Nonlinear vibrations and instability of a bistable shallow reticulated truss

In recent years, a growing interest has been observed on the static and dynamic behavior of multi-stable structures. In this paper, a shallow bistable regular pyramidal truss composed of n equally spaced bars is investigated, allowing a deeper understanding of the static and dynamic buckling of several shallow structures, from carbon nanostructures to large geodesic domes. First an exact formulation, considering large displacements, is presented and the nonlinear equations of motion for a nodal load acting along a generic oblique direction are derived. Based on this formulation, the parametric buckling analysis of the truss under vertical static load as a function of the shallowness parameter is conducted and the equations of motion of the preloaded structure are derived. Then, the fundamental vibration frequency and nonlinear frequency-amplitude relations are obtained as a function of the static preload. The structure exhibits a two-well potential function whose geometry controls the dynamics of the preloaded structure. A detailed parametric analysis of the preloaded truss under harmonic excitation is then conducted. Emphasis is placed on the influence of the load and geometric parameters on the coexisting stable and unstable dynamic responses associated with the two-well potential function. Detailed bifurcations analyses using continuation techniques show the existence of several in-well and cross-well stable and unstable coexisting solutions. Also they permit the identification of the stability boundaries under different loading conditions. These coexisting solutions lead to topologically involved portraits of the attractor-basin response. Thus, an analysis of the evolution and erosion of the basins of attraction is performed. In a dynamic environment excitation, noise has a palpable influence on the system response. So, in the last part of the paper the influence of random noise on the topology of the coexisting basins of attraction of the system is investigated. Notwithstanding the complex nonlinear dynamics of this class of bistable structures, the analysis of the truss global dynamics may provide hints for safe structural design and novel applications of multi-stable structures.