Assuming that both a periodic exciting force (e.g., due to an unbalance force) and a constant force (for example, due to the gravitational) are acting on a non-linear vibrating system, the 1/2-th subharmonic vibrations of the system with a hard characteristic of Duffing type are investigated by making use of an averaging procedure. Furthermore, it is shown that the subharmonic oscillations of order 1/2 can take place in the same system, even if there is only the periodic force as the driving force. They are compared with the results obtained by numerical integration from the same system, carried out with the Runge-Kutta-Gill method. As a result, it appears that the present theory gives relatively satisfactory results in calculating these vibrations.