When a 1D simple harmonic oscillator is under the action of the perturbation that decays exponentially with the increase of time，the energy of the system can increase monotonically or non-monotonically，depends mainly on the ratio of the period of the oscillation and the characteristic time of the perturbation. Once the ratio is small，i.e.，it is much less than 1，the increasing is monotonic; however，which is large，i.e.，it is much greater than 1，the increasing is oscillating with diminishing amplitudes. The physical mechanism is proposed in the following. That the ratio is small means that the perturbation ends practically within one period of the unperturbed system，so the probability of the system in the every energy level cannot change any more. That the ratio is large means that the perturbation needs many periods　of the unperturbed system before it ends practically，so the probability of the unperturbed system in the every energy level can change alternatively. Since this physical mechanism holds universally for the perturbation decays on time，the system studied in present paper serves as an illustration. In addition，we have found that once the perturbation acts，the system starts to deviate from the initial state in such a manner that is independent of the characteristic time of the perturbation at very short time of the action.