Considering the dc case first (), we see from Fig.F.35 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
The loop on the right in Fig.F.35 receives and adds to that. Since , we see it is linearly growing in amplitude. For example, if (with ), we obtain , or
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:
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.41-F.43) verifies this to be the case.
WD Mass-Spring Oscillator at Half the Sampling Rate