Analysis of nonlinear dynamics of a curved beam under gravity bias

Geometrical design endows the curved beam with asymmetric bistability without prestress, contributing to its widespread applications in integrated forming and microscale fabrication. However, the gravity bias inherent in these applications is rarely considered, undermining the accurate prediction of its real-world dynamics. To address this, a gravity-biased nonlinear dynamic model is established via Hamilton's principle and reduced to a Duffing-type equation with a constant term using Galerkin's method. Numerical simulations show that both the interwell oscillations and the contained vacillating behavior manifest as V-shaped regions in the excitation amplitude-frequency parameter space. Bifurcation analysis reveals that the left and right boundaries of the vacillating region correspond to saddle-node and period-doubling bifurcations, respectively, both of which can be theoretically predicted. Furthermore, gravity bias is shown to deepen the lower potential well. For an upward-curved beam, it enlarges the excitation range for interwell motion when initiated from the upper equilibrium, while narrowing the range when starting from the lower one. The influence of key geometric parameters is also systematically examined. Base excitation experiments were conducted to equivalently simulate distributed force excitation, and the results validate the theoretical predictions. This work elucidates the effects of practical environmental conditions and structural parameters on the nonlinear dynamics of curved beams, thereby providing a foundation for the design of mechanical metamaterials and related engineered systems.