$2\mathrm{.}19$ Derive the mathematical model for the small$-$motions mechanical system of

Figure $2\mathrm{.}45.$ Determine the equivalent natural frequency, the damping ratio,

$k_1^{\text{'}}=k_1(\frac{L}{2L}{)}^2=\frac{k_1}{4}$

FIGURE $2\mathrm{.}45$