**E12**

**Second Order Systems**

Today we looked at second order systems of the form:

where ω_{0} is the
natural frequency and ζ is the
damping ratio.

The step response of such a function is given by

One thing that is apparent is that the time is always scaled by
the natural frequency so we get terms of ω_{0}t.
This means that as ω_{0} changes,
the speed of the response changes, but not the shape. In other words ω_{0}
acts to dilate or contract the sequence. This is shown below (the Matlab
Script that generated these system is SecondOrder.m)

**The effect of varying the
natural frequency**

**ζ=0.25, ω _{0} varies.
**

Note that as ω_{0 }varies
it just changes the time scale of the response, the shape doesn't change.
Note also that the poles move away from the origin in a straight line as ω_{0 }varies
(below). For underdamped systems the angle at which
poles are located is determined by ζ,
the distance from the origin is equal to ω_{0}.

**The effect of varying the
damping factor**

If ζ changes, the shape will change. For small ζ we get many oscillation. As ζ increases we get fewer oscillation. Critical damping (no oscillations) occurs when ζ=1. A system is overdamped when ζ>1. It is underdamped if 0<ζ<1.

**ω _{0}=1,
ζ varies.
**

To reiterate, for underdamped systems the angle
at which poles are located is determined by ζ,
the distance from the origin is equal to ω_{0}.
Therefore as ζ changes, the angle
changes and a circular arc is described by the poles with a radius equal to ω_{0}=1.
When ζ=1 the system is
critically damped. When ζ>1
the system is critically damped. When ζ=0
the system is undamped.

**Adding a Zero at the Origin
to the Transfer Function**

Adding a zero to the transfer function caused the final value of the step response to go to zero. We developed this in terms of the Laplace Transform, and by making the argument that multiplication by s (a zero at the origin) is equivalent to differentiation.