Long Answer Problem $1$

A researcher is considering a system to hold a dragonfly of mass $\backslash ($ m $\backslash )$ in place during an experiment. The system produces an upward force $\backslash ($ f$=\backslash$alpha x $\backslash )$ where $\backslash (\backslash$alpha $\backslash )$ is a positive constant and $\backslash ($ x $\backslash )$ is the dragonfly's position. The dragonfly produces an upward force $\backslash ($ d $\backslash ).$

a$)$ If the input is the dragonfly's force $\backslash ($ d $\backslash )$ and the output is the dragonfly's position $\backslash ($ x $\backslash ),$ find the stability of the system. Be sure to justify your answer.

b$)$ Will the system hold the dragonfly in place? Be sure to justify your answer.

c$)$ Let the dragonfly's force be $\backslash ($ d$=\backslash$beta w $\backslash )$ where $\backslash (\backslash$beta $\backslash )$ is a positive constant and $\backslash ($ w $\backslash )$ is the speed of the dragonfly's wings. Draw the block diagram and find the transfer function if the overall input is $\backslash ($ w $\backslash )$ and the overall output is $\backslash ($ x $\backslash ).$ Your block diagram should have two boxes in it$.$