Bistable buckled beam: Elastica modeling and analysis of static actuation

Abstract

We propose the modeling of a buckled elastic slender beam based on elastica approach. The model accounts for large rotations of the beam cross-section and rather large elastic displacements. Moreover, the model incorporates the extensibility of the elastic beam. The nonlinear nature of the model is used to amplify the transition from one stable position of the buckled beam to the other one. Such mechanical structure is said bistable. The bistable beam is simply supported at each of its ends and is subject to a transverse force applied at a point of the beam. The emphasis is placed especially on the bistable mechanism response caused by the applied force. The stability of the buckled beam is investigated in details and the diagram of the applied force of actuation as function of the midpoint displacement is discussed according to the applied force location. The snap-through phenomenon scenario is analyzed. The switching from one stable state to the other one occurs passing through an instability region in which the second buckling mode is involved. For rather small end shortening of the beam, the post buckling behavior is studied by reducing the solution of the complete elastica model to the first two buckling modes. The reduced model allows us to discuss the switching path in terms of energy required and stability properties of the bistable mechanism. Numerical algorithms are developed in order to solve the strongly nonlinear problem.