Rocking bridges

[Robin Hufford]

Ron Overs,
     There is no need, as far as I am concerned for us to agree or disagree,
please, both you and the other advocates of the rocking  point of view,  believe
as your level of  conviction, presumably involuntary, dictates - as do I.   Assent
or the lack of it is truly meaningless to me and I, by no means consider myself an
expert on any matter but rather wish simply to continue to learn and, hopefully,
improve my understanding thereby.  The unfortunate fact, as you say, of the
necessity of earning a living combined with trepidation concerning the likely
drain on time, has kept me from making the arguments I am  now making  on this
list concerning the behavior of soundboards, rocking bridges and other,
apparently, well-established and many times repeated  conventional opinions.  The
view I propose,  contrary as it is, has no doubt, been disturbing to some, perhaps
more than should have been the case,  but in the overall scheme of life the fact
of this disruption is just another matter of trifling signifance.
     Not to put words in your mouth but I think your views and  those of Ron
Nossman and Del Fandrich have been widely expressed as to the analysis of  the
mechanical forces at work on the soundboard, bridge and strings which you allude
to below.  These involve calculation of the forces and displacements using vector
methods as you also indicate below:  a part of this is the application of these
methods to calculate the varying downbearing load on the bridge and soundboard
produced as you and others say, through cyclic behavior of the strings;
calculations would indicate flexion of the bridge under the strings as being the
principal motion along with that of  "fore and aft:" and otherwise as Del has
indicated. The use of  deflection mechanics and various forms of the flexure
formula are employed enabling one to calculate, approximately,  these deflections
or to design new soundboards,  rib scalings, string scalings, bridges designs,
etc. etc.   Inherent in this model is the notion of motion(!) at the bridge, its
interactive behavior with the strings,(compliance); and the idea that the bridge
then moves the soundboard hence sound.  This method is presented as the best thing
since sliced bread, entirely novel and of such import as to completely supersede
the efforts of all previous periods of piano work.
     I too, once thought, at least for a while, in these terms.  However, at least
for me, subsequent reflection and, as I have said, certain  "troublesome
questions"  eventually led me seek another view.  An aspect of some of  these
"troublesome questions" have been expressed in the form of experiments suggested
both by myself and J Delacour.  They were, apparently not very troublesome to
some, judging from the responses they elicited.    I am no mathematician, nor a
physicist, but rather an intense, inveterate,  student of  a wide variety of
subjects and claim no authority to render final prouncements on these or other
matters but in my opinion the model of soundboard behavior indicated above, which
I will call the deflection model, is fundamentally flawed and in error;  the
conclusions drawn from it also are in error. The predictive value of this model,
applied to piano soundboards, the sound they produce, their deterioration or
improvment over time, and other matters,  is thoroughly  in doubt and, at least in
my opinion, substantially contrary to what one finds in the real world of pianos
around us, a point, not to repeat myself,  I gently tried to make in my post of
the last month entitled "Confessions of a Soundboard Heretic".
     The deflexion model represents, in its application, at least as far as I am
able to determine, a reasonably accurate method of determining  resulting
deflections of the soundboard  under load,  that is the equilibrium point of the
things of piano design;  merely a few of these are:  the results of  changes in
ribbing, changes in string scales,  heights and placements of bridges, all the
convoluted relationships of mass and stiffness in wood, the effect of the rim,
and many, many other things -  that is the whole panoply of physical properties of
piano design and is undeniably useful applied in this fashion.
      However, the use  of this method  in  the analysis of the dynamics of energy
transfer from string to bridge/soundboard and from soundboard to air,  and the
resultant  vibration of members of the system , is inadequate, except in  the most
vague and diffuse sense, to well describe the processes involved.  It is the
nature of the load itself in the system, in this case a piano, which determines
how best to approach such an analysis.
      The rate of loading is a critical matter to these issues. Relatively slowly
applied loads result in  stress/strain relationships that  are susceptible to
analysis using the methods of statics which the deflection model employs; it then
works well.  However, in rapid loading such as, I believe,  occurs in connection
with a vibrating string transfering energy, that is an energy load,  into a
bridge/soundboard assembly, these stress/strain relationships are not the same,
the analysis is incorrect and its conclusions such as bridge rocking, bridge
motion, however phrased,  are flawed, even if they can be calculated.    It is, to
be sure, a valid use of statics to consider this loading as a static interaction,
certainly with respect to the actual behavior of the soundboard assembly as the
strings come under tension,  the analysis would then involve forces and
deflections.  However, this method, when used in connection with the vibrating
strings, should such use occur,  would then be called the equivalent-static-load
method  and would be better being replaced by the energy load method for the
dynamic loading which occurs when the string is set into vibration: this  which
requires methods that are somewhat different.   To quote from THE RESISTANCE OF
MATERIALS by Seely, second edition, "the dimensions of the resisting member and
the properties of the material in the member that give it maximum resistance to an
energy load are quite different from those that give the member maximum resistance
to a static load".   This matter has been thoroughly confused, as far as I can
see, for some years now on this list.
     When I undertook to criticize such a widely accepted view of the behavior of
the soundboard/string interaction at the time claims were made implying a sort of
generalized motion existed at this point I did so aware of the distinctions made
above and I immediately,  purposefully,  attempted to lead the discussion in the
direction indicated above, yet no takers.  Subsequently, I have repeatedly couched
my questions in terms that lead to these relations, yet no takers.  Therefore, I
would suggest to all that are unaware of these distinctions that arise from the
nature of loading,  that they take in hand any one of the many texts on Strength
of Materials, Strain of Materials, Resilience of Materials, etc., settle in with
it on a cold winter's night and appreciate the difference between static loading
and energy or impact loading.  Then this discussion could then proceed past the
point of  quibling over the significant or  insignificant  nature of motion at the
string/bridge interface.
      To do so will couch the terms of this discussion, at least as far as I can
see, in terms that are more relevant to the technical questions at hand these
being the usual stress and strain relationships but now used in a dynamic, not
static, context.
 Regards     Robin Hufford


Yes sorry Robin, but we do have to earn a living also from time to time.

>There is far more than merely the
>"tension difference" inhibiting your rocking motion.  The  forces exerted on
>the bridge which itself is stiff expressed as  the downbearing load or to use
>Del's term "downforce" are counterbalanced by  the resisting forces
>provided by
>the soundboard assemply including the ribs, the strain of crownd,
>the rim,  and
>so forth.

True, but the apparent stiffness of the bridge will not prevent it
from rocking as some have asserted (particularly if the bridge height
is not too low). A bridge will 'rock' just as a relatively stiff rim
will flex when the case is leaned upon (I have tested this with dial
gauges previously - and written about it on the list). Similarly, an
engine block will flex when subjected to a moderate hand force. For
those of you who are "doubting Thomas's", confirm this with an engine
reconditioner. Place a bore gauge in the bore of a freshly re-bored
cylinder (in order to test this the bore must be true), the bore
gauge will be capable of supporting its own weight in the bore. Now
flex the engine block (by hand) across the cylinder bore and
perpendicular to the orientation of the bore gauge. The gauge will
then fall out of the cylinder. One could conclude from this that
rigidity is and will always be relative. Therefore, when the
vibrating string goes through a cycle, its tension (which varies
slightly as the speaking length is offset from its resting position)
will cause the bridge to flex slightly backwards and forwards (in a
vector direction parallel to the axis of the speaking length) in
response to the speaking length deflection also. Because the vector
force on the sound board panel is a product of the string tension
times the SIN of the string deflection angle, the downbearing force
will vary similarly to that of the speaking length during the cycle.
Therefore, the board will respond to the position of the speaking
length string segment at each point in the cycle. This is I believe
the most important physical factor which causes the sound board to
respond to the vibration of the speaking length segment.

Now Robin, I do not at this time have numbers to support my
philosophy here. Previously, I didn't feel it necessary to produce a
set just to understand how the process works. Similarly, I suspect
that Charles Darwin had a strong idea of the theory of evolution well
in advance of his voyage on the Beagle (I have read 'The Origin of
Species' but it was a while back). His long held view was merely
confirmed in an ever increasing way as he gathered more information.
I have carried the 'rocking' theory with me for at least fifteen
years (long before I ever knew Ron N and Del existed), and all the
while, thinking about the many different pianos we've rebuilt (as
each one has come along) has re-inforced my view. Even Bösendorfer
seem to be demonstrating an understanding of the principle in their
later pianos by undercutting their dog-leg breaks to allow a more
uniform bridge stiffness. The 'dog-leg's wider footprint, if not
undercut, will tend to 'close' the sound at the breaks (this is just
one more piece of evidence which supports the bridge-rocking theory -
perhaps not to everybody's satisfaction but ah well). I have undercut
dog-legs on many occasions, only to find that the sound improved to
my subjective ears. But please don't accept this idea if you don't
want to. I am not prepared to debate it ad nauseum, since I'm
currently designing two new grand pianos. But when I get some numbers
I will advise the list of my findings.