Consider a simple model of a vehicle moving over a bumpy road as illustrated in the following figure. Assume that the vehicle vibrates only in the vertical direction, the stiffness and damping effects of the tire can be neglected, and the tire has good traction and never leaves the road surface.
The free body diagram of this moving-base system can be illustrated as,
The equation of motion is thus,
Suppose that the vehicle is traveling at a constant speed, v, and the road roughness can be approximated by the equation,
The road roughness can then be rewritten in terms of time (instead of position),
The harmonic moving base system is then equivalent to a harmonic vibration excitation with the equation of motion,
Since we seek the steady state solution for this problem (there are no "initial conditions" to prescribe), the displacement solution is just the particular solution for this problem,
Note that if we had initial conditions, then we would need to also find the complimentary solution and weight the sum of the complimentary and particular solutions such that the initial conditions were satisfied. However, due to the damping in this system, the complimentary solution would die away exponentially and after a period of time only the particular solution (i.e. steady state solution) would remain.