Soundboard Deflection and Pitch Change / was Downbearing

[Ric Brekne]

Hi all

A few posts back Ron N repeated in passing something that he's had up a 
few times that  I think more then a few of us have not looked closely 
enough at... and perhaps because on the surface perhaps its far to easy 
to take some things more for granted then we should. In any case this 
applies to me in this instance. 

For as long as I can remember I've been under the impression that 
soundboard deflection changes substantially enough due to climatic 
changes to account for at least some significant amount of the seasonal 
pitch changes and tuning instability.  At this point... I'm not so sure 
about that at all any more. In fact it <<looks>> like the contribution 
to pitch change from soundboard deflection is nearly insignificant... 
almost negligible !

Let me illustrate..given the following, and by all means check my 
figurings... (for the moment disregard the width of the bridge and deal 
in simple triangle trig)

- an undeflected string tension of 160 lbs.
- string diameter of 0,8 mm.
- front length of 50 mm.
- back length of 25 mm.

This yields a front length frequency of roughly 4248.88 Hz.  f = SQRT((T 
* 398 *10^6)/(L^2d^2))

If you then deflect this string 1 mm upwards you get a string deflection 
angle of a whopping 3.46 ¤,  a downwards force of 9.59 lbs, and a 
frequency of 4248.98 hz. Thats only a change of 0.106 hz.... at note 88 
or there abouts.  Even a 2 mm deflection would'nt increase the frequency 
of the string more then 0.42 hz and that would at the same time cause a 
string deflection angle of 6.87 ¤ !! and a downbearing force of just 
over 19 lbs... for just one string ! You'd be quickly over 3000 lbs of 
total downbearing force on the soundboard...

If these figures are correct... then clearly soundboard deflection can 
nearly be ignored when it comes to pitch changes.  Because in order for 
the soundboard to deflect the strings enough to make a significant 
change in frequency, it would have to create such gargantuan deflection 
angles that the resulting downbearing force would be waaaaaaaay  to 
great for the soundboard assembly to bear.  Even if you figure a total 
downbearing force of 900 lbs... distributed evenly over roughly 240 
strings... thats about 3.75 lbs a string. A single 1 mm of deflection 
for the above string. more then doubles that.

As the strings get longer.. the increase in string deflection angles and 
downwards force for increase in pitch gets worse. A 1000 mm string of 
1.1 mm diameter with the same 160 lbs undeflected tension will only 
increase 0.016 hz in frequency for a 1 mm deflection... and that 
deflection costs  6.56 lbs in terms of downwards force on the 
bridge/soundboard and results in a 2.44 ¤ string deflection angle.

Ok.. this (if I've figured things right)  raises several questions in my 
mind right off.  First and obviously... how does one account for 
seasonal pitch changes then ?!? Secondly... the bulk of the downwards 
pressure is on the back edge of the bridge.  A real string deflection 
spaces the two triangles in the above example so that tho string tension 
may remain the same (theoretically) across the whole length of the 
bridge... the downwards forces applied at the bridge itself are a result 
of the angles at each edge.  And if the bridge surface is parallel to 
the undeflected string plane... the downwards force would be in the 
neighborhood of 75 % on the back edge... which I cant imagine would do 
the bridge / soundboard assembly any good in terms of long term stress.  
I.e. the question in my last post... what do you do to deal with that 
situation when you are setting up a new bridge and downbearing ?  Then 
thirdly... and this goes back to the origional post that started me off 
on this train... why should the bridge as a second class lever cause any 
concern ?  I mean... if a soundboard can't in fact cause enough change 
in string deflection to even effect a significant change in ptich to 
begin with... then why worry about the seemingly low changes in 
downwards force distributed across the whole of the bridge ??

More questions to be sure.  And I would be very greatfull for some 
answers / discussion... :)

Cheers
RicB