more on ribs and beam stuff

[Delwin D Fandrich]

Stephen Birkett wrote:

> 4. Arched beam with ends clamped. This is the closest you can get to a
> glued-in soundboard with a 1-D model. Here the behaviour is different - in
> response to a local force. Try it. Bend a thin plank arched upwards by
> forcing it between two blocks clamped to the bench. Now push down at one
> side. It is initially hard to push (like #2) then suddenly much easier as
> the other side pops up and it chooses the shape of minimum potential
> energy. A 2-D version like a real glued-in soundboard, even a simple
> square or circular shape, would exhibit the same sort of behaviour
> locally, but more complex. In fact, too complex for simple calcs. This
> would explain Del's observations of a non-linear deflection response to an
> applied force. I would be interested to see the data from those
> experiments Del.
>
> Stephen

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By now it should be obvious that the piano soundboard is not just a rib.  Nor is
the piano soundboard assembly a one-dimensional model for a two-dimensional
system.  Nor is it a single-dimensional, edge-clamped, wave-carrying plate.
Notwithstanding that I have called it such myself from time to time.

On every piano soundboard there are a number of wood ribs of varying
cross-section ratios, sizes, shapes and lengths.  These ribs are glued to a wood
panel which also is of varying size, thickness and stiffness characteristics.
This assembly is further stiffened by a rather massive set of bridges attached
to the top.

To further complicate matters, this whole assembly is not flat but is curved in
some more or less deliberate manner.  In fact, the ribs themselves may start out
flat or curved.  The panel may be under edge compression -- or not.  Although we
like to think that the curve -- or 'crown' -- is of some uniform radius, it is
not.  The curvature varies across the span of the assembly.  This whole assembly
is not simply supported but, as Stephen noted, it most closely resembles a
clamped edge vibrating plate.  But it is not completely so.  The ribs are not of
uniform height all the way to the rim.  They are 'feathered' out to a height of
something less than 6 to 8 mm.  The soundboard panel is also often thinned out
around all or part of its parameter.

Nor does the soundboard assembly function as an 'arch.'  It is very tempting to
go off in this direction because the modern piano soundboard does start out with
some crown.  (At least most of the soundboards installed in pianos from roughly
the 1860's and 1870's on started out with some crown built into them.  With a
few odd exceptions I have no personal knowledge of pianos that were built much
earlier than that.)  And it is glued -- 'clamped' -- to a more-or-less rigid
rim.  But, the piano rim is not all THAT rigid.  All piano rims have some give
to them.  Even with the M&H spider.  Nor is the material that the soundboard is
made -- wood -- of all THAT stiff.  Wood is too compressible, even along the
grain of the rib.  The rims flex too much.  And, once strung, there is far too
much distortion in the shape of the soundboard -- i.e., the shape is no longer a
constant radius.  It most often resembles a very elongated 'm.'

I suppose one would have to acknowledge the theoretical potential for the
arch-like behavior described by Stephen, but in real life?  No.  Neither life,
nor the piano soundboard assembly is that simple.  I've not yet seen any
'analysis' or a  'model' of the soundboard assembly that made much sense when it
was actually compared to the real world piano soundboard assembly.

Having said all that, it is still possible to examine the soundboard assembly,
study it, test it, think about it and form at least a few conclusions that can
help us design and build them with more or less predictable structural behavior
and acoustic characteristics.  By treating the rib as an end-supported,
center-loaded beam we can at least roughly predict its deflection
characteristics when loaded by the string downbearing.  The prediction at least
gives us a starting point when it comes to sizing a particular rib for a
particular soundboard using a particular scale.  Is it a perfect prediction?
No.  Of course not.  But, until something better comes along, it's better than
nothing.  Especially when the mathematical prediction is coupled with a certain
amount of common sense and thoughtful experience.  (This rib design methodology,
by the way, has been used for at least 40 or 50 years, by the way.  Possibly
longer.)

When designing a soundboard and rib set for a production piano, there exists the
luxury of sampling a number of pianos and fine tuning the rib dimensions
accordingly.  When designing a soundboard and rib set for a piano being
remanufactured on a one-time basis, things get a bit trickier.  The designing
technician needs all the help he can get.

I'll be doing another set of deflection measurements on another new soundboard
design in the next couple of months.  I may or may not publish those results.
I've not yet decided.

Del