Forced oscillations of the string under conditions of 'sonic vacuum'

We present the results of analytical study of the significant regularities which are inherent to forced nonlinear oscillations of a string with uniformly distributed discrete masses, without its preliminary stretching. It was found recently that a corresponding autonomous system admits a series of nonlinear normal modes with a lot of possible intermodal resonances and that similar synchronized solutions can exist in the presence of a periodic external field also. The paper is devoted to theoretical explanation of numerical data relating to one of possible scenarios of intermodal interaction which was numerically revealed earlier. This is unidirectional energy flow from unstable nonlinear normal mode to nonlinear normal modes with higher wavenumbers under the conditions of sonic vacuum. The mechanism of such a scenario has not yet been clarified contrary to alternative mechanisms consisting in almost simultaneous energy flow to all nonlinear normal modes with breaking the above-mentioned conditions of sonic vacuum. We begin with a description of single-mode manifolds and then show that consideration of arbitrary double mode manifolds is sufficient for solution of the problem. Because of this, the two-modal equations of motion can be reduced to a linear equation which describes a perturbation of initially excited nonlinear normal mode of the forced system in the conditions of sonic vacuum. We have found analytical representation (in the parametric space) of the thresholds for all possible energy transfers corresponding to unidirectional energy flow from unstable nonlinear normal modes. The analytical results are in a good agreement with previous numerical calculations. This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.