10 Problem B2.3 OGATA 4ed (Laplace Transform)

Laplace.T	Root-locus	Transient response	Frecuency response

10 Problem B2.3 OGATA 4ed (Laplace Transform)

Let us calculate the Laplace transform of the following functions.
$\begin{displaymath}f(t)=sen(w\cdot t)\cdot cos(w\cdot t)\end{displaymath}$

Solucion:
Let's change the function as follows:

$\begin{displaymath}f(t)=\frac{sen(2\cdot w\cdot t)}{2}\end{displaymath}$

Using the Laplace transform:

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

With what we get:

$\begin{displaymath}F(s)=\frac{w}{s^{2}+{(2\cdot w)}^{2}}\end{displaymath}$
$\begin{displaymath}f(t)=t\cdot e^{-t}\cdot sen(5\cdot t)\end{displaymath}$

Solucion:
We will use the following properties and transforms

$\begin{displaymath}L(t^{n}\cdot f(t))=(-1)^{n}\cdot \frac{d^{n}}{ds}(F(s))\end{displaymath}$

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

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

With what we get:

$\begin{displaymath}F(s)=(-1)\cdot \frac{d}{ds}\left( \frac{5}{(s+1)^{2}+5^{2}} \right)=\frac{10\cdot (s+1)}{((s+1)^2+5^{2})^{2}}\end{displaymath}$
Program with Scilab.
t=0:0.1:5;
ft=t.*exp(-t).*sin(5*t);
s=%s;
fs=10*(s+1)/((s+1)^2+5^2)^2;
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;