Broadband non-reciprocity with robust signal integrity in a triangle-shaped nonlinear 1D metamaterial

Abstract

In this paper, we propose and numerically study a nonlinear, asymmetric, passive metamaterial that achieves giant non-reciprocity with (i) broadband frequency operation and (ii) robust signal integrity. Previous studies have shown that nonlinearity and geometric asymmetry are necessary to break reciprocity passively. Herein, we employ strongly nonlinear coupling, triangle-shaped asymmetric cell topology, and spatial periodicity to break reciprocity with minimal frequency distortion. To investigate the nonlinear band structure of this system, we propose a new representation, namely a wavenumber–frequency–amplitude band structure, where amplitude-dependent dispersion is quantitatively computed and analyzed. Additionally, we observe and document the new nonlinear phenomenon of time-delayed wave transmission, whereby wave propagation in one direction is initially impeded and resumes only after a duration delay. Based on numerical evidence, we construct a nonlinear reduced-order model (ROM) to further study this phenomenon and show that it is caused by energy accumulation, instability, and a transition between distinct branches of certain nonlinear normal modes of the ROM. The implications and possible practical applications of our findings are discussed.

Appendix

I.:

Linear dispersion analysis

The dispersion relationship of a linear periodic structure can be pursued by solving the eigenvalue problem of the equations of motion with Bloch boundary conditions [39]. Here, we reconsider Eq. 2 with a detailed study as follows,

$$\begin{aligned} \left[ K\left( \mu \right) -\omega ^{2}M \right] \tilde{\hbox {u}}\left( \mu \right) e^{in\mu }=0, \end{aligned}$$

(A.1)

where M and $K(\mu )$ are mass and stiffness matrix, respectively, and $\mu$ denotes the wavenumber. Without loss of generality, we express the non-trivial displacement vector as $\tilde{\hbox {u}}\left( \mu \right) e^{in\mu }$, where n denotes the unit cell of interest. The mass matrix contains diagonal entries corresponding to the individual masses, and the stiffness matrix $K(\mu )$ is computed from,

$$\begin{aligned} K\left( \mu \right) =\sum \nolimits _{m=-1,0,+1} {(e^{im\mu }k_{m})} , \end{aligned}$$

(A.2)

where $m=-1,\, 0,+1$ represents three adjacent unit cells: $n-1,\, n$, and $n+1$, and $k_{m}$ denotes the conventional stiffness matrix describing the spring interaction within the representative unit cell and with its neighbors. The term ${e}^{im\mu }$ further modifies the stiffness matrix with Bloch boundary conditions. Thus, by sweeping the wavenumber from 0 to $\pi$ and solving the eigenvalue problem in term of $\omega ^{2}$, we obtain the linear dispersion relationship in the first Brillion zone.

II.:

Dispersion curve fitting

In numerical simulations, for each input amplitude, we sweep the frequency of the harmonic input from 0 to 22 rad/s, with an increment of 0.1 rad/s. Propagation constants are then measured from each corresponding response. The scatter points are displayed on the frequency–wavenumber plane, and a curve fit is conducted as follows:

(a):

The noise points are recognized based on their dominant frequency (i.e., if the dominant frequency over the selected duration of response is not the sending frequency, the transmitted signal is considered as noise).

(b):

Branches (in our case, only the acoustic and the first optical branches) are recognized manually. Different dispersion curves are curve-fitted individually.

(c):

A 9-degree polynomial curve fit (“poly9” in MATLAB) is selected as the main fit function.

(d):

For the acoustic branch, to ensure a zero-group-velocity condition at $\, \mu =\pi$, we let $x=\mathrm {sin}(\frac{\mu }{2})$, and $y={\upomega }$, and we then find the curve fit coefficients.

(e):

For the first optical branch, to ensure zero-group velocity condition at $\mu =0,\, \pi$, we let $x=\mathrm {cos}(\mu )$, and $y={\upomega }$, and then find the curve fit coefficients.

An example scatter-point diagram with the corresponding curve fit result is shown in Fig. 11.

Fig. 11

Scatter points and curve fit. L–R and R–L propagation constants are marked in blue and red, respectively. Four branches (acoustic and optical branches for L–R and R–L wave transmission) are curve-fitted and plotted individually

Fig. 12

NNM result. The NNM frequency–energy relationship

III.:

Nonlinear normal modes

For an n degree of freedom Hamiltonian system, there are two fundamental theorems by Lyapunov and Weinstein, which indicate n distinct periodic solution(s) near each stable equilibrium at fixed energy level. These solutions are called nonlinear normal modes (NNMs) and can be related to the classic linear vibration modes [36]. By sweeping the energy and solving the equations of motion (see below), we are able to describe the mode frequency and the corresponding mode shape as a function of energy. The resultant NNM result is revealed in Fig. 12.

$$\begin{aligned}&m_{6}\ddot{x}_{6}+k_{2}\, \left( {3x}_{6}-x_{7} \right) +k_\mathrm{NL}\left( x_{6}-x_{7} \right) ^{3}+{2k}_\mathrm{NL}x_{6}^{3}\nonumber \\&\quad +\,c\left( 3\dot{x}_{6}-\dot{x}_{7} \right) =0 \end{aligned}$$

(A.3)

$$\begin{aligned}&m_{7}\ddot{x}_{7}+k_{1}\, x_{7}+k_\mathrm{NL}\left( x_{7}-x_{6} \right) ^{3}+k_{2}\, \left( x_{7}-x_{6} \right) \nonumber \\&\quad +\,c\left( 2\dot{x}_{7}-\dot{x}_{6} \right) =0. \end{aligned}$$

(A.4)

In the low energy regime (near ${10}^{-8}$ J), the frequencies of the nonlinear modes are 23.06 rad/s and 35.08 rad/s, respectively, which is close to the linear natural frequencies of the model. As energy grows, the frequency increases, and Mode 1 experiences a bifurcation at the energy level of approximately ${10}^{-5}$ J.

For further investigation and easier comparison with the full-order model, we replace the horizontal axis of the NNM result from energy to relative displacement between masses 6 and 7, illustrated in Fig. 8a. The modification is motivated by the continuous excitation of the system, which makes it challenging to capture the fluctuating energy and conduct further comparison.