Free or natural oscillations occur when a system is given an initial displacement and then allowed to oscillate without any external influence. Examples of free oscillations are the motion of a simple pendulum and a spring-mass system. In a free oscillation, the system's natural frequency determines the period of oscillation, which remains constant as long as the system is undisturbed.
Forced oscillations occur when an external force is applied to a system, causing it to oscillate with a frequency that is different from its natural frequency. Examples of forced oscillations include a swing pushed by a person. In a forced oscillation, the system's response is determined by the properties of the system and the frequency and amplitude of the external force. The amplitude of the system's response is maximum at the resonance frequency, which is when the external frequency matches the system's natural frequency.
Damped oscillations occur when a system loses energy over time due to dissipative forces like friction, resistance, etc. Examples of damped oscillations include the motion of a pendulum with air resistance, a mass-spring system with friction etc. In a damped oscillation, the amplitude of the motion decreases over time due to the dissipation of energy. The damping factor determines the rate at which the amplitude decreases, and the motion can be classified as underdamped, critically damped, or overdamped depending on the damping factor.