I have been doing some work on this for a while now and I am confused by the very first point.
Looking at the bode plot it starts off at 50dB and has a slope of -20d/dec. I am not sure how to handle this first section.
My transfer function so far is:
I know how to find K and zeta but in my equation the value for a alludes me. My initial assumption is to take a=.00001, something close to 0.
As in class we have only ever gotten bode plots with an initial section which is a constant, if anyone can tell me a way to reasonably handle the first second I would be appreciative.
Best Answer
The piece that you're looking for is $$ T(j\omega) = \frac{K}{j\omega} $$ This has a denominator that looks like \$1 / \omega\$, so the slope of the gain is -20 dB / decade. It also has a phase of -90\$^\circ\$, which matches your phase plot at low frequencies.
You can find \$K\$ by looking at any point on the transfer function. For instance, at \$\omega = 0.01\$, you see a gain of 10 dB, so $$ |T(j0.01)| = \frac{K}{0.01} = 100 K = 10^{10\text{ dB} / 20\text{ dB}} \approx 3.16 $$ so \$K \approx 3.16 \times 10^{-2}\$.
Obviously, your transfer function has more terms in it, but it seems like you have a good handle on those.