Question

An automobile is modeled as a single-degree-of-freedom system vibrating in the vertical direction. It is driven along a road whose elevation varies sinusoidally.The distance from peak to trough is 0.2m and the distance along the road between the peaks is 35m. If the natural frequency of the automobile is 2 Hz and thedamping ratio of the shock absorbers is 0.15, determine the amplitude of vibration of the automobile at a speed of 60 Km/hr. If the speed of the automobile isvaried, find the most unfavorable speed for the passengers.

Answer 1

Calculate the time taken by the car to complete one cycle as follows:

Calculate the excitation frequency from the following relation:

Convert the unit of natural frequency into rad/s as follows:

Calculate frequency ratio (r) from the following relation:

Substitute 2.992 rad/s for and 12.5664 rad/s for .

For the given conditions use the following expression for amplitude of vibration of car:

…… (1)

Here, Y is amplitude of disturbance caused due to variation in road and is damping ratio.

Substitute 0.1 m for Y, 0.2381 for r and 0.15 for .

Thus, the amplitude of vibration of the automobile is .

At most unfavorable speed the ratio of amplitudes will be maximum.

From equation (1) obtain the following relation:

Differentiate right hand side of above expression with respect to r and equate it to zero obtain the value of r at which is maximum.

From above equation values of are

Substitute 0.15 for in following expression to obtain the value of r.

Therefore,

Take the positive value of to obtain the value of r.

When , calculate the value of excitation frequency as follows:

Substitute 12.5664 rad/s for .

Calculate the time taken by the car to complete one cycle when is 12.3038 rad/s as follows:

Calculate the unfavorable speed (s) of the car from the following relation:

Substitute 0.5107 s for .

Thus, the most unfavorable speed of the automobile for the passengers is .