Rocking bridges

[Knabe]

Bravo Robin!

Finally, some engineering sense brought into the discussion. Excellent
insights!

Thanks, Robin, for getting a decent start in this direction.

The key to modeling any complex system is to break it down into a series of
smaller components that can be modeled.

A piano is really an oscillating system. Oscillators are some of the most
complex objects from both a physical and a mathematical standpoint. And to
make matters worse (from an analysis standpoint), there are multiple
oscillators and multiple resonating components throughout the piano. Some of
the most important characteristics that determine how a piano sounds have to
do with how energy is transferred among all the resonating components and
how each of the oscillators and resonators decays.

What this discussion is about is how this all happens. It seems to me that a
good physicist or mechanical engineer should be able to create a conceptual
model that breaks the piano down into a system of simpler oscillators,
resonators and couplings that could describe most of the behavior of the
sound of a piano. Unfortunately, I'm an EE, not an ME, so I can't do it
(yet).

Here's my preliminary list of simpler systems to model. Others can probably
suggest a better component list than this one, but it's a start.

Vibrating string
Coupling to the sound board via the bridge
Soundboard (with boundary conditions)
Soundboard coupling to the air

Then of course, the hammer/ string impact must be modeled to define the
initial impulse injected into the system.

The goal is to come up with a simple mathematical model that describes the
first order sound production of a piano. From there, one refines the model
to add second, third and fourth order effects to make the model more
accurate, and to make it useful in trading off design parameters.

Best regards,
Doug Knabe
(Not related to my knowledge, but I haven't tried to trace it..)
Piano guy in learning...
mailto:dkanbe@airmail.net

-----Original Message-----
From: owner-pianotech@ptg.org [mailto:owner-pianotech@ptg.org]On Behalf
Of Robin Hufford
Sent: Monday, December 31, 2001 12:15 AM
To: pianotech@ptg.org
Subject: Re: Rocking bridges

>     In the function of the piano critical distinctions must be made if the
analysis is to be accurate.  Some of these, in particular, are:  the
mechanical
stabilization of the vibrations of the string through creation of boundary
conditions, that is string terminations,  which must  impose conditions that
force
the string to  vibrate at a stable, constant frequency, as nearly as
possible; the
acquisition of the energy by the string, which. also is a problem in dynamic
loading,  the transduction of the energy of the vibrating string, its
dispersion to
and radiation from the soundboard.  These processes  are best  described by
methods
appropriate to the nauture of loading, which is, as I  say,  a distinction
long
absent here and in the PTG Journal.
>      Should the distinctions arising from the nature of loading be given
their
due,  then it will be seen that the acquisition and  termination/ reflection
of
energy in the string is a dynamic problem and the static loading method is
inadequate for its description, , the transduction of this energy is a
dynamic
problem; the reflection of superposition of this energy in the soundboard is
also a
dynamic problem and finally, the radiation of sound from the board into the
air is
a problem partaking of  both aspects: those that are dynamic problems are
best
analized by the energy load method.

SNIP

>      On the one hand one must consider  the vibratory behavior of the
soundboard/piano system; on the other one must consider the mechanism of
transfer
of energy from the strings to the radiating system. These functions are
distinct.
I have called the transfer function, if you will,   stress transduction.
Label it
as you will, it nevertheless exists.   In all of  the examples offered  by
you and
your co-proponents  a gradual load is applied hence its relevance to the
loading
of the bridge/soundboard by the strings is questionable as are the
conclusions
drawn thereby.

SNIP

>  Mechanical loading of the soundboard
system by the strings and the ensuing deflection is distinct from the
dynamic
loading applied by the string under vibration.  This is the critical point
and if
properly understood the local stress and deformation of the areas in
contact, that
is the localized  mutual strain of the string, bridge and bridge pin is not
synonymous with a generalized, bodily, physical, substantial motion such as
I take
rocking must be for it to be capable of physically, substantially moving the
soundboard as you claim.   In point of fact not only is it not necessary to
energize the system but would be detrimental if it existed in any
appreciable
degree, something, not to be repetitious, I have asserted before.

SNIP

>A better measure of these relations is the
one used in energy loading and that is the modulus of resilience which is
half of
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
piano
soundboard assemblies are profound as it can be used as a predictor for the
absorbtion 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 the methods of static loading.     Static loading, whether
flexion
or axial, manner 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 on the maximum
unit-stress,
s, but also (1) on the distribution of the stress through the body, since
the
energy absorbed by a given unit of volume is ((the modulus of resilience is
quoted,rh)), and hence depends on the degree to which that VOLUME ((caps
mine rh))
is stressed and,(2) 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 the 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 rim 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 absorbtive 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.

SNIP

> Regards, Robin Hufford