•The FBEH owns a variable potential energy function.•The threshold of snap-through for FBEH is much lower than that of BEH.•The FBEH could realize snap-through more easily and create large output voltages. In this paper, aim at improving the performance of bi-stable energy harvester (BEH), we proposed a flexible bi-stable energy harvester (FBEH). The FBEH is composed of two elastic beams: one is the piezoelectric cantilever beam with a tip magnet and the other is the clamped-clamped beam with a mid-magnet. The dynamic behavior of FBEH was studied, and the results show that the FBEH owning a variable potential energy function is beneficial for snap-through. The governing equations of FBEH are derived by energy principles. Then its dynamic response under random excitation was studied. The results prove that the FBEH not only has a smaller threshold for snap-through, but also can generate a larger power output for the excitation intensity larger than the critical one. Validation experiments were designed and carried out. The experimental results are in good agreement with the simulation ones.