Introduction
A worm gear is used when a large speed reduction ratio is required between crossed axis shafts which do not intersect. A basic helical gear can be used but the power which can be transmitted is low. A worm drive consists of a large diameter worm wheel with a worm screw meshing with teeth on the periphery of the worm wheel. The worm is similar to a screw and the worm wheel is similar to a section of a nut. As the worm is rotated the wormwheel is caused to rotate due to the screw like action of the worm. The size of the worm gearset is generally based on the centre distance between the worm and the wormwheel.
If the worm gears are machined basically as crossed helical gears the result is a highly stress point contact gear. However normally the wormwheel is cut with a concave as opposed to a straight width. This is called a single envelope worm gearset. If the worm is machined with a concave profile to effectively wrap around the wormwheel the gearset is called a double enveloping worm gearset and has the highest power capacity for the size. Single enveloping gearsets require accurate alignment of the wormwheel to ensure full line tooth contact. Double enveloping gearsets require accurate alignment of both the worm and the wormwheel to obtain maximum face contact.
The worm is shown with the worm above the wormwheel. The gearset can also be arranged with the worm below the wormwheel. Other alignments are used less frequently.
Nomenclature
As can be seen in the above view a section through the axis of the worm and the centre of the gear shows that , at this plane, the meshing teeth and thread section is similar to a spur gear and has the same features
α_{n} = Normal pressure angle = 20^{o} as standard
γ = Worm lead angle = (180 /π ) tan^{1} (z _{1} / q)(deg) ..Note: for α _{n}= 20^{o} γ should be less than 25^{o}
b _{a} = Effective face width of worm wheel. About 2.m √ (q +1) (mm)
b _{l} = Length of worm wheel. About 14.m. (mm)
c = clearance c _{min} = 0,2.m cos γ , c _{max} = 0,25.m cos γ (mm)
d _{1} = Ref dia of worm (Pitch dia of worm (m)) = q.m (mm)
d _{a.1} = Tip diameter of worm = d _{1} + 2.h _{a.1} (mm)
d _{2} = Ref dia of worm wheel (Pitch dia of wormwheel) =( p _{x}.z/π ) = 2.a  d _{1} (mm)
d _{a.2} = Tip dia worm wheel (mm)
h _{a.1} = Worm Thread addendum = m (mm)
h _{f.1} = Worm Thread dedendum , min = m.(2,2 cos γ  1 ) , max = m.(2,25 cos γ  1 )(mm)
m = Axial module = p _{x} /π (mm)
m _{n} = Normal module = m cos γ(mm)
M _{1} = Worm torque (Nm)
M _{2} = Worm wheel torque (Nm)
n _{1} = Rotational speed of worm (revs /min)
n _{2} = Rotational speed of wormwheel (revs /min)
p _{x} = Axial pitch of of worm threads and circular pitch of wheel teeth ..the pitch between adjacent threads = π. m. (mm)
p _{n} = Normal pitch of of worm threads and gear teeth (m)
q = diameter factor selected from (6 6,5 7 7,5 8 8,5 9 10 11 12 13 14 17 20 )
p _{z} = Lead of worm = p _{x}. z _{1} (mm).. Distance the thread advances in one rev'n of the worm. For a 2start worm the lead = 2 . p _{x}
R _{g} = Reduction Ratio
q = Worm diameter factor = d _{1} / m  (Allows module to be applied to worm )
μ = coefficient of friction
η= Efficiency
V_{s} = Wormgear sliding velocity ( m/s)
z _{1} = Number of threads (starts) on worm
z _{2} = Number of teeth on wormwheel
Worm gear design parameters
Worm gears provide a normal single reduction range of 5:1 to 751. The pitch line velocity is ideally up to 30 m/s. The efficiency of a worm gear ranges from 98% for the lowest ratios to 20% for the highest ratios. As the frictional heat generation is generally high the worm box is designed disperse heat to the surroundings and lubrication is and essential requirement. Worm gears are quiet in operation. Worm gears at the higher ratios are inherently self locking  the worm can drive the gear but the gear cannot drive the worm. A worm gear can provide a 50:1 speed reduction but not a 1:50 speed increase....(In practice a worm should not be used a braking device for safety linked systems e.g hoists. . Some material and operating conditions can result in a wormgear backsliding )
The worm gear action is a sliding action which results in significant frictional losses. The ideal combination of gear materials is for a case hardened alloy steel worm (ground finished) with a phosphor bronze gear. Other combinations are used for gears with comparatively light loads.
Specifications
BS721 Pt2 1983 Specification for worm gearing � Metric units.
This standard is current (2004) and provides information on tooth form, dimensions of gearing, tolerances for four classes of gears according to function and accuracy, calculation of load capacity and information to be given on drawings.
Worm Gear Designation
Very simply a pair of worm gears can be defined by designation of the number of threads in the worm ,the number of teeth on the wormwheel, the diameter factor and the axial module i.e z1,z2, q, m .
This information together with the centre distance ( a ) is enough to enable calculation of and any dimension of a worm gear using the formulea available.
Worm teeth Profile
The sketch below shows the normal (not axial) worm tooth profile as indicated in BS 7212 for unit axial module (m = 1mm) other module teeth are in proportion e.g. 2mm module teeth are 2 times larger
Typical axial modules values (m) used for worm gears are
0,5 0,6 0,8 1,0 1,25 1,6 2,0 2,5 3,15 4,0 5,0 6,3 8,0 10,0 12,5 16,0 20,0 25,0 32,0 40,0 50,0
Materials used for gears



Material  Notes  applications 
Worm 
Acetal / Nylon  Low Cost, low duty  Toys, domestic appliances, instruments 
Cast Iron  Excellent machinability, medium friction.  Used infrequently in modern machinery 
Carbon Steel  Low cost, reasonable strength  Power gears with medium rating. 
Hardened Steel  High strength, good durability  Power gears with high rating for extended life 
Wormwheel 
Acetal /Nylon  Low Cost, low duty  Toys, domestic appliances, instruments 
Phos Bronze  Reasonable strength, low friction and good compatibility with steel  Normal material for worm gears with reasonable efficiency 
Cast Iron  Excellent machinability, medium friction.  Used infrequently in modern machinery 
Design of a Worm Gear
The following notes relate to the principles in BS 7212
Method associated with AGMA are shown below..
Initial sizing of worm gear.. (Mechanical)
1) Initial information generally Torque required (Nm), Input speed(rpm), Output speed (rpm).
2) Select Materials for worm and wormwheel.
3) Calculate Ratio (R _{g})
4) Estimate a = Center distance (mm)
5) Set z _{1} = Nearest number to (7 + 2,4 SQRT (a) ) /R _{g }
6) Set z _{2} = Next number < R _{g } . z _{1}
7) Using the value of estimated centre distance (a) and No of gear teeth ( z _{2} )obtain a value for q from the table below
8) d _{1} = q.m (select) ..
9) d _{2} = 2.a  d _{1}
10) Select a wormwheel face width b _{a} (minimum =2*m*SQRT(q+1))
11) Calculate the permissible output torques for strength (M _{b_1} and wear M _{c_1} )
12) Apply the relevent duty factors to the allowable torque and the actual torque
13) Compare the actual values to the permissible values and repeat process if necessary
14) Determine the friction coefficient and calculate the efficiency.
15) Calculate the Power out and the power in and the input torque
6) Complete design of gearbox including design of shafts, lubrication, and casing ensuring sufficient heat transfer area to remove waste heat.
Initial sizing of worm gear.. (Thermal)
Worm gears are often limited not by the strength of the teeth but by the heat generated by the low efficiency. It is necessary therefore to determine the heat generated by the gears = (Input power  Output power). The worm gearbox must have lubricant to remove the heat from the teeth in contact and sufficient area on the external surfaces to distibute the generated heat to the local environment. This requires completing an approximate heat transfer calculation. If the heat lost to the environment is insufficient then the gears should be adjusted (more starts, larger gears) or the box geometry should be adjusted, or the worm shaft could include a fan to induced forced air flow heat loss.
Formulae
The reduction ratio of a worm gear ( R _{g} )
R _{g} = z _{2} / z _{1}
eg a 30 tooth wheel meshing with a 2 start worm has a reduction of 15
Tangential force on worm ( F _{wt} )= axial force on wormwheel
F _{wt} = F _{ga} = 2.M _{1} / d _{1}
Axial force on worm ( F _{wa} ) = Tangential force on gear
F _{wa} = F _{gt} = F _{wt}.[ (cos α _{n}  μ tan γ ) / (cos α _{n} . tan γ + μ ) ]
Output torque ( M _{2} ) = Tangential force on wormwheel * Wormwheel reference diameter /2
M _{2} = F _{gt}* d _{2} / 2
Relationship between the Worm Tangential Force F _{wt} and the Gear Tangential force F _{gt}
F _{wt} = F _{gt}.[ (cos α _{n} . tan γ + μ ) / (cos α _{n}  μ tan γ ) ]
Relationship between the output torque M _{2}and the input torque M _{1}
M _{2} = ( M _{1}. d _{2} / d _{1} ).[ (cos α _{n}  μ tan γ ) / (cos α _{n} . tan γ + μ ) ]
Separating Force on wormgearwheel ( F _{s} )
F _{s} = F _{wt}.[ (sin α _{n} ) / (cos α _{n} . sin γ + μ .cos γ ) ]
Sliding velocity ( V _{s} )...(m/s)
V _{s} (m/s ) = 0,00005236. d _{1}. n _{1} sec γ
= 0,00005235.m.n (z _{1}^{2} + q ^{2} ) ^{1/2}
Peripheral velocity of wormwheel ( V _{p}) (m/s)
V _{p} = 0,00005236,d _{2}. n _{2}
Friction Coefficient
Cast Iron and Phosphor Bronze .. Table x 1,15
Cast Iron and Cast Iron.. Table x 1,33
Quenched Steel and Aluminum Alloy..Table x 1,33
Steel and Steel..Table x 2
Friction coefficients  For Case Hardened Steel Worm / Phos Bros Wheel
Sliding Speed  Friction Coefficient  Sliding Speed  Friction Coefficient 
m/s  μ  m/s  μ 
   
0  0,145  1,5  0,038 
0,001  0,12  2  0,033 
0,01  0,11  5  0,023 
0,05  0,09  8  0,02 
0,1  0,08  10  0,018 
0,2  0,07  15  0,017 
0,5  0,055  20  0,016 
1  0,044  30  0,016 
Efficiency of Worm Gear
The efficiency of the worm gear is determined by dividing the output Torque M2 with friction = μ by the output torque with zero losses i.e μ = 0
First cancelling [( M _{1}. d _{2} / d _{1} ) / M _{1}. d _{2} / d _{1} ) ] = 1
Denominator = [(cos α _{n} / (cos α _{n} . tan γ ] = cot γ
η = [(cos α _{n}  μ tan γ ) / (cos α _{n} . tan γ + μ ) ] / cot γ
= [(cos α _{n}  μ .tan γ ) / (cos α _{n} + μ .cot γ )]
Graph showing worm gear efficiency related to gear lead angle ( γ )
Worm Design /Gear Wear / Strength Equations to BS721
Note: For designing worm gears to AGMA codes
AGMA method of Designing Worm Gears
The information below relates to BS721 Pt2 1983 Specification for worm gearing � Metric units. BS721 provides average design values reflecting the experience of specialist gear manufacturers. The methods have been refined by addition of various application and duty factors as used. Generally wear is the critical factor..
Permissible Load for Strength
The permissible torque (M in Nm) on the gear teeth is obtained by use of the equation
M _{b} = 0,0018 X _{b.2}σ _{bm.2}. m. l _{f.2}. d _{2}.
( example 87,1 Nm = 0,0018 x 0,48 x 63 x 20 x 80 )
X _{b.2} = speed factor for bending (Worm wheel ).. See Below
σ _{bm.2} = Bending stress factor for Worm wheel.. See Table below
l _{f.2} = length of root of Worm Wheel tooth
d _{2} = Reference diameter of worm wheel
m = axial module
γ = Lead angle
Permissible Torque for Wear
The permissible torque (M in Nm) on the gear teeth is obtained by use of the equation
M _{c} = 0,00191 X _{c.2}σ _{cm.2}.Z. d _{2}^{1,8}. m
( example 33,42 Nm = 0,00191 x 0,3234 x 6,7 x 1,5157 x 80^{1,8} x 2 )
X _{c.2} = Speed factor for wear ( Worm wheel )
σ _{cm.2} = Surface stress factor for Worm wheel
Z = Zone factor.
Length of root of worm wheel tooth
Radius of the root = R _{r}= d _{1} /2 + h _{ha,1} (= m) + c(= 0,25.m.cos γ )
R _{r}= d _{1} /2 + m(1 +0,25 cosγ)
l _{f.2} = 2.R _{r}.sin^{1} (2.R _{r} / b _{a})
Note: angle from sin^{1}(function) is in radians...
Speed Factor for Bending
This is a metric conversion from an imperial formula..
X _{b.2} = speed factor for bending = 0,521(V) ^{0,2}
V= Pitch circle velocity =0,00005236*d _{2}.n _{2} (m/s)
The table below is derived from a graph in BS 721. I cannot see how this works as a small worm has a smaller diameter compared to a large worm and a lower speed which is not reflected in using the RPM.
Table of speed factors for bending
RPM (n_{2)}  X _{b.2}  RPM (n_{2)}  X _{b.2} 
1  0,62  600  0,3 
10  0,56  1000  0,27 
20  0,52  2000  0,23 
60  0,44  4000  0,18 
100  0,42  6000  0,16 
200  0,37  8000  0,14 
400  0,33  10000  0,13 
Additional factors
The formula for the acceptable torque for wear should be modified to allow additional factors which affect the Allowable torque M _{c}
M _{c2} = M _{c}. Z _{L}. Z _{M}.Z _{R} / K _{C}
The torque on the wormwheel as calculated using the duty requirements (M _{e}) must be less than the acceptable torque M _{c2} for a duty of 27000 hours with uniform loading. For loading other than this then M _{e} should be modified as follows
M _{e2} = M _{e}. K _{S}* K _{H}
Thus
uniform load < 27000 hours (10 years) M _{e} ≤ M _{c2}
Other conditions M _{e2} ≤ M _{c2}
Factors used in equations
Lubrication (Z _{L})..
Z _{L} = 1 if correct oil with antiscoring additive else a lower value should be selected
Lubricant (Z _{M})..
Z _{L} = 1 for Oil bath lubrication at V _{s} < 10 m /s
Z _{L} = 0,815 Oil bath lubrication at 10 m/s < V _{s} < 14 m /s
Z _{L} = 1 Forced circulation lubrication
Surface roughness (Z _{R} ) ..
Z _{R} = 1 if Worm Surface Texture < 3μ m and Wormwheel < 12 μ m
else use less than 1
Tooth contact factor (K _{C}
This relates to the quality and rigidity of gears . Use 1 for first estimate
K _{C} = 1 For grade A gears with > 40% height and > 50% width contact
= 1,3  1,4 For grade A gears with > 30% height and > 35% width contact
= 1,51,7 For grade A gears with > 20% height and > 20% width contact
Starting factor (K _{S}) ..
K _{S} =1 for < 2 Starts per hour
=1,07 for 2 5 Starts per hour
=1,13 for 510 Starts per hour
=1,18 more than 10 Starts per hour
Time / Duty factor (K _{H}) ..
K _{H} for 27000 hours life (10 years) with uniform driver and driven loads
For other conditions see table below
Tables for use with BS 721 equations
Speed Factors
X _{c.2} = K _{V} .K _{R}
Note: This table is not based on the graph in BS 7212 (figure 7) it is based on another more easy to follow graph. At low values of sliding velocity and RPM it agrees closely with BS 721. At higher speed velocities it gives a lower value (e.g at 20m/s 600 RPM the value from this table for X _{c.2} is about 80% of the value in BS 7212
Table of Worm Gear Speed Factors
Note sliding speed = V_{s} and Rotating speed = n_{2} (Wormwheel)
Sliding speed  K _{V}  Rotating Speed  K _{R} 
m/s   rpm  
0  1  0,5  0,98 
0,1  0,75  1  0,96 
0,2  0,68  2  0,92 
0,5  0,6  10  0,8 
1  0,55  20  0,73 
2  0,5  50  0,63 
5  0,42  100  0,55 
10  0,34  200  0,46 
20  0,24  500  0,35 
30  0,16  600  0,33 
Stress Factors
Table of Worm Gear Stress Factors
 Other metal
(Worm)  P.B.  C.I.  0,4%
C.Steel  0,55%
C.Steel  C.Steel
Case. H'd 
Metal
(Wormwheel)  Bending
(σ_{bm} )  Wear ( σ _{cm} ) 
MPa  MPa 
Phosphor Bronze
Centrifugal cast  69   8,3  8,3  9,0  15,2 
Phosphor Bronze
Sand Cast Chilled  63   6,2  6,2  6,9  12,4 
Phosphor Bronze
Sand Cast  49   4,6  4,6  5,3  10,3 
Grey Cast Iron  40  6,2  4,1  4,1  4,1  5,2 
0,4% Carbon steel  138  10,7  6,9    
0,55% Carbon steel  173  15,2  8,3    
Carbon Steel
(Case hardened)  276  48,3  30,3    15,2 
Zone Factor (Z)
If b _{a} < 2,3 (q +1)^{1/2} Then Z = (Basic Zone factor ) . b _{a} /2 (q +1)^{1/2}
If b _{a} > 2,3 (q +1)^{1/2} Then Z = (Basic Zone factor ) .1,15
Table of Basic Zone Factors
 q 
z1  6  6,5  7  7,5  8  8,5  9  9,5  10  11  12  13  14  17  20 
1  1,045  1,048  1,052  1,065  1,084  1,107  1,128  1,137  1,143  1,16  1,202  1,26  1,318  1,402  1,508 
2  0,991  1,028  1,055  1,099  1,144  1,183  1,214  1,223  1,231  1,25  1,28  1,32  1,36  1,447  1,575 
3  0,822  0,89  0,989  1,109  1,209  1,26  1,305  1,333  1,35  1,365  1,393  1,422  1,442  1,532  1,674 
4  0,826  0,83  0,981  1,098  1,204  1,701  1,38  1,428  1,46  1,49  1,515  1,545  1,57  1,666  1,798 
5  0,947  0,991  1,05  1,122  1,216  1,315  1,417  1,49  1,55  1,61  1,632*  1,652  1,675  1,765  1,886 
6  1,131  1,145  1,172  1,22  1,287  1,35  1,438  1,521  1,588  1,625  1,694  1,714  1,733  1,818  1,928 
7    1,316  1,34  1,37  1,405  1,452  1,54  1,614  1,704  1,725  1,74  1,76  1,846  1,98 
8      1,437  1,462  1,5  1,557  1,623  1,715  1,738  1,753  1,778  1,868  1,96 
9        1573  1,604  1,648  1,72  1,743  1,767  1,79  1,88  1,97 
10          1,68  1,728  1,748  1,773  1,798  1,888  1,98 
11           1,732  1,753  1,777  1,802  1,892  1,987 
12            1,76  1,78  1,806  1,895  1,992 
13             1,784  1,806  1,898  1,998 
14              1,811  1,9  2 
Duty Factor
Duty  time Factor K _{H}
Impact from Prime mover  Expected life
hours  K _{H} 
Impact From Load 
Uniform Load  Medium Impact  Strong impact 
Uniform Load
Motor Turbine Hydraulic motor  1500  0,8  0,9  1 
5000  0,9  1  1,25 
27000  1  1,25  1,5 
60000  1,25  1,5  1,75 
Light impact
multicylinder engine  1500  0,9  1  1,25 
5000  1  1,25  1,5 
27000  1,25  1,5  1,75 
60000  1,5  1,75  2 
Medium Impact
Single cylinder engine  1500  1  1,25  1,5 
5000  1,25  1,5  1,75 
27000  1,5  1,75  2 
60000  1,75  2  2,25 
Worm q value selection
The table below allows selection of q value which provides a reasonably efficient worm design. The recommended centre distance value "a" (mm)is listed for each q value against a range of z _{2} (teeth number values). The table has been produced by reference to the relevant plot in BS 721
Example
If the number of teeth on the gear is selected as 45 and the centre distance is 300 mm then a q value for the worm would be about 7.5
Important note: This table provides reasonable values for all worm speeds. However at worm speeds below 300 rpm a separate plot is provided in BS721 which produces more accurate q values. At these lower speeds the resulting q values are approximately 1.5 higher than the values from this table. The above example at less than 300rpm should be increased to about 9
Table for optimum q value selection
Number of Teeth On Worm Gear (z _{2}) 
q  20  25  30  35  40  45  50  55  60  65  70  75  80 
6  150  250  380  520  700         
6.5  100  150  250  350  480  660        
7  70  110  170  250  350  470  620  700      
7.5  50  80  120  180  240  330  420  550  670     
8  25  50  80  120  180  230  300  380  470  570  700   
8.5   28  90  130  130  180  220  280  350  420  500  600  700 
9    40  70  100  130  170  220  280  330  400  450  520 
9.5    25  50  70  100  120  150  200  230  300  350  400 
10     26  55  80  100  130  160  200  230  270  320 
11     25  28  55  75  100  130  150  180  220  250 
12       28  45  52  80  100  130  150  100 
13         27  45  52  75  90  105 
AGMA method of Designing Worm Gears
The AGMA method is provided here because it is relatively easy to use and convenient AGMA is all imperial and so I have used conversion values so all calculations can be completed in metric units..
Good proportions indicate that for a centre to centre distance = C the mean worm dia d _{1} is within the range
Imperial (inches)
( C ^{0,875} / 3 ) ≤ d _{1} ≤ ( C ^{0,875} / 1,6 )
Metric ( mm)
( C ^{0,875} / 2 ) ≤ d _{1} ≤ ( C ^{0,875} / 1,07 )
The acceptable tangential load (W _{t}) _{all}
(W _{t}) _{all} = C _{s}. d _{2}^{0,8} .b _{a} .C _{m} .C _{v} . (0,0132) (N)
The formula will result in a life of over 25000 hours with a case hardened alloy steel worm and a phosphor bronze wheel
C _{s} = Materials factor
b _{a} = Effective face width of gearwheel = actual face width. but not to exceed 0,67 . d _{1}
C _{m} = Ratio factor
C _{v} = Velocity factor
Modified Lewis equation for stress induced in worm gear teeth .
σ _{a} = W _{t} / ( p _{n}. b _{a}. y )(N)
W _{t} = Worm gear tangential Force (N)
y = 0,125 for a normal pressure angle α _{ n } = 20^{o}
The friction force = W _{f}
W _{f} = f.W _{t} / (. cos φ _{ n} ) (N)
γ = worm lead angle at mean diameter
α _{ n } = normal pressure angle
The sliding velocity = V _{s}
V _{s} = π .n _{1}. d _{1} / (60,000 )
d _{1 }= mean dia of worm (mm)
n _{1 }= rotational speed of worm (revs/min)
The torque generated γ at the worm gear = M _{b} (Nm)
T _{G} = W _{t} .d _{1} / 2000
The required friction heat loss from the worm gearbox
H _{loss } = P _{in} ( 1  η )
η = gear efficiency as above.
C _{s} values
C _{s} = 270 + 0,0063(C )^{3}... for C ≤ 76mm ....Else
C _{s} (Sand cast gears ) = 1000 for d _{1} ≤ 64 mm ...else... 1860  477 log (d _{1} )
C _{s} (Chilled cast gears ) = 1000 for d _{1} ≤ 200 mm ...else ... 2052 456 log (d _{1} )
C _{s} (Centrifugally cast gears ) = 1000 for d _{1} ≤ 635 mm ...else ... 1503  180 log (d _{1} )
C _{m} values
N_{G} = Number of teeth on worm gear.
N_{W} = Number of stards on worm gear.
m_{G} = gear ration = N_{G} /N_{W}
C _{v} values
C _{v} (V _{s} > 3,56 m/s ) = 0,659 exp (0,2167 V _{s} )
C _{v} (3,56 m/s ≤ V _{s} < 15,24 m/s ) = 0,652 (V _{s}) ^{0,571} )
C _{v} (V _{s} > 15,24 m/s ) = 1,098.( V _{s} ) ^{0,774} )
f values
f (V
_{s} = 0) = 0,15
f (0 < V
_{s} ≤ 0,06 m/s ) = 0,124 exp (2,234 ( V
_{s} )
^{0,645}
f (V
_{s} > 0,06 m/s ) = 0,103 exp (1,1855 ( V
_{s} ) )
^{0,450} ) +0,012
http://www.roymech.co.uk/Useful_Tables/Drive/Worm_Gears.html