Enter the gear rack ratio (defined here as tooth density, in teeth per unit length of rack travel), the number of teeth on the pinion gear, and the pitch radius of the pinion gear into the calculator to determine the missing variable.
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Gear Rack Ratio Formula
The following formula is used to calculate the gear rack ratio as defined on this page: the number of pinion teeth per unit of rack travel (tooth density). Use the pitch radius of the pinion for best accuracy.
R = T / (2 * π * r)
Variables:
- R is the gear rack ratio (tooth density), in teeth per unit length (e.g., teeth/m, teeth/in)
- T is the number of teeth on the pinion gear
- r is the pitch radius of the pinion gear (a length)
To calculate the gear rack ratio, divide the number of teeth on the pinion gear by the pitch circumference (2πr). The inverse of this value is often more intuitive: the linear travel per tooth is 1/R = 2πr/T.
What is a Gear Rack Ratio?
In a rack-and-pinion system, rotation of the pinion produces linear motion of the rack. On this page, the “gear rack ratio” refers to a tooth density: how many pinion teeth correspond to a given amount of rack travel (for example, teeth per meter or teeth per inch). This is the inverse of the linear pitch (travel per tooth). Note that the mechanical advantage (force at the rack for a given pinion torque) is primarily set by the pitch radius via F = τ / r; tooth count mainly affects spacing/resolution and gear size rather than force by itself when the radius is fixed.
How to Calculate Gear Rack Ratio?
The following steps outline how to calculate the Gear Rack Ratio.
- First, determine the number of teeth on the pinion gear (T).
- Next, determine the pitch radius of the pinion gear (r).
- Next, calculate the gear rack ratio (tooth density) using the formula R = T / (2 * π * r).
- Optionally, compute the linear travel per tooth as 1/R = 2πr/T.
- After inserting the values and calculating the result, check your answer with the calculator above.
Example Problem :
Use the following variables as an example problem to test your knowledge.
Number of teeth on the pinion gear (T) = 20
Pitch radius of the pinion gear (r) = 0.05 m. Then R = 20 / (2π × 0.05) ≈ 63.6620 teeth/m (≈ 1.6170 teeth/in), and travel per tooth = 1/R ≈ 0.015708 m = 15.708 mm/tooth.