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

The equation of motion in this case is

It is easy to verify by direct substitution that the particular solution is

so the total solution is

(7.5)

If the initial conditions are

then and so we get

(7.6)

which is illustrated below.

We can interpret this response as an oscillation with amplitude about an equilibrium position that is increasing linearly with time.