The Hammer Model

• 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, $f$, is equally applied to the spring and to the mass:
  $$f = f_k = f_m.$$

• The driving point velocity, $v$, is equal to the sum of the spring compression velocity and the mass velocity,
  $$v = v_k + v_m.$$

• By Hooke's law, the restoring force of the spring is given by
  $$f_{k}(t) = k x(t) \hspace{0.2in} \Longrightarrow \hspace{0.2in} \frac{df_{k}(t)}{dt} = k v_{k}(t),$$
  where $x(t)$ is the compression distance of the spring and $k$ is the spring stiffness constant.

• By Newton's law, the force of the mass is
  $$f_{m}(t) = m \frac{dv_{m}(t)}{dt} - m v_{0} \delta(t),$$
  where $m v_{0} \delta(t)$ represents an initial momentum impulse setting the mass in motion with velocity $v_{0}$.

• Both of these force equations can be expressed in terms of their Laplace transforms and written as a driving-point admittance:
  $$\begin{eqnarray*} V(s) &=& V_{k}(s) + V_{m}(s) \\ &=& \frac{s}{k} F_{k}(s) + \f... ...&=& \left(\frac{s^{2} + k/m}{k s}\right) F(s) + \frac{v_{0}}{s}. \end{eqnarray*}$$
  The admittance function, $(s^{2 }+ k / m) / k s$, is the steady-state response of the system with zeros at $\pm \sqrt{k/m}$. The $v_{0} / s$ term represents the transient effect of the momentum impulse applied at time zero.