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 [472]. 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

where corresponds to a linear spring. The spring constant linearized about zero displacement is

which, for , approaches zero as . 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 , resulting from the damping 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 in addition to the tapering the stiffness as the spring compression approaches zero. One natural choice would be

so that approaches zero at the same rate as . It would be interesting to estimate for the spring and damper from measured data. In the absence of such data, is easy to compute (requiring a single multiplication). More generally, an interpolated lookup of 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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University