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Increase in the reliability and quality of worm cylindrical gears widely applied in transmissions of various bench and lifting and transport equipment is an important scientific and technical task. The wellknown solutions include increase in dimensions, appliance of highimpact and durable materials, increase in the accuracy and fabricability of gears, but they are quite expensive and not always effective. The innovative way for solution is to decrease the load in the contact zone while passing to multipair contact and to increase the fabricability due to the unification of the initial and generating worms. In spite of the differences, the tasks given are solved within the same methodological framework – through the modification of the rack profile. Modification of the depth and angles allows supporting the multipair contact followed by the increase in strength and resourcefulness of the gear; shape modification allows the unification of initial and generation worms and the variety reduction of gear cutting tools.
The multipair contact in the gear is possible only if the contact ratio being equal to the acting engagement line/gear pitch ratio is more than two, which is:
ε_{s} > 2. (1)
The condition (1) is fulfilled with the modification of initial gear data in computer model analyses, and the unification of initial and generating worms is fulfilled through the replacement of the initial worm with the convolute one which is identical to the worm milling cutter for involute gear. To make this, we define the geometry of the initial worm hob for the new dimensions of the worm which correspond to the dimensions of the worm milling cutter chosen. The additional condition can be the demand to preserve the interaxle distance α_{w}, gear ratio u, number of worm screws z_{1}, worm pressure angle α and module m. The programmethodical complexes for worm hob geometry and strength calculations are used as the software. The parameters of the multipair gear rack profile are chosen in accordance to the RF patent data for double, three and fourtooth worm hobs [1], [3], [2].
Similar to the cylindrical tooth gears [4] increase in gear contact frequency is due to the increase of factor computed value, which is defined with:
, (2)
where , (3)
tgϒ = z_{1}/ q; (4)
α, ϒ, z_{1}, α_{t}, h_{a}^{*}, x. q — accordingly the profile angle, dividing angle of lap lift, a number of worm approach, profile face angle, head height and worm offset factors and worm diameter factor.
The value εs depends on the value of module m, and depends mostly on the values of parameters of the initial shape α and ha offset factor x. As an example in the tables 1 and 2 one can see the values of the factors εs for worm hobs with constant values z_{1}, q, m for two and threetooth gear in accordance to the patents [1] and [3]. The calculation has been done with gear geometry computer programs with the following initial data: m = 1 mm; z_{1} = 1; z_{2} = 10 ÷ 40; x = 0 ÷ 0,5; q=10; α=20º (twotooth contact) и α=17,5° (threetooth contact); c*=0,2. For all the gear variants the geometry quality is satisfactory.
Table 1 Factor values εs for doublepair gears
z_{2}  x  h_{a}^{*}  ε_{s}  z_{2}  x  h_{a}^{*}  ε_{s} 
10 
0 
1,2

1,980 
30 
0 
1,3 
2,324 
10 
0,1 
1,2

2,027 
30 
0,1 
1,3 
2,355 
10 
0,2 
1,2

2,072 
30 
0,2 
1,3 
2,384 
10 
0,3 
1,2

2,109 
30 
0,3 
1,3 
2,403 
10 
0,4 
1,2

2,150 
30 
0,4 
1,3 
2,430 
10 
0,5 
1,2 
2,190 
30 
0,5 
1,3 
2,455 
20 
0 
1,25 
2,176 
40 
0, 
1,35 
2,452 
20 
0,1 
1,25 
2,212 
40 
0,1 
1,35 
2,479 
20 
0,2 
1,25 
2,248 
40 
0,2 
1,35 
2,505 
20 
0,3 
1,25 
2,273 
40 
0,3 
1,35 
2,520 
20 
0,4 
1,25 
2,305 
40 
0,4 
1,35 
2,543 
20 
0,5 
1,25 
2,335 
40 
0,5 
1,35 
2,564 
Table 2. Factor values εs for threetooth contact gears
z_{2}  x  h_{a}^{*}  ε_{s}  z_{2}  x  h_{a}^{*}  ε_{s} 
10 
0 
1,65 
2,877 
30 
0 
1,75 
3,346 
10 
0,1 
1,65 
2,940 
30 
0,1 
1,75 
3,392 
10 
0,2 
1,65 
3,002 
30 
0,2 
1,75 
3,437 
10 
0,3 
1,65 
3,063 
30 
0,3 
1,75 
3,481 
10 
0,4 
1,65 
3,112 
30 
0,4 
1,75 
3,511 
10 
0,5 
1,65 
3,170 
30 
0,5 
1,75 
3,552 
20 
0 
1,70 
3,146 
40 
0 
1,80 
3,515 
20 
0,1 
1,70 
3,199 
40 
0,1 
1,80 
3,556 
20 
0,2 
1,70 
3,251 
40 
0,2 
1,80 
3,597 
20 
0,3 
1,70 
3,302 
40 
0,3 
1,80 
3,636 
20 
0,4 
1,70 
3,339 
40 
0,4 
1,80 
3,660 
20 
0,5 
1,70 
3,386 
40 
0,5 
1,80 
3,696 
As it is seen from the tables 1 and 2 if the head height is increased and the profile angle is decreased, the value of gear contact n is increased. In particular cases the needed values of contact ratio may be obtained with zero values of the offset factor. In general case the value of gear contact may be increased to any prescribed number. With a standard initial shape multipair gear is not possible. As the contact ration does not depend on the module value, the table data may be used to design two and threetooth gears.
For the unification of the drive worm and the driven tool one should use the existing worm milling cutters with the necessary parameter values of the rack profile and module. To solve the task, it is necessary to change the geometry of the drive worm hob in accordance to the dimensions of the chosen worm milling cutter, in particular, under the condition of preservation of the gear basic parameters: interaxle distance, gear ratio and a number of worm screws.
To do so, one defines the outer and reference diameters of the new worm d_{α1} and d_{1}:
d_{α1 }= d_{α0} — 2c^{*}m; (5)
d_{1 }= d_{α1} — 2h_{a}^{*}m, (6),
where d_{α1} — the outer diameter of the worm milling cutter.
Then, one defines the new values of the worm diameter factor q and the worm offset factor x:
; (7)
. (8)
Then, with the new data received the worm hob geometry is calculated and its new executive and control dimensions are defined. Under the condition that the new gear meets the quality criteria for gearing and assembling, the next step is to test the gear for contact and bending strength. If the strength indexes meet the demanded criteria we get the worm hob with the unified worm. Otherwise a new worm milling cutter is chosen and the process of recalculation is repeated, or a special worm milling cutter is used to construct the worm tooth gear in accordance to the initial dimensions of the worm. In some cases standard worm milling cutters of GOST 932480 having factor of the tooth profile height of the worm milling cutter h_{0*}=2,5 may be used. It allows having a guaranteed twotooth gearing in the real gear. The table 3 shows the example of comparative data for worm hobs with one and twotooth gear, one of them having a worm unified with the standard worm milling cutter. The initial data for worm tooth gears: singlepair gear is with the Archimede’s worm; standard shape; doubletooth gear is with a convolute worm; α=20º, h_{a*}=1,3; c*=0,2; doublepair (unified) gear is with convolute worm; α=20º, h_{a*}=1,3; c*=0,1; wheel rim width b2=50 mm; number of worm rotations n1=14 1/min; operating regime is intermittent; rotation torque on the wheel axis is T2=140 Н·м; material is bronze BrO10F1; working stress on the wheel teeth: contact strength  σHP=290 mPa; on the bend  σFP=165 mPA. The strength of the worm hob is equal to the strength of the worm wheel as the weakest point of the gear. The worm wheel itself is helical which allows calculating the wheel strength with the method of strength prediction for helical gears of the outer gearing [5]. The parameters of the worm milling cutter with GOST 9324–80: module m0=2,75 mm; dα0=71 mm; the profile angle α=20º; the profile height h0=6,88; the profile head height hα0= 3,44.
Table 3. Comparative data for single and doubletooth contact worm hobs
Worm hob parameters

Original gear 
Unified doubletooth contact gear


singletooth contact 
doubletooth contact 

Worm type 
Archimede's 
convolute 
convolute 
Normal module, mm 
3 
3 
2,75 
Interaxial distance, mm 
96 
96 
96 
Number of worm threads 
1 
1 
1 
Number of worm wheel teeth 
46 
46 
47 
Worm diameter factor 
18 
18 
23,32 
Worm shift factor 
0 
0 
0,25 
Worm reference diameter, mm 
54 
54 
64,13 
Wheel reference diameter, mm 
138 
138 
129,25 
Worm thread addenda diameters, mm 
60 
61,8 
70,45 
Well teeth top diameters, mm 
144 
145,8 
134,2 
Worm thread depth, mm 
6,6 
8,6 
6,88 
Reference angle of worm thread pitches 
3 10’47'' 
3 10’47'' 
2 27′ 19′′ 
Chordal reference thread thickness, mm 
4,71 
4,71 
4,32 
Depth to the worm thread chord, mm 
3,0 
3,9 
3,16 
Contact ratio 
1,86 
2,38 
2,18 
Contact stresses on the wheel teeth, MPa 
287 
207 
242 
Bending stresses on the wheel teeth, MPa 
45 
30 
35 
As it is seen from the Table 3, comparing to a singlepair worm hob, doublepair gears are characterized by significantly smaller stresses – by 1.4 and 1.2 times smaller by contact, correspondingly, and by 1.5 and 1.3 times by bend. If we don’t take into account a uniform gear, virtually the same results are obtained at the transition to a doublepair engagement for cylindrical gears. Higher contact and bend stresses characteristic for a uniform gear in comparison with the other doublepair gear are explained by the fact that this gear has lower module and offset factor values —2.75 and 2.18 instead of 3 and 2.38, correspondingly.
The uniform gear advantage is the opportunity to use the available tool. The lower is the difference between the initial worm diameter and the one of the selected mill, the lower will be the change in engagement parameters and increase in contact and bending stresses at teeth in such cases, i.e. the difference between d_{α1} and d_{α0} shall be reduced to minimum. In particular, it is necessary to provide that the diameter factors for a worm and worm hob q_{0 } are the same or differ by an admissible small value. The factor q_{0} values for the applied mills are defined in the same way (6)(7):
d_{0 = }d_{α0 }— 2 • (h_{a}^{*} + c) • m_{0}; (9)
, (10)
where d_{0 }— reference gear hob diameter m_{0}=m.
Factor values q_{0} for standard gear hubs with a module m_{0} = 1–10 mm obtained on the basis of (9), (10), are given in the Table 4.
Table 4. Factor values for standard gear hubs
m_{0}  q_{0 }  m_{0}  q_{0 }  m_{0}  q_{0 } 
1 
37,5 
2,75 
23,0 
5,5 
17,5 
1,125 
42,0 
3,0 
24,0 
6,0 
16,0 
1,25 
37,5 
3,25 
22,0 
6,5 
15,5 
1,375 
33,5 
3,5 
20,0 
7,0 
14,0 
1,5 
39,5 
3,75 
21,5 
8,0 
13,0 
1,75 
33,5 
4,0 
20,0 
9,0 
13,0 
2,0 
29,0 
4,25 
18,5 
10,0 
12,5 
2,25 
29,0 
4,5 
17,5 
11,0 
12,0 
2,5 
25,5 
5,0 
17,5 
12,0 
11,5 
As it is seen from the Table 4, the monotoneness of the factor q_{0} diminishing and the module m_{0} increase is not always observed which indicates the opportunity and necessity to introduce additional values d_{α0} to reduce the difference between the hob and worm diameters. In compliance with GOST 2144–76, for the key worm hob parameters each module m value correlates with various values of the factor q. Taking this fact into account, when choosing a hob one shall try to comply with the following condition:
q_{0 }= q (11)
The less will be the difference between worm and gear hob diameters, the higher will be the multicontact gear strength properties.
1. The design and engineering synthesis of worm hobs with a transition to multipaired engagement and unification of the initial worm and generating tool dimensions provides for the enhancement of the gear strength indicators and improve the efficiency of their production by means of tooling cost reduction.
2. Increase in the variety of worm hobs with the purpose of diminishing the difference between the worm and mill diameters results in gear quality higher indicators.
3. Worm hob design with an arbitrary npaired engagement is generally possible only with a nonstandard initial and generating contour.
4. The standard tool application is possible only at the manufacture of doublepair gears.
5. In general, the unification of the initial worm and a generating tool is viable at a relatively small difference of their diameters (not more than 10–13 mm) and the diameter factor values (not more than 7).
6. The opportunities for the implementation of multipaired worm hobs with any n value are limited only by the relevance of such gear application.
7. The degree of gear quality increase largely depends on the opportunities to increase the engagement pairing and to reduce the difference between the worm and tool diameters due to the enlargement of worm hob variety.
8. Introducing multipaired and unified worm hobs allow for the increase in the level of the reliability and quality of the drive with worm hobs for many types of engineering systems, for example, lifting and transport ones, machine one, etc.
V. Z.Melnikov, Cand. Tech. Sci., MSIU
1. Taratynov O.V., Melnikov V.Z., Klepikov V.V. Worm Cylindrical Gears with Doublepair Engagement. Patent of the Russian Federation for the Utility Model No. 125643. Bull. No. 7 of 10.03.2013.
2. Melnikov V.Z. Worm Cylindrical Gears with FourPaired Engagement. Patent of the Russian Federation for the Utility Model No. 135751. Bull. No. 35 of 20.12.2013.
3. Melnikov V.Z. Worm Cylindrical Gears with TriPaired Engagement. Patent of the Russian Federation for the Invention No. 2529076. Bull. № 27 of 27.09.2014.
4. Melnikov V.Z. Synthesis of Gears with Arbitrary NPaired Engagement. Bulletin of Mechanical Engineering. 2010. No. 4, p. 29–31.
5. GOST 21354 — 87. Cylindrical Involute External Gears. Strength Calculation. M.: Publisher of Standards, 1988. 128 p.