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The science
What you see. A stationary double-acting steam engine: a boiler supplies steam to a cylinder where a piston is pushed alternately from each end; a cross-head and connecting rod turn the crank, a flywheel smooths the pulses, and a Watt flyball governor on a belt drive trims the steam.
Slider-crank. With crank radius r, rod length l and crank angle θ the piston pin sits at x = r cosθ + √(l² − r² sin²θ), so dx/dθ = −r sinθ − r² sinθ cosθ / √(l² − r² sin²θ). Virtual work gives the crank torque T = F dx/dθ with F = A (Pcrank end − Phead end).
Steam cycle in each cylinder end. Admission at boiler pressure up to the cut-off sc, adiabatic expansion P Vγ = const (γ = 1.3, a usual value for wet-to-dry steam), exhaust at the back pressure, then compression of the clearance steam from sk. The work per revolution of one end has the closed form PsA sc + PsVc[1 − (Vc/Ve)γ−1]/(γ−1) − PbA(S − sk) − PbVk[(Vk/V0)γ−1 − 1]/(γ−1); the page integrates the simulated pressure curve and compares.
Flywheel and governor. I dω/dt = T(θ) − Tload − Tfriction with I = m R² of the rim, Coulomb plus viscous friction, solved with fourth-order Runge-Kutta. The Watt governor balls of arm length L spinning at ω hang at cosφ = g/(ω² L) (height h = g/ω²); the cut-off is changed in proportion to the height error. Like every proportional governor it lets the speed droop under load and can hunt when the gain is high.
Honest simplifications. Valve events are idealised (instant admission and release, no steam-chest pressure drop except a smooth speed-dependent throttling factor 1/(1+(vp/4 m/s)²) chosen by hand), the piston-rod area and heat exchange with the cylinder walls are ignored, γ is constant, steam density comes from the ideal gas law with a compressibility factor of 0.93 and the Antoine saturation line, and the slide valve in the model moves schematically. Friction constants are plausible values, not measurements. It is a teaching model, not a design tool.
Checks. (1) numerical work of a cylinder end equals the closed form to 1e-5 relative; (2) crank-slider position and derivative equal an independent constraint solve and a numerical derivative; (3) the engine energy ledger (cylinders = load + friction + flywheel kinetic energy) closes;