Mass–spring system

Interactive simulation of a mass–spring system that allows users to explore free, damped, forced, and nonlinear oscillations by adjusting mass, linear and nonlinear spring constants, damping, friction, and periodic external forces.

Drag the block while the simulation is stopped at t = 0 to set its initial position, or adjust the parameters in the panels. The "Graph" panel shows position, velocity, and external force over time—click a legend entry to show or hide a curve, or enable "Phase space" with exactly two selected curves to plot one against the other.

This simulation numerically solves the ODE using the fourth-order Runge–Kutta method with integration step $h$ $$ m\ddot{x} + \gamma\dot{x} + kx + k_2x^2 + k_3x^3 = F_1(t) + F_2(t), $$ where $k_2$ and $k_3$ are the nonlinear terms of the spring force, and $F_1(t)$ and $F_2(t)$ are external forces (sinusoidal or periodic square pulses).

Marco P. M. de Souza