an analysis of downbearing, etc.

Mark Kinsler kinsler33@hotmail.com
Mon, 02 Feb 2004 08:28:54 -0500


I don't know if postings from a non- (or not-yet) technician are permitted 
here, but I believe I can contribute a bit of engineering insight here that 
could clear things up.

Mechanical impedance is the result of the springiness and inertia of a 
mechanical part.  Your sounding board has some inertia due to its mass.  It 
also has some springiness.  Wood (and most other materials in a piano) will 
push back in direct proportion to the amount of deflection applied to them. 
(Exceptions: felt and buckskin, by design.)

The sounding board's springiness derives from the behavior of wood in 
compression and bending.  From an engineering standpoint, a crowned sounding 
board that's glued at its rim to a rigid piano frame is an arch.  When you 
push down on the crown of an arch, the ends of the arch try to spread.  
(This force is due to the downbearing of the strings on the bridge.)  Since 
the arch 'ends' (in this case, the glued rim of the sounding board) cannot 
move, the material of the arch is compressed.  Wood in compression behaves 
like a spring.

The situation is really a bit more complex than this because of the 
stiffening ribs.  These experience bending stress, but also behave like a 
spring.  It doesn't matter very much from a purely physical point of view, 
but since the wood in a crowned sounding board is under compression, we get 
a somewhat stronger structure since wood is a bit stronger in compression 
than in tension, especially across the grain.

I don't think I've told anyone here anything new thus far.  But there's a 
bit more.

If the impedance of two coupled mechanical parts--say the string and the 
sounding board--are different, any energy applied from part #1 will not be 
completely transferred to part #2.  What happens to the energy that's not 
transferred?  It isn't lost: it is _reflected_ back into part #1.

In a piano, the impedance of the string is different from that of the 
sounding board.  Thus part of the energy imparted by the hammer to the 
string is reflected back into the string.  We want this to happen: it's what 
makes the string keep vibrating.  If all of the energy from the string went 
into the sounding board, we'd hear only a dull thump instead of that 
splendid ringing sound.

The piano is a highly efficient system. (System=chain of parts.) The energy 
that's reflected back into the spring is not lost, but stored in the 
vibrating string.  It is transferred to the sounding board gradually, on 
each vibration.

Could the sounding board have a negative crown (like a dish), and the 
downbearing be "upbearing?" From a physical standpoint, yes.  There would be 
no difference in the behavior of the system, though the bridge would peel 
off the sounding board pretty quickly and the design of the bridge pins 
would be interesting.

Or, in another possible configuration, could the sounding board have a 
negative crown and the downbearing still exert force downward?  Again yes.  
(The bridge would have to be rather high.)  The sounding board's wood would 
be in tension and the rim of the sounding board would tend to be pulled away 
from the frame, but again from a purely physical standpoint the system would 
work about like a normal piano.  Structurally, of course, this configuration 
would be a disaster, but the sounding board and strings would behave pretty 
much normally if the whole works didn't peel apart.

The relationship of the sounding board to the outside air is a bit more 
complex than the relationship of string to sounding board.  Ideally, we want 
the impedance of the sounding board to match that of the outside air.  This 
makes the piano loud enough to hear.  And the sounding board does just that: 
it's an 'impedance matching' device that tries to match the impedance of the 
strings to that of the air.  (Side note: the bell of a brass instrument is 
also an impedance matching device.)

>From an efficiency standpoint, we do not want energy from the air to be 
reflected back into the sounding board.  From an artist/craftsman 
standpoint, we _do_ want this to happen, because that's part of what gives 
the piano its tone.

It's also worth examining the resonant frequency of the strings vs. that of 
the sounding board.  The resonant frequency of each string is obvious to 
anyone.  The resonant frequency of the mounted sounding board is known to 
the technician, who can determine it when the piano is unstrung: only one 
note sung into the unstrung piano will resonate loudly, and that's the 
resonant frequency of the sounding board.  (This is the simplest case: a 
real sounding board will resonate at several different frequencies.  The 
resonant frequency of the sounding board will also change somewhat under the 
force and mass of the strings.)

We really don't want the resonant frequency of the sounding board to be a 
large factor in the behavior of a piano.  Ideally, the sounding board should 
be forced to vibrate at the frequency of the struck strings.  In practice, 
however, this is not the case.  I don't know if the terms are equivalent 
from one sort of instrument to another, but in the bowed string instruments, 
we get what's called a 'wolf' tone when the string's resonant frequency 
approaches that of the sounding board.  On the 'cello, this occurs at the F 
played on the C string.  I have read about 'wolf' scales on the piano, and 
perhaps this is the same phenomenon.

So as it happens, the resonant frequency of the piano sounding board is a 
factor in its behavior, but it's not necessarily a bad thing: it contributes 
to the overall tone of the instrument, as do all the other 'imperfections.'

I have read the other posts in this thread, and it's clear that everyone has 
a good understanding of these concepts.  However, descriptions, terms, and 
experiences vary, thus making the various contributions appear inconsistent. 
  What I've given here are the terms used in engineering and physics.  I 
gladly defer to the experience of the craftsmen who read this list.  A far 
better treatment of the matter is given in an old Scientific American 
article called 'The Physics of the Piano.'  I believe that the complete 
reference is given in Art Reblitz' textbook.

Mark Kinsler
512 E Mulberry St. Lancaster, Ohio USA 43130 740-687-6368
http://home.earthlink.net/~mkinsler1

_________________________________________________________________
Learn how to choose, serve, and enjoy wine at Wine @ MSN. 
http://wine.msn.com/


This PTG archive page provided courtesy of Moy Piano Service, LLC