Worm gears

January 14, 2025

Intro

▶ 0:00

Worm gears transmit power between non-intersecting shafts, and achieve a very high gear reduction ratio in a very compact space.

Modeling the worm

▶ 0:15

Think of the worm as a rack that has been revolved around an axis, except the teeth are revolved around in a helix instead of in circles.

Parameters

▶ 0:45


\(module\)		2mm
\(length\) (in teeth)		10
\(pressure angle\)		20°
\(starts\)		4
\(diameter\)		40mm

New component: worm

▶ 1:06

New sketch on the vertical (xz) plane

▶ 1:14

This sketch looks much like the sketch in the rack video:

A: line thin right 1
X1: line thin color gray down 50% from 2mm below A.w
X2: line thin color gray down 50% from 2mm below A.e
text "diameter" "2" at (1/2,6/8)
line thin from 2mm right of previous text.w to 2mm left of previous text.e
line thin color gray from previous text.w left until even with X1 ->
line thin color gray from previous text.e right until even with X2 ->

B: line from A.w up 2
line thin color gray left 70% from 2mm left of B.n
line same from 2mm left of B.s
text "length ×" "module ×" "π" at (7/8,1/2)
line thin color gray from previous text.n up until even with B.n ->
line thin color gray from previous text.s down until even with B.s ->

Sweep

▶ 1:45

Surface → Sweep, of type “Single Path”


profile		horizontal line
path		vertical line
twist angle		\( length / starts \times 360° \)


/*
 These splines are all unfortunately ugly hacks.
 I drew over a background screencap of the helix
 and eyeballed things.
*/

spline from 0,-1.06 thickness 0.2mm color green \
  then go 0.05 heading 323 \
  then go 0.1 heading 310 \
  then go 0.2 heading 298 \
  then go 0.45 heading 282

spline from 0,-.48 thickness 0.2mm color green \
  then go 0.05 heading 217 \
  then go 0.1 heading 230 \
  then go 0.2 heading 242 \
  then go 0.45 heading 258

spline from 0,0.255 thickness 0.2mm color green \
  then go 0.05 heading 323 \
  then go 0.1 heading 310 \
  then go 0.2 heading 298 \
  then go 0.45 heading 282

spline from 0,0.840 thickness 0.2mm color green \
  then go 0.05 heading 217 \
  then go 0.1 heading 230 \
  then go 0.2 heading 242 \
  then go 0.45 heading 258

spline from 0, 1.57 thickness 0.2mm color green \
  then go 0.05 heading 323 \
  then go 0.1 heading 310 \
  then go 0.2 heading 298 \

line from 0,-1.65 then go 1.1 heading 72.2 color lightblue dashed
line from last.end then go 0.2 heading 72.2 color lightblue ->
line from 0,-1.65 then go 1.3 heading 107.8 color lightblue ->

VERT: line from 0,-1.65 to 0,1.67 color black
line up 0.3 -> color lightblue

HORIZ: line from VERT.s right 0.63 down 0.20 color black

spline from HORIZ.end color green \
  then go 0.3 heading 60 \
  then go 0.2 heading 45 \
  then go 0.1 heading 25 \
  then go 0.1 heading 357 \
  then go 0.1 heading 330 \
  then go 0.2 heading 311 \
  then go 0.3 heading 297 \
  then go 0.62 heading 285

spline from last.end color green thickness 0.2mm dashed \
  then go 0.4 heading 280 \
  then go 0.4 heading 274

spline from -0.75,-0.765 color green \
  then go 0.17 heading 275 \
  then go 0.17 heading 265 \
  then go 0.01 s \
  then go 0.17 heading 95 \
  then go 0.5 heading 86 \
  then go 0.4 heading 81 \
  then go 0.4 heading 77 \
  then go 0.3 heading 71 \
  then go 0.25 heading 63 \
  then go 0.2 heading 49 \
  then go 0.1 heading 30 \
  then go 0.1 heading 0 \
  then go 0.1 heading 335 \
  then go 0.2 heading 314 \
  then go 0.25 heading 297 \
  then go 0.3 heading 290 \
  then go 0.4 heading 284

spline from last.end color green thickness 0.2mm dashed \
  then go 0.4 heading 280 \
  then go 0.4 heading 272

spline from -0.75,0.55 color green \
  then go 0.17 heading 275 \
  then go 0.17 heading 265 \
  then go 0.01 s \
  then go 0.17 heading 95 \
  then go 0.5 heading 86 \
  then go 0.4 heading 81 \
  then go 0.4 heading 77 \
  then go 0.3 heading 71 \
  then go 0.25 heading 63 \
  then go 0.2 heading 49 \
  then go 0.1 heading 30 \
  then go 0.1 heading 0 \
  then go 0.1 heading 335 \
  then go 0.2 heading 314 \
  then go 0.25 heading 297 \
  then go 0.3 heading 290 \
  then go 0.3 heading 286 \
  then go 0.3 heading 279 \
  then go 0.4 heading 278

line to VERT.end color green

The only part we’re interested in is that outer helical path. Since we want to have a total length of 10 teeth, and four starts, each start will twist 2½ times.

Tooth profile

▶ 2:38

Create a sketch on the vertical (xz) plane, and project in the horizontal line we used for the sweep profile, so we can align the tooth profile with it.

▶ 2:57

The tooth profile itself is just a trapezoid drawn with lines and several constraints.


A: line go 1 heading 110;
B: line go down 0.5;
C: line go 1 heading 250
D: line to first line.start

E: line from 3/5  \
  to 2/5 
dot at E

X1: line thin color gray from 2mm below D.start then down 50%
X2: line thin color gray from 2mm below E.end then down until even with last line.end
X3: line thin color gray from 5.4mm below B.end then down until even with last line.end
X4: line thin color gray from 4mm right of E.start then right 0.8
X5: line thin color gray from 4mm right of E.end then right 0.8

text "1.5 ×" small "module" small at 1/2 <3mm above X1.end, 3mm above X2.end>
line thin color gray from previous text left until even with X1 ->
line thin color gray from previous text right until even with X2 ->

text "1 ×" small "module" small at 1/2 <3mm above X2.end, 3mm above X3.end>
line thin color gray from previous text left until even with X2 ->
line thin color gray from previous text right until even with X3 ->

text "pi" small "×" small "module/2" small at 1/2 <4mm left of X4.end,4mm left of X5.end>
line thin color gray from previous text.n up until even with X4 ->
line thin color gray from previous text.s down until even with X5 ->

line thin color gray dashed from last dot left until even with 50mm left of D
line thin color gray dashed from last.end up 30mm

spline thin color gray dashed from last dot \
  up 1.5mm \
  then go 5mm heading 310 \
  then go 10mm heading 290 \
  then go 20mm heading 285 \
  then go 40mm heading 280

spline thin color gray from 0.3 heading 0 from C.end \
 to 0.3 heading 10 from C.end \
 to 0.3 heading 20 from C.end \
 to 0.3 heading 30 from C.end \
 to 0.3 heading 40 from C.end \
 to 0.3 heading 50 from C.end \
 to 0.3 heading 60 from C.end \
 to 0.3 heading 70 from C.end

text "90° -" small "pressure" small "angle" small at 6mm heading 45 from last

line thin from 0.05 heading 20 from A to 0.05 heading 200 from A
line thin from 0.03 heading 110 from last.start to 0.03 heading 110 from last.end

line thin from 0.05 heading 340 from C to 0.05 heading 160 from C
line thin from 0.03 heading 70 from last.start to 0.03 heading 70 from last.end

dashwid = 0.02

	spline thin color gray dotted from 1/4  \
 then go 0.5 heading 20

text at 0.7mm heading 20 from last.end "pitch line" ljust small

Helical sweep

▶ 3:50

Solid → Sweep → “path + guide rail sweep”

The profile is the tooth profile we just drew, the path is the vertical line, and the guide rail is the helix. (Make sure “chain selection” is off when selecting these.)

This gives us a single start for the worm gear:

# This is an abuse of pikchr.

line from 1.2921,6.5444 to 2,5.9023 \
  then to 2,0.78764 \
  then to 2.7697,0.11964 \
  color green dashed

line from 1.1904,6.5503 to 1.3819,6.2473 \
  then to 1.3819,6.682 \
  then to 1.1904,6.6879 \
  close

line from 1.6391,6.2333 to 1.3699,6.2473

spline from 1.3041,6.5503 \
  then to 1.5135,6.5563 \
  then to 2.4526,6.4427 \
  then to 3.5115,5.9222 \
  then to 3.9302,5.1805 \
  then to 3.1645,4.2413 \
  then to 1.0469,3.8225 \
  then to 0.02991,4.1815 \
  then to 0.69392,4.5523 \
  then to 2.1775,4.4865 \
  then to 3.5713,3.8943 \
  then to 3.9362,2.9731 \
  then to 2.8415,2.0818 \
  then to 1.0349,1.8245 \
  then to 0.17946,2.0638 \
  then to 0.28714,2.3629 \
  then to 1.2203,2.5244 \
  then to 2.333,2.3509 \
  then to 3.0987,2.0638 \
  then to 3.7627,1.5494 \
  then to 3.7926,0.84347 \
  then to 3.2782,0.37687 \
  then to 2.9432,0.20339 \
  then to 2.7697,0.11964 color green dotted

spline from 2.006,4.5224 \
  then to 1.7168,4.5942 \
  then to 1.2502,4.6481 \
  then to 0.51446,4.5882 \
  then to 0.4666,4.2951 \
  then to 1.5733,4.2054 \
  then to 2.991,4.5882 \
  then to 3.7029,5.4616 \
  then to 3.0449,6.2991 \
  then to 1.8006,6.664 \
  then to 1.3878,6.682

spline from 2.652,0.4327 \
  then to 2.8175,0.52044 \
  then to 3.2722,0.8016 \
  then to 3.5892,1.34 \
  then to 3.3918,1.8066 \
  then to 3.1645,2.016 \
  then to 3.0249,2.1196

spline from 1.996,2.4905 \
  then to 1.7288,2.5543 \
  then to 1.2622,2.6082 \
  then to 0.52642,2.5484 \
  then to 0.47856,2.2552 \
  then to 1.5852,2.1655 \
  then to 2.997,2.5663 \
  then to 3.6072,3.2602 \
  then to 3.4038,3.8405 \
  then to 3.1765,4.0499 \
  then to 3.0349,4.1575

spline from 2.002,2.5324 \
  then to 1.8365,2.5723 \
  then to 1.5972,2.6082 \
  then to 0.9332,2.6201 \
  then to 0.32303,2.5184 \
  then to 0.089731,2.3868 \
  then to 0,2.1595 \
  then to 0.80758,1.8066 \
  then to 2.7817,2.0758 \
  then to 3.9242,2.7936 \
  then to 3.984,3.5952 \
  then to 3.4337,4.0738 \
  then to 3.1725,4.2233

spline from 2,4.5643 \
  then to 1.8365,4.6062 \
  then to 1.5972,4.6421 \
  then to 0.9332,4.654 \
  then to 0.32303,4.5523 \
  then to 0.089731,4.4207 \
  then to 0.017946,4.1755 \
  then to 0.83151,3.8465 \
  then to 2.7817,4.1097 \
  then to 3.9242,4.8275 \
  then to 4,5.5872 \
  then to 3.4816,6.1256 \
  then to 2.8156,6.4447 \
  then to 2.1635,6.6122 \
  then to 1.7866,6.66 \
  then to 1.5494,6.672

spline from 0.053838,4.2951 \
  then to 0.053838,4.002 \
  then to 0.90927,3.7507 \
  then to 2.2971,3.8704 \
  then to 3.4277,4.3071 \
  then to 3.9701,4.8933 \
  then to 3.9681,5.3579

spline from 0.047856,2.2313 \
  then to 0.047856,1.9621 \
  then to 0.90329,1.7109 \
  then to 2.2911,1.8305 \
  then to 3.4217,2.2672 \
  then to 3.9661,2.8475 \
  then to 3.9561,3.328

spline from 2.8315,0.0019939 \
  then to 3.0628,0.083749 \
  then to 3.5474,0.34098 \
  then to 3.9661,0.82951 \
  then to 3.9661,1.2602

spline from 2.8295,0.11964 \
  then to 2.985,0.18544 \
  then to 3.4397,0.41276 \
  then to 3.996,0.93121 \
  then to 3.9302,1.6092 \
  then to 3.4397,2.0279 \
  then to 3.1805,2.1854

line from 2.8295,0 to 2.65,0.0059821 \
 then to 2.65,0.43669 \
 then to 2.8295,0.11765 \
 close

Go into the worm gear, and hide the unneeded surface body.

Circular pattern

▶ 4:23

Select our sweep and do a circular pattern with Object type Bodies, around the central axis, with Quantity \( starts \):

# This is an even worse abuse of pikchr. It was simply not designed for this!

line from 1.2558,6.7968 to 1.416,6.6899

line from 1.416,6.6899 to 1.416,6.316

line from 1.416,6.316 to 1.2499,6.6781

line from 0.18351,5.2912 to 0.59303,5.5227

line from 0.59303,5.5227 to 0.59303,5.1488

line from 0.59303,5.1488 to 0.18351,5.1725

line from 3.8257,6.1756 to 3.4102,6.2033

line from 3.4083,6.2013 to 3.4043,5.8313

line from 2.6308,0.39675 to 2.7576,0.11277

line from 2.7534,4.67 to 2.7534,4.5394

line from 2.5833,5.0321 to 2.7534,4.67

line from 2.7576,0.11277 to 2.7516,0

line from 2.7516,0 to 2.5833,0.10395

line from 2.5854,0.10683 to 2.5854,0.38578

line from 0.18173,0.62319 to 0.18443,0.74189

line from 0.18766,0.74189 to 0.44558,0.89027

spline from 1.416,6.686 \
  then to 1.5466,6.6978 \
  then to 2.0274,6.6563 \
  then to 2.6743,6.4486 \
  then to 3.2737,6.0094 \
  then to 3.4518,5.6533 \
  then to 3.4933,5.5405

spline from 1.2499,6.7968 \
  then to 1.3804,6.8086 \
  then to 1.873,6.7987 \
  then to 2.425,6.6978 \
  then to 2.9948,6.4842 \
  then to 3.3212,6.2705 \
  then to 3.4102,6.1993

spline from 0.58512,5.5168 \
  then to 0.53112,5.5976 \
  then to 0.4599,5.7579 \
  then to 0.5964,6.0427 \
  then to 1.2552,6.2623 \
  then to 2.0564,6.1971 \
  then to 2.6381,6.0071 \
  then to 3.1307,5.6748 \
  then to 3.3503,5.384 \
  then to 3.4043,5.309

spline from 1.4185,6.313 \
  then to 1.6143,6.313 \
  then to 2.0654,6.2536 \
  then to 2.7064,6.0518 \
  then to 3.1693,5.6957 \
  then to 3.3771,5.4346 \
  then to 3.428,5.3446

spline from 0.56383,5.5652 \
  then to 0.64099,5.6305 \
  then to 0.94368,5.8026 \
  then to 1.7034,5.8797 \
  then to 2.4868,5.6898 \
  then to 2.9438,5.4346 \
  then to 3.1931,5.2091 \
  then to 3.2797,5.1072

spline from 0.59945,5.5177 \
  then to 0.64693,5.5652 \
  then to 0.93775,5.7314 \
  then to 1.6678,5.8144 \
  then to 2.4156,5.6483 \
  then to 2.8845,5.399 \
  then to 3.1634,5.1675 \
  then to 3.2559,5.0716

spline from 0.56383,5.1141 \
  then to 0.61132,5.1616 \
  then to 0.92588,5.3396 \
  then to 1.6084,5.4287 \
  then to 2.374,5.2743 \
  then to 2.8785,5.0132 \
  then to 3.0566,4.8767

spline from 0.61132,5.0726 \
  then to 0.6588,5.12 \
  then to 0.99116,5.2862 \
  then to 1.6381,5.3574 \
  then to 2.4037,5.1972 \
  then to 2.8726,4.942 \
  then to 3.0091,4.847

spline from 0.18153,5.2972 \
  then to 0.1187,5.3574 \
  then to 0,5.5889 \
  then to 0.16025,5.9866 \
  then to 0.73002,6.2714 \
  then to 1.2286,6.3486 \
  then to 1.3888,6.3545

spline from 3.4117,6.2018 \
  then to 3.4473,6.1306 \
  then to 3.4888,6.0594 \
  then to 3.5423,5.5786 \
  then to 3.2871,5.0504 \
  then to 2.8479,4.724 \
  then to 2.7529,4.6705

spline from 2.5867,5.0326 \
  then to 2.4739,4.9856 \
  then to 2.0288,4.8551 \
  then to 1.4649,4.7902 \
  then to 0.84764,4.8911 \
  then to 0.58056,5.081 \
  then to 0.53901,5.1522

spline from 2.741,4.6646 \
  then to 2.6283,4.6177 \
  then to 2.1831,4.4871 \
  then to 1.5941,4.4111 \
  then to 0.95908,4.4645 \
  then to 0.43679,4.6248 \
  then to 0.051006,4.9453 \
  then to 0.039136,5.2361 \
  then to 0.068811,5.3192

spline from 0.039091,5.5308 \
  then to 0.033156,5.418 \
  then to 0.11031,5.2578 \
  then to 0.18351,5.1725

spline from 2.747,4.54 \
  then to 2.6342,4.493 \
  then to 2.1891,4.3624 \
  then to 1.6001,4.2865 \
  then to 0.96501,4.3399 \
  then to 0.44272,4.5001 \
  then to 0.056941,4.8206 \
  then to 0.045026,5.0086 \
  then to 0.045026,5.0858

spline from 3.8271,6.1721 \
  then to 3.8628,6.1009 \
  then to 3.905,5.9969 \
  then to 4,5.5458 \
  then to 3.7863,4.9523 \
  then to 3.2063,4.3661 \
  then to 1.9215,3.9251 \
  then to 0.58988,4.0521 \
  then to 0.044699,4.503 \
  then to 0.050634,4.7463 \
  then to 0.06844,4.8531

define $upper_spline {
  spline from ($1, $2) \
  then right 0.007 down 0.3332 \
  then left 0.1543 down 0.4984 \
  then left 0.6935 down 0.6542 \
  then left 1.2701 down 0.3924 \
  then left 1.26535 up 0.1425 \
  then left 0.55673 up 0.4569 \
  then right 0.011871 up 0.2738 \
  then right 0.029675 up 0.0712
}

define $lower_spline {
  spline from ($1, $2) \
  then down 0.1662 \
  then left 0.0837   down 0.3372 \
  then left 0.3536   down 0.552 \
  then left 1.0683   down 0.5816 \
  then left 1.3295   down 0.1009 \
  then left 0.80027  up 0.2848 \
  then left 0.279745 up 0.3204 \
  then left 0.017929 up 0.1297 \
  then right 0.01187   up 0.0771
}

$lower_spline(3.9662, 5.5504)
$upper_spline(3.9662, 5.3427)
$lower_spline(3.9662, 5.0993)
$upper_spline(3.9662, 4.8797)
$lower_spline(3.9662, 4.6423)
$upper_spline(3.9662, 4.4287)
$lower_spline(3.9662, 4.1794)
$upper_spline(3.9662, 3.9776)
$lower_spline(3.9662, 3.7283)
$upper_spline(3.9662, 3.5206)
$lower_spline(3.9662, 3.2713)
$upper_spline(3.9662, 3.0636)
$lower_spline(3.9662, 2.8202)
$upper_spline(3.9662, 2.6125)
$lower_spline(3.9662, 2.3632)

line from 1.2499,6.6721 to 1.2499,6.7968

spline from 3.4043,5.8313 \
  then to 3.4676,5.6928 \
  then to 3.5032,5.5504

line from 0.18351,5.2972 to 0.18351,5.1745

spline from 3.9595,1.7281 \
  then to 3.9655,1.6687 \
  then to 3.9595,1.3423 \
  then to 3.7689,0.82214 \
  then to 3.3237,0.40668 \
  then to 2.8667,0.16334 \
  then to 2.7599,0.11586

spline from 3.9602,1.9122 \
  then to 3.9602,1.746 \
  then to 3.8766,1.4087 \
  then to 3.5229,0.85677 \
  then to 2.4546,0.27513 \
  then to 1.1693,0.17521 \
  then to 0.42739,0.41262 \
  then to 0.23153,0.57286 \
  then to 0.18405,0.62628

spline from 3.9543,2.1555 \
  then to 3.9612,1.8223 \
  then to 3.8069,1.324 \
  then to 3.1135,0.66976 \
  then to 1.8103,0.27611 \
  then to 0.7004,0.41262 \
  then to 0.23747,0.68563 \
  then to 0.18999,0.74498

spline from 3.9602,1.4552 \
  then to 3.9602,1.289 \
  then to 3.8766,0.95174 \
  then to 3.5255,0.40668 \
  then to 2.8489,0.038704 \
  then to 2.7539,0.0030933

Worm gear body

▶ 4:46

Create a sketch on the horizontal (xy) plane:


circle width 2;
line from last.w to last.e thin <-> "diameter - 2½ ✕ module" above

Extrude → Two Sides:


Above:		\( (length\_in\_teeth + 1) \times module \times \pi \)
Below:		\( module \times \pi \)