The Cardan straight-line mechanism uses a circle rolling inside a fixed circle of twice its diameter. A point on the smaller circle traces a diameter of the larger one. A geared version uses an internal ring with exactly twice the pinion tooth count; the calculator uses that exact relationship.
Comparing the Cardan output with actuator travel
A Cardan tracing point follows a cosine displacement as the carrier rotates at constant speed. A direct linear actuator produces a different travel profile. Compare required displacement, speed and loading before choosing either arrangement or connecting them through an additional linkage.
With leverage, force is traded against travel. A bracket change that improves force can also increase required stroke, slow the output motion, or move the actuator into a poor mounting angle.
The mechanism may also change speed as the linkage angle changes. A constant actuator extension speed does not always create constant output speed, so timing should be checked at the actual output point.
- 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.
Cardan Straight-line Mechanism Interactive Calculator
Set the ring pitch diameter and an even ring tooth count. The pinion has half the diameter and half the teeth. Watch its front-face tracing pin stay on the horizontal diameter while the carrier revolves and the pinion counter-rotates.
Equation Used
- The ring is fixed and its pitch diameter is twice the pinion pitch diameter.
- An even ring tooth count gives an integer pinion tooth count; both use the same module.
- The tracing point lies exactly on the pinion pitch circle.
- Rigid parts and ideal rolling engagement; no backlash or elastic deflection.
- Rounded tooth outlines show count and motion, not an involute profile or validated manufacturable gear pair.
- Animation speed is illustrative; the view rescales the ring to preserve readability.
The fixed ring has twice the pinion’s pitch diameter and tooth count. A tracing pin on the smaller pitch circle follows a straight diameter of the ring. Change ring diameter to change the stroke, or tooth count to compare the illustrated gear pair.
Two rotations produce a straight path
The geometric construction is also known as a Tusi couple. Linda Hall Library documents a historical paper model in its copy of Copernicus’s book: a half-size circle moves inside the larger circle and produces rectilinear motion. The animation here implements that geometry as a geared teaching model.
Let the ring pitch radius be R and the pinion pitch radius r = R/2. At carrier angle θ, the pinion center is at (r cos θ, r sin θ). The tracing pin is at angle −θ relative to that moving center, adding (r cos θ, −r sin θ). The vertical components cancel, leaving (R cos θ, 0).
The pinion turns once backward relative to the fixed frame during one forward carrier revolution. Relative to the carrier itself, that is two backward turns. The fixed frame and rotating carrier therefore give different descriptions of the same pinion motion.
The blue carrier supports the pinion spindle in front of the gear face. The orange tracing pin is on its front face, at the pitch radius. No external slider is needed to make the ideal point trace a line.
Exploring exact straight-line geometry
This calculator compares ring pitch diameter, pinion pitch diameter, stroke and tooth counts for an ideal 2:1 pair. Changing ring diameter changes the dimensional stroke. Changing tooth count changes the displayed gears while preserving the 2:1 geometry.
It is a kinematic demonstration rather than a specification for a compressor, actuator or production gear train. Loads, speed ratings, tooth interference, bearings and lubrication are not entered or solved here.
Position and stroke
With ring pitch diameter D, the pinion pitch diameter is D/2 and the carrier radius is D/4. The tracing pin position relative to the ring center is x = (D/2) cos θ and y = 0. Its extreme positions are −D/2 and +D/2, so total stroke is D.
The ring-to-pinion tooth ratio is Nr/Np = 2. This ratio is a geometry result, not a 2:1 speed reduction relative to the fixed frame. At carrier angular speed ω, the pinion spins at −ω relative to the frame and at −2ω relative to the carrier.
With no RPM input, playback uses an illustrative constant carrier speed. The ideal position follows a cosine through one outward-and-return stroke per revolution.
Read the default 100 mm example
A 100 mm ring pitch diameter gives a 50 mm pinion pitch diameter, 25 mm carrier radius and 100 mm total stroke. A 60-tooth ring requires a 30-tooth pinion, with common module 100/60 ≈ 1.667 mm.
At carrier angles 0°, 90°, 180° and 270°, the pin lies at x = +50, 0, −50 and 0 mm respectively. Its transverse coordinate remains zero. Doubling D doubles these distances; changing the tooth count alone leaves them unchanged.
Exact geometry and physical gears
The straight path requires the half-size rolling circle and a tracing point on that circle. If the tracing-point radius changes while the rolling-circle ratio stays 2:1, the two transverse components no longer cancel. Changing the gear ratio is a different modification and generally gives another hypocycloidal path.
The rounded tooth outlines in the illustration make the entered tooth count visible. They are not tooth-cutting data. Selecting counts here does not validate undercut, internal-gear interference, contact stress or manufacturing clearances.
Questions about the Cardan mechanism
Does the pinion rotate once or twice backward?
Once per carrier revolution relative to the fixed frame; twice relative to the rotating carrier. Both descriptions require their reference frame.
Why is the ring tooth slider limited to even values?
The pinion must have exactly half as many teeth, so an even ring count keeps the pinion count an integer.
Does changing tooth count change the stroke?
Not at fixed ring pitch diameter. The ideal stroke equals that diameter, while tooth count determines the common module.
Is the tooth drawing suitable for manufacture?
No. It shows counts and ideal engagement using rounded outlines. Detailed gear design and interference checks are outside this calculator.
Primary geometric reference
Linda Hall Library: Nasīr al-Dīn al-Tūsī, including the library’s historical paper Tusi-couple model. It establishes the half-size rolling-circle construction and straight-line trace. The geared housing and carrier in this illustration are schematic teaching additions.
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
