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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 rackandpinion 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
$a=\frac{v}{{t}_{b}}$Eq. 2
${F}_{u}=\frac{m\cdot g+m\cdot a}{1,000}\text{}(for\text{}lifting\text{}axle)$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/s^{2}
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/s^{2}
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).
LifeTime 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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