Non-reciprocal bilinear structures

Structures that display non-reciprocal behavior resulting from nonlinear effects are presented. The nonlinear mechanism is bilinear stiffness, also known as bimodular elastic response. Specific realizations considered are in the form of bilinear springs in series making a finite degree of freedom uni-dimensional nonlinear structure. Recall that a system is reciprocal if FB uB,A = FA uA,B where is uB,A (respectively uA,B) is the displacement at point B (A) resulting from a force FA applied at A (FB applied at B). We say the system is fully non-reciprocal if the reciprocity relationship is violated for any sign of the applied forcing, positive or negative, compressive or tensile. Perhaps the simplest system displaying full non-reciprocity is a two degree-of-freedom spring-mass-spring-mass-spring structure, fixed at both ends. We first describe the static and low frequency behavior, illustrating full non-reciprocity. Similar systems with many bilinear springs and masses display nonlinear traveling wave effects, including pulse spreading and shock formation depending on whether the leading edge of the incident pulse is compressive or tensile, respectively. The talk will also discuss fabrication and optimization of the bilinear springs using additive manufacturing. Work supported by NSF-EFRI.