Transient Vibrations: Response of Spring–Mass System to a Ramp Function

Ramp Function
The equation of motion in this case is
It is easy to verify by direct substitution that the particular solution is
(7.5)
If the initial conditions are
(7.6)
which is illustrated below.

Response to Ramp Function with x(0) = ˙x(0) = 0
We can interpret this response as an oscillation with amplitude about an equilibrium position that is increasing linearly with time.