Lattices with internal resonator defects

We consider a variety of settings involving chains with one or more defects stemming from the introduction of nodes bearing internal resonators. Motivated by experimental results in woodpile elastic lattices with one or two defects, we consider a variety of different theoretical scenarios. These include multidefect chains and their ability to transmit, reflect, and especially trap energy. Moreover, they involve defects which are spatially separated and either statically or, more effectively, dynamically enable the confinement of energy between the separated defects. Wherever possible, comparisons of the experiments with numerical simulations as well as with theoretical intuition are also offered, to provide a justification for the observed findings.

Figure 1

(a) Illustration of experimental setup involving the woodpile chain, the striker leading to the formation of the wave and the laser Doppler vibrometer enabling its probing. (b) Digital image of the woodpile chain with a single-defect rod. The horizontal posture of the rods is supported by soft polyurethane foam inserted between the rods. The effect of the foam on wave propagation is negligible due to the orders-of-magnitude lower stiffness of the foam compared to the quartz cylinders [19].

Figure 2

Solitary wave propagation in the woodpile chain with a single defect (panels (a)–(c)) and two separated defects (panels (d)–(f)) in experiments and numerical simulations. See the main text for details. Panels (a) and (d) show space-time velocity evolution plots obtained from experiments with one and two defects, respectively, and panels (b) and (e) show the corresponding results of the numerical simulations. Panels (c) and (f) compare experimental and numerical evolution of the normalized kinetic energy fraction in the chain with a single and double defect rods, respectively.

Figure 3

Space-time evolution of velocities in numerical simulations with $\epsilon =10$, $\kappa =1$ and defect lengths 3 ($L=2$, left panel) and 7 ($L=6$, right panel). In the second case the two edge defects are visibly more separated demonstrating a wider defect zone.

Figure 4

The reflected ($R$), transmitted ($T$), and trapped ($E_{\mathrm{tr}}$) energy fractions as functions of the mass $\epsilon$ with $\kappa =1$ for the cases of (a) single defect ($L=0$), (b) three adjacent defects ($L=2$), (c) six adjacent defects ($L=5$), and (d) seven adjacent defects ($L=6$).

Figure 5

The global maximum (over the mass of the resonator $\epsilon$) of the trapped energy as a function of the length $L$ of the “defective” region (bearing the resonators), with $\kappa =1$.

Figure 6

Contour plot of the trapped energy for various ${\epsilon }_1$ and ${\epsilon }_2$ ratios, as a two-dimensional function of both variables (for fixed $\kappa =1$). The left panel is for a finer grid spacing and smaller yet important (in terms of energy trapping) region of the two-dimensional parameter space, while the right panel represents a coarser but extended grid. The left panel shows the domain ${[0.1,1]}^2$ with an $\epsilon$ increment of $\mathrm{\Delta }\epsilon =0.01$ and the right panel shows the domain ${[1,100]}^2$ with a $\mathrm{\Delta }\epsilon =1$.

Figure 7

Space-time velocity plot for the $({\epsilon }_1,{\epsilon }_2)$ combinations that yield the maximal trapping. The left panel corresponds to the (ordered) pair $({\epsilon }_1,{\epsilon }_2)=(0.67,0.54)$, and the right panel to $({\epsilon }_1,{\epsilon }_2)=(13,100)$. In the left panel we see that the major contributor to the trapped energy is the energy emanating from the resonator nodes, as opposed to the right panel where we see that the major contributor to the trapped energy is the motion of a single solitary wave bouncing back and forth between the beads. In addition, in the right panel we can also discern slightly smaller solitary waves emanating from the ringing vibrations of the two defects, as indicated by the faint blue lines.

Figure 8

Left panel: plot of the semianalytical approximation of $E_{\text{tr}}\left(\epsilon \right)$ discussed in the context of Fig. 6 for ${\epsilon }_1={\epsilon }_2=\epsilon$. Middle and right panels: plot of $E_{\text{tr}}({\epsilon }_1,{\epsilon }_2)$ compared with a zoomed in panel of Fig. 6. The middle panel is the analytical approximation while the right panel represents the numerics.

Figure 9

Space-time velocity plot for the system with two defects of equal mass: ${\epsilon }_1={\epsilon }_2=10$. The first defect, at $d_1=0$, is dynamically switched on with $\kappa \left(t\right)$ given by (13), where $k_1=0$, $k_2=2.5,t_0=40,\tau =0.1$. The second defect, at $d_2=20$, has fixed $\kappa =2.5$. Observe that this setting achieves the trapping of a significant fraction of the solitary wave's energy, in the sense of it bouncing back and forth between the two defect sites.