1.7: Numerical Experiment (Quadratic Roots)
There is a version of the quadratic equation that will arise over and over again in your study of electrical and mechanical systems
\[s^2+2ξω_0s+ω^2_0=0 \nonumber \]
For reasons that can only become clear as you continue your study of engineering, the parameter \(ω_0\) is called a resonant frequency, and the parameter \(ξ≥0\) is called a damping factor. In this experiment, you will begin by
- finding the “underdamped” range of values \(ξ≥0\) for which the roots \(s_1\) and \(s_2\) are complex;
- finding the “critically damped” value of \(ξ≥0\) that makes the roots \(s_1\) and \(s_2\) equal; and
- finding the “overdamped” range of values \(ξ≥0\) for which \(s_1\) and \(s_2\) are real.
- For each of these ranges, find the analytical solution for \(s_{1,2}\) as a function of \(ω_0\) and \(ξ\); write your solutions in Cartesian and polar forms and present your results as
\[s_{1,2} = \begin{cases} & 0≤ξ≤ξ_c 0 \\& ξ=ξ_c \\& ξ≥ξ_c\end{cases} \nonumber \]
where \(ξ_c\) is the critically damped value of \(ξ\). Write a MATLAB program that computes and plots \(s_{1,2}\) for \(ω_0\) fixed at \(ω_0=1\) and \(ξ\) variable between 0.0 and 2.0 in steps of 0.1. Interpret all of your findings.
Now organize the coefficients of the polynomial \(s^2+2ξs+1\) into the array \([12ξ1]\). 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 \(s^2+2ξs+1\) as \(ξ\) ranges from 0.0 to 2.0. Note that r is a 1×2 array of complex numbers. You should observe the Figure. We call this “half circle and line” the locus of roots for the quadratic equation or the “root locus” in shorthand