Newton$-$Euler Method

Problem $22\mathrm{.}1\mathrm{.}1$

Determine the equations of motion for the horizontal bar shown, which is supported by two equal springs of stiffness $k$ and one damper with coefficient $C,$ and which has mass $m$ and moment of inertia about its center $I_G.$ Use generalized coordinates $y$ for the vertical displacement of the center of the bar, positive upwards, and $$ for the rotation of the bar, positive clockwise. Assume that center of the bar does not move horizontally and that the rotation is small enough that the horizontal motion of all other points along the bar $($i$.$e$.,$ the attachment points of the springs and damper$)$ can be neglected. Express your answer as both two individual $($coupled$)$ equations and as a $22$ matrix equation.

Figure $22\mathrm{.}6\mathrm{.}1$: A uniform bar in a plan is held by two spring and a damper and the bar can both translate and rotate. Taken from...