A machine and its foundation weigh is $10(L)kN.$ The spring constant and the damping ratio of the soil supporting the foundation may be taken as $(L){10}^{3}k\frac{N}{m}$ and $c=(L){10}^{2}kN-\frac{s}{m}.$ Forced vibration of the foundation is caused by a force that can be expressed as determine:

$Q(kN)=(L)sin2t$

Determine:

$($a$)$ amplitude of motion during machine vibration,

$($b$)$ maximum dynamic force transmitted to the subgrade during machine vibration.

$($c$)$ the type of system and its frequency after turning off the machine, and

$($d$)$ the ratio of two successive amplitudes after turning off the machine.