Question 17.1: Calculate the natural frequency of the systems shown in Fig....

(a) Force in each spring = 2W.

\text { Deflection of weight } W, \delta_{s t}=2 \text { (deflection of spring } 1+\text { deflection of spring 2) } .

=2\left(\frac{2 W}{k_{1}}+\frac{2 W}{k_{2}}\right) .

=4 W\left(\frac{k_{1}+k_{2}}{k_{1} k_{2}}\right) .

Natural frequency,        \omega_{n}=\sqrt{\frac{g}{\delta_{s t}}}=\sqrt{\frac{g k_{1} k_{2}}{4 W\left(k_{1}+k_{2}\right)}} .

=\sqrt{\frac{k_{1} k_{2}}{4 m\left(k_{1}+k_{2}\right)}} rad / s .

(b) Force in spring 1 = W.

\text { Force in spring } 2=\frac{W}{2} .

\text { Deflection of } W, \delta_{s t}=\text { Deflection of spring } 1+\text { Deflection of spring } 2 .

=\frac{W}{k_{1}}+\frac{1}{2}\left(\frac{W}{2} \times \frac{1}{k_{2}}\right) .

=W\left[\frac{1}{k_{1}}+\frac{1}{4 k_{2}}\right] .

=W\left(\frac{k_{1}+4 k_{2}}{4 k_{1} k_{2}}\right) .

\omega_{n}=\sqrt{\frac{g}{\delta_{s t}}}=\sqrt{\frac{4 k_{1} k_{2}}{m\left(k_{1}+4 k_{2}\right)}} rad / s .