Any completely isolated system exhibiting simple harmonic motion will carry on swinging/vibrating for ever! It conserves mechanical energy.

There will be continuous interchange of kinetic to potential energy and back again.

For example,

In a spring-mass system, the the energy conversion is between kinetic energy and the potential energy stored in a stretched/compressed spring.
In a pendulum, the energy is converted between kinetic energy and gravitational potential energy.

BUT in the real world it is impossible to isolate the system - friction with supports and surrounding air molecules drains energy from the swinging/bouncing system into the environment - the total mechanical energy left in the system therefore gradually decreases.

Energy is still conserved!

BUT energy within the system is being drained to another place.

Damping is the term used to describe a means by which energy drain from the oscillating system can be occomplished.

Remember 'damping' is just a euphemism for frictional forces like air-resistance. Once there is 'damping', the system is no longer conservative.

SHM systems under damping

Damping changes the behaviour of S.H.M. systems. Certain features of the oscillation (amplitude etc.) are dependent on the extent of damping.

An undamped system gives `normal' S.H.M. - a sinusoidal trace - with constant amplitude - no energy transfer out of the sytem

In a lightly-damped system, the amplitude of oscillation decreases slowly as time goes on. An example of such a system would be a pendulum in a clock - air resistance causes the swing's amplitude to decrease with time. the system suffers a steady drain of energy.

A heavily-damped system moves slowly until coming to rest. It does not oscillate but the governing equations are still the same. A spring-mass system immersed in a bath of very viscous liquid would be heavily-damped. Energy is drained from the system so that it takes longer to reach zero.

Critical damping causes a system's amplitude to reach zero (and stay there) in the shortest time possible. Shock absorbers on cars are critically damped.

(Also see resonance)