Gear Design Equations and Formula
Circular Pitches and Equivalent Diametral Pitches Table

Gear Design and Selection Application

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Spur Gear design formula for geometry, pitch, tooth clearance and critical functional data.

Calculate
When Defined
Formula
Diametral Pitch (P)
Pitch Diameter (PD) and the Number of Teeth (N)
P = N / PD
Diametral Pitch (P)
Circular Pitch (p)
P = 3.1416/p
Diametral Pitch (P)
Outside Diameter (OD) and the Number of Teeth (N)
P = (N+2)/OD
Pitch Diameter (PD) .
Number of teeth (N) and the Diametral Pitch (P)
PD = N/DP
Outside Diameter (OD)
Number of teeth (N) and the Diametral Pitch (P)
OD = (N+2)/DP
Number of Teeth (N).
Pitch Diameter (D) and the Diametral Pitch (P)
N = PD * DP
Addendum (a) .
Diametral Pitch (P)
a = 1/DP
Dedendum (d)
Whole Depth and Addendum
d = hw - a
Tooth Thickness (t) at the Pitch Diameter
Diametral Pitch (P)
t = 1.5708/P
Working Depth (WD).
Addendum
WD = 2(a)
Center Distance (C)
Normal Diametral Pitch (P) and the Number of Teeth in Both Gears
C = (N1+N2)/2P
Center Distance (C)
Pitch Diameters of both gears
C = (PD1+PD2)/2
Circular Pitch (p)
Diametral Pitch (P)
p = 3.1416 / P
Whole Depth (hw) for 20 Pitch & finer .
Diametral Pitch (P)
hw = 2.2/P + .002
Whole Depth (hw) for Coarser than 20 Pitch .
Diametral Pitch (P)
hw = 2.157/P

Equations Tooth Parts, 20-and 25-degree Involute Full-depth Teeth ANSI Coarse Pitch Spur Gear Tooth Forms ANSI B6.1

Find
Diametral Pitch, P, Known
Circular Pitch, p, Known
Addendum
a = 1.000 / P
a = 0.3183 x p
Dedendum (Preferred)
b = 1.250 / P
b = 0.3979 x p

(Shaved or Ground Teeth)a

b = 1.350 / P
b = 0.4297 x p
Working Depth
hk = 2.000 / P
hk = 0.6366 x p
Whole Depth (Preferred)
ht = 2.250 / P
ht = 0.7162 x p

(Shaved or Ground Teeth)

ht = 2.350 / P
ht = 0.7480 x p
Clearance (Preferred)b
c = 0.250 / P
c = 0.0796 x p

a When gears are preshave cut on a gear shaper the dedendum will usually need to be increased to 1.40/P to allow for the higher fillet trochoid produced by the shaper cutter. This is of particular importance on gears of few teeth or if the gear blank configuration requires the use of a small diameter shaper cutter, in which case the dedendum may need to be increased to as much as 1.45/P. This should be avoided on highly loaded gears where the consequently reduced J factor will increase gear tooth stress excessively.
b A minimum clearance of 0.157/P may be used for the basic 20-degree and 25-degree pressure angle rack in the case of shallow root sections and use of existing hobs or cutters.

Helical Gear

Calculate
When Defined
Formula
Normal D.P. (Pn)
Transverse D.P.(P) and the Helix Angle (A)
Pn= P / cos A
Normal D.P. (Pn)
Number of teeth (N) and the Helix Angle (A)
Pn= N / (D X cos A)
Pitch Diameter (D)
Number of teeth (N), the Normal Diametral Pitch and the Helix Angle (A)
D = N / (Pn X cos A)
Outside Diameter (OD)
Pitch Diameter (D) and the Addendum (a)
OD = D + (2 X a)
Outside Diameter (OD)
Normal Diametral Pitch (P) and the Pitch Diameter (D)
OD = D + 2/Pn
Helix Angle (A) for Parallel Shaft Drive
Number of Teeth (N), Pitch Diameter (D) and the Diametral Pitch (P)
Cos A = N / (Pn X D)
Addendum (a)
Normal Diametral Pitch (Pn)
a = 1 / Pn
Lead (L)
Pitch Diameter (D) and Pitch Helix Angle
L = (pi*D) / Tan

Circular Pitches and Equivalent Diametral Pitches Table

Circular Pitch
Diametral Pitch
Module
Arc Thickness
of Tooth on Pitch Line
Addendum
Working Depth
of Tooth
Dedendum or Depth
of Space Below Pitch Line
Whole Depth
of Tooth
4 0.7854 32.3402 2.0000 1.2732 2.5464 1.4732 2.7464
3 - 1/2 0.8976 28.2581 1.7500 1.1140 2.2281 1.2890 2.4031
3 1.0472 24.2552 1.5000 0.9549 1.9098 1.1049 2.0598
2 - 3/4 1.1424 22.2339 1.3750 0.8753 1.7506 1.0128 1.8881
2 - 1/2 1.2566 20.2117 1.2500 0.7957 1.5915 0.9207 1.7165
2 - 1/4 1.3963 18.1913 1.1250 0.7162 1.4323 0.8287 1.5448
2 1.5708 16.1701 1.0000 0.6366 1.2732 0.7366 1.3732
1 - 7/8 1.6755 15.1595 0.9375 0.5968 1.1937 0.6906 1.2874
1 - 3/4 1.7952 14.1488 0.8750 0.5570 1.1141 0.6445 1.2016
1 - 5/8 1.9333 13.1382 0.8125 0.5173 1.0345 0.5985 1.1158
1 - 1/2 2.0944 12.1276 0.7500 0.4775 0.9549 0.5525 1.0299
1 - 7/16 2.1855 11.6223 0.7187 0.4576 0.9151 0.5294 0.9870
1 - 3/8 2.2848 11.1169 0.6875 0.4377 0.8754 0.5064 0.9441
1 - 5/16 2.3936 10.6116 0.6562 0.4178 0.8356 0.4834 0.9012
1 - 1/4 2.5133 10.1062 0.6250 0.3979 0.7958 0.4604 0.8583
1 - 3/16 2.6456 9.6010 0.5937 0.3780 0.7560 0.4374 0.8154
1 - 1/8 2.7925 9.0958 0.5625 0.3581 0.7162 0.4143 0.7724
1 - 1/16 2.9568 8.5904 0.5312 0.3382 0.6764 0.3913 0.7295
1 3.1416 8.0851 0.5000 0.3183 0.6366 0.3683 0.6866
15/16 3.3510 7.5798 0.4687 0.2984 0.5968 0.3453 0.6437
7/8 3.5904 7.0744 0.4375 0.2785 0.5570 0.3223 0.6007
13/16 3.8666 6.5692 0.4062 0.2586 0.5173 0.2993 0.5579
3/4 4.1888 6.0639 0.3750 0.2387 0.4775 0.2762 0.5150
11/16 4.5696 5.5586 0.3437 0.2189 0.4377 0.2532 0.4720
2/3 4.7124 5.3903 0.3333 0.2122 0.4244 0.2455 0.4577
5/8 5.0265 5.0532 0.3125 0.1989 0.3979 0.2301 0.4291
9/16 5.5851 4.5479 0.2812 0.1790 0.3581 0.2071 0.3862
1/2 6.2832 4.0426 0.2500 0.1592 0.3183 0.1842 0.3433
7/16 7.1808 3.5373 0.2187 0.1393 0.2785 0.1611 0.3003
2/5 7.8540 3.2340 0.2000 0.1273 0.2546 0.1473 0.2746
3/8 8.3776 3.0319 0.1875 0.1194 0.2387 0.1381 0.2575
1/3 9.4248 2.6947 0.1666 0.1061 0.2122 0.1228 0.2289
5/16 10.0531 2.5266 0.1562 0.0995 0.1989 0.1151 0.2146
2/7 10.9956 2.3100 0.1429 0.0909 0.1819 0.1052 0.1962
1/4 12.5664 2.0213 0.1250 0.0796 0.1591 0.0921 0.1716
2/9 14.1372 1.7967 0.1111 0.0707 0.1415 0.0818 0.1526
1/5 15.7080 1.6170 0.1000 0.0637 0.1273 0.0737 0.1373
3/16 16.7552 1.5160 0.0937 0.0597 0.1194 0.0690 0.1287
1/6 18.8496 .5053 0.0833 0.0531 0.1061 0.0614 0.1144

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