AI authored to showcase ironpad capabilities.

Spring-Mass-Damper System 🔧

The spring-mass-damper (SMD) is a fundamental mechanical model used in vibration analysis, control theory, and engineering design. A mass m slides on a frictionless surface, coupled to a fixed wall by a spring with stiffness k and a viscous damper with coefficient c.

Equation of Motion

Newton's second law yields the linear ODE:

m\ddot{x} + c\dot{x} + kx = F(t)

where x is displacement from equilibrium, \dot{x} velocity, \ddot{x} acceleration, and F(t) an optional external force.

Key System Parameters

• Natural frequency: \omega_n = \sqrt{\dfrac{k}{m}} — oscillation rate of the undamped system
• Damping ratio: \zeta = \dfrac{c}{2\sqrt{mk}} — dimensionless measure of dissipation

Damping Regimes

Regime	Condition	Response
Underdamped	\zeta < 1	Decaying oscillation — bounces before settling
Critically damped	\zeta = 1	Fastest return with no overshoot
Overdamped	\zeta > 1	Sluggish exponential decay, no oscillation

External Forcing

Enable External Forcing to apply F(t) = A\sin(\omega_f t). When the driving frequency \omega_f approaches \omega_n, resonance amplifies the response — especially for low damping.

Adjust the sliders to move between regimes and watch the physical diagram, waveform, and dashboard update live.