The Cardan straight-line mechanism is a gear-based linkage that converts rotary motion into exact straight-line motion using a small gear rolling inside an internal gear of exactly twice its diameter. The key component is the inner pinion, which carries a tracing pin on its pitch circle — that pin traces a perfectly straight line across the centre of the larger ring gear. It exists to eliminate the approximations of four-bar straight-line linkages like Watt's or Chebyshev's, giving true linear motion without prismatic guides. You see it in scotch yokes, sine generators, and certain hypocycloid drives.
Cardan Straight-line Mechanism Interactive Calculator
Vary ring pitch diameter and tooth count to see the exact 2:1 Cardan gear geometry, stroke, pinion size, and animated straight-line pin motion.
Equation Used
The Cardan mechanism uses a pinion rolling inside an internal ring gear with an exact 2:1 pitch diameter ratio. With the tracing pin on the pinion pitch circle, the pin position is sinusoidal along one diameter and the peak-to-peak stroke equals the ring pitch diameter.
- Internal ring gear pitch diameter is exactly twice the pinion pitch diameter.
- Tracing pin is located on the pinion pitch circle.
- Backlash, tooth errors, and bushing play are ignored.
- Ring tooth count is even so the required pinion tooth count is an integer.
How the Cardan Straight-line Mechanism Works
The mechanism works on a property of hypocycloids first formalised by Cardano in the 16th century: when a circle rolls inside a circle of exactly twice its diameter, every point on the smaller circle's circumference traces a straight line — a degenerate hypocycloid. These are called the Cardan circles, and the same geometry is sometimes called a Tusi couple after the 13th-century Persian astronomer Nasir al-Din al-Tusi, who used it to decompose circular motion into linear motion. The 2:1 ratio is exact and non-negotiable. If the ring gear has 60 teeth, the pinion must have exactly 30. Run 31 teeth and the tracing point sweeps a narrow ellipse instead of a line. Run 29 and you get an ellipse the other way. There is no tolerance band on the tooth count itself — only on the gear cutting accuracy.
The tracing pin sits on the pitch circle of the inner pinion. As the pinion rolls inside the ring gear under the influence of a carrier arm, the pin oscillates back and forth along a fixed diameter of the ring. The pinion spins at twice the carrier rate but in the opposite direction, and that exact 2:1 counter-rotation is what cancels the orbital component of the pin's motion, leaving only linear oscillation. If you notice the pin tracing a thin ellipse rather than a clean line, you have one of three problems: gear backlash letting the pinion lag the carrier, a tracing pin that sits slightly inside or outside the pitch circle, or worn carrier-arm bushings letting the pinion centre wander. Backlash above roughly 0.05 mm at the gear mesh shows up as visible line-thickening at the ends of stroke, where the pin reverses direction and the mesh reloads.
The stroke length equals the diameter of the ring gear's pitch circle. A 100 mm pitch diameter ring gives 100 mm peak-to-peak stroke. The motion follows a pure sinusoid in time when the carrier rotates at constant speed — which is why the Cardan mechanism is sometimes called a sine generator and forms the kinematic basis of the Scotch yoke.
Key Components
- Internal Ring Gear: The fixed outer gear with internal teeth on a pitch circle of diameter D. The pinion rolls inside it. Tooth count must be exactly twice the pinion tooth count — typical builds use 60/30 or 80/40 pairings with AGMA Q10 or better tooth quality to keep mesh backlash under 0.05 mm.
- Inner Pinion (Rolling Gear): Half the diameter of the ring gear, with exactly half the tooth count. Carries the tracing pin on its pitch circle. The pinion rotates twice for every one rotation of the carrier arm and in the opposite direction — that counter-rotation is what produces the straight-line output.
- Carrier Arm (Crank): Drives the pinion centre around the ring gear axis at the input rotation rate. The arm length equals one-quarter of the ring gear pitch diameter. Bushing clearance on the carrier pivot must stay under 0.02 mm or the pinion centre wanders and the trace thickens into an ellipse.
- Tracing Pin: Mounted on the pinion's pitch circle. The radial position of this pin is critical — it must sit exactly on the pitch circle, not on an arbitrary radius. Position it at 0.95 of pitch radius and you trace a narrow ellipse rather than a line.
- Output Slider or Yoke: Couples the tracing pin to the load via a slot or pin-and-socket. In a Scotch yoke implementation the yoke runs in linear guides and converts the pin's motion to pure translation of an external rod.
Real-World Applications of the Cardan Straight-line Mechanism
You see the Cardan straight-line mechanism wherever exact linear motion must come from a rotating shaft without using prismatic slides, ball screws, or approximating four-bar linkages. The 2:1 hypocycloid geometry is the kinematic backbone of the Scotch yoke, and it appears in sine-wave generators, certain reciprocating pumps, and hypocycloid drives where designers exploit the property to convert orbital motion into pure translation. It earns its place when you need genuine straight-line output (not a Chebyshev or Watt approximation) and you can accept the gear-train cost.
- Marine engines: The Bourke engine and several Scotch yoke marine compressors use the Cardan principle to convert piston motion into smooth sinusoidal crank rotation without a connecting rod side-load.
- Astronomical instruments: Tusi-couple geometry — kinematically identical to the Cardan straight-line mechanism — appeared in al-Tusi's lunar models and later in Copernican planetary diagrams to decompose circular into linear motion.
- Mechanical sine generators: Analogue computers and harmonic synthesisers like the Kelvin tide-predicting machine used 2:1 hypocycloid drives to generate clean sinusoidal signals from a constant-speed input shaft.
- Reciprocating compressors: Scotch yoke compressors used in LNG service (Ariel and Burckhardt have made variants) rely on the same hypocycloid geometry to keep the piston rod purely axial, eliminating side-load wear on the cylinder liner.
- Stamping and punching machines: Hypocycloid press drives use a 2:1 internal gear pair to deliver high force at bottom-dead-centre with zero side-thrust on the ram, used in coining presses and small-format blanking lines.
- Educational kinematics: University demonstration rigs — including the classic MIT mechanisms display — use a 60/30 tooth Cardan setup to show the hypocycloid degenerating into a straight line.
The Formula Behind the Cardan Straight-line Mechanism
The position of the tracing pin along its straight-line path is a pure cosine of the carrier angle, and stroke equals the ring gear's pitch diameter. At low carrier speeds — say 10 RPM on a benchtop demonstrator — the motion is slow enough that gear backlash and bushing slop dominate the trace quality, and you'll see line-thickening at stroke ends. At nominal speeds in the 60-200 RPM range the mechanism runs in its sweet spot, where mesh forces stay loaded in one direction long enough to suppress backlash. Push past 600 RPM on an industrial Scotch yoke compressor and pinion inertia plus mesh excitation start to generate audible whine and accelerated tooth wear. The formula below tells you stroke and instantaneous position; the speed range tells you when the result actually matches the formula.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| x(θ) | Position of the tracing pin along the straight-line path, measured from ring centre | mm | in |
| D | Pitch diameter of the internal ring gear (equals peak-to-peak stroke) | mm | in |
| θ | Carrier arm rotation angle from the line of motion | rad | rad |
| Stroke | Total peak-to-peak linear travel of the tracing pin | mm | in |
Worked Example: Cardan Straight-line Mechanism in a benchtop sine-wave calibration rig
You are building a benchtop sine-wave calibration rig for a vibration-sensor lab in Eindhoven. The rig must drive an accelerometer head through a known sinusoidal displacement so the lab can verify sensor linearity against an interferometric reference. You spec a Cardan straight-line mechanism with a 120 mm pitch diameter ring gear (60 teeth) and a 60 mm pinion (30 teeth), driven by a brushless DC servo. Target peak-to-peak stroke is 120 mm and the lab wants to characterise sensors across 0.5 Hz to 5 Hz drive frequency.
Given
- D = 120 mm
- Ring teeth ZR = 60 teeth
- Pinion teeth ZP = 30 teeth
- Frequency range f = 0.5 to 5 Hz
Solution
Step 1 — confirm the gear ratio is exactly 2:1, otherwise you don't have a Cardan mechanism at all:
Step 2 — compute peak-to-peak stroke from the ring pitch diameter. Stroke equals D directly because the pin traverses a full diameter:
Step 3 — compute peak velocity at the nominal mid-range frequency of 2 Hz. Carrier angular speed is ω = 2π × f, and peak linear velocity is (D/2) × ω:
Step 4 — at the low end of the lab's range, 0.5 Hz, peak velocity drops to 0.188 m/s. This is slow enough that the accelerometer head moves visibly back and forth like a metronome, and any backlash in the 60/30 mesh shows up as a sharp blip at the stroke ends. Below 0.3 Hz you'll start to see the trace thicken on a laser interferometer because the mesh teeth unload at reversal.
Step 5 — at the high end, 5 Hz, peak velocity reaches 1.88 m/s and peak acceleration is (D/2) × ω2 = 0.060 × (2π × 5)2 = 59.2 m/s2, or roughly 6 g. That's enough to demand a balanced counterweight on the carrier arm, and pinion bearing loads scale with ω2 so a 10× frequency increase from 0.5 to 5 Hz is a 100× bearing load increase.
Result
At the nominal 2 Hz drive frequency the rig delivers 120 mm peak-to-peak stroke and 0. 754 m/s peak velocity. To the lab technician this feels like a smooth, continuous oscillation — fast enough that you can't follow individual cycles by eye but slow enough to mount a contact-type displacement reference without contact bounce. At 0.5 Hz the rig creeps at 0.188 m/s and reversal-blip is the dominant error source; at 5 Hz it reaches 1.88 m/s and inertia-driven mesh loading dominates, with the sweet spot for sub-micron linearity sitting between 1 and 3 Hz. If the laser interferometer reports the trace deviating from a pure sinusoid, the most likely causes are: (1) the tracing pin sitting 0.1-0.3 mm off the pinion pitch circle, which converts the line into a thin ellipse; (2) gear mesh backlash above 0.05 mm at the 60/30 pair, visible as flat-tops at the displacement extremes; or (3) a carrier-arm bushing worn beyond 0.02 mm radial clearance, which lets the pinion centre orbit and adds a small circular component to the trace.
Choosing the Cardan Straight-line Mechanism: Pros and Cons
The Cardan straight-line mechanism delivers exact linear motion, but you pay for it with gear-train cost and the noise/wear penalty of running internal gears at speed. Compare it against the two practical alternatives a designer reaches for when straight-line motion is the goal: a Scotch yoke (which is mechanically simpler but uses a sliding pin in a slotted yoke) and a Chebyshev linkage (which is purely pin-jointed but only approximates a straight line).
| Property | Cardan Straight-Line Mechanism | Scotch Yoke | Chebyshev Linkage |
|---|---|---|---|
| Linear accuracy | Exact (geometric) | Exact (geometric) | Approximate, ±0.4% of stroke deviation typical |
| Practical speed range | 50-600 RPM | 100-1500 RPM | 30-300 RPM |
| Cost (relative) | High — precision internal gear pair | Low — slot and pin | Low — pin-jointed bars only |
| Wear interface | Gear mesh + bearings | Pin-in-slot sliding contact | Pin joints only |
| Maintenance interval | 5,000-10,000 hr (gear lube) | 500-2,000 hr (slot wear) | 10,000+ hr (pin bushings) |
| Stroke vs. footprint | Stroke = ring pitch diameter (compact) | Stroke = 2× crank radius | Stroke ≈ 2× coupler length (large footprint) |
| Side-load on output | Zero (pure axial) | Zero (pure axial) | Small but non-zero |
| Best application fit | Sine generators, hypocycloid presses | Reciprocating compressors, pumps | Walking robots, demonstrators |
Frequently Asked Questions About Cardan Straight-line Mechanism
The gear ratio is only one of three things that must be exact. The tracing pin must also sit precisely on the pinion's pitch circle — not on the addendum circle, not on a tooth root, but on the pitch circle. If the pin is offset by even 0.2 mm on a 60 mm pinion, the hypocycloid no longer degenerates and you get a narrow ellipse with minor axis roughly equal to twice the offset.
Check the pin radius with a dial indicator while rotating the pinion in isolation — it should sit at exactly Dpinion/2. The other common cause is the ring gear's centre not coinciding with the carrier rotation axis; if the assembly stack-up puts those two axes 0.1 mm apart, you'll see the same ellipse signature.
No. The straight-line property is unique to the 2:1 ratio. At 4:1 the tracing pin draws a four-cusped astroid (a star-shaped hypocycloid), at 3:1 it draws a deltoid, and at any non-integer ratio it draws a complex rosette pattern. Cardano's theorem is specifically that a circle rolling inside a circle of twice its diameter produces a degenerate hypocycloid — a straight line. Change the ratio and the geometry fundamentally changes.
If you need a straight line with different stroke, scale the entire mechanism — make the ring bigger or smaller, but keep the 2:1 ratio.
For a compressor under about 5 kW, a Scotch yoke wins almost every time. Both deliver pure sinusoidal motion with zero piston-rod side-load, but the Scotch yoke uses a single sliding pin and slot — far cheaper to manufacture than a precision internal gear pair. The Cardan mechanism only earns its place when you specifically need the gear-driven version for sealing reasons (no sliding pin exposed to process fluid) or when you're running fast enough that slot wear becomes the limiting factor.
Rule of thumb: below 1500 RPM and 5 kW, use a Scotch yoke. Above that, or in dirty/abrasive service where the slot would wear, the Cardan version becomes attractive.
At low speeds the gear mesh load isn't high enough to keep teeth firmly engaged on one flank. Each time the tracing pin reverses at end-of-stroke, the inertia direction flips and the mesh momentarily unloads, then reloads on the opposite tooth flank. That backlash crossover is audible as a tick or rattle, and it's worst at low speeds where there's no centrifugal mesh-loading to mask it.
Two fixes: tighten the gear centre distance to reduce backlash below 0.03 mm, or add a small preload spring on the carrier arm to keep mesh load directional. Industrial Scotch yoke compressors handle this with split anti-backlash gears.
The carrier arm pivot bearings see the reaction force from accelerating the pinion plus the tracing pin's load reaction. At nominal speed with light loads, this is dominated by pinion inertia: Fbearing ≈ mpinion × (D/2) × ω2. For a 0.5 kg pinion on a 60 mm carrier radius at 5 Hz, that's about 30 N — modest.
The trap is that the load scales with ω2, so doubling the drive frequency quadruples bearing force. For high-speed builds (above 600 RPM) the pinion bearing is usually the life-limiting component, not the gear teeth. Size it for L10 life at the maximum continuous speed, not the nominal.
Yes, but you give up the natural input speed advantage. When you drive the carrier, the pinion runs at 2× input speed in the opposite direction. When you drive the pinion directly, the carrier runs at half pinion speed — meaning you need a faster, lower-torque input motor to get the same output stroke rate.
The bigger issue is reaction torque: driving the pinion puts the full input torque through the gear mesh continuously, where driving the carrier shares load between the carrier pivot and the gear mesh. For high-torque applications (presses, compressors), drive the carrier. For low-torque sine generators where motor selection is easy, either works.
References & Further Reading
- Wikipedia contributors. Tusi couple. Wikipedia
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