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Table 4.22 presents the relationships among worm and worm wheel with regard to axial plane, transverse plane, normalplane, module, pressure angle, pitch and lead.
Fig. 4.16 Cylindrical worm (Righthand)
Table 4.22 The relations of cross sections of worm gear pairs
Worm   
Axial plane  Normal plane  Transverse plane 



Transverse plane  Normal plane  Axial plane 
Worm wheel   
Reference to Figure 4.16 can help the understanding of the relationships in Table 4.22. They are similar to the relationsin Formulas (4.16) and (4.17) in that the helix angle β be substituted by (90 deg – γ). We can consider thata worm with lead angle γ is almost the same as a helical gear with helix angle (90 deg – γ).
(1) Axial Module Worm Gear Pair
Table 4.23 presents the equations, for dimensions shown in Figure 4.16, for worm gears with axial module, mx, andnormal pressure angle αn=20°.
Fig. 4.17 Dimentions of cylindrical worm gear pair
Table 4.23 The calculations for an axial module system worm gear pair
No.  Item  Symbol  Formula  Example  
Worm (1)  Wheel (2)     
1  Axial module  mx  Set Value  3  
2  Normal pressure angle  ( αn )  ( 20 deg )   
3  No. of threads,
no. of teeth  z  Double Thread (R)  30 (R)  
4  Coefficient of Profile shift  zt2  –  0  
5  Reference diameter  d1
d2  ( Qmx) NOTE1
z2mx  44.000  90.000 
6  Reference cylinder lead
angle  γ 
 7.76517 deg  
7  Center distance  a 
 67.000  
8  Addendum  ha1
ha2  1.00 mx
( 1.00 + xt2 ) mx  3.000  3.000 
9  Tooth depth  h  2.25 mx  6.750  
10  Tip diameter  da1
da2  d1 + 2ha1
d2 + 2ha2 + mx NOTE2  50.000  99.000 
11  Throat diameter  dt  d2 + 2ha2  –  96.000 
12  Throat surface radius  ri 
 –  19.000 
13  Root diameter  df1
df2  da1 – 2h
dt – 2h  36.500  82.500 
NOTE 1.
Diameter factor, Q, means reference diameter of worm, d1, over axial module, mx.
Q=d1 / mx
NOTE 2.
There are several calculation methods of worm wheel tip diameter da2 besides those in Table 4.25.
NOTE 3.
The facewidth of worm, b1, would be sufficient if: b1=πmx (4.5 + 0.02z2)
NOTE 4.
Effective facewidth of worm wheel bw=
So the actual facewidth of b2 ≧ bw + 1.5mx would be enough.
(2) Normal Module System Worm Gear Pair
The equations for normal module system worm gears are based on a normal module, mn, and normal pressure angle, αn=20°. See Table 4.24.
Table 4.24 The calculations for a normal module system worm gear pair
No.  Item  Symbol  Formula  Example  
Worm (1)  Wheel (2)     
1  Normal module  mn  Set Value  3  
2  Normal pressure angle  αn  ( 20 deg )   
3  No. of threads,
No. of teeth  z  Double (R)  30 (R)  
4  Reference diameter of worm  d1  44.000  –  
5  Normal profile shift coefficient  xn2  –  – 0.1414  
6  Reference cylinder lead
angle  γ 
 7.83748 deg  
7  Reference diameter of
worm wheel  d2 
 –  90.8486 
8  Center distance  a 
 67.000  
9  Addendum  ha1
ha2  1.00 mn
( 1.00 + xn2 ) mn  3.000  2.5758 
10  Tooth depth  h  2.25 mn  6.75  
11  Tip diameter  da1
da2  d1 + 2ha1
dt2 + 2ha1 + mn  50.000  99.000 
12  Throat diameter  dt  d2 + 2ha2  –  96.000 
13  Throat surface radius  ri 
 –  19.000 
14  Root diameter  df1
df2  da1 – 2h
dt – 2h  36.500  82.500 
NOTE : All notes are the same as those of Table 4.23.
(3) Crowning of the Tooth
Crowning is critically important to worm gears. Not only can it eliminate abnormal tooth contact due to incorrectassembly, but it also provides for the forming of an oil film, which enhances the lubrication effect of the mesh. Thiscan favorably impact endurance and transmission efficiency of the worm mesh. There are four methods of crowning wormgear pair :
(a) Cut Worm Wheel with a Hob Cutter of Greater Reference Diameter than the Worm.
A crownless worm wheel results when it is made by using a hob that has an identical pitch diameter as that of theworm. This crownless worm wheel is very difficult to assemble correctly. Proper tooth contact and a complete oil filmare usually not possible.
However, it is relatively easy to obtain a crowned worm wheel by cutting it with a hob whose reference diameter isslightly larger than that of the worm.
This is shown in Figure 4.18. This creates teeth contact in the center region with space for oil film formation.
Fig.4.18 The method of using a greater diameter hob
(b) Recut With Hob Center Position Adjustment.
The first step is to cut the worm wheel at standard center distance. This results in no crowning. Then the worm wheelis finished with the same hob by recutting with the hob axis shifted parallel to the worm wheel axis by ±Δh. This resultsin a crowning effect, shown in Figure 4.19.
Fig.4.19 Offsetting up or down
(c) Hob Axis Inclining Δθ From Standard Position.
In standard cutting, the hob axis is oriented at the proper angle to the worm wheel axis. After that, the hob axisis shifted slightly left and then right, Δθ, in a plane parallel to the worm wheel axis, to cut a crown effect on theworm wheel tooth.
This is shown in Figure 4.20. Only method (a) is popular. Methods (b) and (c) are seldom used.
Fig. 4.20 Inclining right or left
(d) Use a Worm with a Larger Pressure Angle than the Worm Wheel.
This is a very complex method, both theoretically and practically. Usually, the crowning is done to the worm wheel,but in this method the modification is on the worm. That is, to change the pressure angle and pitch of the worm withoutchanging base pitch, in accordance with the relationships shown in Equations 4.25 :
In order to raise the pressure angle from before change, αwx, to after change, αx , it is necessary to increase theaxial pitch, pwx, to a new value, px, per Equation (4.25). The amount of crowning is represented as the space betweenthe worm and worm wheel at the meshing point A in Figure 4.22. This amount may be approximated by the following equation:
Where
d1 : Reference diameter of worm
k : Factor from Table 4.25 and Figure 4.21
Table 4.25 The value of factor k
Axial pressure angle αx
Fig. 4.21 The value of factor (k)