Dispersion of bending waves in rods with periodic parameters

An analytical-numerical method is developed for analysis of the geometric dispersion of bending waves in a rod with bending rigidity and mass per unit length varying periodically along its axis. The Bernoulli-Euler equations in the case of harmonic vibrations are reduced to a Hamiltonian system of the longitudinal coordinate. The general solution is constructed. It is expressed in terms of the matriciant of the sq stem over one period, multipliers, and the eigenvectors of monodromy matrices. A technique is developed to determine the wave propagation constant as a function of the frequency, and the conditions of wave blanking and transmission are established. The results of solution and analysis of specific problems are presented.