Slider-Crank (Exact Kinematics and Gas Torque)

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Take a four-bar linkage and push one ground pivot off to infinity: the rocker that used to swing on an arc now slides in a straight line, and the four-bar becomes a slider-crank — crank, connecting rod, piston. It is the mechanism that turns the up-and-down of a piston into the round-and-round of a crankshaft (and back): every internal-combustion engine, every reciprocating compressor and pump, every old steam locomotive’s driving wheel. One degree of freedom — set the crank angle $\theta$ and the whole mechanism is decided.

Unlike its four-bar parent, the in-line slider-crank assembles on a single branch. The parent’s open/crossed pair came from a quadratic; here the position problem is a right triangle — rod length $l$ as the hypotenuse, the crank pin’s transverse offset $r\sin\theta$ as one leg, and the rod’s axial reach $q=\sqrt{l^2-r^2\sin^2\theta}$ as the other — with only the positive root physical. The piston sits at

$x = r\cos\theta + \sqrt{l^2 - r^2\sin^2\theta}$

measured from the crankshaft axis: $x=r+l$ at top dead centre, $x=l-r$ half a turn later, a stroke of exactly $2r$ no matter how long the rod.

Two passes: motion, then force

The page carries the two analyses an engine designer runs in turn, as its two configurations:

• Kinematics. Spin the crank at a constant speed $\omega$ and the piston velocity and acceleration are just $v=\omega\,\mathrm{d}x/\mathrm{d}\theta$ and $a=\omega^2\,\mathrm{d}^2x/\mathrm{d}\theta^2$ — space derivatives scaled by $\omega$, never a time integral. The results are not sinusoids: the piston spends longer near bottom dead centre than near top, and the acceleration carries a twice-per-revolution term that a simple counterweight can never balance. Alongside the exact velocity and acceleration sits the classic two-term approximation $s\approx r(1-\cos\theta)+\frac{r^2}{4l}(1-\cos 2\theta)$, shown with its own error readout so you can watch it drift — negligible at real engine proportions ($r/l\approx\frac14$ to $\frac13$), visibly wrong once the rod gets short.
• Force path. The connecting rod is not aligned with the cylinder; it tilts by the obliquity angle $\phi$, where $\sin\phi=(r/l)\sin\theta$. Push on the piston with a gas force $F$ and that tilt does two things: it drives the rod force up to $F/\cos\phi$ (with the leftover $F\tan\phi$ shoved sideways into the cylinder wall as side thrust), and it turns the piston force into a crank torque $T=F r\,\sin(\theta+\phi)/\cos\phi$. That torque is zero at both dead centres and peaks near mid-stroke — the ripple every engine smooths with a flywheel.

Each configuration hands you the knobs for that pass and holds the other driver — gas force $F$ in the kinematic pass, crank speed $\omega$ in the force pass — at a labelled reference value, so the full quasi-static state is always on screen.

There is no material picker here, and that is the point: kinematics and the quasi-static force path are pure geometry. Steel, aluminium, or titanium, the piston follows the same path and the same gas force makes the same torque. Material re-enters the moment inertia and stiffness matter — the crank’s own whirl and torsional vibration, the rod’s buckling, the bearing loads — which is where the combined bending + torsion shaft and the torsional oscillator pick the story back up.

Try it

Governing relations

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

How it fails

A slider-crank, like its four-bar parent, mostly misbehaves kinematically and dynamically long before a link breaks — and several of its troubles are written directly into the obliquity angle $\phi$ this page computes.

• Side thrust and bore wear. The connecting-rod force $F/\cos\phi$ is never aligned with the cylinder; the leftover $F\tan\phi$ presses the piston sideways against the bore, reversing direction each stroke. That is the “piston slap” that wears cylinder walls oval and the reason engines keep $l/r$ generous — a longer rod means a smaller $\phi$ and less side thrust. Drive $r/l$ past about $0.5$ on this page and the warning fires for exactly this reason.
• Secondary imbalance. The exact acceleration carries a twice-per-revolution term — the $\frac{r}{l}\cos 2\theta$ piece of $a\approx -r\omega^2(\cos\theta+\frac{r}{l}\cos 2\theta)$. A rotating counterweight on the crank can cancel the once-per-rev shaking force, but nothing rotating at crank speed cancels a force at twice crank speed. This “secondary” shake is why inline-four engines carry balance shafts spinning at $2\omega$, and it grows with $r/l$.
• Dead centres. At top and bottom dead centre the crank torque $T=Fr\sin(\theta+\phi)/\cos\phi$ is zero: the gas force points straight through the crank axis with no moment arm. An engine coasts through on flywheel inertia, but a mechanism that must start against load from a dead centre cannot — the same toggle geometry that locking pliers exploit on purpose.

The torque ripple, and why the flywheel exists

The single clearest lesson of the force pass is that crank torque is not constant. Over one revolution $T(\theta)$ swings from zero at a dead centre to a mid-stroke peak and back — and in a real engine it goes negative over the compression stroke, when the crank drives the piston rather than the other way round. A load, meanwhile, wants steady torque. The mismatch between the engine’s lumpy delivery and the load’s steady demand is stored and released twice a revolution by a flywheel: its rotational inertia soaks up the excess near the torque peak and gives it back through the dead centres, holding the speed fluctuation to a chosen coefficient.

Sizing that flywheel means integrating the torque curve — accumulating the energy $\int(T-T_\text{mean})\,\mathrm{d}\theta$ between the speed extremes to get the required inertia. This page refuses to do that honestly: it evaluates $T$ at the crank angle you set, exactly, but it does not integrate over the cycle, because time- and angle-integration of a varying quantity is a dynamics capability the catalog has deliberately not built yet. So the flywheel story is told here and sized on the flywheel page — the torque ripple is the input that page’s energy method consumes, and wiring one THING’s output into the next is the job of the chaining work still to come. Naming the hand-off rather than faking the integral is the honest version.

Where material finally re-enters

Everything on this page is geometry and quasi-static force, so nothing here depends on what the parts are made of. The failures that do depend on material live one link away:

• Crankshaft torsional vibration. The crankshaft is not rigid — it is a torsional spring carrying disks (the cranks and flywheel), and the very torque ripple above feeds it at multiples of crank speed. Hit a torsional natural frequency and the shaft winds and unwinds enough to fail in fatigue; big engines fit tuned dampers for precisely this. That is the torsional oscillator story, and its stiffness and inertia carry $G$ and $\rho$.
• Rod buckling and bearing loads. The connecting rod is a column under the compressive force $F/\cos\phi$ computed here, alternating with tension — a fatigue-and-buckling problem the combined bending + torsion shaft and column pages own. The rod force and its obliquity are the loads those analyses start from.

So the slider-crank’s own page stops at the clean, material-blind skeleton: where the piston is, how fast it moves, and what torque a given gas force makes. Every failure that needs a modulus, a density, or a yield strength is deliberately handed to the THING that carries it.

Sources

• Norton, R. L., Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 5th ed., McGraw-Hill, 2012 — slider-crank position, velocity, and acceleration analysis (the exact closed forms and the r/l two-term series for piston motion) and engine dynamics (the gas-force torque with connecting-rod obliquity, T = F·r·sin(θ+φ)/cosφ).
• Uicker, J. J., Pennock, G. R., & Shigley, J. E., Theory of Machines and Mechanisms, 5th ed., Oxford University Press, 2017 — slider-crank position/velocity/acceleration and static force analysis; used as an independent cross-check on the kinematic closed forms and the gas-torque expression.