Homework F$.5$

For the planar VTOL system, note that the system naturally divides into so$-$called

longitudinal dynamics $($up$-$down motion$)$ with total force tilde$($F$)$ as the input and alti$-$

tude tilde$($h$)$ as the output, and lateral dynamics $($side$-$to$-$side motion$)$ with tilde$(\backslash$tau $)$ as the input

and tilde$($z$)$ as the output, with tilde$(\backslash$theta $)$ as an intermediate variable.

$($a$)$ Start with the linearized equations of motion and use the Laplace transform to

convert the equations of motion to the s$-$domain.

$($b$)$ For the longitudinal dynamics, find the transfer function from the input tilde$($F$)($s$)$

to the output tilde$($H$)($s$).$

$($c$)$ For the lateral dynamics, find the transfer function from the input tilde$(\backslash$tau $)($s$)$ to the

intermediate state tilde$(\backslash$Theta $)($s$)$ and the output tilde$($Z$)($s$).$ Find the transfer function from

tilde$(\backslash$Theta $)($s$)$ to tilde$($Z$)($s$).$

$($d$)$ Draw a block diagram of the open loop longitudinal and lateral systems.