  This study investigates the programming of elastic wave propagation bandgaps
in a kirigami material by intentionally sequencing its constitutive mechanical
bits. Such sequencing exploits the multi-stable nature of the stretched
kirigami, allowing each mechanical bit to switch between two stable equilibria
with different external shapes (aka. "(0)" and "(1)" states). Therefore, by
designing the sequence of (0) and (1) bits, one can fundamentally change the
underlying periodicity and thus program the phononic bandgap frequencies. To
this end, this study develops an algorithm to identify the unique periodicities
generated by assembling "$n$-bit strings" consisting of $n$ mechanical bits.
Based on a simplified geometry of these $n$-bit strings, this study also
formulates a theory to uncover the rich mapping between input sequencing and
output bandgaps. The theoretical prediction and experiment results confirm that
the (0) and (1) bit sequencing is a versatile approach to programming the
phonic bandgap frequencies. Moreover, one can additionally fine-tune the
bandgaps by adjusting the global stretch. Overall, the results of this study
elucidate new strategies for programming the dynamic responses of architected
material systems.
