Linear and Nonlinear Elastic Waves in Magnetogranular Chains

We study a magnetogranular chain composed of stainless-steel beads which are placed inside an appropriately designed periodic magnetic field. The latter provides attractive forces between the particles, leading to a stable structure, free of mechanical boundaries. Using a scanning laser-based probe at the individual-particle level, we observe experimentally the propagation of longitudinal as well as transverse-rotational waves. In addition, we obtain the dispersion band diagram. In the linear regime, these observations are well supported by a mass-spring model that takes into account both a normal and a shear mechanical coupling between the beads, considering translational and rotational degrees of freedom. In the weakly nonlinear regime, we present experimental results including beating in the amplitude of the second harmonic for the longitudinal waves and propagation under oblique driving excitation. A theoretical model that takes into account the Hertzian contact mechanics, dissipation, and the finite size of the system captures well the results for second-harmonic generation with longitudinal waves. This magnetosensitive system offers great freedom to design complex waveguide geometries where the interplay between geometry, wave polarization, and nonlinearity may pave the way toward the development of advanced signal-processing elastic devices.

Figure 1

Schematic illustration of the magnetogranular phononic crystal, where a magnetic field, created by the permanent magnets, induces attractive forces between the particles.

Figure 2

(a) Schematic illustration of the contact between two elastic particles subjected to normal forces $F_0$ and transverse forces $T_0$. (b) Schematic presentation of the normal, shear, and rotational interactions.

Figure 3

(a) Dispersion relation for an infinite granular chain. The cyclic frequency $\omega$ is normalized by the linear longitudinal cutoff cyclic frequency ${\omega }_c=\sqrt{4K_N/m}$, namely $\mathrm{\Omega }=\omega /{\omega }_c$, and the lattice constant is $a=2R$. (b) Relative amplitude (expressed as a percentage) of transverse motion ($u$) and rotation ($\varphi$) in the upper $TR$ branch as a function of the normalized frequency. (c),(d) Normal mode of the upper $TR$ branch for a chain of 20 beads, assuming periodic boundary conditions at (c) $k=0$ and (d) $k=\pi /a$, for the transverse and rotational components.

Figure 4

(a) Diagram of the instrumentation. (b) Photograph of the 38-bead chain on top of the rubber substrate, with permanent magnets inside it. (c) Schematic illustration of the measurement positions of the laser vibrometers.

Figure 5

One period of the “haversine” electric-signal excitation used to derive the dispersion curves, in (a) temporal and (b) frequency domain. (c) Experimental and (d) numerical velocities (${\dot{u}}_x$) for longitudinal-wave propagation of a pulse as a function of time on a chain of 38 beads. (e) Experimental and (f) numerical dispersion curves with $F_0=1.8$ N and ${\tau }_x=2.4$ ms (${\delta }_0=325$ nm). Analytical dispersion curves are superimposed.

Figure 6

(a) Experimental and (b) numerical velocities (${\dot{u}}_y$) for transverse-wave propagation of a pulse as a function of time in a chain of 38 beads. (c) Experimental and (d) numerical dispersion curves with $F_0=1.4$ N and ${\tau }_y=2$ ms. Analytical dispersion curves are superimposed.

Figure 7

(a) Experimental spatiotemporal diagram of the velocity ${\dot{u}}_x$ of the particles during the propagation of a sine wave with a frequency of 3 kHz ($\omega /{\omega }_c=0.42$). (b) Numerical spatiotemporal diagram of the velocity ${\dot{u}}_x$ with $F_0=1.55$ N and ${\tau }_x=2.4$ ms. (c) Velocity ${\dot{u}}_x$ as a function of time for (left) the first bead and (right) the last bead of the chain.

Figure 8

(a) Experimental spatiotemporal diagram of the velocity ${\dot{u}}_y$ of the particles during the propagation of a sine wave with a central frequency of 8 kHz. (b) Transverse part of the numerical spatiotemporal diagram with $F_0=1.55$ N and ${\tau }_y=0.93$ ms. (c) Velocity ${\dot{u}}_y$ and (d) angular velocity ${\dot{\varphi }}_z$ as a function of time for (left) the first bead and (right) the last bead of the chain.

Figure 9

Comparison of the theoretical result obtained from Eq. (20) (lines) and a numerical simulation of Eq. (8a) (circles). Normalized amplitude of the fundamental (top) and second harmonic (bottom) for $\epsilon =0.05$ and $\mathrm{\Omega }=0.41$. The black circles and blue lines correspond to a dissipative term $\tau =1.5$ ms, while the red circles and orange lines correspond to $\tau =5$ ms. The vertical dashed lines depict the analytical predictions for the disappearance of the second-harmonic contribution obtained from Eq. (21).

Figure 10

(a) Spatiotemporal representation of the numerical displacement of the particles as a function of time. (b) Normalized amplitude of the fundamental and second harmonic for $\mathrm{\Omega }=0.42$ and $\epsilon =0.1$ when a reflected wave is present. The solid lines correspond to the analytical solution obtained from Eq. (20), and the markers to the numerical results. The blue circles and lines correspond to a lossless case, and the black diamonds and lines to a dissipative case with $\tau =10$ ms. Gray zones correspond to the presence of reflection during the analysis. The vertical dashed lines are analytical predictions for the disappearance of the second-harmonic contribution obtained from Eq. (21).

Figure 11

Experimental, numerical, and analytical harmonic generation for driving frequencies of (a) 3 kHz (${\omega }_1/{\omega }_c=0.41$), (b) 5 kHz (${\omega }_1/{\omega }_c=0.69$), (c) 3.2 kHz (${\omega }_1/{\omega }_c=0.44$), and (d) 3.4 kHz (${\omega }_1/{\omega }_c=0.47$). The black circles, red diamonds, and solid lines correspond to numerical, experimental, and analytical results, respectively. The vertical dashed lines are given by Eq. (21). Gray zones correspond to the experimental presence of reflection during the analysis.

Figure 12

Experimental and numerical harmonic generation for a driving frequency of 4 kHz (${\omega }_2/{\omega }_c=0.58$). The longitudinal polarization ($u_x$ component) and the transverse polarization ($u_y$ component) are presented in (a) and (b), respectively. The black circles correspond to numerics and the red diamonds to experimental results. Gray zones correspond to the experimental presence of reflection during the analysis.