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3.7: Numerical Experiment (Quadratic Roots)

  • Page ID
    10089
  • 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

    rootsQuadratic.PNG