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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 [470]. 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

$\displaystyle 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

$\displaystyle 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

$\displaystyle \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.


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Piano String Wave Equation
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Digitization of the Damped-Spring Plectrum