Doubling the Length of a Crank Stroke: How the Rack-and-Pinion Mechanism Works

Follow the complete drive chain

The crank pin revolves about a fixed bearing. A constant-length rod joins that pin to a horizontally guided carrier. The pinion spins freely on the carrier spindle, meshing below with the fixed rack and above with the output rack.

Rolling on the fixed rack adds tangential movement to the translating pinion centre. At the upper pitch contact the two contributions add, so output displacement and velocity are twice those of the carrier.

The diagram exposes the driving crank implied by Hiscox’s movement 882 and Brown’s movement 118. The rod and carrier supports are outboard of the rack teeth; they are drawn in front in this side view. A separate upper rail guides the output member. Housings and axial offsets are schematic rather than manufacturing dimensions.

Separate crank throw from output travel

Use the radius slider to set the crank geometry. Carrier travel is twice that radius; the rack doubler then multiplies travel by two again. The rod-length slider changes how displacement and velocity vary during the revolution, while the two stroke endpoints remain unchanged.

The pitch diameter sets the number of pinion revolutions during each stroke. The shown pinion uses 3 mm module and an even tooth count; changing diameter does not change the output-to-carrier travel ratio.

This is a geometry and speed calculator. It does not select a motor, establish torque capacity or predict a dwell at either reversal. A standard crank-slider stops instantaneously at each endpoint without a finite dwell interval.

Exact finite-rod closure

For crank radius r, connecting-rod pin distance L and crank angle θ, carrier position from the fixed crank centre is b=r cosθ+√(L²−r²sin²θ). The model requires L greater than r, as ensured by the offered input ranges.

Displacement from the left endpoint is x=b−(L−r). Carrier stroke is 2r; upper-rack displacement is 2x and its full stroke is 4r. The animation begins with θ=180°, at the left endpoint.

For angular speed ω=2πn/60, carrier speed is ω[−r sinθ−r²sinθ cosθ/√(L²−r²sin²θ)]. The upper-rack speed is twice this expression. Positive speed means rightward motion. The second term is the finite-rod correction omitted by a sinusoidal approximation.

Peak output speed magnitude is sampled at 1440 positions per crank revolution. Pinion turns per one-way stroke are 2r/(πD). Pinion angle changes by x/R, where R=D/2; that rolling constraint keeps both racks in step.

A 40 mm crank gives 160 mm output stroke

With a 40 mm crank radius, the carrier travels 80 mm and the upper rack travels 160 mm. The default 240 mm connecting rod makes the motion close to sinusoidal, but its finite length still changes the live and peak speed results.

At 60 rpm, the crank makes one revolution per second. The simple infinite-rod estimate of peak output speed is about 502.7 mm/s; the finite-rod calculation gives a slightly larger peak. The live speed is zero at either stroke endpoint.

The 72 mm pinion makes about 0.354 revolution during each one-way 80 mm carrier stroke. It reverses rotation on the return stroke even though the crank continues rotating in the same direction.

Geometry rather than a machinery rating

A longer rod reduces the departure from sinusoidal carrier motion. A shorter rod increases obliquity and changes the velocity curve. Neither setting establishes an allowable load or operating speed.

The ideal rack stage doubles speed and halves transmitted force relative to its carrier. Input motor torque additionally depends on crank position, load and inertia; those quantities are not calculated here.

The model excludes backlash, compliance, friction and dynamic stresses. Rods, guides and gear roots are simplified, and projected parts may overlap because their axial separation is not drawn. Playback is ten times slower than the entered crank speed.

Crank stroke-doubler questions

Is output stroke twice the crank radius?

No. The carrier already travels twice the radius. The rack stage doubles that, producing four times the radius at the output.

Why does connecting-rod length change speed but not stroke?

The two straight-line endpoint positions differ by 2r for any L greater than r. Between those endpoints, the changing rod angle alters the carrier speed.

Does the maximum speed occur exactly at a quarter turn?

Not generally with a finite rod. The calculator samples the exact derivative throughout a revolution rather than assuming the peak is at 90°.

Does the pinion rotate continuously?

No. It rolls forward and backward with the carrier, reversing each half-cycle.

What does zero rpm do?

It stops automatic motion and makes the live and peak speed results zero. Manual position inspection remains available.