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Topological dynamics of a singing saw

5 May 2022

New work demonstrates a connection between the dynamics of thin shells and topological insulators.

A carpenters saw along side a violin bow and drumstick. Then two images of the saw bent in one-direction and in two-directions
Credit: Adapted from S. Shankar, P. Bryde, L. Mahadevan, Proc. Natl. Acad. Sci. USA 119, e2117241119 (2022)

A carpenter’s saw, shown in panel a in the figure above, is unusual in that when manually contorted, it can be bowed as a musical instrument. For a note to emerge when the saw is bowed or struck with a mallet, its blade should not be bent into a constant-curvature J shape (panel b), but rather into an S shape (panel c), whose curvature changes sign. The geometric distortion lets the saw “sing.” Bowing the blade near the sweet spot in the curve of the S-shaped blade produces clear, sustained notes, whereas bowing it far from that spot deadens them. To characterize what accounts for that behavior, previous researchers had analyzed the vibrational modes of a thin elastic shell and shown that localized and long-lived vibrational eigenmodes emerge at the inflection line in a shell with spatially varying curvature. But a deeper understanding of the localization has remained elusive.

In recent work, Harvard University’s Suraj Shankar (a junior fellow), Petur Bryde (a graduate student), and L. Mahadevan (their research adviser) asked themselves how geometry can be used to trap and insulate those acoustic modes from dissipative decay. Musicians have used the fact that although pitch can be varied by changing the curvature of the saw, the sustained quality of a note is largely indifferent to the actual curvature profile or to details of the clamps, so long as an inflection line exists.

The lack of sensitivity to those details convinced the Harvard researchers that the localized mode responsible for the saw’s sonority must have a topological origin. Using a combination of theoretical analysis, simulations, and experiments, they demonstrated that varying the spatial curvature in a thin shell can localize vibrational modes at inflection lines. The analysis showed that the inflection line behaves like an internal “edge” with the vibrational mode topologically trapped, akin to an edge state in topological insulators. (See “Topological insulators: from graphene to gyroscopes,” Physics Today online, 27 November 2018.)

Two plots of normalized amplitude over frequency for the different shapes of the saw. The first showing a peak at 85 Hz and the other at just below 595 Hz
Credit: Adapted from S. Shankar, P. Bryde, L. Mahadevan, Proc. Natl. Acad. Sci. USA 119, e2117241119 (2022)

In plots of the density of states, the researchers calculated the distribution of those modes as a function of frequency for both a constant-curvature shell and for an S-shaped shell. Whereas the relevant modes remain spectrally gapped and delocalized over the entire saw in the constant-curvature case, new ones appear within the spectral gap that opens in the S-shaped case. The midgap modes turn out to be trapped in the interior, and the localized mode loses energy slowly through the clamped boundaries of the saw. In the second figure, for instance, notice the differences in a saw’s Q factor—a measure of how efficiently the resonator stores energy—when it has a J shape (panel a) and when it has an S shape (panel b). Although the plots are normalized, the S-shaped saw exhibits a dramatic 15-fold enhancement in Q factor of its peak localized mode (marked with the dashed line) over delocalized modes in the J shape.

Because the researchers’ findings are independent of scale and material, they apply equally well to designing nanoscale electromechanical resonators that may be used as sensors or electronic components. Whether the object in question is a macroscopic metal sheet or an atomically thin graphene ribbon, topological modes can be tuned—forced to remain vibrationally isolated and decay extremely slowly—by the simple act of changing the object’s shape. (S. Shankar, P. Bryde, L. Mahadevan, Proc. Natl. Acad. Sci. USA 119, e2117241119, 2022.)

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