The banding waves in the beam with periodically located point masses

The paper is concerned with stationary oscillations of a one-dimensional, elastic infinite beam (Bernoulli beam) with periodic system of point masses. All processes are assumed to be time-dependent harmonic. The infinite beam and its isolated periodicity cell are considered. The problem is solved in a rigorous mathematical statement. A faithful representation of the energy flux in an infinite periodic system is found. The asymptotics of pass and stop band boundaries and the energy flux in the infinite periodic system is explored. The effect of relative energy flux attenuation in the first pass band is analyzed when comparing with other pass bands with increasing masses of inertial inclusions. The dependence of the character and «heterogeneity degree» of the wave process in the beam on the position of a corresponding wave number with respect to pass bands is considered. The effect is explored in the analysis of vibrations corresponding to different pass and stop bands. The modes of free vibrations of a periodic cell in the case of its asymmetry are analyzed with special attention to edge effects on the basis of the parameters of the problem. The position of point masses is considered with respect to the knots and pseudo-knots of standing and propagating waves in the infinite system and to the knots of a standing wave in the isolated periodicity cell depending on the problem parameters. These results can be used for studying and designing periodic structures with a given interval of natural frequencies (in stop or pass bands), as well as for analyzing edge effects in them. This can be achieved by using the symmetry properties of the cell and by matching boundary conditions.