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Am I correct in thinking that for a given harmonic oscillator, with constant magnitude driving force, the resultant amplitude of the steady-state motion will be generally lower above resonance than below.

If this is the case then is there a reason for this.

Qmechanic
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Benjamin Rogers-Newsome
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1 Answers1

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As the frequency of the driver tends to zero the amplitude of the steady state oscillations tend to the amplitude of the driver.
Think of it as the mass on the end of a spring with you moving the top of the spring up and down.
If you move the top of the spring very, very slowly the mass at the end of the spring will follow your movements and have almost the same amplitude as your movement and also only lag behind your movement by a very small amount.
In other words the motion of the mass will be almost in phase with the movement of your hand and have almost the same amplitude as your hand.

At the other end of the scale if you move the top of the spring up and down very rapidly then the mass at the ned of the spring just cannot keep up with the movement of your hand.
The mass starts going down and then your hand starts to move up when the mass has moved down hardly at all before the mass responds to the upward movement.
However soon after the mass starts to move up a small amount the top of the spring is now moving down.
At high frequencies well away from resonance the movement of the mass is almost $\pi$ behind the movement of your hand and the amplitude of motion of the mass is very much smaller than the amplitude of motion at the top of the spring.

So the graph of the amplitude of the mass (driven) against the frequency

enter image description here

Note that this is for amplitude resonance.

For a plot of maximum speed of mass against frequency of the top of the spring the maximum velocity of the mass would tend to zero at both very low and very high frequencies relative to the resonant frequency.
When the speed of the mass is a maximum you have velocity resonance.

This is also true of the energy resonance and means that the energy of the mass at the end of the spring tends towards zero at low and high frequencies.

Farcher
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