Problem B$-2-2$: Three mechanical systems. Study the three mechanical system models shown schemat$-$

ically below comprised of ideal mass, spring, and damper elements connected in different ways. Note that

the indicated rigid frame has negligible mass, and motion is constrained to the vertical direction. Assume

that the three ideal elements have linear constitutive relations. Include effects of gravity in your model. For

each case, do the following:

$($a$)$ Build a free body diagram $($FBD$)$

$($b$)$ Identify the forces applied on the mass

$($c$)$ Show that the three systems have the same mathematical model, which you should derive first in state

space form $(1$st order equations$)($d$)$ Convert the state equations a single $2$nd order ODE.

Figure B$-2-2$: Three different mass$-$spring$-$damper schematics