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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

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Eq. 1

${\displaystyle a=\frac{v}{t_b}}$

Eq. 2

${\displaystyle F_u=\frac{m\cdot <!--\; \cdot \; -->g+m\cdot <!--\; \cdot \; -->a}{1,000}(forliftingaxle)}$

Eq. 3

${\displaystyle F_u=\frac{m\cdot <!--\; \cdot \; -->g\cdot <!--\; \cdot \; -->\mathrm{\mu <!--\; \mu \; -->}+m\cdot <!--\; \cdot \; -->a}{1,000}}$

Eq. 4

${\displaystyle F_{uperm}=\frac{F_{uTab}}{K_A\cdot <!--\; \cdot \; -->S_B\cdot <!--\; \cdot \; -->f_n\cdot <!--\; \cdot \; -->L_{KHB}}}$

Where:

a, rack acceleration is in m/s²
F_u, is in kN
F_uperm, is in kN
F_utab = load carrying capability of rack pinion kN
m = mass in kg
v = speed in m/s
t_b = seconds
g = gravity =9.81 m/s²
µ = constant
K

The Condition F_u < F_uperm Must be Fulfilled

Load factor K_A

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 S_B

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

Life-Time Factor f_n

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

Linear Load Distribution Factor L_KHß

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

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

L_KHß = 1.2 for preloaded bearings on the output shaft

L_KHß = 1.5 for unpreloaded bearings on the output shaft

Source:

Kent's Mechanical Engineers Handbook, 1952

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