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Pluck Modeling

The piano-hammer model of the previous section can also be configured as a plectrum by making the mass and damping small or zero, and by releasing the string when the contact force exceeds some threshold $ f_{\mbox{\tiny max}}$ . That is, to a first approximation, a plectrum can be modeled as a spring (linear or nonlinear) that disengages when either it is far from the string or a maximum spring-force is exceeded. To avoid discontinuities when the plectrum and string engage/disengage, it is good to taper both the damping and spring-constant to zero at the point of contact (as shown below).

Starting with the piano-hammer impedance of Eq.$ \,$ (9.19) and setting the mass $ m$ to infinity (the plectrum holder is immovable), we define the plectrum impedance as

$\displaystyle R_p(s) \isdefs \mu+\frac{k}{s} \eqsp \frac{\mu s + k}{s}. \protect$ (10.22)

The force-wave reflectance of impedance $ R_p(s)$ in Eq.$ \,$ (9.22), as seen from the string, may be computed exactly as in §9.3.1:

$\displaystyle \hat{\rho}_f(s)$ $\displaystyle =$ $\displaystyle \frac{\mbox{Impedance Step}}{\mbox{Impedance Sum}}
\eqsp \frac{[R_p(s)+R]-R}{[R_p(s)+R]+R}
\eqsp \frac{R_p(s)}{R_p(s)+2R}$  
  $\displaystyle =$ $\displaystyle \frac{\mu}{\mu+2R} \cdot
\frac{s+\frac{k}{\mu}}{s+\frac{k}{\mu+2R}}
\protect$ (10.23)

If the spring damping is much greater than twice the string wave impedance ($ \mu\gg 2R$ ), then the plectrum looks like a rigid termination to the string (force reflectance $ \hat{\rho}_f(s)=1$ ), which makes physical sense.

Again following §9.3.1, the transmittance for force waves is given by

$\displaystyle \hat{\tau}_f(s) = 1+\hat{\rho}_f(s),
$

and for velocity and displacement waves, the reflectance and transmittance are respectively $ -\hat{\rho}_f(s)$ and $ 1-\hat{\rho}_f(s)$ .

If the damping $ \mu $ is set to zero, i.e., if the plectrum is to be modeled as a simple linear spring, then the impedance becomes $ R_k(s) = k/s$ , and the force-wave reflectance becomes [129]

$\displaystyle \hat{\rho}_f(s) \eqsp \frac{\frac{k}{2R}}{s+\frac{k}{2R}}. \protect$ (10.24)



Subsections
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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA