The previous article in this series introduced the “idealized” action components. These are
sufficient for expressing much of the physics, but with regards to Key Return (KR) and “total force”, in
particular, it is sometimes convenient to work with an equivalent “leveraged see-saw” mathematical
model. The three “rearward” masses of such a model, representing the three major components (wippen,
shank, and hammerhead) that rotate at different angular speeds than the keystick, will be described first.
The front, “key” portion of the model is unchanged from that of the idealized Piano Key Action (PKA).
This is followed by a description of the three types of movement “used” by the model in making
predictions. An equation for KR is briefly developed. A “family” of closely-related PKA’s is then
analyzed, using both Key Force One (KFO) measurements and the model. This helps to establish
legitimacy of both the model and the inertia and “key sluggishness” measurements made by the KFO.
The Mathematical Model
A virtual “leveraged see-saw” can be carefully set up so that both its static forces and inertial
properties duplicate those of a given piano action. Because the KR parameter is governed by these same
static forces and inertial properties, it is also predicted by the model. The extra physics required for the
latter involves coupling the KR parameter (an elapsed time) to the upward acceleration of the Application
Point (AP) during the upward movement. This is done by assuming – very reasonably – that a constant
upward acceleration is occurring during an unhindered Key Return movement, and then employing an
appropriate kinematic equation.
The full “see-saw” model is shown in Figure 4, with the AP to the right. As stated, the rightmost
portion of the model (the green segment in the figure) is identical to the idealized key shown in Figure 1
of the last article. The distance from the pivot P to the AP is “L”. A moment or torque “Mspr” is shown,
representing any spring or magnetic forces that may exist – against the keystick or wippen  for the
purpose of reducing static force at the finger. A long, rigid, massless member is attached to the left of the
“key” segment. It exactly locates masses representing the action components  hammerhead, hammer
shank, and wippen  that rotate faster than the keystick. The point masses representing these components
must be carefully determined, along with their exact locations, so they produce the same “static” force
component at the AP and the same inertia component as their counterparts in the idealized and actual
PKA. Formulas will be given, expressing each of these three masses – and their locations  in terms of
the idealized PKA parameters. Once each mass is calculated and located in the model, its moment of
inertia is simply the product of its mass and the square of its distance from P.
More on the Mathematical Model
By equating both the reflected inertia and the static force of the idealized wippen to the inertia and
static force created by the mathematical model’s “wippen” point mass, one can show that the mass mwmod
and location rwmod of the “wippen” point mass are as given in Table 1. With the model created, the
moment of inertia of the wippen mass is simply the product of mwmod and the square of rwmod
By equating both the reflected inertia and the static force component of the idealized hammer
shank to the inertia and static force component created by the mathematical model’s “shank” point mass,
one can show that the mass mshmod and location rshmod of the point mass are as given in Table 1. In the
mathematical model, the moment of inertia of the shank mass is simply the product of mshmod and the
square of rshmod.
By equating both the reflected inertia and the static force of the idealized hammerhead to the
inertia and static force created by the “hammerhead point mass” of the mathematical model, one can show
that the mass mhmod and location rhmod for this point mass are as given in Table 1. Because the
hammerhead is part of the same “local group” as the shank, these expressions look quite similar to those
given for the shank. Note how the product of the expression for mhmod and the square of rhmod is (mhact)
(Rh)2(MR)2, identical to the reflected inertia equation derived in the Idealized Hammerhead section of the
last article.
A Note on the Action Ratio
Since the Action Ratio varies somewhat throughout the stroke, it should be either: (a) averaged
across a reasonable stroke, or (b) determined at a point approximately halfway along a “full” stroke. It is
very often the case that the “hammer axis to hammerhead center of mass” line is approximately horizontal
when the key is approximately halfway down. If this is the case, then the product (AR)(L) equals the
product (Rh)(MR), and the expression for mhmod in Table 1 becomes mhact. Similarly, mshmod becomes
three-fourths of mshact.
Using the Mathematical Model to Calculate IK, BF, ATF, and KR
Once the values of an idealized action are plugged into the formulas in Table 1, one can easily use
the mathematical model to calculate the moment of inertia of each major component, as was demonstrated
above, using well-known formulas. The IK is obtained by simply summing these components’ inertias.
No “free body” force diagrams are necessary. One does not even need the “seesaw” mathematical model
to determine the IK. As described earlier, one can simply calculate the individual idealized components’
reflected inertias and sum them. To calculate the BF, Average Total Force (ATF) and KR from the
model, it is convenient to utilize one of three corresponding force diagrams on the model. Each diagram
corresponds to a particular type of movement of the mechanism. These diagrams, and their usage, are now
described.
The Force Diagrams, and More on Using the Model
The three types of movement necessary for determining BF, ATF, and KR from the model are
constant-speed, constant downward acceleration, and unhindered upward acceleration, respectively.
Figure 5(a) is a “free body” force diagram representing a constant-speed movement of the mathematical
model’s AP, in either the upward or downward direction, where friction is set to zero. With friction
absent, the speed could even be considered zero. By summing moments about P, and setting them equal
to zero, one can easily get the BF in this situation. In the diagrams, “g” is the acceleration due to gravity,
0.00981 mm/ms2. Figure 5(b) is a “free body” force diagram representing an accelerated downstroke of
the model, with the constant acceleration at the AP fully known and controlled, and the reaction force at
the AP being the ATF. One then applies Newton’s Second Law to this situation. The gravitational
“moments” (torques), about the pivot P, of all the various masses in the model, along with that due to
frictional force “FF” acting at the AP, that due to the ATF, and that due to Mspr (if applicable), are
summed and set equal to the product of IK and the known angular acceleration. One may use this
equation to solve for the ATF, directly from the model. Finally, Figure 5(c) is a “free body” force
diagram representing a Key Return movement of the mathematical model, where the AP is allowed to rise
unhindered by some known distance, with the total elapsed time for the movement being the KR, in ms.
As mentioned earlier, kinematics is used to tie the KR value to the upward acceleration of the AP. As in
the accelerated downstroke situation, the gravitational “moments” of all the masses in the model are
summed. A frictional moment also exists from frictional force FF, but it now acts downwardly, opposing
the upward motion. The sum of all these moments, and any due to Mspr, is then set equal to the product of
IK and angular acceleration. This is used to solve for angular acceleration, which is tied directly –
through kinematics  to the KR value. KR is thus also predicted directly by the model. The equation for
KR is:
Eq. 2
where “k” is a factor that depends on some simple geometric parameters, including the upward distance
travelled.
Calculating the Inertia and Key Return of Typical Piano Actions,
and Verification with KFO Measurements
The TMA “trick” for Measuring IK
For convenience, Eq. 1 from the previous article is given here.
Eq. 1
Years of testing and development have taught the author to incorporate a helpful trick during “total
force/IK” measurements. In order for Eq. 1 to consistently yield accurate values for IK, the (ATF – ADF)
term should be maximized. With this term appearing in the numerator, inevitable inaccuracies in those
measured values affect the resulting IK value. However, if the term is maximized, such errors have less
effect, and the resulting IK value becomes much more accurate. One way of doing this is by minimizing
ADF. For all configurations herein, a known mass is temporarily added to the front of the key, before the
ATF measurements are made. This is referred to as the TMA version of the action/model, where TMA
stands for “temporary mass added”. This significantly decreases the ADF, resulting in a much more
accurate usage of Eq. 1. The “temporary mass” version of ADF will be specified as ADFTMA. It can
either be directly measured with the temporary mass attached, or can be easily “converted” from a
“normal” ADF measurement, taken without the temporary mass. Similarly, the “temporary mass” version
of BF will be called BFTMA. The direct measurement of IK (through use of Eq. 1) on the TMA action is
easily converted to the “normal” IK by subtracting the product of the temporary mass and the square of its
distance from the pivot. If a TMA model or action is being described, the temporary mass is represented
as a mass “mTMA”, at location “rTMA”. While there may be instances herein where KR is also measured
with the TMA mass attached, it should normally be thought of as relating to the normal, “non TMA”
action. If measured with TMA, it will be specifically designated KRTMA.
A Word about Friction
The frictional force that is traditionally measured is “constant speed” friction. However, when the
key is accelerated, the various reaction forces  at the various contact/pivot points of the action  tend to
increase above their constant-speed values. These larger reaction forces, during an accelerated
downstroke, often lead to an additional “accelerated friction” component. To account for this, I
established a unitless Friction Sensitivity Factor, which specifies the sensitivity of the friction to these
acceleration forces. It changes Eq. 1 slightly, by subtracting yet another term – accelerated friction – from
the “ATF – ADF” term. In essence, some of the “total force” experienced is going towards additional
friction, rather than towards accelerating the mechanism. A Friction Sensitivity Factor of 1.0 means an
inertia force equal to ADF would double the overall friction at the AP. A factor of 0.5 means an inertia
force equal to ADF would increase friction by 50%. A factor of zero means the friction is unchanged by
acceleration. For simplicity’s sake, I will leave Eq. 1 as is, and simply adjust it “behind the scenes”,
based on an announced Friction Sensitivity Factor. This factor can often be considered constant across all
same-colored keys of a given piano action.
The Tests versus the Mathematical Model
A “family” of piano actions, fairly representative of typical grand PKA’s, was analyzed. Each
family member differs from the others only in its hammer mass and/or “front mass”. The three family
members are given the names MC1, MC2, and MC3, with MC implying “mass configuration”. Table 2
lists the relevant idealized action parameters for each of the family members. Hammerhead mass is
shown as “mhact”, while the keylead mass to be varied is “mld2”. There are no springs or magnets acting
in the pre-letoff (PLO) region of the action, so Mspr in Figures 4 and 5 is zero. Certain keystick
parameters, not listed in the table, are now given. Pieces A and B are less than a half-inch thick (into
page), while piece C – the wide part near the AP  is much thicker. The masses of A, B, and C are 13.4,
57.5, and 17.1 grams, respectively, their total being 88 grams, or “mkey”. The centers of mass of A, B, and
C are 188, 26.5, and 213.5 mm from P, the latter two being to the right of P in Figure 1. The lengths of A,
B, and C are 102 mm, 327 mm, and 45 mm, respectively. The Ikey,P value in Table 2 is the total moment
of inertia of A, B, and C about P. As shown in the table, the center of gravity of the A, B, C combination
is 30 mm to the right of P. The moment of inertia of the keystick, including the backcheck and capstan, is
2.07 x 106 gmm2. Regarding the wippen, it is very similar to the idealized wippen described above, with
mjack, mbeam and rjack having approximate values of 4 grams, 16 grams and 100 mm, respectively. MRwip is
1.6. Using the equations from that section, this produces an Iwip,loc of 93,333 gmm2, and a reflected inertia
at the key of 239,000 gmm2.
Table 3 shows the mathematical model point mass locations and mass values for the hammerhead,
shank, and wippen, for each configuration. These are obtained directly from Table 1, recalling that all
idealized PKA values are known beforehand. This is done for both the TMA and non-TMA situations.
The BF, AIF, IK, and KR, predicted by the models, are also shown in Table 3. The results of the KFO
measurements, on the TMA version of each configuration, are shown in the rightmost portion of the table.
These are the “measured” values of BF, ATF, IK (using ATF and Eq. 1), and KR. All measurements –
including ADF/BF and KR  were made with a TMA mass of 10 grams, added 243 mm in front of P.
While BF and KR could have been measured on the non-TMA actions, they were only measured on the
TMA versions here. In looking at the TMA results of any configuration, one sees good agreement
between model and measurements, for both the IK and KR parameters. For example, the MC1
configuration resulted in “model” and measured values for IK of 2.06 x 107 and 2.01 x 107, respectively.
For Key Return, the values were 153.3 and 148.5. In both cases, good agreement is achieved. The same
can be said for the other two configurations. However, in looking at predicted changes versus measured
changes – from one configuration to another – the results are not quite as good, at least regarding Key
Return. In changing from MC1 to MC3, consisting of adding 4.5 grams to the hammerhead and 29 grams
out near the A.P., the model predicts changes in IK and Key Return of 8.8 x 106 and 74.8, respectively.
The measurements show corresponding changes of 8.2 x 106 and 87.9. The change in IK is thus predicted
within 7 percent of measurements. However, the Key Return change is only predicted within 16 percent,
in this example. Regarding BF, it is apparent from Table 3 that its measured values agree very well with
the model’s predicted values. This is true for all three configurations. A value of 0.5 was used for the
Friction Sensitivity Factor.
For the normal, “non-TMA” versions of both the MC1 and MC3 configurations, the contributions
that all five components make towards the IK, based on the mathematical model, are shown in Figure 6.
This is the same figure that was shown at the end of the previous article. Even though 29 grams are added
out near the AP, the percentage of the IK due to keyleads increases from 6.3% to only 9.6%. Even with
this keylead addition, the percentage due to the hammerhead increases from 71.5% to 74.9%, due to its
own 4.5 gram increase. While not shown in the figures or tables, additional insight is gained by changing
MC1 in the other direction. That is, decreasing the 8.9 gram hammer mass significantly. If it is
decreased to 5 grams, while also reducing mld1 (from 23 grams to 2 grams) to maintain a 43 grams-force
Balance Force, the Inertia at the Key drops from 2 x 107 g mm2 to only 1.26 x 107 g mm2. The keylead
contribution becomes negligible, with the shank and key contributions each jumping to about 17%.
However, the 5-gram hammerhead still accounts for 64% of the Inertia at the Key in this situation! Thus,
no matter if considering typical treble keys or bass keys, the hammerhead tends to account for anywhere
from 64% to 80% of the IK. The 80% values tend to occur in keys where heavy hammers are statically
balanced by keyleads added closer to the balance rail. In such cases, the hammerhead will account for
nearly 80% of the IK, with the keylead percentage down below 5%. It is also the case that, no matter if
the hammerhead is light or heavy, the contribution towards the IK of the combination of hammerhead and
hammer shank tends to stay between 81% and 87%.
Note that Frictional Force is also shown in Table 3, but only for the measurements on the real
action. This is because there is no mechanism in the model to actually predict frictional force. The
model’s purpose in life is to predict values of IK, KR, and – to a lesser extent – BF and ATF.
In the next article, the model will be further employed to vary both hammer mass and “front mass”
over wide ranges, so their impact on the IK and KR is readily seen. As a preview, Figures 7 and 8 are
shown here. These figures show the effect of varying either hammer mass or “front mass” independently.
Other graphs will be provided in the next article, where hammer mass and front mass are varied
simultaneously, thus maintaining a constant Balance Force.