Oscillators composed of incommensurate graphene sheets have been investigated by molecular dynamics simulations. The oscillation frequencies can reach tens of gigahertz range and depend on the surface energy of the bilayer graphene and the oscillatory amplitude. We demonstrate the tunability of such an oscillator in terms of frequency and friction by its varying geometric parameters. Exploration of the damping mechanism by combining the autocorrelation function theory and the direct atomistic simulations reveals that the friction force is proportional to the velocity of oscillatory motion. The results should help optimize the design of graphene-based nanoelectromechanical devices.