Modelling the propagation of harmonic longitudinal waves in discretely inhomogeneous linearly elastic rods

Introduction. The regularities of the propagation of harmonic longitudinal waves in discretely inhomogeneous linearly elastic rods are investigated. The relevance of the study is due to the problem of controlled transfer and localization of mechanical energy in engineering systems. The aim of the work is to develop a general analytical solution for the wave field in semi-infinite discretely inhomogeneous rods consisting of an arbitrary number of layers, as well as to show that the choice of the sequence of layers, their thicknesses, and the contrasts of mechanical parameters (Young’s modulus, density, and acoustic impedance) makes it possible to control the amplitude-frequency characteristics and to create zones of amplification and attenuation of vibrations in prescribed frequency ranges.

Materials and methods. A general analytical solution is proposed for rods composed of a finite number of layers with piecewise constant parameters. In each layer, the field is represented as a superposition of counter-propagating traveling waves, and at the interfaces the continuity conditions for displacements and normal stresses are satisfied. This leads to a matrix description (the transfer, or impedance, matrix method), which makes it possible to compute complex amplitudes in the layers, obtain reflection/transmission coefficients for a given excitation frequency ω, and construct amplitude-frequency characteristics at an arbitrary point of the rod. A procedure is presented for numerical finite element modelling of discretely inhomogeneous rod models in the ANSYS Mechanical APDL software package.

Results. It is shown that discrete material inhomogeneity makes it possible to purposefully shape the amplitude–frequency characteristics and control wave processes by creating zones of amplification or attenuation of vibrations. Using a three-layer rod as an example, the dependences of vibration amplitudes on material parameters (different wave propagation velocities in the medium) and on the frequency of external excitation are presented. Numerical verification against the analytical solution has been carried out, confirming the correctness of the modelling procedure.

Conclusions. The obtained results open up prospects for practical applications in solving engineering problems, including the design of seismic barriers, waveguides, and filters with prescribed dynamic properties that increase the resistance of structures to vibrational and seismic action.

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