Feathering

Since the pluck model is linear, the parameters are not signal-dependent. As a result, when the string and spring separate, there is a discontinuous change in the reflection and transmission coefficients. In practice, it is useful to ``feather'' the switch-over from one model to the next [473]. In this instance, one appealing choice is to introduce a nonlinear spring, as is commonly used for piano-hammer models (see §9.3.2 for details).

Let the nonlinear spring model take the form

$$f_k(y_d) = k y_d^p,$$

where $p=1$ corresponds to a linear spring. The spring constant linearized about zero displacement $y_d$ is

$$k(y_d) = f^\prime_k(y_d) = pk y_d^{p-1}$$

which, for $p>1$ , approaches zero as $y_d\to0$ . In other words, the spring-constant itself goes to zero with its displacement, instead of remaining a constant. This behavior serves to ``feather'' contact and release with the string. We see from Eq.(9.23) above that, as displacement goes to zero, the reflectance approaches a frequency-independent reflection coefficient $\hat{\rho}_f=\mu/(\mu+2r)$ , resulting from the damping $\mu$ that remains in the spring model. As a result, there is still a discontinuity when the spring disengages from the string.

The foregoing suggests a nonlinear tapering of the damping $\mu$ in addition to the tapering the stiffness $k$ as the spring compression approaches zero. One natural choice would be

$$\mu(y_d) = p\mu y_d^{p-1}$$

so that $\mu(y_d)$ approaches zero at the same rate as $k(y_d)$ . It would be interesting to estimate $p$ for the spring and damper from measured data. In the absence of such data, $p=2$ is easy to compute (requiring a single multiplication). More generally, an interpolated lookup of $y_d^p$ values can be used.

In summary, the engagement and disengagement of the plucking system can be ``feathered'' by a nonlinear spring and damper in the plectrum model.