Consider the LTV Vought A$-7$ data indicated below for three flight conditions where H denotes altitude and V denotes velocity. $($Introduction to Aircraft Flight Mechanics, T$.$ R$.$ Yechout, $2003).$ Also consider the pure rolling model indicated below $.$

${}^{}-L_{{}^{}}{}^{}=L_{{}_A}{}_A$

with

$L_{{}^{}}=-\frac{4\mathrm{.}4011}{s},L_{{}_A}=+\frac{28\mathrm{.}461}{s^2}$ for $(H,V{)}_1=(00,000ft,670f\frac{t}{s})$

$L_{{}^{}}=-\frac{1\mathrm{.}2951}{s},L_{{}_A}=+\frac{7\mathrm{.}8591}{s^2}$ for $(H,V{)}_2=(35,000ft,584f\frac{t}{s})$

$L_{{}^{}}=-\frac{2\mathrm{.}9561}{s},L_{{}_A}=+\frac{14\mathrm{.}241}{s^2}$ for $(H,V{)}_3=(35,000ft,876f\frac{t}{s})$

a$)$ Compute decay rate $,$ time constant $,$ and amplitude $$ for an aileron step input ${}_A^{**}=+3$ deg for flight condition condition $(H,V{)}_2.$

b$)$ Consider the A$-7$ and associated flight condition to be Class IV and Category B$.$ Use the table below to determine the roll dynamics flying quality level $($Introduction to Aircraft Flight Mechanics, T$.$ R$.$ Yechout, $2003)$ for flight condition $($H$,$V$)2-$

$\backslash$table$[[$Table $7\mathrm{.}10,$Mavinum roll mode time consiant $($seconds$)],[\backslash$table$[[$Flight phase$],[$category$]],$Class,Level I,Level $2,$Level $3,,,],[$A$,,,,,,,],[,$I and IV$,1\mathrm{.}0,1\mathrm{.}4,10,,,],[$B$,$II and III,$1\mathrm{.}4,3\mathrm{.}0,10,,,],[$C$,$AII,$1\mathrm{.}4,3\mathrm{.}0,10,,,],[,$I, II$-$C$,$ and IV$,1\mathrm{.}0,1\mathrm{.}4,10,,,],[,$II$-$L and III,$1\mathrm{.}4,3\mathrm{.}0,d10,,,]]$

c$)$ Plot ${}^{}(t)$ vs$.t$ and $(t)$ vs$.t$ for an aileron step input ${}_A^{**}=+3$deg and $0t10s$ for all three flight conditions in overlay fashion.

d$)$ Which flight condition exhibits the slowest ${}^{}(t)$ response quickness?

Explain why this flight condition is the slowest to respond.

e$)$ Plot ${}^{}(t)$ vs$.t$ and $(t)$ vs$.t$ for the aileron rectangular input signal indicated below for all three flight conditions in overlay fashion.