Group $1$

Consider the following structural model that represents an

engine at the free end of a cantilever beam. The total mass of

the engine is $M=50$ ton. When the engine is working there is

an eccentric mass $m=0\mathrm{.}1$ ton that rotates with a constant

speed of $240$ rpm $($rotations per minute$)$ along a

circumference of radius Disregard the mass and axial

deformability of the beam.

$EI={10}^{6}kN{m}^2$

$M=50$ ton

$m=0\mathrm{.}1$ ton

$1-$ Assume that i$)$ an initial vertical displacement of $2$ cm is

imposed on the engine and it starts vibrating freely in the

vertical direction, and ii$)$ the maximum displacement in the

following cycle is $1$ cm $.$ Calculate the relative damping factor

$2-$ Calculate the damped angular frequency $($if you didn't

solve the question $1,$ assume $=10\%)$

$3-$ Calculate the maximum vertical displacement of the

engine when it is working and vibrates in the vertical

direction. The engine working is equivalent to the application

of a vertical force of $F=mm{r}^2*sin(wt)$