Question

A slender uniform rigid bar of mass 'm' is hinged at 'O' and supported by two springs with stiffness 3k and k and a damper with damping coefficient c as shown in figure. For the system to be critically damped, the ratio of $\frac{c}{\sqrt{km}}$ should be________

• A. $4\sqrt{7}$
• B. 2
• C. 4
• D. $2\sqrt{7}$

Answer 1

The correct option is A $4\sqrt{7}$



$I=\frac{M{l}^2}{12}+m{\left(\frac{l}{4}\right)}^2=\frac{m{l}^2}{12}+\frac{m{l}^2}{16}=\frac{7m{l}^2}{48}$

Total torque about hinge 'O'

$I\ddot{\theta }+c{\left(\frac{l}{4}\right)}^2\dot{\theta }+[k\frac{9l^2}{16}+3k\frac{l^2}{16}]\theta =0$

$\frac{7m{l}^2}{48}\ddot{\theta }+\frac{c{l}^2}{16}\ddot{\theta }+\frac{12k{l}^2}{16}\theta =0$

$\frac{7m{l}^2}{3}\ddot{\theta }+\left(c{l}^2\right)\dot{\theta }+\left(12k{l}^2\right)\theta =0$

$\ddot{\theta }+\left(\frac{3c}{7m}\right)\dot{\theta }+\left(\frac{36k}{7m}\right)\theta =0$

$[{\omega }_n=\sqrt{\frac{36k}{7m}}=6\sqrt{\frac{k}{7m}}]$

$2\xi {\omega }_n=\frac{3c}{7m}$

For critical damping
$\xi =1$
$\Rightarrow 2{\omega }_n=\frac{3c}{7m}$

$\Rightarrow 2\times 6\sqrt{\frac{k}{7m}}=\frac{3c}{7m}$

$\Rightarrow 12=\frac{3c}{\sqrt{km}\sqrt{7}}$

$\Rightarrow \frac{c}{\sqrt{km}}=4\sqrt{7}$