Frequency Shifting Analysis of a Simply Supported Beam Composed of an Axially Embedded Periodic Two-Phased Array

Abstract

Due to high demand for the safe operation of structures, structural members with the potential to shift their natural frequencies/specific-stiffness to satisfy design requirements or avoid the occurrence of dynamical instability are of great importance. Hence, the purpose of this study is to present a methodology to evaluate the impact arising from the number of layer and properties of laminas on the natural frequencies of a passive periodic-array beam. The periodic array is assumed to be composed of two different materials, basic and embedded laminas, which are periodically embedded into the beam. The problem is solved by treating the periodic-array beam as a single-phase nonhomogeneous continuum. The variation of material properties in the axial direction is expressed by a Fourier series half-range expansion with a wavelength equal to the average space between the two materials. The Fourier-series-based approach and Newtonian mechanics are employed in the analysis. The result of the study indicates that due to the nonhomogeneity of the periodic array’s materials, the symmetry of the system’s coefficient matrix strongly depends on the quantity of laminas. If there are more than a few periodic layers, the number of off-diagonal entries in the matrix is low and can be neglected when compared with the diagonal elements; the problem reduces to a self-adjoint eigenvalue problem. It shows that the periodic array can be treated as a tuning parameter to vary the natural frequency of the beam; the frequency shifting range depends on the ratio between the mechanical properties of the two materials. This implies that if periodic laminas are integrated into structures, it gives structures the potential to prolong their useful life by adjusting their natural frequencies to reduce the possibility of occurrence of dynamic instability, such as resonance.