Mass and Dashpot in Series

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The wave impedance in a propagation medium relates the force-related component of the wave motion to the displacement-related component of the motion. — Click for http://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

A delay-free loop is a feedback loop containing no delays. It therefore cannot be implemented in that form. Typically a unit delay is inserted, or system is rederived in some other way that avoids the delay-free loop. — Click for https://ccrma.stanford.edu/~jos/DigitizingNewton/Delay_Free_Loops.html

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

A digital filter can be visualized as a 'black box' that accepts a sequence of numbers and emits a new sequence of numbers. In digital audio signal processing applications, such number sequences usually represent sounds. For example, digital filters are used to implement graphic equalizers and other digital audio effects. — Click for https://ccrma.stanford.edu/~jos/filters/

The wave impedance in a propagation medium is the force variable (such as pressure for acoustic waves) divided by the velocity variable. In ideal plane waves and traveling waves, the wave impedance is a real, positive number. For spherical waves in air, the wave impedance is different at each frequency. — Click for https://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

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Wave Digital Filters (WDF) digitize analog circuits element-by-element using the bilinear transform and virtual traveling waves. — Click for https://ccrma.stanford.edu/~jos/pasp/Wave_Digital_Filters.html

In physics, a wave is an oscillation that propagates through a medium (space-time, gas, fluid, or solid) — Click for https://en.wikipedia.org/wiki/Wave

The elementary impedance element in mechanics is the dashpot. — Click for http://ccrma.stanford.edu/~jos/pasp/Dashpot.html

A force is required to change the momentum of an object. In the absence of external forces, momentum is conserved. For a mass m in flight, the momentum is m v, where v denotes the velocity of the mass. Newton's second law, F = m a, says that force equals mass times acceleration, i.e., force equals the time derivative of momentum. — Click for https://scienceworld.wolfram.com/physics/Force.html

A waveguide restricts wave propagation to a particular subspace, which is usually a line. Vibrating strings, woodwinds, and transmission lines are examples of one-dimensional waveguides. — Click for http://en.wikipedia.org/wiki/Waveguide

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Mass and Dashpot in Series

This is our first example illustrating a series connection of wave digital elements. Figure F.28 gives the physical scenario of a simple mass-dashpot system, and Fig.F.29 shows the equivalent circuit. Replacing element voltages and currents in the equivalent circuit by wave variables in an infinitesimal waveguides produces Fig.F.30.

**Figure F.28:** External force driving a mass which in turn drives a dashpot terminated on the other end by a rigid wall.
$\includegraphics{eps/massdash}$

**Figure F.29:** Electrical equivalent circuit of the mass and dashpot system of Fig.F.28.
$\includegraphics{eps/massdashec}$

**Figure F.30:** Intermediate wave-variable model of the mass and dashpot of Fig.F.29.
$\includegraphics{eps/massdashdt}$

**Figure F.31:** Wave digital filter for an ideal force source in parallel with the series combination of a mass and dashpot $\mu$ . The parallel and series adaptors are joined at an impedance which is calculated to suppress reflection from port 1 of the series adaptor.
$\includegraphics{eps/massdashjunc}$

The system can be described as an ideal force source

connected in parallel with the series connection of mass

and dashpot $\mu$ . Figure F.31 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

necessary to make input 1 of the series adaptor reflection free. From Eq.(F.44), we require

The parallel adaptor, viewed alone, is equivalent to a force source driving impedance $R=m+\mu$ . It is therefore realizable as in Fig.F.22 with the wave digital spring replaced by the mass-dashpot assembly in Fig.F.31. However, we can also carry out a quick analysis to verify this: The alpha parameters are

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

Let's now calculate the internals of the series adaptor in Fig.F.31. From Eq.(F.33), the beta parameters are

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

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

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.32 shows a wave flow diagram of the computations derived, together with the result of elementary simplifications.

**Figure F.32:** Wave flow diagram for the WDF modeling an ideal force source in parallel with the series combination of a mass and dashpot $\mu$ .
$\includegraphics{eps/massdashwdf}$

Because the difference of the two coefficients in Fig.F.32 is 1, we can easily derive the one-multiply form in Fig.F.33.

**Figure F.33:** One-multiply form of the WDF in Fig.F.32.
$\includegraphics{eps/massdashwdfom}$

Mass and Dashpot in Series

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University