Bode plots solve frequency domain problems   Every child who has
ever held a spring upright knows that tugging on the top end
causes the bottom end to start bouncing and that repeated
tugging keeps those oscillations going. Some may notice that
even though both ends always oscillate at the same frequency,
the bottom end bounces higher at some frequencies than at
others. Truly gifted children might even notice that the bottom
end oscillates out of sync with the top end and lagsfurther and
further behind as the frequency increases.  Engineers know that
many mechanical, electrical, and chemical processes with energy-
storing components behave the same way. A step change at the
input end causes decaying oscillations at the output. A
sinusoidal input, on the other hand, causes a sinusoidal output
with an amplitude and a lag time that depend on the frequency of
the oscillations. Not coincidentally, these two phenomena are
related and form the basis of the frequency domain techniques
that are fundamental to the analy