Question 17.5: Calculate the natural frequency of the system shown in Fig.1...

Force in spring 1,    F_{1}=m g .

Force in spring 2,    F_{2}=\frac{m g b}{a} .

Deflection of mass m,

\delta_{s t}=\frac{F_{1}}{k_{1}}+\frac{F_{2}}{k_{2}} \times \frac{b}{a} .

=\frac{m g}{k_{1}}+\frac{m g b}{k_{2} a} \times \frac{b}{a} .

=m g\left[\frac{1}{k_{1}}+\left(\frac{b}{a}\right)^{2} \times \frac{1}{k_{2}}\right]=m g\left[\frac{k_{1} b^{2}+k_{2} a^{2}}{k_{1} k_{2} a^{2}}\right] .

Natural frequency,    \omega_{n}=\sqrt{\frac{g}{\delta_{s t}}} .

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