Question 17.30: For the system shown in Fig.17.54, determine the natural fre...

The equation of motion is:

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

Comparing with the standard equation of motion, we have

I=m(a+b)^{2} .

c_{t e}=c a^{2} .

q_{e}=k a^{2} .

Undamped natural frequency, \omega_{n}=\sqrt{\frac{q_{e}}{I}} .

=\sqrt{\frac{k a^{2}}{m(a+b)^{2}}} .

=\frac{a}{(a+b)} \sqrt{\frac{k}{m}} rad / s .

\text { Critical damping coefficient, } c_{t c}=2 I \omega_{n} .

=2 m(a+b)^{2}\left[\left(\frac{a}{a+b}\right) \sqrt{\frac{k}{m}}\right] .

=2 a(a+b) \sqrt{k m} .

Damping factor,        \zeta=\frac{c_{t e}}{c_{t c}} .

=\frac{c a^{2}}{2 a(a+b) \sqrt{k m}} .

=\frac{c a}{2(a+b) \sqrt{k m}} .

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

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

=\frac{a}{2(a+b)^{2} m} \sqrt{4(a+b)^{2} k m-c^{2} a^{2}} .