Frequency estimate and adjustment of composite beams with small periodicity

It is the first time for us to investigate the estimation and adjustment of the natural frequencies for a composite beam with small periodicity. An approach is presented in which the two-scale asymptotic expansion method is used to establish a homogenized model of the original heterogeneous beam. The accuracy of the homogenized model to predict the natural frequencies is checked by comparing with a fine finite element simulation of the original beam. In the case that each repetitive unit is composed of two different materials and the volume ratio of each material is given, the maximum and minimum frequencies of the composite beam are determined. Then it is shown that the beam can be designed to have particular frequencies within these frequency limits through a proper selection of the material distribution law in the unit. A concrete example is given when the material distribution function is assumed to be piecewise linear.