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Bilinear Transform of the Ideal Mass

Starting with the driving point impedance

$\displaystyle R(s) \isdef \frac{F(s)}{V(s)} = ms

the bilinear transform gives the digital impedance

$\displaystyle \frac{F_d(z)}{V_d(z)} \isdef
R_d(z) = R\left(c\frac{1-z^{-1}}{1+z^{-1}}\right)
= mc\frac{1-z^{-1}}{1+z^{-1}}

Multiplying out

$\displaystyle F_d(z) + z^{-1}F_d(z) = mc V_d(z) - mc z^{-1}V_d(z)

and taking the inverse $ z$ transform gives

$\displaystyle f_n + f_{n-1} = mc \left(v_n - v_{n-1}\right)


$\displaystyle \zbox{v_n = v_{n-1} + \frac{1}{mc}\left(f_n + f_{n-1}\right)}

(The $ f_{n-1}$ term is new relative to the FDA.)

Can check: Equivalent to trapezoid rule for numerical integration

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Download SMAC03S.pdf
Download SMAC03S_2up.pdf

``Recent Developments in Musical Sound Synthesis Based on a Physical Model'', by Julius O. Smith III, (Stockholm Musical Acoustics Conference (SMAC-03), August 6--9, 2003).
Copyright © 2006-02-19 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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