Edge States and Topological Pumping in Spatially Modulated Elastic Lattices

Spatial stiffness modulations defined by the sampling of a two-dimensional surface provide one-dimensional elastic lattices with topological properties that are usually attributed to two-dimensional crystals. The cyclic modulation of the stiffness defines a family of lattices whose Bloch eigenmodes accumulate a phase quantified by integer valued Chern numbers. Nontrivial gaps are spanned by edge modes in finite lattices whose location is determined by the phase of the stiffness modulation. These observations drive the implementation of a topological pump in the form of an array of continuous elastic beams coupled through a distributed stiffness. Adiabatic stiffness modulations along the beams’ length lead to the transition of localized states from one boundary, to the bulk and, finally, to the opposite boundary. The first demonstration of topological pumping in a continuous elastic system opens new possibilities for its implementation on elastic substrates supporting surface acoustic waves, or to structural components designed to steer waves or isolate vibrations.

Figure 1

(a) Spatially modulated 1D lattices with modulated stiffness $k_n=k_0[1+\alpha \cos (2\pi np/q+\varphi )]$. (b) Surface $\mathcal{S}(x,\varphi )=\cos (2\pi \tau x+\varphi )$ generating the stiffness constants by sampling at $x_n=n$. Black lines: stiffness variation with $\varphi$ at a lattice site; blue lines: cross sections at ${\varphi }_r=(2\pi r/3)$ ($r\in [0,3]$) with the red dots showing the sampled stiffness values.

Figure 2

Dispersion properties for $p/q=1/3$, $\alpha =0.6$ lattice. (a) Dispersion surfaces as a function of $\mu$ and $\varphi$ showing three bulk bands and two band gaps. Corresponding Chern numbers and gap labels are added for convenience. (b) Natural frequencies of a commensurate finite chain of 60 masses and their variation in terms of $\varphi$ (black lines) superimposed to the bulk bands (shaded gray regions), where edge modes (red lines) span the gaps. (c) Magnitude of the topological mode in the second gap and its the transition from right to left localization resulting from the variation of $\varphi$.

Figure 3

Dispersion properties for system of coupled beams. (a) Schematic of the lattice of coupled beams: thick black lines show the beams in a notional deformed configuration, while the red lines correspond to the undeformed state. The thin black horizontal lines depict the distributed springs of constant ${\gamma }_n$ coupling the beams. (b) Dispersion $\mathrm{\Omega }(\varphi ,{\beta }_y)$ for array comprising $N=15$ identical beams with stiffness parameters $p/q=1/3$, $\alpha =0.6$. (c) Wave number ${\beta }_y$ as function of $\varphi$ for fixed frequency $\mathrm{\Omega }=2.1$, corresponding to the cross section illustrated by the blue plane in (b). The contours represent the Fourier decomposition of the displacement field $|\widehat{W}(\tau ,{\beta }_y)|$, revealing the adiabatic evolution experienced by the waves.

Figure 4

Topological pumping through adiabatic evolution of the edge state. Snapshots of three time instants illustrate the transition from left-localized, to bulk and finally to right-localized mode as the wave propagates along the $y$ direction. Deformed configurations show the absolute value of beam deflections $|w_n(y,t)|$, represented along the vertical direction and also quantified by the associated color map for ease of visualization.