The effects of the constant excitation on the local bifurcation of the periodic solutions in the 1:2 internal resonant systems were analyzed based on the singularity theory. It is shown that the constant excitation make influence only when there exist some nonlinear terms, in the oscillator with lower frequency. Besides acting as main bifurcation parameter, the constant excitation, together with coefficients of some nonlinear terms, may change the values of unfolding parameters and the type of the bifurcation. Under the non-degenerate cases, the effect of the third order terms can be neglected.