Nonlinear vibrational-state excitation and piezoelectric energy conversion in harmonically driven granular chains

This article explores the excitation of different vibrational states in a spatially extended dynamical system through theory and experiment. As a prototypical example, we consider a one-dimensional packing of spherical particles (a so-called granular chain) that is subject to harmonic boundary excitation. The combination of the multimodal nature of the system and the strong coupling between the particles due to the nonlinear Hertzian contact force leads to broad regions in frequency where different vibrational states are possible. In certain parametric regions, we demonstrate that the nonlinear Schrödinger equation predicts the corresponding modes fairly well. The electromechanical model we apply predicts accurately the conversion from the obtained mechanical energy to the electrical energy observed in experiments.

Figure 1

Schematic of the experimental setup. Upper inset: Digital image of the system. Lower inset: Schematic of the piezoelectric ceramic (PZT)–embedded sensor bead.

Figure 2

(a) Voltage amplitude of time-periodic solutions of Eq. (7) vs driving frequency and a fixed driving amplitude of $a=0.11\mu \mathrm{m}$ (blue, red, and green lines). The color is associated with the stability properties of the respective solutions (see details in the text). Experimental voltages shown (black symbols with error bars) represent the average amplitude of the voltage over the final 10 ms of a 100-ms run. Each experimental run was initially at rest. Three experiments per marker were performed to obtain statistics. (b) Zoom-in on (a) in the 7.1- to 7.6-kHz region. Thick red error bars correspond to an experiment where the actuation frequency was decreased dynamically (every $0.05$ s). Values shown represent the average voltage amplitude over the final $0.01$ s before the frequency was updated. The linear response is also shown (light-gray line). (c) Voltage response for an amplitude of $a=0.11\mu \mathrm{m}$ and $f_b=7.32$ kHz with a short amplitude burst where the maximum amplitude is increased by a factor of 1.5 (solid black line).

Figure 3

Bifurcation diagrams corresponding to the voltage as a function of the actuation frequency $f_b$ and for various values of the actuation amplitude—$a=0.05\mu \mathrm{m},a=0.07\mu \mathrm{m}$, and $a=0.1\mu \mathrm{m}$—are represented by smooth curves, in increasing order (see the arrows). Note that blue segments correspond to stable parametric regions, while red and green segments correspond to real and oscillatory unstable parametric regions, respectively (see text).

Figure 4

(a) Voltage amplitude of time-periodic solutions of Eq. (7) vs driving frequency and a fixed driving amplitude of $a=0.12\mu \mathrm{m}$ (blue, red, and green lines). The color is associated with the stability properties of the respective solution (see details in the text). Black symbols correspond to a simulation where the actuation frequency was decreased dynamically (every $0.05$ s) starting at $f_b=7.5$ kHz. Values shown represent the average voltage amplitude over the final $0.01$ s before the frequency was updated. (b) Time series of the voltage response. For clarity, only local maxima of the time series are shown.

Figure 5

(a) Continuation in driving amplitude $a$ of time-periodic solutions of Eq. (1) for a fixed driving frequency of $f_b=7.29$ kHz and various dissipation values. The maximum velocity of the 10th particle is shown. Saddle-node (SN) bifurcations which progressively approach the NLS limit as $a\to 0$ (for $\tau$ sufficiently large) are labeled with arrows. (b) Comparison of the NLS approximation (red diamonds) and corresponding time-periodic solution of Eq. (7) (green squares) with a low dissipation associated with $\tau =0.7\mathrm{s}$. (c) Steady-state amplitude of the voltage as predicted by Eq. (22) (red circles) and the actual values obtained through Eq. (7) (green squares) using the experimentally determined value of the parameter, $\tau =0.5$ ms.