The wave blocking techniques for wave based problems such as vibration, noise and earthquake applications have been continuously studied. Among them, artificial structures such as metamaterials that have high potential in wave-based engineering are expected to be one of the effective solutions. The band gap of metamaterials composed of periodic arrangements of unit cells is one of the key physics in metamaterials; the wave is gradually attenuated as it passes through each unit cell, and it does not propagate in a certain frequency band called the band gap.
The metasurface, which is formed by arranging a single unitary structure whose scale is less than subwavelength, can change the phase shift of the waves. In theory, by using the functionality to control the phase shift of the waves, it is possible to adjust the wave propagation as desired. The proposed metasurface has two main features; first, it changes the boundary condition from the free to the fixed condition, that is, the practically fixed boundary condition can be realized.
In general, reflected waves are in phase with respect to incident waves if the waves encounter free boundary surfaces. In other words, there is a displacement equal to twice the amplitude of the incident waves on the free surfaces. However, the proposed elastic metasurface converts the phase shift of the reflected waves as 𝜋 so that the reflected waves are out of phase with respect to the incident waves.
Our removable elastic metasurface is expected to be a solution to wave-based problems by attaching it to free surfaces within or around already fabricated mechanical systems and structures.
Introduction
Theoretical & Mathematical Development
Before deriving the wave equation, the following equations commonly used in dynamics are defined in an elastic body. In this equation, 𝜆 and 𝜇 Lame parameters, Lame modulus and Shear modulus respectively, are defined as 𝐸 and 𝑣 as (2-8) and (2-9). When the body force is zero, the equation of motion is defined in terms of displacement vector as (2-10) and (2-11).
When incident waves arrive at a boundary surface, reflected waves occur and propagate as shown in Fig. The incident P waves and reflected P waves have the same angle, and the incident SV waves and reflected SV waves also have the same angle, 𝛼 > 𝛽. In these equations, I and R stand for the incident and reflected waves, respectively, and 𝐴 is the amplitude of the waves.
When incident shear waves reach a boundary surface, reflected waves occur and propagate as shown in Fig. The total displacement and the differential value of the total displacement on a boundary surface are shown as (2-40) and (2-41). The elastic body is infinite in the y-direction and has infinite resonators to form metasurfaces on a boundary surface as shown in Fig.
To distinguish between area and amplitude and apply tensor notation, 𝑣𝐼 and 𝑣𝑅 are redefined as (2-47) and (2-48). Therefore, 𝑣𝑇𝑜𝑡𝑎𝑙 is also redefined as (2-49) and divided into real term and imaginary term by Euler's formula as indicated (2-50). In other words, free boundary surface can be converted to virtually fixed boundary metasurface, when 𝜔 = 𝜔𝑛.
Numerical Modeling
In this model, it is possible to separately identify incident waves, reflected waves and total displacement at a boundary surface. Similar to the fixed boundary condition, the reflected waves have opposite phase to the incident waves at the practically fixed boundary as shown in Fig. These results show that it is possible to convert the free boundary surface into a virtually fixed boundary metasurface by arranging the resonators on the surface.
In this model, the total displacement at boundary 1 is measured according to the change in the number of resonators from 1 to 9 and the frequency from 0.1 kHz to 20 kHz. It means that the free boundary surface is converted into an almost solid boundary metasurface by placing finite resonators. In this model it is also possible to separately identify incident waves, reflected waves and total displacement at an interface.
When the resonators are attached to the free boundary surface of the elastic body, the feasibility of the VFB model can be confirmed by comparing the displacement on the surface. The numerical result when there is only one elastic body is the displacement on the free boundary surface. 3-13(b), the VFB metasurface is formed by attaching the resonators to the free boundary surface of the elastic body.
The total displacement at the boundary surface is measured at the red point with respect to the change in frequency from 3 kHz to 7 kHz. 3-14 (a), the total displacement on the VFB metasurface is much smaller than the free boundary surface in certain frequency bands, considering that the results are in logarithm. In these figures, the total displacement at the VFB metasurface is almost zero at the surface due to the resonators.
Experimental validation
For ease of fabrication, aluminum plates with mounting holes are assembled with M4 bolts as shown in Figure 4-3. A vibration isolation system (optical table) is used to improve the accuracy of the experiment. The linear motion guide (LM) and its additional height adjustment block are shown in Figure 4-5.
Holes for M5 bolts are for assembly with LM guide and holes for M6 bolts are for assembly with optical table. The function generator is used to determine the shape and frequency of the input signal. In addition to the input signal, the result measured by the laser vibrometer is also identified in the oscilloscope.
The vibration exciter (Shaker) is used for shear excitation on the upper surface of the elastic body. The total displacement on the upper plate of the elastic body is measured by the laser vibrometer sensor. The sensor measures the result using the type of measurement specified in the controller, and the result is delivered to the oscilloscope.
Similar to the numerical results, the total displacement on the VFB metasurface is much smaller than the free boundary surface at specific frequency bands considering that the results are in log scale.
Conclusion
Acknowledgement
