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8 Problem B2.1 OGATA 4ed(Laplace Transform)

Let us calculate the Laplace transform of the following functions.





  1. \begin{displaymath}f(t)=e^{-0.4\cdot t}\cdot cos(12\cdot t)\end{displaymath}


    Solucion:
    We will use the following properties and transforms

    \begin{displaymath}L(e^{-a\cdot t}\cdot f(t))=F(s+a)\end{displaymath}



    \begin{displaymath}L(cos(w\cdot t))=\frac{s}{s^{2}+w^{2}}\end{displaymath}


    With what we get:

    \begin{displaymath}F(s)=\frac{(s+0.4)}{(s+0.4)^{2}+12^{2}}\end{displaymath}


    Let's check the result with Scilab

    t=0:0.5:20;
    ft=exp(-0.4*t).*cos(12*t);
    s=%s;
    numfs=s+0.4;
    denfs=(s+0.4)^2+12^2;
    fs=syslin('c',numfs/denfs);
    fs1=csim('impulse',t,fs);
    subplot(2,1,1);
    plot2d(t,ft,2);
    xtitle('Phrasing');
    xgrid;
    subplot(2,1,2);
    plot2d(t,fs1,1);
    xtitle('Solution');
    xgrid;
    
    Image B2-1a








  2. \begin{displaymath}f(t)=\sin(4\cdot t+\frac{\pi}{3})\end{displaymath}


    Solucion:
    We will decompose the function as follows:

    \begin{displaymath}f(t)=sen(4\cdot t)\cdot cos(\frac{\pi}{3})+cos(4\cdot t)\cdot sen(\frac{\pi}{3})\end{displaymath}


    Laplace transform of us would be:

    \begin{displaymath}F(s)=\frac{4}{s^{2}+4^{2}}\cdot cos(\frac{\pi}{3})+ \frac{s}{s^{2}+4^{2}}\cdot sen(\frac{\pi}{3})\end{displaymath}


    Let's check the result with Scilab.

    t=0:0.5:50;
    ft=sin(4*t+(%pi/3));
    s=%s;
    fs=(4/(s^2+4^2))*cos(%pi/3)+(s/(s^2+4^2))*sin(%pi/3);
    fs2=syslin('c',fs);
    fs1=csim('impulse',t,fs2);
    clf;
    subplot(2,1,1);
    plot2d(t,ft,2);
    xtitle('Phrasing');
    xgrid;
    subplot(2,1,2);
    plot2d(t,fs1,1);
    xtitle('Solution');
    xgrid;
    
    Image B2-1b