Flexural Waves In Thin Plates Research Articles

Ribbed elastic metasurface with lateral scalability for flexural wave manipulation

With the goal of engineering applications, the scalability and structural portability of elastic metasurfaces have become significant and attractive. However, most existing elastic metasurfaces cannot be arbitrarily scalable, which will inevitably limit their applications. Here, a novel conceptual design of ribbed elastic metasurfaces (REMs) is proposed for anomalous wavefront manipulation of flexural waves in thin plates. Ribbed subunits, as functional subunits of REMs, are characterized by simple constructions. The phase shift mechanism for the flexural waves across the ribbed subunits is revealed by analyzing the lowest dispersion bands. The analytical model for ribbed subunits is also established based on the equivalent orthotropic plate theory and transfer matrix method (TMM) to accurately predict the phase shift and amplitude of the transmitted waves. In addition, lateral scalability can be observed when the ratio of rib width to subunit width is constant, owing to the constant equivalent bending stiffness. By considering the lateral scalability of subunits, the continuous adjustment of the refraction angle can be perfectly achieved. The deflecting and focusing functionalities of the designed REMs are numerically and experimentally demonstrated. Both simulated and experimental results are consistent with theoretical results. Since ribbed plates are typical load-bearing structures in engineering, our design has broad application prospects, including but not limited to vibration control, energy harvesting, and structural health monitoring.

Broadband metasurfaces for steering flexural waves in thin plates: A topology optimization approach

Flexural vibrations in thin plates are prevalent in practical engineering applications, and significant efforts have been dedicated to controlling these vibrations in recent years. However, steering of broadband flexural vibrations/waves in thin plates remains challenging. In this study, we propose a topology optimization-based technique to design broadband metasurfaces to steer flexural waves in plate-like structures. Broadband performance is achieved by formulating fitness functions that incorporate results at uniformly distributed sampling frequencies within the target frequency range. As an illustration, two refractive metasurfaces with a relative bandwidth of 20% and 60% are designed and characterized numerically. The first metasurface is further fabricated using 3D printing and experimentally tested under two different signal excitations. Both numerical simulations and experiments validate the efficacy of our designs. In addition, we conduct a comparative analysis with two counterparts designed through experience-based methods, highlighting the evident advantages of employing topology optimization for broadband metasurface design. This work proposes a topology optimization-based approach for designing broadband elastic metasurfaces.

Achromatic ribbed elastic meta-structure for ultra-broadband flexural wave manipulation

Elastic metasurfaces have shown remarkable ability in exotic wavefront manipulations based on the generalized Snell’s law (GSL), stimulating great promise for many potential applications, such as elastic wave signal processing, vibration control, and energy harvesting. However, elastic metasurfaces suffer from strong narrow-band confinement and dispersion modulation due to the inherent velocity-dispersive properties and the dispersion of the accumulated phase during propagation. To address this issue, we propose a novel conceptual design paradigm of achromatic ribbed elastic meta-structures (AREMs) beyond the narrow-band limit for the ultra-broadband manipulation of flexural waves in thin plates. We establish an analytical model for ribbed subunits to accurately solve the phase shift and amplitude of the transmitted waves. In addition, we reveal the relationship between broadband achromatic functionality and subunit phase shift dispersion. We theoretically design and numerically demonstrate three flexural wave manipulation functionalities in an ultra-broadband frequency range of 1–10 kHz, including flexural wave deflection, focusing, and splitting, with a relative bandwidth of 163.6 %. Finally, we experimentally verify the ultra-broadband flexural wave manipulation capabilities of AREMs. Our study may open new horizons for ultra-broadband high-efficiency achromatic and structurally simplified flexural wave-based devices, with promising extensions to other elastic wave modes.

Rendering Dynamic Source Motion in Surface Haptics via Wave Focusing.

Emerging surface haptic technologies can display localized haptic feedback anywhere on a touch surface by focusing mechanical waves generated via sparse arrays of actuators. However, rendering complex haptic scenes with such displays is challenging due to the infinite number of physical degrees of freedom intrinsic to such continuum mechanical systems. Here, we present computational focusing methods for rendering dynamic tactile sources. They can be applied to a variety of surface haptic devices and media, including those that exploit flexural waves in thin plates and solid waves in elastic media. We describe an efficient rendering technique based on time-reversal of waves emitted from a moving source and motion path discretization. We combine these with intensity regularization methods that reduce focusing artifacts, improve power output, and increase dynamic range. We demonstrate the utility of this approach in experiments with a surface display that uses elastic wave focusing to render dynamic sources, achieving millimeter-scale resolution in experiments. Results of a behavioral experiment show that participants could readily feel and interpret rendered source motion, attaining 99% accuracy across a wide range of motion speeds.

Compact functional elastic waveguides based on confined mode

Waveguides realized by defected phononic crystals or topological metamaterials possess strong abilities of energy flux rectifying, but suffer from great challenges in practical applications due to their bulky sizes, complicated configurations or narrow working band. Here we present a compact and frequency-robust waveguide squeezed by two layers of metagratings with only one identical unit cell, which we call metafence, to efficiently guide flexural waves in thin plates. Apart from the guided modes, a confined mode is numerically and experimentally observed by designing the waveguide width below a tailored threshold. Excitingly, this confined mode can be converted into a guided mode by easily introducing a couple of obstacles. Such feasible and efficient mode steering allows us to demonstrate robust total-wave-trapping and switchable directional guiding. The designed prototype may enrich the scope of nontrivial modes of elastic waveguides and expand the pathway of tunable energy routing scenarios.

Active exterior cloaking of an inclusion with evanescent multipole devices for flexural waves in thin plates.

We present an active exterior cloak for flexural waves propagating in a Kirchhoff plate of infinite extent. The evanescent multipole devices are characterized by Macdonald functions of the required order, which, assuming time-harmonic vibrations, are solutions of the fourth-order biharmonic equation. It is shown that in the region of interfering waves, which emanate from the devices, a field is recreated which cancels the incident wave to yield a region of ‘stillness’. An inclusion is then positioned in this region for further investigation, with additional attention given to the boundary condition.This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)’.

High-efficiency wavefront manipulation in thin plates using elastic metasurfaces beyond the generalized Snell’s law

Elastic metasurfaces have shown immense potential in exotic wavefront manipulations based on the generalized Snell’s law (GSL), stimulating applications in various engineering fields such as vibration and noise reduction, structural health monitoring and signal processing. One recognized bottleneck is that the GSL can lead to limited efficiencies of metasurfaces at steep deflection angles, which thereby arouses intense efforts in realizing perfect efficiencies, however, mainly in acoustics and electromagnetics. In this paper, we propose a paradigm to design high-efficiency elastic metasurfaces beyond the GSL efficiency limit, for the refraction manipulations of flexural waves in thin plates. We first find that the theoretical efficiency limits of anomalous refractions of flexural waves approach to zero at large deflection angles, but are higher than those of acoustic and electromagnetic waves due to the inherent evanescent mode contributing to a better impedance matching. A genetic algorithm-based inverse method is then proposed to design metasurfaces with almost perfect efficiencies (>95%), by engineering the nonlocal coupling between neighboring unit cells, which works well for the coarse assembly of only two unit cells in one period. Furthermore, by introducing bianisotropy, we demonstrate advanced high-efficiency functions such as the asymmetric transmission and spatial filtering. This work may offer insights towards the design of high-efficiency and structurally simplified wave-based devices.

A subwavelength sinusoidally-shaped phononic beam structures-based metasurface for flexural wave steering

In this work, we propose a method to design subwavelength metasurfaces for flexural waves in thin plates based on sinusoidally-shaped phononic beam structures. In the proposed method, the effective bending stiffness of sinusoidally-shaped phononic beam structures is tuned, and the effective phase velocity of the flexural waves is modulated subsequently by adjusting two key geometry parameters, namely, the amplitude of the sinusoidal profiles and lattice constants of the phononic beam structures. Compared with the previous methods, which modulate a single design parameter, the proposed method is a two-parameter modulating method. Thus, it not only maintains the functionality of metasurfaces, but also is much more flexible when obtaining a high wave transmittance. In addition, the mass of functional units of metasurfaces designed by the proposed method is not relevant to the design parameters, which is particularly important in weight-critical applications. To validate the feasibility of the proposed design method, the transmission spectra of sinusoidally-shaped phononic beam structures are modeled using the finite element method, and results are compared with experiments of 3D-printed samples. In addition, two representative metasurfaces, refractive and focusing metasurfaces, are designed with an available minimum transmission coefficient of 0.93 for functional units. Numerical evaluation of these metasurfaces’ performances confirms the effectiveness of the proposed method. This work paves a novel way to design elastic metasurfaces to manipulate flexural waves in thin plates and demonstrates a potential new direction to design metasurfaces for other wave types.

Steering Flexural Waves by Amplitude-Shift Elastic Metasurfaces

Abstract As 2D materials with subwavelength thicknesses, elastic metasurfaces show remarkable abilities to manipulate elastic waves at will through artificial boundary conditions. However, current elastic metasurfaces are still far away from arbitrary wave manipulations since they just play a role of phase compensator. Herein, we present the next generation of elastic metasurfaces by incorporating amplitude discontinuities as an additional degree of freedom. A general theory predicting target wave fields steered by metasurfaces is proposed by modifying the Huygens–Fresnel principle. As examples, two amplitude-shift metasurfaces concerning flexural waves in thin plates are carried out: one is to transform a cylindrical wave into a Gaussian beam by elaborating both amplitude and phase shifts, and the other one is to focus incident waves by metasurfaces of amplitude modulations only. These examples coincide well over theoretical calculations, numerical simulations, and experimental tests. This work may underlie the design of metasurfaces with complete control over guided elastic waves and may extend to more sophisticated applications, such as analog signal processing and holographic imaging.

Metaclusters for the Full Control of Mechanical Waves

We present a method for the control of waves based on inverse multiple scattering theory. Conceived as a generalization of the concept of metagrating, we call metaclusters to a finite set of scatterers whose position and properties are obtained by inverse design once we have defined their response to some external incident field. The particular focus is on designing passive metaclusters that do not require an external source of energy. The method is applied to the propagation of flexural waves in thin plates, and to the design of far-field patterns, although its generalization to acoustic or electromagnetic waves is straightforward. Numerical examples are presented to the design of uni- and bidirectional ``anomalous scatterers,'' which will bend the scattering energy along a specific direction, ``odd pole'' scatterers, whose radiation pattern presents an odd number of poles, and to the generation of vortical patterns. Finally, some considerations about the optimal design of these metaclusters are discussed.

Vibration control of flexural waves in thin plates by 3D-printed metasurfaces

Steering flexural waves in thin plates benefits several applications, including vibration energy harvesting and micro-electro-mechanical systems (MEMS). This study proposes a technique using metasurfaces based on U-shaped phononic beams to realize the flexural wave control. This technique demonstrates the use of metasurfaces made of the same material as that of the host plates wherein the flexural waves are to be controlled, which significantly facilitates the fabrication of metasurfaces. The total height H of the U-shaped beams is selected to change the effective bending stiffness and consequently tune the effective phase velocity of flexural waves in the U-shaped phononic beams. Numerical simulations verify that the effective phase velocity of flexural waves in the U-shaped phononic beams is highly dependent on H. A macro-model is proposed to estimate the effective phase velocities. The phase shift and transmittance of flexural waves transmitted via U-shaped phononic beams with different values of H are calculated. A full 2π phase shift can be obtained by a change in H within an appropriate range while maintaining the transmittance at a relatively high level. As illustration examples, two subwavelength metasurfaces (refraction and focusing) for flexural waves in thin plates are designed, 3D-printed using VeroPureWhite, and tested. All experimental measurements support our proposed technique for the manipulation of flexural waves in thin elastic plates by metasurfaces.

Scattering theory and cancellation of gravity-flexural waves of floating plates

We combine theories of scattering for linearized water waves and flexural waves in thin plates to characterize and achieve control of water wave scattering using floating plates. This requires manipulating a sixth-order partial differential equation with appropriate boundary conditions of the velocity potential. Making use of multipole expansions, we reduce the scattering problem to a linear algebraic system. The response of a floating plate in the quasistatic limit simplifies, considering a distinct behavior for water and flexural waves. Unlike similar studies in electromagnetics and acoustics, scattering of gravity-flexural waves is dominated by the zeroth-order multipole term and this results in non-vanishing scattering cross-section also in the zero-frequency limit. Potential applications lie in floating structures manipulating ocean waves.

Estimation of the Propagation of Flexural Waves in Thin Plates Using a Single Low-Cost Sensor

This paper demonstrates how flexural wave propagation in a thin plate can be modeled by estimating the combined effect of the excitation source signal and the impulse response of the ultrasonic sensor. The wave propagation in the plate is modeled using the wave equation for the flexural wave mode. A theoretical model for flexural wave propagation in thin plates has been derived, and it has been compared with measurements excited by tapping gently on the surface. The combined effects of the excitation source signal and the impulse response of the low-cost piezoelectric sensor are modeled using finite-impulse response and/or infinite-impulse response filters. Thereafter, the performances of the selected filters are compared on estimating the wave propagation in a thin quartz glass plate. Results indicate that the most accurate estimation of wave propagation has been obtained using a linear phase filter which attributes all dispersions to the flexural wave.

Metasurface constituted by thin composite beams to steer flexural waves in thin plates

We report a novel approach to control flexural waves in thin plates using metasurfaces constituted of an array of parallel arranged composite beams with their neutral planes the same as that of the host plate. The composite beams are composed of two connecting parts made of different materials, and have a thickness identical to that of the host plate. To steer flexural waves in thin plates, a rectangular zone is subtracted from the thin plate and is then filled with the designed metasurface. The time delay of flexural waves in each composite beam of the metasurface is tuned through the varying length of the two connecting components, while keeping the total length fixed. To quantitatively evaluate the time delay in each composite beam, a theoretical model for analyzing the phase of the transmitted flexural waves is developed based on both Mindlin plate theory and Timoshenko beam theory. To control the flexural waves at will, each composite beam in the metasurface is delicately designed according to the proposed theoretical model. For illustrative purposes, the refracted and focusing metasurfaces are designed and numerically validated.

Broadband Bending of Flexural Waves: Acoustic Shapes and Patterns

Directing and controlling flexural waves in thin plates along a curved trajectory over a broad frequency range is a significant challenge that has various applications in imaging, cloaking, wave focusing, and wireless power transfer circumventing obstacles. To date, all studies appeared controlling elastic waves in structures using periodic arrays of inclusions where these structures are narrowband either because scattering is efficient over a small frequency range, or the arrangements exploit Bragg scattering bandgaps, which themselves are narrowband. Here, we design and experimentally test a wave-bending structure in a thin plate by smoothly varying the plate’s rigidity (and thus its phase velocity). The proposed structures are (i) broadband, since the approach is frequency-independent and does not require bandgaps, and (ii) capable of bending elastic waves along convex trajectories with an arbitrary curvature.

Controlling flexural waves in thin plates by using transformation acoustic metamaterials

In this study, we design periodic grille structures on a single homogenous thin plate to achieve anisotropic acoustic metamaterials that can control flexural waves. The metamaterials can achieve the bending control of flexural waves in a thin plate at will by designing only one dimension in the thickness direction, which makes it easier to use this metamaterial to design transformation acoustic devices. The numerical simulation results show that the metamaterials can accurately control the bending waves over a wide frequency range. The experimental results verify the validity of the theoretical analysis. This research provides a more practical theoretical method of controlling flexural waves in thin-plate structures.

Analysis of flexural wave cloaks

This work presents a comprehensive study of the cloak for bending waves theoretically proposed by Farhat et al. [see Phys. Rev. Lett. 103, 024301 (2009)] and later on experimentally realized by Stenger et al. [see Phys. Rev. Lett. 108, 014301 (2012)]. This study uses a semi-analytical approach, the multilayer scattering method, which is based in the Kirchoff-Love wave equation for flexural waves in thin plates. Our approach was unable to reproduce the predicted behavior of the theoretically proposed cloak. This disagreement is here explained in terms of the simplified wave equation employed in the cloak design, which employed unusual boundary conditions for the cloaking shell. However, our approach reproduces fairly well the measured displacement maps for the fabricated cloak, indicating the validity of our approach. Also, the cloak quality has been here analyzed using the so called averaged visibility and the scattering cross section. The results obtained from both analysis let us to conclude that there is room for further improvements of this type of flexural wave cloak by using better design procedures.

Linear transformation method to control flexural waves in thin plates.

In this paper, the linear transformation method (LTM) to control flexural waves propagating in thin plates is presented. Unlike earlier studies, only a small number of homogeneous materials with no requirement of in-plane forces or pre-stress are needed, which tremendously simplifies the implementation of devices for flexural waves. An invisibility cloak with homogeneous materials is studied to confirm the validity of the present approach, and to show its imperfection due to impedance mismatch at interfaces. Required materials can be further simplified as layered isotropic materials using the effective medium theory. Finally, the LTM can be extended to the case of flexural waves propagating in anisotropic thin plates. The present method opens a promising avenue toward the realization of advanced structured shields and other devices.

Flexural waves focusing through shunted piezoelectric patches

In this paper, we designed and analyzed a piezo-lens to focus flexural waves in thin plates. The piezo-lens is comprised of a host plate and piezoelectric arrays bonded on the surfaces of the plate. The piezoelectric patches are shunted with negative capacitance circuits. The effective refractive indexes inside the piezo-lens are designed to fit a hyperbolic secant distribution by tuning the negative capacitance values. A homogenized model of a piezo-mechanical system is adopted in the designing process of the piezo-lens. The wave focusing effect is studied by the finite element method. Numerical results show that the piezo-lens can focus flexural waves by bending their trajectories, and is effective in a large frequency band. The piezo-lens has the ability to focus flexural waves at different locations by tuning the shunting negative capacitance values. The piezo-lens is shown to be effective for flexural waves generated by different types of sources.

Experiments on cloaking in optics, thermodynamics and mechanics.

Spatial coordinate transformations can be used to transform boundaries, material parameters or discrete lattices. We discuss fundamental constraints in regard to cloaking and review our corresponding experiments in optics, thermodynamics and mechanics. For example, we emphasize three-dimensional broadband visible-frequency carpet cloaking, transient thermal cloaking, three-dimensional omnidirectional macroscopic broadband cloaking for diffuse light throughout the entire visible range, cloaking for flexural waves in thin plates and three-dimensional elasto-static core-shell cloaking using pentamode mechanical metamaterials.