Question

If the roots of a second order characteristic equation are given $s_{1,2}=-3\pm j2$, then the values of damping ratio $\xi$ and the damped natural frequency ${\omega }_d$ are respectively

• A. $\left(\frac{1}{\sqrt{13}}and\sqrt{13}rad/\sec \right)$
• B. $\left(\frac{3}{\sqrt{13}}and2rad/\sec \right)$
• C. $\left(\frac{3}{\sqrt{13}}and\sqrt{13}rad/\sec \right)$
• D. $\left(\frac{1}{\sqrt{13}}and2rad/\sec \right)$

Answer 1

The correct option is B

$\left(\frac{3}{\sqrt{13}}and2rad/\sec \right)$

(b)

The roots of the characteristic equation are given by,

$s_{1,2}=-\xi {\omega }_n\pm j{\omega }_n\sqrt{1-{\xi }^2}$

Here, $-\xi {\omega }_n=-3$

$\Rightarrow$ ${\omega }_n=\frac{3}{\xi }$ ...$\left(1\right)$

and ${\omega }_n\sqrt{1-{\xi }^2}=2={\omega }_d$ ...$\left(2\right)$

By putting the value of $w_n$ in equation (2),we get

$\frac{3}{\xi }\times \sqrt{(1-{\xi }^2)}=2$

$\frac{9}{{\xi }^2}\times (1-{\xi }^2)=4$

or $9(1-{\xi }^2)=4{\xi }^2$

$9-9{\xi }^2-4{\xi }^2=0$

or $13{\xi }^2=9$

or $\xi =\sqrt{\frac{9}{13}}=\frac{3}{\sqrt{13}}$