I suspect that worrying about types of damping, non-linear terms, and
using Lagrangians is way off base for this high school student. I think
referencing a source on solving the normal DSHM equations would be
sufficient.
I would recommend:
MECHANICS, by Symon, Addison Wesley, 1960 (also 1971), p 47 +
or a book by Becker -- I can't find mine -- who borrowed it??
or I suspect most classical mechanics books (Marion?) etc. will handle
the solution particulars of the resulting DSHM differential equation.