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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Virtual Musical Instruments
       String Excitation
          Pluck Modeling

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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 [128]

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


Subsections
Previous: Piano Hammer Mass
Next: Digital Waveguide Plucked-String Model

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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