Defect modes in one-dimensional granular crystals

We study the vibrational spectra of one-dimensional statically compressed granular crystals (arrays of elastic particles in contact) containing light-mass defects. We focus on the prototypical settings of one or two spherical defects (particles of smaller radii) interspersed in a chain of larger uniform spherical particles. We present a systematic measurement, using continuous noise, of the near-linear frequency spectrum within the spatial vicinity of the defect(s). Using this technique, we identify the frequencies of the localized defect modes as a function of the defect size and the position of the defects relative to each other. We also compare the experimentally determined frequencies with those obtained by numerical eigenanalysis and by analytical expressions based on few-site considerations. These approximate analytical expressions, based on normal-mode analysis, are found to be in excellent agreement with numerics for a wide range of mass ratios. We also observe that the experimentally measured frequencies of the localized defect modes are uniformly upshifted, compared to the numerically and theoretically predicted values.

Figure 1

(a) Schematic diagram of the experimental setup for the homogeneous chain with a single-defect configuration. (b) Experimental transfer functions (as defined in the “single defect” section) for a granular crystal with a static load of $F_0=20$ N and a defect bead of mass $m=5.73$ g located at site $n_{\text{def}}=2$. Blue (dark gray) [red (light gray)] curves correspond to the transfer function obtained from the force signal of a sensor particle placed at $n=4$ [$n=20$]. The diamond marker is the identified defect mode. The triangle marker is the identified upper acoustic cutoff mode. The vertical black dashed (solid) line is the theoretically predicted defect-mode frequency (acoustic band cutoff frequency).

Figure 2

Frequency of the defect mode, with the defect bead placed at $n_{\text{def}}=2$, as a function of mass ratio $m/M$. Solid blue line (dark gray, closed diamonds) corresponds to experiments, solid black line (open diamonds) to numerically obtained eigenfrequencies [see Eq. (3)], and green dashed line (light gray, x markers) to the analytical prediction of the three-bead approximation [see Eq. (4)]. The error bars account for statistical errors on the measured frequencies and are $\pm 2\sigma$. Inset: The normalized defect mode for $\frac{m}{M}=0.2$.

Figure 3

(a) Experimental transfer functions for a granular crystal with two defect beads of mass ratio $\frac{m}{M}=0.2$ located at $n_{\text{def}}=2$ and $n_{\text{def}}=3$ (in contact). The blue (dark gray) [red (light gray)] curve corresponds to the transfer function obtained from the force signal of a sensor placed at $n=4$ [$n=20$]. (b) Frequencies of the defect modes as a function of the distance between them. The solid line denotes experimental data, the dashed line the numerically obtained eigenfrequncies, and the x markers the frequencies from the analytical expressions of Eqs. (5, 6). (c, d) The normalized defect-mode shapes corresponding to the defect modes identified in (a) with frequency of the same marker type.