The effects of rotary oscillation on the unsteady laminar flow past a circular cylinder. are numerically investigated in the present study. The numerical solutions for the 20 Wavier-Stokes equation are obtained using a finite volume method Tn the framework of an overlapping grid system. The vortex formation behind a circular cylinder and the hydrodynamics of wake flows for different rotary oscillation conditions are analyzed from the results of numerical simulation. The lock-on region is defined as the region that the natural shedding frequency due to the Karmann Vortex shedding and the forcing frequency due to the forced oscillating a cylinder are nearly same, and the quasi-periodic states are observed around that region. At the intersection between lock-on and non-lock-on region the shedding frequency is bifurcated. After the bifurcation, one frequency fellows the forcing frequency($S_f$) and the other returns to the natural shedding frequency($St_0$). in the quasi-periodic states, the variation of magnitudes and relevant phase changes of $C_L$ with forcing phase are examined.