Machine Design II
Prof. K.Gopinath & Prof. M.M.Mayuram
Module 2 - GEARS
Lecture 15 – WORM GEARS
Contents
15.1 Worm gears –an introduction
15.2 Worm gears - geometry and nomenclature
15.3 Worm gears- tooth force analysis
15.4 Worm gears-bending stress analysis
15.5 Worm gears-permissible bending stress
15.6 Worm gears- contact stress analysis
15.7 Worm gears- permissible contact stress
15.8 Worm gears -Thermal analysis
15.1 INTRODUCTION
Worm gears are used for transmitting power between two non-parallel, non-intersecting
shafts. High gear ratios of 200:1 can be got.
(b)
(a)
Fig.15.1 (a) Single enveloping worm gear, (b) Double enveloping worm gear.
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Machine Design II
Prof. K.Gopinath & Prof. M.M.Mayuram
Fig.15.2 The cut section of a worm gearbox with fins and fan for cooling
15.2 GEOMETRY AND NOMENCLATURE
Fig. 15.3 Nomenclature of a single enveloping worm gear
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Prof. K.Gopinath & Prof. M.M.Mayuram
a. The geometry of a worm is similar to that of a power screw. Rotation of the worm
simulates a linearly advancing involute rack, Fig.15.3
b. The geometry of a worm gear is similar to that of a helical gear, except that the
teeth are curved to envelop the worm.
c. Enveloping the gear gives a greater area of contact but requires extremely
precise mounting.
1. As with a spur or helical gear, the pitch diameter of a worm gear is related to its
circular pitch and number of teeth Z by the formula
d2 
Z2 p
π
(15.1)
2. When the angle is 90 between the nonintersecting shafts, the worm lead angle 
is equal to the gear helix angle. Angles  and  have the same hand.
3. The pitch diameter of a worm is not a function of its number of threads, Z 1 .
4. This means that the velocity ratio of a worm gear set is determined by the ratio of
gear teeth to worm threads; it is not equal to the ratio of gear and worm
diameters.
ω1 Z2
=
ω2 Z1
(15.2)
5. Worm gears usually have at least 24 teeth, and the number of gear teeth plus
worm threads should be more than 40:
Z 1 + Z 2 > 40
(15.3)
6. A worm of any pitch diameter can be made with any number of threads and any
axial pitch.
7. For maximum power transmitting capacity, the pitch diameter of the worm should
normally be related to the shaft center distance by the following equation
C0.875
C0.875
 d1 
3.0
1.7
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(15.4)
Machine Design II
Prof. K.Gopinath & Prof. M.M.Mayuram
8. Integral worms cut directly on the shaft can, of course, have a smaller diameter
than that of shell worms, which are made separately.
9. Shell worms are bored to slip over the shaft and are driven by splines, key, or
pin.
10. Strength considerations seldom permit a shell worm to have a pitch diameter less
than
d 1 = 2.4p + 1.1
(15.5)
11. The face width of the gear should not exceed half the worm outside diameter.
b ≤ 0.5 d a1
(15.6)
12. Lead angle λ, Lead L, and worm pitch diameter d 1 have the following relationship in connection with the screw threads.
tan λ =
L
πd1
(15.7)
13. To avoid interference, pressure angles are commonly related to the worm lead
angle as indicated in Table 15.1.
Table 15.1 Maximum worm lead angle and worm gear Lewis form factor for
various pressure angles
Pressure Angle
Maximum Lead
Lewis form factor
Modified Lewis
Φn
Angle λ (degrees)
y
form factor Y
14.5
15
0.100
0.314
20
25
0.125
0.393
25
35
0.150
0.473
30
45
0.175
0.550
(Degrees)
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Machine Design II
Prof. K.Gopinath & Prof. M.M.Mayuram
Table 15.2 Frequently used standard values of module and axial pitch of worm or
circular pitch of gear p in mm:
Module m mm
Axial pitch p mm
Module m mm
2.0
2.5
3.15
6.283
7.854
9.896
8
10
12.5
4.0
5.0
6.3
12.566 15.708 19.792
16
20
Axial pitch p mm 25.133 31.416 39.270 50.625 62.832
b) Values of addendum and tooth depth often conform generally to helical gear
practice but they may be strongly influenced by manufacturing considerations.
c) The load capacity and durability of worm gears can be significantly increased by
modifying the design to give predominantly “recess action” i.e. the angle of
approach would be made small or zero and the angle of recess larger.
d) The axial pitch for different standard modules are given Table 15.2
15.3 FORCE ANALYSIS
Fig. 15.4 Worm gear force analysis
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Prof. K.Gopinath & Prof. M.M.Mayuram
a) The tangential, axial, and radial force components acting on a worm and gear are
illustrated in the Fig. 15.4
b) For the usual 90 shaft angle, the worm tangential force is equal to the gear axial
force and vice versa.
F 1t = F 2a
(15.8)
F 2t = F 1a
(15.9)
c) The worm and gear radial or separating forces are also equal,
F 1r = F 2r
(15.10)
If the power and speed of either the input or output are known, the tangential force
acting on this member can be found from equation
F1t =
1000 W
V
(15.11)
1. In the Fig. 15.4, the driving member is a clockwise-rotating right hand worm.
2. The force directions shown can readily be visualized by thinking of the worm as a
right hand screw being turned so as to pull the “nut” (worm gear tooth) towards
the “screw head”.
3. Force directions for other combinations of worm hand and direction of rotation
can be similarly visualized.
15.3.1 Thrust Force Analysis.
The thrust force direction for various worm and worm wheel drive conditions are shown
in Fig. 15.6
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Prof. K.Gopinath & Prof. M.M.Mayuram
(a)
(b)
Fig.15.6 (a) and (b) Worm gears thrust force analysis
Indian Institute of Technology Madras
Machine Design II
Prof. K.Gopinath & Prof. M.M.Mayuram
The thread angle λ of a screw thread corresponds to the pressure angle φ n of the worm.
We can apply the force, efficiency, and self-locking equations of power screw directly to
a worm and gear set. These equations are derived below with reference to the worm
and gear geometry. Figs.15.7 to 15.9 show in detail the forces acting on the gear.
Components of the normal tooth force are shown solid. Components of the friction force
are shown with the dashed lines.
Fig. 15.7 Forces on the worm gear tooth
Fig. 15.8 Worm driving
Indian Institute of Technology Madras
Machine Design II
Prof. K.Gopinath & Prof. M.M.Mayuram
Fig. 15.9 illustrates the same directions of rotation but with the torque direction reversed
(i.e., gear driving). Then contact shifts to the other side of the gear tooth, and the normal
load reverses.
Fig.15.9 Gear driving (Same direction of rotation)
The friction force is always directed to oppose the sliding motion. The driving worm is
rotating clockwise:
F2t =F1a = Fn cosφ n cos λ -f Fn sin λ
(15.12)
F1t = F2a = Fn cosφ n sin λ +f F n cos λ
(15.13)
F2r = F1r = Fn sinφ n
(15.14)
Combining eqns. (15.12) with (15.13), we have:
F
2t = cosφ n cos λ - f sin λ
F
cosφ n sin λ + f cos λ
1t
(15.15)
Combining eqns. (15.12) with (15.14) and (15.13) with (15.14), we have:
Indian Institute of Technology Madras
Machine Design II
Prof. K.Gopinath & Prof. M.M.Mayuram
F2r =F1r =F2t
sinφ n
cosφ n cos λ - f sin λ
= F1t
sinφ n
cosφ n sin λ + f cos λ
(15.16)
15.4 KINEMATICS
The relationship between worm tangential velocity, gear tangential velocity, and sliding
velocity is,
V2
= tanλ
V1
(15.17)
15.5 EFFICIENCY
Efficiency η is the ratio of work out to work in. For the usual case of the worm serving as
input member,
(15.18)
The overall efficiency of a worm gear is a little lower because of friction losses in the
bearings and shaft seals, and because of “churning” of the lubricating oil.
15.6 FRICTION ANALYSIS
The coefficient of friction, f, varies widely depending on variables such as the gear
materials, lubricant, temperature, surface finishes, accuracy of mounting, and sliding
velocity. The typical coefficient of friction of well lubricated worm gears is given in Fig.
15.10.
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Prof. K.Gopinath & Prof. M.M.Mayuram
Fig. 15.10 Friction of well lubricated worm gears, A for cast iron worm and gear
and B for case hardened steel worm and phosphor bronze worm gear
The sliding velocity Vs is related to the worm and gear pitch line velocities and to the
worm lead angle by
Vs =
V1
V
= 2
cosλ sinλ
(15.19)
Fig.15.11 Velocity components in worm gearing
F1 t  F
n
cos 
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n
s in  - f F
n
cos 
(1 5 .2 0 )
Machine Design II
Prof. K.Gopinath & Prof. M.M.Mayuram
a) Eqn. 15.20 shows that with a sufficiently high coefficient of friction, the gear
tangential force becomes zero, and the gear set “self-locks” or does not “overhaul.”
b) With this condition, no amount of worm torque can produce motion.
c) Self-locking occurs, if at all, with the gear driving.
d) This is desirable in many cases and helps in holding the load from reversing,
similar to a self-locking power screw.
The worm gear set self-locks if this force goes to zero, which happens if
f  cos n tan 
(15.21)
A worm gear set can be always overhauling or never overhauling, depending on the
selected value coefficient of friction (i.e., λ and to a lesser extent on φ n ).
15.7 BENDING AND SURFACE FATIGUE STRENGTHS
Worm gear capacity is often limited not by fatigue strength but by cooling capacity. The
total gear tooth load F d is the product of nominal load F t and factors accounting for
impact from tooth inaccuracies and deflections, misalignment, etc.). F d must be less
than the strength the bending fatigue and surface fatigue strengths F b and F w The total
tooth load is called the dynamic load F d , the bending fatigue limiting load is called
strength capacity F b , and the surface fatigue limiting load is called the wear capacity F w .
For satisfactory performance,
Fb ≥ Fd
(15. 21)
and
Fw ≥ F d
(15.22)
The “dynamic load” is estimated by multiplying the nominal value of gear tangential
force by velocity factor “K v ” given in the following Fig.15.
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 6.1+ V2 
(15.23)

 6.1 
Adapting the Lewis equation to the gear teeth, we have
Fd = F2t K v = F2t 
Fb =[ b ] bpy = [ b ] bmY
(15.24)
Where, [σ b ] is the permissible bending stress in bending fatigue, in MPa, Table 15.3
Table 15.3 Permissible stress in bending fatigue, in MPa0.5
Material of the gear
[σ b ] MPa
Centrifugally cast Cu-Sn bronze
23.5
Aluminum alloys Al-Si alloy
11.3
Zn alloy
7.5
Cast iron
11.8
b – is the face width in mm ≤ 0.5 d a1
p – is the axial pitch in mm, Table 15.2
m – is module in mm, Table 15.2
y – is the Lewis form factor, Table 15.1
Y – is modified Lewis form factor, Table 15.1
By assuming the presence of an adequate supply of appropriate lubricant, the following
equation suggested by Buckingham may be used for wear strength calculations
Fw =d 2 b K w
(15.25)
F w – Maximum allowable value of dynamic load under surface fatigue condition.
d g - Pitch diameter of the gear.
b - Face width of the gear.
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Prof. K.Gopinath & Prof. M.M.Mayuram
K w - A material and geometry factor with values empirically determined from the Table
15.4.
Table 15.4 Worm Gear Wear Factors K w
Material
K w (MPa)
Worm
Gear
<10
<25
>25
Steel, 250 BHN
Bronze
0.414
0.518
0.621
Hardened steel
Bronze
0.552
0.690
0.828
Chill-cast Bronze
0.828
1.036
1.243
Bronze
1.036
1.277
1.553
(Surface 500
BHN)
Cast iron
15.8 THERMAL CAPACITY
The continuous rated capacity of a worm gear set is often limited by the ability of the
housing to dissipate friction heat without developing excessive gear and lubricant
temperatures. Normally, oil temperature must not exceed about 200ºF (93oC) for
satisfactory operation. The fundamental relationship between temperature rise and rate
of heat dissipation used for journal bearings does hold good for worm gearbox.
H = CH A  T0 -Ta 
(15.26)
Where H – Time rate of heat dissipation (Nm/sec)
C H – Heat transfer coefficient (Nm/sec/m2/ºC)
A – Housing external surface area (m2)
T o – Oil temperature (º C)
T a – Ambiant air temperature (º C)
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Surface area of A for conventional housing designs may be roughly estimated from the
Eqn 15.27,
A =14.75 C1.7
(15.27)
Where A is in m2 and C (the distance between the shafts) is in m.
Housing surface area can be made far greater than the above equation value by
incorporating cooling fins. Rough estimates of C can be taken from the following
Fig.15.12.
Fig.15.12 Influence of worm speed on heat transfer
15.9 DESIGN GUIDELINES
The design guidelines for choosing the lead angle, pressure angle, addendum
dedendum, helix angle and the minimum number of teeth on the worm gear are given in
Tables 15.5 to 15.8.
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Prof. K.Gopinath & Prof. M.M.Mayuram
Table 15.5 Recommended pressure angles and tooth depths for worm gearing
Lead angle λ in
degrees
Pressure angle φ n
in degrees
Addendum h a in
mm
Dedendum h f in
mm
0-15
14.5
0.3683 p
0.3683 p
15-30
20
0.3683 p
0.3683
30-35
25
0.2865 p
0.331 p
35-40
25
0.2546 p
0.2947 p
40-45
30
0.2228 p
0.2578 p
Table 15.6 Efficiency of worm GEAR set for f = 0.05
Helix angle Efficiency Helix angle Efficiency Helix angle Efficiency
Ψ in O
η in %
Ψ in O
η in %
Ψ in O
η in %
1.0
25.2
7.5
71.2
20.0
86.0
2.5
46.8
10.0
76.8
25.0
88.0
5.0
62.6
15.0
82.7
30.0
89.2
Table 15.7 Minimum number of teeth in the worm gear
Pressure angle φ n
14.5o 17.5o 20o
22.5o 25o
27.5o 30o
Z 2 minimum
40
17
12
27
21
14
Table 15.8 Maximum lead angle for normal pressure angle
Normal Pressure angle φ n
14.5o
20o
25o
30o
Maximum lead angle λ max
16 o
25 o
35 o
45 o
------------------------
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