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 . 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
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
In summary, the engagement and disengagement of the plucking system can be ``feathered'' by a nonlinear spring and damper in the plectrum model.
Piano String Wave Equation
Digitization of the Damped-Spring Plectrum