10.3 Calculation for Cylindrical worm gear pair strength
Gear strength calculation formula for Cylindrical worm gear pair JGMA 405-01 (1978)
1. Applicable range (Common)

1.1 This standard is applied to Worm gear pair withfollowing ranges and shaft angle 90 ° for powertransfer used in general industrial machinery.
Axial module: 1 ～ 25 mm
Reference diameter of Worm wheel: Below 900 mm
Sliding velocity: Below 30 m/s
Revolving velocity of Worm wheel: Below 600 min-1
Tooth profile: Stipulated in JIS B1723(Cylindrical worm gear
pair)
Material: Refer to Table 7
1.2. This standard is used for calculating Allowableload from given dimension of Cylindrical worm gear
pair or is used for determining suitable dimensions of
Cylindrical worm gear pair from given load.

HerebyT2 : Nominal torque (kgf・m) for Worm wheel
T1 : Nominal torque (kgf・m) for Worm gear
Ft : Nominal Tangential load (kgf ) on the Reference pitch circle for Worm wheel
d2 : R eference pitch diameter (mm) for Wormwheel
u : Gear ratioη R : Transfer efficiency of Worm gear when Wormgear is driver (excludes bearing loss and mixer
loss of lubricating oil)μ : Friction factor｛Refer to (3) of 3.2｝
α n : Normal reference pressure angle(°)(2) When Worm wheel is driver (speed increment)T2 =Ft d 2(kgf • m) ............................................ Same as (2)
2000

T1 =

T 2η 1 F t d 2η 1=
(kgf • m) ............................................. (5)
u
2000u

η1 =

2. Definition

Gear strength of Cylindrical worm gear pair is Allowable load for Surface durability.

3. Basic conversion formula and numericalvaluetanγ −µ

cos α n§µ ·tanγ ¨1 + tanγ¸
cos α n ¹©

................................................. (6)Herebyη 1 : Transfer efficiency of Worm gear pair when
Worm wheel is driver (excludes bearing loss
and mixer loss of lubricating oil).

3.1 Sliding velocityν s (m/s)

υs =

d 1n1
19100cosγ ...................................................................... (1)Herebyd1 : Reference pitch diameter of Worm gear (mm)n1 : Revolving velocity of Worm gear (min-1)
γ : Reference pitch cylindrical lead angle (°)
3.2 Torque, Tangential load and Efficiency(1) When Worm gear is driver (speed reduction)
T2 =Ft d 2(kgf • m) ............................................................... (2)
2000

T1 =

Ft d 2
T2=(kgf • m) . .......................................... (3)uηR 2000uηR§µ ·tanγ ¨1 − tanγ
¸
cos
αn ¹
©
ηR =
tanγ +

148µcos α n................................................. (4)(3) Numerical value of friction factor μ
Obtain Friction factorμ from Fig. 1 of sliding velocity when engaged with Worm gear with Case harden
and ground or Worm wheel with phosphor bronze.
Remark 1. Friction factor for engagement with othermaterials.
Due to insufficient data, values of Friction factor are
difficult to stipulate. Therefore Reference table 1 proposed by H.E Merritt is adopted for reference.

Reference table 1 Friction factor μ for different materials combination
Materials

Value of μ

Casting iron and phosphor bronze

1.15 times value of Fig. 1

Casting iron and Casting iron1.33 times value of Fig. 1

Hardened steel and Aluminium

1.33 times value of Fig. 1

Steel and Steel

2.0 times value of Fig. 1

Fig. 1 Friction factor
0.150
0.120
0.100
0.090
0.080

Friction factor M

0.070
0.060
0.050

0.040

0.030

0.020
0.015
0.012

0 0.001 0.01

0.05 0.1

0.2

0.4 0.6

1

1.5

2

3

4

5

6

7

8 9 1012

14 16 18 20 22 24 26

30

Sliding velocity Us (m/s)

4. Calculation formula of Allowable loadfor Surface durability
4.1 Basic load capacity calculation
Calculate Basic load capacity for Surface durability from given dimensions and material of Cylindricalworm gear pair using following calculation formula.
Allowable tangential load Ftlim (kg・f)
F t lim = 3.82 K υ K n S clim Zd 2 0 .8 mx

ZLZMZR........................... (7)KC

Allowable Torque for Worm wheel T2lim (kgf・m)
T 2lim = 0.00191 K υ K n S clim Zd 21.8 mx

ZLZMZR.................... (8)KC

Herebyd2 : R eference pitch diameter (mm) for Wormwheel
mx : Axial module (mm)Z : Zone factor
Kυ : Sliding velocity factor
Kn : Revolving speed factor
ZL : lubricating oil factor
ZM : Lubrication factor
ZR : Roughness factor
Kc : Tooth contact factor
Sclim : Allowable stress factor for Surface durability

4.2 Equivalent load calculation
Basic load capacity from formulas (7) and (8) is thelimit of Tangential load and torque to withstand
26,000 hours of usage when in a non-impact environment. It is considered non impact if number of starts
per hour is under 2 times and starting impact torqueis below 200% of rated torque (1). However, if suchcondition is not met, calculate Equivalent load andcompare with basic load capacity. In other words,
when expected life is more or less than 26,000 hours
with impact conditions applied. Starting toque islarger than above. Calculation method for Equivalentload is as follow.
Note(1)	This is torque for Worm wheel when prime
mover (or load) performs rated load operation.

149

Equivalent tangential load Fte (kgf)
Fte = F t K h K s .......................................................................... (9)5.2 Zone factor Z
Calculate Zone factor from (1) and (2) using Table 3.

Virtual torque of Worm wheel T2e (kgf・m)
T 2 e = T 2 K h K s ....................................................................... (10)(1) W hen , b 2 < 2.3mx Q + 1 use value in Table 3 multiplied by

HerebyFt : Nominal tangential load on the Pitch circle ofWorm wheel (kgf)
T2 : Nominal torque of Worm wheel (kgf・m)
Ks : Starting factor (Refer to 5.9)
Kh : Time factor (Refer to 5.10)b2
as value for Z.
2mx Q + 1
(2) When b 2 ҈ 2.3mx Q + 1 , use value in Table 3 multiplied by 1.15 as value for Z.
Herebyd1 ·§Q: Diameter
quotient
Q : Diameter¸quotient ¨Q =mx ¹©

Zw : Number of thread for Worm gear

4.3 Load definition(1) When non-impact, expected life is 26,000 hours.
It should meet the following conditions.
Ft ҇ F t lim . .............................................................................. (11)T 2 ҇ T 2lim . .............................................................................. (12)(2) Other than above cases,
it should meet the following conditions.
Fte ҇ F t lim ............................................................................... (13)T 2 e ҇ T 2lim ............................................................................... (14)Remark: For fluctuating load, use total torque T2c to

5.3 Sliding velocity Kυ
Obtain Sliding velocity factor based on Sliding velocity from Fig. 3.
5.4 Revolving velocity factor Kn
Obtain Revolving velocity factor based on Revolving
speed of Worm wheel from Fig. 4.
5.5 Lubricating oil factor ZL

define load based on formulas (10) and (12)instead of T 2. Calculation method of T 2c is

As long as lubricating oil with proper viscosity containing extreme additives is used, ZL = 1.0.
If bearing is used in Worm gear pair equipment orcompelled to use lubricating oil with thin viscosity. ZL
is less than 1.0.
Remark: Viscosity
There are many recommended viscosity values fromdifferent sources for proper lubricating oil. However,there is no consensus. Recommended mean values
are collected from sources and shown in Referencetable 2.

found at「Calculation of Fluctuating load」
in Reference table 4 (page 153).

5. How to calculate each factor for Surface
durability from calculation formula

Factors used for Surface durability calculation formulas mentioned above are stipulated below.
5.1 Facewidth of Worm wheel b2 (mm)Refer to Fig. 2 for Facewidth of Worm wheel.

Fig. 2 Facewidth of Worm wheelb2

b2

Table 3. Base value of Zone factor
Q

7

7.5

8

8.5

9

9.5

10

11

12

13

14

1

1.052

1.065

1.084

1.107

1.128

1.137

1.143

1.160

1.202

1.260

1.318

2

1.055

1.099

1.144

1.183

1.114

1.223

1.231

1.250

1.280

1.320

1.360

3

0.989

1.109

1.209

1.266

1.305

1.333

1.350

1.365

1.393

1.422

1.442

4

0.981

1.098

1.204

1.301

1.380

1.428

1.460

1.490

1.515

1.545

1.570

ZW

150

Reference table 2 Recommended dynamic viscosity
Unit cSt/37.8°C
Operating oil temperature

Sliding velocity m/s

Max. oil temperature

Starting oil temperature

Below 2.5

Above 2.5 to below 5

Above 5

0 C° to below 10°C

‒10°C to below 0C°
Above 0°C

110 - 130

110 - 130

110 - 130

110 - 150

110 - 150

110 - 150

10 C° to below 30°C
30 C° to below 55°C
55 C° to below 80°C

Above 0°C

200 - 245

150 - 200

150 - 200〃

350 - 510

245 - 350

200 - 245〃

510 - 780

350 - 510

245 - 350

80 C° to below 100°C〃

900 - 1100

510 - 780

350 - 510

Fig. 3 Sliding velocity factor
1.0
0.9
0.8

0.6
0.5
0.4
0.3
0.2
0.1

0 0.001 0.01

0.05 0.1 0.2

0.4 0.6

1

2

3

4

5

6

7

8 9 10 11 12 14 16 18 20 22 24 26

30

Sliding velocity factor KU (m/s)

Fig. 4 Revolving velocity factor
1.0

0.9

0.8
Revolving velocity factor Kn

Sliding velocity factor KG

0.7

0.7

0.6

0.5

0.4

0.3

0

0.1 0.5 1 2

4

10

20
40 60 80 100
200
Revolving velocity of Worm wheel (min - ¹)

300

400 500

600

151

5.6 Lubrication factor ZM
Obtain Lubrication factor from Table 4.

Tooth bearing for classification equivalent to A in JIS
B 1741 (tooth bearing) will be Kc = 1.0. ....................... (16)Value of Kc for classification B and C is larger than 1.0.
Reference table 3: Shows JIS Tooth bearing ratio andapproximate values of Kc.

Table 4. Lubrication factor ZM
Sliding velocity m/s

Below 10

Above 10,
below 14

Above 14

Oil bath lubrication

1.0

0.85

-

Forced lubrication

1.0

1.0

1.0

5.9 Starting factor Ks
Starting factor is stipulated below(1) Obtain value from Table 5 if the starting torque isbelow 200% of rated torque.
(2) If Starting torque exceeds 200% of rated torque,value of K s = 1.0. With starting torque to be as
maximum, then calculate fluctuating load (refer toTable 4) to calculate total load.

5.7 Roughness factor ZR
Roughness factor is determined with consideration
based on influence on Pitting and Wearing to flank ofWorm gear and Worm wheel. Due to insufficient data,
ZR = 1.0 is adopted at the moment............................... (15)
However, Surface roughness is to be below 3S for
Worm gear and below 12S for Worm wheel.
If Surface roughness is rougher than above, Roughness factor Z R should be lower than 1.0.

5.10 Time factor Kh
Obtain Time factor from Table 6 using expected lifespan and extent of impact. Use interpolation when
expected lifespan is between the values in the below
Table.

5.8 Tooth bearing Kc
Quality of Tooth bearing has large influence on load
capacity. Due to insufficient data at the moment,

Reference table 3 Classification of Tooth bearing and approximate value of Kc
Ratio of tooth bearing

Classification

Kc

Tooth trace direction

Direction of tooth depthA

Above 50% of length of effective trace direction

Above 40% of effective tooth depth

1.0

BAbove 35% of length of effective trace direction

Above 30% of effective tooth depth

1.3 - 1.4

C

Above 20% of length of effective trace direction

Above 20% of effective tooth depth

1.5 - 1.7

Remark: Conditions for tooth bearing from JIS B1741
Table 5. Starting factor Ks
Number of start times per hour

Below 2 times

2 - 4 times

5 - 9 times

Above 10 times

Ks

1.0

1.07

1.13

1.18

Table 6. Time factor K hKhImpact from prime mover side

Uniform load
(Motor, Turbine, Hydraulicmotor and others)

Light impact(Multiple cylinder engine)

Expected lifespan
Uniform load

Medium impact

Heavy impact

1,500 hours

0.80

0.90

1.0

5,000 hours

0.90

1.0

1.25

26,000 hours(1)

1.0

1.25

1.50

60,000 hours

1.25

1.50

1.75

1,500 hours

0.90

1.0

1.25

1.0

1.25

1.50

5,000 hours
26,000 hours(1)

60,000 hours

Medium impact(Cylinder engine)

Impact from load

1.25

1.50

1.75

1.50

1.75

2.0

1,500 hours

1.0

1.25

1.50

5,000 hours

1.25

1.50

1.75

26,000 hours (1)

1.50

1.70

2.0

60,000 hours

1.75

2.0

2.25

Note (1) Operating 10 hours a day for 260 days per a year is equivalent to 10 years and above.

152

5.11 Allowable stress factor Sclim
Table 7 shows Allowable stress factor and limits of sand burning sliding speed for Surface durability.
Table 7. Allowable stress factor Sclim for Surface durability
Material of Worm wheel

Phosphor bronze centrifugal casting

Phosphor bronze chill casting

Phosphor bronze sand casting or Forging

Aluminum bronze

BronzeGraphite flake high strength casting
Gray iron casting (Pearlite quality)

Material of Worm gear

Sclim

Limits of sand burning sliding velocity (1) m/s

Alloyed steel with Case hardening

1.55

30

Alloyed steel HB400

1.34

20

Alloyed steel HB250

1.12

10

Alloyed steel with Case hardening

1.27

30

Alloyed steel HB400

1.05

20

Alloyed steel HB250

0.88

10

Alloyed steel with Case hardening

1.05

30

Alloyed steel HB400

0.84

20

Alloyed steel HB250

0.70

10

Alloyed steel with Case hardening

0.84

20

Alloyed steel HB400

0.67

15

Alloyed steel HB250

0.56

10

Alloyed steel HB400

0.49

8

Alloyed steel HB250

0.42

5

Same material as Worm wheel but withhigher hardness.

0.70

5

Phosphor bronze casting and Forging

0.63

2.5

Same material as Worm wheel but withhigher hardness.

0.42

2.5

Note (1): V alues of Sclim in the table 7 is maximum sliding velocity applicable. Even if used below a calculated load,
there is risk of sand burning if the sliding velocity exceeds this limit.
Remark 4 Calculation for Fluctuating load
(1)For combination of uniform torque with different
revolving speeds,
When maximum nominal action T21(1) operates Worm
wheel at U 1 seconds per 1 cycle, smaller nominal
torque T22, T23… at U2, U3… seconds and mean revolving speed is n21, n22, n23, … . calculate Equivalent timeper 1 cycle based on T21 and n21 using below formula.
Ue = U 1 + U 2

3

3

n 22 § T 22 ·n 23 § T 23 · . .................
¨
¸ +U3
¨
¸ +
(R1)
n 21 © T 21 ¹
n 21 © T 21 ¹

HerebyUe: Equivalent time (per 1 cycle) (s) basedon T21 and n21.
....
: m ean revolving velocity of Worm
n21n22 n23
wheel (min-1)
T21,T22, T23 .... : Nominal torque of Worm wheel		(kgf・m)Therefore Total equivalent time within 26,000 hours is
as follow,
Uec =

Ue
× (Total number of cycle within 26,000 hours) ...... (R2)3600

Hereby, Total equivalent time per 26,000 hours basedon Uec・T21 and n21.

T 2 c = T 21Kh' .......................................................................... (R3)
Hereby,T2c : Total sum of torques, T 21, T22, T23… (kgf・m)
Kh’ : Factor taken from Reference table 4. If Uec ismedian value, use interpolation.
Reference table 4 Kh’
Uee

Kh

Uee

Kh’

500 hours

0.77

5,000 hours

0.90

1,000 hours

0.79

10,000 hours

0.92

2,000 hours

0.81

25,000 hours

1.0

3,000 hours

0.84

26,000 hours

1.0

Note (1) : This table does not include torque peakwith instantaneous change. Please use calculation formula from (2) for such types of
torque peak.
Remark: When 1 cycle of the fluctuating load exactly
matches one revolution of a Worm wheel,
the largest torque always fall on only 1 specific tooth of the Worm wheel. Therefore
calculation formula for fluctuating load is
not applied. Calculated maximum torque isapplied continuously to the whole expected
lifespan.

Calculate Total torque from Uec and Reference table 4
using the following formula.

153

Determine dimensions of Worm gear based on calculated Total torque T2c from formula (R3) from (a) and(b).
(a) Non impact, expected lifespan is 26,000 hours. It
is considered non impact if number of starts per houris under 2 times and starting impact torque is below
200% of rated torque.
Detemine dimensions for worm gear pair in accordance with following relation.
T 2 c ҇ T 2lim ............................................................................. (R4)
HerebyT2lim : Allowable torque for Worm wheel (kgf・m) tomatch with revolution velocity n21 for Worm
wheel.
(b) When life is about 26,000 hours, impact conditions
and number of start is above 2 times per hour. Design
dimensions for Worm gear pair to form following relation.
T 2 c K h K s ҇ T 2lim ................................................................... (R5)
Hereby T2lim : Allowable torque for Worm wheel (kgf・m)
to match with revolution velocity n21 for Worm wheel.
(2)	For combination of Peak torque and Flat torquewhen starting {Refer to 5.9 number(2)}.
	Value of peak T 21 during start reaches steadyspeed of operation after acceleration time of Ua
seconds. If constant driving and torque are designated as T22, Equivalent action time U1e (s) is using
following calculation.

2
Ua § T 22 ·­° § T 22 · ½°
¸ ¾ ....................................... (R6)¸®1 + ¨¨1 +4 © T 21 ¹°̄ © T 21 ¹ °¿Calculation of U 1e’(Torque peak equivalent action
time per hour) with N times of start per hour is
U 1e' = NU 1e .......................................................................... (R7)
Actual time is NU 1e.
When such peak torque acts NUa seconds per hour,
steady torque T22 and Uniform torque T23,T24…acts for
U2, U3, U4 seconds. When each mean revolution velocity is n21, n22, n23, n24, calculation of Equivalent timeUe(s) per hour is by following formula,

U 1e =

Ue = U 1e' + U 2

3

However, standard revolving speed n21 is the average value of peak torque between starting and end.
Therefore, from standstill to reach n21 is calculated by
n21= n’21 / 2. T 21 is standard torque. (Refer to Reference
Fig. 1)
Total Virtual time in 26,000 hours is as follows.
Uec =

Ue
× 26000 ........................................................... (R9)3,600

HerebyUec : T otal equivalent time (h) per 26,000 hours
based on T21 and n21.
This Uec is equivalent to Uec of formula (R2) of previous
item (1). Dimensions of Worm gear pair can be determined from formula (R3), (R4) or (R5) of (1) but Ks to
be 1.0.

Reference Fig. 1 Conditions of Peak and Uniform torque

T21

Peak TorqueT23 (n23min -1)
T22 (n22min -1)
T24 (n24min -1)

Other types ofUniform torque
Ua

154

3

n 22 § T 22 ·n 23 § T 23 ·
¨
¸ +U3
¨
¸ + . ................. (R8)n 21 © T 21 ¹
n 21 © T 21 ¹

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