A Spring Cyclograph is a draughting instrument that traces decaying spiral and rosette curves by coupling a pen arm to one or more spring-driven oscillators whose amplitude shrinks as the spring unwinds. The Newton & Co. pattern sold to British technical schools through the 1890s is the best-known example. It exists to give draughtsmen a repeatable way to draw logarithmic and cycloidal curves that compasses cannot generate. The result is a single continuous spiral or rosette plotted to roughly ±0.2 mm over a 150 mm field.
Spring Cyclograph Interactive Calculator
Vary the initial radius, damping, angular frequency, and elapsed time to see the decaying spiral radius and trace position.
Equation Used
The calculator applies the article's radius decay relation for a spring cyclograph: the pen starts at initial radius R0 and the radius shrinks exponentially as damping zeta and angular frequency omega act over time t.
- Pen motion follows a logarithmic spiral with exponential radial decay.
- Angular frequency is treated as constant over the plotted interval.
- Damping combines spring energy loss, pen drag, and pivot friction into zeta.
Inside the Spring Cyclograph
The Spring Cyclograph belongs to the same family as the harmonograph, but it lives on a drawing board rather than a tabletop. A flat clock spring or a pair of helical springs drives a pivoting arm that carries the pen. As the spring unwinds, two things happen at the same time: the arm sweeps in a near-circular path, and the radius of that path decays because the restoring force falls off with each cycle. That decay is what makes the curve a spiral rather than a closed circle. If you tune the damping correctly you get a logarithmic spiral; tune it wrong and you get either a flat-decay near-circle or a rapid in-spiral that crashes to the centre in three turns.
The geometry is set by three things — the spring rate k, the effective arm length L, and the damping coefficient c that comes from pen-on-paper friction plus pivot bushing drag. Get the bushing tolerance wrong and the whole instrument goes off. The pivot bore must run a slip fit on the post — typically 3.00 mm post into a 3.05 mm reamed brass bushing. Open it up to 3.15 mm and the arm wobbles laterally, putting a visible 0.3 mm tremor on every loop. Close it to 3.02 mm and the spring stalls before the curve closes.
Failure modes are predictable. The spring takes a set after a few hundred cycles and the decay rate changes — you'll see the outer loops grow tighter than the inner ones, which is the signature of a fatigued spring. Pen drag is the other big one. A nib that's too wet drops the damping coefficient unpredictably, and you get a curve that wanders off the predicted log spiral by 1-2 mm at the outer turn.
Key Components
- Driving spring: A flat spiral clock spring or paired helical tension springs storing the energy that drives the oscillation. Spring rate is typically 0.8 to 1.2 N·mm/rad on bench instruments; outside that band the decay either stalls or runs away.
- Pen arm: A balanced brass or aluminium arm pivoting on a hardened steel post. Effective length usually 80-150 mm. The arm must be balanced about the pivot to within 0.5 g·mm, otherwise gravity adds a once-per-revolution heartbeat in the trace.
- Pivot bushing: A reamed brass bushing running a slip fit over the post — 3.00 mm post into a 3.05 mm bushing is the standard fit on Stanley and Newton patterns. Wear past 3.10 mm shows up as a 0.3 mm radial tremor.
- Pen carrier: A sprung clip holding a ruling pen, mapping pen, or fine drafting pencil. The carrier applies 15-25 g of down-force — more than that and the pen drags hard enough to throttle the spring early.
- Adjustable counterweight: Slides along the back of the pen arm to trim out-of-balance moments and to set the natural frequency. Moving it 10 mm shifts ω by roughly 4%.
- Drawing board mount: A weighted base or screw clamp that fixes the pivot post to the board. Any base movement above ~50 µm during a run shows up directly in the trace as a step or kink.
Where the Spring Cyclograph Is Used
The Spring Cyclograph never had a single industrial home — it lived between draughting offices, technical schools, and ornamental designers who needed repeatable spiral curves. Anywhere you need a decaying logarithmic spiral, a cycloidal rosette, or a damped Lissajous figure plotted onto paper, this is the instrument. Modern users tend to be restorers, model engineers, and designers reproducing 19th-century ornament where compass-and-French-curve methods cannot reproduce the original geometry.
- Banknote and security printing: Reproducing guilloché-style decaying rosettes for engraved plate work — Bradbury Wilkinson historically used spring-driven rose engines and cyclographs to lay out background patterns.
- Watch dial engraving: Drawing the master spiral for a damaskeened or engine-turned dial face before transfer to the rose engine. Used at small workshops in La Chaux-de-Fonds for one-off restoration dials.
- Architectural ornament: Laying out volute spirals for cast iron Corinthian capital reproductions, where a true logarithmic spiral is required and a compass approximation looks wrong.
- Technical education: Newton & Co. supplied Spring Cyclographs to UK Mechanics' Institutes and the City and Guilds drawing classes from the 1880s through the 1920s to demonstrate damped harmonic motion graphically.
- Antique instrument restoration: Replotting missing scale spirals on early 20th-century slide rules and circular calculators where the original master plate has been lost.
- Textile and wallpaper design: Generating master rosette and spiral motifs for repeating-pattern transfer rollers, particularly at Morris & Co. successor workshops working on heritage reproduction.
The Formula Behind the Spring Cyclograph
The pen tip traces a decaying sinusoid on each axis, and the radius from the pivot follows an exponential decay set by the damping ratio. At the low end of the typical range — light damping, ζ ≈ 0.02 — the spiral takes 50+ turns to close and the outer loops sit almost on top of each other, which is what you want for a fine guilloché background. At the high end, ζ ≈ 0.15, the curve closes in 6-8 turns and reads as a bold rosette. The sweet spot for legible drawing-office work sits around ζ = 0.05, giving roughly 15-20 visible turns over a 150 mm field.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| r(t) | Pen tip radius from pivot at time t | m | in |
| R0 | Initial radius — the starting amplitude when the spring is fully wound | m | in |
| ζ | Damping ratio — friction plus pen drag divided by critical damping | dimensionless | dimensionless |
| ω | Natural angular frequency of the spring-arm system | rad/s | rad/s |
| t | Elapsed time from release of the spring | s | s |
Worked Example: Spring Cyclograph in a heritage typefoundry in Antwerp
A heritage typefoundry in Antwerp is reproducing the spiral ornament from a 1908 Plantin specimen page using a restored Newton & Co. Spring Cyclograph. The starting radius is set to 75 mm, the natural angular frequency measured by stopwatch over 20 cycles is 4.2 rad/s, and the team wants the outer loop to decay to roughly 5 mm over 30 seconds of run time. They need to know what damping ratio to dial in by adjusting the pen down-force.
Given
- R0 = 75 mm
- r(t) = 5 mm (target)
- ω = 4.2 rad/s
- t = 30 s
Solution
Step 1 — rearrange the decay equation to solve for ζ:
Step 2 — plug in the nominal target values, R0 = 75 mm decaying to r = 5 mm in 30 s at ω = 4.2 rad/s:
That puts the instrument firmly in the lightly-damped regime, which is exactly where Plantin-era spiral ornament lives — many fine, closely-spaced loops.
Step 3 — check the low end of the typical operating range, ζ = 0.01 (almost no pen drag, very wet ink):
At this damping the spiral barely shrinks — the outer loops crowd into a 50 mm-wide band and the design reads as a thick ring rather than a spiral. The drawing looks muddy at arm's length.
Step 4 — check the high end, ζ = 0.08 (heavy pen, dry nib, tight bushing):
The pen crashes to the centre in under 8 seconds and the spring still has half its wind left. You'd see 4-5 loops and then a dot — useless for a typefoundry ornament but acceptable for a quick decorative flourish.
Result
The required damping ratio is ζ ≈ 0. 0215, which the operator dials in by trimming pen down-force to roughly 18 g and using a medium-flow ink. At this nominal setting the curve fills the 75 mm field with about 20 visible loops over 30 seconds — the classic Plantin look. Drop ζ toward 0.01 and the spiral barely decays, leaving a muddy ring; push it past 0.08 and the pen crashes to centre in 8 seconds with only 4-5 loops drawn. If your measured outer-to-inner ratio comes out wrong, check three things in order: (1) the spring may have taken a set, raising the effective decay rate even at correct down-force — a quick test is to time the unloaded period, which should match the nameplate ω within 3%; (2) the pen carrier spring may have weakened, dropping down-force below 15 g and undershooting the target damping; (3) the pivot bushing may have worn past 3.10 mm, adding a once-per-revolution radial wobble that masquerades as variable damping in the trace.
Spring Cyclograph vs Alternatives
The Spring Cyclograph competes with two other ways of getting a controlled spiral onto paper — a true harmonograph and a CNC pen plotter. They are not interchangeable. Each one wins on different axes.
| Property | Spring Cyclograph | Harmonograph (pendulum) | CNC pen plotter |
|---|---|---|---|
| Trace accuracy over 150 mm field | ±0.2 mm | ±0.3 mm | ±0.05 mm |
| Setup time per drawing | 2-5 minutes | 10-20 minutes | 30-60 minutes (file prep) |
| Curve types possible | Decaying spirals, simple rosettes | Lissajous, complex damped rosettes, beat patterns | Any plottable curve |
| Cost (working instrument) | £200-£800 antique | £300-£1500 antique or kit | £400-£3000 new |
| Repeatability of identical curve | Moderate — depends on spring set | Poor — sensitive to release | Excellent — bit-identical |
| Skill required | Low — wind and release | Moderate — release timing matters | High — CAD and toolpath |
| Mains power needed | No | No | Yes |
Frequently Asked Questions About Spring Cyclograph
That's almost always a counterweight balance issue, not a spring problem. If the pen arm has a residual moment about the pivot, gravity pulls harder on one half of each revolution than the other. The result is an eccentricity that's worst at full wind and decays as the spring unwinds, because the gravity moment becomes a larger fraction of the dwindling spring torque.
Quick check: hold the arm horizontal with the spring released and see if it drifts. It should sit motionless. Trim the slider counterweight until it does. On Newton-pattern instruments the trim range is about ±15 mm of slider travel, which corresponds to roughly 0.3 g·mm of moment.
You get a true logarithmic spiral only if the damping is purely viscous and proportional to velocity. Pen-on-paper friction is not — it has a Coulomb (static) component that dominates at low velocities near the centre of the spiral. The outer loops trace a clean log spiral; the inner few loops deviate, typically pulling tight 1-2 mm earlier than the equation predicts.
If you need mathematical purity for the inner turns, switch to a ruling pen with a polished agate tip running on calendered paper — that drops the Coulomb component below 5% of total drag and the deviation falls under 0.3 mm.
Pick on the curve family you actually need. The Spring Cyclograph gives you single-frequency decaying spirals and simple rosettes — clean, predictable, and quick to set up. A harmonograph gives you two-frequency Lissajous and beat patterns, which the cyclograph cannot produce at all because it has only one oscillator.
Rule of thumb: if your reference drawing has obvious self-intersecting beat loops or three-fold symmetry that doesn't repeat exactly, it came off a harmonograph. If it's a clean inward spiral or a simple n-petal rosette, a cyclograph made it and a cyclograph will reproduce it.
That's stick-slip in the pivot bushing. As the spring torque drops with each cycle, it eventually falls below the static friction threshold of the bushing for part of the swing. The arm sticks for a few milliseconds, then slips, then sticks again. You see this as a series of tiny radial steps in the inner loops.
The fix is to clean the bushing and post with isopropanol, then apply a single drop of light clock oil — Moebius 8000 or equivalent. Do not over-oil; excess oil migrates onto the paper and changes pen drag. If the jitter persists after lubrication, the bushing has worn oval and needs reaming and re-bushing.
Humidity changes the friction coefficient between pen and paper by 15-20% across a typical 30% to 70% RH swing. On damp days the paper fibres swell, the pen drags harder, and ζ rises — your spiral closes faster. On dry days the opposite happens.
If you need day-to-day repeatability, condition the paper for 24 hours in the drawing room before use, and time the period of one full unloaded swing as a calibration check. A 5% shift in measured period is your cue to retrim down-force before committing to the final draw.
Not without redesigning the spring. Arm length scales linearly but the moment of inertia scales with the cube of length, which crashes the natural frequency and makes the trace painfully slow. Tripling the arm from 150 mm to 500 mm drops ω by roughly a factor of 5 unless you also stiffen the spring by 25× — and at that point the spring is hard to wind by hand and the bushing loads exceed what a brass-on-steel slip fit will tolerate.
Practitioners working at 500 mm fields universally switch to a CNC plotter or a motorised rose engine. The Spring Cyclograph's natural sweet spot is 100-200 mm.
References & Further Reading
- Wikipedia contributors. Harmonograph. Wikipedia
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