37

The general solution for all harmonics higher than the first is similar to (31)-(33), with the corresponding substitution of equivalent forces and the multiplier at . Due to this, the resonance lines of each harmonic consist of the lines of natural resonances located before the boundary frequency of this harmonic, and of the introduced resonances from the lower harmonics, which are located between the natural boundary frequency of this harmonic and that of the first harmonic. The general structure of solutions for higher harmonics enables us to conclude that the condition of convergence of the series (8) out of the resonance area will be similar to the before-obtained condition for the third harmonic (29). In the resonance area, the amplitudes of harmonics follow the resonance curves of all lower harmonics and introduce their own resonance frequencies. So the convergence of series (8) on the whole for an elastic non-resistant line will be determined in fact only by the areas out of resonances. In the resonance areas all harmonics will turn into infinity at the frequencies corresponding to the frequencies of the previous harmonics.

We can mark that Landau, studying the anharmonic vibrations, also came to the recurrent relation, when expanded the Lagrange function up to the third-power terms. For example, for normal vibrations in the second approximation he obtained

where  ⁽¹⁾ and _a⁽²⁾ are the normal coordinates of the first and second approximations correspondingly [22, p. 110]. However, in the absence of exact analytical solutions for the first approximation (the first harmonic) his further investigation was limited by seeking the combinational frequencies of higher harmonics. Due to this, in Landau's approach it is impossible to reveal the fact of the boundary frequency decrease with the growing number of harmonic in the presence of three vibration regimes for each harmonic, neither to reveal the introduced resonances, nor to reveal the analytical dependence of the vibration amplitudes of harmonics on the parameters of studied system, nor to present the spectral expansion as a functional series. As opposite to this, in the recurrent relationship basing on the exact analytical solutions, the equivalent forces are not reduced to the first and second derivatives of normal co-ordinates, but depend only on the amplitude shifts of lower harmonics, and the resonance vibrations of higher harmonics have much more complex structure than the simplified concept in the form of combinational frequencies. It makes the spectral analysis of nonlinear dynamical process much more exact and complete. The main distinction is that in frames of presented method we have not to solve the original problem for each harmonic. It is sufficient to have general solution of the linear system of equations of the type (39) to find recurrently the functional regularities for any harmonic. Furthermore, this is convenient in the view of numerical programming.

4. Prospects of this method development

As the obtained solution is common for all harmonics of a dynamical process, it allows on the basis of the exact analytical solution of the linear problem to obtain sequentially the spectral dispersion of the momentary shift of each element of a nonlinear elastic system in the analytical form. The revealed recurrent relationship is not limited by a specifically chosen simple modelled system. It would be sufficient to look at the broad complex of modelling differential equations presented in [9]- [16], to realise that they have a similar structure. It means, however complex model of an elastic line we investigate, if we approach to the modelling in the view of complete description of a specific model, which allows us to obtain the exact analytical solutions for the corresponding linear model, we will certainly obtain a sequential series of systems of linear equations for the harmonics of process. The reason is, the stiffness coefficient expansion in powers will certainly contain a linear part, which will form the linear system of equations for each harmonic of the spectral dispersion. Still, the power terms of spectral dispersion in any case will depend only on the momentary shifts of the lower harmonics, so they can be presented in the form of equivalent forces connecting the dynamical processes in the studied harmonic with the lower harmonics.

The range of validity of the obtained recurrent relationships is not limited by the stiffness coefficient non-linearity. The method works fine both in presence of nonlinear resistance, in its expansion into the power series in terms of shift velocity of the line elements, and in case of complex spectral composition of an external force. The same, the line can be heterogeneous and can contain, e.g., nonlinear constraints only in some part of elastic connections. In this case, as we can see from the consideration, the equivalent forces remain only in those equations of systems of all harmonics which describe the indicated nonlinear constraints, and the general structure of solution remains constant.

The main advantage of the obtained solutions is their exactness. As we showed in the course of solving, in finding the harmonics of dynamical process we did not use the asymptotic approximations, nor conditions of smallness, except the standard approximation of the stiffness coefficient by the power series. For each harmonic the solution was found by the exact analytical method, since in each case, without lessening generalisation, the problem was reduced to the linear system of equations with exact analytical solutions. As a result, we have obtained the functional series, which is not necessarily descending, and when out of the resonance, the inequality (29) determines its convergence. Furthermore, we have to take into account that in an ideal line the resonance amplitude is infinite, and general decrease of the amplitude with the growing number of a harmonic does not effect on its value. The additional feature is that in the general solution of ideal lines, the density of resonance frequencies grows with the lowering frequencies, and we can prove that at = 0 this density turns to infinity. Such problem is easy to get over, when taking into account the resistance present in every real elastic system. In this case, due to the finite amplitude of resonance [11], the resonance density growth is compensated by an abrupt decrease of their amplitude, due to which the infinite spectral series can be limited by the conditions of given accuracy of the solution.

Finally, we would like to mention one more property of the obtained solutions. As we saw in the previous item, in order to find sequentially the spectral harmonics, it is sufficient to have the solution for one linear problem corresponding to that non-linear. Thus, the problem of sequential finding the higher harmonics of the vibration process does not get complicated with the growing number of a harmonic, since, in order to find any high harmonic, it is sufficient to have a general algorithm of solution, substituting to it sequentially the corresponding values of equivalent forces and the number of harmonic. This is the advantage provided by the analytical methods, as described by Cherepennikov [18] to whom we referred in the introduction.