Making the Felt Nonlinear

• As originally discussed, the felt is modeled as a nonlinear spring with a spring “constant” that increases as the compression of the spring increases. In this way, the spring stiffness constant $k(x_{k})$ is dependent on $x_{k}$.

• Because $x_{k} = f_{k}/k$ and because the driving-point force on the mass and spring system is equal to $f_{k}$, we can write
  $$\begin{eqnarray*} x_{k} &=& (1/k)(f_{h}^{-} + f_{h}^{+}) \\ &=& (R_{h}/k)(v_{h}^{-} - v_{h}^{+}). \end{eqnarray*}$$

• The above expression can then be written in terms of $a_{0}$ as
  $$x_{k} = \frac{1-a_{0}}{2 \alpha} (v_{h}^{-} - v_{h}^{+}).$$

• At any time in the model, the value of $x_k$ can be used to lookup or determine an appropriate value for the spring constant $k$.