A vehicle of 500 kg supported by a transmission system compo

A vehicle of 500 kg supported by a transmission system composed of a spring of stiffness of 400 kN/m and a viscous damper of damping constant 3000 N middot s/m is moving along a rough road. The road profile is assumed to be sinusoidal as illustrated in the figure. Suppose the vehicle moves at a constant speed of 52 m/s. Determine (a) the maximum vertical acceleration of the vehicle; (b) the critical speed the driver should avoid.

Solution

Given:

A (amplitude) = 0.01 m

Mass (m) = 500 kg = 500 x 9.81 N = 4903 N

Spring Stiffness (k) = 400 x 1000 N/m

Damping Constant () = 3000 N.s/m

Speed (V) = 52 m/s

Wavelength () = 2.5 m

Ans:

Angular frequency () is related to spring constant & mass by formula

= Square Root (k/m)

= (400*1000/4903)

= 9.03 Hz

As acceleration is proportional to displacement in simple harmonic motion, magnitude of maximum acceleration can be defined as

A (max) = ² x A

= 9.03² x .01

= 0.13 m/s² (Maximum Acceleration)

Critical Speed at natural frequency () is given by

V = x / (2)

= 9.03 x 2.5 / (2)

= 3.6 m/s (Critical Speed)