Damped Harmonic Oscillation as an LTI
This is a classic dynamics problem. I think you do not have to go through the convolution to solve these type of problem. Instead, you can draw the frequency response function (FRF) of the system and multiply with the FRF of the forcing function to obtain the resulting FRF. Then the resulting FRF can be transformed into the time domain using the inverse Fourier transform. You can also solve it by doing the convolution. Then you have to use the Duhamel's integral to solve this type of problem.

The link is attached.

http://people.duke.edu/~hpgavin/cee541/sdof-dyn.pdf