$($show work and the step$-$by$-$step procedure on how to develop $($derive$)$ all the dimensional derivatives in the Z and M equations$)$ Problem $3$:

Develop expressions for the dimensional stability and control derivatives needed in

the ${}^{}$ and $q^{}$ perturbation equations.

As in class, first develop expressions for $m$ and $f_z$ using partial expansion, then

derive expressions for the partial derivatives in terms of non$-$dimensional coefficients,

finally defining the dimensional derivatives.

Recall that we did the $u^{}($X$-$force$)$ equation in class, starting from:

$u^{}=-$gcos${}_1+\frac{1}{(m)}[f_x+f_{T_x}]$

Neglecting $f_{T_x}$ and defining dimensional derivatives give the following equation:

$u^{}-\{\frac{-\frac{C_{D_u}+2C_{D_1}}{b}ar(q)(S)}{\frac{?}{mU}{?}_{(}()()1)}\}u-\{\frac{-\frac{C_{D_{}}+2C_{L_1}}{b}ar(q)(S)}{(m)}\}-$gcos${}_1=\{\frac{-\frac{C_{D_e}}{b}ar(q)(S)}{(m)}\}e$

$u^{}+x_{u}u+x_{}-$gcos${}_1=x_{e}e$

Hints:

In the ${}^{}$ equation, you will need to develop the expressions for the following dimensional

derivatives: $Z_u,Z_{},Z_{{}^{}},Z_q,$ and $Z_e.$

In the $q^{}$ equation, you will need to develop the expressions for the following dimensional

derivatives: $M_u,M_{},M_{{}^{}},M_q,$ and $M_e.$