The fuel spray seriously affects the combustion efficiency and emission performance in the internal combustion engines. Due to the turbulent motion with strong transience and pulsation, it is significant to study the spray characteristics of a single droplet under the time-dependent acceleration for improving the accuracy of the spray model of the secondary atomization. In this paper, we placed a droplet on a diaphragm that vibrates in the vertical direction to investigate the spray characteristics of the droplet under the sinusoidal inertial force. The diameters of the atomized sub-droplets under different experimental parameters were measured by the shadowgraph shooting technique with a high-speed camera. The results show that the droplet volume affects the vibration mode of the droplet surface waves, and has little effect on the diameter of the atomized sub-droplets. The average diameter decreases as the surface tension coefficient reduces. The liquid viscosity has little effect on the average diameter of the sub-droplets. As the acceleration amplitude increases, more sub-droplets are ejected and the size distribution range of these sub-droplets becomes wide, but the average diameter increases a little. As the frequency of the inertial force increases, the average diameter declines first and then slows down, which agrees well with Lang's equation. Thus, we can ignore the effects of the droplet volume, liquid viscosity and the acceleration amplitude, and use Lang's equation in the spray model to calculate the diameters of the sub-droplets. As the surface tension coefficient and viscosity increase, the acceleration threshold of the droplet atomization increases. The frequency and droplet volume affect the threshold by changing the spherical mode-number of the droplet surface waves.