In class, a two$-$dimensional model was presented with concentrated stiffness values $K_F$ and $K_E.$

For algebraic simplification purposes, the assumptions made in class considered the airfoil to be symmetric, elininating the terms invalving e$\_0.$

a$)$ Derive the equations that rebate the farce of lift per unit of span length $L^{\text{'}}$ and the tatal moment per unit of appled torque about the elastic axis MEE, considering a two$-$dimencional aiffoil model.

b$)$ Implement an iterstive model to determine the torque $$ and deffection $z$ in a non$-$symmetric

model now subject to aerodynamic velocity $U_{ow.}.$ Here, you will define and adjust the parmeters that

will impact velocity.

c$)$ Plot the evolution of pasition $z$ and torque $$ for each itteration, given a predetermined value of

velocity $U_{}.$ Ensure, for this velocity value, that the dynamic balance is accieved in the iteration

process, adfusting the deffection and torque values as necessary.

d$)$ Determine the maximum value of divergence speed based on the andytical result from chass and

compare it to the ressult obtained from your implemented model in the previous step.

Implement the model of an airfoil with specific masx inertia and present the specific mass per span

as $h$

Complete the model you implamentad in question $1,$ now considering a control device with a length

$E_b$

a$)$ Implement modets of $a_2$ and $b_2$ as a function of the relative length $E$ of the control device and

plat the graphs of $a_2$ and $b_2$ versus $E.$

b$)$ Compare the values you obtained for lft relative to the length of the control device in the case of

a rigid wing and the value obtained in your flecible model $L_{\text{f}\text{l}\text{a}\text{x}}^{\text{'}}$

c$)$ Calculate the efficiency $$ of control as a function of the freestream velocity $U_{}-$

Based on the parameters of your model, determine the relationship between divergence speed and

the reversal speed of controls.

Implement a dynamic model with two degrees of freedom as presented in class $($where one includes

a moment of inertia relative to axis A of elastic torsion $[$Ae$].$ For th$\text{'}$s model, define the center of

mass of the section, add the moment of inertia relative to the CG$,$ and add the mass per unit span

defined in the previsus step.

Using the matrices you hwwe defined, estimste the natural frequencies and damping coefficients. To

do sa$,$ use the mass and darnping matrices $[C]=[{}_1]+[{}_2],$ where ${}_1$ and ${}_2$ are adjustment

parameter.

For the model without darnping, chaose initisl conditions for $z$ and $$ and perform the integration of

the system using the Predictor$-$Correctar method. Discass the influence of integration time on the

methods' stability.

Simulate the results and determine the oscilistory modes of the system, both in terms of time and

stability. Compare the syatem's behavior when a contral surface is added to the efastic model.

Compare the aerodjynamic response of the model with and without damping and present your

condtusions.

Calculate the value of the damping coefficient for a given ${}_c(r)$ for the interval $kG()darrk=0\mathrm{.}050.$ Plot

the resilts $as$ a function $ofk$ and $G().$

Calculate the damping value for $an$ acrodynamic darr fness cocfficient $ofk=0\mathrm{.}05.$