Nonlinear vibrations and mode interactions for a continuous rod with microstructure

Natural longitudinal vibrations of a periodically heterogeneous rod embedded into a nonlinear elastic matrix are considered. The governing macroscopic dynamical equation was obtained by the higher-order asymptotic homogenization method. An asymptotic solution is developed by the method of multiple time scales. The effects of internal resonances and modes coupling are predicted. The specific objective of the paper is to analyse how the presence of the microstructure influences on the processes of mode interactions. It is shown that depending on a scaling relation between the amplitude of the vibrations and the size of the unit cell different scenarios of the modes coupling can be realized.