The slider-crank
Rotation becomes stroke: crank, connecting rod and piston. The same principle is at work in every piston engine and every piston pump.
Derivation & calculation
Wanted: the piston travel x as a function of the crank angle φ. Crank (radius r) and connecting rod (length l) form a triangle; the piston runs along the cylinder axis.
the cos 2φ term is the “obliquity” effect
Example: at φ = 90°, cos φ = 0, but cos 2φ = −1 – so the piston is not in the middle but closer to bottom dead centre. That is exactly what the animation above shows.
Gears & gear ratio
Two meshing gears turn in opposite directions; their circumferences roll on each other without slip. The ratio of the numbers of teeth is the gear ratio.
Derivation & calculation
Two gears mesh without slip. At the point of contact (the pitch point, marked in orange) both have exactly the same circumferential speed v.
Example z₁ = 14, z₂ = 28: i = 2. 100 min⁻¹ becomes 50 min⁻¹, and the torque doubles. This is exactly how a gearbox turns a fast, weak motor into a slow, strong drive.
The damped oscillation
Mass, spring, damper – the basic model of every oscillation. With little damping it keeps swinging for a long time; in the critically damped case it returns to rest fastest without overshooting.
Derivation & calculation
Newton for the mass m: the spring pulls back with −k·x, the damper brakes with −c·ẋ (proportional to the velocity ẋ).
Example k = 40 N/m, m = 1 kg → ω₀ = 6.3 rad/s (≈ 1 Hz). With c = 1.5 Ns/m, ζ = 0.12 – lightly damped, it keeps oscillating for a long time.
Kármán vortex street
When fluid flows around a cylinder, vortices shed alternately from the top and bottom. This vortex street makes chimneys sway and power lines sing.
The physics behind it
The base flow here follows the potential flow around a circular cylinder – which is why the streamlines split cleanly and close again behind the cylinder:
u = U·( 1 − R²·(x²−y²)/r⁴ ) , v = −U·2R²·x·y/r⁴r = distance from the cylinder centre, R = radius
Fourier: waves from circles
Every periodic function is a sum of sine waves – visually: rotating circles linked together. The tip of the chain draws the curve.
Derivation & calculation
Idea: every periodic function is a sum of pure sine waves – a fundamental plus overtones. A rotating circle produces exactly one sine (its height above the centre).
Exactly this decomposition is behind MP3, JPEG and every vibration analysis: split a signal into its frequencies and leave out what does not matter.
The double pendulum
Hang two pendulums one below the other, and simple mechanics turns into chaos. Two almost identical starting angles diverge completely after a short time.
Why chaos?
The upper and lower pendulums pull on each other – their motions are coupled. The Lagrangian formalism yields two linked differential equations for the angles φ₁, φ₂.
a non-linear expression with sin(φ₁ − φ₂)
The Otto cycle
Compress, ignite, expand, exhaust. In the p-V diagram the state runs in a loop – the enclosed area is the work gained.
Derivation & calculation
Efficiency is the share of the heat supplied that turns into work: η = W/Qin. For the ideal Otto cycle (two isentropic + two isochoric steps) almost everything cancels – what remains is a surprisingly simple formula:
The area enclosed by the loop in the p-V diagram is directly the useful work per cycle. Larger area = more work.
Motor meets load: the operating point
A drive settles where motor torque and load torque are equal. As long as the motor delivers more than the load needs, the flywheel accelerates.
Derivation & calculation
Everything revolves around a torque balance at the flywheel (moment of inertia J):
J · dω/dt = Mmotor(n) − Mload(n)A large flywheel (large J) runs up sluggishly but holds the speed steady. If the load exceeds the breakdown torque, the motor “stalls” – it never gets up to speed at all.
The jet engine
Intake, compression, combustion, exhaust. The large fan at the front pushes cold air past the hot core; the core supplies the energy.
Derivation & calculation
Thrust comes from a change in momentum: the engine draws in air at vin and expels it faster at vout – by Newton (actio = reactio):
F = ṁ · ( vout − vin )ṁ = air mass drawn in per second
Astonishing facts
Engineering facts that are hard to believe – and that still work reliably every single day.
On a finger-sized blade
Each turbine blade of a jet engine produces as much power as a Formula 1 engine – while withstanding a centrifugal force as if a fully loaded coach were hanging from it.
Hotter than its melting point
The gas in the turbine is hotter than the melting point of the blade metal (~1300 °C). The blade survives only thanks to fine cooling-air channels inside and a ceramic thermal barrier coating.
The blade without grain boundaries
A turbine blade is cast as a single metal crystal. Grain boundaries would “creep” under heat and centrifugal force – the single crystal holds.
Hot before anything burns
Compression alone squeezes the air in the engine to ~40 times its pressure and heats it to around 600 °C – before a single drop of kerosene has burned.
The fan shovels air
The large, cold fan at the front moves more than a tonne of air – per second. And it provides ~85 % of the thrust, not the hot core.
Saturn V, the moon rocket
At lift-off it burned 13 tonnes of propellant per second. Its power output was roughly that of 85 Hoover Dams at once.
The wind turbine blade tip
The rotor turns at a leisurely pace, yet the tip of a large rotor blade races through the air at around 300 km/h – hence the “whoosh” in the wind.
The Eiffel Tower in summer
Heat expands steel: on hot days the Eiffel Tower is around 15 cm taller than in winter. That is why every bridge needs expansion joints.
How far a wing bends
In the ultimate load test, the wing of a wide-body jet is bent upwards by several metres – up to 150 % of the maximum flight load before it fails.
Concrete that is still curing
Without built-in cooling pipes, the concrete of the Hoover Dam would still not have cooled down today – the residual heat would have taken around 125 years to dissipate.
What I built with what
An overview of the languages and tools behind the projects. Hovering over a language or a project highlights its connections.
All models are deliberately simplified – meant for playing with and understanding, not for design calculations. The models run entirely in the browser; inputs are not stored.