SHM MULTIPLE damping coefficents practical help

[dfx]

Right, I'm about to open a big can of worms:
I'm investigating eddy current damping of Simple Harmonic motion and I'm going to be investigating how the damping coefficient varies with various factors including the distance of the magnetic field (irrelevant, just informing). A "magnetic tunnel" creates eddy currents which damp the metallic glider on an air track. I hope the setup is clear.
However, since I am using a glider on an air track and a motion sensor, I've to attach a target on the glider which the motion sensor tracks (to record the simple harmonic motion). The problem is that this target is of such a shape that it'll offer plenty of air resistance (it's a flat rectangular sheet, perpendicular to the plane of motion). So the problem is that air resistance will no longer be negligible, but has to be reasonably taken into account. Hence attempting to find the damping coefficienet with air resistance being offered, will give an incorrect value (since I am purely interested in the value of the damping coefficient due to eddy current damping from the magnetic field).
So, my proposal is, say I found the damping coefficient with just air resistance (no magnetic tunnel), and then with the magnetic field AND air resistance, how would I get the damping coefficient due to the magnetic field ONLY?
i.e. if I wanted the damping coefficient due to eddy currents (magnetic field) ONLY, would I SUBTRACT the damping coefficient of AIR from damping coefficient of (air + eddy currents)?? (i.e. to get an accurate value, so as to remove the effect of air resistance damping from the value). Basically, are there any known theories/mathematics for "combined damping"?
Secondly, this is slightly beyond my physics course but since I have done differential eqns e.t.c I'm fairly comfortable with the mathematics involved. I will be mainly investigating an underdamped system, and so will be sketching a graph of log (maximum displacements from the SHM sinusoidal curve) against time and getting the gradient to find the value of the damping coefficient. However I believe it's not SOLELY the value of the gradient which gives the damping coefficient - apparently you have you equate it to something? The problem is this is slightly beyond my course however seems doable. Could someone please advise on how to get the damping coefficient from the gradient, as there various different D.E s describing the motion and research is only further confusing.

edit: Could you confirm that this would be the equation for underdamped motion:

https://www.physicsforums.com/latex_images/44/444951-1.png

And hence plotting ln(x) against t would yield a slope equal to -b/2m? (and then obviously just proceed to solve for b)

I hope someone can help. There is practically nothing on this (multiple damping coefficients) on the internet! Feedback/help much appreciated.

 

[dfx]

*bump* :( anybody?

 

[Astronuc]

When one is exploring two factors - e.g. simultaneous damping by magnetic field and air resistance - then it is necessary to try to reduce one as low as possible in order to determine the separate effect of the other.

Is it possible to turn off the magnetic field and simply measure a damping effect of the air.

Otherwise, try to reduce the size of the target used as a motion sensor. Ideally, one could run the same experiment in a vacuum chamber thus eliminating the air resistance.

Here is some background, which you might already have from other sources - http://hyperphysics.phy-astr.gsu.edu/hbase/oscda.html

http://hyperphysics.phy-astr.gsu.edu/hbase/permot.html#permot

I think the challenge here is that the damping coefficient is the sum of two separate coefficients.

 

[dfx]

Is it possible to turn off the magnetic field and simply measure a damping effect of the air.
Ideally, one could run the same experiment in a vacuum chamber thus eliminating the air resistance.

Thanks for that. I was intending to run the experiment without the magnetic field and with the magnetic field ( + air resistance). Is there any mathematics to manipulate the two i.e. as you said, the challenge is the fact that it is a sum of two dampers - is it literally a "sum" (in which case I could simply subtract the effect of air from the effect air + magnetism) or is it far more complicated?

Unfortunately I cannot create anything remotely close to a vacuum chamber at high school lol.

Thanks once again.

 

[Astronuc]

As long as the damping functions are linear functions of velocity, then that should be OK, i.e. as long as the damping functions are $c_1\,\dot{x}$ and $c_2\,\dot{x}$ in the equation presented at hyperphysics, then one can use $(c_1 + c_2)\,\dot{x}$.

Also, one must look at whether the combined damping makes the system overdamped vs underdamped. It maybe that the damping due to air resistance is much greater than the damping due to the magnetic field, and perhaps the system is overdamped due to air resistance anyway. I can't say without knowing the specifics of your system.

 

[dfx]

Awesome. I think I'll ignore the possibility of critical/overdamping as from preliminary testing its highly unlikely to happen.

One final question: So I would go about equating the gradient of the graphs obtained (ln(x) against t) to -c/2m and solve for c to obtain the damping coefficient, right? Do the implications of what you said (simply summing the two damping coefficients) carry down the line to this stage? What I mean is, practically, after solving for c from the graph, could I just add/subtract the coefficients?

edit: the reason I'm asking is because the graph involves taking logarithms so that might potentially have an effect because of the maths of logs? Just wondering.

Thanks very much, much appreciated.

 

[Astronuc]

You might have difficulty with a semilog plot if the system is an underdamped SHM.

Taking ln f(x), where f(x) = $A\,e^{-bt/2m}\,sin(\omega t\,+\,\theta)$ would yield

$ln A\,-\,bt/2m\,+\,ln(sin(\omega t\,+\,\theta))$.

As for adding the damping coefficients, I would recommend comparing the solutions of the differential equation for one coefficient and then with the sum.

I do have a question, with regard to the system. How did you determine that the motion, which seems to be linear motion, is determined by SHM, which implies a restorative force proportional to displacement from some equilibrium point?

 

[dfx]

Well the experimental setup is essentially an airtrack with a glider attached to springs on both ends - the sort of classical SHM air track setup. This would imply acceleration is directly proportional to the displacement, which in turn implies SHM right? Also I have read on several places that eddy current damping is viscous if that's of any relevance to your question.

"As for adding the damping coefficients, I would recommend comparing the solutions of the differential equation for one coefficient and then with the sum."

I will be continuing with the investigation after Christmas and I will try to spend some time working out the differential equation for the combined damping and compare as you suggested. The problem is the Physics course I'm currently on is High School level and in England that is fairly basic, unfortunately lol. However I guess what I am doing is rather ambitious and I've been coerced several times into a far simpler investigation but I definitely want to do this. I'll come back on the combined d.e. and post it up.

As usual, your advice much appreciated.