Geometric control of topological dynamics in a singing saw

The saw is modeled as a very thin rectangular elastic shell (thickness $h\ll W<L$, where W, L are the width and length of the strip) made of a material with Young’s modulus Y, Poisson ratio ν, and density ρ (Fig. 1E). Its geometry is characterized by a spatially varying curvature tensor (second fundamental form) $\mathbf{b}(\mathbf{x})$, where $\mathbf{x}=(x,y)$ is the spatial coordinate in the plane. As the saw is bent only along the (long) x axis, $b_{xx}(x)\equiv b(x)$ is the sole nonvanishing curvature. To describe its dynamical response, we take advantage of its slenderness and treat the saw as a thin elastic shell that can be bent, stretched, sheared, and twisted. Before moving to a computational model that accounts for these modes of deformation as well as real boundary conditions, to gain some insight into the problem and expose the topological nature of elastic waves, it is instructive to instead consider a simplified description valid for shallow shells with slowly varying curvature.

In a thin shallow shell ($h\left|b_{ij}\right|\ll 1$), as bending is energetically cheaper than stretching (30), shear becomes negligible ($\mathbf{Q}\approx \mathbf{0}$; Fig. 1E), and in-plane deformations propagate much more rapidly (at the speed of sound $c=\sqrt{Y/\rho }$) so that the depth-averaged stresses can be assumed to equilibrated, i.e., ${\partial }_j{\sigma }_{ij}=0$ (6, 31). In this limit, using the solution of these equations in terms of the Airy stress function χ (${\sigma }_{ij}={\mathcal{P}}_{ij}{\nabla }^2\chi$, where ${\mathcal{P}}_{ij}={\delta }_{ij}-{\partial }_i{\partial }_j/{\nabla }^2$ is a projection operator; SI Appendix, section 3), the in-plane geometric compatibility relation and the linearized dynamical equations for transverse motions can be written as (7, 32)

Here f is the out-of-plane deflection of the shell (Fig. 1E) and the bending rigidity $\kappa =Y{h}^3/[12(1-{\nu }^2)]$. Crucially, in-plane and flexural (out-of-plane) modes remain geometrically coupled in the presence of curvature even in the linearized setting (Eqs. 1 and 2). For a shell bent with constant curvature along the x axis, i.e., a section of a uniform cylinder, $b(x)=b_0$ is a constant. In the bulk of the system, disregarding boundaries, we can Fourier transform Eqs. 1 and 2 using the solution ansatz $f=f_{\mathbf{q}}{e}^{-i\omega t+i\mathbf{q}·\mathbf{x}}$ to obtain the dispersion relation for flexural waves to be ${\omega }_{\pm }(\mathbf{q})=\pm \sqrt{(\kappa /\rho h)q^4+c^2{b}_0^2{(q_y/q)}^4}$ (Fig. 2A), where $q=\left|\mathbf{q}\right|$. When q_y = 0, i.e., the sheet is undeformed in the transverse direction, it remains developable (with generators that run parallel to the y direction), and the bending waves are gapless, i.e., $\omega \to 0$ as $q\to 0$. However, when $q_y\ne 0$, a finite frequency gap $\sim c\left|b_0\right|$ (in addition to finite q_y corrections) controlled by the speed of sound and the curvature of the shell emerges as $q\to 0$ (Fig. 2A). Intuitively, this arises due to the geometric coupling between bending and stretching deformations in a curved shell which leads to an effective stiffening that forbids wave propagation below a frequency threshold. Similar spectral gaps appear in curved filaments and doubly curved shells as well (31, 33).

For the S-shaped saw, curvature scales of $b\sim$0.4 to $0.8 {\text{m}}^{-1}$ are easily achievable (as in Fig. 1 B and C), while the typical sound speed in steel is $c\sim$5 to $6\times {10}^3$m/s so that the frequency gap is of order 2 to $5$kHz. Comparing these estimates to the spectrogram in Fig. 1D (further quantified in Fig. 3) suggests that the localized mode excited upon bowing the S-shaped saw (Fig. 1C) lies within the frequency gap. The J-shaped saw (Fig. 1B) also vibrates at low frequencies (compared to the gap) when struck, presumably through the q_y = 0 branch of delocalized flexural modes, although higher frequencies above the band gap can be excited by careful bowing (SI Appendix, Fig. S1 A and B).

To unveil the topological structure of the vibration spectrum of the saw, we cast the second-order dynamical equations (Eqs. 1 and 2) in terms of first-order equations by taking the square root of the dynamical matrix (14, 34). Focusing on the flexural modes alone, we obtain a Schrödinger-like equation for the transverse deflections of a shallow shell (SI Appendix, section 3),where $\mathrm{\Psi }=(c{\mathcal{D}}^{†}f,\hspace{0.17em}i{\partial }_{t}f)$ and $\mathcal{D}=i\sqrt{\kappa /Yh}{\nabla }^2+b_{ij}{\mathcal{P}}_{ij}$ and $†$ represents the conjugate transpose. The eigenvalues of the effective Hamiltonian $\mathcal{H}$ are given by the previously derived (${\omega }_{\pm }(\mathbf{q})$), and its complex eigenvectors ${\mathrm{\Psi }}_{\pm }(\mathbf{q})$ encode the topology of the band structure. The singularities in the arbitrary phase of the eigenvectors signals nontrivial band topology. To understand the phase of eigenvectors along the saw’s long direction, we can consider fixing the transverse wave vector $q_y\ne 0$, leading to an effective one-dimensional (1D) system along the x axis. Then the obstruction to continuously define the phase of the eigenvectors at every q_x in Fourier space while respecting all the symmetries of the problem is quantified by the 1D Berry connection $\mathcal{A}(q_x)=i{\displaystyle {\sum }_{n=\pm }{\mathrm{\Psi }}_n}{(q_x)}^{†}{\partial }_{q_x}{\mathrm{\Psi }}_n(q_x)$ (the q_y dependence is suppressed) (35, 36). However, what are the symmetries of our elastodynamic system?

One important symmetry is that imposed by classical time-reversal invariance in a passive, reciprocal material ($\mathcal{C}$: $x\to x,\hspace{0.17em}t\to -t,\hspace{0.17em}\mathrm{\Psi }\to {\mathrm{\Psi }}^{*}$; SI Appendix, section 3), which maps forward moving waves into backward moving ones and guarantees that eigenmodes appear in complex-conjugate pairs (34). A second symmetry special to the saw is an emergent spatial reflection symmetry in the local tangent plane (Π: $x\to -x,\hspace{0.17em}t\to t,\hspace{0.17em}\mathrm{\Psi }\to \mathrm{\Psi }$; SI Appendix, section 3), which originates from the uniaxial nature of the prescribed curvature along the x axis and the insensitivity of bending to the orientation of the local tangent plane, a symmetry that is inherited from 3D rotational invariance. The latter is easily seen by noting that the bending energy only involves an even number of gradients via ${\nabla }^{2}f$. Upon simultaneously enforcing both dynamical and spatial symmetries, a new topological obstruction posed by curvature emerges and is quantified by a ${\mathbf{Z}}_2$ index (SI Appendix, section 3),similar to topological insulators with crystalline symmetries (37–39). $\text{Pf}[\mathbf{W}]$ denotes the Pfaffian of the antisymmetric overlap matrix $W_{ij}(q_x)={\mathrm{\Psi }}_i{(q_x)}^{†}\mathcal{C}\mathrm{\Pi }{\mathrm{\Psi }}_j(q_x)$ ($i,j=\pm$). We note that unlike the mechanical Su–Schrieffer–Heeger chain (14) that exhibits a topological polarization in 1D, the emergent tangent-plane spatial reflection symmetry in our problem forces this polarization to vanish (SI Appendix, section 3).

As we work in the continuum, only differences in the topological invariant are well defined independent of microscopic details. Across an interface at which curvature changes sign, i.e., a curvature domain wall, the jump in the topological invariant is given bywhere $b_{<}$ and $b_{>}$ are the curvature on either side of the interface (SI Appendix, section 3). This expression directly demonstrates that the two oppositely curved sections of the saw behave as topologically nontrivial bulk systems, with a $\mathrm{\Delta }\nu =1$, that meet at the inflection line that functions as an internal edge. As a result, nontrivial band topology underlies the emergence of the localized midgap mode, endowing it with robustness against details of the curvature profile and weakly nonlinear deformations (SI Appendix, section 3).

We test these predictions by numerically computing the eigenmodes of a finite elastic strip of length $L=1$m, width $W=0.25$m, and thickness $h={10}^{-3}$m. For our shell model, we move away from the Kirchhoff model for shells and account for the kinematics associated with shear in addition to those associated with bending and stretching, as they effectively reduce the numerical ill-conditioning commonly seen in high-order continuum theories for slender plates and shells while allowing for numerical methods that require less smoothness and are easier to implement (SI Appendix); together, these allow for better computational accuracy. This framework forms the basis for the Naghdi shell model (40) (see SI Appendix, section 2, for details) and accounts for an in-plane displacement vector along the midsurface $\mathbf{u}(\mathbf{x},t)$, an out-of-plane deflection $f(\mathbf{x},t)$ normal to the shell, and an additional rotation $\mathit{\theta }(\mathbf{x},t)$ of the local normal itself (Fig. 1E). These modes of deformations lead to depth-averaged stress resultants associated with stretching ($\mathit{\sigma }$), bending (M), and shear (Q) as shown in Fig. 1E. The resulting covariant nonlinear shell theory along with inertial Newtonian dynamics provides an accurate and computationally tractable description of the elastodynamics of thin shells (Fig. 1E and SI Appendix, section 2). To highlight the topological robustness of our results, in our calculations we vary both the boundary conditions and curvature profiles imposed on the saw.

In Fig. 2B, the distribution of eigenmodes as a function of frequency is shown in the integrated density of states for a constant curvature shell, $b(x)=b_0$ (dashed lines), and an S-shaped shell with a smooth curvature profile $b(x)=b_0\tanh (x/\ell )$ (solid lines) that varies over a width $\ell$ near the inflection point at x = 0 (i.e., a curvature domain wall). In both cases, the ends of the strip are kept clamped, and the spectra are calculated using an open-source code based on the finite element method (41, 42). As the curvature of the S shape approaches a constant $\pm b_0$ far from the origin, the bulk spectral gap and delocalized modes match that of the constant curvature case. Flexural modes that vary at most linearly in the y direction* (labeled by discrete mode numbers m = 0, 1 due to the lack of translational invariance) correspond to linearized isometries; they delocalize over the entire ribbon (Fig. 2 C, Top and Middle) and populate states all the way to zero frequency, i.e., with a gapless spectrum. This is true for both constant curvature (dashed blue line, Fig. 2B) and the S-shaped shell (solid blue line, Fig. 2B) as these bulk modes are unaffected by curvature. In contrast, all other modes that bend in both directions ($m\ge 2$) are generically gapped for a constant curvature profile (dashed red line, Fig. 2B) as expected. However, for the S shape, in addition to the gapped bulk modes, a new mode appears within the spectral gap (solid red line, Fig. 2B). This midgap state (shown here for m = 2) is a localized mode that is trapped in the neighborhood of the inflection line (Fig. 2 C, Bottom). For increasing mode number $m\ge 2$, similar topological modes appear within the bulk bandgap, with growing localization lengths (Fig. 2 D, Inset) and higher frequencies (Fig. 2D), as predicted analytically (SI Appendix, section 3). Qualitatively, the presence of an inflection line in the S-shaped saw makes it geometrically soft there; the generators of cylindrical modes are now along the length of the saw, and the curved regions on either side that are geometrically stiff serve to insulate the soft internal edge from the real clamped edges.

Of particular note is that the localized modes, unlike the extended states, are virtually unaffected by the boundaries and the conditions there (see SI Appendix, Fig. S2A, for eigenmodes in a strip with asymmetric boundary conditions where the left edge is clamped and the right edge is free). Spatial gradients in curvature, however, do impact the extent of localization. We demonstrate this using a piecewise continuous curvature profile that has a constant linear gradient $b\prime$ over a length $\ell$ across the origin and adopts a constant curvature outside this region. By varying both the curvature gradient $b\prime$ and the length scale $\ell$, we can tune the localization of the lowest topological mode (same as Fig. 2 C, Bottom), quantified by the inverse participation ratio $\text{IPR}=\int \text{d}\mathbf{x}|f(\mathbf{x}){|}^4/{(\int \text{d}\mathbf{x}|f(\mathbf{x}){|}^2)}^2$ (Fig. 2E). Strong localization (high IPR) is quickly achieved for sharp gradients in curvature ($\text{IPR}\propto \sqrt{|b\prime |/h}$;SI Appendix, section 3) as long as the length scale of curvature variation is not too small ($\ell /L\ge 0.1$, Fig. 2E), corresponding to a diffuse domain wall. In the opposite limit of $\ell \to 0$ for $b\prime \ell =b_0$ fixed, i.e., a sharp domain wall with a discontinuous curvature profile $b(x)=b_0\hspace{0.17em}\text{sgn}(x)$, strong localization persists (SI Appendix, Fig. S3), consistent with our topological prediction and demonstrating the ease of geometric control of localization.

The boundary insensitivity of topologically localized modes has important dynamic consequences that can be harnessed to produce high-quality resonators. The primary mode of dissipation in the saw, as in nanoelectromechanical devices (43), is through substrate or anchoring losses at the boundary. Internal dissipation mechanisms (from, e.g., plasticity, thermoelastic effects, and radiation losses), although present, are considerably weaker and neglected here. To model dissipative dynamics, we retain clamped boundary conditions on the left end and augment the right boundary to include a restoring spring (k) and dissipative friction (γ) for both the in-plane forces and bending moments (Fig. 3 A, Inset, and SI Appendix, section 2). Informed by Fig. 2E, we choose a linear curvature profile spanning the entire length of the shell to obtain a strongly localized mode. Upon driving the shell into steady oscillations, with a periodic point force applied at the inflection point (x = 0; Fig. 3A, red curve), we see an extremely sharp resonance peak right at the frequency of the first localized mode (Fig. 3A). In contrast, when the shell is driven closer to the boundary ($x=0.4L$; Fig. 3A, black curve), the response is at least six orders of magnitude weaker as the localized mode is not excited and only the delocalized modes contribute. Localization hence protects the mode from dissipative decay, unlike extended states that dampen rapidly through the boundaries. We further quantify this using a Q factor computed from undriven relaxation of the shell initialized in a given eigenmode (SI Appendix, section 2). Ultrahigh values of $\text{Q}\approx {10}^5$ to 10⁶ are easily attained when a localized mode is excited (Fig. 3B, red), well over the Q factor of all other modes (Fig. 3B, blue). Similar results are obtained for other curvature profiles as well, such as a sigmoid curve (SI Appendix, Fig. S2B).

To compare these computational results with experiments, we perform ringdown measurements on a musical saw (see SI Appendix, section 1, for details) clamped in both the J shape (Fig. 1B) and the S shape (Fig. 1C). As indicated by Eq. 5, the key distinguishing feature of the S-shaped saw (compared to the J shape) is the presence of an inflection line (curvature domain wall) that engenders a well-localized domain wall mode capable of sustaining long-lived oscillations. The normalized Fourier spectra and exponential decay (τ) of the signal envelope are shown in Fig. 3C (J shape) and Fig. 3D (S shape) with the dominant frequency (ω₀) marked. We find a factor ∼15 enhancement in the Q factor ($\text{Q}={\omega }_0\tau /2$) for the S-shaped saw ($\text{Q}\approx 150$; Fig. 3 D, Left) over the J shape ($\text{Q}\approx 10$; Fig. 3 C, Left). We emphasize that this significant quality factor improvement, although not as dramatic as the numerically computed Q factors (Fig. 3B), is still striking given the initial impulse (mallet strike for J shape and bow for S shape; see SI Appendix, Fig. S1, for other cases) excites an uncontrolled range of frequencies and other sources of energy loss including internal damping are presumably also present.

The wooden handle of the musical saw (Wentworth) was clamped onto an optical table, while the tapered end of the blade was attached to a sliding metal bracket mounted onto a vertical guide rail. Cork discs (around $2$cm in diameter and 0.5 to 1 cm in thickness) were used to cushion and softly support the clamped end of the blade. This helped damp out oscillations at the saw end and reduced any high-frequency ringing arising from direct metal-on-metal contact. The saw blade was bent into two configurations, a J shape (Fig. 1B) and an S shape (Fig. 1C), and manually either struck with a mallet or bowed with a violin bow at the straight edge, both near the center of the blade. The blade was allowed to freely ringdown postexcitation. The audio was recorded using a USB microphone (Fifine technology, K669-K669B, sampling frequency $f_s=44.1$kHz) placed near the saw and analyzed using the software Audacity.

Multiple measurements of the ringdown signal, each lasting 5 to 6 s, were made with a gap of a few seconds between runs. A separate 10 to 15 s audio sample with a stationary saw was used as a template to filter any background noise using the in-built noise reduction functionality in Audacity. The denoised audio samples were then analyzed using a custom Python code. Both left and right (stereo) channels are strongly correlated with each other, so we simply averaged the two to get the signal for each run. Upon using a Hann window and Fourier transforming each signal, we binned the frequency axis with a bin size of $\mathrm{\Delta }\omega =5$Hz and averaged the normalized (by the maximum) magnitude of the Fourier transform over different runs (N = 26: J shape, mallet; N = 28: S shape, bow).

The average spectrum (normalized) is plotted in Fig. 3 C and D, Top, with the shaded region corresponding to the SE over the independent runs. The spectrograms in Fig. 1 D, Middle and Right, were computed for individual audio signals using matplotlib’s specgram function with options NFFT $=512$Hz (number of fast Fourier transform data points per block), pad_to $=8,192$Hz, and noverlap $=256$Hz. In order to compute the decay time of the sound, we normalized each time series by its maximum (in magnitude) and lined them up so t = 0 is at the maximum of the signal. We averaged the absolute value of the temporally aligned signals over independent runs and performed an additional moving average over a time step $\mathrm{\Delta }t=0.025$s to smooth out all the high-frequency oscillations, leaving behind only the required envelope. This smoothed average curve (once again normalized by its maximum) is shown in dark blue in Fig. 3 C and D, Bottom. A similar calculation and smoothing is also done for the SE computed over independent runs and is plotted as the shaded region about the average. The smoothed average time series is then fit to an exponential function with a constant offset using SciPy’s in-built nonlinear curve fitting function. The errors on our estimate for the dominant frequency (ω₀) and the decay time (τ) arise primarily from the chosen resolution of our smoothing windows ($\mathrm{\Delta }\omega ,\hspace{0.17em}\mathrm{\Delta }t$) as other sources of measurement error are much smaller. We have nonetheless checked that our choice of the window size ($\mathrm{\Delta }\omega ,\hspace{0.17em}\mathrm{\Delta }t$) is optimum as changing it by small amounts does not affect our results, but decreasing $\mathrm{\Delta }t$ by an order of magnitude significantly degrades the exponential fit.