Alivaimennettu

Alivaimennettu is a Finnish term used to describe a system in which the damping is insufficient to prevent oscillatory motion after a disturbance. In a linear single-degree-of-freedom system described by m x'' + c x' + k x = f(t), the damping ratio is zeta = c / (2 sqrt(mk)). The system is underdamped when 0 < zeta < 1. The natural frequency is omega_n = sqrt(k/m) and the damped frequency is omega_d = omega_n sqrt(1 - zeta^2).

With a step or impulse input, an underdamped system responds with oscillations whose amplitude decays roughly as e^{-zeta omega_n t}. The response exhibits overshoot and a finite settling time, which depends on zeta and omega_n. As zeta approaches 1 from below, overshoot diminishes but the settling time increases; for zeta in practice, a balance is sought to achieve a quick yet controlled response.

Applications and relevance: underdamped behavior is common in mechanical and structural engineering, such as vehicle suspensions, mechanical springs and dampers, and buildings or bridges subjected to dynamic loading. It also appears in electrical circuits like RLC networks. Understanding the underdamped regime helps engineers predict oscillations, design appropriate damping levels, and compare with critically damped (zeta = 1) or overdamped (zeta > 1) cases. Measurement of zeta can be done from impulse or step responses using methods like log decrement or frequency-domain analysis.