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Mass and Dashpot in Series

This is our first example illustrating a series connection of wave digital elements. Figure F.26 gives the physical scenario of a simple mass-dashpot system, and Fig.F.27 shows the equivalent circuit. Replacing element voltages and currents in the equivalent circuit by wave variables in an infinitesimal waveguides produces Fig.F.28.

Figure F.26: External force driving a mass which in turn drives a dashpot terminated on the other end by a rigid wall.

Figure: Electrical equivalent circuit of the mass and dashpot system of Fig.F.26.

Figure: Intermediate wave-variable model of the mass and dashpot of Fig.F.27.

Figure F.29: Wave digital filter for an ideal force source in parallel with the series combination of a mass $ m$ and dashpot $ \mu $ . The parallel and series adaptors are joined at an impedance $ R$ which is calculated to suppress reflection from port 1 of the series adaptor.

The system can be described as an ideal force source $ f(t)$ connected in parallel with the series connection of mass $ m$ and dashpot $ \mu $ . Figure F.29 illustrates the resulting wave digital filter. Note that the ports are now numbered for reference. Two more symbols are introduced in this figure: (1) the horizontal line with a dot in the middle indicating a series adaptor, and (2) the indication of a reflection-free port on input 1 of the series adaptor (signal $ f^{{+}}_1(n)$ ). Recall that a reflection-free port is always necessary when connecting two adaptors together, to avoid creating a delay-free loop.

Let's first calculate the impedance $ R$ necessary to make input 1 of the series adaptor reflection free. From Eq.$ \,$ (F.34), we require

$\displaystyle R = m + \mu

That is, the impedance of the reflection-free port must equal the series combination of all other port impedances meeting at the junction.

The parallel adaptor, viewed alone, is equivalent to a force source driving impedance $ R=m+\mu$ . It is therefore realizable as in Fig.F.20 with the wave digital spring replaced by the mass-dashpot assembly in Fig.F.29. However, we can also carry out a quick analysis to verify this: The alpha parameters are

\alpha_1 &\isdef & \frac{2\Gamma _1}{\Gamma _1+\Gamma _2}
= \frac{2\cdot\infty}{\infty+\frac{1}{m}+\frac{1}{\mu}}
= 2\\
\alpha_2 &\isdef & \frac{2\Gamma _2}{\Gamma _1+\Gamma _2}
= \frac{2\left(\frac{1}{m}+\frac{1}{\mu}\right)}{\infty+\left(\frac{1}{m}+\frac{1}{\mu}\right)}
= 0

Therefore, the reflection coefficient seen at port 1 of the parallel adaptor is $ \rho = \alpha_1 - 1 = 1$ , and the Kelly-Lochbaum scattering junction depicted in Fig.F.20 is verified.

Let's now calculate the internals of the series adaptor in Fig.F.29. From Eq.$ \,$ (F.23), the beta parameters are

\beta_1 &\isdef & \frac{2R_1}{R_1+R_2+R_3}
= \frac{2(m+\mu)}{(m+\mu)+m+\mu}
= 1\\
\beta_2 &\isdef & \frac{2R_2}{R_1+R_2+R_3}
= \frac{2\mu}{(m+\mu)+m+\mu}
= \frac{\mu}{m+\mu}\\
\beta_3 &\isdef & \frac{2R_3}{R_1+R_2+R_3}
= \frac{2m}{(m+\mu)+m+\mu}
= \frac{m}{m+\mu}

Following Eq.$ \,$ (F.27), the series adaptor computes

f^{{+}}_J(n) &=& f^{{+}}_1(n)+f^{{+}}_2(n)+f^{{+}}_3(n)
= f(n) + f^{{+}}_3(n)
\\ [5pt]
f^{{-}}_1(n) &=& f^{{+}}_1(n) - \beta_1f^{{+}}_J(n)
= f^{{+}}_3(n)\\ [5pt]
f^{{-}}_3(n) &=& f^{{+}}_3(n) - \beta_3f^{{+}}_J(n)\\
&=& (1-\beta_3)f^{{+}}_3(n) - \beta_3 f(n)\\
&=& \frac{\mu}{m+\mu}f^{{+}}_3(n) - \frac{m}{m+\mu} f(n)

We do not need to explicitly compute $ f^{{-}}_2(n)$ because it goes into a purely resistive impedance $ \mu $ and produces no return wave. For the same reason, $ f^{{+}}_2(n)\equiv\message{CHANGE eqv TO equiv IN SOURCE}0$ . Figure F.30 shows a wave flow diagram of the computations derived, together with the result of elementary simplifications.

Figure F.30: Wave flow diagram for the WDF modeling an ideal force source in parallel with the series combination of a mass $ m$ and dashpot $ \mu $ .

Because the difference of the two coefficients in Fig.F.30 is 1, we can easily derive the one-multiply form in Fig.F.31.

Figure: One-multiply form of the WDF in Fig.F.30.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-10-15 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University