massspring

A mass-spring system is a simple mechanical model consisting of a mass m attached to a spring with stiffness k, optionally damped by a dashpot. The spring provides a linear restoring force F = -k x proportional to displacement x from equilibrium. The model is a standard lumped-parameter, one-degree-of-freedom oscillator, used to study vibrations and dynamic response.

In the absence of external forcing, and neglecting damping, the motion follows m x¨ + k x = 0, yielding simple harmonic motion with natural frequency ω_n = sqrt(k/m) and period T = 2π/ω_n. When damping is included (coefficient c), the equation becomes m x¨ + c x˙ + k x = 0, with damping ratio ζ = c / (2 sqrt(k m)).

If ζ < 1, the system is underdamped and x(t) decays while oscillating at the damped frequency ω_d = ω_n sqrt(1 - ζ^2). At ζ ≥ 1, it is critically or overdamped, returning to equilibrium without oscillation. The quality factor Q = ω_n m / c characterizes the sharpness of resonance.

With an external forcing F_ext(t) = F0 cos(ω t), the equation is m x¨ + c x˙ + k x = F0 cos(ω t). The steady-state response has amplitude X(ω) = F0 / sqrt((k - m ω^2)^2 + (c ω)^2) and exhibits resonance near ω ≈ ω_n when damping is small.

Variants include multi-degree-of-freedom systems, nonlinear springs, or including dry friction. The mass-spring model underpins studies in mechanical engineering, vehicle suspensions, civil engineering, and MEMS resonators, serving as a fundamental teaching and design tool due to its simplicity and analytic tractability.