Internal resonances and modes interactions in non-linear vibrations of viscoelastic heterogeneous solids

The aim of the paper is to study how viscous damping influences mode coupling in non-linear vibrations of microstructured solids. As an illustrative example, natural longitudinal vibrations of a layered heterogeneous medium are considered. The macroscopic dynamic equation is obtained by asymptotic homogenisation. The input continuous problem is analysed using a spatial discretisation procedure. An asymptotic solution is developed by the method of multiple time scales and the fourth-order Runge-Kutta method is employed for numerical simulations. Internal resonances and energy transfers between the vibrating modes are predicted and analysed. The conditions for possible truncation of the original infinite system are discussed. The obtained numerical and analytical results are in good agreement.