Multistability, symmetry and geometric conservation in eightfold waterbomb origami

The nonlinearities inherent in the mechanics of origami make it a rich design space for multistable structures and mechanical metamaterials. Here, we investigate the multistability of a classic origami base: the symmetric eightfold waterbomb. We prove that the waterbomb is bistable for certain crease properties, and derive bounds on, and closed-form approximations of, its stable states. We introduce a simplified form of the waterbomb kinematics and present a design procedure for tuning the depth and the symmetry/asymmetry of its energy wells. By incorporating the concept of pretensioned torsional springs, we also demonstrate the existence of tristable cases for the waterbomb fold pattern. We then apply the analysis of a single waterbomb to study quasi-one-dimensional arrays of waterbombs, where we discover a conserved geometric-kinematic quantity in which the number of popped-up and popped-down vertices is determined uniquely through analysis of the origami structure's boundaries. This culminates with a discussion of how the quasi-one-dimensional array may be designed to achieve stable states with various degeneracies, kinematics and gaps between energy levels. Collectively, this work presents an alternative approach for characterizing origami multistability properties and reveals an origami design motif that has potential applications in physically reconfigurable structures, mechanical energy absorption and metamaterials.