OPTIONAL: Challenge Problem

Consider the hanging crane modeled as a thin rod of mass $m_P,$ mass

moment of inertia about one end of the $rod{I}_O,$ and half$-$length $L,$

which is connected to a trolley of mass $m_T$ through a pivot point $O.$

The trolley is rolling without slip on a ceiling rail and is connected to

a wall through a spring of stiffness coefficient $k$ and through a

damper of damper coefficient $b.$ The pendulum's rotation angle is

$(t)$ and the trolley's horizontal displacement is $x(t).$ A horizontal

input force $u(t)$ is applied to the trolley.

$($a$)$ Let $A$ be a point fixed in inertial frame $($e$.$g$.,$ any point on the

wall$)$ and let vec$(AC)$ be the vector between points A and $C.$ Show

that the inertial acceleration of the rod's center of mass $C$ along

the horizontal inertial frame vector vec$(i)$ and along the vertical

direction vec$(k)$ can be expressed as:

$\frac{{?}^{\text{I}}{d}^2\text{v}\text{e}\text{c}(AC)}{d{t}^2}=[(x^{})+L({}^{})cos-L{}^{}{?}^{2}sin]vec(i)+[L({}^{})sin+L{}^{}{?}^{2}cos]vec(k)$

$($b$)$ Derive non$-$linear EOMs for this mechanical system.