A $200$ kg mass is distributed along the centre of a light, flexible floor as shown in Figure Q$2)($i$).$ The floor can be assumed to deform as a built$-$in beam $($i$.$e$.$ ends are fixed$)$ and bending is mainly resisted by $5$ timber joists with cross$-$sectional dimensions shown in Figure Q$2($ii$).$ When load is applied to the floor, it can be assumed that each joist will deflect by the same amount. The Young's Modulus of the timber is $10$ GPa and the damping ratio for the floor is $3\%.$

a$)$ Calculate the stiffness of the floor.

b$)$ What is the magnitude of the undamped natural frequency of the floor?

c$)$ Determine the damping coefficient for the system

d$)$ State the equation of motion of the system

e$)$ If the mass is suddenly removed, sketch a graph of the approximate shape of the displacement response for at least $3$ oscillations $($assume that the negative direction is downwards$).$

Dimensions in mm

$($i$)$

$($ii$)$

Figure Q$2$