Note: Clearly state any assumptions made in the initial part of the solution. You are free to use hand calculations, coding, or any finite element analysis $($FEA$)$ software to estimate the results. Please attach associated theory and reference for the same. Ensure proper interpretation of trends and provide a comprehensive analysis of the findings. Results with graphical and mathematical$/$logical inference will give you additional brownie points.

For a simplified analysis of vertical vibrations in an airplane, a$$ model as illustrated in the accompanying figure $($Fig$.1),$ is employed to characterize the flight dynamics. In this model, the three masses represent the respective masses of the two wings $($ m$1=$ m $)$ and $($ m$3=$ m$)$ and the fuselage $($ m$2=8$m $).$ The wing stiffnesses are assumed to be symmetrical and can be approximated as cantilever beams.

$3.$ Optimal Wing Dimensions:

$-($a$)$ Given the wings have a rectangular cross$-$section with thickness $($t$)$ and width $($b$),$ determine the optimal width$-$to$-$thickness ratio to achieve the higher flexible first natural frequencies for the airplane.

$-($b$)$ Explain how the geometric parameters affect the natural frequency. You can create representative plots to illustrate the relationship between geometric parameters and natural frequencies.

$-($c$)$ Discuss how the environment of the airplane might affect the natural frequencies. Identify the factors in the environment that could influence these frequencies.

$-($d$)$ Are all the natural frequencies critical for the system? Discuss how to determine which natural frequencies are critical and why they are important for the system's stability and performance.