Lezhneva, “Free bending vibrations of a beam of variable length,” Uch. Zap. Perm. Gos. Univ., No. 156, 143–150 (1966).
L. Wang and Y. Zhao, “Multiple internal resonances and non-planar dynamics of shallow suspended cables to the harmonic excitations,” J. Sound Vib., No. 1–2, 1–14 (2009).
Y. Zhao and L. Wang, “On the symmetric modal interaction of the suspended cable: Three to one internal resonance,” J. Sound Vib., No. 4–5, 1073–1093 (2006).
V. L. Litvinov and V. N. Anisimov, “Application of the Kantorovich–Galerkin method for solving boundary value problems with conditions on moving boundaries,” Izv. Ross. Akad. Nauk Mekh. Tverd. Tela, No. 2, 70–77 (2018).
A. Berlioz and C.-H. Lamarque, “A non-linear model for the dynamics of an inclined cable,” J. Sound Vib. 279, 619–639 (2005).
S. H. Sandilo and W. T. van Horssen, “On variable length induced vibrations of a vertical string,” J. Sound Vib. 333, 2432–2449 (2014).
W. Zhang and Y. Tang, “Global dynamics of the cable under combined parametrical and external excitations,” Int. J. Nonlinear Mech. 37, 505–526 (2002).
L. Faravelli, C. Fuggini, and F. Ubertini, “Toward a hybrid control solution for cable dynamics: Theoretical prediction and experimental validation,” Struct. Control Health Monit. 17, 386–403 (2010).
A. I. Vesnitskii, Waves in Systems with Moving Boundaries and Loads (Fizmatlit, Moscow, 2001) [in Russian].
V. N. Anisimov, V. L. Litvinov, and I. V. Korpen, “On one method of obtaining an analytical solution of the wave equation describing the oscillations of systems with moving boundaries,” Vestn. Samar. Gos. Tekh. Univ. Ser. Fiz. Mat. Nauki 3 (28), 145–151 (2012).
A. I. Vesnitskii, “Inverse problem for a one-dimensional resonator changing its dimensions with time,” Izv. Vyssh. Uch. Zaved. Radiofiz. 10, 1538–1542 (1971).
K. A. Barsukov and G. A. Grigoryan, “On the theory of a waveguide with movable boundaries,” Izv. Radiofiz. 2, 280–285 (1976).
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Litvinov, V.L., Litvinova, K.V. An Approximate Method for Solving Boundary Value Problems with Moving Boundaries by Reduction to Integro-Differential Equations. Comput. Math. and Math. Phys. 62, 945–954 (2022). https://doi.org/10.1134/S0965542522060112
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DOI: https://doi.org/10.1134/S0965542522060112