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In the musical acoustics literature, the piano hammer is classically
modeled as a nonlinear spring
[495,63,179,76,60,488,165].10.14Specifically, the piano-hammer damping in Fig.9.22 is
typically approximated by
, and the spring
is
nonlinear and memoryless according to a simple power
law:
where
for a linear spring, and generally
for pianos. A
fairly complete model across the piano keyboard (based on acoustic
piano measurements) is as follows [489]:
The upward force applied to the string by the hammer is therefore
 |
(10.20) |
This force is balanced at all times by the downward string force
(string tension times slope difference), exactly as analyzed in
§9.3.1 above.
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