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.37), 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}
= \...
= 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.26), the beta parameters are

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

Following Eq.$ \,$(F.30), the series adaptor computes

f^{{+}}_J(n) &=& f^{{+}}_1(n)+f^{{+}}_2(n)+f^{{+}}_3(n)
= f(...
&=& \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.

Checking the WDF against the Analog Equivalent Circuit

Let's check our result by comparing the transfer function from the input force to the force on the mass in both the discrete- and continuous-time cases.

For the discrete-time case, we have

$\displaystyle H_m(z) \isdef \frac{F_3(z)}{F(z)}
= \frac{F^{+}_3(z) + F^{-}_3(z)}{F(z)}
= (1-z^{-1}) \frac{F^{-}_3(z)}{F(z)}

where the last simplification comes from the mass reflectance relation $ F^{+}_3(z) = -z^{-1}F^{-}_3(z)$. (Note that we are using the standard traveling-wave notation for the adaptor, so that the $ \pm$ signs are swapped relative to element-centric notation.)

We now need $ F^{-}_3(z)/F(z)$. To simplify notation, define the two coefficients as

a &=& \frac{m}{m+\mu}\\
b &=& \frac{\mu}{m+\mu}

From Figure F.30, we can write

F^{-}_3(z) &=& -a\left[F(z)-z^{-1}F^{-}_3(z)\right] + b\left[-...
F^{-}_3(z) &=& -a\frac{F(z)}{1-(a-b)z^{-1}F^{-}_3(z)}

Thus, the desired transfer function is

$\displaystyle H_m(z) = -a \frac{1-z^{-1}}{1-(a-b)z^{-1}}
= -\frac{m}{m+\mu} \frac{1-z^{-1}}{1-\left(\frac{m-\mu}{m+\mu}\right)z^{-1}}

We now wish to compare this result to the bilinear transform of the corresponding analog transfer function. From Figure F.27, we can recognize the mass and dashpot as voltage divider:

$\displaystyle H^a_m(s) = \frac{ms}{ms+\mu}

Applying the bilinear transform yields

H^a_m\left(\frac{1-z^{-1}}{1+z^{-1}}\right) &=& \frac{m\left(\...
...{-1}}{1 - \left(\frac{m-\mu}{m+\mu}\right)z^{-1}}\\
&=& H_m(z)

Thus, we have verified that the force transfer-function from the driving force to the mass is identical in the discrete- and continuous-time models, except for the bilinear transform frequency warping in the discrete-time case.

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Wave Digital Mass-Spring Oscillator
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Spring and Free Mass