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For external spur gears, tooth-count ratio determines the inverse speed ratio and direction reversal.
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Formula
i dimensionless; ω in rad/s; z dimensionless; d in m
| Symbol | Meaning | Unit |
|---|---|---|
| Ratio defined as input speed over output speed | dimensionless | |
| ω1, ω2 | Angular speeds of pinion 1 and gear 2 | rad/s |
| z1, z2 | Tooth counts | dimensionless |
| d1, d2 | Pitch diameters | m |
i=\frac{\omega_1}{\omega_2}=\frac{z_2}{z_1}=\frac{d_2}{d_1}For two external spur gears without slip, pitch-circle tangential speeds are equal and opposite, so ω1/ω2=z2/z1=d2/d1 in magnitude. The sign of directed angular velocity captures reversal and is often reported separately from positive ratio i. Backlash, elastic deformation and losses affect real systems. Internal gears have a different direction relationship.
For two external spur gears without slip, pitch-circle tangential speeds are equal and opposite, so ω1/ω2=z2/z1=d2/d1 in magnitude. The sign of directed angular velocity captures reversal and is often reported separately from positive ratio i. Backlash, elastic deformation and losses affect real systems. Internal gears have a different direction relationship.
For external spur gears, tooth-count ratio determines the inverse speed ratio and direction reversal.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Ratio of an ideal spur-gear pair” and check its unit and magnitude.
A pinion with z1=20 drives a gear with z2=60: i=3, output speed is one-third and direction reverses.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: i dimensionless; ω in rad/s; z dimensionless; d in m.
Formula cluster
For external spur gears, tooth-count ratio determines the inverse speed ratio and direction reversal.
A pinion with z1=20 drives a gear with z2=60: i=3, output speed is one-third and direction reverses.
Determine ideal speed ratio and describe a spur-gear pair.
Here is how to work through a typical Ratio of an ideal spur-gear pair (i = ω₁/ω₂ = z₂/z₁ = d₂/d₁) task step by step:
Task
Work through the numerical example for “Ratio of an ideal spur-gear pair” and check its unit and magnitude.
Solution path
A pinion with z1=20 drives a gear with z2=60: i=3, output speed is one-third and direction reverses.
For an external gear pair, speed is inversely proportional to tooth count. A larger output gear turns more slowly; if z2/z1=3, output speed is one-third input speed. The two gears also reverse direction. The ratio is ideal at pitch circles. The tooth-count relation assumes ideal meshing gears with compatible pitch geometry. The positive ratio uses speed magnitudes; external gears rotate in opposite directions. Torque multiplication is reduced by losses and is not exact for a real gear train. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
In an ideal lossless gear, power Tω is conserved. Reducing speed by a factor of three can ideally raise torque by a factor of three. Real bearing and tooth losses reduce output power. Tooth forces, lubrication and strength also limit usable torque. The tooth-count relation assumes ideal meshing gears with compatible pitch geometry. The positive ratio uses speed magnitudes; external gears rotate in opposite directions. Torque multiplication is reduced by losses and is not exact for a real gear train. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
If angular velocities are directed quantities, external gears rotate oppositely and ω2/ω1 is negative. Many references define ratio as a positive magnitude and state reversal separately. Choose a convention before combining multiple gear stages. The tooth-count relation assumes ideal meshing gears with compatible pitch geometry. The positive ratio uses speed magnitudes; external gears rotate in opposite directions. Torque multiplication is reduced by losses and is not exact for a real gear train. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
Write the quantity and unit from the variable list next to every value: i dimensionless; ω in rad/s; z dimensionless; d in m. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example A pinion with z1=20 drives a gear with z2=60: i=3, output speed is one-third and direction reverses. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. The tooth-count relation assumes ideal meshing gears with compatible pitch geometry. The positive ratio uses speed magnitudes; external gears rotate in opposite directions. Torque multiplication is reduced by losses and is not exact for a real gear train. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
The background states this starting condition: For two external spur gears without slip, pitch-circle tangential speeds are equal and opposite, so ω1/ω2=z2/z1=d2/d1 in magnitude.. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. The tooth-count relation assumes ideal meshing gears with compatible pitch geometry. The positive ratio uses speed magnitudes; external gears rotate in opposite directions. Torque multiplication is reduced by losses and is not exact for a real gear train. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
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