The forced oscillations occur at the frequency of the external force and not at the natural frequency of the vibrating system. However, the amplitude of the oscillation depends on the relationship between the natural frequency and the frequency of the applied force. Below figure shows the amplitude of the forced vibrations as the driving frequency is varied in the vicinity of the natural frequency w

(a) As the driving frequency w approaches zero, the damping will be small. So the amplitude of the forced oscillation increases rapidly as w'' approaches w and reaches its maximum when w'' = w.
(b) As the driving frequency w approaches infinity, the damping will be large. So the amplitude does not increase nearly as rapidly near resonance, and for the largest damping the resonant frequency is even displaced slightly from the natural frequency.