Physical Perspective on Repeated Poles in Mass-Spring System

In the physical system, dc and infinite frequency are in fact strange cases. In the case of dc, for example, a nonzero constant force implies that the mass $m$ is under constant acceleration. It is therefore the case that its velocity is linearly growing. Our simulation predicts this, since, using Eq.(F.60) and Eq.(F.59),

$$\begin{eqnarray*} v_m(n) &=& \frac{f^{{+}}_m(n)}{m} - \frac{f^{{-}}_m(n)}{m} = \frac{1}{m} \left[x_2(n+1) + x_2(n)\right] \\ &=& \frac{1}{m} \left[2(n+1) + 2n\right]x_0 = \frac{1}{m} (4 n x_0 + 2 x_0). \end{eqnarray*}$$

The dc term $2x_0/m$ is therefore accompanied by a linearly growing term $2nx_0/m$ in the physical mass velocity. It is therefore unavoidable that we have some means of producing an unbounded, linearly growing output variable.