The piano hammer, or a felt mallet, can be modeled as a non-linear spring and mass system.
The spring represents the felt portion of the hammer or mallet and its compliance is considered to vary with felt compression. In this way, the felt is modeled as a non-linear spring with a stiffness “constant” that increases as it is compressed.
First consider a linear mass and spring system as shown in Fig. 11. The mass is assumed to have an initial velocity at time zero.
Figure 11:
A mass and spring force diagram.
Since the mass and spring are in parallel, the driving force applied on the left side of the spring, , is equally applied to the spring and to the mass:
The driving point velocity, , is equal to the sum of the spring compression velocity and the mass velocity,
By Hooke's law, the restoring force of the spring is given by
where is the compression distance of the spring and is the spring stiffness constant.
By Newton's law, the force of the mass is
where
represents an initial momentum impulse setting the mass in motion with velocity .
Both of these force equations can be expressed in terms of their Laplace transforms and written as a driving-point admittance:
The admittance function,
, is the steady-state response of the system with zeros at
. The term represents the transient effect of the momentum impulse applied at time zero.