Simple harmonic motion implies that no energy is lost as a system oscillates. In reality, energy does get lost - this is known as damping. Damped harmonic motion accounts for the energy loss.

If the resistive force is opposite to, and proportional to, the velocity the equation giving x as a function of time is:

x(t) = A e^-bt/2m cos(wt)

If there is no damping the solution reduces to the simple harmonic motion solution. Turn the damping on and the negative exponential means that the oscillations decrease as time goes by. The larger the damping parameter b the faster the oscillations die away.