Question 17.2: Determine the natural frequency of free vibrations of the sy...

Force in spring 1,    F_{1}=\frac{W b}{a+b} .

Force in spring 2,    F_{2}=\frac{W a}{a+b} .

Deflection of spring 1,  x_{1}=\frac{F_{1}}{k_{1}}=\frac{W b}{k_{1}(a+b)} .

\text { Deflection of spring } 2, x_{2}=\frac{F_{2}}{k_{2}}=\frac{W a}{k_{2}(a+b)} .

\text { Deflection of point } c, \quad \delta=x_{1}+\left(\frac{a}{a+b}\right)\left(x_{2}-x_{1}\right) .

=\frac{W b}{k_{1}(a+b)}+\left(\frac{a}{a+b}\right)\left[\frac{W a}{k_{2}(a+b)}-\frac{W b}{k_{1}(a+b)}\right] .

=\frac{W b}{k_{1}(a+b)}+\frac{W a}{(a+b)^{2}}\left[\frac{a}{k_{2}}-\frac{b}{k_{1}}\right] .

=\frac{W}{(a+b)^{2}}\left[\frac{b(a+b)}{k_{1}}+\frac{a^{2}}{k_{2}}-\frac{a b}{k_{1}}\right] .

=\frac{W}{(a+b)^{2}}\left[\frac{\{b(a+b)-a b\} k_{2}+a^{2} k_{1}}{k_{1} k_{2}}\right] .

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

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