For large enough values of total harmonic distortion, the
improvement with feedback can be seen on an oscilloscope. The
upper waveform is the input signal, the lower waveform is the
output. As the total harmonic distortion drops from 43% to
13%, there is a marked improvement in the waveform’s shape. We
also calculated the typical accelerations, using the
oscilloscope to look at the accelerometer’s output. The output
signal ranged from -2.2 V to +2.3 V. The accelerometer puts
out 38 mV/g, so the accelerations range from approximately
-58g to 60g.

In an effort to model the system, we also looked at the
system’s frequency response, as seen in Figure 5 of Appendix
C. The effect of the speaker enclosure can again be seen at
490 rad/sec, where the response dips suddenly. Using
techniques from ENGS 52, we found that the system can best be
approximated with the transfer function:

The slope of the frequency response for low frequencies is
60 dB per decade, which corresponds to an s3 term in the
numerator. We then estimated the gain K for which , leading
to K = 1.3x10-8. The peak at 345 rad/sec and the dip at 490
rad/sec correspond to second order terms in the denominator
and numerator, respectively. Some tweaking with the damping
factors resulted in a nice fit with our experimental data. For
the second half of the curve (w > 600 rad/sec), we at first
assumed another second order term in the denominator for the
peak at 900 rad/sec, followed by another in the denominator
for the slight downturn at 1450 rad/sec. This didn’t work very
well. A closer look at the response curve revealed that there
was a slight change in slope at 600 rad/sec. By adding a first
order term in the denominator, we improved the curve’s shape,
but we were still unable to find the correct damping factors
for the peaks at 900 rad/sec and 1450 rad/sec. We then decided
to try squaring the second order term for 900 rad/sec. This
resulted in the desired shape of the curve, and we tinkered
with the damping factors until obtaining the final curve shown
in Figure 5.