Vibrations of long repetitive structures by a double scale asymptotic method

In this paper, an asymptotic two-scale method is developed for solving vibration problem of<br />long periodic structures. Such eigenmodes appear as a slow modulations of a periodic one. For those, the<br />present method splits the vibration problem into two small problems at each order. The first one is a<br />periodic problem and is posed on a few basic cells. The second is an amplitude equation to be satisfied<br />by the envelope of the eigenmode. In this way, one can avoid the discretisation of the whole structure.<br />Applying the Floquet method, the boundary conditions of the global problem are determined for any order<br />of the asymptotic expansions.