# 3.7: 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

$s^2+2ξω_0s+ω^2_0=0$

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

1. finding the “underdamped” range of values $$ξ≥0$$ for which the roots $$s_1$$ and $$s_2$$ are complex;
2. finding the “critically damped” value of $$ξ≥0$$ that makes the roots $$s_1$$ and $$s_2$$ equal; and
3. finding the “overdamped” range of values $$ξ≥0$$ for which $$s_1$$ and $$s_2$$ are real.
4. 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

$s_{1,2} = \begin{cases} & 0≤ξ≤ξ_c 0 \\& ξ=ξ_c \\& ξ≥ξ_c\end{cases}$

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

Code $$\PageIndex{1}$$ (MATLAB):

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