Problem $(35$ points$)$:

A machine of mass $M$ rests on a massless clastic floor, which is hinged at the wall, as shown in Figure $10($a$).$ A shaker having total mass $m,$ and carrying two rotating unbalanced masses produces a vertical harmonic force $ml{}^{2}sint,$ where the frequency of rotation may be varied. The floor is made of steel $($Young$\text{'}$s modulus, $E=200$GPa $)$ with the cross$-$section of $b=40cm$ wide, $h=2cm$ thick, and $L=4m$ long; the total mass of the machine and shaker is $M_{er}=100kg$; and the forcing magnitude is $500$ N when the operational speed of the shaker is at $300$ rpm $.$

In the equivalent$/$simplified spring$-$damper$-$mass system $($see Figure $10($b$)),$ the equivalent mass $(M_{eq})$ and stiffness $(K_{eq})$ are, respectively, equal to: $K_{eq}\frac{48EI}{L^3},I\frac{8h^2}{12},$ and $M_{eq}=m_3+M.$

$1-$ Using Figure $10($b$),$ determine the equation of motion using Newton's second law. $(5$ points$)$

$2-$ Determine the natural frequency of this system. $(2$ points$)$

$3-$ For an undamped system $(c=0),$ what is the resulting particular solution amplitude of the machine$-$shaker setup due to rotating unbalance? $(8$ points$)$

$4-$ For a damped system, the resonant amplitude of the system is $x=3,41cm,$ determine the damping coefficient cof this system. $(10$ points$)$

$5-$ Suppose an additional component is rigidly attached to the machine. This component adds an additional mass of $25$ kg $.$ Determine the particular solution amplitude of oscillation for the undamped system with this additional mass. $(10$ points$)$