Metastable modular metastructures for on-demand reconfiguration of band structures and nonreciprocal wave propagation

We present an approach to achieve adaptable band structures and nonreciprocal wave propagation by exploring and exploiting the concept of metastable modular metastructures. Through studying the dynamics of wave propagation in a chain composed of finite metastable modules, we provide experimental and analytical results on nonreciprocal wave propagation and unveil the underlying mechanisms that facilitate such unidirectional energy transmission. In addition, we demonstrate that via transitioning among the numerous metastable states, the proposed metastructure is endowed with a large number of bandgap reconfiguration possibilities. As a result, we illustrate that unprecedented adaptable nonreciprocal wave propagation can be realized using the metastable modular metastructure. Overall, this research elucidates the rich dynamics attainable through the combinations of periodicity, nonlinearity, spatial asymmetry, and metastability and creates a class of adaptive structural and material systems capable of realizing tunable bandgaps and nonreciprocal wave transmissions.

Figure 1

(a) Characteristic force displacement profile of a bistable element. (b) Characteristic force displacement profile of a metastable module. (c, d) Top view and corresponding schematic of the experiment setup. (e) Conceptual diagrams of metastable module assembled in series under forward (excitation from left) and backward (excitation from right) actuations.

Figure 2

Schematic and discrete mass spring representation of an $N$ metastable module assembled in series. A periodic unit cell, in this case same as the metastable module, is highlighted with red dashed box.

Figure 3

(a) Reaction force profile as a function global displacement for a 10 module metastable chain. Points A, B are two different configurations for the same global topology realized by internal configuration switching. (b, c) Corresponding band structures for configurations A and B. Passband are within $[0,{\omega }_{A1}], [{\omega }_{A2},{\omega }_{A3}]$ and $[0,{\omega }_{B1}], [{\omega }_{B2},{\omega }_{B3}]$, respectively, demonstrating the massive band structures tunability via internal configuration switching.

Figure 4

(a) Top view of experiment setup with two different configurations. Two configurations can be reconfigured by switching between metastable states. Periodic cell of each configuration is highlighted with dashed box. (b) FRF of output displacement over input displacement as input excitation frequency changes for configuration I (red) and configuration II (blue) with small excitation level. Frequency 15 Hz (triangle) is in the passband for configuration II and stopband for configuration I and 20 Hz (star) is in the stopband for both configurations. (c) Time history of output displacement with input RMS displacement 0.2 mm and input frequency at 15 Hz. Signal changes in situ from propagating to nonpropagating as configuration changes from configuration II to I. (d) Transmittance ratio (TR) for forward (circles connected with solid line) and backward (triangles connected with dashed line) actuation for configuration I with input frequency at 15 Hz. Shaded area denotes the region of nonreciprocal wave propagation. (e) Transmittance ratio (TR) for configurations I and II under forward actuation at 20 Hz (circles connected with solid lines), demonstrating the adaptiveness of supratransmission.

Figure 5

Steady-state displacements of input (magenta), output (cyan), and internal (gray) masses for configuration A shown in Fig. 3 under forward [(a) and (e)] and backward [(c) and (g)] actuation with excitation frequency ${\omega }_d=1.15$ and excitation amplitude $\delta =0.1$ [(a) and (c)] and $\delta =0.5$ [(e) and (g)]. (b, d, f, h) Frequency domain analysis of corresponding velocities. Red dashed lines are band structure boundaries frequencies for configuration A predicted from linear analysis and green solid line is the input driving frequency. Time $t$ is normalized with respect to the natural frequency ${\omega }_1=\sqrt{\frac{k_1}{m_1}}$.

Figure 6

(a, b) Transmittance ratio (TR) for metastructure under forward (solid line) and backward (dashed line) actuation for configuration A (red) and B (blue) with shaded areas denoting parameter space for nonreciprocal wave propagation. (a) Varying input amplitude with fixed frequency ${\omega }_d=0.95$ and (b) Varying input frequency with fixed amplitude $\delta =0.5$. Gray area indicates the region of nonreciprocal wave transmission. (c) Contour plot on transmittance ratio (vs. input frequency and amplitude) of forward and backward actuation for configurations A and B.