Nonlinear resonances and energy transfer in finite granular chains

In the present work we test experimentally and compute numerically the stability and dynamics of harmonically driven monoatomic granular chains composed of an increasing number of particles $N(N=1–50)$. In particular, we investigate the inherent effects of dissipation and finite size on the evolution of bifurcation instabilities in the statically compressed case. The findings of the study suggest that the nonlinear bifurcation phenomena, which arise due to finite size, can be useful for efficient energy transfer away from the drive frequency in transmitted waves.

Figure 1

(a) Schematic of the experimental setup where the chain's length is varied between 1 and 50 beads. For one and two bead systems there was no embedded sensor. (b)–(d) The experimental bifurcation dynamics in a 15 bead chain statically compressed at 8 N and driven at $7.3\mathrm{kHz}$. (b) The linear transfer function measured using a white noise excitation. The dotted line at $6.8\mathrm{kHz}$ indicates the band cutoff frequency measured at the half power point of the last peak. The drive frequency $\left(7.3\mathrm{kHz}\right)$ for the force time series in (c) is therefore in the band gap. (c) The force time series measured at the end of the chain shows how the bifurcation results in the amplitude growth and stabilization. (d) The power spectral density (PSD) of the red portion of the force signal in (c) shows that energy is transferred from the drive frequency, ${\mathrm{f}}_{\mathrm{d}}=7.3\mathrm{kHz}$, to two new frequencies ${\mathrm{f}}_{\mathrm{N}}$. We study how this bifurcation results from the finite size of a 1D system.

Figure 2

Color maps of the experimentally measured rms velocity (mm/s) of single bead (a) and two bead (b) systems as a function of the drive amplitude and frequency. The velocity is measured in the second bead for the two bead system. The dotted line in (a), which starts at $4.05\mathrm{kHz}$ for low amplitudes and decreases in frequency at higher amplitudes, indicates the maximum of the resonance at each drive amplitude. This clearly displays the mode softening to lower frequencies as the amplitude of the excitation increases. The insets show cross sections at selected drive amplitudes. The horizontal axis of the insets is the same frequency axes as each corresponding panel. The asymmetry and the mode softening is a result of the nonlinear Hertzian contact interaction. The measurements are taken using a lock-in amplifier to reduce noise. In addition, the low amplitude response is used to estimate the dissipation coefficients used in the one and two bead computational results. Panels (c) and (d) are the computational counterparts to (a) and (b). The system depends sensitively on the initial compression $F_0$ and the diagrams are fit to have the same linear (low amplitude) frequency as the experimental plots. This corresponds to a 8.67 N static compression for the single bead and 4.36 N for two beads.

Figure 3

The experimental nonlinear resonance and bifurcation behavior of a single bead driven at $6.85\mathrm{kHz}$. (a),(b) The velocity of the bead (a) before and (b) after the bifurcation. (c) The maximum velocity measured at each drive amplitude. The two crosses indicate the drive amplitudes for the time series in (a), red lower left cross, and (b) right lower left cross. (d) The corresponding PSD of two time series, showing the dominant subharmonic frequency at ${\mathit{f}}_{\mathit{d}}/2$ for the excitation above the bifurcation amplitude. The dotted lines indicate the drive frequency (right) and new subharmonic (left) frequency. (e) the Poincaré section of the dynamics of the bead before (red, central points) and after (blue, side points) the bifurcation. The splitting of the section from one point to two points is characteristic of a period doubling subharmonic bifurcation. Panels (f) and (g) are the computational plots that correspond to the experimental panels (c) and (d). (g) The PSD clearly shows that a subharmonic bifurcation occurs after the critical amplitude is crossed.

Figure 4

Experimental nonlinear resonance and quasiperiodic bifurcation behavior in a system of two beads driven at $6.94\mathrm{kHz}$. (a),(b) The velocity of the second bead (a) before and (b) after the bifurcation. (c) The corresponding PSD of two time series, showing the new frequencies $f_{N1}$ and $f_{N2}$ supported by the nonlinearity of the system, where $f_{N1}+f_{N2}=f_d$. The PSD of the time series clearly shows that energy is transferred from the drive frequency $f_d$ to the two new frequencies. (d) Poincaré sections of the dynamics of the second bead before (red, central point) and after (blue, surrounding points) the bifurcation. This Poincaré shows the classic intersection of a torus and a plane for quasiperiodic dynamics. The finite number of points is due to the finite length of the signal. (e) The PSD of the computational time series taken at the drive amplitudes indicated in (f) and using the same parameters as measured during the experimental runs. The lower left cross indicates the drive amplitude chosen before the bifurcation, and the upper right cross is at a higher amplitude after the bifurcation.

Figure 5

The experimentally measured bifurcation tongues observed in (a) one bead and (b) two bead systems. The color scale corresponds to maximum velocity amplitude (dB), and it demonstrates that as the mode moves further from its linear frequency, the change in dynamics becomes more drastic. The numerically calculated tongue edge is plotted directly on top of the experimental data as closely spaced solid black dots. Panels (c) and (d) show the computational results for one and two bead chains, respectively (the units are nondimensional). The solid points indicate where a bifurcation has occurred (i.e., a FM has left the unit circle). Red points (at the higher frequency portions for each tongue) indicate gaps have opened between beads. The dashed rectangles indicate the parameter range for the experimental measurements in (a) and (b). The vertical dotted lines correspond to the Floquet diagrams in (e)–(g). We show the unit circle to guide the eye.

Figure 6

The interplay between finite size and dissipation. The points indicate a critical bifurcation amplitude, calculated using numeric. In (a) we hold the dissipation of the system constant $(Q=27)$ and vary the size of the system. The individual tongues begin to overlap and the bifurcations begin to occur at higher amplitudes. In (b) the finite size $(N=15)$ is held constant and the dissipation is varied. For lower dissipations the bifurcation tongues start at lower amplitude. All units shown are nondimensional.