Nonlinearity especially from aerodynamic coefficients in high orders has a significant effect on projectile dynamics. Its investigation has been hindered in the conventional analysis by the complexity in nonlinear motion equations and the lack of appropriate analysis tools. Therefore, the widely used method of normal forms was introduced for the analysis of projectile angular motion. Considering the second order damping and the seventh order static moment terms, the normal form of the angular motion was derived and thus the universal analytical solution of the angle of attack is obtained, which is verified to show good agreement with the numerical integration results over a wide range of angle of attack and also demonstrates its being applicable to the undamped case and the cases with lower or higher order of static moment. In addition, the obtained relationship between initial conditions can give a conventional but simple determination of the region of attraction to the origin. Also, the amplitude equation combined with the equilibrium analysis provides a accurate prediction for the existence and stability of limit cycle in angular motion.