Interaction of traveling waves with mass-with-mass defects within a Hertzian chain

We study the dynamic response of a granular chain of particles with a resonant inclusion (i.e., a particle attached to a harmonic oscillator, or a mass-with-mass defect). We focus on the response of granular chains excited by an impulse, with no static precompression. We find that the presence of the harmonic oscillator can be used to tune the transmitted and reflected energy of a mechanical pulse by adjusting the ratio between the harmonic resonator mass and the bead mass. Furthermore, we find that this system has the capability of asymptotically trapping energy, a feature that is not present in granular chains containing other types of defects. Finally, we study the limits of low and high resonator mass, and the structure of the reflected and transmitted pulses.

Figure 1

Diagram of a granular chain containing a mass-with-mass defect.

Figure 2

Suggested experimental implementation of the mass-with-mass defect. (a) Defect bead with a ring resonator that provides the mass with mass. (b) Part of the suggested experimental setup containing the MwM defect.

Figure 3

Left panel: the different fractions of the energy as functions of the mass ratio $\epsilon =m_2/m_1$ at $\kappa =1$. Right panel: trapped portion of the energy. The transmitted energy fraction $T$ of the right part of the chain is presented by the red dash-dotted line, the reflected part of the energy $R$ is given by the green dashed line, while the nonvanishing (for $\epsilon >0$) trapped fraction of the energy is shown by the blue solid line.

Figure 4

The results of simulations at $\epsilon =0.1$. The top left panel presents the space-time ($n$-$t$) evolution of the contour plot of the field ${\dot{u}}_n\left(t\right)$. The thick blue and green lines in the bottom left panel show the quantities ${(u_{n-1}-u_n)}_{+}$ and ${(u_n-u_{n+1})}_{+}$, respectively, to give a sense of when (after the impact) gap openings arise. The top right panel presents the displacements of the central sites; the cyan dash-dotted line corresponds to $u_{-1}\left(t\right)$, the thick blue solid line to $u_0\left(t\right)$, the green solid line to $u_1\left(t\right)$, and the red dashed line to $v_0\left(t\right)$. The same pattern is followed for the corresponding velocities in the bottom right plot. Here $\kappa =1$.

Figure 5

The same properties as in Fig. 4 but now for the case of $\epsilon =10000$. Here $\kappa =1$.

Figure 6

The same properties as in Fig. 4 but now for the case of $\epsilon =10$. Here, it is the lighter mass $m_1$ executing the fast-scale oscillations, while opening a gap from the site to its left and presenting a cascade of alternating gap openings and compression events with its right neighbor, which, in turn, lead to the successive emission of progressively smaller-amplitude (and thus slower) waves within the solitary wave train that emerges in the space-time contour plot. Here $\kappa =1$.

Figure 7

The case of $\epsilon =0.5$, close to the value at which the maximal trapping occurs. The same diagnostics as for $\epsilon =0.1$ in the case of Fig. 4 are shown. In this case the results reveal vibration over the fast scale of both $u_0$ and $v_0$ at the defect site and formation of only a second wave traveling to the right. Here $\kappa =1$.

Figure 8

Transmitted, reflected, and trapped portions of the energy as functions of $\epsilon$ at different values of $\kappa$ in the case of the three-bead system (13). Observe that in addition to the existence of a maximum for a particular $\epsilon =m_2/m_1$, the energy fractions approach constant values at large $\epsilon$. Both features are shared by the infinite-chain case.

Figure 9

The same diagnostics as those of Fig. 3 (transmitted $T$, reflected $R$, and trapped $1-T-R$ fractions of the energy) as functions of the mass ratio, but now for the case of two adjacent mass defects (left) and that of three adjacent mass defects (right). Here $\kappa =1$.

Figure 10

The space-time ($n$-$t$) evolution of the contour plot of the field ${\dot{u}}_n\left(t\right)$ for the case of two adjacent defects, at $n=0$ and $n=1$ (left panel), and two defects that are spaced widely apart, at $n=0$ and $n=50$ (right panel). A larger chain, $-150\le n\le 250$, was used in these simulations. Here $\epsilon =5$ and $\kappa =1$.

Figure 11

The time derivative of the difference $w_0\left(t\right)=u_0\left(t\right)-v_0\left(t\right)$ given by the direct numerical computation (black) and the analytical approximation (green) for $\epsilon =0.005$ and $\epsilon =0.025$. The insets zoom in on the regions inside the rectangles.

Figure 12

Numerical solution of (18) and (19) at $\epsilon =0.01$ and $\epsilon =0.1$.

Figure 13

Comparison of the numerical solution of (18) and (19) at $\epsilon =0.01$ (colored solid curves) and the two-timing approximation up to $O\left(\epsilon \right)$ (black dashed curves).