On Optimum Design and Periodicity of Band-gap Structures

Abstract

A band-gap structure usually consists of a periodic distribution of elastic materials or segments, where the propagation of waves is impeded or significantly suppressed for a range of external excitation frequencies. Maximization of the band-gap is therefore an obvious objective for optimum design. This problem is sometimes formulated by optimizing a parameterized design model which assumes multiple periodicity in the design. However, it is shown in the present paper that such an a priori assumption is not necessary since, in general, just the maximization of the gap between two consecutive eigenfrequencies leads to significant design periodicity.

Hence, it is the aim of this paper to apply the method presented in the preceding paper Olhoff and Du (2013B) to maximize gaps between two consecutive eigenfrequencies by shape optimization of transversely vibrating Bernoulli-Euler beams without damping, and to present and study the associated creation of periodicity in the optimized beam designs.

In the end of the present paper, in order to study the band-gap for travelling waves, a repeated inner segment of the optimized beams is analyzed using Floquet theory and the waveguide finite element (WFE) method. Finally, the frequency response is computed for the optimized beams when these are subjected to an external time-harmonic loading with different excitation frequencies, in order to investigate the attenuation levels in prescribed frequency band-gaps. The results demonstrate that there is almost perfect correlation between the band-gap size/location of the emerging band structure and the size/location of the corresponding eigenfrequency gap in the finite structure.