Analysis of a weakly-nonlinear periodic structure and its finite counterpart

The question of applicability of the Floquet theory to the weakly non-linear structures is considered. Usually, in the references, numerical solutions to the non-linear spring-mass problem are considered. Despite the amount of the numerical experiments considered, there is still no analytical solution. Moreover, Floquet theory expansion to the non-linear structures is still discussed. Floquet theorem is only applicable to linear problems. The structures without transitional symmetry cannot be considered within the Floquet theory frame. As an attempt to expand the Floquet theorem to the non-linear case, the problem of the arbitrary acoustical waveguides joined with the non-linear stiffness spring is considered. Such a structure has not the transitional symmetry. However, the problem described above can be considered within the Floquet theory with some adjustments. From the other hand, one could consider the eigenfrequency problem for a finite counterpart. The problem does not have a restriction on the properties of the waveguide. One could consider the waveguide within the two frameworks, described above and assess the stop-bands for the two different problems. To illustrate this approach, the problem of the stop-band analysis the weakly non-linear problem is considered. In the paper solution illustrated with the rods connected with the non-linear stiffness, spring is considered.