Eigenfrequencies and eigenmodes of a beam with periodically continuously varying spatial properties

A dispersion relation of the considered non-uniform periodic structure is obtained, and values of the modulation amplitudes at which frequency bandgaps arise are determined. It is shown that eigenfrequencies of the beam can lie within the bandgaps, and that such eigenfrequencies can be considerably affected by modulation. It is revealed that there is an abrupt shift in the effect of modulation on eigenfrequencies, and that modulations of the beam mass per unit length and of the beam stiffness affect them oppositely. It is shown that eigenmodes having a period close to the period of modulation comprise a long-wave component; this illustrates the capacity of non-uniform structures to sustain long-wave oscillations on comparatively high frequencies.