Nonlinear vibrations of periodic beams

Geometrically nonlinear vibrations of beams with properties periodically varying along the axis are investigated. The tolerance method of averaging differential operators with highly oscillating coefficients is applied to obtain governing equations with constant coefficients. The proposed model describes dynamics of the beam with the effect of microstructure size. In an example, an analysis of undamped forced nonlinear vibrations of the periodic beam
is shown. Moreover, the results obtained for undamped free vibrations of periodic beams by the tolerance model are justified by those results from the finite element method. These results can be used as a benchmark in similar problems.