Droplet spreading behavior on a substrate is closely bound up with the wettability of the substrate, and plays a critical role in many industrial applications, such as lubrication, painting, coating, and mineral flotation. In this paper, a dynamical model of droplet spreading on a smooth substrate is established through a mechanical analysis. According to the lubrication approximation theory and Navier-Stokes equation, a general nonlinear evolution equation or equations are derived, including the momentum equation, the continuity equation, and the evolution equation of film thickness. We adopt numerical methods to solve these equations, and also quantitatively analyze the relation among film thickness, spreading radius, speed of wetting contact line and time in detail. The results show that the droplet spreading process is mainly divided into two phases, namely expansion phase and contraction phase. Moreover, the spreading process is along with mutual transformation among surface energy, kinetic energy, and different kinds of potential energies. In addition, the final spreading radius Rf of droplet is determined by the inherent wettability of solid surface, and the collapse effect, which emerges at t=0.006 s in the spreading process, is related to Laplace pressure difference of curved liquid surface. Finally, by controlling the droplet size, we obtain the scaling law of droplet spreading radius with time, which approximately meets R ～ t1/7. The scaling law is validated both experimentally and numerically. The results of this study are expected to enhance our knowledge of the movement of wetting contact line and also provide some guidance for the wetting theory.