Damping and resonance

In summary: The problem seems a bit unclear and confusing, but that would be the most logical way to approach it.
  • #1
Poetria
267
42

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.?
 
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  • #2
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.
 
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  • #3
Poetria said:
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 ?
 
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  • #4
BvU said:
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.
 
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  • #5
Orodruin said:
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. :)
 
  • #6
I suspect the idea is to look at x''+b*x'+k*x=y
with y=cos(omega*t)
 
  • #7
BvU said:
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.
 
  • #8
BvU said:
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.,
$$
x'' + b x' + kx = k \cos(\omega t) - b \omega \sin(\omega t).
$$
 
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  • #9
Which would be relevant if ##b## is time dependent. Am I to interpret
Poetria said:
b - damping constant slowly increases.
as "find the steady state solution with ##b## constant and look at how the resonance peak depends on ##b## "?
 
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  • #10
BvU said:
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.
 
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1. What is damping?

Damping is the gradual decrease in the amplitude of a waveform or oscillation due to the dissipation of energy. It is often used to describe the reduction of mechanical vibrations.

2. How does damping affect resonance?

Damping can affect resonance by reducing the amplitude of the oscillation, causing it to take longer to reach its peak and decreasing the frequency at which resonance occurs. This can also lead to a decrease in the quality factor, which measures the efficiency of energy transfer in a resonant system.

3. What are the different types of damping?

The three main types of damping are viscous or external damping, structural or internal damping, and coulomb or dry friction damping. Viscous damping is caused by a fluid or air resistance, structural damping is due to the internal friction of materials, and coulomb damping is caused by the friction between two surfaces.

4. How is damping measured?

Damping can be measured using the damping factor, which is the ratio of the actual damping in a system to the critical damping. It can also be measured using the quality factor, which is the inverse of the damping factor.

5. Why is damping important in engineering and science?

Damping plays a crucial role in engineering and science as it helps to control and reduce unwanted vibrations in systems. It is also important for ensuring the stability and safety of structures, such as buildings and bridges, and for improving the performance of machines and equipment.

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