Nonlinear Hysteretic Torsional Waves

We theoretically study and experimentally report the propagation of nonlinear hysteretic torsional pulses in a vertical granular chain made of cm-scale, self-hanged magnetic beads. As predicted by contact mechanics, the torsional coupling between two beads is found to be nonlinear hysteretic. This results in a nonlinear pulse distortion essentially different from the distortion predicted by classical nonlinearities and in a complex dynamic response depending on the history of the wave particle angular velocity. Both are consistent with the predictions of purely hysteretic nonlinear elasticity and the Preisach-Mayergoyz hysteresis model, providing the opportunity to study the phenomenon of nonlinear dynamic hysteresis in the absence of other types of material nonlinearities. The proposed configuration reveals a plethora of interesting phenomena including giant amplitude-dependent attenuation, short-term memory, as well as dispersive properties. Thus, it could find interesting applications in nonlinear wave control devices such as strong amplitude-dependent filters.

Figure 1

(a) Experimental configuration for the torsional wave propagation (pure rotation of the beads around the $z$ axis of the chain). (b) Signal acceleration at bead 15 versus frequency for a 15-bead chain. (c) Dispersion relation retrieved from the $p$ resonance frequencies of the chain.

Figure 2

Magnitude of the acceleration transfer function (detected acceleration $A_1$ at bead 1 over the acceleration $A_0$ at bead 0) of a “two-beads–one-contact” system for different excitation levels, experiment (a) and theory (b). (c) Relative resonance frequency shift as a function of the detected resonance amplitude (normalized to the one at the maximum excitation level). (d) Setup for the single contact characterization in torsion.

Figure 3

Experimental angular particle velocity ${\overline{\theta }}_v={\theta }_v/{\theta }_0$ at bead 1 for the smallest excitation amplitude and its approximation by a linear piecewise function. $T_n$’s are the positions of the signal zeros in time normalized by $t_0=5.77\mathrm{ms}$ so that $T_1=1$, ${\tau }_n$’s are the extrema positions in time, and $S_n$’s are the extrema amplitudes normalized by ${\theta }_0$ so that $S_1=1$.

Figure 4

(a) Experimental angular velocity signals for different values of $\xi$: at bead 1 ($\xi =0$ dashed line) and at bead 18 for different excitation amplitudes ($\xi =0.5$, 1, 1.3, 1.8, and 2.4 for $h=800$). (b) Corresponding theoretical angular velocity signals. (c) Experimental (symbols) and theoretical (lines) amplitudes of the four first signal extrema versus $\xi$. (d) Experimental (symbols) and theoretical (lines) durations of the three first signal phases versus $\xi$. Error bar limits represent simulated results for $h=640$ and 960.