Question

your work! All MATLAB code must be included in this .pdf file. Problem One Analyzing Roots in the Complex Plane 1. Match the lines in the first graph with the roots in the second graph: Use the legend of the first graph to fill in A,B,C, or D for each line. 2. For roots "A": Estimate 1) the damped natural frequency, wd, (in rad/s) and 2) the damping ratio, C? (Hint: 41,2 = -(wn #wai) 1.5 0.5 0 -0.5 -10 -1.5 0 0.5 1.5 2 25 X Imag 25 А 20 15 X 10 5 Real 01 D 0 -10 X - 15 -20 -25 -16 -12-10 -14 -4 8 2 8 2

Answer 1

1.

C and D are Real. No oscillation. So, the two bottom curves are C and D.

The value of pole C is higher because it is on the left side of D. This means, its steady-state value is lower(higher value in the denominator) than D. In the figure, Yellow is below the purple curve at the end.

So, Yellow - C. Purple - D.

The two upper graphs that have an oscillatory response, in the beginning, belong to A and B since the roots A and B have imaginary parts.

The Blue graph damps out quickly. So, it has a higher damping ratio.

You can see below that Line Joining B to origin makes a higher angle with horizontal. This means it has lower damping than A.

So, Blue - A. Orange - B.

25 Imag 20 15 B 10 5 Real D -5 Х -20 -25 -16 -12 -10 02

2.

From the figure, we can see that the poles of A are:

= CPM Fum- - 15.8 + 27j

-Swn = –15.8; wd = wn = wnV1 - 52 = 27 V1 - 52 = 27/15.8 = 1.71 5 5 = 0.5048