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Rack and Pinion Drive Calculations

Rack and Pinion Drive Calculations

A rack and pinion is a type of linear actuator that comprises a circular gear (the pinion) engaging a linear gear (the rack). Together, they convert between rotational motion and linear motion. Rotating the pinion causes the rack to be driven in a line. Conversely, moving the rack linearly will cause the pinion to rotate. A rack-and-pinion drive can use both straight and helical gears.

Formulas for Determining the Tangential Force on the rack and pinion gears

Eq. 1

$a=\frac{v}{{t}_{b}}$

Eq. 2

Eq. 3

${F}_{u}=\frac{m\cdot g\cdot \mu +m\cdot a}{1,000}$

Eq. 4

${F}_{uperm}=\frac{{F}_{uTab}}{{K}_{A}\cdot {S}_{B}\cdot {f}_{n}\cdot {L}_{KHB}}$

Where:

a, rack acceleration is in m/s2
Fu, is in kN
Fuperm, is in kN
Futab = load carrying capability of rack pinion kN
m = mass in kg
v = speed in m/s
tb = seconds
g = gravity =9.81 m/s2
µ = constant
K

The Condition Fu < Fuperm Must be Fulfilled

 Drive Uniform Loading Medium Shocks Heavy Shocks Uniform - 1.25 1.75 Light Shocks 1.25 1.50 2.00 Medium Chocks 1.50 1.75 2.25

Safety Coefficient SB

The safety coefficient should be allowed for according to experience (SB = 1.1 to 1.4).

Life-Time Factor fn

considering of the peripheral speed of the pinion and lubrication.

 Lubrication Continuous Daily Monthly Peripheral Speed of Gearing m/sec m/min 0.5 30 0.85 0.95 From 3 to 10 1.0 60 0.95 1.10 1.5 90 1.00 1.20 2.0 120 1.05 1.30 3.0 180 1.10 1.50 5.0 300 1.25 1.90

The linear load distribution factor considers the contact stress, while it describes unintegrated load distribution over the tooth width (LKHß = √KHß).

LKHß = 1.1 for counter bearing, e.g. Torque Supporter

LKHß = 1.2 for preloaded bearings on the output shaft

LKHß = 1.5 for unpreloaded bearings on the output shaft

Source:

Kent's Mechanical Engineers Handbook, 1952

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