...
running...
How to use
Side view of a tracked excavator: boom, stick and bucket are driven by two hydraulic cylinders (plus a bucket-curl circuit). The bars at the top left show the oil pressure each circuit needs to hold the arm still against gravity and the payload; the red line is the relief valve, which is the most the pump circuit can deliver. The top right gauge shows the tip-over safety factor about the front edge of the tracks (dashed red line).
Try: turn off the automatic cycle and drag the boom, stick and bucket sliders. Reach far with a full bucket and watch the boom pressure and the tip-over factor. Raise the payload until the relief valve opens (red) or the safety factor falls below 1. Increase the cylinder bore and the same load needs less pressure.
The science
Hydraulic force. Pascal's law: pressure is transmitted equally, so a cylinder pushes withF = p A, A_bore = pi D^2/4 (push), A_annulus = pi (D^2 - d^2)/4 (pull)350 bar on a 140 mm bore gives about 539 kN.
Lever arm and virtual work. A cylinder attached at radius r from a pin gives a torque of F times the perpendicular distance from the pin to the cylinder line. The same torque follows from virtual work: tau = F dL/d(theta), where L is the cylinder length. The simulator computes the lever arm geometrically and verifies it against a finite difference of L.
Static equilibrium. To hold a pose each cylinder must cancel the gravity torque of everything beyond its pin: F = -(sum m_i g x_i)/lever. Because the lever arm changes with angle, the required pressure is not proportional to the load alone, and some poses are much harder than others.
Tip-over. About the front edge of the tracks, safety factor = (stabilising moment of body, counterweight and rear arm parts)/(overturning moment of the parts ahead of the edge). Below 1 the machine tips.
Simplified: static (no inertia or dynamic loads), a 2D side view, assumed link masses and centres of mass (boom 1200 kg, stick 600 kg, bucket 400 kg, body and tracks 20 t), idealised cylinder attachment points, a constant effective bucket-linkage lever arm of 0.32 m, no friction, and no soil resistance except the payload. The tooth breakout figure is only the stick-cylinder limit, not a measured digging force.
What is checked and why you can trust it
Every time the page loads, and whenever you press Re-run check, the simulation evaluates its own mathematics against an independent reference, such as a closed-form formula, a finite-difference derivative or an exhaustive count. Each line above reports PASS or FAIL together with the measured numbers, so you can see by how much it agrees. Results are also published on window.__msChecks for automated testing. The check never uses the same code path as the animation it validates, otherwise it would only confirm itself.
How to read a failure: a FAIL line means the measured discrepancy exceeded the stated tolerance for the current slider values. Try Reset, then move the sliders back toward their defaults; if a check still fails at the defaults it is a bug worth reporting. Tolerances are chosen from the numerical method itself, never tuned to make a test pass: round-off level (1e-9) for exact algebra, about 1e-3 for finite differences and Monte Carlo estimates with a few hundred thousand samples, and a documented margin for stochastic learning runs.
Reproducibility: the particle and pen simulations use the browser random generator for initial positions, so exact pixels differ between visits, but the checks depend only on mathematics and not on those positions. Fixed wave tables (curl noise) and fixed grammars (L-system) are deterministic.
This is an educational model. It shows the mechanism clearly, but it is deliberately simplified; the science panel lists what was left out.