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But the main distinction of the investigated model is the incoincidence of the resonance frequencies for longitudinal and transversal vibration components. Due to it, in one and the same elastic line, with one and the same angle of kink and one and the same external force inclination, dependently on frequency the longitudinal, inclined or transversal waves can propagate. No one of the above considered models had such feature. Can the variation or asymptotic techniques, that ground on the finding the small variation of the elastic line parameters, reveal these resonance peaks, taking as a basis an ideal elastic line? The more that for an ideal line the conventional solutions are not exact and determined enough? Naturally, they cannot. The solutions for an ideal line contain no information about possible transformations. Neither on the basis of the modelling system of differential equations nor on the basis of solutions one can reconstruct so substantially transformed multipliers and summands. So in the solutions for vibrant systems only transitions from the complex to the simpler models will be correct, never vice versa. And only having the exact analytical solutions for the specific class of elastic lines or for that generalising, one can investigate quite reliably the vibration pattern in them. This is the principal advantage which the exact analytical solutions give.

With equal stiffness coefficients the solutions essentially simplify. None the less, in this case the vibration patterns have their interesting features. In Fig. 11 we show the vibration pattern in an elastic line having equal longitudinal and transversal stiffness coefficients, under inclined external force, at positive (a) and negative (b) kink angles. One can see that the vibration pattern before and after the kink is different, though the kink does not effect on the solutions in this class of problems. Here we run into the influence of transition between the reference systems. Transiting through the kink point, we automatically transit into another reference system. With it the amplitudes of longitudinal and transversal components (and it means, the propagation pattern) varies. To compare, in Fig. 12 we show the vibration diagram in the kinked line with the inequal stiffness coefficients. One can see that the vibration pattern has changed. In particular, in the kink domain the vibration amplitude has increased comparing with the domains out of the kink. In this domain the dynamical cutoffs have appeared that lead, as is known, to the additional destructive stress in an elastic line.

An interesting case of the kinked elastic lines are closed lines. An external force in them acts simultaneously in both directions from the application point and the reflecting bounds are absent. And it is obviously that the resulting vibrations are the consequence of the superposition of many waves propagating in both directions of the closed line. The diagrams of such vibrations are presented in Fig. 13. Their main feature is the coincidence of the vibration directions in the entire line with the direction of an external force action. This feature is inherent in all vibration processes in lines having equal longitudinal and transversal stiffness coefficients. Only their inequality violates this rule. This property retains with the limiting process to the distributed lines.

Summing up, we would like to mark that the spectrum of solutions obtained with the help of our new non-matrix method is quite wide and cannot be described in frames of one brief survey. The more that now when we have solved the problem of closed elastic lines we opened a direct possibility to transit from 1D to 2D and 3D systems. Many of presented models can be complicated, on the basis of many models the exact analytical solutions can be obtained for alike systems. And due to our method, the approximate methods to solve the nonlinear problems can gain a powerful additional pulse, since with the help of our method there appears a possibility to use as the base the models more close to the nonlinear problems being under our consideration.