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Considering the dc case first (
), we see from Fig.F.37
that the state variable
will circulate unchanged in the
isolated loop on the left. Let's call this value
. Then the physical force on the spring is always equal to
![$\displaystyle f_k(n) = f^{{+}}_k(n) + f^{{-}}_k(n) = 2x_1(n) = 2 x_0. \qquad\hbox{(spring force, dc case)} \protect$](img5115.png) |
(F.58) |
The loop on the right in Fig.F.37 receives
and adds
to that. Since
, we see it is
linearly growing in amplitude. For example, if
(with
), we obtain
, or
![$\displaystyle x_2(n) = 2 n x_0, \quad n=0,1,2,3,\ldots\,. \protect$](img5121.png) |
(F.59) |
At first, this result might appear to contradict conservation of
energy, since the state amplitude seems to be growing without bound.
However, the physical force is fortunately better behaved:
![$\displaystyle f_m(n) = f^{{+}}_m(n) + f^{{-}}_m(n) = x_2(n+1) - x_2(n) = 2x_0. \protect$](img5122.png) |
(F.60) |
Since the spring and mass are connected in parallel, it must be the
true that they are subjected to the same physical force at all times.
Comparing Equations (F.58-F.60) verifies this to be the case.
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