Combination crank-motion curves are output paths traced by a point on a compound linkage built from two or more crank-driven four-bar stages connected in series. Each stage adds its own coupler curve to the previous one, so the final point follows a sum-of-motions trajectory that no single crank can produce. Designers use them to generate dwells, figure-eights, D-shaped strokes, and approximate straight lines in packaging, textile, and assembly machinery — the kind of motion a cam could do, but with rolling-pair joints that survive millions of cycles at 300 RPM.
Combination Crank-motion Curves Interactive Calculator
Vary the crank ratio, phase lead, cycle time, and primary revolutions to see the coupled timing and vector-sum motion curve.
Equation Used
The worked example shows a secondary crank geared 2:1 with a 90 degree phase lead over a 4 s cycle. This calculator applies the timing relation theta_s = ratio*theta_p + phi and computes how many turns and rpm each crank makes during the cycle. The canvas illustrates the related vector-sum idea, P_out = P_2g + r_p, using normalized vectors.
- Secondary crank is geared to the primary crank with a fixed ratio.
- Cranks rotate at constant speed over the cycle.
- Canvas uses normalized link vectors to illustrate the path shape, not a dimensioned linkage synthesis.
Operating Principle of the Combination Crank-motion Curves
Take a single crank-rocker four-bar. Pick a point on the coupler link — not the pin centres, somewhere out on the body of the link — and trace where it goes as the crank rotates. You get a closed curve. That curve is the coupler curve. Hrones and Nelson published an atlas of about 7,000 of these in 1951, and that atlas is still how a lot of designers pick a starting geometry. A single coupler curve gives you one shape per linkage. Useful, but limited.
Combination crank-motion curves come from stacking. You drive a second four-bar from a point on the first coupler — meaning the second linkage's ground pivot is itself moving along the first coupler curve, or the second crank is geared to the first at a non-1:1 ratio. The output point on the second coupler then traces the vector sum of both motions. Change the phase angle between cranks, change the ratio, change the link lengths, and the output curve changes shape dramatically. You can synthesise dwells where the output point sits nearly stationary for 60° to 90° of input rotation, or generate near-straight strokes 4 to 6 times longer than the driving crank radius.
Tolerances bite hard here. Pin-joint clearance of 0.05 mm in a single four-bar is invisible. Stack two linkages and that clearance compounds — the output point can wander 0.2 to 0.3 mm off the theoretical curve, which is the difference between a clean product transfer and a jammed machine. Common failure modes are coupler-link flex under inertia loads at speed, pin-bushing wear that shifts the phase relationship between stages, and crank-shaft misalignment between the two driven cranks when they share a timing belt rather than a rigid gear train.
Key Components
- Primary crank: The driven input, typically rotating at constant speed between 60 and 600 RPM. Its radius sets the base scale of the whole motion — double the crank radius and you double every dimension of the final output curve.
- Primary coupler link: The floating link of the first four-bar. Carries the tracing point or the mounting pivot for the second stage. Must be stiff in bending — a deflection of 0.1 mm at the trace point at 300 RPM throws the output curve off by visible amounts.
- Secondary crank: The input to the stacked second four-bar. Driven from the primary shaft through a timing belt, gear pair, or chain at a defined ratio — commonly 1:1, 2:1, or 3:1. The phase angle between primary and secondary cranks is the single most powerful tuning parameter.
- Secondary coupler with output point: Carries the final trace point that follows the combination curve. Position of this point on the link body — typically described by two offset dimensions from the coupler's pin centres — determines whether you get a dwell, a loop, or a near-straight segment.
- Phase coupling: The geared, belted, or chained linkage between the two cranks. Backlash here directly becomes phase error. A 1° phase drift in a 2:1 system shifts the dwell location by roughly 2° on the output, which on a 200 mm stroke is around 3 mm of position error.
- Ground frame: Holds both fixed pivots in fixed relationship. Frame flex of 0.5 mm under cycle loading is enough to kill dwell repeatability — this is why combination linkages live on cast-iron sub-frames, not welded sheet steel.
Industries That Rely on the Combination Crank-motion Curves
Combination crank-motion curves earn their keep wherever you need a non-trivial repeating path at high cycle rates with low maintenance. Cams do the same job but wear, need lubrication, and lose accuracy as the follower roller pits. Stacked linkages are pin joints — bushings or needle bearings — and at 300 to 600 cycles per minute they outlast cam-follower systems by 5 to 10× before any rebuild. The catch is that you can't change the curve without changing hardware. So they show up where the motion is fixed by product geometry and the machine runs that one product for years.
- Packaging machinery: The intermittent film-feed mechanism on a Bosch SVE 2520 vertical form-fill-seal machine uses a stacked four-bar to generate a dwell-then-pull motion that holds the film stationary during sealing then advances it 200 mm in roughly 90° of crank rotation.
- Textile machinery: The needle bar drive on a Karl Mayer HKS 3-M warp knitting machine uses a combination crank curve to give the needles a near-straight vertical stroke at top of cycle and a precisely shaped lay-in path at the bottom, running at 2,200 RPM.
- Sewing machines: The feed-dog motion on a Pfaff 1245 industrial walking-foot machine combines two cranks to produce the four-motion feed: lift, advance, drop, return — all from a single input shaft.
- Glass container manufacturing: The invert and revert mechanism on an Emhart IS forming machine uses paired crank linkages to flip parisons through 180° with a controlled dwell at each end of the swing.
- Printing presses: The gripper bar transfer on a Heidelberg Speedmaster CD 102 sheet-fed press uses a stacked linkage to accelerate, decelerate, and dwell the grippers during sheet handoff at 15,000 sheets per hour.
- Assembly automation: Pick-and-place units on Adept SmartMotion-style two-axis linkage robots generate D-shaped pick paths using combination crank motion rather than separate servo axes — cheaper and faster for fixed-pitch component placement.
The Formula Behind the Combination Crank-motion Curves
There is no single closed-form equation for an arbitrary combination curve, but the position of the output point is the vector sum of the position of the secondary ground pivot (which is itself a point on the primary coupler) and the offset from that pivot to the output trace point on the secondary coupler. At the low end of typical operating ranges — say a 50 mm primary crank at 60 RPM driving a 2:1 secondary — the curve is a clean, slow-traced shape ideal for inspection. At the high end — 200 mm crank, 600 RPM — coupler inertia dominates and the actual traced path deviates from the kinematic prediction by 1 to 3% because the links flex under their own acceleration. The sweet spot for most production machines lands at 150 to 300 RPM where the geometry still rules but you're getting useful cycle rate.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| Pout(θ) | Position vector of the output trace point as a function of primary crank angle θ | mm | in |
| P2g(θ) | Position of the secondary linkage ground pivot (a point on the primary coupler curve) | mm | in |
| R2(θ + φ) | Rotation matrix of the secondary coupler at phase-shifted angle (θ + φ) | dimensionless | dimensionless |
| rp | Offset vector from secondary coupler pin to the output trace point | mm | in |
| φ | Phase angle between primary and secondary cranks | rad or ° | rad or ° |
| n | Crank speed ratio (secondary RPM / primary RPM) | dimensionless | dimensionless |
Worked Example: Combination Crank-motion Curves in a chocolate enrober's wire belt feed
You are designing the stacked crank linkage for the wire-belt indexing drive on a Sollich Turbotemper enrobing line at a confectionery plant in Lübeck Germany. The belt must dwell for 0.4 seconds while the chocolate curtain coats the pieces, then advance 120 mm in the remaining cycle time. Primary crank radius is 80 mm. Secondary crank runs at 2× primary speed via a toothed belt, with phase angle φ = 90°. You want to predict belt advance per cycle at three operating points and check whether dwell time holds at the high end of the speed range.
Given
- r1 = 80 mm (primary crank radius)
- rp = 60 mm (output offset on secondary coupler)
- n = 2 (secondary:primary ratio)
- φ = 90 ° (phase angle)
- Stroke target = 120 mm advance per cycle
- Dwell target = 0.4 s
Solution
Step 1 — at nominal 90 RPM primary, compute cycle period:
The dwell portion is roughly the segment of the combination curve where the output velocity drops below 10% of mean. With r1 = 80 mm, rp = 60 mm, n = 2, φ = 90°, geometric synthesis gives a dwell arc covering about 110° of input rotation.
That falls short of the 0.4 s target — you need to slow the primary down to hit the dwell window. Step 2 — at the low end of the typical operating range, 60 RPM:
Still short. Drop further to 45 RPM:
That hits the coating window. Belt advance per cycle stays at 120 mm regardless of speed because the geometry is fixed — only time changes. Step 3 — at the high end, 180 RPM, dwell collapses to 0.102 s, which is too short for the chocolate curtain to settle. You'd see streaking and incomplete coverage on the trailing edge of every piece. The curve geometry is right; the speed range is what bounds the application.
Result
Belt advance is 120 mm per cycle at every speed, but useful dwell time is only achievable below 50 RPM — call it 0. 4 s of dwell at 45 RPM. At 90 RPM nominal you get 0.20 s dwell which won't coat reliably; at 180 RPM dwell falls to 0.10 s and chocolate streaks visibly on the trailing edge of each piece. If your measured dwell is shorter than predicted, check three things in order: (1) phase angle drift caused by toothed-belt stretch on the secondary drive — a 2 mm belt elongation on a 200 mm centre distance shifts φ by roughly 4° and shortens the dwell arc by about 12°, (2) flex in the secondary coupler link if it's machined from aluminium thinner than 12 mm, which under inertia loading bends and shaves dwell on the trailing side, and (3) play in the secondary ground-pivot bushing — anything above 0.08 mm radial clearance lets the trace point wander off the synthesised curve.
Choosing the Combination Crank-motion Curves: Pros and Cons
Combination crank-motion curves compete with cams, servo-driven multi-axis systems, and Geneva drives for the same job — generating non-uniform repeating motion. Each option wins on different axes, and the right pick depends on cycle rate, accuracy budget, and how often the motion profile needs to change.
| Property | Combination crank linkage | Cam-follower system | Servo multi-axis |
|---|---|---|---|
| Max practical cycle rate | 600 RPM | 1,200 RPM | 300 RPM (limited by servo bandwidth) |
| Path repeatability | ±0.05 mm with hardened bushings | ±0.02 mm new, ±0.2 mm worn | ±0.01 mm with closed-loop control |
| Cost per axis (typical machine build) | $800–$2,500 hardware | $1,500–$5,000 (cam grinding dominates) | $4,000–$12,000 (servo + drive + controller) |
| Lifespan before rebuild | 50–100 million cycles | 10–30 million cycles before re-grinding | 20,000–40,000 service hours on bearings |
| Lubrication requirement | Sealed bushings, often grease-for-life | Continuous oil bath or grease feed | Sealed servo, gearbox oil change every 5,000 hr |
| Path reconfigurability | None — fixed by hardware | None — must regrind cam | Full — software change only |
| Synthesis complexity | High — Burmester theory or atlas search | Medium — direct profile generation | Low — point-to-point programming |
Frequently Asked Questions About Combination Crank-motion Curves
The kinematic synthesis assumes rigid links and zero-clearance joints. Real linkages have both pin-joint clearance and link compliance. At 300 RPM and above, the inertia force on a coupler link can be 10 to 50 times its static weight — enough to bend an aluminium coupler 0.1 to 0.3 mm in the direction of acceleration. Worse, it bends one way during the acceleration phase and the other way during deceleration, so the trace point traverses an envelope rather than a line.
Quick check: instrument the trace point with a dial indicator while turning the machine over by hand at zero speed. If the path matches simulation by hand but drifts at speed, you have a stiffness or inertia problem — usually solved by switching the secondary coupler to steel or by reducing its mass at the trace-point end.
Phase angle is the most powerful single tuning parameter and there's no closed-form way to pick it. Start with φ = 90° as a default — it gives the most asymmetric output curve, which is usually what you want for dwell-and-stroke motions. φ = 0° and φ = 180° produce symmetric figure-eights, useful for some weaving applications but rarely for indexing.
From there, sweep φ in 15° increments in your CAD or kinematic simulator and plot the resulting curve. The dwell length and stroke length trade against each other along that sweep, and the right choice is the one that matches your product cycle. Lock φ mechanically with a keyed coupling, not a setscrew — phase drift of more than 2° on most production geometries kills dwell accuracy.
Decide on cycle rate and changeover frequency. If you run one product geometry for years and the machine cycles above 200 per minute, the linkage wins on cost, reliability, and energy use — a stacked four-bar burns maybe 100 W to do what two servos burn 800 W to do. If you change product every shift or run below 60 cycles per minute, the servo wins because you can reprogram in software and don't pay the synthesis cost on every product change.
The middle ground is a fixed linkage whose primary crank is driven by a servo. You keep the geometry-defined output curve but add programmable timing — useful for synchronising to upstream conveyors that drift in speed.
Thermal growth in the link lengths. Aluminium has a coefficient of thermal expansion of about 23 µm/m/°C. A 300 mm coupler link that runs 30°C above ambient grows by roughly 0.2 mm — enough to shift the dwell arc location by 1 to 2° of input rotation, and enough to slide the dwell window off the position where your downstream process needs it.
Steel links shift about a third as much. If thermal stability matters, use steel for all links longer than 150 mm, and let the machine reach thermal equilibrium (typically 30 to 60 minutes of running) before fine-tuning phase angle. Don't tune cold — you'll re-tune at lunchtime.
Within 20% on link length ratios and within 30° on coupler-point location is usually close enough to converge in a numerical solver. The atlas is for finding a curve family that's roughly the right shape — a near-straight segment, a flat-bottomed D, a single-loop figure-eight, whatever you need. Once you pick the family, optimisation routines (most modern kinematic packages have one) will refine the link lengths to hit your target points.
If you start with the wrong curve family, no amount of optimisation rescues it — the solver converges to a local minimum that doesn't match your requirement. Spend the time browsing the atlas before you turn on the optimiser.
Almost always one of two things: pin-joint wear or phase-coupling backlash. Output overshoot specifically points to backlash in the drive between primary and secondary cranks — a worn timing belt or a chain that's stretched lets the secondary crank lag on acceleration and overshoot on deceleration. The output point then traces a curve with little hooks at each end of the stroke instead of clean transitions.
Diagnostic check: lock the primary crank, try to rotate the secondary crank by hand. Anything more than 1° of lost motion is your culprit. Replace the timing belt or re-tension the chain before chasing other causes.
References & Further Reading
- Wikipedia contributors. Four-bar linkage. Wikipedia
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