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To my understanding when the driving frequency is equal to the natural/resonant frequency of the object, there is constructive interference between the oscillations of the object and driving force. This causes the amplitude of the oscillations to increase. However if the driving frequency is not equal to the resonant frequency there is some periods of constructive interference and some periods of destructive interference, thus there is no net change effect on the amplitude of oscillations.

What I don't get is why doesn't the driving frequency have to be EXACTLY equal to the resonant frequency? If the driving frequency is exactly equal to the resonant frequency it will always be constructively interfering. However if the driving frequency is out by jut a small amount, to begin with it would interfere constructively as before (yes)?. But because the period is shorter (or longer) it would soon go out of phase, and then would it not be destructively interfering and, much like when the driving frequency is completely different to the resonant frequency, have no net effect on the amplitude?

Thanks in advance, if this question is poorly worded please ask for clarification and I'll try my best.

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    $\begingroup$ Your confusion might arise from thinking of these oscillations as waves that can interfere. For example, if I am driving the oscillation of a spring on a mass, my force and the motion of the mass are not "interfering". The oscillation of the mass is determined by my force. It is not oscillating at some period and then my force comes along with a separate period that can be in or out of phase with the mass. $\endgroup$ – Aaron Stevens Jul 9 '18 at 17:26
  • $\begingroup$ Energy is transferred in any event. But when the driving frequency matches the system resonance, the energy is transferred with maximum efficiency. $\endgroup$ – docscience Jul 9 '18 at 22:38
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This is an elaboration of Aaron Stevens's comment. When you analyse what's going on mathematically (starting by applying Newton's second law) you find that if the oscillatory system (a pendulum, say) is initially at rest, the oscillations that you get when you first start to apply the periodic driving force are the superposition (sum) of oscillations at the natural frequency and oscillations at the forcing frequency. The first are due to the sudden disturbance of the oscillatory system and die away exponentially. The second remain as long as the forcing frequency is applied. They are what we usually mean by 'forced oscillations'. And their amplitude is greater the closer the forcing frequency is to the system's natural frequency.

The point of all this is that it's probably misleading to think of the oscillations at the forcing frequency as being a reinforcement of the natural oscillations. The natural oscillations die away!

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Systems can be driven to large amplitudes by a "near resonant" driving force if the sharpness (a.k.a. "Q") of the system is low. This is because a low Q resonator has a broadened resonant peak which responds to a spread of frequencies to either side of resonance and not just to the exact frequency of resonance.

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