Traveling-Wave Decomposition of the Lumped System

• The lumped driving-point admittance of the mass-spring system can be defined in terms of wave variables in a manner analogous to distributed systems.

• An applied force can be written as $f = f_{\mbox{in}} + f_{\mbox{out}}$, where $f_{\mbox{in}}$ is viewed as a force wave traveling into the admittance and $f_{\mbox{out}}$ is viewed as a force wave traveling out of the admittance.

• In a similar way, we can define $v = v_{\mbox{in}} + v_{\mbox{out}}$.

• In distributed systems, waves have a finite propagation speed and force and velocity wave components can be related by a wave impedance, which is defined by the parameters of the medium of travel.

• In lumped systems, however, traveling waves may only be understood to travel instantaneously and there is no particular medium. But an arbitrary reference impedance, $R_{h}$, can still be defined such that:
  $$F_{\mbox{in}} = R_{h} V_{\mbox{in}} \hspace{0.2in} F_{\mbox{out}} = -R_{h} V_{\mbox{out}}.$$

• Making this change of variables, the velocity transfer function can be written
  $$V_{\mbox{out}}(s) = \left(\frac{s^{2} - (k/R_{h})s + k/m}{s^{2} +... ...k/m}\right) V_{\mbox{in}}(s) + \frac{v_{0} k/R_{h}}{s^{2} + (k/R_{h})s + k/m}.$$

• Using the bilinear transform to convert from a continuous-time system to a discrete-time system, we obtain
  $$V_{\mbox{out}}(z) = \left(\frac{b_{0} + b_{1} z^{-1} + z^{-2}}{1 ... ...+ mk\alpha + kR_{h})} (1 + 2 z^{-1} + z^{-2})}{1 + b_{1}z^{-1} + b_{0}z^{-2}},$$
  where the continuous-time impulse function mapping has been bandlimited as $\delta(t) \rightarrow \delta(n)/T_{s}$ and the coefficients are defined as
  $$\begin{eqnarray*} b_{0} &\ensuremath{\stackrel{\Delta}{=}}& \frac{mR_{h} \alpha^... ...R_{h} -mR_{h}\alpha^{2})}{mR_{h}\alpha^{2} + mk\alpha + kR_{h}}. \end{eqnarray*}$$

• Because the reference impedance $R_{h}$ is arbitrary, it is possible to choose a value such that the coefficient $b_{0} = 0$, as
  $$R_{h} = \frac{mk\alpha}{m\alpha^{2} + k}.$$

• In this way, a unit of delay can be factored out of the second-order transfer function, such that
  $$V_{\mbox{out}}(z) = z^{-1} H(z) V_{\mbox{in}}(z) + G(z) v_{0},$$
  where
  $$\begin{eqnarray*} H(z) &=& \frac{a_{0} + z^{-1}}{1 + a_{0} z^{-1}}, \\ G(z) &=&... ...pha T_{s}}\right) \frac{1 + 2 z^{-1} + z^{-2}}{1 + a_{0}z^{-1}}, \end{eqnarray*}$$
  with $a_{0}$ given by
  $$a_{0} = \frac{k - m\alpha^{2}}{k + m\alpha^{2}}.$$

• It is then possible to rewrite $R_{h}$ in terms of $a_{0}$, eliminating $k$, as
  $$R_{h} = \frac{m\alpha}{2}(1 + a_{0}).$$