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.
The system can be described as an ideal force source connected in parallel with the series connection of mass and dashpot . 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 ). Recall that a reflection-free port is always necessary when connecting two adaptors together, to avoid creating a delay-free loop.
The parallel adaptor, viewed alone, is equivalent to a force source driving impedance . 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
Following Eq.(F.30), the series adaptor computes
We do not need to explicitly compute because it goes into a purely resistive impedance and produces no return wave. For the same reason, . Figure F.30 shows a wave flow diagram of the computations derived, together with the result of elementary simplifications.
For the discrete-time case, we have
We now need . To simplify notation, define the two coefficients as
From Figure F.30, we can write
Thus, the desired transfer function is
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.
Wave Digital Mass-Spring Oscillator
Spring and Free Mass