Modal Analysis (was Negative bearing (long))

[Phillip L Ford]

On Tue, 4 Dec 2001 20:18:34   
 Delwin D Fandrich wrote:
>
> The sand
>technique is a common
>> method of visualizing the mode shapes of a vibrating body.  This is one of
>the practical
>> techniques I mentioned that violin makers use.
>
>I can't see any advantage to this. Other than as a teaching aid to help
>understand how the soundboard works. It is only going to illustrate at what
>point the soundboard resonances come in and what their shapes are. Again,
>this is with an unloaded board. Everything is going to change once the
>strings go on and their up to pitch.
>
>Del
>
Yes it does illustrate the soundboard resonance frequencies and shapes.  In the case of
the violin that we have been using as an example the maker knows that to get a good 
sounding violin he needs to have a top  something like the following (this is just an example -
these are not real numbers since I'm not a violin maker.  But if it's important I could in
fact look this information up.):
1.  A first mode at 249 hz in a symmetric diaphragm like shape.
2.  A second mode at 468 hz in an unsymmetric mode with a node line centered more
or less at the bridge line.
3.  Etc., etc. with other modes.

This is just for the top.  He will have something similar for the belly.  These modes are NOT
the same modes the top will have in the assembled violin.  But a relationship between the
unassembled top and the assembled violin has been established.  That is the point.  Perhaps
to have a good sounding violin you need an assembled first top mode that is 350 hz in an 
assymetrical shape centered around the soundpost (once again a fictitious example).  It
has been established through experiment and lab test that to arrive at that you need the
top modes shown above.  It has also been established through experiment and test how the
top plate modes respond to changes in the top thicknessing, arching, etc.  So, you make
your top to your basic design (violin makers use very precise patterns for arching and
thicknessing their tops) which you know from past experience will get you close to your
desired result.  You test your top using the sand method and you arrive at  mode frequencies
and shapes that don't match exactly what you're trying to achieve.  Now you have some
guidelines about where to thin and how much.
This is a very powerful tool.  It means that you can lay out and make your top and adjust it
for slight variations in geometry and material properties and know that when you assemble
the violin it will sound right.  You would never have been able to do this if all you did was test
the assembled violin.  You have to relate the modal properties of the parts to the modal
properties of the whole.  Otherwise how do you know what to change and how much?

To establish this relationship would be time consuming and tedious.  As I see it the process
would be something like this:

1.  Find some good sounding pianos.  Test their modes in the assembled condition to establish
some sort of pattern for modal  frequencies and shapes that good sounding pianos seem
to share.
2.  Build lots of soundboards with different parameters which you change one at a time:
different board thickness, different grain angle, different rib dimensions, different rib spacing,
etc.  Measure the modal frequencies and shapes of these boards.  Install these boards
in a piano and build it up.  Test the modal frequencies and shapes of the boards in
these pianos.  See how they compare to your good sounding pianos in 1.
3.  Try to establish a relationship between the modal frequencies and shapes of the uninstalled
boards and the modal frequencies and shapes of the assembled piano.
4.  Determine what arrangement of the parameters (what board thickness, what rib spacing,
etc.) gives the required modal frequencies and shapes for the uninstalled board.  Use this
as your basic soundboard design.
5.  Test some uninstalled boards to see how varying parameters of the board changes the
modal frequencies and shapes of the uninstalled board.  Use this information to tweak your
basic design (which is going to vary somewhat from board to board because of changes in
material, slight differences in rib dimensions, etc.) to the required modal frequencies and
shapes before installing the boards in your factory.

Phil F.

--
Phillip Ford
Piano Service & Restoration
1777 Yosemite Ave - 215
San Francisco, CA  94124