Large-Angle Pendulum - 3D Test Rig

Machining the rig...

Presets

Live readouts

Graphs

Angle vs time with linear twin

Phase portrait

Kinetic, potential and total energy

Period vs amplitude

Poincare section (drive on)

Numerical checks

running...

How to use

Pick a preset or set the length, gravity, release angle and losses. The grey twin is the small-angle (linear) pendulum started from the same state: watch it fall out of step with the real one at large amplitude. The period graph shows the exact curve and your measured cycles. Switch the drive on to explore resonance and chaos. Space pauses, R restarts.

Drag to orbit, wheel to zoom. The bob is drawn 1.7x larger than its physical radius for visibility. The red needle on the pivot dial shows the drive torque A cos(W t). The rig, dial, protractor and textures were modelled and baked in Blender; Three.js r160 animates every part.

The science

A point bob of mass m on a rigid massless rod of length L, angle θ from the downward vertical, obeys

θ'' = -(g/L) sin θ - b θ' - k_d |θ'| θ' + A cos(W t), k_d = ρ C_d A_bob L / (2 m)

The first term is the exact restoring torque; the textbook model replaces sin θ by θ, which gives a period 2π sqrt(L/g) that does not depend on amplitude. The true undamped period depends on the release angle θ0 through the complete elliptic integral of the first kind:

T = 4 sqrt(L/g) K(k), k = sin(θ0 / 2), T ≈ T0 (1 + θ0²/16 + 11 θ0⁴/3072 + ...)

At 90 degrees T is 18% longer than T0; as θ0 approaches 180 degrees T diverges logarithmically because the bob lingers near the unstable upright position. K is evaluated here with the arithmetic-geometric mean, which converges to machine precision in a handful of steps.

Losses and drive. b is a viscous pivot friction; for small angles the amplitude then decays as exp(-b t / 2). The quadratic term is air drag on the sphere (C_d = 0.47, frontal area π r²); it dominates for a light, large bob. With a periodic drive and damping the same equation is the classic driven pendulum, which is chaotic in some windows: with g = L = 1, b = 0.5 and W = 2/3 the motion is periodic at small A and chaotic near A = 1.2. The Poincare section samples (θ, θ') once per drive period; a chaotic orbit fills a fractal set of dots instead of a few points.

Rod tension follows from the radial equation T = m (g cos θ + L θ'²), so it is largest at the bottom of the swing and, for the over-the-top preset, nearly zero at the top. The bob is a point mass for the dynamics; its radius only sets the drag area and the drawing.

Checks. (1) The measured period (successive zero crossings, RK4, step 0.5 ms) at 10, 90 and 150 degrees is compared with 4 sqrt(L/g) K(sin(θ0/2)); (2) the 0.5 degree period with 2π sqrt(L/g) (the 5e-6 difference is real finite-amplitude physics); (3) the three-term series with the exact value at 60 degrees; (4) the relative energy drift of RK4 over 20 s; (5) the small-angle damping envelope with exp(-b t / 2). The integrator menu lets you see explicit Euler gain energy and semi-implicit Euler stay bounded.

Simplifications. Rigid massless rod, point-mass dynamics, planar motion, constant g, no rod drag, pivot friction idealised as viscous.