Our oscillating system has
a certain amount of energy stored that is permanently exchanged between
kinetic and potential energy. This is the only energy balance found in
an undamped free harmonic oscillator, where the summation of those energies
is always constant. For the damped free oscillator, the initial energy
(potential, kinetic or both) is gradually lost to the environment through
the action of the frictional force that we have supposed to be proportional
to the velocity. In this case the oscillator will finally stop.