BIZOMAT MAKEOS · Mechanics
Rack and Pinion Axis Sizing
What pinion torque, motor torque and gear ratio does this rack-and-pinion axis need, and are the teeth strong enough?
Sizing of a rack-and-pinion linear axis (gantry, transfer or vertical axis): forces from friction, gravity on an incline, external load and acceleration, pinion pitch diameter, torque and speed, reducer ratio for the motor speed, motor torque including rotor inertia, and the Lewis tooth bending stress with the velocity factor (as the Lewis bending tool) against an allowable.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Load and motion | ||
| Moving mass | kg | Mass of the carriage and load. |
| Guide friction | Friction coefficient of the guides. | |
| External force | N | Process force opposing motion. |
| Axis inclination from horizontal | ° | Angle of the axis from horizontal (90° for a vertical axis; no counterbalance assumed). |
| Maximum speed | m/s | Top speed. |
| Acceleration time | s | Time to reach top speed. |
| Pinion and drive | ||
| Module | mm | Pinion and rack module. |
| Pinion teeth | Teeth on the pinion. | |
| Face width | mm | Face width in mesh. |
| Lewis form factor | Lewis form factor for the pinion tooth count. | |
| Tooth quality | choice | Manufacturing quality, for the Barth velocity factor. |
| Allowable bending stress | MPa | Allowable bending stress for the rack and pinion material. |
| Rack and pinion efficiency | % | Efficiency of the rack and pinion mesh. |
| Reducer ratio | Reducer ratio between motor and pinion. | |
| Reducer efficiency | % | Reducer efficiency. |
| Motor inertia | kg·m² | Motor rotor inertia. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
Load and motion
What this means: Moving mass
- What it is
- Total moving mass: the carriage, payload and anything else that travels with the axis.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0.000001 kg to 1000000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 300 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Friction, gravity and acceleration forces are all proportional to mass.
What this means: Guide friction
- What it is
- Friction coefficient of the linear guides or slideways carrying the carriage, from the guide maker’s data.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 1. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.05 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: External force
- What it is
- Process force acting against the motion, such as a cutting or pushing force, added to friction and gravity.
- Units and format
- Enter a number in N: newtons. The form accepts 0 N to 10000000 N. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Axis inclination from horizontal
- What it is
- Angle of the axis from horizontal: 0 for a horizontal axis, 90 for a vertical one. Gravity adds to the drive force as the angle rises.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0° to 90°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- The load is assumed to be lifted with no counterbalance.
What this means: Maximum speed
- What it is
- Top linear travel speed of the carriage, which sets the pinion and motor speeds.
- Units and format
- Enter a number in m/s: metres per second. The form accepts 0.000001 m/s to 20 m/s. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.5 m/s as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Acceleration time
- What it is
- Time taken to accelerate from standstill to top speed.
- Units and format
- Enter a number in s: seconds. The form accepts 0.000001 s to 100 s. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.3 s as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A shorter time raises the acceleration force and the peak motor torque.
Pinion and drive
What this means: Module
- What it is
- Gear module of the rack and pinion: pitch diameter ÷ number of teeth, which sets the tooth size.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 50 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Module in millimetres, not diametral pitch; module = 25.4 ÷ diametral pitch.
What this means: Pinion teeth
- What it is
- Number of teeth on the pinion; with the module it sets the pitch diameter.
- Units and format
- Enter a plain number with no unit. The form accepts 10 to 200. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More teeth give a larger pinion, more torque per newton of force and a lower pinion speed.
What this means: Face width
- What it is
- Width of the teeth in mesh, measured across the face parallel to the pinion axis.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 500 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Tooth bending stress is inversely proportional to face width.
- Get it right
- Use the narrower of the rack and pinion face widths.
What this means: Lewis form factor
- What it is
- Lewis form factor for the pinion’s tooth shape and number of teeth, from a gear reference table.
- Units and format
- Enter a plain number with no unit. The form accepts 0.1 to 0.7. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.32 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Tooth bending stress is inversely proportional to Y.
- Get it right
- Use the capital-Y Lewis factor (Y = π × y) that matches σ = F ÷ (b m Y), not the lower-case y; mixing them gives a stress π times off.
What this means: Tooth quality
- What it is
- How the teeth are made and finished. It selects the Barth velocity factor that raises the tooth load with pitch-line speed.
- Units and format
- Choose one of 4 options from the list: Cast; Cut; Hobbed or shaped; Ground.
- Example
- The form opens with “Ground” selected as a worked example. Choose the option that matches your case.
What this means: Allowable bending stress
- What it is
- Allowable bending stress for the rack and pinion tooth material and heat treatment, from the gear maker or a gear standard.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 2000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Rack and pinion efficiency
- What it is
- Efficiency of the rack and pinion mesh, from its maker.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 50% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 97% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Reducer ratio
- What it is
- Speed reduction of the gearbox between the motor and the pinion, motor speed ÷ pinion speed.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 1000. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Reducer efficiency
- What it is
- Efficiency of the gearbox between the motor and the pinion, from the gearbox maker.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 30% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 95% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Motor inertia
- What it is
- Rotor inertia of the motor from its data sheet; the motor must accelerate it along with the load.
- Units and format
- Enter a number in kg·m²: kilogram square metres (mass moment of inertia). The form accepts 0 kg·m² to 100 kg·m². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.002 kg·m² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- kg·m², not kg·cm²: 1 kg·cm² = 0.0001 kg·m².
Method and formulas
- Running force = m g (sin α + μ cos α) + F_ext; accelerating force = running + m a.
- Pinion d = m z; pinion torque = F d ÷ 2 ÷ η_rp; pinion speed = v ÷ π d (as the rack and pinion tool).
- Motor torque = pinion torque ÷ (i η_g) + J_m α_motor; motor speed = pinion speed × i.
- Lewis: σ = K_v F ÷ (b m Y) with the Barth factor for the tooth quality (as the Lewis bending tool).
- Example: 300 kg at 1.5 m/s with a module 3, 20-tooth pinion needs 9.3467 N·m peak at the motor.
Limitations
- Vertical axes need a counterbalance or brake check beyond this; gravity is included only as a running force.
- Lewis is a first check; use ISO 6336 for final rack ratings.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size motor, reducer and pinion for a long-travel axis
- Check rack teeth for a heavier carriage
- Plan a vertical rack drive with counterbalance
Who it is for
- Machine designers
- Automation engineers
- Design engineers
What you get
- Axial forces running and accelerating
- Pinion and motor torque and speed
- Tooth bending stress
Interpreting the result
- A larger pinion lowers tooth load but needs more motor torque or a higher ratio.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2131–2140, 2219. Topic reference; no source tables or recommended settings are reproduced.
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