An automobile running on 4 wheels is about to pass over a speed bump as shown in the insert and the analytical model in fugure 4.32(a). the velocity of the vehicle at the time is 10 km/hr.the vehicle is supported by a suspension system that consists of one coilover per wheel(a coilover is a combination of a shock absorber, or a damper, with a coil spring wrapping over its outside. the analytical model of the suspension system is illustrated in figure 4.32(b). the mass of the automobile carried by each wheel is 250 kg. the spring constant of the coilover k=50000 N/m with a damping coefficient c. the damping effect by the flexible tires of the wheels is assumed to be negligible. determine the following:

(a) Estimate the required time in seconds for either the front wheels or rear wheels to pass the speed bump. this time is considered to be small to justify the assumption that the disturbance to the suspension system of the vehicle by the speed bump is instantaneous. the cross section of the speed bump is :d=6 cm and w= 18 cm in figure 4.32(a)

(b) the value of minimum damping coefficent c of the shock absorber for non-oscillatory motion of the vehicle after passing the speed bump.

(c) the expression for the amplitutude of vibration of the mass after the vehicle runs over the speed bump, and also the maxumum amplitutde of vibration of the mass.

(d) the time at which the amplitude of vibration to be 1mm from the inception of the vibration.