Damping an Oscillatory System through 'Derivative' Control
Problem 6.2 deals with a system 'unopposed by friction'; that is,
without mechanical damping.
As covered in Lecture #13 (Oct-26-00), we can create
something which "looks like damping" by incorporating derivative
action in our control of the system.
The two methods we covered in class were:
Rate Feedback Control
and
Proportional-plus-Derivative (PD) Control
General Methodology:
Read the course notes entitled "Proportional, derivative and integral control"
(from Oct-26-00) which review for you the solutions for the natural frequency
and damping ratio of a 2nd-order system (e.g. your CLOSED-LOOP transfer function).
Derive the closed-loop transfer function of your system
(with the given rate feedback or PD control incorporated into
the whole system).
Place your resulting CLOSED-LOOP TF equation in the form:
(w_n^2) / (s^2 + 2*zeta*w_n*s + w_n^2)
By setting a desired zeta and w_n, you can then solve for Gx and Gv.
You can repeat this methodology for the POLES of the PD-controlled
system.
(Note that you will also create a zero with PD control but are
asked to solve for the system poles. Thus, you will examine the
denominator of the closed-loop transfer function.)