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All existing methods join the solutions for different parts of a system, basing on the continuity conditions. And the methods are highly sensitive to the attempts to vary some parameters of the system in the course of solution; this causes the necessity to calculate anew. Besides, high accuracy of many methods is commonly known to be illusory [6, p. 317].

The approaches are limited by the matrix, integral and asymptotic methods. Together with the firmly established practice to give additionally the initial and boundary conditions for a generalised system of differential equations, it brings an insurmountable problem, when "the presence of irregular boundaries in the majority of practical problems disables to construct the analytical solutions of differential equations, and the numerical techniques became the only possible means to obtain quite accurate and detailed results" [7, p. 12]. Whilst the numerical techniques, with all their simplicity and broad applicability, are completely useless when we need to process them logically either mathematically. A great sensitivity to the chosen step is their another demerit [8, p. 9], since two nearest points of the system can behave very different, and we always risk to omit an important feature of behaviour.

However the main shortcoming of the numerical techniques is the absence of a reliable analytical formalism which, if having been found as the basis of calculations, would factually pre-determine the quality of obtained numerical solutions. "As a rule, the search of solutions was carried out by different techniques (Chesare, Krylov-Bogolyubov, through the variable action-angle etc.) for different cases, with the expansion of sin x and 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" [3, p. 36].

The analytical record of solutions enable us to avoid all these shortcomings. It is much more simple and accurate, gives good understanding of the process in local sections and as the whole, it is easily programmable, well lessens the laborious calculations and saves the operative memory of machine [8, p. 9]. We will show in this paper that just the absence of exact analytical solutions for the whole complex of linear problems was up to now the cause, why the conventional solutions for nonlinear problems were limited to the values of natural frequencies of the modelled system. This disabled studying of the regularities of processes in the form achievable analytically, when we yield the solutions in an analytical form.

The necessity to get over the listed problems has led us to the necessity to develop a new original non-matrix method to obtain exact analytical solutions, which already now is able to solve a broad circle of problems for 1D elastic lines (such as mechanical shafts), for bound systems (as the joint of a table-land and mountain ridge, or as the mountain ridge bend being the concentrators of seismic destroying power), for closed-loop systems (allowing, for example, to model the dynamical processes in a wheel), for complex elastic systems containing the resonance sub-systems, etc. We also have successfully applied our method for calculation of mismatched electric filters and checked it experimentally for the electric circuit of six mismatched ladder filters (as this is beyond the conventional calculation ability, it ensured the purity of check). The experiment gave us the diagrams coinciding with those calculated up to trifles that are usually thought the noises of experiment, but appeared the regular small peaks.

Up to now our method had a great demerit - it was developed only for linear systems, while just nonlinear systems are most difficult in calculation. The target of this paper is to fill in this gap. We will make use of the advantages of exact analytical solutions studied in [9]- [16] and extend these solutions into the area of nonlinear dynamics, neglecting the smallness of non-linearity of constraints. We will not base on the linear solutions and will not try to obtain the solution for the nonlinear system of differential equations, varying the linear solution, but we will try to transfer to the nonlinear mechanics the basic principle grounded on the clear specification of the features of model. The cause of our intention is, as we showed in the mentioned papers, if we completely take into account the features of specific model in the modelling system of differential equations, the additional initial and boundary conditions become excessive. The boundary conditions are reflected in the features of the very system of differential equations, and the initial conditions are determined by the pattern of external affection for forced vibrations, either by the features of vibrations of the selected element in case of free vibrations. With it, it will be unnecessary to join the solutions; it enables to achieve the maximal determinacy of solutions for the specific model of elastic system. In its turn, this enables to pass to the nonlinear dynamics of processes, keeping the continuous analytical relation with the linear solutions