Frequency-doubling effect in acoustic reflection by a nonlinear, architected rotating-square metasurface

Nonlinear acoustic metamaterials offer the potential to enhance wave control opportunities beyond those already demonstrated via dispersion engineering in linear metamaterials. Managing the nonlinearities of a dynamic elastic system, however, remains a challenge, and the need now exists for new strategies to model and design these wave nonlinearities. Inspired by recent research on soft architected rotating-square structures, we propose herein a design for a nonlinear elastic metasurface with the capability to achieve nonlinear acoustic wave reflection control. The designed metasurface is composed of a single layer of rotating squares connected to thin and highly deformable ligaments placed between a rigid plate and a wall. It is shown that during the process of reflection at normal incidence, most of the incoming fundamental wave energy can be converted into the second harmonic wave. A conversion coefficient of approximately 0.8 towards the second harmonic is derived with a reflection coefficient of $<0.05$ at the incoming fundamental frequency. The theoretical results obtained using the harmonic balance method for a monochromatic pump source are confirmed by time-domain simulations for wave packets. The reported design of a nonlinear acoustic metasurface can be extended to a large family of architected structures, thus opening new avenues for realistic metasurface designs that provide for nonlinear or amplitude-dependent wave tailoring.

Figure 1

Nonlinear metasurface design: (a) Single layer of periodically arranged rigid squares sandwiched between a moving rigid plate and a fixed wall, with elastic springs running between all the elements. The propagation medium (1) in front of the designed metasurface is assumed to be semi-infinite. (b) The metasurface unit cell is composed of two identical squares with elementary mass $m$. The front rigid plate has an elementary mass $2m_0$. (c) Due to symmetry, taking into account the motion of just one square of mass $m$ and the face plate with a mass per unit length of $m_0$ is demonstrated to be sufficient for the considered reflection problem at normal incidence.

Figure 2

Eigenfrequencies and eigenvectors of the considered metasurface. In the linear dissipative regime, three resonance frequencies ${\omega }_i$ ($i=1,2,3$) are presented (a), related to either a translation-dominated movement, denoted ${\omega }_u$, or a rotation-dominated movement denoted ${\omega }_{\theta }$, or a combination of both. When the resonance frequency condition ${\omega }_u=2{\omega }_{\theta }$ is satisfied, the mass ratio ${\alpha }_m$ is determined for different values of initial angles of rotation ${\theta }_0$ and stiffnesses $K_s$ and $K_{\theta }$ (b). The ratio of $\theta$ to $U_1$ is examined as a function of ${\theta }_0$ and $K_s=K_{\theta }$ as well, at resonance frequencies ${\omega }_{\theta }$ and ${\omega }_u$ in panels (c) and (d), respectively. The optimal value range of stiffness and initial angle of rotation is indicated by the white dotted line in both panels (c) and (d).

Figure 3

Absolute reflection coefficients of the fundamental and second harmonic components, denoted $R_1$ and $R_2$, respectively, as a function of both the dimensionless impedance parameter $\gamma$ and the normalized excitation frequency detuning $\mathrm{\Delta }\mathrm{\Omega }$. The latter is defined as the difference between the excitation frequency $\omega$ and the linear resonance frequency ${\omega }_{\theta }$, subsequently normalized by ${\omega }_0$, i.e., $\mathrm{\Delta }\mathrm{\Omega }=(\omega -{\omega }_{\theta })/{\omega }_0$. When the input intensity is relatively weak, with a magnitude $A_{\mathrm{inc}}=5\times {10}^{-6}$, the required excitation detuning is less ($\mathrm{\Delta }\mathrm{\Omega }=-1\times {10}^{-4}$), as the maximum value of $R_2$ exceeds 0.4 (a), (b). Whereas with a stronger source of magnitude $A_{\mathrm{inc}}=5\times {10}^{-5}$, a frequency detuning of around $\mathrm{\Delta }\mathrm{\Omega }=-1.7\times {10}^{-3}$ is needed to totally absorb $R_1$, which does not alter the amplitude of the second harmonic $R_2$ to reach a maximum value of nearly 0.8 (c), (d).

Figure 4

Kinetic energy $E_{\mathrm{kinetic}}$ of the metasurface at various excitation levels, from a linear configuration with $A_{\mathrm{inc}}={10}^{-7}$ to a weakly nonlinear configuration with $A_{\mathrm{inc}}=5\times {10}^{-6}$ and a highly nonlinear configuration $A_{\mathrm{inc}}=5\times {10}^{-5}$, respectively, for the cases of (a) excitation frequencies $\omega$ close to the linear rotation-dominated resonance frequency ${\omega }_{\theta }$, and (b) excitation frequencies $\omega$ close to the linear translation-dominated resonance frequency ${\omega }_u$ verifying ${\omega }_u=2{\omega }_{\theta }$. During the kinetic energy test, the metasurface is excited, at each excitation frequency, by 1000 periods of a sine signal. The dotted black line in panel (a) indicates the resonance shift under excitation $A_{\mathrm{inc}}=5\times {10}^{-5}$, which corresponds exactly to the optimal excitation detuning introduced in Fig. 3.

Figure 5

Absolute reflection coefficient of fundamental ($R_1$) and second harmonics ($R_2$), as investigated by varying the excitation amplitude from a linear level ($A_{\mathrm{inc}}={10}^{-7}$) to a nonlinear level ($A_{\mathrm{inc}}={10}^{-4}$). Frequency detuning is introduced in order to eliminate reflection of the fundamental wave at the desired excitation amplitude, such that (a) $A_{\mathrm{inc}}=5\times {10}^{-6}$ and (b) $A_{\mathrm{inc}}=5\times {10}^{-5}$, with the impedance parameter defined as $\gamma =0.008$ and $\gamma =0.0195$, respectively.

Figure 6

Theoretical and numerical results for the nonlinear metasurface reflection, as obtained with the Harmonic Balance Method (HBM) and the fourth-order Runge-Kutta (RK4) method, respectively, for the case of a relatively weak excitation with amplitude $A_{\mathrm{inc}}=5\times {10}^{-6}$ (a), (b), and for the case of a stronger excitation with amplitude $A_{\mathrm{inc}}=5\times 0^{-5}$ (c), (d). The frequency axes are normalized by the detuned excitation frequency. By considering a wave packet source with characteristic width $N_T=4000T$, the RK4 results are compared to the theoretical HBM results. Magnitudes of short-term Fourier transforms taken at fundamental and second harmonic frequencies, by showing the temporal variation in the reflected wave spectrum (a), (c), and by exploring the Fourier transform of the central 1000 periods of the reflected temporal wave (b), (d), respectively.

Figure 7

Optimal frequency conversion effect achieved for various physical properties of the proposed metasurface, under an excitation level of $A_{\mathrm{inc}}=5\times {10}^{-5}$. The maximum absolute value of the reflection coefficient of the second harmonic $R_2$ is identified by varying the impedance parameter and excitation frequency detuning simultaneously, for stiffnesses in the range $K_s=K_{\theta }\in (0,0.1)$ and an initial angle of rotation set at ${10}^{\circ }$ and ${20}^{\circ }$ in panel (a), and for an initial angle of rotation in the range ${\theta }_0\in (0^{\circ },{30}^{\circ })$ and stiffnesses set at 0.02 and 0.06 in panel (b). Shaded zones indicate the optimal ranges of initial angle of rotation and stiffnesses, i.e., $K_s=K_{\theta }\in (0,0.04)$ and ${\theta }_0\in (3^{\circ },{15}^{\circ })$, thus yielding an efficient second harmonic reflection with $R_2$ greater than 0.4. The results presented have been output by HBM.

Figure 8

Optimal frequency conversion effect achieved for various metasurface unit cell shapes, i.e., for different inertial moments $\alpha$ of rotating elements. The maximum conversion is determined as a function of $\alpha$ over the range of $[0.02,0.66]$ by varying the impedance parameter $\gamma$, excitation frequency detuning $\mathrm{\Delta }\mathrm{\Omega }$, and stiffness $K_s=K_{\theta }$ simultaneously. In order to lessen the calculation burden, the initial angle of rotation has been set at ${10}^{\circ }$ and $8^{\circ }$ respectively, as these values are found to be favorable for producing the desired reflection over the entire inertial moment range.