A vertical vibratory system is shown in Figure $2.y_1(t)$ and $y_2(t)$ are

prescribed motions of the base and top $($both are massless$),$ respectively.

i$)$ Derive the equation of motion of the system by using the force

balance method.

ii$)$ Determine the natural frequency and damping factor of the system.

HINT: The spring and damper forces are related to the relative displacement and relative

velocity, respectively. Figure $2$