The guyed tower shown in Figure $(1)$ is fixed at the base and supported by two guys. The height of tower is $=30.$ The guys are hinged at the base at a distance $=10$ measured from the tower base. The tower cross section is WWF$2000$x$732,$ and the guy diameter is $20$mm$.$ The modulus of elasticity of tower and guys are $2\mathrm{.}06\backslash$times $105$ MPa and $1\mathrm{.}97\backslash$times $105$ MPa, respectively. The specific gravity of steel is $7850/3.$ The tower is holding a machine at its top having a mass m equal to $350\backslash$times $103.$Assume the machine vibrates with a harmonic excitation in the form of $(=0)$ where $0=300,$ and $=0\mathrm{.}2.$ The structural damping is $0\mathrm{.}2\%$:a$)$Calculate the natural frequency of the structure assuming SDOF system, where the mass is lumped at the top point of tower and no prestress force is applied to the guys.b$)$Evaluate and draw the time history response $($displacement and acceleration$)$ due to the harmonic excitation using the closed$-$form solution.c$)$Calculate the effective damping ratio of an external damping device needed to maintain the amplitude of vibration to be less than H$/500.$d$)$Calculate the maximum stresses in the WWF section due to the harmonic excitation $($with the addition of the external damping device$).$e$)$If the guys are prestressed, recalculate requirements $($a$),($b$)$ and $($c$),$ and the prestress force needed to maintain no slack in the guys.

Fig. $(1)$