What is relaxation time of an under damped osccilator?

The relaxation time of an underdamped oscillator is the time it takes for the oscillator's amplitude to decrease to approximately 37% (1/e) of its initial value due to damping. In mathematical terms, it is often associated with the damping coefficient and the natural frequency of the system. The specific relaxation time can be determined using the damping ratio, which characterizes how quickly the system returns to equilibrium after being disturbed. In underdamped systems, oscillations occur while the amplitude gradually diminishes over time.