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Calculation of basic parameters and geometric dimensions of ordinary cylindrical worm gearing

1. basic parameters:

(1) modulus m and pressure angle α：

in the middle plane, in order to ensure the correct engagement of worm and worm gear transmission, the axial modulus MA1 and pressure angle of the worm α A1 shall be equal to the normal modulus MT2 and pressure angle of worm gear respectively α T2, i.e.

ma1=mt2=m α a1= α T2

the relationship between worm axial pressure angle and normal pressure angle is:

tg α a=tg α n/cos γ

where: γ- Lead angle

(2) the graduation circle diameter D1 and diameter coefficient q

in order to ensure the correct engagement between the worm and the worm gear, the worm hob with the same size as the worm should be used to process the worm gear. Because of the same module, there can be many different worm diameters, which makes it necessary to equip many worm gear hobs to adapt to different worm diameters. Obviously, this is uneconomical

in order to reduce the number of worm gear hobs and facilitate the standardization of hobs, a certain number of worm indexing circle diameters D1 are specified for each standard module, and the ratio of the indexing circle diameter to the module is called the worm diameter coefficient Q, that is:

q=d1/m

the commonly used standard module M and the worm indexing circle diameter D1 and the diameter coefficient Q, see the matching table

(3) the number of worm heads Z1 and the number of worm gear teeth z2

the number of worm heads can be selected according to the required transmission ratio and efficiency. Generally, Z1 =, and it is recommended that Z1 = 1, 2, 4, 6

the selection principle is: when a large transmission ratio is required or a large torque is required to be transmitted, Z1 is taken as the small value; Z1 ＝ 1 when the transmission is required to be self-locking; If high transmission efficiency is required, or high-speed transmission is required, Z1 shall be taken as the larger value

the number of worm gear teeth affects the stability of operation and is subject to two restrictions: the minimum number of teeth should avoid undercutting and interference. In theory, z2min should be ≥ 17, but when Z2 ＜ 26, the meshing area will be significantly reduced, affecting the stability. When Z2 ≥ 30, more than two pairs of teeth can always be meshed, so it is generally specified that Z2 ＞ 28. On the other hand, Z2 should not be too much. When Z2> 80 (for power transmission), the diameter of the worm gear will increase too much. In terms of structure, it is necessary to increase the span between the two support points of the worm, which will affect the stiffness and meshing accuracy of the worm shaft; For a worm gear with a certain diameter, if Z2 is obtained too much, the modulus m will be greatly reduced, which will affect the bending strength of the gear teeth; Therefore, for power transmission, the commonly used range is Z2 ≈. For transmission of motion, Z2 can reach 200, 300, or even 1000. See the table below for the recommended values of Z1 and Z2

i=z2/z1z1z2 ≈ 5629-317-15. Bayer's trademark 429-6114-30229-6129-82129-82

(4) lead angle γ

the forming principle of the worm is the same as that of the screw, so the relationship between the axial tooth pitch PA of the worm and the lead PZ of the worm is PZ = z1pa. It can be seen from the following figure:

tan γ= PZ/π D1 = z1pa/π D1 = z1m/D1 = Z1/q

lead angle γ The range is 3.5 ° - 33 °. The lead angle is related to the efficiency. When the lead angle is large, the efficiency is high, usually γ＝ 15°-30°。 And multi head worm is mostly used. However, the lead angle is too large, so it is difficult to turn the worm. Small lead angle, low efficiency, but self-locking, usually γ＝ 3.5 ° - 4.5 °

5) transmission ratio i

transmission ratio i=n active 1/n driven 2

worm is active in deceleration movement

i=n1/n2=z2/z1 =u

where: N1 - worm speed; N2- worm gear speed

for power worm drive with reduced speed, 5 ≤ u ≤ 70 is usually taken, and 15 ≤ u ≤ 50 is preferred; Speed increase transmission 5 ≤ u ≤ 15

basic dimensions and parameters of ordinary cylindrical worm and its matching table with worm gear parameters

2 characteristics of worm drive displacement

worm drive displacement

standard

positive displacement

negative displacement

modified worm drive according to different application occasions, one of the following two displacement modes can be selected in places with low humidity

1) before and after the modification, the number of teeth of the worm gear remains unchanged (z2 '= Z2), and the center distance of the worm gear changes (a' ≠ a), as shown in figures a and C. The calculation formula of the center distance is as follows:

a '= a+x2m= (d1+d2+2x2m)/2

2) before and after the modification, the center distance of the worm gear remains unchanged (a' = a), and the number of teeth of the worm gear changes (z2'≠ Z2), as shown in figures D and E, Z2'is calculated as follows:

since A' = a, z2'= zx2

worm transmission displacement:

3 calculation of geometric dimensions of ordinary cylindrical worm transmission

calculation formula of basic geometric dimensions of ordinary cylindrical worm transmission:

description of calculation formula of name code center distance aa= (d1+d2+ 2x2m)/2 select the number of worm heads as required Z1 select the number of worm gears as required Z2 determine the tooth shape angle as per the transmission ratio aaa=20. Or an=20. Determine the modulus according to the worm type mm=ma=mn/cosr ring selection stiffness testing machine according to the regulations is widely used to determine the ring stiffness of thermoplastic pipes and fiberglass pipes with annular cross-section. Take the transmission ratio ii=n1/n2 worm as the active, and select the tooth ratio uu=z2/z1 according to the regulations when the worm is active, I=u worm gear modification coefficient x2x2=a/m- (d1+d2)/2m the worm should be cleaned with the antirust oil applied during packaging. Diameter coefficient =d1/m worm axial tooth pitch papa= m worm lead pzpz= mz1 worm indexing circle diameter d1d1 = MQ select the worm tooth top circle diameter as required. Diameter of the worm tooth top circle da1da1=d1+2ha1=d1+2ha*m worm tooth root circle straight diameter df1df1=dhf1=da-2 (ha*m+c) top clearance cc=c*m as required involute worm tooth root circle diameter db1db1=r/tgrb=mz1/tgrb

worm tooth top height h a1ha1=ha*m=1/2 (da1-d1) according to the specified worm tooth root height hf1hf1= (ha*+c*) m=1/2 (da1-df1) worm tooth height h 1h1=hf1+ha1=1/2 (da1+df1) worm lead angle rtgr=mz1/d1=z1/q involute worm base circle lead angle rbcosrb=san worm tooth width B1 see table 11-4. Determine the worm gear graduation circle diameter by design d2d2= mz2=2a-dx2 M worm gear throat circle diameter da2da2=d2+2ha2 worm gear root circle diameter df2df2=dha2 worm gear tooth top height h a2ha2=1/2 (da2-d2) =m (ha*+x2) worm gear tooth root height h F2 hf2=1/2 (d2-df2) =m (ha*-x2+c*) worm gear tooth height H 2 h2=ha2+hf2=1/2 (da2-df2) worm gear throat parent circle radius rg2rg2=a-1/2 (DA2) worm gear tooth width B2 determined by design wide angle =2arcsin (b2/d1) worm axial tooth thickness sasa=1/2 (m) Worm normal tooth thickness snsn=sr worm gear tooth thickness st determine worm pitch circle diameter d1'd1'=d1+2x2m=m (q+2x2) worm pitch circle diameter

d2'd2'=d2

(end)

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