Figure a$-$c shows the structure to be analyzed in this problem. Note that $AB$ and $BC$ have the same material and cross section, but different length. Neglect axial deformation in your analysis.

a$.$ Determine the mass and stiffness matrix of the structure presented in figure a$.$ Use three degrees of freedom.

b$.$ The structure is modified by constraining the horisontal movement of $C,$ see figure $b.$ The system can now be described by two degrees of freedom. Determine the natural frequencies and sketch the mode shapes of the modified structure.

c$.$ The structure in figure $b$ is assumed to have Rayleigh damping, with $1\%$ damping at $1$ and $5Hz.$ Determine the modal damping in the first and second mode and establish the mass normalized modal damping matrix in the equation below. Assume that $\sqrt[\phantom{2}]{E\frac{\text{I}}{m}{L}^4}=2ra\frac{d}{s}.$

$q^{}+\left[\begin{array}{cc}?& ?\\ ?& ?\end{array}\right]q^{}+q=0$

The structure in figure $b$ should support an electrical engine ontop of $BC.$ The engine introduces a harmonic point load $p(t)=p_{0}sin(t)$ located at $x,$ see figure c$.$ The engine runs at rotation speed very close to the second natural frequency.

d$.$ What is your main concern regarding this engine in relation to the dynamic performance of the structure? Sketch the Argand diagram for a mode in this state. Discuss possible solutions to improvelavoid issues in the presented case.

e$.$ Your boss ignores your concerns, you are stuck with the designed structure and the engine. Suggest placement of the electrical engine along the length axis of $BC(x$ in figure $c)$ to minimize issues regarding dynamic performance of the structure. Sketch and justify your suggestion, make reasonable assumptions where necessary. Note, you are not allowed to place the engine at the supports $B$ or $C.$