An Elliptical Crank is a two-slider linkage in which a rigid coupler bar carries two pivots that ride in perpendicular slots, so any selected point on the bar traces a true ellipse. Unlike a standard slider-crank, which converts rotation into reciprocating linear motion, the Elliptical Crank converts rotation into a controlled curved path. We use it where a single drive shaft must produce coordinated horizontal and vertical motion at the same time — packaging tuckers, ellipsographs, and harmonic shaker tables all rely on it to deliver smooth, repeatable elliptical paths from one input.
Elliptical Crank Interactive Calculator
Vary the ellipse semi-axes, phase angle, and cycle time to see the tracing point position and motion speed of an elliptical crank.
Equation Used
The tracing point of a perpendicular two-slider elliptical crank follows a true ellipse. The semi-axis a sets the horizontal amplitude, b sets the vertical amplitude, theta is the crank phase, and T is the time for one complete cycle.
- Slots are exactly perpendicular at 90 deg.
- Coupler bar is rigid and slider clearance is neglected.
- a and b are the ellipse semi-axes measured from the tracing point geometry.
- Cycle time is constant over one full revolution.
How the Elliptical Crank Works
The geometry is older than most people realise — it's the Trammel of Archimedes dressed up as a powered linkage. You have a rigid bar with two pin joints, A and B, set a fixed distance apart. Pin A rides in a horizontal slot, pin B rides in a vertical slot, and the slots cross at 90°. Drive any point on the bar (or extend the bar past one of the pins) and that point traces an ellipse whose semi-axes are set purely by the distances from your chosen point to pins A and B. No cams, no gears — just two prismatic joints and one coupler.
Why build it this way? Because the elliptical path falls out of the geometry for free. If you wanted the same coupler curve from a four-bar linkage you'd be tuning link lengths against a Hrones-Nelson atlas and still getting an approximate ellipse. Here you get a mathematically exact one, and you can re-tune the major and minor axes just by moving the tracing point along the bar. That's why ellipsograph drafting tools used this layout for over 200 years.
Tolerance matters more than people expect. The two slots must be perpendicular to within about 0.1° on a precision build — any more and the path stops being a true ellipse and starts looking like a lopsided egg. Slot clearance on the sliders should sit around 0.02–0.05 mm; loose sliders rattle at the slot ends where velocity reverses, and you'll hear a tick-tick-tick at every quarter revolution. The most common failure mode is wear at the slot ends, because the slider dwells there momentarily as it changes direction. If you notice the trace drifting after a few thousand cycles, check the slot ends first — they wear into shallow pockets that pull the path off-shape.
Key Components
- Coupler Bar: The rigid link carrying the two slider pins and the tracing point. Length between pins sets one of the ellipse semi-axes. Stiffness matters — any bending under load distorts the path, so we typically size the bar for less than 0.05 mm deflection at peak inertial load.
- Slider A (horizontal): Constrains pin A to move only along the horizontal slot. Slot straightness should hold to about 0.02 mm over the full travel; deviations show up directly as ellipse distortion. Bronze or PTFE-lined bushings are standard for moderate-speed industrial use.
- Slider B (vertical): Constrains pin B to move only along the vertical slot, perpendicular to slot A. The 90° between slots is the single most critical assembly tolerance — hold it to 0.1° or better. Cross-slot squareness gauges or a precision square against a ground reference face is how you check it.
- Tracing Point / Output Pin: The point on the coupler whose path is the working ellipse. Its distance from pin A sets the major semi-axis (a), distance from pin B sets the minor semi-axis (b). Move it past pin A and you get a long, thin ellipse; place it between A and B and you get a tighter loop.
- Drive Crank or Belt Pulley: Inputs rotary motion, typically through a short crank attached to one of the sliders or to the tracing point itself. Drive speed sets cycle rate; for packaging applications 60–180 RPM is typical, with the upper limit fixed by inertial reversal forces at the slot ends.
Industries That Rely on the Elliptical Crank
The Elliptical Crank shows up wherever a single rotary input has to produce coordinated motion in two axes — and where a true ellipse, not an approximation, is required. You'll find it in classroom ellipsographs, packaging machines that need tuck-and-lift motion in one stroke, harmonic vibrators tuned to elliptical orbits, and certain textile and printing applications where ink or thread has to follow a closed loop without hesitation.
- Drafting & Education: The Stanley Acme ellipsograph and similar drafting tools use the trammel layout to draw exact ellipses for technical drawings — still sold as classroom kinematic demonstration units by companies like Gear Educational Systems.
- Packaging: Carton-tucker drives on machines such as the Bosch Doboy stripack flow-wrappers use elliptical-path linkages to push and lift carton flaps in one continuous motion, eliminating the dwell of a pure cam-follower setup.
- Textile Machinery: Embroidery pantograph heads on older Tajima TMFX-series machines used elliptical-crank style trammel arms to convert a single drive shaft into the needle's small-orbit feed motion.
- Vibratory Equipment: Elliptical-orbit screen shakers built by Derrick Corporation for mineral processing use a forced two-slider geometry to generate tuned elliptical motion that conveys oversize material along the deck while drainage continues.
- Lab & Test Equipment: Orbital shaker platforms in biology labs — for example the New Brunswick Innova series — use elliptical crank or eccentric-trammel mechanisms to swirl flasks at 25–400 RPM with a defined orbit diameter.
- Printing: Ink-fountain duct rollers on legacy Heidelberg sheet-fed presses use elliptical linkage motion to oscillate the ink doctor blade, distributing ink across the roller width as it transfers.
The Formula Behind the Elliptical Crank
The defining equation gives you the path of any tracing point on the coupler bar as a function of crank angle θ. What you really want from it is a feel for how the ellipse changes as you slide the tracing point along the bar — at one end of the typical operating range you get a near-circular path, at the other end you get a long thin ellipse approaching a straight line, and the sweet spot for packaging or shaker work sits around a 2:1 axis ratio where the motion has clear directional bias without slamming the sliders at reversal.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| x, y | Coordinates of the tracing point at crank angle θ | mm | in |
| a | Major semi-axis — distance from tracing point to pin riding the perpendicular slot | mm | in |
| b | Minor semi-axis — distance from tracing point to the other pin | mm | in |
| θ | Crank rotation angle from the major axis | rad | rad |
Worked Example: Elliptical Crank in a pharmaceutical blister-pack lidding station
Sizing the elliptical-crank lid-applicator drive on a Uhlmann BEC 300 blister-line lidding station. The applicator finger has to lower onto the foil lid, sweep 40 mm horizontally to seat the lid against the registration stop, then lift clear before the next blister web indexes. We want a 40 mm horizontal stroke and a 20 mm vertical lift, running at 90 cycles per minute nominal. Coupler bar length between sliders is 60 mm, tracing point set 40 mm out from the vertical-slot pin.
Given
- a = 40 mm
- b = 20 mm
- N = 90 cycles/min
- Lbar = 60 mm
Solution
Step 1 — convert nominal cycle rate to angular velocity in rad/s:
Step 2 — compute peak tracing-point velocity at the major-axis crossing (θ = 90°), which is the worst case for slider B:
Step 3 — at the low end of typical lidding-line speed, 60 cycles/min, the same geometry gives:
That's a calm, deliberate motion — the foil settles cleanly and you can watch the sweep with the naked eye. At the nominal 90 cycles/min, 0.377 m/s feels brisk but controlled, and the foil-to-stop seating is still clean. Push the crank to the high end of the typical range, 150 cycles/min:
At 0.628 m/s the slider reversal forces climb with the square of speed, and on a 60 mm coupler with steel pins you'll start to hear the slot-end impact as a metallic tick. Above roughly 130 cycles/min the foil lid begins to flutter on approach because the boundary layer of air pulled along by the applicator finger lifts the foil off the registration stop.
Result
Nominal peak applicator velocity is 0. 377 m/s at 90 cycles per minute, with a 40 mm horizontal stroke and 20 mm vertical lift per cycle. At 60 cycles/min the motion looks slow and deliberate — easy to inspect on a setup pass; at 150 cycles/min it's running at the practical ceiling for this geometry, with audible slot-end ticking and foil flutter starting to appear. If you measure peak velocity below the predicted 0.377 m/s, the three usual culprits are: (1) coupler bar flexing under inertial load — check for visible bow at peak crank angle, fix by going to a stiffer bar section; (2) slot-A and slot-B not square to within 0.1°, which collapses the major axis and shortens stroke; (3) drive-belt slip on the input pulley showing up as a 5–10% velocity shortfall that gets worse as the belt warms up.
When to Use a Elliptical Crank and When Not To
The Elliptical Crank competes with a handful of other mechanisms whenever a designer needs a closed curved path from rotary input. Each alternative wins on one or two engineering dimensions and loses on others — the choice usually comes down to whether you need a true ellipse or just any closed loop, and how fast you need to run.
| Property | Elliptical Crank | Four-bar Linkage (coupler curve) | Cam-and-Follower |
|---|---|---|---|
| Path accuracy | True ellipse, mathematically exact | Approximate ellipse, deviates 1–3% from ideal | Any path, accuracy set by cam grinding (±0.01 mm typical) |
| Practical max speed (RPM) | 120–180 RPM, slot-end inertia limited | 300–600 RPM, pin-joint limited | 600–1200 RPM, follower bounce limited |
| Build cost (relative) | Low — two slots, one bar, two pins | Low–medium — four pins, four links | High — precision-ground cam profile |
| Tunability after build | High — slide tracing point along bar to retune axes | Low — must re-make links to retune | None — re-grind or replace cam |
| Wear failure mode | Slot-end pocketing at reversal points | Pin-bushing wear, gradual path drift | Cam surface pitting, follower wear |
| Best application fit | Moderate-speed coordinated 2-axis motion needing exact ellipse | Complex coupler curves where shape matters more than precision | High-speed precision motion with arbitrary profile |
Frequently Asked Questions About Elliptical Crank
The slots aren't square to each other. The math assumes exactly 90° between slot A and slot B — every 1° of skew tilts the principal axes of the ellipse by roughly 0.5° and makes one quadrant noticeably fatter than the opposite one.
Check it with a precision square against a ground reference edge on each slot, or measure the slot end-points with a height gauge. On a fabricated steel frame I usually find the error at the welding stage, not the machining stage — heat distortion pulls the slots out of square by 0.3–0.5° on anything bigger than a 200 mm frame.
Ask whether you actually need a true ellipse or just a closed loop with a similar shape. If the operation tolerates 1–3% path deviation — most carton tuckers do — a four-bar runs faster and with fewer wear surfaces. If you need an exact ellipse for harmonic balancing, ink distribution, or repeatable seating against a fixed stop, the elliptical crank wins.
The other deciding factor is tunability. If you'll need to adjust the major or minor axis after the machine is in service, the elliptical crank lets you do it by sliding the tracing point along the coupler bar. A four-bar requires re-machined links.
You've got slot-end pocketing. The slider dwells momentarily at each reversal point because velocity passes through zero there, and that dwell concentrates load on a tiny patch of slot wall. Over thousands of cycles it wears a shallow pocket, and once the pocket is deep enough the slider drops into it and back out at each pass — that's the tick.
Pull the sliders and inspect the slot ends with a straightedge. If the pocket is under 0.05 mm deep you can sometimes lap it flat; deeper than that and you're better off case-hardening the slot or fitting hardened insert plates at the reversal zones.
For a typical 40–80 mm semi-axis industrial build with steel pins and bronze slider bushings, 150 cycles/min is the practical ceiling. The limit isn't the formula — it's slider reversal inertia. Force at the slot ends scales with ω², so doubling the speed quadruples the impact load.
If you need to push past 150 cycles/min, switch the sliders to needle-roller cam followers running in hardened slot inserts, and balance the coupler bar by adding a counterweight on the opposite side of the drive crank. With those two changes I've seen 220 cycles/min on a 50 mm semi-axis build, but you'll be replacing slot inserts every 6 months.
It changes shape. As long as the tracing point sits on the line through the two pins, the path is always a true ellipse — but the semi-axes are set by the perpendicular distances from the tracing point to each slot. Move the point outside the pin pair and one semi-axis grows past the slot length, giving you a long, narrow ellipse.
Push it far enough out and the ellipse approaches a straight line — that's actually how trammel-based straight-line generators work. So the same hardware can give you a near-circle, a working ellipse, or a near-straight-line just by repositioning one bolt on the coupler bar.
Almost always it's coupler bar flex. The bar is in compression-tension cycling at ω², and any bending shortens the effective distance between the tracing point and the slider pins, which shrinks both ellipse axes proportionally.
Quick check: clamp a dial indicator against the middle of the bar and watch it through one full cycle. If you see more than 0.1 mm of mid-span deflection on a 60 mm bar, you need a stiffer section — go from a flat bar to a square or T-section, or shorten the unsupported length by adding a centre support that doesn't interfere with slot motion.
You can, and it actually simplifies the build, but you lose force capability through the dead-centre positions. When the driven slider is at either end of its slot, its velocity is zero and the input torque has no mechanical advantage on the system — the linkage will stall there if loaded.
The fix is either to drive both sliders through a 90°-phased pair (rare, mechanically awkward), or accept the dead centres and add enough flywheel inertia on the input shaft to coast through them. For lightly loaded applications like ellipsographs it's fine; for anything pushing real work into the ellipse, drive a crank arm attached to the tracing point instead.
References & Further Reading
- Wikipedia contributors. Trammel of Archimedes. Wikipedia
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