Abstract
Structures with periodically arranged resonators have attracted attention within vibroacoustics recently. These periodic structures attenuate wave propagation in a defined frequency range. Engineers can tune the resonators to produce structures with a desired band gap behavior. The majority of the established stop band materials is produced by additive manufacturing today. While promising scientific results have been achieved with this method, it is not yet fit for industrial applications. In order to develop new manufacturing approaches and to close the gap between science and industry, the influence of uncertainties of periodic structures has to be quantified. In this study, the influence of geometrical uncertainties on periodic structures is investigated. The authors develop a finite element model of an epoxy plate with beam resonators for this purpose. In a parameter study, the influence of some parameters is identified. Afterwards, the behavior of stop band material with uncertain input parameters is studied using spectral stochastic methods. Having predefined probability density functions, the generalized polynomial chaos expansion is used to propagate the uncertainty of these parameters. The stop band behavior is accordingly represented by the generalized polynomial chaos having unknown deterministic coefficients. A collocation-based stochastic simulation is used to estimate the coefficients employing the deterministic finite element model as a black-box. The results show the necessity of regarding uncertainties in periodic structures. The performed stochastic simulations are suitable to define manufacturing tolerances for the production of stop band material.
Highlights
During recent years, acoustic metamaterials received growing attention due to their possibility to gain acoustic properties which can not be found in nature
Summarizing the results found in literature, stop band materials which are achieved with periodic structures of tuned resonators are a suitable solution for vibroacoustic applications
It seems that the probability density function (PDF) for the band gap center positions are identical for the three parameters considered as single uncertain inputs due to the fact that they have the same magnitude of the gradient
Summary
Acoustic metamaterials received growing attention due to their possibility to gain acoustic properties which can not be found in nature. Summarizing the results found in literature, stop band materials which are achieved with periodic structures of tuned resonators are a suitable solution for vibroacoustic applications. In [19], the wave finite element method is employed to investigate a structure with one-dimensional periodicity which is locally perturbed. The employed method to calculate the dispersion relations in periodic structures applying the Bloch theorem is described. Time harmonic waves with angular frequency x a response variable w can be expressed by w 1⁄4 WeiðxtÀkxxÀkyyÞ; ð1Þ with W describing the wave mode through the thickness of the structure and kx; ky being the components of the wavenumber in x and y direction This can be obtained by employing finite element methods.
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