Time-Periodic Stiffness Modulation in Elastic Metamaterials for Selective Wave Filtering: Theory and Experiment

Elastic waveguides with time-modulated stiffness feature a frequency-periodic dispersion spectrum, where branches merge at multiple integers of half the modulation frequency and over a finite wave number range. In this range, frequency becomes complex, with its real part remaining constant. The vanishing group velocity associated with these flat bands leads to frequency-selective reflection at an interface between a nonmodulated medium and a time-modulated one, which converts a broadband input into a narrow-band output centered at the half modulation frequency. This behavior is illustrated in an elastic waveguide in transverse motion, where modulation is implemented experimentally by an array of piezoelectric patches shunted through a negative electrical capacitance controlled by a switching circuit. The switching schedule defines the modulation frequency and allows the selection of the output frequency. This implementation is suitable for the investigation of numerous properties of time-space modulated elastic metamaterials, such as nonreciprocity and one-way propagation, and can lead to the implementation of novel functionalities for acoustic wave devices operating on piezoelectric substrates.

Figure 1

Frequency-periodic dispersion diagrams for a time-modulated waveguide (beam in bending). Black and red lines respectively denote the real part and (the negative of) the imaginary part of frequency. (a) ${\alpha }_m\to 0$: the intersection between the $n=0$ and $n=-1$ occurs at frequency ${\omega }_m/2$ and ${\kappa }^{*}$; the imaginary component is nil as a result of the vanishing modulation amplitude. (b) ${\alpha }_m=0.4$: merging of the dispersion branches at $r{\omega }_m/2$ and corresponding nonzero imaginary frequency. The $\kappa$-band-gap range $\kappa \in [{\kappa }^{-},{\kappa }^{+}]$ predicted by Eq. (5) is highlighted by the shaded blue region.

Figure 2

Concept of a single-port system converting a broadband input into a narrow-band output through time-modulation. An incoming broadband wave is converted into a narrow-band output at a frequency defined by the modulation frequency ${\omega }_m=2\pi /T_m$, which can be used as a tuning parameter for selecting the output frequency content.

Figure 3

Numerical results for the response of the single-port system for three modulation frequencies ($f_m=10$, 12, 15 kHz). (a) Single point FTs show that a broadband input (black solid line) is converted into narrow-band outputs centered at $f_m/2$: $f_m=10\mathrm{kHz}$ (red dashed line), $f_m=12\mathrm{kHz}$ (blue solid line), and $f_m=15\mathrm{kHz}$ (green dash-dotted line). (b)–(d) Normalized 2D-FT magnitude $|\widehat{\mathcal{W}}(\kappa ,\omega )|$ associated with the wave field $w(x,t)$ shows narrow-band frequency reflection of the reflected waves in the $\kappa <0$ half plane at $f_m/2$, which is in agreement with the location of the flat branches predicted theoretically (gray lines).

Figure 4

Experimental setup for observation of the time-modulation of the stiffness in a beam through negative capacitance shunts and switches. (a) The beam is equipped with 11 pairs of piezoelectric patches, each connected to a NC circuit. (b) A switch opens and closes the patch-NC circuit series with a periodic law, inducing the stiffness to vary between two values with period $T_m$.

Figure 5

Experimental results for the response of the beam waveguide for three modulation frequencies ($f_m=10$, 12, 15 kHz). (a) Single point FTs show the conversion of a broadband input (black solid line) into narrow-band outputs centered at $f_m/2$: $f_m=10\mathrm{kHz}$ (red dashed line), $f_m=12\mathrm{kHz}$ (blue solid line), and $f_m=15\mathrm{kHz}$ (green dash-dotted line). (b)–(d) Normalized 2D-FT magnitude $|\widehat{\mathcal{W}}(\kappa ,\omega )|$ associated with the wave field $w(x,t)$ shows a narrow-band frequency reflection of the reflected waves in the $\kappa <0$ half plane at $f_m/2$, which is in agreement with the location of the flat branches predicted theoretically (gray lines).