EngineerHub.orgWhat this simulator teaches
A pendulum is simple enough to visualize but rich enough to demonstrate nonlinear dynamics, energy conservation, damping, resonance and normal modes. Drag the bob in the free-motion views or change the controls and watch the equations become geometry.
nonlinear motionsmall-angle limitenergydampingresonancecoupled modesphase spacephysical pendulum
Select pendulum model6 animated learning modes
Ideal pendulum controls
Ideal nonlinear pendulum
Full sin θ dynamics with exact period comparisonIdeal pendulum — motion, forces and energy
The left side shows the moving pendulum and instantaneous vectors. The right side plots recent motion; drag the bob to choose a new release angle.
θ¨ + (g/L) sinθ = 0
Nonlinear motionComparison / second pendulumEnergy / driveReference / approximation
Four ideas that explain pendulums
01 / Restoring torqueGravity produces a tangential restoring acceleration proportional to sin θ, not exactly to θ. The system is therefore nonlinear at large angles.
02 / Small-angle limitWhen |θ| is small, sin θ ≈ θ and the pendulum behaves like simple harmonic motion with T ≈ 2π√(L/g).
03 / Energy exchangeIn the ideal model, gravitational potential energy is highest at the turning points while kinetic energy is highest at the bottom.
04 / Resonance & modesDamping removes energy; periodic forcing can replace it. Coupled pendulums add normal modes and allow energy to move between oscillators.
Model assumptions & limits
The bob modes use a point mass on a massless rigid link. This permits large-angle motion without the slack-string limitation of a literal cord. In driven and coupled modes the forcing and coupling inputs are normalized angular terms, so bob mass cancels from the motion equations and is intentionally not exposed as a control.
Nonlinear motion is integrated with a fourth-order Runge–Kutta method using small internal time steps. The animation frame rate does not set the physics time step.
For an ideal release from rest, the finite-amplitude period uses the complete elliptic integral: T = 4√(L/g) K(sin(θ₀/2)). K is evaluated numerically by the arithmetic–geometric mean.
The animation integrates the nonlinear driven equation. The displayed steady-state amplitude and resonance-peak location are corresponding linear small-angle results and are explicitly labelled as approximations; strong nonlinear motion can shift and broaden the response.
The pair uses equal pendulums with an ideal angular coupling term κ(θ₁−θ₂). Normal-mode frequencies shown on the page are undamped small-angle values; the live motion retains sinθ gravity and the selected damping.
Background & FAQ
No. Mass cancels from the ideal equation of motion. Real pendulums can show mass-dependent differences indirectly through air drag, bearing friction, flexibility or geometry.
The approximation replaces sinθ with θ. At larger angles the restoring acceleration is weaker than that linear approximation predicts, so the real pendulum takes longer to complete a cycle.
For an undamped release from rest it is fastest at the bottom, where gravitational potential energy is minimum and kinetic energy is maximum.
The damping mode writes the viscous term as 2ζω₀θ̇. ζ<1 is underdamped, ζ=1 is critically damped and ζ>1 is overdamped in the small-angle linearized system.
When forcing repeatedly adds energy with favorable timing, the response grows until damping and nonlinear effects balance the input. The strongest response is near the system's natural frequency for weak damping and small oscillations.
A physical pendulum has distributed mass. Its rotational inertia about the pivot matters, so its small-angle period is T=2π√(Iₚ/(mgd)), where d is the pivot-to-center-of-mass distance.