Bevel and worm gear force analysis
Ref: Mechanical Engineering Design, Second Edition by Joseph Edward Shigley
Worm Gear
p. 543-550
As shown in figure 12-16 on page 544 the forces exerted on the worm gear are given by
𝑊 𝑥 = 𝑊𝑐𝑜𝑠(𝜙𝑛 ) sin(𝜆) + 𝜇 cos(𝜆)
𝑊 𝑦 = 𝑊𝑠𝑖𝑛(𝜙𝑛 )
𝑊 𝑧 = 𝑊𝑐𝑜𝑠(𝜙𝑛 ) cos(𝜆) − 𝜇 sin(𝜆)
Where 𝜙𝑛 is the pressure angle, 𝜆 is the lead angle and µ is the coefficient of friction. For our worm gear
𝜙𝑛 = 0.253 radians and 𝜆 = 0.0977 radians. An unknown is the coefficient of friction between a nickel
plated stainless steel worm and a molybdenum worm gear. We know it takes, as an upper limit 100
in.lbs to rotate the jaw with the tubes clamped. We also know it takes 6 in.lbs to rotate the jaw through
the worm. Also, the gear reduction through the worm is 80. So the efficiency of the worm is
𝜂=
𝑊𝑡 (without friction) 100/80
=
= 0.2.
𝑊𝑡 (with friction)
6
Using equation 12-26 and solving for the coefficient of friction we find
𝜇=
cos(𝜙𝑛 ) − 𝜂 cos(𝜙𝑛 )
= 0.3.
tan(𝜆) + 𝜂 cot(𝜆)
From the torque and the pitch radius of the worm we can find the tangential force on the worm:
𝑊𝑤𝑡 = 𝑊 𝑥 =
6 in.lbs
= 24 lbs.
𝑟 = 0.25 in
From this we can find the total force and the other two force components:
𝑊 𝑥 = 𝑊𝑐𝑜𝑠(𝜙𝑛 ) sin(𝜆) + 𝜇 cos(𝜆) = 24 lbs
𝑊 = 64 lbs.
And so
𝑊 𝑦 = 𝑊𝑠𝑖𝑛(𝜙𝑛 ) = 16 lbs
𝑊 𝑧 = 𝑊𝑐𝑜𝑠(𝜙𝑛 ) cos(𝜆) − 𝜇 sin(𝜆) = 60 lbs.
The thrust on the worm is 60 lbs and the transverse component is 𝑊𝑡𝑟𝑎𝑛𝑠 = √242 + 162 = 29 lbs.
Bevel Gear
p. 555-558
The transmitted load through a bevel gear is given by
𝑊𝑡 =
𝑇
𝑟𝑎𝑣
Where W_t is the transmitted load tangent to the gear radius, T is the torque and r_av is the pitch
radius.
The torque transmitted through the worm gear is 6 in.lbs. This torque is transmitted through the Geneva
Mechanism with a driven wheel minimum radius of 0.25 in and the driver having a radius of 0.75 in. The
torque on the bevel is therefore,
1
𝑇 = 6 in.lbs (
) (0.75 in) = 18 in.lbs
0.25 in.
The average pitch radius for the gear is 7/16 in. So,
𝑊𝑡 =
18
= 41 lbs.
7/16
The total force on the gear also has two other components, radial (W_r) and axial (W_a) as shown in
figure 12.23 p. 557 in the above referenced book. They are determined by
𝑊𝑟 = 𝑊𝑡 tan(𝛾) cos(Γ)
𝑊𝑎 = 𝑊𝑡 tan(𝛾) sin(Γ)
 being the pinion pitch angle and  being the gear pitch angle.
tan(𝛾) =
𝑁𝑝
𝑁𝐺
tan(Γ) =
𝑁𝐺
𝑁𝑝
Where N_G and N_p are the gear and pinion teeth numbers.
16
𝛾 = arctan ( ) = 0.464 radians
32
Γ = arctan (
32
) = 1.11 radians
16
The resultant forces are
𝑊𝑟 = 41 tan(0.464) cos(1.11) = 9.1 lbs
𝑊𝑎 = 41 tan(0.464) sin(1.11) = 18.3 lbs
The pinion is 90 degrees to the gear and so the radial and axial forces are swapped. So, the thrust force
on the pinion axel is 9.1 lbs. The transverse force is
𝑊trans = √412 + 18.32 = 45 lbs.