An inverse method to determine the dispersion curves of periodic structures based on wave superposition

Phononic crystals and acoustic metamaterials have unique properties, such as the existence of band gaps, which give them huge potential in many applications, such as vibration isolation, acoustic cloaking, acoustic lensing, and more. Many methods have been proposed to determine the band structure of these materials but almost all require a model of the structure. In this paper, an inverse method to calculate the band structure of one dimensional periodic structures based on Bloch wave boundary conditions and wave superposition is introduced. The proposed method only requires the frequency responses measured at a small number of points within the structure. This allows the band structures to be determined experimentally using simple equipment, like a shaker and accelerometers. The band structure of a simple bi-material beam was calculated in this study as a demonstration of the method, and the results were found to be in agreement with calculations made using the transfer matrix method. The proposed method was then extended to predict the response of a finite periodic bi-material beam with arbitrary boundary conditions using only the band structure and components of the eigenvectors; some resonance peaks were observed within the band gaps and these were found to be caused by the reflection of the waves at the boundaries. The effects of the number of unit cells on the transmissibility of a beam were investigated. It was found that the transmissibilities within the band gaps can be estimated to be directly proportional to the number of unit cells. Lastly, an attempt was made to extend the method to two and three dimensional periodic structures and the wave superposition method was found to be able to measure a portion of the dispersion surface of two dimensional structures with a fair degree of accuracy, especially at lower bands. Errors and scatter are present at high frequencies caused by more waves significantly affecting the responses of the system. This issue can be alleviated by taking measurements further away from the boundaries or increasing the number of waves considered. However, the key limitation of the method for two- and three-dimensional periodic structure is that it can only measure a portion of the dispersion surface or volume