The vortex-induced vibrations of a circular cylinder at low Reynolds (Re) numbers are simulated by applying a method of the two-dimensional computational fluid dynamics coupled with the structural dynamics based on the multi-physics. The fluid solver is first tested on the case of a fixed cylinder at Re$leq$160, and shows a good agreement with the previous high-resolution numerical results. The present study then reports on the detailed findings concerning the vibrations of an elastic cylinder with two degrees of translational freedom for a number of cases in which Re is fixed at 200, a reduced damping parameter Sg=0.01, 0.1, 1.0, 10.0 and the mass ratio M$^$*/ = 1, 10.