Dynamics task $2$

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.

FIGURE

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.

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$1.$ Determine the Equations of motion for the above simplified model of airplane :

$$ Derive the equations of motion for the airplane, incorporating the dynamics of the wings. Assume any necessary variables to formulate these equations and state them clearly at the beginning. How many degrees of freedom does the above system have? What are the above types of simplifications referred to$$ in dynamics?

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$2.$ Analyze Mode Shapes and Frequencies:

$$ Using the derived equations, identify and depict the mode shapes. Classify the types of modes and calculate the natural frequencies of the system. Properly interpret the significance of each mode and frequency in the context of the airplane's$$ vibrations.