Phononic crystals can control wave propagation through the spatial periodicity of their constituents. For example, band gaps, negative refraction, and topological edge states allow for vibration isolation and filtering, super resolution, and energy localization, respectively. However, most phononic crystal designs assume Cartesian plane wave fronts, an assumption only valid for the far field. In the near field, the wave front takes a cylindrical profile (in 2D), which propagates radially from the source. Even if the medium’s material properties are invariant to radial translations, the spatial part of the equations of motion that describe these radially-propagating fronts yield an ordinary differential equation with coefficients that fail to be periodic. This inhibits the application of Floquet theory, which is the main reason why there are only limited studies dealing with phononic crystals for radial waves.  In fact, current studies that deal with radial waves involve periodic materials with homogenous layers, for which the transmission is radially-dependent, or require anisotropic mass densities, which are challenging to physically achieve. Here, instead, we introduce a layered material with spatially varying material properties inside each layer, which we physically realize using additive manufacturing methods, to obtain a phononic crystal design for near-field and radially-propagating vibrations. 

In this study, we introduce a torsional phononic crystal (TPC) to control radially propagating torsional waves, typical of rotating and reciprocating machinery and often used for sensing. The TPC is composed of a set of radial rings, and the material properties inside each ring are isotropic but vary non-periodically with radius. By selecting how the material properties vary with radius inside each layer, we can reduce the spatial part of the near field equation of motion to an ordinary differential equation with periodic coefficients. Under this framework, we can apply Floquet theory to calculate the dispersion relation for radial torsional waves of the TPC. In fact, the dispersion relation of the TPC is analogous to that of a one-dimensional infinite Cartesian phononic crystal. We further validate this point by comparing finite element simulation transmission results for the TPC to the transfer matrix method results for a finite Cartesian phononic crystal. Interestingly, the TPC is not geometrically periodic (due to the radial variation of material properties) but it exhibits phononic crystal behavior in response to torsional waves. Dynamic behavior of the TPC is invariant to radial translations, which is further proof of its phononic crystal character. We finally show feasibility of this concept by fabricating a TPC with multi-material additive manufacturing. We experimentally validate its dynamic behavior using a laser Doppler vibrometer and a set-up that converts linear motion of a shaker into torsional excitation. This study provides a preliminary framework for the design of non-Cartesian phononic crystals. This framework can be extended to other wave polarizations, propagation directions, dimensions, and wave front profiles, opening the possibility to extrapolate other well-known phenomena of Cartesian phononic crystals to non-planar wave fronts and near-field vibrations. 


A Continuous Phononic Crystal for Near Field Torsional Waves

Division / Sponsors: 
Aerospace, Bioengineering, Applied Mechanics, Design Engineering Division, Advanced Energy Systems, Manufacturing Engineering Division, Noise Control and Acoustics, Nondestructive Evaluation, Diagnosis, and Prognosis (NDPD), Nuclear Engineering Division, Process Industries Division, Fluids Engineering, Heat Transfer, Materials, Microelectromechanical Systems, Safety Engineering and Risk Analysis, Computers & Information in Engineering


Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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