Balancing

An off-centre mass generates F = mω²r rotating with the shaft, so the bearings see a force changing direction once per revolution. The ω² term means it quadruples for every doubling of speed.

A single gram misplaced at the rim of a turbine wheel becomes tonnes of force at operating speed, which is why high-speed rotors are balanced to fractions of a gram-millimetre and the same imbalance is harmless at low speed.

Static balance requires Σmr = 0, tested on knife edges. But two equal masses 180° apart in different planes along the shaft cancel as forces and form a couple — the rotor sits still on knife edges and rocks violently when it spins.

That makes the static test misleading rather than merely incomplete. Dynamic balance requires both Σmr = 0 and Σmrl = 0, which is two vector conditions and therefore two correction planes — and any distribution of unbalanced masses can be corrected by exactly two.

A reciprocating mass produces a force along one line, not a rotating one. A rotating counterweight cancels it at the stroke ends and creates an equal force perpendicular to it at mid-stroke — so a single-cylinder engine cannot be balanced, only compromised.

The connecting rod's swing adds a secondary component at twice crank speed, smaller and much harder to cancel — which needs shafts geared to twice engine speed. That is what a balancer shaft in a large four-cylinder engine does.

Adding cylinders lets one piston's inertia force cancel another's, and the firing order decides which cancel. An inline six is inherently balanced in primary and secondary forces and couples — nothing is added to achieve it.

A V12 is two inline sixes, which is why both are so smooth. Engine layout is a balancing decision before it is a packaging one.