Question 17.29: Determine the damped natural frequency of the system shown i...

The equation of motion is:

m l^{2} \ddot{\theta}+c b^{2} \dot{\theta}+m g l \theta+k a^{2} \theta=0 .

\ddot{\theta}+\left(\frac{c b^{2}}{m l^{2}}\right) \dot{\theta}+\left(\frac{m g l+k a^{2}}{m l^{2}}\right) \theta=0 .

Comparing with the standard equation of motion for the spring dashpot system, we have

\ddot{\theta}+2 \zeta \omega_{n} \dot{\theta}+\omega_{n}^{2} \ddot{\theta}=0 .

Natural frequency,            \omega_{n}=\sqrt{\frac{m g l+k a^{2}}{m l^{2}}} rad / s .

2 \zeta \omega_{n}=\frac{c b^{2}}{m l^{2}} .

Damping factor,      \zeta=\frac{c b^{2}}{2} \cdot \sqrt{\frac{1}{m l^{2}\left(m g l+k a^{2}\right)}} .

Damped natural frequency,    \omega_{d}=\omega_{n} \sqrt{1-\zeta^{2}} .

\omega_{d}=\sqrt{\frac{m g l+k a^{2}}{m l^{2}}} \times \sqrt{1-\frac{c^{2} b^{4}}{4} \times \frac{1}{m l^{2}\left(m g l+k a^{2}\right)}} .

=\sqrt{\frac{m g l+k a^{2}}{m l^{2}}} \times \sqrt{\frac{4 m l^{2}\left(m g l+k a^{2}\right)-c^{2} b^{4}}{4 m l^{2}\left(m g l+k a^{2}\right)}} .

=\frac{1}{2 m l^{2}} \sqrt{4 m l^{2}\left(m g l+k a^{2}\right)-c^{2} b^{4}} .

=\sqrt{\frac{m g l+k a^{2}}{m l^{2}}}-\frac{c^{2} b^{4}}{4 m^{2} l^{4}} rad / s .