Inertia in the Grand Piano Action

** **

Copyright
2007

Roy Mallory

A static analysis of
the grand piano action is simple and much has been written about it. Piano actions are typically set for a
specific down weight—the amount of force it takes to slowly depress a key. But the down weight only tells part of
the story. By definition, a pianist
interacts with the piano action by moving the keys. Therefore, the force felt by the
pianist, and the speed with which he must depress the key to create a certain
volume is highly dependent on the dynamics of the action. All the principal components of the
action, the key, the wippen, and the hammer, are rotating levers, so to analyze
the action requires the physics of rotational dynamics

** **

Sophisticated
analyses of the piano action are available, but such analyses, despite their
accuracy and ability to model subtle effects, fail in some respects. Their density and mathematical
complexity limit their accessibility, and in their description of minutiae, the
larger, fundamental issues involved are often somewhat obscured. As will be shown, the inertias of the main parts of the action—the keystick, wippen, and hammer assembly—are so different that even a simple analysis such as presented here will be sufficient to elucidate their relative contributions

The principal
simplifications are in the treatment of all action components as rigid bodies,
and all pivots as perfect, with no friction and with only one degree of
freedom. Other simplifications will
be described as they are introduced.
Although the errors of the model presented here are probably on the order
of several percent instead of several tenths of a percent, the relative
magnitude of the contributions of the major action components to the dynamics of
the action will be revealed in a meaningful way.

The math required
for the analysis is simple algebra, and the formulae required for the analysis
are simple and few. Let’s start by
showing the relationship between the parameters of linear motion, which are more
familiar to most people, and those of rotational
motion^{1}.

## Linear Motion (units) |
## Rotational Motion (units) | ||

Displacement |
x
(M) |
Angular
displacement |
q (radians) |

Velocity |
v
(M/sec) |
Angular
velocity |
w (radians /
sec) |

Acceleration |
a
(M/sec |
Angular
acceleration |
a (radians / sec |

Mass |
M
(kG) |
Inertia |
I (kG
m |

Force |
F=Ma (
newtons) |
Torque |
T=Ia (newton M) |

Kinetic
energy |
1/2Mv |
Kinetic
energy |
1/2Iw |

The inertia of a point
mass rotating at some radius around an axis of rotation is just its mass times
the square of the radius. A piano
action, however, consists mostly of pivoting rods. The inertia of a thin rod pivoting
around one end is its total mass times it length squared divided by 3.

The key, the wippen, and
the hammer assembly are interconnected by different ratios. That is, rotating the key through some
angle moves the wippen through a different angle, and the wippen moves the
hammer assembly through yet a different angle. A question that must be answered is how
the inertias of the wippen and hammer assembly are felt by the pianist at the
point where the pianist’s finger touches the key. Inertia is reflected through
interconnected parts by the square of their rotational ratio. The simplest example to consider may be
that of two gears. Let’s say that
when gear one rotates one turn, gear two rotates two turns, for a ratio of
2:1. The inertia of gear two, as
felt at gear one, is the square of this ratio, or four times.

Now, we will apply this
relationship to the piano action.
Below are two levers, lever 1, pivoted from its left end, and lever 2,
pivoted at its right end. A small
button at the free end of lever 1 causes it to push on lever two. These two levers could represent the key
and wippen, or the wippen and hammer assembly.

If the right end of
lever 1 moves up by a small distance Dy, the left end of lever
2 moves up by the same distance.
Using the small-angle approximation, the angular movement of lever 1 is
Dy/L1, and the angular
movement of lever 2 is Dy/L2. Therefore, the ratio of their angular
motions is:

_{}

So, the inertia of lever
2, as reflected to lever 1, is just (L1^{ } / L2)^{2}. I should note that a second
approximation was made to derive this simple formula. As levers 1 and 2 rotate, their contact
points, and therefore L1 and L2 vary.
In order to keep the math from getting messy indeed, the slight changes
in L1 and L2 are ignored. We will
find that the inertias of the action components as felt by the pianist are so
different that simplifications such as this will not obscure the lessons that
can be drawn from the results.

Now, on to the
analysis. Both a static and dynamic
analysis is performed, and in order to supply typical values for the various
action parameters, the action parts from middle C of a Steinway B were weighed
and measured. Note that the use of
mixed units may appear to be sloppy, but it was done for a reason—units most
familiar to US piano technicians were used. All results are expressed in metric,
however. Also, weights, which
should be expressed in units of force, are expressed as mass because of the
almost universal habit of doing so.
Only when necessary are they converted.