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There is a version of the quadratic equation that will arise over and
over again in your study of electrical and mechanical systems:
s2+2ξω0s+ω02=0.s2+2ξω0s+ω02=0.
(1)For reasons that can only become clear as you continue your study of engineering, the parameter
ω
0
ω
0
is called a resonant frequency, and the parameter ξ≥0ξ≥0 is called a damping factor. In this experiment, you will begin by
- finding the “underdamped” range of values ξ≥0ξ≥0 for which the roots
s1 and s2 are complex;
- finding the “critically damped” value of ξ≥0ξ≥0 that makes the roots s1
and s2 equal; and
- finding the “overdamped” range of values ξ≥0ξ≥0 for which s1 and s2
are real.
- For each of these ranges,
find the analytical solution for s1,2s1,2 as a function of ω0 and ξ; write
your solutions in Cartesian and polar forms and present your results
as
s
1
,
2
=
,
0
≤
ξ
≤
ξ
c
,
ξ
=
ξ
c
,
ξ
≥
ξ
c
s
1
,
2
=
,
0
≤
ξ
≤
ξ
c
,
ξ
=
ξ
c
,
ξ
≥
ξ
c
(2)
Now organize the coefficients of the polynomial s2+2ξs+1s2+2ξs+1 into the
array [12ξ1][12ξ1]. Imbed the MATLAB instructions
r=roots([1 2*e 1]);
plot(real(r(1)),imag(r(1)),'o')
plot(real(r(2)),imag(r(2)),'o')
in a for loop to compute and plot the roots of s2+2ξs+1s2+2ξs+1 as ξξ ranges from
0.0 to 2.0. Note that rr is a 1×21×2 array of complex numbers. You should
observe Figure 1. We call this “half circle and line” the locus of roots for
the quadratic equation or the “root locus” in shorthand.
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