Advanced11 min read7 / 9

Inertia forces & engine balance

Why does a straight-six feel like silk and an inline-four buzz through the steering wheel? Not combustion — inertia. Every piston is violently accelerated and reversed twice per revolution, and unless the cylinders are arranged so those forces cancel, the whole engine shakes on its mounts. Balance is pure geometry, and we can do all of it with one equation from the last article.

The shaking force

Newton’s third law: whatever force accelerates the piston, the piston applies back to the engine block. Per cylinder, along its bore axis, using the harmonic form of the acceleration:

F(\theta) \approx m\,r\,\omega^2\left(\underbrace{\cos\theta}_{\text{primary}} + \underbrace{\lambda\cos 2\theta}_{\text{secondary}}\right)

(1)

$m$: reciprocating mass: piston + rings + pin + ~⅓ of the rod [kg]
$m r \omega^2$: one cylinder’s peak primary force — the demo’s reference bar [N]
$\lambda$: rod ratio r/L ≈ 0.3, the secondary’s relative size [–]

A multi-cylinder engine is a sum of these, one per cylinder, each evaluated at its own phase: cylinder $k$ contributes $F(\theta + \delta_k)$ along its own bore direction, where $\delta_k$ is its crank-throw offset. Balance is the art of choosing the $\delta_k$ (and bank angles) so the sum vanishes. Two frequencies must cancel independently: primaries at crank speed, secondaries at twice crank speed.

Case study: the inline-four

The standard I4 crank is flat: throws at 0°, 180°, 180°, 0°. Sum the primaries —

\sum \cos\theta_k = \cos\theta + \cos(\theta{+}180^\circ) + \cos(\theta{+}180^\circ) + \cos\theta = 0

(2)

— perfect cancellation: two pistons rise exactly as two fall. The layout’s mirror symmetry kills the rocking couple too. But look at the secondaries. Since $\cos 2(\theta + 180^\circ) = \cos(2\theta + 360^\circ) = \cos 2\theta$ , a 180° crank offset does nothing at double frequency:

\sum \lambda\cos 2\theta_k = 4\,\lambda\cos 2\theta \;\neq\; 0

(3)

Fig. 1. The inline-four's ledger. Dashed: the two opposite pairs of primary forces, cancelling perfectly. Solid red: all four secondaries in phase, adding to a 4λ·cos2θ shake at twice engine speed.

The other layouts, same arithmetic

Straight-six. Throws at 0-120-240-240-120-0°. Three phases 120° apart sum to zero — and so do their doubles at 240° spacing: $\sum\cos\theta_k = \sum\cos 2\theta_k = 0$ . Primary, secondary, and (by the mirror-image crank) both rocking couples all vanish identically. Nothing needs balancing; this is the silkiness BMW built a brand on, and the reason the demo’s I6 balance meter simply sits at zero.
Boxer-four. Opposed pistons move outward and inward together, cancelling primaries and secondaries along the bore axes. Only a small rocking couple survives, because opposed cylinders must be offset slightly along the crank to clear each other.
60° V6. Even 120° firing and zero net shaking force, but the two three-cylinder banks act like a see-saw: a primary rocking couple (a torque that pitches the engine end-over-end) remains — why most V6s carry one balance shaft.
Cross-plane V8. Throws at 0-90-270-180°: the 90° spacing scrambles the secondaries to zero ( $\cos 2\theta$ terms land 180° apart in pairs) and the 90° bank angle lets simple crank counterweights absorb the rotating primary couple. Result: V8 smoothness — plus the uneven bank-to-bank exhaust order that makes the burble.

Force vs couple — read the demo’s meter correctly

Cancelling the net force means the block doesn’t translate. But if cylinder 1 pushes up at the front while cylinder 6 pushes down at the back, the net force is zero and the engine still pitches — a couple, force × lever arm along the crank. The Cylinder Configurations demo computes both: the force vector at the block’s centre and the couple index along the crank axis. A layout is only truly smooth when both stay flat — which of the five presets manages it, you now know before pressing play.