Nonlinear transverse vibrations of semi-infinite cable with consideration paid to resistance

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Authors:

P.Ya. Pukach, Cand. Sci. (Phys.-Math.), Associate Professor, Lviv Polytechnic National University, Associate Professor of Higher Mathematics Department, Lviv, Ukraine.

I.V. Kuzio, Dr. Sci. (Tech.), Professor, Lviv Polytechnic National University, Head of Department of Mechanics and Mechanical Engineering Automation, Lviv, Ukraine.

Abstract:

 

Purpose. To study the methods of solution of the problem of nonlinear transverse vibrations of elastic long-length body under the force of resistance in unbounded domain. Such problems have applications in various tech­nical systems, for example, vibration of pipelines, railways, long bridges, electric lines, optical fibres. The unboundedness of the area creates more fundamental difficulties in the study of the problem. For the models of nonlinear oscillations under consideration there are no general analytical techniques of determination of the dynamic characteristics of the oscillation process.

 

Methodology. The qualitative study of semi-infinite cable vibrations under the forces of resistance is based on the general principles of the theory of nonlinear boundary value problems, the method of monotony and Galerkin method.

 

Findings. We have suggested using qualitative methods of the theory of nonlinear boundary value problems to obtain the problem solution correctness conditions (existence and uniqueness of the solution). The required conditions of the correctness of the solution have been obtained for the mathematical models of nonlinear systems (sufficient conditions of the existence and uniqueness in the class of locally integrable functions).

 

Originality. We have generalized the methods of study of nonlinear problems for new class of oscillation systems in unbounded domains. We have substantiated the correctness of the solution of specified mathematical model, which has practical applications in real engineering oscillation systems.

 

Practical value. The technique allows proving the correctness of the model solution, and gives the opportunity to apply various approximate methods. 

 

References:

 

 

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4. Ghayesh, M.H., (2010), “Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation”, Int. J. Non-Lin. Mech., Vol. 45, pp. 382–394.

 

 

5. Demeio, L. and Lenci, S., (2007), “Forced nonlinear oscillations of semi-infinite cables and beams resting on a unilateral elastic substrate“, Nonlinear Dynamics, Vol. 49, pp. 203–215.

 

 

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7. Lavrenyuk, S.P. and Pukach, P.Ya., (2007), “Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables”, Ukrainian Mathematical Journal, Vol. 59, no.11, pp. 1708–1718.

 

 

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Pukach, P.Ya., (2004), “Mixed problem in unbounded domain for weakly nonlinear hyperbolic equation with growing coefficients”, Matematychni metody i fizyko-mekhanichni polya, vol. 47, no. 4, pp. 149–154.

 

 

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Lions, J.L. (2002), Nekotorye metody resheniya nelineynykh krayevykh zadach [Some Methods For Solving Nonlinear Boundary Value Problems], Translated by О.А. Оleynik, Editorial URSS, Moscow, Russia.

 

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ISSN (print) 2071-2227,
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