A crank disk carries a pin that slides inside an upright rocker. The rocker turns about a fixed pivot near its lower end. Its toothed sector drives a horizontal rack back and forth, while a rope from its upper end passes over a pulley to raise and lower a weight.
Crank-disk with Toothed Sector and Rack Interactive Calculator
Set the crank radius and distance between its center and the rocker pivot. Choose the sector’s tooth size and equivalent full-circle tooth count. The calculator derives the rocker swing and rack stroke, while the animation keeps the crank pin inside its slot and the sector engaged with the rack.
Equation Used
- Rigid crank and rocker; frictionless pin sliding in a straight slot.
- Crank center directly above the rocker pivot; r<h throughout the offered input ranges.
- Sector and rack share module and an illustrative 20-degree tooth profile.
- Sector covers the full swing with extra end teeth; no disengagement or finite dwell.
- No force, backlash, inertia, tooth-strength or rope-elasticity calculation.
The rocker drives the rack back and forth while the teeth remain engaged. The equivalent tooth count specifies the full gear from which the displayed sector is cut; it is not the number of visible teeth.
One crank, two reciprocating outputs
The disk pin circles the crank center. Because that pin stays inside the rocker slot, the rocker must turn toward it while allowing it to slide along the slot. The lower pivot stays fixed.
The toothed sector is rigidly attached to the rocker and remains engaged with the guided rack. Reversing rocker motion reverses rack motion. The upper end carries the rope, so its changing distance from the fixed pulley changes the suspended tail length.
The crank pin slides within the rocker slot throughout each revolution. Both the rack and suspended weight reverse as the rocker changes direction.
Converting rotary input to reciprocation
This arrangement illustrates how a rotating crank can produce both a guided horizontal motion and a rope-driven vertical motion. The two outputs follow different displacement laws but share one rocker. The illustration is a motion study rather than a rated lifting device.
Deriving the rack stroke
Place the rocker pivot at the origin and the crank center a distance h above it. A crank radius r smaller than h produces a maximum rocker angle α = asin(r/h). The total swing is 2α.
For equivalent full-circle tooth count z and module m, sector pitch radius R = zm/2. Rack displacement from its center is proportional to rocker angle, so full rack stroke is 2Rα with α in radians. Circular tooth pitch is πm.
The tooth-count input describes an equivalent complete gear, ensuring a consistent pitch radius and integer tooth pattern. Only the arc needed for the rocker’s travel is drawn.
Example: a 40 mm crank
With r=40 mm and h=200 mm, the rocker swings about 23.07 degrees in total. An equivalent 60-tooth sector at module 2 has a 60 mm pitch radius. Its rack stroke is about 24.16 mm, and tooth pitch is about 6.283 mm.
A larger crank radius increases swing and stroke. Moving the crank center farther from the rocker pivot reduces both. Tooth count and module set the pitch radius and therefore rack travel for that same angular swing.
Reversal is not dwell
The rack pauses instantaneously when its direction changes, but there is no finite dwell interval. The gear sector stays meshed throughout the cycle. The rope model keeps a fixed total length by following the moving tangent, pulley arc and vertical tail.
Loads, backlash, tooth stress, bearing clearance and rope tension are not evaluated. The displayed involute tooth outlines omit root fillets and should not be treated as manufacturing profiles. The upper lever extension and pulley dimensions are chosen for a clear schematic.
Crank-sector questions
Why does the pin slide along the bar?
Its distance from the rocker pivot changes during the crank revolution, while the slot constrains its direction.
Does the rack disengage during a dwell?
No. This mechanism remains engaged and reciprocates continuously, with instantaneous reversals at its limits.
What determines rack stroke?
The ratio of crank radius to pivot spacing sets rocker swing. The sector pitch radius converts that swing into linear rack travel.
Why enter a full-gear tooth count for a sector?
It defines the pitch radius and tooth spacing of the circle from which the sector is cut.
Does the weight require a separate drive?
No. Its rope is attached to the same rocker, with a fixed pulley changing the direction of the free tail.
Historical reference
Henry T. Brown, Five Hundred and Seven Mechanical Movements, figure 282, printed page 70. Movement 282 description.
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