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by Full-Measure Response, Inc.
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Enabling your customers to experience the ultimate in keyboard continuity
The Power of the Key Force One Dyna-StatTM Equations
     The complicated mechanism that is the grand piano key action has been
mathematically analyzed in many different ways here at KeyForce 1.   The
goal in this has long been to not only develop an accurate way of compre-
hensively balancing the 88 notes of a piano action, but to actually do so in
cost-effective manner
.   Many dozens of pages of calculations, scattered
across both space and time, were created along the way.  Many of the results
were inevitably discarded, but certain guiding physical principles eventually
emerged.  An offshoot of all this was a rigorous verification of the age-old Down
Weight, UpWeight, Friction equation, and its extension into the world of con-
tinuous forces and displacements.  That was discussed at length in U.S. patent
8,049,090, along with physical means of making these improved methods a
reality.  While these methods certainly improved the accuracy of the longstand-
ing Down Weight, Up Weight and Balance Weight numbers, they had little to
say about the world of
key action balancing
.  In the context
of key action balancing,
is herein taken to mean satisfying
both dynamic (inertial) and "static" equations.
     Solving the equations is not just something that should be done because
it can be, or because it sounds cool or...gasp...geeky.  It should be done because
the result is an incredibly powerful mathematical model for each note across
the keyboard.  Each model has unprecedented predictive capabilities, ultimately
resulting in a nonobvious note-specific prescription for mass changes required
to achieve true key action balancing, both statically and dynamically!
     Enter three primary equations that - taken together - fully govern the behav-
ior of the piano key mechanism.  These equations are:
     Looking at the equations above, one sees that the only "reaction force" terms
are BF in eq. P and F
in eq. Q.  These are both measured noninvasively with
the Key Force 1 machine.  There are no component-level reaction force terms
such as Strike Weight or Front Weight.  Rather, BF and F
are ensconced
solidly in equations chock full of mass-based terms.  Equation XY has no
reaction force terms, and simply expresses the true, effective moment of
inertia in terms of the various masses and locations.  Having these important
properties allows the three individual equations to "talk amongst themselves"
as it were, and thus be solved simultaneously.  Note that Eq. P is the purely
static equation, while the other two equations are the dynamic equations. The
Dyna-Stat process not only brings much-needed dynamic/inertial equations to
the key balancing fight, it brings them in a way that allows the entire shabang to
be solved, and solved in a note-specific manner!
The goal was not simply to form equations that describe each key mechanism,
but for these equations to have certain properties that would ultimately allow
them to be solved together.  This simultaneous solving was really the holy grail
that would allow tremendous predictive power to be brought to bear on real,
measured piano actions.  The important two properties the equations would
have to exhibit for this "holy grail" to be obtained are:
(a)  any "contact" or "reaction force" terms must be measurable at
the mechanism level, with everything in its assembled state, and
(b)  the terms of the equations - with the exception of any reaction
force terms - must be mass-based...expressed in terms of one or
more masses.
The Key Return Time (KRT) Switcheroo
     As I detailed in my 4-Part piano action inertia series in early 2014 issues of
the PTG Journal, this business of accelerating the key and measuring the
resulting total (and inertial) force is fraught with difficulty.  As I indicated
there, the equation-solving process could also be done using the KRT equation,
rather than the inertial force/acceleration equation.  The reason is because both
the inertial force equation and the KRT equation contain the all-important
Inertia at the Key (IK) term.  Thus, while there are four potential equations
available for comprehensive action balancing, two of these are not independent,
and can be switched.  In essence, our development of the Key Return Time pa-
rameter - and its analytical formulation - opened up a more easily accessible
"back door" into solving real-world
key balancing problems.
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