Instead of solving ordinary differential equations (ODEs), the damped simple harmonic motion (SHM)
is surveyed qualitatively from basic mechanics and quantitatively by the instrumentality of a graph
of velocity against displacement. In this way, the condition ##IMG##
[http://ej.iop.org/images/0031-9120/51/2/025006/pedaa05c5ieqn001.gif] {$bgeqslant sqrt{4mk}~$} for
the occurrence of the non-oscillating critical damping and heavy-damping is derived. Besides, we
prove in the under-damping, the oscillation is isochronous and the diminishing amplitude satisfies a
rule of ‘constant ratio’. All are done on a non-ODE basis.