underdamped

Underdamped describes a damping regime in which an oscillator returns to equilibrium by oscillating with a gradually decreasing amplitude. It occurs when the damping is light enough that the system’s motion is oscillatory rather than monotonic.

In a standard second-order damped oscillator, the equation of motion is m x'' + c x' + k x = 0, where m is mass, c is damping coefficient, and k is stiffness. A convenient measure is the damping ratio ζ = c / (2 sqrt(mk)). The system is underdamped when 0 ≤ ζ < 1. It is critically damped at ζ = 1 and overdamped when ζ > 1. For the underdamped case, the characteristic roots are complex: -ζ ω_n ± i ω_d, where ω_n = sqrt(k/m) is the natural frequency and ω_d = ω_n sqrt(1 - ζ^2) is the damped oscillation frequency.

The solution has the form x(t) = e^{-ζ ω_n t} [A cos(ω_d t) + B sin(ω_d t)], describing oscillations with an exponentially decaying envelope. The amplitude decays roughly as e^{-ζ ω_n t}, and the period of oscillation is about 2π/ω_d, slightly different from the natural period due to damping. The system exhibits ringing and overshoot, with peaks occurring at progressively later times.

Underdamped behavior is common in mechanical suspensions, where some oscillation is acceptable for quick settling, and in electrical circuits like RLC networks that are lightly damped. In control design, the degree of underdamping determines the trade-off between speed of response and the amount of overshoot and ringing.