Static and dynamic nonlinear behavior and instabilities of a multistable system consisting of two shallow coupled arches

Abstract

Increased attention has been lately given to multistable systems that use the snap-through buckling of each unit to attain a desired hysteretic force-deformation response. Many of these systems consist of chains of bistable elements flexibly or rigidly connected and respond to an applied load or displacement in a progressive manner. However, little is known of their nonlinear static and dynamic nonlinear behavior and stability. This paper investigates the nonlinear response of multistable structure composed of two simply-supported or clamped buckled beams connected in series by a stiff frame. The resulting multistable system is subject to a transverse force applied at the top of the upper arch. The Lagrangian of the system is obtained and the nonlinear equilibrium equations and equations of motion are derived by the Ritz approach applying the energy stability criterion and Hamilton's principle respectively. Both local and global analyses are numerically conducted to clarify the nonlinear dynamics of the multistable model. Applying brute force and continuation techniques, the coexisting stable and unstable nonlinear equilibrium paths are obtained and the bifurcation points identified. This analysis is aided by the investigation of the potential energy landscape and the basins of attraction of the damped system. The influence of the rise-to-span ratio is clarified and the influence of different ratios on the energy required for switching from one stable state to another, escaping through a saddle channel, is discussed. When subjected to a harmonic excitation the multistable system exhibits coexisting bifurcation sequences starting at each potential well, which can merge leading to large amplitude cross-well motions. The diverse resonance regions are investigated using bifurcation diagrams, phase-space portraits, Poincaré maps and basins of attraction. This study can be used as a basis for the nonlinear analysis of more complex multistable structures constituted by several bistable units.