Massspringdamper

A mass-spring-damper is a canonical mechanical system used to model vertical or lateral vibrations. It consists of a rigid mass m connected to a fixed support by a linear spring with stiffness k and a viscous damper with damping coefficient c. In the standard single-degree-of-freedom configuration, the spring and damper act in parallel between the mass and the base. The model assumes linear components and small displacements.

If x(t) denotes the displacement of the mass from its static equilibrium, and F(t) is an external force applied to the mass, the equation of motion is m x''(t) + c x'(t) + k x(t) = F(t). In the absence of forcing (F=0), the system has a natural frequency ω_n = sqrt(k/m) and, for nonzero damping, a damping ratio ζ = c / (2 sqrt(k m)).

The response characteristics are: underdamped (0 < ζ < 1), which oscillates with an envelope exp(-ζ ω_n t); critically damped (ζ = 1), which returns to equilibrium without oscillation in the minimum time; and overdamped (ζ > 1), which returns without oscillation but more slowly than the critically damped case.

Forcing produces a particular solution in addition to the transient. A sinusoidal force F(t)=F0 cos(ω t) yields a steady-state response with amplitude and phase determined by the frequency response function H(i ω) = 1 / (k - m ω^2 + i c ω).

Applications include modeling mechanical vibrations in structures and vehicles, designing suspension systems, vibration isolation, and aiding control system design where predictable dynamic behavior is required.