Static and dynamic non-reciprocity in bi-linear structures

Non-reciprocal acoustic and elastic wave propagation has been shown to enable a plethora of effects analogous to phenomena seen in quantum physics and electromagnetics, such as immunity from back-scattering, unidirectional band gaps, and topologically protected states. These phenomena are of interest because they could lead to the design of direction-dependent acoustic devices that could be used to augment acoustic sensing and transmitting capabilities and may provide insight in the design of materials and structures for vibration and impact isolation. In the present work, we show that static and dynamic non-reciprocity can be achieved in structures composed of bi-linear springs, which have different, amplitude-independent moduli in tension and compression, by investigating the non-reciprocal response of a simple bi-linear structure composed of three springs and two masses in series. Non-reciprocity is demonstrated by calculating the response of each mass to forcing applied to the opposite mass, i.e. by applying Betti’s reciprocity theorem to analytical force-displacement relationships in the static sense, and by numerically calculating the response to harmonic forcing in the dynamic sense. Non-reciprocity can be identified via examination of the frequency response curves, making this system a promising candidate for experiments and the design of non-reciprocal devices.