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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Virtual Musical Instruments
       String Excitation
          Pluck Modeling
             Incorporating Control Motion

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Incorporating Control Motion

Let $ y_m(t)$ denote the vertical position of the mass in Fig.9.22. (We still assume $ m=\infty$.) We can think of $ y_m$ as the position of the control point on the plectrum, e.g., the position of the ``pinch-point'' holding the plectrum while plucking the string. In a harpsichord, $ y_m$ can be considered the jack position [347].

Also denote by $ L$ the rest length of the spring $ k$ in Fig.9.22, and let $ y_e \isdeftext y_m+L$ denote the position of the ``end'' of the spring while not in contact with the string. Then the plectrum makes contact with the string when

$\displaystyle y_e(t) \ge y(t) $

where $ y(t)$ denotes string vertical position at the plucking point $ x_p$. This may be called the collision detection equation.

Let the subscripts $ 1$ and $ 2$ each denote one side of the scattering system, as indicated in Fig.9.23. Then, for example, $ y_1=y_1^-+y_1^+$ is the displacement of the string on the left (side $ 1$) of plucking point, and $ y_2$ is on the right side of $ x_p$ (but still located at point $ x_p$). By continuity of the string, we have

$\displaystyle y(t)\eqsp y_1(t)\eqsp y_2(t). $

When the spring engages the string ($ y_e > y$) and begins to compress, the upward force on the string at the contact point is given by

$\displaystyle f_k \eqsp k\cdot (y_e-y) $

where again $ y_e=y_m+L$. The force $ f_k$ is applied given $ y_e\ge y$ (spring is in contact with string) and given $ f_k < f_{\mbox{\tiny max}}$ (the force at which the pluck releases in a simple max-force model).10.15 For $ y_e< y$ or $ f_k \ge f_{\mbox{\tiny max}}$ the applied force is zero and the entire plucking system disappears to leave $ y_1^- = y_2^-$ and $ y_2^+=y_1^+$, or equivalently, the force reflectance becomes $ \hat{\rho}_f=0$ and the transmittance becomes $ \hat{\tau}_f=1$.

During contact, force equilibrium at the plucking point requires (cf. §9.3.1)

$\displaystyle 0 \eqsp f_1+f_k-f_2 \protect$ (10.25)

where $ f_i \isdeftext -Ky'_i \isdeftext -K\partial y_i/\partial x$ as usual (§6.1), with $ K$ denoting the string tension. Using Ohm's laws for traveling-wave components (p. [*]), we have

$\displaystyle f_i \eqsp f_i^++f_i^- \eqsp Rv_i^+ - Rv_i^-, $

where $ R=\sqrt{K\epsilon }$ denotes the string wave impedance (p. [*]). Solving Eq.$ \,$(9.25) for the velocity at the plucking point yields

$\displaystyle v \eqsp v_1^+ + v_2^- + \frac{1}{2R} f_k, $

or, for displacement waves,

$\displaystyle y \eqsp y_1^+ + y_2^- + \frac{1}{2R} \int_t f_k. \protect$ (10.26)

Substituting $ f_k = k\cdot (y_e-y)$ and taking the Laplace transform yields

$\displaystyle Y(s) \eqsp Y_1^+(s) + Y_2^-(s) + \frac{1}{2R} \frac{F_k(s)}{s} \eqsp Y_1^+(s) + Y_2^-(s) + \frac{k}{2Rs}\left[Y_e(s) - Y(s)\right]. $

Solving for $ Y(s)$ and recognizing the force reflectance $ \hat{\rho}_f(s)$ gives

\begin{eqnarray*} Y(s) &=& \left[1-\hat{\rho}_f(s)\right]\cdot \left[Y_1^+(s)+Y... ...)\cdot \left\{Y_e(s) - \left[Y_1^+(s)+Y_2^-(s)\right]\right\}, \end{eqnarray*}

where, as first noted at Eq.$ \,$(9.24) above,

$\displaystyle \hat{\rho}_f(s) \isdefs \frac{\left(\frac{k}{s}+R\right) - R}{\l... ... \frac{\frac{k}{s}}{2R+\frac{k}{s}} \eqsp \frac{\frac{k}{2R}}{s+\frac{k}{2R}}. $

We can thus formulate a one-filter scattering junction as follows:

\begin{eqnarray*} Y_d^+ &=& Y_e - \left(Y_1^+ + Y_2^-\right)\\ [5pt] Y_1^- &=& Y... ...\hat{\rho}_f Y_d^+\\ [5pt] Y_2^+ &=& Y_1^+ + \hat{\rho}_f Y_d^+. \end{eqnarray*}

This system is diagrammed in Fig.9.24. The manipulation of the minus signs relative to Fig.9.23 makes it convenient for restricting $ y_d^+(t)$ to positive values only (as shown in the figure), corresponding to the plectrum engaging the string going up. This uses the approximation $ y_1(t)=y_2(t)\approx y_1^+(t)+y_2^-(t)$, which is exact when $ \hat{\rho}_f=0$, i.e., when the plectrum does not affect the string displacement at the current time. It is therefore exact at the time of collision and also applicable just after release. Similarly, $ y_d^+(t)>f_{\mbox{\tiny max}}/k$ can be used to trigger a release of the string from the plectrum.

Figure 9.24: Instantaneous spring displacement-wave scattering model driven by the spring edge $ y_e(t)=y_m(t)+L$.
\includegraphics{eps/uscatii}

Previous: Digital Waveguide Plucked-String Model
Next: Successive Pluck Collision Detection

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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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