Active acoustic switches using two-dimensional granular crystals

We employ numerical simulations to study active transistor-like switches made from two-dimensional (2D) granular crystals containing two types of grains with the same size but different masses. We tune the mass contrast and arrangement of the grains to maximize the width of the frequency band gap in the device. The input signal is applied to a single grain on one side of the device, and the output signal is measured from another grain on the other side of the device. Changing the size of one or many grains tunes the pressure, which controls the vibrational response of the device. Switching between the on and off states is achieved using two mechanisms: (1) pressure-induced switching where the interparticle contact network is the same in the on and off states and (2) switching through contact breaking. In general, the performance of the acoustic switch, as captured by the gain ratio and switching time between the on and off states, is better for pressure-induced switching. We show that in these acoustic switches the gain ratio between the on and off states can be larger than ${10}^4$ and the switching time (multiplied by the driving frequency) is comparable to that obtained recently for sonic crystals and less than that for photonic transistor-like switches. Since the self-assembly of grains with different masses into 2D granular crystals is challenging, we describe simulations of circular grains with small circular knobs placed symmetrically around the perimeter mixed with circular grains without knobs. Using umbrella sampling techniques, we show that grains with six knobs most efficiently form the hexagonal crystals that yield the largest frequency band gap. Using the simulation results, we estimate the time required for vibration experiments to generate granular crystals of millimeter-sized steel beads with maximal band gaps.

Figure 1

(a) A schematic of a metal-oxide-semiconductor field-effect transistor (MOSFET) with gate (G), source (S), and drain (D) ports and (b) a schematic of a switch made from a 2D granular crystal with three ports for the (1) output, (2) control, and (3) input signals.

Figure 2

Mechanically stable packings of $N=100$ disks with the same size, two different masses, and mass ratio $m_L/m_S=10$ arranged on a hexagonal lattice with periodic and fixed boundary conditions in the $x$ and $y$ directions, respectively. In (a), the system is homogeneous with $N_L=0$ (dark blue) and $N_S=N$ (light blue). In (b), we set $N_L=25$ and $N_S=75$. The first row contains all small masses. In the second row, the large and small masses alternate. The third row alternates between large and small masses, and this order repeats for a total of 10 rows. In (c), $N_L=50$ and $N_S=50$ and large and small masses are distributed randomly on the hexagonal lattice. Panel (d) is similar to (b) except inverted with $N_L=75$ and $N_S=25$.

Figure 3

Eigenfrequencies of the mass-weighted dynamical matrix ${\omega }_k$, sorted in ascending order and indexed by $k$, for the $N=100$ configurations in Fig. 2. (a) Circles, (b) $\mathsf{X}\mathrm{s}$, (c) plusses, and (d) squares with periodic and fixed boundary conditions in the $x$ and $y$ directions, respectively. $w$ indicates the maximum band gap in the eigenfrequency spectrum.

Figure 4

The width $w$ of the maximum band gap in the eigenfrequency spectrum of the mass-weighted dynamical matrix for the configuration in Fig. 2 (and inset) as a function of the mass ratio $m_L/m_S$.

Figure 5

(a) Spectrum of eigenfrequencies of the mass-weighted dynamical matrix sorted in ascending order with index $k$ for systems with $N=100$ disks, $m_L/m_S=10$, and arranged on a hexagonal lattice in the optimal configuration in Fig. 2 at pressure $p={10}^{-3}$ (circles) and 1 ($\mathsf{X}\mathrm{s}$). The dashed line indicates a driving frequency at which the acoustic switch can operate. (b) The Fourier transform of the velocity correlation function $D\left(\omega \right)$ for the mechanically stable packing in Fig. 2 at pressure $p={10}^{-4}$ after adding velocities to all grains such that the eigenfrequencies of the mass-weighted dynamical matrix are included with equipartition of the total kinetic energy $K_0$. The color scale from dark red to violet represents decreasing $D\left(\omega \right)$ on a linear scale.

Figure 6

(a) An illustration of a three port acoustic switch with fixed, flat boundary conditions in the $y$ direction and periodic boundary conditions in the $x$ direction. The device includes $N=30$ disks (with $N_L=21$ (dark), $N_S=9$ (light), and $m_L/m_S=10$) arranged on a hexagonal lattice. The solid white lines indicate the $N_c=90$ distinct contacts between disks. Disk 3 is the input port, indicating where the system will be driven. The gain of the system is measured via the output port, labeled disk 1. The switch can be turned on and off by varying the pressure of the system through port 2, e.g., by changing the size of a single disk or all disks in the system. Here, the device changes from pressure $p={10}^{-6}$ (dashed outline) to ${10}^{-1}$ (solid outline) when all disks increase in size. (b) Illustration of the device in (a) at $p={10}^{-6}$ with disk 3 driven at $A_0={10}^{-6}$ and frequency ${\omega }_0=16.0$, which causes contact breaking. In this snapshot, the device has four fewer contacts than in (a). The central grain with the dashed outline provides the pressure control when we use single-particle control for port 2.

Figure 7

The eigenfrequencies of the mass-weighted dynamical matrix plotted in increasing order with index $k$ for the acoustic device in Fig. 6 at pressure $p={10}^{-1}$ (the “on” state, circles) and $3.2\times {10}^{-2}$ (the “off” state, $\mathsf{X}\mathrm{s}$). The horizontal line at $\omega =14.9$ indicates a potential driving frequency that yields a large gain ratio between the on and off states.

Figure 8

(a) The Fourier transform $F_1\left(\omega \right)$ of the $x$ displacement of disk 1 for the acoustic device with pressure $p={10}^{-1}$ (dashed line) and $3.2\times {10}^{-2}$ (solid line) obtained by driving disk 3 sinusoidally with amplitude $A_0={10}^{-6}$ and frequency ${\omega }_0=14.9$. The dotted line shows the Fourier transform $F_3\left(\omega \right)$ of the $x$ displacement of the input disk 3. (b) The gain $\mathrm{G}\left({\omega }_0\right)$ [defined in Eq. (4)] plotted as a function of the driving amplitude $A_0$ with driving frequency ${\omega }_0=14.9$ for the device at pressure $p={10}^{-1}$ (open circles) and $3.2\times {10}^{-2}$ ($\mathsf{X}\mathrm{s}$).

Figure 9

(a) The gain $G\left({\omega }_0\right)$ for the acoustic device as a function of pressure $p$ for three values of the driving frequency ${\omega }_0=13.1$ (circles), 14.9 ($\mathsf{X}\mathrm{s}$), and 15.7 (triangles). (b) The gain $G({\omega }_0+\mathrm{\Delta }\omega )$ over a small frequency range $\mathrm{\Delta }\omega$ near the driving frequency ${\omega }_0=14.9$. The inset is a close-up of the gain to within ${10}^{-3}$ of ${\omega }_0$. (c) The gain ratio $G_{\mathrm{on}}\left({\omega }_0\right)/G_{\mathrm{off}}\left({\omega }_0\right)$ as a function of the normalized change in pressure between the on and off states, $(p_{\mathrm{on}}-p_{\mathrm{off}})/p_{\mathrm{on}}$, for $p_{\mathrm{on}}=3.2\times {10}^{-1}$ (circles), ${10}^{-1}$ ($\mathsf{X}\mathrm{s}$), and $3.2\times {10}^{-2}$ (triangles) and the sizes of all particles are changed to control the pressure. The vertical dashed lines indicate the value of $(p_{\mathrm{on}}-p_{\mathrm{off}})/p_{\mathrm{on}}$ at which contacts would begin breaking if the size of only a single particle was changed to control the pressure. For all data, the driving amplitude is $A_0={10}^{-6}$ and the damping parameter $b={10}^{-3}$.

Figure 10

The gain ratio $G_{\mathrm{on}}\left({\omega }_0\right)/G_{\mathrm{off}}\left({\omega }_0\right)$ between the on and off states versus the damping parameter $b$ at fixed driving frequency ${\omega }_0=14.9$ for pressure-induced switching (solid line) and 16.0 for switching with contact breaking (dashed line). The inset shows the switching time ${\omega }_0{t}_s^1/2\pi$ from the on to the off state (open circles) and ${\omega }_0{t}_s^2/2\pi$ from the off to the on state ($\mathsf{X}\mathrm{s}$) versus $b$ for the same systems in the main panel.

Figure 11

(a) The Fourier transform $F_1\left({\omega }_0\right)$ of the $x$ displacement of disk 1 as a function of time ${\omega }_0(t-t^{*})/2\pi$ when switching the device at time $t^{*}$ from the “on” (pressure $p={10}^{-1}$) to “off” ($p=3.2\times {10}^{-2}$) states (circles) and vice versa ($\mathsf{X}\mathrm{s}$) using a damping coefficient $b={10}^{-2}$. The horizontal dotted line indicates the geometric mean ${\overline{F}}_1\left({\omega }_0\right)$ of the on and off values of $F_1\left({\omega }_0\right)$. The switching times $t_s$ are obtained by finding when $F_1\left({\omega }_0\right)$ crosses ${\overline{F}}_1\left({\omega }_0\right)$. (b) The Fourier transform $F_1\left({\omega }_0\right)$ (open circles and left axes labels) of the $x$ displacement of disk 1 as a function of time $t/\mathrm{\Delta }t$ (after reaching an initial steady state at $t=0$) during continuous switching of the device between the “on” and “off” states using $b={10}^{-2}$. The pressure of the device (dashed line and right axes labels) follows a square-wave signal with $\mathrm{\Delta }t/t_s\approx 3.7$. (c) Same as (b) except $\mathrm{\Delta }t/t_s\approx 0.7$.

Figure 12

(a) The Fourier transform $F_1\left(\omega \right)$ of the $x$ displacement of disk 1 for the device with pressure $p={10}^{-6}$ (dashed line) and ${10}^{-8}$ (solid line) obtained by driving disk 3 sinusoidally with amplitude $A_0=3.2\times {10}^{-7}$ and frequency ${\omega }_0=16.0$, using damping coefficient $b={10}^{-3}$. The dotted line shows the Fourier transform $F_3\left(\omega \right)$ of the $x$ displacement of the input disk 3. (b) The gain $\mathrm{G}\left({\omega }_0\right)$ [defined in Eq. (4)] plotted as a function of the driving amplitude $A_0$ with driving frequency ${\omega }_0=16.0$ for a device at pressure $p={10}^{-6}$ (dashed line) and ${10}^{-8}$ (solid line), using damping parameter $b={10}^{-3}$. The vertical dotted line indicates the amplitude of the driving $A_0=3.2\times {10}^{-7}$ in (a).

Figure 13

(a) The Fourier transform $F_1\left({\omega }_0\right)$ of the $x$ displacement of disk 1 as a function of time ${\omega }_0(t-t^{*})/2\pi$ when switching the device from the “on” (pressure $p={10}^{-6}$) to “off” ($p={10}^{-8}$) states (circles) and vice versa ($\mathsf{X}\mathrm{s}$) at time $t^{*}$ using a damping coefficient $b={10}^{-3}$. The driving frequency and amplitude are ${\omega }_0=16.0$ and $A_0=3.2\times {10}^{-7}$. (b) The Fourier transform $F_1\left({\omega }_0\right)$ of the $x$ displacement of disk 1 (circles and left axes labels) as a function of time $t/\mathrm{\Delta }t$ when continuously switching the device between the “on” and “off” states using $b={10}^{-3}$. The dashed line shows the pressure of the device (right axes labels), which has a square-wave form with $\mathrm{\Delta }t/t_s\approx 2.7$ (where $t_s$ is the time for the device to switch from the off to the on states). (c) Same as (b) except $\mathrm{\Delta }t/t_s\approx 0.68$. For (a)–(c), the off and on states possess different interparticle contact networks.

Figure 14

The normalized change of pressure $(p_{\mathrm{on}}-p_{\mathrm{off}})/p_{\mathrm{on}}$, where $p_{\mathrm{on}}$ and $p_{\mathrm{off}}$ are the pressures in the on and off states, respectively, as a function of the normalized change in the size $({\sigma }_{\mathrm{on}}-{\sigma }_{\mathrm{off}})/{\sigma }_{\mathrm{on}}$ of a single control particle for pressures $p_{\mathrm{on}}={10}^{-2}$ (circles), $3.2\times {10}^{-2}$ ($\mathsf{X}\mathrm{s}$), and ${10}^{-1}$ (triangles). The vertical dashed lines (from left to right) indicate the change in size above which the control particle loses a contact with neighboring particles for $p_{\mathrm{on}}={10}^{-2}, 3.2\times {10}^{-2}$, and ${10}^{-1}$.

Figure 15

(a) The gain $\mathrm{G}\left({\omega }_0\right)$ [defined in Eq. (4)] plotted as a function of the driving amplitude $A_0$ at fixed driving frequency ${\omega }_0=16.0$ for a device with a single control particle at $\mathrm{\Delta }\sigma /\sigma =0$ (dashed line), $5.1\times {10}^{-7}$ (circles), $5.4\times {10}^{-7}$ ($\mathsf{X}\mathrm{s}$), and $5.5\times {10}^{-7}$ (triangles), using damping parameter $b={10}^{-3}$. (b) The Fourier transform $F_1\left({\omega }_0\right)$ of the $x$ displacement of disk 1 as a function of time ${\omega }_0(t-t^{*})/2\pi$ when switching the device from the on ($\mathrm{\Delta }\sigma /\sigma =0$) to off ($\mathrm{\Delta }\sigma /\sigma =5.4\times {10}^{-7}$) state (circles) and vice versa ($\mathsf{X}\mathrm{s}$) at time $t^{*}$ using a single control particle and damping parameter $b={10}^{-3}$. The driving frequency and amplitude are ${\omega }_0=16.0$ and $A_0=6.3\times {10}^{-8}$, respectively.

Figure 16

Circular particles with different numbers of circular knobs, (a) $n=0$, (b) 3, and 6, placed symmetrically around the perimeter of the particle. The central disk has diameter $\sigma$ and the circular knobs have diameter ${\sigma }_k=(2\sqrt{3}/3-1)\sigma$.

Figure 17

(a) Snapshot of a disk configuration generated using the accelerated discrete element method simulations after four rounds of acceleration. The light disks have $n=6$ knobs and the dark disks have zero knobs. The inherent structures with zero kinetic energy and (b) $N_{kk}=0$ and (c) 7 were obtained after removing the knobs. Contacts between the light-colored grains in (c) are indicated by the dotted lines.

Figure 18

The average maximum difference between adjacent eigenfrequencies $\langle w\rangle$ of the mass-weighted dynamical matrix versus the number of contacts between the grains with knobs $N_{kk}/N$ for a hexagonal packing with $N_L=21$ and $N_S=9$ and mass ratio $m_L/m_S=100$ (circles), 20 ($\mathsf{X}\mathrm{s}$), 10 (triangles), and 5 (squares). The means and standard deviations (error bars) are obtained by averaging over 50 configurations in which the masses of the grains are chosen randomly as either $m_L$ and $m_S$ to yield a given $N_{kk}/N$.

Figure 19

The average $N_{kk}/N$ from inherent structures as a function of the number of rounds of the accelerated MD simulations for the systems in Fig. 17 with $n=0$ (circles), 3 ($\mathsf{X}\mathrm{s}$), and 6 (triangles) knobs. The inset shows a scatter plot of $N_{kk}$ versus the number of rounds for the system with $n=6$ knobs.

Figure 20

The gain $G\left({\omega }_0\right)$ for the acoustic device as a function of pressure $p$ (for a system with no contact breaking) using a total simulation time ${\omega }_{0}T/2\pi ={10}^4$ (circles) and ${10}^5$ (squares). The inset shows the Fourier transforms of the output and input signals, $F_1\left({\omega }_0\right)$ ($\mathsf{X}\mathrm{s}$) and $F_3\left({\omega }_0\right)$ (circles), as a function of ${\omega }_{0}T/2\pi$ for the device with pressure $p={10}^{-1}$. For all data, ${\omega }_0=14.9, A_0={10}^{-6}, b={10}^{-3}$, and the sampling time ${\omega }_0\mathrm{\Delta }/2\pi =5.9\times {10}^{-3}$.

Figure 21

The gain $G\left({\omega }_0\right)$ for the acoustic device as a function of pressure $p$ (for a system with no contact breaking) measured with sampling interval ${\omega }_0\mathrm{\Delta }/2\pi =4.7\times {10}^{-2}$ (circles), $2.4\times {10}^{-2}$ ($\mathsf{X}\mathrm{s}$), $5.8\times {10}^{-3}$ (triangles), and $2.9\times {10}^{-3}$ (squares). The inset shows the Fourier transforms of the output and input signals, $F_1\left({\omega }_0\right)$ ($\mathsf{X}\mathrm{s}$) and $F_3\left({\omega }_0\right)$ (circles), as a function of ${\omega }_0\mathrm{\Delta }/2\pi$ for the device with pressure $p={10}^{-1}$. For all data, ${\omega }_0=14.9, A_0={10}^{-6}, b={10}^{-3}$, and the total simulation time ${\omega }_{0}T/2\pi =5.9\times {10}^4$.

Figure 22

Spectrum of eigenfrequencies for the mass-weighted dynamical matrix for the hexagonal lattice in the inset to Fig. 4 with $N_L=21$ and $N_S=9$ for mass ratios (a) $m_L/m_S=10$, (b) 3, and (c) 1. The horizontal dashed lines indicate the frequencies at which we seek to drive the acoustic switching device. The insets of each panel show the frequency-dependent gain $G\left({\omega }_0\right)$ (ratio of the Fourier transforms of the output and input signals) for the respective mass ratios. For all systems, the pressure $p={10}^{-1}$.

Figure 23

The ratio of the gain $G_{\mathrm{on}}\left({\omega }_0\right)/G_{\mathrm{off}}\left({\omega }_0\right)$ in the on state to that in the off state as a function of the normalized difference in pressure $(p_{\mathrm{on}}-p_{\mathrm{off}})/p_{\mathrm{on}}$ between the on and off states for mass ratios $m_L/m_S=10$ (circles), 3 ($\mathsf{X}\mathrm{s}$), and 1 (triangles). The devices are driven at the frequencies ${\omega }_0$ indicated by the dashed lines in Figs. 22–22.