Question

Consider a state-variable model of a system:

$\left[\begin{array}{c}{\hat{x}}_1\\ {\hat{x}}_2\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -\alpha & -2\beta \end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ \alpha \end{array}\right]r$

$y=\left[10\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

Where y is the output, and r is the input. The damping ratio $\xi$ and the undamped natural frequency ${\omega }_n$ (rad/sec) of the system are given by

• A. $\xi =\frac{\beta }{\sqrt{\alpha }}$ ; ${\omega }_n=\sqrt{\alpha }$
• B. $\xi =\sqrt{\alpha };{\omega }_n=\frac{\beta }{\sqrt{\alpha }}$
• C. $\xi =\frac{\sqrt{\alpha }}{\beta };{\omega }_n=\sqrt{\beta }$
• D. $\xi =\sqrt{\beta };{\omega }_n=\sqrt{\alpha }$

Answer 1

The correct option is A $\xi =\frac{\beta }{\sqrt{\alpha }}$ ; ${\omega }_n=\sqrt{\alpha }$

Characteristic equation is,

$|sI-A|=0$

$|sI-A|=\left[\begin{array}{cc}s& -1\\ \alpha & s+2\beta \end{array}\right]$

= $s^2+2s\beta +\alpha =0$

$\therefore$ ${\omega }_n^2=\alpha$

${\omega }_n=\sqrt{\alpha }$

$2\xi {\omega }_n=2\beta$

$\xi =\frac{\beta }{\sqrt{\alpha }}$