Selective Mode Conversion and Rainbow Trapping via Graded Elastic Waveguides

We experimentally achieve wave-mode conversion and rainbow trapping in an elastic waveguide loaded with an array of resonators. Rainbow trapping is a phenomenon that induces wave confinement as a result of a spatial variation of the wave velocity, here promoted by gently varying the length of consecutive resonators. By breaking the geometrical symmetry of the waveguide, we combine the wave-speed reduction with a reflection mechanism that mode converts flexural waves impinging on the array into torsional waves traveling in opposite directions. The framework presented herein may open opportunities in the context of wave manipulation through the realization of structural components with concurrent wave-conversion and energy-trapping capabilities.

Figure 1

(a) A graphical representation of the waveguide, which is made of a homogeneous beam rigidly connected to a graded array of resonators that activates the rainbow effect. Due to the broken geometrical symmetry, the waveguide operates as a converter between input $A_0^{+}$ and output $T_0^{-}$ wave modes. (b) Idealization of the conversion mechanism: the system can be conceptually interpreted as two distinct elastic waveguides supporting $A_0$ and $T_0$ wave modes, which are coupled using an array of resonators. (c) A schematic of the reference cell, which consists of identical resonator pairs separated by a non-null distance $\xi$, along with the employed reference system.

Figure 2

(a) The dispersion surface $\omega \left(\mu ,l\right)/2\pi$. For ease of visualization, only the first three dispersion surfaces are displayed. The color is proportional to the polarization factor $p$, which allows us to discriminate between rotation $\varphi$ and vertical displacement. Due to the broken symmetry and local resonance, the second and third dispersion surfaces are separated by a frequency gap. The isofrequency line, highlighted in black, represents the operating frequency of the structure, while the colored version is reported in (b). (c) The group velocity $c_g$ upon varying the length of the resonators $l$ with change step $\mathrm{\Delta }l=0.315\mathrm{mm}$, for the first two dispersion bands. In the figure, the group-velocity line associated with the target dispersion branch is highlighted with darker dots. Such a target branch corresponds to the dispersion tangent to the isofrequency line displayed in (b). The group velocity $c_g\left(l\right)$ computed for the target frequency is superimposed on the colored plot with black dots. (d) The dispersion relation $\omega \left(\mu \right)$ upon varying $l$. (d) The schematic representation with coloured stars illustrates the conversion mechanism that takes place in correspondance of the target resonator. The system is excited through the $A_0^{+}$ wave mode, shown by the red star (i), which is wave number transformed to the cyan star (ii), due to the grading. As the wave reaches the zero-group-velocity region within the second Brillouin zone, it is backscattered to the cyan star (iii) by a quantity equal to the reciprocal lattice vector ${\mu }_m$ and wave number transformed to the back-traveling $T_0^{-}$ wave mode marked with the blue star (iv). The black dots represent the numerical solution computed in comsol Multiphysics.

Figure 3

(a) The numerical displacement field measured at different time instants for an input wave mode $A_0^{+}$, along with the corresponding dispersion relations for a waveguide endowed with absorbing boundaries. The diagrams show that the input wave is reflected and converted from $A_0^{+}$ to $T_0^{-}$ in time. (b)–(d) The spectrograms display the temporal evolution of the spectral content for (b) a waveguide with the graded array of resonators; (c) a waveguide without the initial part of the grading, i.e., with only resonators 7–9; and (d) a homogeneous waveguide, without the array of resonators. The excitation signal is shown alongside the spectrogram. The input wave is confined, trapped, and delayed only if the complete array of resonators is present. (e)–(g) The corresponding out-of-plane displacement field filtered to separate the contribution of the $A_0$ wave mode.

Figure 4

The experimental setup employed to measure wave propagation. The waveguide is excited using an electrodynamic shaker (a), while the opposite side is clamped (b). Wave propagation is measured on the surface of the waveguide through a Polytec 3D scanner laser doppler vibrometer (c), which is able to separate the 3D velocity field. The top view of the graded array is also provided (d).

Figure 5

(a) The experimental velocity field and the corresponding dispersion relation measured at different time instants. The (b) experimental and (c) numerical spectrograms, illustrating the evolution of the spectral content in time. The experimental and numerical results are characterized by a comparable temporal delay for the $A_0^{-}$ wave mode, while different conversion amplitudes are observed, due to the presence of imperfections and misalignments that enhance the torsional coupling.