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Dale
There is just such a relationship and it is no a mystery, although,
as far as I can tell it is neither comprehended nor taken into account
in the flexural view of soundboard function. This factor is precisely
what I suggested be taken into account in a post put up three years ago
during the debate on the behavior of soundboards, entitled Rocking
Bridges, Dec 30, 2001, in commentary on the Modulus of Resilience,
which was ignored, or misunderstood as an impedance matter which is not
the case. In my opinion, one should view the soundboard, as I have
repeatedly urged, not through the prism of deflection mechanics or
cyclic static pressures but, rather, as an energy absorbtive,
concentrating and transmitting medium, the energy absorbed being the
output of the string, which is a pressure excitation at the terminations.
In my opinion, (and, I am going to drop this phrase through the
remainer of this post although it all should be taken with this
qualification) the soundboard should be seen as a device which has
several functions. These functions themselves are not necessarily
complementary and, in fact, are perhaps somewhat contradictary. How
they are adjusted, vis a vis, one another is the particular solution
found by any given design approach. At one and the same time, the
board, bridge and ribs together, must be stiff enough to ensure loop
stability on the strings which motion of the terminations past certain
limits would preclude; at the same time, it must absorb this energy,
which is just sound, another problem in and of itself;, it must then
concentrate the sound in ways that build up the amplitude and, finally,
transfer momentum out of the system as acoustic radiation.
I will not repeat here the many arguments I have made for the
nature of motion at the bridge and the energy loading that occurs there,
they are, likely, well known.
What follows will be a synoptic treatment of this entire question
which will be published in substantially greater detail later this year,
elsewhere.
There are critical distinctions that arise, as I said in the post
referred to above, from the nature of loading. These were dismissed as
mere impedance issues. Not so.
The absorption by the soundboard of this energy is a function of
its energy resistance. Quoting from the post referred to above, which I
will then elaborate upon:
"The approach taken by your school of thought is generally, as far
as I can tell, expressed in terms of mass and stiffness, flexion, and
the ratio of stress to strain, that is the modulus of elasticity.(here I
can't render appropriate notation due to the limitations of the keyboard
I am using); These are the terms of deflection mechanics, among
others. When applied to the transfer relations between string and
bridge they are inadequate. A better measure of the relations is the
one used in energy loading and that is the modulus of resilience which
is half the quotient of the square of the stress to the modulus of
elasticity. Although the modulus of resilience is in fact a measure of
how much energy is absorbed per unit volume of the material when the
material is stressed to the proportional limit, its implications for the
design and manufacture or remanufacture of soundboards are profound as
it can be used as a predictor for the absorbion of energy or energy
resistance of a member and therefore models the transfer relations
between string and bridge, among others.
Critical implications of the modulus of resilience and energy
loading arise in comparison to those of static loading. Static loading,
whether flexion or axial depends upon the maximum stress developed,
energy loading is substantially different, (quoting from Seely) " the
resistance... of the bar((bridge, rh) to an energy load......depends not
only the maximum unit-stress, s, but also, (1) on the distribution of
stress through the body, since the energy absorbed by a given unit
volume is ((the modulus of resilience is quoted, rh)), and hence
depends upon the degree to which that VOLUME (caps mine, rh) is
stressed, and (2),
and on the number of units of volume of material in the bar ((bridge,
rh)). What this means to those that have not grasped it is that the
transfer relations between string and bridge/soundboard are a function
of the VOLUME and the DISTRIBUTION of stress in the bridge itself, and
not simply the stiffness and mass. The undercutting of the bridge,
thinning of soundboards, tapering of ribs, inner rib angles, etc. are
in fact methods of volume and stress control the purpose of which is to
equalize the stress distribution in the material and thereby optimize
its energy absorptive capacity or control its energy resistance. As far
as I can see, this should be a matter dear to the heart of anyone
attempting to design, remanufacture, or otherwise modify a piano
soundboard.
To further quote from Seely, "...show that the material in a beam
having a constant cross-section is inefficient in absorbing energy.
For example,........a rectangular beam, when loaded at mid-span with a
concentrated load, can absorb only one-ninth as much energy as the same
beam could absorb if all the material in the beam were stressed to the
same degree." The requirement for stress-equalization, hence control of
energy resistance, can be expressed as taper of ribbing, undercutting of
bridges, notching of struts, etc.
It is absolutely critical to understand that energy absorption
under dynamic loading, as indicated above, is functionally different
from that of static loading, one being dependant upon the maximum stress
developed, the other the nature of the stress distribution, a more
complex formulation requiring cognizance of the volume and stress
together. This is, at the least, one important relationship between
mass/stiffness/soundboard area which fundamentally influences the tonal
qualities of an instrument, to use Ron O.'s words.
It is often maintained, erroneously in my view, that the loudness
or softness of a given note is some function of an "impedance" problem,
and that, generally, this is true for the entire system. A much better
view would be to see the entire piano structure as part of a completely
whole, organic system, coupled in a dynamic manner, loaded with acoustic
energy, and subjected to a forced vibration. The energy of these
vibrations may find sinks where it is lost through excessive damping,
or, it may superpose in ways which build it up in the soundboard which,
itself, is the greatest sink of all. One can evaluate the soundfield in
a piano soundboard, the rim, or the plate through various means. A
simple way is to use the mechanic's stethoscope I suggested several
years ago and explore the distribution of sound. The sound produced
by the string is distributed to a greater or lesser degree, throughout
the entire piano structure, which itself is also coupled to the floor,
air, and, generally, the world. Piano design has attempted to control
the distribution and superposition of these forced vibrations,
particularly by attempting to control energy absorption, or its
inverse, energy resistance, in the soundboard, bridge, ribs and rim,
using just the principles described above, whether conscious or not.
The sound does indeed traveld, as structure-borne-sound, through
the entirety of the system, that is all components of the piano but,
particularly through the soundboard, rim and plate. Good design will
attempt to direct sound back into the soundboard where it may assist in
building up the sound pressure level. The acoustic dowel is a design
feature that attempts to facilitate this process. This is, regardless of
any outlandish sales claims arising from this process, the dreaded
"Circle of Sound", and, as such, is a real process. That such things
will happen is a commonplace notion, just taken for granted, a complete
given, and is the norm in sonic analysis. It is astonishing that
technicians, who really should know better, confuse this process, one
indubitably real, with their antagonism for what may be exaggerated
claims by certain factories.
The modulus of resilience is a measure, as I have indicated before,
of the amount of energy a structure may absorb up to the proportional
limit and is, in a way, inadequate for the structure-borne- sound found
in a piano, as we are not looking to take the energy level to such a
point. Its usefulness, however, lies in the perspective it affords.
That is, what are the capabilties of a medium which influence its
ability to absorb and transmit energy, in this case, acoustic energy and
how does one maximize this.
The soundboard can be made more effective at acquiring energy from
the string, and, further, reacquiring it numerous times from the rim and
plate by control of energy resistance and stress distribution, and, in
particular, the equality of stress distribution. Consider an unribbed
soundboard: it has a kind of moisture induced stress to some degree or
the other. Dry it to some point and rib it, either by rib crowning or
compression crowning, a new level of stress, glue it in, now a different
distribution and then press it down by string bearing a further change
in stress. I think it plainly evident piano design has evolved methods
to impart certain stresses into the system for several reasons, for
example, equalization for purposes of acoustic absorption, but also
mechanical reasons such as the need to maintain tuning stability, and
string termination.
Crown, downbearing pressure, board thinning, ribbing, rib-tapering
and inner rim angling achieve a number of these objectives
simultaneously. That is, they can all be made to work together to give
the best chance for equality of stress distribution. If terminations
are to be secure there must be some offset allowed, were there no need
for it acoustically, to counteract the relaxation, some degree of
compression set, in my opinion much smaller than generally claimed,
and plastic reponse of the board after loading by the strings. This is
an utterly paramount, particularly as regards terminations, but,
nevertheless mechanical consideration in a new soundboard, but which, as
most know, must be accomplished effectively if there is to be a
functional soundboard for any length of time after manufacture. This is
another consideration in design discussions, which seems to have been
generally disregarded here on the list.
As I have said above these factors convienently serve control of
energy resistance, itself the heart of acoustic function, which
modulates the nature, along with reflection and superposition, of the
coupled string/soundboard/ rim/case system as well. Where energy
resistance is lessened the system easily absorbs energy from the string
and feeds this energy right back into the vibrating string itself, the
two become a dynamic whole. This, again, is a kind of circle of sound.
It is easily seen that it differs entirely from attributing power and
sustain solely to the degree of transmittivity and reflectivity
resulting from wave activity at an impedance discontinuity which is
expressed by the impedance ratio of the two media. Obviously, the
interplay of these variables, alone, affords a considerable range of
design flexibility, as long as energy resistance is controlled, which,
again, requires equalization of stress distribution, that is
manipulation of both volume and stress levels in a coordinated
fashion. As volume varies as the cube slight changes in dimension,
for example, the soundboard, ribbing, or rib taper, may cause
substantial effects, equalities or inequalities, in the stress
distribution, for better or worse insofar as absorption is concerned.
The board, of course is highly anistropic, which requires
structural alterations the purposes of which are also those of energy
control, such as board thinning, ribs and rib tapering. These, along
with downbearing pressure allow for some level of equalization of
stress. It is entirely possible, as crown lessens, where such does
occur, over time, that these changes actually result in more, rather
than less, equalization, with a probable result being a better sound,
and this may account for the better sound some find in old boards. I
don't urge this as a mechanism I am certain of but, merely, a possible
explanation. Ribbing, with or without crown, lessens the anistropy of
the board. As the speed of sound is much greater along the grain the
ribs, crossing the grain as they do, in at least one functional sense,
lessen this anistropy by providing a sound path which allows the sound
to more effectively travel into the board, where it does it's
superpositional thing, than it could do by simply crossing the grain,
arriving late and attenuated. As it is late, I will not, at the
moment take up the last of the functions I indicated, which is acoustic
radiation from the board itself.
Regards, Robin Hufford
Erwinspiano@aol.com wrote:
> Ron
> Yes & I happen to agree with you. Mysteries are after
> all............still mysteries?
> Dale
>
> I strongly suspect that there is some sort of important relationship
> between mass/stiffness/soundboard area which fundamentally influences
> the tonal qualities of an instrument. Please don't ask me to
> elaborate on this matter at this time. This theory remains just that,
> at present.
>
> Ron O.
>
>
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