What Do the Parameters b and n Represent in Damped Harmonic Motion?

[romd]

Concerning damped harmonic motion (eg. mass on a spring, using cardboard discs as dampers); for the equation (below) of the graph describing the effect of different sized dampers on the time taken for amplitude of oscillations to halve, what would b (y-intercept) and n (gradient) represent? (A=area of damper; T=time taken for amplitude to halve)

$$T=b.A^n$$
$$ln(T)=n.ln(A) + ln(b)$$

Thanks

 

[Andy Resnick]

Blech... I vaguely recall this stuff from a control engineering class.

Wikipedia has some useful background:
http://en.wikipedia.org/wiki/Damping

What helps me is to consider limiting cases- the area of the damper going to zero, or infinity, for example. Let's first consider a damper of unit area: A^n is always 1 then. Then b is linearly related to the 'damping time', and is probably connected to the 'damping ratio' of a damped oscillator.

That leaves 'n', which is the effect of varying the area of the damper. Question- how does varying the area change the damping time? is it a linear relationship (n=1)? nonlinear (n>1)? sublinear (n<1)? I don't know the answer, but that's what 'n' represents.

 

[romd]

Thanks for the reply!

Now, I have a value for both constants, and both a graph of T against A and one of log(T) against log(A). This is for coursework, and having neglected to find spring constant k or any other potentially useful information, I am having trouble making my interpretation and conclusions 'worthwhile'- other than stating vague implications of the values of b and n. With the data I have would it be possible to find a complete equation for the motion of the spring? Thanks

 

[Andy Resnick]

Well, you may be able to figure out actual values for 'n' and 'b'... What exactly is the purpose of the lab?

 

[romd]

I have values but without n and b directly representing anything physical I'm finding it hard to go very in-depth in analysis; for alevel coursework I think more than just a few lines would be needed. The aim was to investigate the effect of damping on SHM of a spring-mass system- I left it vague because at the time didn't know how I would go about it

 

[Andy Resnick]

Oh. That's a different question than what you initially asked, I think. How did you arrive at the equation in your original post? Was it given to you, or did you guess using Excel or something?

 

[romd]

It was given to me, as an equation for the graph. Initially I though b and n would represent something physical, but it seems they don't, at least not directly - eg. you said b would be connected to the damping coefficient, but without knowing how I can't write much on it.

 

[Andy Resnick]

Hmmm.. Well, when I get ambiguous comments from reviewers, my strategy is to first re-state the comment as best I can in terms that I do understand, and then provide a response. Sometimes that works, sometimes it doesn't.

So, think about how the damping occurs in your system- demonstrate you understand a damped oscillator. Then, think about what a power-law (T ~ A^n) means physically- that would be impressive to talk about- and then try and relate the two. For example- the area of the disk is related to the mass of the damper, and so is related to 'b' as well: can you re-write the power law in terms of the mass of the damper?