Alternating Circular Motion is rotary motion that reverses direction periodically — the output shaft swings clockwise, stops, swings counter-clockwise, and repeats while the input keeps turning one way. The reversal comes from a linkage, gear, or cam that maps continuous input rotation onto a bounded angular output. We use it where a machine needs a controlled back-and-forth swing without a separate reversing motor — sewing machine feed dogs, washing machine agitators, and windshield wipers all rely on it.
Alternating Circular Motion Interactive Calculator
Vary crank length and machining error to estimate the rocker swing-angle shift and see the crank-rocker motion respond.
Equation Used
This calculator uses the article tolerance example: a 0.5 mm crank-length error on a 25 mm crank shifts rocker swing by about 2-3 degrees. The result is scaled linearly with crank error for quick design checks.
- Rule is scaled from the article tolerance example for a crank-rocker linkage.
- Small crank-length errors are assumed to create approximately linear swing-angle shifts.
- Applies as a quick tolerance estimate, not a full four-bar kinematic synthesis.
How the Alternating Circular Motion Works
Alternating Circular Motion, also called Alternate Circular Motion in older shop manuals and kinematics texts, takes a continuously rotating input and forces the output to oscillate between two angular limits. The trick is geometry. A crank-rocker four-bar linkage, a Scotch yoke driving a sector gear, or a mangle wheel with a guided pin all do the same job — they re-map a full 360° input revolution onto a bounded swing angle, typically anywhere from 30° to 270° at the output. The output reverses every half input cycle, so output frequency equals input frequency, but the angular velocity reverses sign twice per revolution.
Why design it this way? Because a separate reversing motor is expensive, slow to reverse, and rough on the drivetrain. A passive linkage reverses with zero electrical switching and zero stall current. The motion profile is also smooth — angular velocity follows a near-sinusoidal curve through each swing, which means low jerk at the reversal points if the link lengths are picked correctly.
Tolerances bite hard here. If you're building a crank-rocker and the crank length is off by even 0.5 mm on a 25 mm crank, the rocker swing angle shifts by 2-3° and the dwell at the end of stroke disappears. Worn pivot bushings are the most common failure — once radial play exceeds about 0.15 mm, the output starts chattering at each direction reversal because the linkage briefly free-wheels through the slop. You'll hear it as a tick-tick at the reversal points before you see it on the output.
Key Components
- Driver Crank: The continuously rotating input link, driven by a motor or geartrain. Length sets the throw — on a typical crank-rocker, crank length runs 15-40% of the rocker length. A 0.5 mm error on a 25 mm crank shifts swing angle by 2-3°.
- Coupler Link: Connects the crank to the rocker and transfers force. Length must satisfy the Grashof condition (s + l ≤ p + q) for the crank to fully rotate while the rocker oscillates. Get this wrong and the mechanism locks at a singular position.
- Rocker / Output Link: The oscillating output that swings between two angular limits. Swing amplitude is set by the link length ratios; typical designs run 60-120° peak-to-peak. Mounted on a precision pivot — bushing radial play above 0.15 mm causes audible chatter at reversal.
- Pivot Bearings: Support the crank and rocker pivots. Need to handle full reversal load twice per cycle — fatigue life is the limiting factor. Sealed needle bearings or oil-impregnated bronze bushings are standard; ball bearings struggle with the small reciprocating arc and brinell quickly.
- Frame / Ground Link: The fixed reference that locates the two pivot centres. Centre-to-centre distance tolerance of ±0.05 mm is typical for industrial linkages — drift of 0.2 mm changes the swing angle and breaks symmetry between the two reversal points.
Industries That Rely on the Alternating Circular Motion
Alternating Circular Motion shows up wherever a machine needs controlled angular oscillation from a one-way motor. The reason is simple: motors love spinning one direction, and reversal logic adds cost, control complexity, and mechanical shock. Industries from textiles to automotive to agriculture have settled on linkage-based or cam-based oscillators because they're passive, cheap, and last decades. Here's where you'll find it.
- Sewing Machinery: The feed dog drive on a Juki DDL-8700 industrial lockstitch machine uses an Alternate Circular Motion linkage off the lower shaft to swing the feed dog back and forth in a 4-5 mm stroke at up to 5,500 stitches per minute.
- Automotive: Windshield wiper transmissions on passenger vehicles — Bosch and Valeo systems convert continuous motor rotation into a 70-110° wiper sweep using a crank-and-link mechanism mounted directly on the wiper motor output.
- Home Appliances: Top-loading washing machine agitators, including the legacy Whirlpool direct-drive design, use a sector-gear oscillator to swing the agitator approximately 210° back and forth at 60-65 cycles per minute.
- Machine Tools: Shaper machines like the Atlas 7B use a Whitworth quick-return mechanism — a variant of Alternating Circular Motion — to convert spindle rotation into the cutting-stroke swing of the ram.
- Agricultural Equipment: Sickle bar mowers on John Deere and Kuhn hay cutters convert PTO rotation into the oscillating cutter-bar drive via a wobble-box or pitman linkage running at 800-1,000 cycles per minute.
- Robotics & Education: Tabletop kinetic sculptures and STEM kits like the Tamiya Mechanical Beetle use small geared DC motors with crank-rocker linkages to produce the leg-swing motion from a single rotating input.
The Formula Behind the Alternating Circular Motion
The core question is: given a crank radius and link lengths, what's the rocker swing angle θswing? At the low end of the typical design range — short crank relative to rocker — you get a tight oscillation around 30-45°, useful for things like a wiper near the centre of its sweep. At the nominal design range, swing sits around 60-90°, which is the sweet spot for sewing feed dogs and agitators. Push the crank length toward the rocker length and you can hit 180°+ swings, but the motion gets violent at the reversal points and pivot loads spike. The formula below gives the swing angle directly from the four link lengths of a crank-rocker.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| θswing | Total angular swing of the rocker (output) from one extreme to the other | degrees (°) | degrees (°) |
| r | Rocker (output link) length | mm | in |
| g | Ground link length (pivot-to-pivot distance on the frame) | mm | in |
| l | Coupler link length | mm | in |
| c | Crank (input link) length | mm | in |
Worked Example: Alternating Circular Motion in an industrial sewing machine feed dog drive
You're designing the feed dog oscillator for a Juki-style industrial lockstitch head. The lower shaft spins at 2,500 RPM nominal, and you need the feed dog rocker to swing through roughly 35° to give a 4 mm stitch length at the dog's tip radius. Your linkage layout uses crank c = 6 mm, coupler l = 32 mm, rocker r = 28 mm, ground g = 30 mm. Verify the swing angle and check what happens if you scale the crank for shorter and longer stitch lengths.
Given
- c = 6 mm
- l = 32 mm
- r = 28 mm
- g = 30 mm
- Ninput = 2500 RPM
Solution
Step 1 — compute the rocker angle at the far extreme of the swing (crank pointing away from the rocker pivot, so coupler + crank line up):
Step 2 — compute the rocker angle at the near extreme (crank pointing toward the rocker pivot):
Step 3 — nominal swing angle is the difference:
That's a clean 28.7° swing at nominal crank length. At the low end of typical operating range — drop crank to c = 3 mm for a 2 mm short stitch — repeating the calc gives θswing,low ≈ 14.0°. Barely a flicker at the rocker; the feed dog moves the fabric in tiny increments suited for fine stitching on shirt collars. At the high end — push crank to c = 10 mm for maximum stitch length — you get θswing,high ≈ 49°, which is right at the edge of where Grashof condition starts to limit and pivot reversal forces climb steeply.
Push crank past 12 mm and the linkage approaches its singularity — the coupler and rocker briefly align, output speed spikes, and you'll snap a coupler pin within a few hundred hours. The 6-10 mm window is the sweet spot.
Result
Nominal swing angle is 28. 7°, which gives roughly a 4 mm stitch length at the feed dog tip radius — exactly what a Juki DDL-8700 produces at its mid-range stitch setting. At the low end (3 mm crank) you get 14° swing for fine 2 mm stitches, and at the high end (10 mm crank) you get 49° swing for coarse 7 mm stitches; the sweet spot is the 5-8 mm crank window where pivot loads stay manageable and the Grashof margin holds. If your measured swing comes out 5° below predicted, the most common causes are: (1) ground-link pivot centres drifted from spec by more than 0.2 mm (check the frame casting against drawing), (2) the coupler is the wrong length because the rod-end thread engagement was set short during assembly, or (3) the crank pin is loose in the eccentric, allowing it to migrate inward under load and shorten effective throw.
Alternating Circular Motion vs Alternatives
Alternating Circular Motion isn't the only way to get oscillating rotary output. You can get there with a Scotch yoke, a Geneva drive run in reverse, a servo motor with reversal logic, or a cam-driven sector gear. Each has a different cost, speed ceiling, and lifespan. Here's how the crank-rocker variant of Alternate Circular Motion compares with the two most common alternatives.
| Property | Crank-Rocker (Alternating Circular Motion) | Scotch Yoke Oscillator | Servo Motor with Reversal |
|---|---|---|---|
| Maximum cycle frequency | Up to 100 Hz (6,000 RPM input) | Up to 50 Hz — yoke slot wear limits speed | Up to 20 Hz practical for full reversal |
| Swing angle range | 30° to 270° depending on link ratios | Fixed at 180° peak-to-peak | Arbitrary, software-set 0-360° |
| Cost per unit (production volume) | $3-15 in stamped or cast form | $8-25 (precision yoke slot needed) | $80-300 with driver electronics |
| Lifespan at rated load | 10,000-30,000 hours with bushing service | 3,000-8,000 hours — yoke slot wears | 20,000+ hours, bearing-limited |
| Velocity profile at reversal | Smooth sinusoidal, low jerk | Pure sinusoidal, lowest jerk | Trapezoidal or s-curve, programmable |
| Failure mode | Pivot bushing wear, coupler pin shear | Yoke slot scoring, pin galling | Driver overcurrent, encoder drift |
| Best application fit | High-cycle low-cost (sewing, wipers) | Medium-cycle pumps and saws | Variable-profile motion, robotics |
Frequently Asked Questions About Alternating Circular Motion
That's the time ratio asymmetry inherent to non-symmetric four-bar linkages — the crank covers different angular distances during the forward swing versus the return swing because the coupler-and-rocker geometry isn't mirror-symmetric about the crank axis. It's actually a feature in quick-return mechanisms but a problem in feed dog drives.
If you need symmetric swings, the ground link must equal the coupler length within about 0.5%. Check the dimension between pivot centres — a 0.3 mm drift on a 30 mm ground link is enough to push the swing 2-3° off symmetry. The other cause is a bent crank pin from a previous overload event; pull the crank and check pin perpendicularity with a square.
Yes — they're the same mechanism. Older texts and some Asian manufacturers' shop manuals use Alternate Circular Motion, while modern kinematics literature standardised on Alternating Circular Motion. Both refer to a rotary input producing a bounded back-and-forth rotary output, typically via a crank-rocker, mangle wheel, or sector-gear oscillator.
Decide on three things: required swing angle, required velocity smoothness at reversal, and budget. A Scotch yoke gives you exactly 180° swing with pure sinusoidal velocity — beautiful for low-jerk applications like optical scanners — but the swing is fixed by geometry. A crank-rocker gives you anything from 30° to 270° by tuning link ratios, but velocity at reversal isn't perfectly sinusoidal.
Rule of thumb: if you need anything other than 180° swing, use a crank-rocker. If you need 180° and you're cycling above 30 Hz, the yoke wins on lifespan because it has fewer pivots. Below 10 Hz with 180° swing it's a coin flip — go with whichever is cheaper to fabricate.
Two likely causes that aren't bushing wear. First — coupler-rocker angle approaches a singularity at the swing extremes. If your transmission angle (the angle between coupler and rocker) drops below 30° at either extreme, force transmission collapses and the linkage briefly stalls before snapping through. Recompute the transmission angle at both swing limits and increase coupler length if either is below 40°.
Second — input shaft compliance. If the motor mount or input shaft has any torsional give, the crank lags slightly at peak-load reversal then catches up suddenly. Stiffen the input mount or add a flywheel on the input shaft to smooth the torque spike.
Yes, but inertia becomes the dominant design constraint, not link strength. Industrial sewing heads run feed dog crank-rockers at 5,500 RPM input, but the link masses are tiny — typically under 5 grams for the coupler. Above 5,000 RPM, peak inertial torque at reversal can exceed the steady-state torque by 5-10×.
Practical limit: keep coupler mass under 10 grams per 1,000 RPM of input speed, and keep crank radius under 8 mm. Past that, you need to balance the crank with a counterweight to avoid hammering the input bearing every revolution.
That's bedding-in of the pivot bushings. Bronze bushings and needle bearings both shed a few microns of material in the first hundred hours of running, especially with reversing loads — the radial clearance grows by 20-50 µm. On a short crank, that translates directly to lost effective stroke and 1-2° of swing-angle drift.
The fix is straightforward: use the linkage in a run-in jig at 50% load for the first 20 hours, then re-shim or adjust the rocker stop. After 100 hours the wear rate drops by an order of magnitude and the swing stabilises until the bushings reach end-of-life around 15,000-30,000 hours.
References & Further Reading
- Wikipedia contributors. Four-bar linkage. Wikipedia
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