An eccentric cone change-velocity mechanism (form 1) is a cam-and-eccentric device that uses a rotating tapered cone with an offset axis to drive a follower at a non-uniform output velocity from a constant input speed. The configuration appears in early 20th-century textile and printing machinery catalogues, including the work cited in Henry T. Brown's 507 Mechanical Movements. The cone's eccentricity sweeps the follower closer and farther from the input axis through each revolution, so output speed rises and falls predictably within one rotation. Builders use it where a smooth, programmable change-of-velocity profile is needed without electronic control.
Eccentric Cone Change-velocity Interactive Calculator
Vary cone radius, eccentricity, input speed, and shaft angle to see the changing effective radius and output surface velocity.
Equation Used
The eccentric cone changes output velocity by changing the effective follower radius during each revolution. The calculator uses the article relation R(theta) = R0 + e*cos(theta), then converts that instantaneous radius to surface speed at the selected input RPM.
FIRGELLI Automations - Interactive Mechanism Calculators.
- Follower measures radius along the eccentric direction shown by the article end-view.
- Output surface velocity is proportional to effective radius at constant input RPM.
- Eccentricity is less than the base radius so the minimum radius remains positive.
Operating Principle of the Eccentric Cone Change-velocity (form 1)
The mechanism rests on one geometric trick — a cone whose rotational axis is offset from its geometric centerline. As the cone spins, a follower riding against its slanted surface sees a continuously varying radius. That radius variation is what produces the change-velocity output. If the input shaft turns at a steady 100 RPM, the follower output traces a sinusoidal-like velocity curve over each revolution, peaking when the follower contacts the cone at maximum effective radius and dropping when it sees the minimum.
Why a cone and not a flat disk? The taper lets you adjust the velocity ratio by sliding the follower axially along the cone's length. Move the follower toward the wide end and you get a larger swing in radius — bigger velocity variation per revolution. Slide it toward the narrow end and the output smooths out toward near-uniform motion. That axial adjustment is the design knob. In a typical textile build the cone might run 80 to 200 mm long with a half-angle of 8° to 15°, giving you a velocity ratio range of roughly 1.4:1 to 3:1 across the slide stroke.
Tolerance matters more than people expect. The eccentricity offset must hold within ±0.05 mm of nominal — drift beyond that and the follower starts hammering at the high-radius point, which you'll hear as a periodic tick at input frequency. Surface finish on the cone face needs Ra below 0.8 µm or the follower wears a groove that locks the velocity profile to one position and kills the axial adjustability. Common failure modes are bearing wear at the offset journal (which lets the cone wobble and adds an unwanted second-harmonic to the output), and follower spring fatigue (which lets the follower lift off near top dead center and chops the velocity curve in half).
Key Components
- Eccentric Cone Body: The rotating tapered cylinder, machined with its rotational axis offset 2 to 8 mm from the geometric centerline. Material is typically hardened tool steel — 58 HRC minimum — because the contact patch with the follower sees concentrated Hertzian stress on every revolution.
- Sliding Follower: A roller or flat-faced follower that rides the cone surface. The follower carriage slides axially along a guide rail parallel to the cone axis, and that axial position sets the velocity ratio. Carriage backlash should stay under 0.1 mm or the velocity profile drifts revolution-to-revolution.
- Adjustment Lead Screw: Drives the follower carriage along the cone length. A 1 mm pitch fine-thread screw gives roughly 0.5° of velocity-ratio resolution per quarter turn — fine enough to dial in production speeds on a textile take-up without overshooting.
- Return Spring or Counter-cam: Keeps the follower in positive contact with the cone face. Spring rate must produce at least 1.5× the peak inertial lift force at maximum input RPM, otherwise the follower lifts off near the high-radius point and the output velocity profile clips.
- Output Linkage: Converts follower displacement into useful motion — usually a rocker arm or a connecting rod driving a sliding rack. Pivot bushings at this stage want clearances under 0.05 mm because slop here multiplies through the lever ratio and shows up as positional jitter on the driven element.
Who Uses the Eccentric Cone Change-velocity (form 1)
The eccentric cone change-velocity shows up wherever a machine needs a programmable, mechanically reliable variable-speed output without going to electronic servo control. It's a 19th and early-20th century solution that survives in modern equipment because it's cheap, repeatable, and immune to power outages. Where you see it most today: textile take-up drives, web-handling tension compensators, and any process that needs a slow-fast-slow cycle synchronized to a master shaft.
- Textile Machinery: Yarn take-up speed modulation on Saurer-style ring spinning frames, where the cone's velocity profile compensates for changing bobbin diameter as yarn winds on.
- Paper Web Handling: Tension compensation rollers on Voith paper-machine winders, using the cone to ramp roller speed against drum buildup.
- Wire Drawing: Capstan speed adjustment on Niehoff multi-die wire drawing lines, where each successive die needs a different surface speed proportional to drawn diameter.
- Printing: Variable feed roll drives on legacy Heidelberg cylinder presses, modulating sheet feed velocity through the gripper handoff cycle.
- Cable Stranding: Bow-drive speed variation on planetary cable stranders by Cortinovis Machinery, syncing strand lay length to take-up speed.
- Conveying: Surge-control drives on bottle-handling conveyors at lines like Krones bottling, smoothing flow into accumulation tables without servo controls.
The Formula Behind the Eccentric Cone Change-velocity (form 1)
The formula computes the instantaneous output velocity ratio of the follower relative to the input shaft as a function of cone rotation angle and follower axial position. At the low end of the typical axial range, the follower sits near the narrow end of the cone and you get a velocity-ratio swing of maybe 1.2:1 — barely a hint of change-velocity, useful for fine trim. At the high end of the axial range — follower near the wide end — the swing climbs to 3:1 or more, where the output briefly stalls near the minimum radius point. The sweet spot for most textile and web-handling work sits around 1.6:1 to 2:1, where the velocity curve has clear modulation but the follower never approaches lift-off.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| ωout | Output angular velocity of the follower link | rad/s | rad/s |
| ωin | Input angular velocity of the cone shaft | rad/s | rad/s |
| R0 | Mean cone radius at the follower's axial position | mm | in |
| e | Eccentricity offset of cone rotational axis from geometric centerline | mm | in |
| θ | Cone rotation angle measured from maximum-radius position | rad | rad |
| rf | Effective radius of the follower output link pivot | mm | in |
Worked Example: Eccentric Cone Change-velocity (form 1) in a glass-fiber roving winder
You're sizing the eccentric cone change-velocity drive for the traverse-stroke modulator on a Leistritz-style glass-fiber roving winder. Input shaft runs steady at 90 RPM from the line gearbox. The cone has a mean radius R₀ of 40 mm at the nominal follower position, an eccentricity e of 6 mm, and the output rocker pivots at r_f = 50 mm. You need to know the output velocity at the cone's maximum-radius point, the minimum-radius point, and a quarter-turn between, so you can verify the traverse stroke timing matches the package winding angle.
Given
- ωin = 9.42 rad/s (90 RPM)
- R0 = 40 mm
- e = 6 mm
- rf = 50 mm
Solution
Step 1 — compute the nominal velocity ratio at θ = 0° (maximum-radius contact point):
So at the peak of the cycle the output runs at 0.92 × 9.42 = 8.67 rad/s, or about 82.8 RPM. This is the fast portion of the traverse stroke — the yarn guide sweeps quickly across the package face here, laying down the long diagonal pass.
Step 2 — at the low end of the cycle, θ = 180° (minimum-radius contact):
That gives 6.41 rad/s, or 61.2 RPM. This is the slow portion — the guide dwells near the package edge, allowing crossover wraps to lay cleanly without a hard stack-up at the turnaround. The 0.68 ratio is also the lowest point in the cycle, and at this radius the follower spring sees the lightest load — if you've sized the spring for the high point, the low point is automatically safe.
Step 3 — at θ = 90° (quarter turn, the steepest velocity transition):
That's 7.54 rad/s, or 72 RPM, and crucially this is where d(ωout)/dθ is maximum — the follower is accelerating fastest. Inertial loads on the follower carriage spike here. If you push input speed to 120 RPM, that acceleration scales with the square of input speed, so the follower lift-off risk grows fast. Above roughly 110 RPM input on this geometry you'll see the spring start losing contact at θ ≈ 90°, which shows up as a tick at twice the input frequency.
Result
Nominal output velocity sweeps between 61. 2 RPM and 82.8 RPM, with a 1.35:1 ratio swing — a moderate change-velocity profile suitable for clean roving-package crossover. To a builder watching the yarn guide, this looks like a smooth fast-then-slow oscillation with no visible jerk at the turnaround, which is exactly what you want. Drop the input to 60 RPM and you get the same ratio swing scaled down — output runs 40.8 to 55.2 RPM, slow enough to hand-trace the motion through one cycle. Push input to 120 RPM and you reach 81.6 to 110.4 RPM output, but the follower carriage starts approaching lift-off near the 90° transition. If your measured output ratio is closer to 1.0:1 than the predicted 1.35:1, the most common causes are: (1) follower carriage slid off-position because the lead-screw lock nut backed off, (2) cone eccentricity has been ground down by abrasion to under 4 mm — check with a dial indicator on the OD, or (3) the output rocker pivot has worn the bushing oversize, swallowing displacement before it reaches the driven element.
Choosing the Eccentric Cone Change-velocity (form 1): Pros and Cons
The eccentric cone competes against a small handful of other ways to get variable output speed from a constant input. Each option lands differently on cost, controllability, and noise. Pick based on whether you need on-the-fly adjustability, what RPM range you live in, and how much velocity ratio swing the application demands.
| Property | Eccentric Cone Change-Velocity | Non-circular Gear Pair | Servo Motor with Cam Profile |
|---|---|---|---|
| Velocity ratio range | 1.1:1 to 3:1, axially adjustable | Fixed by gear cut, typically 1.2:1 to 2.5:1 | Unlimited, electronically programmable |
| Maximum input RPM | Practical limit ~600 RPM before follower lift-off | Up to 3,000 RPM with proper tooth profile | Up to motor rated speed, often 4,000 RPM+ |
| On-the-fly adjustability | Yes, via lead-screw axial slide | No, ratio fixed at manufacture | Yes, software-defined |
| Cost (mid-size machine) | $400 to $1,200 for the cone-and-follower assembly | $1,500 to $4,000 for ground non-circular gears | $2,500 to $8,000 for servo plus drive plus controller |
| Maintenance interval | Re-grease every 2,000 hours, follower spring inspection at 6,000 hours | Lubrication every 1,000 hours, gear inspection at 10,000 hours | Servo bearing replacement at 20,000 hours, encoder recalibration annually |
| Application fit | Textile, paper, wire — moderate speeds, mechanical sync required | High-speed packaging, printing — fixed ratio acceptable | Anywhere needing reprogrammable profiles or remote control |
| Noise at peak speed | 55 to 70 dB depending on follower spring tuning | 60 to 75 dB from gear mesh | 40 to 55 dB, mostly motor cooling fan |
Frequently Asked Questions About Eccentric Cone Change-velocity (form 1)
You almost certainly have a follower that's rotating to align with the cone surface instead of staying parallel to the cone axis. The follower's contact line must sit perpendicular to the cone's rotational axis, not perpendicular to the tapered surface. When the follower self-aligns to the taper — which happens with spherical or self-aligning roller followers — it tracks the local radius rather than the eccentric offset, so axial position stops mattering.
Fix it by replacing with a flat-faced or fixed-axis cylindrical follower. Diagnostic check: with the machine off, rotate the cone by hand and watch the follower contact mark — if it shifts circumferentially as you slide axially, alignment is wrong.
Eccentricity sets the maximum velocity-ratio swing you can ever achieve, regardless of follower position. 6 mm on a 40 mm mean radius gives you up to 1.35:1 swing — fine for tension compensation and gentle traverse modulation. 10 mm on the same mean radius pushes you to 1.67:1, which is where you want to be for aggressive winding-pattern work or wire-drawing capstan matching.
The cost is dynamic load. Inertial force on the follower scales linearly with eccentricity at constant RPM, so a 10 mm cone needs roughly 1.7× the spring preload of a 6 mm cone to keep contact. If you don't need the swing, don't pay for it — bigger eccentricity also means more bearing load on the offset journal and shorter service life.
Second-harmonic content (a wiggle that repeats twice per cone revolution rather than once) almost always points to bearing journal wear at the offset shaft. As the journal wears oval, the cone wobbles slightly in-plane, adding a 2× component to the radius variation seen by the follower.
Confirm with a dial indicator on the cone OD at the follower position — readings should vary by exactly 2e over one revolution. If you see additional small humps at 90° and 270°, the journal is gone. The other suspect is a bent follower carriage rail letting the carriage rock at the velocity peaks; check rail straightness with a precision level along its length.
It can work closed-loop, but only with a position encoder on the output side, not on the input cone shaft. Because the input-to-output ratio varies through each revolution, encoder counts on the input don't map linearly to output position. You'd need a lookup table that converts input angle to output displacement using the cone geometry, which adds calibration complexity.
For most applications it's not worth it — if you need closed-loop control, you've already crossed the threshold where a servo with a programmable cam profile becomes the better engineering choice. The eccentric cone earns its place specifically when you want mechanical determinism without the encoder stack.
You're hitting follower lift-off resonance. As input speed climbs, the inertial force trying to throw the follower away from the cone face at the velocity-peak point eventually exceeds the spring preload holding it in contact. The follower hops, contacts again at the next valley, and you hear it as periodic chatter at input frequency or 2× input.
Calculate the lift-off RPM as ωlimit = √(Fspring / (mfollower × e)) and stay 20% below it. If you've already hit chatter, the cheap fix is a stiffer return spring; the better fix is a lighter follower or a positive-engagement counter-cam that pushes the follower back instead of relying on a spring.
Ra 0.8 µm is the upper limit for a working surface and only acceptable if the follower is a roller, not flat-faced. A flat-faced follower running on Ra 0.8 µm will polish the cone unevenly along the contact band, gradually changing the local radius and shifting your velocity profile by a few percent over the first 500 hours of operation.
For flat followers, target Ra 0.4 µm or better, hardened to 58 HRC minimum. For rollers, Ra 0.8 µm is fine because contact is line-on-line and rolling rather than sliding. If you see your velocity profile drifting after a few hundred hours, that's surface-finish-driven wear, not a bearing or alignment problem.
References & Further Reading
- Wikipedia contributors. Cam. Wikipedia
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