How Does Increasing Damping Impact Resonance in Forced Vibrations?

[Poetria]

Homework Statement

x''+b*x'+k*x=k*y+b*y' y=cos(omega*t)

k is fixed, b - damping constant slowly increases.

How does increasing the damping constant b affect the resonance peak?

2. The attempt at a solution

Well, I thought the answers:
It significantly decreases the height of the resonant peak.
and
It significantly decreases the frequency at which resonance happens.
are correct.

I have found a useful resource - http://physicsnet.co.uk/a-level-physics-as-a2/further-mechanics/forced-vibrations-resonance/

But these answers were marked as wrong. I have no idea why. What have I missed? Is it possible that this option is correct: It does not significantly change the shape nor the location of the resonance peak.?

 

[Orodruin]

Normally when you consider damping you assume the amplitude of the driving force to be fixed. In this case, you should note that your driving force depends on and increases with b. Think about how that will change the behaviour of the system.

 

[BvU]

x''+b*x'+k*x=k*y+b*y' y=cos(omega*t)

This the literally rendered problem statement ? Why the funny expression on the righthand side ?

 

[Orodruin]

This the literally rendered problem statement ? Why the funny expression on the righthand side ?

I suspect there is a comma missing between y' and y.

 

[Poetria]

I suspect there is a comma missing between y' and y.

Yes, a comma is missing. :( There is also a mathlet but of course I can't copy it.

I have got it right eventually. :)

Many thanks. :)

 

[BvU]

I suspect the idea is to look at x''+b*x'+k*x=y
with y=cos(omega*t)

 

[Poetria]

I suspect the idea is to look at x''+b*x'+k*x=y
with y=cos(omega*t)

Yes, exactly. You can also adjust omega with the mathlet.

 

[Orodruin]

I suspect the idea is to look at x''+b*x'+k*x=y
with y=cos(omega*t)

I strongly suspect that this is not the case and that the problem indeed wants a derivative of $y$ on the right-hand side along with the quoted constants, i.e.,
$$
x'' + b x' + kx = k \cos(\omega t) - b \omega \sin(\omega t).
$$

 

[BvU]

Which would be relevant if $b$ is time dependent. Am I to interpret

b - damping constant slowly increases.

as "find the steady state solution with $b$ constant and look at how the resonance peak depends on $b$ "?

 

[Orodruin]

Which would be relevant if $b$ is time dependent. Am I to interpret
as "find the steady state solution with $b$ constant and look at how the resonance peak depends on $b$ "?

That is how I would interpret it.