Noncircular gears vary their speed ratio as the distances from the shafts to the pitch point change. This calculator examines one position: enter center distance, driver-radius fraction and input speed to find both radii and the opposing output speed. The diagram explains that local relationship; it does not generate a complete rectangular gear pair.
If the machine also uses a linear actuator
This calculator describes rotary gearing and does not size a linear actuator. If the surrounding machine uses one for a separate adjustment or positioning task, select it from that task’s force, stroke, loaded speed and control needs.
For structural members, the actuator load should be applied through brackets that do not introduce unintended bending. Check both static load and the extra load created when the mechanism starts, stops, or binds.
The control method should match the consequence of being wrong. Simple extend/retract switching is fine for many applications, but repeatable stops, synchronized motion, or programmable positions require the right feedback and controller architecture.
- Use loaded speed, not only no-load speed, when timing the motion.
- Confirm voltage, controller rating, and limit-switch behavior before wiring.
- Use feedback when synchronized or repeatable position control is required.
For FIRGELLI actuator control, use the linear actuator wiring diagram generator before wiring the final circuit and compare suitable actuator families in the FIRGELLI linear actuator range. If the machine already has an actuator, the linear actuator replacement finder can help shortlist replacements.
Noncircular-Gear Radius and Speed Calculator
Compare the two shaft-to-contact radii and instantaneous speed ratio at one position. Input rpm sets the output-speed result. The markers use a fixed rocking demonstration and do not reproduce the entered rpm.
Equation Used
- The two shaft centers are fixed at the entered distance C.
- The selected pitch curves are tangent at one common contact point.
- The selected contact rolls ideally without slip.
- The entered radius fraction selects one local station and is not a full transmission function.
- Closed pitch curves, tooth geometry, loads, losses and durability are excluded.
The second diagram follows the same radius inputs. The circles illustrate shaft-to-contact distances, not manufactured gear outlines.
How the contact radii set speed
For two external gears with fixed shaft centers, the pitch point lies between the centers. Let their separation be C and the driver distance to the pitch point be r₁. The mate distance is r₂=C−r₁.
At ideal rolling contact, the two pitch-point velocities agree. The shafts rotate in opposite directions, so their signed angular velocities satisfy ω₁r₁+ω₂r₂=0. A larger driver radius relative to the mate radius therefore gives a larger output-speed magnitude.
In a noncircular pair these radii change with angular position. The circles shown here illustrate one radius combination; they do not describe the complete pitch-curve shapes.
Comparing one operating position
Use the sliders to explore how center distance and radius share affect the instantaneous speed ratio. Increasing center distance scales both radii while keeping their ratio unchanged. Increasing the driver-radius fraction increases output speed for the same input rpm.
This calculation does not provide complete rectangular gear profiles, tooth dimensions or a cyclic speed program. A working noncircular pair needs compatible pitch curves throughout rotation, followed by tooth, clearance and load analysis.
Radius and speed equations
With center distance C and driver fraction f, the radii are r₁=fC and r₂=(1−f)C. The speed-ratio magnitude is r₁/r₂=f/(1−f). For input speed n₁ in rpm, the output card shows the positive magnitude n₂=n₁r₁/r₂; the mate turns oppositely.
Using signed angular velocities in radians per second, ω₂=−ω₁r₁/r₂ and the common pitch-point speed magnitude is |v|=|ω₁|r₁=|ω₂|r₂. These equations apply at the chosen position. They do not determine how the radii change over a complete revolution.
Worked example
For C=120 mm and f=0.40, the selected driver radius is 48 mm and the required mate radius is 72 mm. At 300 rpm input, the ideal instantaneous output is 200 rpm in the opposite direction. This is one contact condition, not a rectangular-gear cycle.
From a speed ratio to a gear design
A useful speed ratio at one position is only one part of a noncircular transmission. The two pitch curves must remain compatible as the gears turn and close after the required cycle. The teeth must also mesh with suitable clearance and contact geometry.
Two rounded rectangular outlines alone do not establish those conditions. This calculator gives the local radius and speed relation, leaving full-cycle motion, tooth profiles, contact stress, losses and durability to a separate gear design.
Noncircular gear questions
Does the diagram show rectangular gear outlines?
No. Its circles show the two shaft-to-contact radii used in the calculation. They are not complete noncircular pitch curves, tooth outlines or local curvature measurements.
Why does the output turn in the opposite direction?
These are external gears. Their angular velocities have opposite signs, while the velocities of their pitch points agree at ideal rolling contact. The output card gives the positive speed magnitude.
Why does center distance not change the speed ratio?
At a fixed radius fraction f, changing C scales both radii equally. Their ratio remains f/(1−f). Moving the fraction slider changes the ratio.
Does input rpm control the animation speed?
No. Input rpm changes the calculated output speed. The moving markers rock at a fixed demonstration rate, even at zero entered rpm, to illustrate the radius ratio. They do not simulate a complete gear revolution.
Can these inputs define a full noncircular gear pair?
No. A complete pair needs compatible pitch curves around the whole revolution and suitable tooth profiles. A radius ratio at one position does not determine that geometry or a load rating.
Authoritative references
- US12287028B2, Noncircular gear pair — fixed center distance and conjugate pitch-curve synthesis from a transmission relationship.
- US6905432B2, Chain drive arrangement — angular-position-dependent rolling curves and equal rolling arc length.
- CN102705448A, Non-circular gear pair with Fourier-function pitch curves — driven displacement obtained by integrating the radius relationship.
References & Further Reading
- Wikipedia contributors. Non-circular gear. Wikipedia
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