1.7: Numerical Experiment (Quadratic Roots)
- Page ID
- 10089
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
