Nonlinear low-to-high-frequency energy cascades in diatomic granular crystals

We study wave propagation in strongly nonlinear one-dimensional diatomic granular crystals under an impact load. Depending on the mass ratio of the “light” to “heavy” beads, this system exhibits rich wave dynamics from highly localized traveling waves to highly dispersive waves featuring strong attenuation. We demonstrate experimentally the nonlinear resonant and antiresonant interactions of particles, and we verify that the nonlinear resonance results in strong wave attenuation, leading to highly efficient nonlinear energy cascading without relying on material damping. In this process, mechanical energy is transferred from low to high frequencies, while propagating waves emerge in both ordered and chaotic waveforms via a distinctive spatial cascading. This energy transfer mechanism from lower to higher frequencies and wave numbers is of particular significance toward the design of novel nonlinear acoustic metamaterials with inherently passive energy redistribution properties.

Figure 1

Schematic of the experimental setup for measuring nonlinear wave propagation in a diatomic granular crystal. The inset shows a cross section of the sensor particle.

Figure 2

Numerical and experimental surface maps for (a),(d) AR2, (b),(e) R1, and (c),(f) R2. The insets in (b) and (c) show an enlarged view of the nonlinear wave beating phenomenon in R1, and regular (I) and irregular (II) forms of the wave tail in R2, respectively. The insets in (d)–(f) illustrate comparisons of three successive (heavy-light-heavy) particles' velocity profiles between numerical [solid blue (heavy) and green (light)] and experimental (dashed red) results.

Figure 3

Frequency and wave-number spectra of particles' velocity profiles in (a),(d) AR2, (b),(e) R1, and (c),(f) R2, obtained from the analysis of the computational results and illustrating the presence of cascades associated with the redistribution of energy among different scales. The wave number is calculated based on the wavelength normalized by the average diameter of the large and small particles.

Figure 4

Normalized energy carried by the primary wave packet in antiresonances and resonances. The inset shows a comparison between the experimental and numerical results. The error bars represent standard deviations based on five tests.

Figure 5

Wavelength of the solitary waves in a diatomic chain at various mass ratios from AR1 to AR8. The solid curve represents the analytical prediction based on Porter et al. [15]. Green squares and red circles are from numerical simulations and experiments, respectively.

Figure 6

Wave-number analysis in the case in which two different profiles with an identical wave number of $\pi /2$ are mixed together in an alternating way. Based on the FFT analysis, (b), (d), and (f) show the wave number of the wave profiles in spatial domains for (a) heavy beads, (c) light beads, and (e) the mixed data of the two, respectively. In (a) and (c), red curves represent continuous signals, while blue circles denote the sampled data at particle positions.

Figure 7

Wave-number analysis in the case in which two different profiles with an identical wave number of $0.48\pi$ are mixed together in an alternating way. Based on the FFT analysis, (b), (d), and (f) show the wave number of the wave profiles in spatial domains for (a) heavy beads, (c) light beads, and (e) the mixed data of the two, respectively. In (a) and (c), red curves represent continuous signals, while blue circles denote the sampled data at particle positions.