Frequency of a damped motion is always the same(constant) but will be less than undamped one. Only the amplitude changes with time.
As you can see only the amplitude changes in a damped vibration.
Applications of damped oscillations or vibrations:
1)Shock absorbers in automobiles and carpet pads are examples of damping devices.It provides a resistive force with reduces the amplitude or strength of the wave that produces the shock.
Shock absorber in car which prevent the car from excessive vibrations.
2)Another similar application of damped oscillation is the car suspension system. The suspension system of a car makes use of damping to make rides less bumpy and more comfortable by counteracting, and hence reducing, the vibrations of the car when it is on the road.
3) Pendulums in mechanicalenvironment-an example is swing.Swing can be considered as a pendulum which works in a mechanical environment with a resistive force which causes damping.It is because of this damping,the swing comes to rest without external opposite force applied.