Problem $33$

The rigid bar with mass per unit length $$ shown at the right is hinged at the left end, where a rotational spring of stiffness $k_R$ is attached. A mass $m$ is attached to the end of the bar to act as a vibration absorber; the motion of the mass is only vertical. Here, the rotational angle of the bar, $(t),$ is taken to be positive counterclockwise, and $u(t)$ is the displacement of the discrete mass $m$ relative to the end of the rigid bar. The bar is excited by a distributed force given by

$20$

$p(x,t)=\{\begin{array}{cc}(2p_0\frac{x}{L})f(t),& 0x\frac{L}{2}\\ 0,& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

For the questions below, assume small rotations.

a$.$ Determine the kinetic and potential energy of the bar$-$mass system in terms of the coordinates $(t)$ and $u(t).$

b$.$ Determine the nonconservative virtual work for the bar$-$mass system.

c$.$ Use Lagrange's equations to derive the equations of motion in terms of the coordinates $(t)$ and $u(t).$

d$.$ Write the equations in the form:

$M{x}^{}+C{x}^{}+Kx=Gf(t),$ where $x={\left[\begin{array}{cc}& u\end{array}\right]}^T$