damped

Damped refers to a reduction in the amplitude of oscillatory motion due to dissipative forces such as friction, viscosity, or electrical resistance. In physics, damping describes how a system returns to equilibrium after a disturbance, with the rate and character of the return determined by the damping mechanism and system parameters.

A common model is the linear damped harmonic oscillator, described by the equation m x'' + c x' + k x = 0, where m is mass, c is the damping coefficient, and k is the restoring stiffness. The natural frequency is omega_n = sqrt(k/m), and the damping ratio is zeta = c / (2 sqrt(mk)). The qualitative behavior depends on zeta: underdamped motion (0 < zeta < 1) oscillates with an exponentially decaying envelope; critically damped motion (zeta = 1) returns to equilibrium as quickly as possible without oscillation; overdamped motion (zeta > 1) returns to equilibrium without oscillating, more slowly than in the critical case.

Damping mechanisms can be categorized by their force law. Viscous damping is proportional to velocity; Coulomb or dry friction provides a constant opposing force; structural or internal damping arises from material deformation and internal friction. In electrical systems, damping corresponds to resistive losses in circuits, such as RC or RLC networks.

Applications of damping include stabilizing structures against wind and seismic forces, reducing vibrations in machinery and vehicle suspensions, shaping the response of audio and electronic filters, and improving the performance of control systems. Damping also influences the quality factor, resonance behavior, and transient response of dynamic systems across engineering and natural sciences.