The sensing mechanism of an optical fiber accelerometer based on the interaction between the frequency shifted two modes in a two-mode fiber with an off-center core is presented. When the fiber is bent, by the action of an acceleration in a specific direction, due to the asymmetry induced by the off-center core, there exists a preferential net strain. The bending causes a tension or a compression field in the fiber and provides a positive or negative acceleration, respectively. The frequency shift is achieved by an collinear acousto-optic interaction in the two-mode fiber. We show both theoretical analysis and experimental results. Several approaches which can enhance the sensitivity are also provided.