P. Pukach, I. Kuzio, M. Sokil

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A b s t r a c t . The aim of paper is to study the solution of the prob lem of nonlinear transverse vibrations of elastic elongated body under the force of resistance in unbounded domain. Such problems have applications in various tech nical systems – vibration of pipelines, railways, long bridges, electric lines, optical fi bers. Unboundedness of the area creates more fundamental diffi culties in the study of the problem. For the considered models of nonlinear os ci llations have no ge ne ral analytical techniques for determining the dy na mic cha racteristics of the oscillatory process. Therefore it is sug ges ted to use qualitative methods of the theory of nonlinear boundary value problems to obtain correct problem solution conditions (existence and uniqueness of the solu tion). In the paper conditions of the correctness of the solu tion of mathematical model for these nonlinear systems (suffi cient conditions of the existence and uniqueness in the class of locally integrable functions) are obtained. Methods of qualitative study of semi-infi nite cable vibrations under the forces of resistance based on general principles of the theory of nonlinear boundary value problems – method of monotony and Galerkin method. Scientifi c novelty of the work lies in particular in the generalization of methods of studying nonlinear problems on a new class of oscillatory systems in unbounded domains, justifying the correctness of the solution with specifi ed mathematical model, which has practical applications in real engineering oscillatory systems. The technique allows not only for proving the correctness of the model solution, but also has an opportunity in its study to apply various approximate methods.

Key words: mathematical model, nonlinear vibrations, nonlinear boundary value problem, Galer kin method, method of monotony, unbounded domain.

Qualitative methods for research of transversal vibrations of semi-infi nite cable under the action of nonlinear resistance forces

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