# Elastic Metamaterial Insulator for Broadband Low-Frequency Flexural Vibration Shielding

Joo Hwan Oh, Shuibao Qi, Yoon Young Kim, Badreddine Assouar

► To cite this version:

Joo Hwan Oh, Shuibao Qi, Yoon Young Kim, Badreddine Assouar. Elastic Metamaterial Insulator for Broadband Low-Frequency Flexural Vibration Shielding. Physical Review Applied, 2017, 8 (5), pp.054034. 10.1103/PhysRevApplied.8.054034. hal-04953601

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Distributed under a Creative Commons Attribution 4.0 International License
Elastic Metamaterial Insulator for Broadband Low-Frequency

Flexural Vibration Shielding

Joo Hwan Oh¹, Shuibao Qi²,³, Yoon Young Kim⁴† and Badreddine Assouar²,³‡

¹ School of Mechanical, Aerospace and Nuclear Engineering, Ulsan National Institute of Science and Technology, UNIST-gil 50, Eonyang-eup, Ulju-gun, Ulsan, 44919, Korea

² University of Lorraine, Institut Jean Lamour, Boulevard des Aiguillettes, BP: 70239, Vandoeuvre-lès-Nancy 54506, France

³ CNRS, Institut Jean Lamour, Vandoeuvre-lès-Nancy F-54506, France

⁴ School of Mechanical and Aerospace engineering, Seoul National University, 599 Gwanak-ro, Gwanak-gu, Seoul, 151-744, Korea

**Abstract**

Achieving stop band over broad band at low frequency range has remained a great scientific challenge in spite of various efforts made using metamaterials or other technologies. In this work, we propose a new idea that creates unprecedented stop band for broad band at low frequency range. The dual mechanism of shear stiffening and rotation softening is initiated here to achieve broad stop band at low frequency range. Through analytical, numerical and experimental studies, we reveal the underlying physical mechanism and confirm the effectiveness of this metamaterial on vibration shielding for flexural elastic wave covering 235 to 4520 Hz, which is the broadest one so far. This work opens up a new avenue for the development of elastic metamaterials with unprecedented performances and functionalities that are highly desirable in many fields such as vibration shielding.

Keywords: Wave Insulation, Elastic Metamaterial, Broadband, Low Frequency

PACS : 46.40.-f, 62.30.+d, 63.20.D-

<footnote>† Corresponding Author (Co-corresponding), Email: yykim@snu.ac.kr, Tel: +82 2 880 7549</footnote>

<footnote>‡ Corresponding Author (Co-corresponding), Email: badreddine.assouar@univ-lorraine.fr, Tel: +33 3 83 68 49 05</footnote>
# I. INTRODUCTION

Recent advances in acoustic and elastic waves have opened a new way in manipulating phonons transfer with artificial media, called metamaterials. From dynamics of artificial unit cell structures, it was shown that the band gap in quantum scale can also be obtained at macroscopic scale for wave insulations. The most well-known structure for the wave insulation is periodic structure mimicking atomic arrangement, called phononic crystals, in which the band gap due to the Bragg scattering can be formed [1-7]. In addition, the resonance-based metamaterials [8-23], of which the effective density or stiffness can be negative around the unit cell's monopolar or dipolar resonance frequencies, have been actively researched. Although these ideas opened a new way of wave insulation and manipulation, there are still limitations in elastic regime that the broad elastic wave insulation is still hard to be achieved at low frequency range. There have been several approaches based on the metasurfaces [23-28] and corrugated structures [29-31] for the low frequency wave manipulation, but still the broad band gap at low frequency range has not been achieved with these ideas. Recently, acoustic [32, 33] or elastic [34, 35] plasmonic-like phenomena have been reported. However, the acoustic plasmonics cannot be applied for elastic case due to the physical difference and that elastic plasmonics require an external fixed boundary condition connected to every unit cell. Also, there have been various researches on inertia amplification [36-40] which can provide very broad stop band at low frequencies. However, it would be much favorable if one can achieve broad stop band at any desired frequency ranges with high designability. To clarify this issue, the typical band structure of metamaterials shown in Fig. 1 (a) is considered. In phononic crystals, the band gaps are formed at the frequency ranges where the wavelength is comparable with the periodicity of crystal. Therefore, band gaps are usually achieved at high frequencies, i.e., both the lower and upper edge frequencies of band
gap (depicted as $\omega_1$ and $\omega_2$ in Fig. 1 (a)) are high. On the other hand, for the band gaps formed by the resonance-based metamaterials, $\omega_1$ and $\omega_2$ can be low, and low frequency band gap can be achieved. However, due to the nature of resonance, the band gaps generally suffer from narrow bandwidth issue. If attainable, very low $\omega_1$ and very high $\omega_2$ can obviously provide the broad wave insulation at low frequency range. However, there has been no method enabling such frequency tuning.

In this research, we propose an elastic metamaterial for the elastic wave insulation at broad low frequency range. The metamaterial proposed in this work, shown in Fig. 1 (b), is periodic structure as in the phononic crystals. However, the uniqueness of our metamaterial is that the lower and upper edge frequencies of the band gap, $\omega_1$ and $\omega_2$, can independently be tuned so that very low $\omega_1$ and very high $\omega_2$ can be simultaneously achieved. Fig. 1 (a) shows the main idea enabling the frequency tuning, the dual mechanism of shear stiffening and rotation softening. Here, we mainly focus on the elastic wave propagating along thin plate with its polarization along the thickness direction (usually called flexural wave). Note that the same idea can also be applied to other elastic wave modes if a mass-spring system is properly designed. Unlike in acoustic or electromagnetic waves, the wave motion is governed by two kinds of stiffnesses in flexural elastic wave, i.e., the vertical shear stiffness and the rotational stiffness. We found that the lower edge frequency $\omega_1$ is governed by the rotational stiffness, while the upper edge frequency $\omega_2$, by the shear stiffness. Therefore, increasing shear stiffness (shear stiffening) and decreasing rotational stiffness (rotational softening) can provide the broad stop band at low frequency range. In our new metamaterial consisting of rigid mass block, the mass-connecting plate, and the set of side bars linking adjacent mass blocks [41, 42], the plate structure is mainly dedicated to the rotational softening while the bar parts are mainly dedicated to the shear stiffening. Therefore, very broad band gap at low
frequency can be achieved with the proposed metamaterial. Numerical and experimental investigations showed that achieved the band gap extends from 235 to 4520 Hz, which is, for the best of the authors' knowledge, the broadest and lowest band gap achieved so far for the flexural elastic wave. Also, the proposed method can provide very high designability, which is very favorable in various applications such as vibration shielding.

## II. DUAL MECHANISM OF SHEAR STIFFENING AND ROTATION SOFTENING

### A. Key idea in realizing the broad band gap

Because the key idea in realizing the broad band gap at low frequencies is the dual mechanism, we will explain why it is critical. Consider a flexural elastic wave propagating through an infinite periodic equivalent mass-spring system with the periodicity of $a$, as shown in Fig. 2. The model in Fig. 2 reflects the fact that flexural elastic waves are governed by vertical shear stiffness and rotational stiffness; the equivalent mass-spring model shown in Fig.2 [41, 43] includes two springs $\alpha$ and $\beta$ that have vertical shear and rotational stiffnesses, respectively. Following the detailed procedures shown in the Supplementary Material [44], the equation of motion for the equivalent mass-spring system of a continuum structure shown in Fig. 2 is given as

$$ -\omega^2 \begin{bmatrix} m w_n \\ I \theta_n \end{bmatrix} = \begin{bmatrix} A & B \\ \text{conj}(B) & C \end{bmatrix} \begin{bmatrix} w_n \\ \theta_n \end{bmatrix}, \quad (1a) $$

where each matrix components A, B and C are defined as

$$ A = \alpha(\exp(-ika) + \exp(ika) - 2), \quad (1b) $$

$$ B = 0.5a^* \alpha(\exp(ika) - \exp(-ika)), \quad (1c) $$

$$ C = \beta(\exp(-ika) + \exp(ika) - 2) - (0.5a^*)^2 \alpha(\exp(-ika) + \exp(ika) + 2). \quad (1d) $$

Based on Eq. (1), one can calculate the lower edge frequency $\omega_l$ and the upper edge frequency $\omega_u$.
frequency $\omega_2$. Since $\omega_1$ corresponds to the wavevector of $ka = \pi$, $\omega_1$ can be analytically calculated by solving Eq. (1) for $\omega$ at $ka = \pi$ and selecting the lower one. In the same manner, $\omega_2$ corresponds to the wavevector of $ka = 0$ (if the optical branch has a positive wavespeed) or $ka = \pi$ (if the optical branch has a negative wavespeed) and thus can be analytically calculated by solving Eq. (1) at $ka = 0$ or $ka = \pi$ and selecting the higher one. The calculated results are

$$ \omega_1 = 2\sqrt{\beta / I} \qquad (2a) $$

$$ \omega_2 = a^* \sqrt{\alpha / I} \quad \text{if } \omega_2 \text{ corresponds to the wavevector of } ka = 0 \qquad (2b) $$

$$ = 2\sqrt{\alpha / m} \quad \text{if } \omega_2 \text{ corresponds to the wavevector of } ka = \pi \qquad (2c) $$

From Eq. (2), one can clearly see that the lower edge frequency of the band gap $\omega_1$ is a function of $\beta$, while the upper edge frequency of the band gap $\omega_2$, a function of $\alpha$. To achieve a broad band gap at low frequency, $\omega_1$ should be low while $\omega_2$ is high, i.e., a small value of $\beta$ and a large value of $\alpha$ are needed to form a broad band gap in a low frequency range.

In general continuum media or structures, however, $\alpha$ and $\beta$ usually increase or decrease simultaneously depending on the thickness, width and Young's modulus of the continuum medium. Thus, if one lowers $\beta$, $\alpha$ also becomes lowered, not increasing the band gap width. This intrinsic coupling behavior makes it difficult to achieve broad band gap in a low frequency range for flexural elastic waves. This is why a new metamaterial is required; a dual mechanism metamaterial possessing very high $\alpha$ (shear stiffening) and very low $\beta$ (rotation softening) simultaneously can achieve extremely broad band gap at low frequency range.
## B. Metamaterial proposition for the dual mechanism

To explain how the dual mechanism can be achieved, we first consider the metamaterial without bar structures shown in the left side of Fig. 3 (a). Here, the plate acts as the springs, and the corresponding spring coefficients, $\beta_{\text{plate}}$ and $\alpha_{\text{plate}}$, are very small due to its thin thickness. As a result, with the metamaterial without bar structure, the lower edge frequency of the band gap $\omega_1$ is very low due to the small value of $\beta_{\text{plate}}$, i.e., the rotation softening can be achieved. However, the broad wave insulation at low frequency range cannot be achieved since the upper edge of the band gap $\omega_2$ is also very low due to very small value of $\alpha_{\text{plate}}$, and thus the band gap, is very narrow.

Now, consider the metamaterial without plate structure shown in the middle of Fig. 3 (a). As shown in the previous research, the metamaterial with crossed hinge-connected bar structure has zero rotational stiffness $\beta_{\text{beam}}=0$, but nonzero vertical stiffness $\alpha_{\text{beam}}$ [41]. Accordingly, the shear stiffening can be achieved by increasing the bar's stiffness; if the bar structure has very high stiffness, the upper edge frequency of the band gap $\omega_2$ will be very high. Also, the lower edge frequency $\omega_1$ remains almost zero value due to the zero rotational stiffness. Unfortunately, it was shown that the metamaterial structure having zero rotational stiffness cannot sustain its own weight under static case and thus it is extremely hard to apply the metamaterial in actual applications. Also, the metamaterial without the plate structure does not have any mechanism to tune the lower edge frequency of the band gap. The metamaterial without the plate structure cannot realize the broad wave insulation at low frequency range.

Finally, by combining two metamaterials as in the right of Fig. 3 (a), a new metamaterial with rotational stiffness $\beta = \beta_{\text{plate}}$ and vertical shear stiffness $\alpha = \alpha_{\text{plate}} + \alpha_{\text{beam}} \sim \alpha_{\text{beam}}$
(since $\alpha_{\text{beam}} \gg \alpha_{\text{plate}}$) can be achieved. In the new metamaterial, the plate provides low rotational stiffness (rotation softening) while the bars, high vertical shear stiffness (shear stiffening). This indicates that each stiffness component can be independently tuned to have very low $\omega_1$ and very high $\omega_2$. Therefore, broad band gap at very low frequency range can be obtained. Furthermore, each frequency can be independently tuned by adjusting the plate and the bar structure, respectively, to obtain any range of band gap.

This point can be better explained if the equivalent mass-spring system is analytically investigated, as shown in Fig. 3 (b). In Fig. 3 (b), the spring coefficient $\gamma$ originates from the stiffness of the connecting bar while the rotational spring coefficient $\beta$, from the bending stiffness of the connecting plate. As a result, the wave dispersion relationship can be expressed as (see the Supplementary Material for details [44])

$$ -\omega^2 \begin{bmatrix} m w_n \\ I \theta_n \end{bmatrix} = \begin{bmatrix} A' & B' \\ \text{conj}(B') & C' \end{bmatrix} \begin{bmatrix} w_n \\ \theta_n \end{bmatrix}, \quad (3a) $$

$$ A' = 2\gamma \sin^2 \Phi (\exp(-ika) + \exp(ika) - 2), \quad (3b) $$

$$ B' = 2\gamma \sin \Phi (b \sin \Phi + d \cos \Phi) (\exp(ika) - \exp(-ika)), \quad (3c) $$

$$ C' = \beta(\exp(-ika) + \exp(ika) - 2) - 2\gamma(b \sin \Phi + d \cos \Phi)^2 (\exp(-ika) + \exp(ika) + 2). \quad (3d) $$

where $\Phi$ is the angle of the bar structure as shown in Fig. 3. Comparing Eq. (3) with Eq. (1), the following equivalences can be observed;

$$ \alpha = 2\gamma \sin^2 \Phi, \quad a^* = 2(b \sin \Phi + d \cos \Phi) / \sin \Phi, \quad \beta = \beta. \quad (4) $$

Eq. (4) reveals a very important fact: the bar stiffness $\gamma$ only affects the shear spring coefficient $\alpha$, while the plate's stiffness, the rotational spring coefficient $\beta$. Therefore, the proposed metamaterial can have the dual mechanism of shear stiffening and rotation softening by designing the metamaterial with small $\beta$ and considerably large $\alpha$, from
which the extremely broad band gap in low frequency range can be achieved. It should be noted that there should be a certain lower limit for the rotational spring coefficient $\beta$; otherwise the metamaterial would not sustain its own weight.

It should be noted that the equivalent mass-spring system shown in Fig. 3 (b) can only be used for low frequency ranges because the mass of bar and plate are not considered. At high frequencies, it was shown that the bar or plate can exhibit dynamic motions due to their own mass and the equivalent mass-spring model shown in Fig. 3 (b) fails to describe the wave motion of the metamaterial. For instance, at frequencies around 6500 Hz, it was shown that the bar structure is under the resonance motion, so that other forces that are not considered in the mass-spring system in Fig. 3 (b) take place. However, at low frequencies as in this work, it was shown that the plate and bar structures are almost in their quasi-static motion, and those masses can be ignored.

### III. NUMERICAL VALIDATION

#### A. Validation of the idea with wave dispersion curves

Based on the analytic investigation, numerical simulations are carried out to validate our idea. The wave dispersion curve of the proposed elastic metamaterial is numerically and analytically calculated. In the numerical simulation, the unit cell structure shown in Fig. 4 (a) is used and the Floquet-Bloch boundary condition [5] was imposed at the sides of the unit cell and the eigenvalue problem was solved for various wavevectors $k$ inside the first Brillouin zone [45, 46]. Especially, the hinge-connection should be properly considered in the simulation. For the analytic calculation, the spring and mass coefficients of the actual metamaterial were analytically or numerically determined to obtain the analytic dispersion curve by using the mass-spring model. The detailed geometry, material properties and
calculation procedure are given in the Supplementary Material [44].

The dispersion curve in Fig. 4 (b), which is obtained for the proposed metamaterial, supports our argument that the proposed metamaterial effectively forms a broad band gap at low frequencies, specifically, between 235.59 and 4521.9 Hz. The normalized bandwidth is 1.802. Also, very good agreements can be observed between the analytically calculated wave dispersion curves and the numerically calculated one, validating our analytic approaches based on the equivalent mass-spring system. In addition, the formation of broad band gap at low frequencies from 235.59 to 4521.9 Hz clearly supports that our idea of the dual mechanism works well as desired.

Since the metamaterial is not symmetric along the thickness direction (due to the bar connection), one may argue that there may exist a coupling between the longitudinal (x-polarized) and flexural (z-polarized) waves. Thus, in Fig. 4 (b), both the wave dispersion curves for longitudinal (x-polarized) and flexural (z-polarized) waves are plotted, while other wave modes such as the shear (y-polarized) and torsional (rotation with respect to the x-axis) are not plotted due to the unit cell's symmetric configuration. As can be seen in the dispersion curve, the coupling effect between the longitudinal and flexural waves does not affect the overall wave dispersion curves at the frequency range of interest. In fact, although the bar structures are not symmetric along the z direction, the coupling effect is shown to be almost negligible.

Fig. 4 (c) shows the mode shapes of the unit cell structure at the lower and upper edge frequency of the band gap, respectively. As can be seen in Fig. 4 (c), the mode shape at the lower edge of the band gap (left) is dominated by the rotational motion of the mass block, while that of the upper edge (right), by the vertical motion of the mass block. This agrees well with our findings shown in the previous section that the lower edge frequency of the band gap is governed by the rotational stiffness, while the upper edge frequency, the vertical
shear stiffness. Also, it can be seen that at the lower edge of the band gap, the plate exhibit large displacements while the bars do not show any displacement. This supports that the rotation softening is mainly from the plate structure. On the other hand, at the upper edge frequency, both the plate and bars exhibit large displacements because the plate also has a vertical shear stiffness. However, since the plate's vertical shear stiffness is very small, it does not highly affect the shear stiffening mechanism. More simulation results with various plate and bar stiffnesses are given in the Supplementary Material [44].

## B. Numerical simulation for the low frequency wave insulation

Fig. 5 (a) is the numerical simulation setting to measure the flexural wave transmission of the proposed elastic metamaterial. For the simulation, we used the metamaterial system consisting of four unit cells arranged along the x direction. The fixed condition was imposed at the left boundary of the system, and the vertical harmonic force was applied at the 2^nd^ mass, as indicated in Fig. 5 (a). Under this condition, the time harmonic simulation is carried out for various frequencies. In addition, only the half of the system was modeled and the symmetric boundary condition along the y direction was imposed to reduce the computational time.

It should be noted that the simulation setting in Fig. 5 (a) is not generally used in measuring the wave transfer in metamaterial; generally, wave based simulations (using the perfectly matched layers or FDTD method) have been considered to measure the wave transfer in the metamaterials. However, in this work, the simulation setting in Fig. 5 (a) is used to correlate the simulation with the experiment; the reasons why such setting is required in experiment will be explained later in the experiment part. Nevertheless, the simulation carried out in this work can also be useful to predict the formation of band gap; if an actuating frequency belongs to the band gap, there will be no wave transfer and the harmonic motion of the 5^th^ mass in Fig. 5 (a) should be very low, too.
Figs. 5 (b-e) show the simulation results for various frequencies of 100, 1000, 3000 and 5500 Hz. At 100 Hz (which belongs to the pass band), both 2^nd^ and 5^th^ masses exhibit large vibration, indicating that there exists wave transfer between each mass. However, at 1000 and 3000 Hz (which belong to the band gap), the 5^th^ mass exhibits very small vibration compared with the 2^nd^ mass. This shows that the wave transfer is highly suppressed and very small wave transfer exists between each mass, i.e., the band gap is formed to suppress the wave transfer. At 5500 Hz (which belongs to the pass band), the 5^th^ mass vibrates again, shows the rotational motion again, indicating that the frequency belongs to the pass band.

To better observe the flexural wave insulation at various frequencies, the vibration transmission is measured for various frequencies by normalizing the motion of the 5^th^ mass with that of the 2^nd^ mass. To calculate the vibration transmission, the vertical displacement at the left and right sides of 2^nd^ and 5^th^ masses were measured for varying excitation frequencies, respectively. Then, the rotations of 2^nd^ and 5^th^ masses were calculated and normalized to evaluate the vibration transmission plotted in Fig. 6 (b). It should be noted that unlike in the general wave simulation, the ratio can exceed 1 since it is based on the vibration simulation. Fig. 6 (b) clearly demonstrates that the vibration transmission is extremely low in the frequency range from 235 to 4520 Hz. This behavior is consistent with the formation of the broad band gap at low frequency for flexural elastic wave, as predicted by the dispersion curve. Such low vibration is almost impossible to be achieved with general materials such as viscoelastic materials; see the simulation results shown in the Supplementary Material [44] that compares the vibration shielding performance of the proposed metamaterial and the conventional rubber beam. This low vibration transmission can be effectively applied in various applications on vibration shielding.

To check whether the stop band frequencies are highly affected by the damping effect, Fig. 6 (b) is prepared with the transmission under various damping coefficients. Fig. 6 (b) shows
that the transmission behavior and the stop band frequencies are little affected by the effect of damping, suggesting the robustness of the proposed metamaterial system for the lossy cases.

Note that instead of the rotation of each mass, the vertical displacement may also be considered to evaluate the wave transfer since the flexural wave is governed by both vertical and rotational motion. However, since we considered the vibration simulation with the fixed boundary condition here, the use of vertical displacement may cause a wrong result. Fig. 6 (c) shows the vibration transmission measured for the ratio of vertical motion. In Fig. 6 (c), there seems to be a pass band around 1000 Hz, which in fact belongs to the band gap as shown in Fig. 5 (c). This is mainly because of the fixed boundary condition imposed at the left side of the metamaterial structure; due to the fixed boundary, the 2^nd^ mass exhibits large rotation but very small vertical displacement at 1000 Hz. In fact, the rotational motion is more dominant than the vertical one in the low frequency ranges, as elucidated in the literatures [41, 43, 45].

# IV. EXPERIMENTAL REALIZATION

## A. Experimental setting

We fabricated the metamaterial system and carried out the experiment as shown in Fig. 7. Aluminum material was used to make all components (masses, bars, rods and plates) of the metamaterial, and the bars are assembled to the mass block by rods to form the hinge connection. The fabrication details of the metamaterial can be found in the Supplementary Material [44].

The experiment was set up to simulate the conditions used for the numerical simulation depicted in Fig. 6 (a); the left side was fixed the piezoelectric actuator was installed at the 2^nd^ mass. Note that the static deformation of the metamaterial sample is shown to be less than 0.1 mm, which can be ignored in the experiments. At the piezoelectric actuator, sine wave is actuated with various frequencies from 0.1 to 6 kHz. After the actuation, the displacement of
each mass is measured with the laser Doppler vibrometer (PSV-500). In the measurement, the following sampling frequencies are used for each frequency range to achieve the best results; 12.5 kHz from 0.1 to 1 kHz, 50 kHz from 1 to 4 kHz, 100 kHz from 4 to 6 kHz. Also, the measurements are carried out at the left and right side of the 2^nd^ and 5^th^ mass, respectively, to measure the rotation of each mass, as in the simulation. After the measurement, the Fourier transformation is carried out to extract the displacement amplitude at the actuating frequency.

It is worth noting that the general experiments have been carried out as follows; actuation is made at the media outside the metamaterial, and reflected and transmitted waves are measured at the outside media with proper methods. However, in this work, we found that the general wave experiments are hard to be applied and may not be accurate. The most critical problem is that in such a thick plate, the actuation is not strong enough that it requires some time to have enough amplitude. Since there exist very small gaps between mass and rod (or rod and bar), small excitation may result in zero wave transmission regardless of the frequency. On the other hand, in our vibration experiment, the measurements are carried out under the steady state and there is enough time for sufficient energy transferring into the metamaterial. Therefore, sufficiently large actuation and fully developed flexural waves inside the metamaterial can be achieved with the current vibration experiment. Although this experiment cannot provide any other data such as effective parameters or dispersion curve of the metamaterial, it can accurately show the formation of band gap, which is the main point focused in this work.

After the measurement, the rotation of the 2^nd^ and 5^th^ masses are calculated. Unlike the case of simulation, a special attention is required here. In the simulation, the rotation of each mass can be readily calculated as $\theta = |U_R - U_L|$. However, since the Fourier transformation used in the experiment provides amplitudes for each frequency component, the
experimentally measured values are always positive, i.e., $|U_R|$ and $|U_L|$ are measured.
Thus, if we calculate the rotation with the difference between the measured values at the left
and right side, the result will be $|U_R|-|U_L|$ which causes a problem if the displacements at
the left and right side have different sign. Therefore, we additionally measured the
displacement at the middle of the mass, $U_M$ in Fig. 7. If the average of the measured value
is similar to $U_M$, $|U_R|+|U_L| \sim 2|U_M|$, $U_R$ and $U_L$ have the same sign and the rotation
can be defined as $\theta=|U_R|-|U_L|$. However, if the average is larger than $U_M$,
$|U_R|+|U_L| > 2|U_M|$, $U_R$ and $U_L$ have the different sign and the rotation should be defined
as $\theta=|U_R|+|U_L|$. Note that no other exceptional case was observed since the mass exhibit
almost rigid-body motion.

## B. Experimental results

Fig. 8 (a) compares the rotation ratio (calculated by the ratio of the rotation of the 5th mass to that of the 2nd mass) that were obtained numerically and experimentally. Here, +, o and × markers indicate the experimental results obtained for different sampling rate. As previously explained, three different sampling rates are used in the experiments since the target frequency range is very broad and a single sampling rate cannot be used: 12.5 kHz from 0.1 to 1 kHz (corresponding to the + marker), 50 kHz from 1 to 4 kHz (corresponding to the o marker) and 100 kHz from 4 to 6 kHz (corresponding to the × marker). To consider any possible loss effect, the experimentally measured loss factor of $1.62e-8 \times f$, where f is the actuation frequency, is used in the simulation.

Even though there are some discrepancies, they agree well in the targeted band gap
frequency range (235 to 4520 Hz), validating the actual formation of such a broad band gap
at low frequency range. At extremely low frequencies, from 0 Hz to 300 Hz, high vibration transmission can be observed, indicating that those frequencies belong to the pass band of the metamaterial. On the other hand, at frequencies from 300 Hz to 4000 Hz, the vibration transmission becomes significantly lowered. This is due to the formation of the stop band – as in the simulation, the formation of the stop band prohibits the flexural wave propagation and thus very small amount of the vibration is transferred from the actuation point (the second mass) to the measurement point (the fifth mass). Finally, at the frequencies higher than 4000 Hz, high vibration transmission is measured, indicating that the frequency range corresponds to the pass band of the metamaterial. Overall, the experiment well captures the broad band gap at low frequencies as predicted by the simulation.

The discrepancy between the experimental result and the simulation result can be found around 5000 Hz, which is caused by the effect of the rods' deformation ignored in simulations. Since almost all regions of the rod are embedded in other parts, the deformation of the rod is mainly a shear one at very small region, indicating that the stiffness of the rod is very high. Furthermore, because almost all deformation takes place at plate structure at low frequencies, the stiffness of the rod can be ignored at low frequencies. However, at high frequencies where the deformation of the bar becomes dominant, the stiffness effect of the rod also affects the overall motion. The peaks in Fig. 8 (a) are due to the formation of standing wave mode inside the metamaterial (since the simulations and experiments are carried out under the steady state), and the stiffness of the aluminum rod decreases the overall stiffness, resulting in the reduced peak frequency in the experimental results. The discrepancy can be reduced if the stiffness of the rod becomes significantly harder than the bar. For instance, if the bar is made of plastic, or if the rod is made of steel, the experimental result will fit much better.

In addition to the rotation ratio, the vertical displacement ratio is also experimentally
calculated and compared in Fig. 8 (b). Good agreements can be observed in the case of
displacement ratio, which well supports the validity of the experimental results.

V. DISCUSSION

In this research, we propose a new idea of wave insulation for broad and low frequency
range by the dual mechanism of shear stiffening and rotation softening. Based on our idea,
we create an elastic metamaterial with a unique configuration: a rigid mass block, a thin mass
-connecting plate and a set of bars that are hinge-connected to the masses. Analytic
investigation showed that the plate alone contributes to the lower edge of the band gap by
rotation softening while the bars largely contributes the shear stiffening that governs the
upper band edge frequency. Numerical and experimental results showed that the realized
band gap is from 235 to 4520 Hz, which is, for the best of the authors' knowledge, the
broadest and lowest band gap realized for flexural elastic wave. Although the numerical and
experimental validations were carried out for 1 dimensional flexural elastic waves, the idea
can also be extended to other wave modes and to 2 or 3 dimensional system. In fact, the idea
can also be applied for other wavemodes such as the longitudinal or shear wavemode if a unit
cell is properly designed so that the longitudinal/shear motion is coupled with additional
rotational motion.

As possible applications, our new idea of the shear stiffening and rotational softening can
effectively be applied in various vibration shielding applications. Usually, in vibration
shielding, it has been a long issue to effectively isolate vibration at very low and broad
frequencies. Moreover, almost all vibration problems are based on the flexural motions. Our
metamaterial can provide extremely effective broad-band flexural vibration shielding at very
low frequency ranges, which cannot be achieved with general materials. Since the lower and
upper edge frequencies of the stop band can be easily tuned, our idea can be readily applied
in various applications such as vibration reduction in bridges or buildings, vibration
engineering in cars and automobile applications, etc. Furthermore, various vibration devices
can be designed, such as vibration waveguide, cavity and vibration focusing, based on the
present research. We believe that our new metamaterial will open a new way in flexural
vibration manipulation at low frequency ranges.

Acknowledgements

This work was supported by the Center for Advanced Meta-Materials (CAMM) funded by
the Ministry of Science, ICT and Future Planning as Global Frontier Project" (CAMM-
2014M3A6B3063711), by the National Research Foundation of Korea (NRF) grant funded
by the Korea government (MSIP) (No. 2017R1C1B1004436), and by the Institut CARNOT –
ICEEL (France). The authors would like to thank Emmanuel Vatoux for the structures
fabrication.

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## Figure Legends

FIG. 1. (a) Schematic illustration of the main idea, dual mechanism of shear stiffening and rotation softening. (b) The proposed elastic metamaterial with the dual mechanism; the bar provides the shear stiffening, while the plate, the rotation softening.

FIG. 2. General mass-spring system of the flexural elastic wave.

FIG. 3. (a) Design of the proposed elastic metamaterial by combining metamaterial with plate and that with bars. (b) Equivalent mass-spring system of the proposed elastic metamaterial for the analytic investigation of the proposed metamaterial.

FIG. 4. (a) A unit cell structure of the proposed elastic metamaterial considered in the numerical simulation. (b) The numerically and analytically calculated wave dispersion curves. (c) Mode shapes of the unit cell structure at the lower (left) and upper (right) edge of the band gap, respectively.

FIG. 5. (a) The metamaterial system considered in the numerical simulation. The simulation results for (b) 100 Hz, (c) 1000 Hz, (d) 3000 Hz and (e) 5500 Hz.

FIG. 6. (a) The metamaterial system considered in the numerical simulation for the wave transfer. Plot of (b) the rotation ratio and (c) the vertical displacement ratio of 5th and 2nd mass.

FIG. 7. Experimental setting for the realization of wave insulation with the proposed elastic
metamaterial.

FIG. 8. Experimentally measured wave transfer in the proposed elastic metamaterial at various frequencies. Plot of (a) the rotation ratio and (b) the vertical displacement ratio of 5th and 2nd mass.
<figure><img src="image_2.png" /><figcaption>FIG. 1. (wide)</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>FIG. 2.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>FIG. 3. (wide)</figcaption></figure>
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>FIG. 4.</figcaption></figure>
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>FIG. 5.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>FIG. 6. (wide)</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>FIG. 7. (wide)</figcaption></figure>
<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>FIG. 8.</figcaption></figure>