This crank substitute uses a planet gear rolling inside a fixed internal ring. At an exact 2:1 pitch-diameter ratio, a pin on the planet pitch circle traces a straight diameter of the ring. One carrier turn produces a full outward-and-return stroke.
The model shows the carrier, meshing gears and actual pin path. Change the tooth count or move the pin radially to see why a direct straight guide no longer fits. The altered pin path remains visible for comparison with the straight-line case.
Related actuator design tools
This internal-gear mechanism converts rotation into pin motion. If you are comparing it with a separate linear-actuator drive, calculate that drive’s force, stroke and mounting geometry; this pin-path calculator does not evaluate actuator loads.
For an actuator-driven linkage, check force throughout the travel, including positions where leverage becomes small.
If the application needs synchronized motion, feedback should be treated as a requirement rather than an accessory. Two open-loop actuators wired together can drift apart when friction, load, or mounting geometry differs between sides.
- Map actuator stroke to the actual output travel before choosing a model.
- Check for binding, changing leverage, and peak load near the ends of travel.
- Decide whether timed motion is acceptable or position feedback is required.
For a FIRGELLI design review, calculate the load case with the FIRGELLI actuator force calculator and plan the switching or controller arrangement with the linear actuator wiring diagram generator. If an actuator or cylinder is already installed, the linear actuator replacement finder helps keep the comparison tied to real dimensions.
Crank Substitute Interactive Calculator
Compare the true pin path at and near an exact 2:1 internal-gear ratio. A guide appears only when the geometry is exactly straight. All spans and peak speed refer to the first carrier turn.
Equation Used
- Fixed 100-tooth ring and same-module planet.
- Rigid gears with no slip or backlash.
- Pin radius equals planet pitch radius plus signed offset.
- First-turn metrics refer to carrier angles 0 to 360 degrees.
Ideal rigid no-slip gears. Direct guide only at 50 planet teeth and zero pin offset. Other ratios keep the true unconstrained pin path.
A straight path from two opposed rotations
The carrier moves the planet center around the fixed ring. No-slip internal gearing makes the planet spin in the opposite direction. With 50 planet teeth against 100 ring teeth, the planet’s absolute spin angle is the negative of the carrier angle.
At that ratio, the vertical components of the planet-center motion and pitch-circle pin motion cancel exactly. Their horizontal components add, giving x=R cosθ and y=0, where R is the ring pitch radius.
A blue guide block is shown only for that ideal geometry. When the selected pin path is not straight, the block is omitted and the orange pin remains free to trace its true path. A rigid guide cannot simply absorb that transverse motion.
Compare an ideal crank substitute with deliberate geometry changes
Ring diameter sets the physical scale. Integer planet teeth alter the gear ratio while preserving the same module with the 100-tooth ring. Pin offset is a signed radial change from the planet pitch radius; positive values move the pin outward.
The model uses illustrative 25° stub involute teeth. It explains engagement and motion, but does not specify a manufactured gear pair, load capacity, backlash or bearing clearance. The output pin and carrier occupy separate axial layers in the front view.
The calculator reports pin X and Y spans over the first carrier revolution, the displacement between its start and end, and peak pin speed over that same turn. A changed ratio can need multiple carrier turns to return to its starting pin position.
General pin path with the selected tooth ratio
Let R=D/2, r=R Np/100, A=R−r, q=A/r and ρ=r+e. Carrier angle is θ, and planet spin relative to the frame is −qθ.
x=A cosθ+ρ cos(qθ); y=A sinθ−ρ sin(qθ)
The derivatives with respect to carrier angle are x′=−A sinθ−ρq sin(qθ) and y′=A cosθ−ρq cos(qθ). Pin speed is √(x′²+y′²) times 2πN/60. The reported extrema are found from these derivatives over 0≤θ≤2π, including the endpoints.
At Np=50, q=1 and r=R/2, so x=(R+e)cosθ and y=−e sinθ. Only e=0 gives a straight line. Otherwise the trace is an ellipse with X span 2(R+e) and Y span 2|e|.
A changed gear ratio generally produces a hypotrochoid, not simply an ellipse. Its first-turn X span must be calculated; it is not automatically the ring diameter.
100 mm ring and a 50-tooth planet
With the default 100 mm ring, 50 planet teeth and zero pin offset, the planet pitch diameter is 50 mm. The output pin travels from +50 mm to −50 mm, giving 100 mm stroke and zero transverse motion.
At 30 carrier rpm, peak pin speed is 157.08 mm/s. Setting pin offset to +0.5 mm changes the X span to 101 mm and the Y span to 1 mm. That modified pin can no longer be connected directly to a perfectly straight guide without changing the mechanism.
Changing the planet count to 48 or 52 changes the spin relationship and leaves a nonzero displacement after one carrier turn. The animation follows the continuous pin path across carrier revolutions.
Geometry and design limits
The 2:1 straight-line construction is described as the underlying known gear arrangement in patent CA2881543A1, in its description. This page models that single gear set, not the patent’s complete multi-gear converter.
Gear strength, manufacturing tolerances, wear, efficiency and permissible speed require data for the actual assembly. The model does not calculate stress, torque capacity, lubrication needs or a guaranteed reduction in package size.
The orange path is the actual geometric pin trace. The separate Y plot automatically magnifies transverse motion for visibility and labels its numerical scale. Zero carrier speed stops rotation and gives zero pin speed.
Crank-substitute questions
Why is the guide block sometimes absent?
The selected pin path is not straight. A rigid straight guide would conflict with that path.
Does every ratio error create an ellipse?
No. A radial pin offset at an exact 2:1 ratio creates an ellipse. A changed gear ratio generally creates a different hypotrochoid.
Why does the pin not return after one carrier turn?
The planet may not have completed an integer number of turns relative to the fixed frame. The integer tooth counts determine when the full configuration repeats.
Does the Y plot exaggerate the actual movement?
It magnifies that component for inspection and shows its own millimeter scale. The main mechanism remains proportionally drawn.
Does zero transverse pin motion prove zero bearing load?
No. This is a geometric result. A force and structural analysis would be needed to determine bearing and guide loads.
Construction and geometry references
CA2881543A1, Crank-less motion converter, description: the known half-diameter planet and fixed internal-ring arrangement. Wolfram MathWorld, Hypocycloid, equations for a point on a circle rolling inside a larger circle and the 2:1 straight-line case.
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