Doubling a revolution on the same shaft is a gear arrangement that makes an output shaft rotate twice for every single revolution of an input shaft mounted on the same centreline. You see it in mechanical car odometers, where a low-speed input shaft drives a coaxial output that ticks at exactly twice the rate. It solves the problem of needing a 1:2 step-up without offsetting the output centreline. The result is a compact, repeatable 2× speed multiplier that fits inside a single bore.
Doubling a Revolution on Same Shaft Interactive Calculator
Vary the input turns and gear tooth counts to see the coaxial output speed multiplication through an idler gear.
Equation Used
The speed multiplication is set by the tooth-count ratio of the coaxial input gear to the smaller loose output gear. The idler gear lets both gears share the same centerline but does not change the final ratio.
- Ideal spur gears with no slip.
- The idler bridges the offset but does not change the tooth-count ratio.
- Input and output gears rotate independently on the same centerline.
- Torque factor is the ideal inverse of speed ratio.
How the Doubling a Revolution on Same Shaft Works
The trick is that you cannot get a 1:2 ratio from a single direct gear pair and still keep the input and output on the same axis — the two gears would have to occupy the same space. So we route the motion through an idler. The input gear sits on the input shaft, drives an offset idler gear, and the idler drives a second gear that is bored loose on the input shaft and carries the output. Both shafts share a centreline, but they spin independently because the output gear runs on a bushing or needle bearing around the input shaft, not keyed to it.
The gear ratio comes from tooth count alone. If the input gear has 40 teeth and the output gear has 20 teeth, with any idler in between, the output turns twice per input revolution. The idler does not change the ratio — it only reverses direction and bridges the offset distance. This is the same step-up gear pair logic used in any gear train, just folded back onto a concentric shaft drive.
Tolerances matter. The bore on the output gear must run on a bushing with no more than about 0.05 mm radial play, otherwise the gear cocks and you get tooth-tip contact instead of flank contact, which wears the teeth in a few hundred hours. Centre distance between the input and idler must hold within about 0.1 mm of nominal, or backlash opens up and the output develops a noticeable lash you can feel by hand. The most common failure mode is the bushing on the output gear seizing because someone ran the assembly dry — once the bushing grabs, the output gear locks to the input shaft and the ratio goes to 1:1 instantly.
Key Components
- Input Gear: Fixed to the input shaft with a key or pin. In a typical 1:2 step-up this gear has twice the tooth count of the output gear — for example 40 teeth driving a 20-tooth output. Module is usually 0.5 to 1.0 in instrument-grade builds.
- Idler Gear: Mounted on a parallel offset shaft, meshes both the input and output gears. Tooth count is irrelevant to the final ratio — pick whatever fits the available centre distance. The idler reverses rotation direction, so input and output spin opposite ways.
- Output Gear: Bored loose around the input shaft and supported on a bushing or needle bearing. Carries the output hub or shaft extension. Bore-to-shaft clearance must stay around 0.02 to 0.05 mm radial — tighter and it binds, looser and the gear cocks under load.
- Coaxial Bushing: Sits between the output gear bore and the input shaft. Usually oil-impregnated bronze or PTFE-lined. This is the part that lets two shafts share a centreline while spinning at different speeds.
- Idler Shaft and Support: A short stub shaft pressed into the housing, carrying the idler on its own bearing. Centre distance from the input axis must hold within about 0.1 mm to keep backlash predictable across the gear pair.
Real-World Applications of the Doubling a Revolution on Same Shaft
You find this layout anywhere a designer needs an output that runs at a different speed from the input but cannot afford the space or alignment cost of an offset output shaft. Mechanical odometers and counters are the classic case, but it shows up in clockwork, instrument drives, and any compact gearbox where the output has to come out of the same hole the input went in. The reason it persists in modern designs is simple — it gives you a clean concentric shaft drive without the cost of a planetary set.
- Automotive Instruments: Mechanical odometer and trip-meter drives in pre-1990s vehicles like the VDO and Smiths instruments fitted to older Land Rovers, where the speedometer cable input drives a coaxial counter wheel at a stepped-up ratio.
- Horology: Centre-seconds clockwork in pocket watches and bench clocks where the seconds hand and minute hand share the same arbor but rotate at different rates — the Hamilton 992B railroad pocket watch uses a related coaxial scheme.
- Industrial Counters: Veeder-Root mechanical pulse counters used on production lines, where an input cam drives a coaxial digit wheel that has to advance at a multiplied rate.
- Marine Hardware: Compact winch handles on Lewmar and Harken self-tailing winches, where a 2-speed gear train uses a concentric output to keep the handle and drum on one axis.
- Scientific Instruments: Coarse-fine adjustment knobs on Leitz and Zeiss benchtop microscopes, where the fine-focus knob drives a coaxial coarse-focus output through a stepped gear pair.
- Hand Tools: Geared hand drills like the Schroder breast drill, where the crank input drives a coaxial chuck output at roughly 2× crank speed.
The Formula Behind the Doubling a Revolution on Same Shaft
The output speed depends only on the ratio of input gear teeth to output gear teeth — the idler drops out of the equation entirely. At the low end of typical instrument use, around 30 RPM input, you get a quiet, light-load drive where backlash dominates measured error. At the high end, around 1,500 RPM input, gear-mesh noise and bushing heating become the limits. The sweet spot for a coaxial 1:2 layout sits between 100 and 600 RPM input, where the bushing runs cool, mesh noise stays low, and tooth flank contact is clean.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| Nout | Output shaft rotational speed | rev/min | RPM |
| Nin | Input shaft rotational speed | rev/min | RPM |
| Tin | Tooth count on the input gear | teeth | teeth |
| Tout | Tooth count on the output gear (bored on input shaft) | teeth | teeth |
Worked Example: Doubling a Revolution on Same Shaft in a coin-sorting machine digit counter
You are designing the digit-counter drive on a benchtop coin-sorting machine similar to the Klopp CM model used in small bank branches. The input cam shaft turns at 200 RPM during a sort run. The counter wheel has to advance at twice that rate so each coin pass produces two distinct digit ticks. You pick a 36-tooth input gear, an 18-tooth output gear bored on the input shaft, and a 24-tooth idler on a 14 mm offset stub.
Given
- Nin = 200 RPM
- Tin = 36 teeth
- Tout = 18 teeth
- Tidler = 24 teeth (does not affect ratio)
Solution
Step 1 — compute the gear ratio from input to output tooth count. The idler is ignored:
Step 2 — at the nominal 200 RPM input, the output speed is:
Step 3 — at the low end of the typical operating range, 50 RPM input (machine warm-up or jog mode):
At 100 RPM the digit wheel ticks slowly enough that an operator can read each digit by eye — useful during setup but useless for production sorting. Backlash in the gear pair shows up here as a visible jitter in digit position because the output wheel is barely loaded.
Step 4 — at the high end, 600 RPM input (peak production sort rate):
At 1200 RPM output, the bronze bushing on the output gear starts heating noticeably after about 20 minutes of continuous run. This is where you stop scaling — beyond about 700 RPM input, the same layout needs a needle bearing instead of a plain bushing or the bushing welds itself to the input shaft.
Result
Nominal output speed is 400 RPM, exactly twice the input. At 400 RPM the digit wheel advances cleanly with no visible jitter and the bushing stays cool indefinitely. The low-end 100 RPM result feels sluggish and shows backlash jitter, while the high-end 1200 RPM result runs hot and noisy — the sweet spot for this layout sits between 300 and 800 RPM output. If you measure 380 RPM instead of 400, the most likely causes are: (1) the output gear bore worn beyond 0.08 mm clearance letting the gear cock and skip a tooth every few revolutions, (2) the idler stub shaft pressed in 0.2 mm off nominal centre distance which opens backlash and lets the output lag under inertia, or (3) a damaged tooth flank on the input gear from a prior jam, which produces a measurable speed dip once per input revolution.
When to Use a Doubling a Revolution on Same Shaft and When Not To
When you need a step-up between two shafts, you have a few real options. The coaxial 1:2 layout wins on packaging but loses on torque density. Here is how it stacks up against the two alternatives a designer normally considers — a simple offset spur pair, and a planetary step-up.
| Property | Coaxial 1:2 with idler | Offset spur pair | Planetary step-up |
|---|---|---|---|
| Output centreline vs input | Same axis | Offset by centre distance | Same axis |
| Typical max input speed | 700 RPM with bushing, 3000 RPM with needle bearing | 5000+ RPM | 10,000+ RPM |
| Backlash (typical assembled) | 0.3° to 0.8° | 0.1° to 0.3° | 0.5° to 1.5° |
| Part count | 3 gears + bushing + idler shaft | 2 gears + 2 shafts | Sun + 3 planets + ring + carrier (6+ parts) |
| Relative cost (small batch) | Low | Lowest | High |
| Torque capacity at 1:2 ratio | Limited by bushing — moderate | High | Very high |
| Best application fit | Counters, instruments, compact concentric drives | General gearboxes where offset is acceptable | High-torque coaxial drives like power tools |
Frequently Asked Questions About Doubling a Revolution on Same Shaft
Because the idler is driven by the input gear and then drives the output gear — its tooth count appears once in the numerator and once in the denominator of the ratio chain, so it cancels mathematically. The idler only does two jobs: bridge the centre distance between input and output, and reverse rotation direction.
What the idler tooth count does change is the centre-distance geometry and the contact ratio at each mesh. Pick an idler too small and you lose contact ratio at both meshes, which raises noise. Pick it too large and you waste space. Most designers size the idler to land the centre distance on a convenient round number.
The 0.05 deficit is almost always elastic windup plus accumulated backlash, not a tooth-count error. If you turn the input slowly by hand and measure output position with an encoder, you will read a true 2.00 ratio. Spin it up under load and the bushing-supported output gear lags slightly because the bronze bushing has measurable torsional compliance and the gear flanks are pushed into mesh at one side only.
Quick check: turn the input shaft 10 full revolutions and count output revolutions to the nearest 1/8 turn. If you get exactly 20 turns ± 1/8, your geometry is fine and the 1.95 is a dynamic measurement artefact.
For a hand tool with low duty cycle and moderate torque — under about 5 Nm output — the coaxial idler layout wins on cost and simplicity. You get the same concentric output as a planetary in roughly a third of the parts and a quarter of the cost.
For a power tool that sees continuous load above 10 Nm or input speeds above 3,000 RPM, switch to a planetary. The bushing on the output gear in the idler layout becomes the limiting part — it heats up, wears, and eventually seizes. Planetaries spread the load across three planet gears and run on rolling-element bearings, so they handle continuous duty without complaint.
Click once per output revolution almost always means a single damaged tooth on the output gear, usually from a momentary overload or a small chip lodged in the mesh. Pull the cover and rotate the output by hand — you will feel a notch at the bad tooth.
Click once per input revolution points at the input gear instead, and is usually a burr on a single tooth from initial assembly. Click at an irregular interval suggests the bushing has worn enough that the output gear is cocking and contacting the idler tooth tip instead of flank — measure radial play at the output hub, and if it exceeds 0.1 mm, replace the bushing before the gear teeth get destroyed.
Yes, the geometry is symmetric — drive the small gear and take output from the large gear and you get a 2:1 reduction with the same concentric layout. The catch is that the formerly-output gear, now bored on the driving shaft, has to carry full input torque through its bushing instead of the lower output torque it saw before.
That doubles the bushing PV product (pressure × velocity), so you typically need to upgrade to a needle bearing or accept a much shorter service life. For step-down duty above about 2 Nm input, most designers move to a proper planetary reducer instead.
For module 0.5 to 1.0 instrument gears, hold centre distance to ±0.05 mm if you want backlash under 0.5° and quiet operation. ±0.1 mm is acceptable for counters and non-precision drives where 1° of backlash does not matter.
The reason it matters more here than in a normal offset gearbox: you have two meshes in series (input-to-idler, idler-to-output), so any centre-distance error stacks. A 0.1 mm error at each mesh produces roughly twice the backlash you would see in a single-mesh offset pair. Bore the housing with both shaft holes in one setup if you can — that is the cheapest way to hold the relationship tight.
References & Further Reading
- Wikipedia contributors. Gear train. Wikipedia
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