Induced voltage vs. string displacement

[Mike Sulzer]

So the problem with equation 13 is more than I thought. I put your correction in and it makes about a 1% difference from what I had.

I do not see a problem with equation 10. What am I missing?

Great Mike!

I've just checked all the equations.
I found all but eq. 13 were OK (except for the sign of the terms that depend on "x" or "y" from equations 10 through 12).
I found a discrepancy in the last term of the numerator in eq. 13.
I still have to recheck it, but I'm attaching a view of the revised spreadsheet (also showing the original and revised equations in readable form) because the amplitude has increased significantly, although still low (but, as I wanted to reproduce McDonald's results, and although I don't know which values he used for "x" and "y", I didn't change his parameters).

The assumption McDonald made, that the string is in a uniform magnetic field, is, as you say, affecting not only the amplitude but the shape of the output signal.

Another nice approach to this problem is reference 14, but it doesn't give much details.

Regards,
Esteban

[ATTACH]35243[/ATTACH]

[elucches]

Hello Mike,

The only problem with eq. 10 and the following ones is the sign, because cos(theta)**2-sin(theta)**2 is, if I'm not mistaken, x**2-y**2.
Of course, the sign doesn't matter at all.

Using the revised version of eq. 13 with McDonald's values, the amplitude is more than doubled (more than tripled when xpeak = ypeak = 0.5 mm) and the shape changes as well.

Regards,
Esteban

[Mike Sulzer]

Hello Mike,

The only problem with eq. 10 and the following ones is the sign, because cos(theta)**2-sin(theta)**2 is, if I'm not mistaken, x**2-y**2.
Of course, the sign doesn't matter at all.

Using the revised version of eq. 13 with McDonald's values, the amplitude is more than doubled (more than tripled when xpeak = ypeak = 0.5 mm) and the shape changes as well.

Regards,
Esteban

OK, yes, I agree that the sign is wrong, and that it does not matter!

Mike

[Mike Sulzer]

The McDonald pickup model assumes a string in a magnetic field that extends forever along the string. This is an approximation; the exact problem is difficult, but analyzing a different approximation can be useful. Suppose the magnetic material were all confined to a small sphere; would we see the same non-linearity? What would be the dependence of of the voltage on coil size?

Assuming a simple symmetrical form has a long history, discussed lightly in the topic of "bovine physics" (https://en.wikipedia.org/wiki/Spherical_cow). So, I say, consider a spherical string...

A circular coil is assumed; voltages resulting from parameters similar to what have been used before in this discussion are chosen, and the results for three different coil radii are shown here


The derivation is in this file . The solution for the potential is simple and can be found on the web, and so the job is already partly done.

This might seem like a very bad approximation, but several "spherical strings" could be used to construct a better model ("beads on a string").

[elucches]

The spherical cows joke is funny; I didn't know it.

I get a message about invalid attachments when I click the links.

[Mike Sulzer]

My previous post with functioning attachments

The McDonald pickup model assumes a string in a magnetic field that extends forever along the string. This is an approximation; the exact problem is difficult, but analyzing a different approximation can be useful. Suppose the magnetic material were all confined to a small sphere; would we see the same non-linearity? What would be the dependence of of the voltage on coil size?

Assuming a simple symmetrical form has a long history, discussed lightly in the topic of "bovine physics" (https://en.wikipedia.org/wiki/Spherical_cow). So, I say, consider a spherical string...

A circular coil is assumed; voltages resulting from parameters similar to what have been used before in this discussion are chosen, and the results for three different coil radii are shown here sphericalStringModel.pdf.

The derivation is in this file pickupVsphcoord.pdf. The solution for the potential is simple and can be found on the web, and so the job is already partly done.

This might seem like a very bad approximation, but several "spherical strings" could be used to construct a better model ("beads on a string").