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Thus, "the old problem remains open, and no available "modern" methods make possible to calculate the real frequencies of a non-linear system. This problem remains unsolved for the applications because in approximations by series, whether converging or only formal, only finite and, generally speaking, quite small number of terms can be calculated. For today we cannot find a way to express the common term and the sum of these series" [13, p. 305]. "Another problem of a great interest is an issue of better understanding of the solution 'in the far, in the near and at the resonance conditions'. When will we really have the most preferable determination of the system resonance?" [13, p. 309].

As we will show in this paper, just the absence of the exact analytical solutions for linear problems was still the principal reason for the conventional solutions to be limited by the values of natural frequencies of a modelling system. This prevents from studying the regularities of a process in a form achievable by the analytical methods with the solutions in analytical form. "The classical vibration theory is grounded on solving the differential equations and on joining the solutions for different parts of the system on the basis of continuity conditions. Any slight change of the system shape makes necessary to calculate all anew. However, disregarding the calculation labour, one can note that the high accuracy of classical theory is illusory" [14, p. 317].

Just the limitations of approaches by the matrix, integral and asymptotic techniques, in junction with the settled practice to give additionally the initial and boundary conditions for a generalised system of differential equations, causes an insurmountable problem, when "the presence of irregular boundaries in the majority of practical problems disables constructing the analytical solutions of differential equations, and the numerical techniques became the only possible means to obtain quite accurate and detailed results" [15, p. 12]. Although "the numerical techniques are quite universal and simple, they inherited a number of imperfections. First of all, these are the solutions obtained in the form of tables, which is especially inconvenient in performing some logic or mathematical operations with them. A great imperfection of the numerical modelling is also its sensitivity to the choice of a step" [16, p. 9].

However the main shortcoming of the numerical techniques is the absence of a reliable analytical formalism which, having been found as the basis of calculations, in fact pre-determines the quality of obtained numerical solutions. "As a rule, the search of a solution was carried out by different techniques (Chesare, Krylov-Bogolyubov, through the variable action-angle etc.) for different cases, with the expansion of sin xand cos x into a series in the orders of x smallness. Such diversity of techniques has impeded the evaluation of particular solutions, the interpretation of obtained results and understanding the reasons of chaos and bifurcation in the systems" [11, p. 36]. Because "the notation of solution as an analytical expression allows to identify, how a solution behaves in accordance with the right part, coefficients and initial data. With the successfully constructed analogue method that takes into account the specific of this class of problems, one can achieve a solution in its most simple form; with it the evaluations of accuracy of an approximate solution may appear more visibly. In the view of applicability of the analytical methods in computing, I would emphasise that due to great informativeness of the values, the number of calculations decreases saving the operative memory" [16, p. 9].

In this work we will make use of the advantages of the exact analytical solutions that we studied in [2]- [9]. We will extend these solutions to the non-linear dynamics, neglecting the condition of smallness and non-linearity of constraints. With it we will not be basing on the linear solutions themselves, varying the linear solution in attempt to obtain a solution for a non-linear system of differential equations by way of varying the linear solution. We will try to transfer the basic principle grounded on the clear specification of the model features into the non-linear mechanics. The cause of our intention is, if we completely take into consideration the features of the specific model in the modelling system of differential equations, then the additional initial and boundary conditions become excessive, as we showed it in the above papers. The boundary conditions are reflected in the features of system of differential equations itself, and the initial conditions are determined by the type of the external force for forced vibrations or by the features of vibrations of a randomly selected element for free vibrations. This makes unnecessary to join the solutions and allows to achieve the maximal determinacy of solutions for a specific model of an elastic line. In its turn, this enables the transition to the non-linear dynamics of processes, retaining the continuous analytical connection with the linear solutions.