# Drawing and meshing involute bevel gears in CAD

1. Feb 28, 2015

### Jarfi

Hello, I urgently need to be able to draw a set of bevel gears for 3d printing. The gears I am currently drawing intersect that is their teeth are penetrating each other when the gear turns.

Currently using http://www.daycounter.com/Calculators/Bevel-Gear-Calculator.phtml to find the numbers, However I cannot seem to find any straightforward information on how to draw correctly meshing teeth.

Is there some involute bevel gear tutorial or text out there, which I can use to create a real CAD drawing of actually meshing involute gear teeth?

2. Mar 1, 2015

### Baluncore

Try these attached files. Ask more specific questions if you want more.

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3. Mar 1, 2015

### Baluncore

Here is some old code written in FreeBASIC, (very similar to MS VB). You can download FB for free.
It will generate the involute profile needed when given fundamental parameters.
Extract the parts you need to generate your gear.
Code (Text):

'=======================================================================
' Generate an involute spur gear from an ideal rack or hob.
' only the general situation without correction of undercut is modelled
'=======================================================================
Dim As Double MM = 3.5  ' MM = 25.4 / DP
Dim As Double PA = 20.  ' pressure angle in degrees
Dim As Integer n = 57 '19.  ' number of teeth

'=======================================================================
Const As Double Pi = 4 * Atn(1)
Const As Double DtoR = Atn(1) / 45
Dim As Double PD = n * MM  ' pitch diameter
Dim As Double BD = PD * Cos(PA * DtoR)  ' base diameter
Dim As Double ad =  1.00 * MM  ' addendum
Dim As Double dd =  1.25 * MM  ' dedendum
Dim As Double OD = PD + 2.00 * MM  ' outside diameter
Dim As Double RD = PD - 2.50 * MM  ' root diameter
Dim As Double CP = Pi * PD / n  ' circular pitch
Dim As Double min = 2/Sin(PA*DtoR)^2  ' minimum n without undercut
Dim As Double BL = 0.04 * MM  ' recommended backlash

'=======================================================================
Print Using "  $.$ metric module"  ; MM
Print Using "  .# pressure angle in degrees"; PA
Print Using "  # number of teeth"  ; n
Print Using ".$# pitch diameter" ; PD Print Using "$$.$# base diameter"  ; BD
Print Using ".$# addendum" ; ad Print Using "$$.$# dedendum"  ; dd
Print Using ".$# outside diameter" ; OD Print Using "$$.$# root diameter"  ; RD
Print Using ".$# circular pitch" ; CP Print Using "$$.$# undercut limit"  ; min
Print Using ".$# backlash" ; BL ' half to each gear ? If min > n Then Print " WARNING: generated teeth will be undercut. " '======================================================================= Dim As Double Rp = PD * 0.5 ' pitch radius Dim As Double Ra = OD * 0.5 ' outer tip radius, Rp + addendum Dim As Double Rb = BD * 0.5 ' base radius Dim As Double Rr = RD * 0.5 ' root radius '----------------------------------------------------------------------- ' find gamma of the pitch point Dim As Double t = Sqr(Rp*Rp - Rb*Rb) / Rb ' pitch point parametric value Dim As Double ux = Cos(t) Dim As Double uy = Sin(t) Dim As Double px = Rb * ( ux + t * uy ) Dim As Double py = Rb * ( uy - t * ux ) print using "gamma for pitch point is$.$# "; Atn(py/px) / DtoR print using "pitch point x=$#.$# y=$#.$#"; px; py ' Circle(px, py), .27, 13,,,,F dim as double tooth = 2 * Pi / n print using "Each tooth is$#.### degrees"; tooth * 180 / Pi

for r as double = Rb to Ra + 1e-6 step (Ra - Rb) / 15  ' n+1 points
Dim As Double t = Sqr(r*r - Rb*Rb) / Rb
Dim As Double ux = Cos(t)
Dim As Double uy = Sin(t)
Dim As Double px = Rb * ( ux + t * uy )
Dim As Double py = Rb * ( uy - t * ux )
print px, py
next r

'=======================================================================
Screenres 1000, 1000, 4
' Window (0.4*Ra, -Ra/2) - (1.2*Ra, Ra/2)
Window (0.4*Ra, -Ra/4) - (1.2*Ra, Ra/4)
Line (-2, 0)-(Ra, 0), 6
Line ( 0,-2)-( 0, 2), 6

'-----------------------------------------------------------------------
' Line ( Ra,-3)-( Ra, 3), 6
'Circle (0,0), Ra, 6
'Circle (0,0), Rp, 6
'Circle (0,0), Rb, 6
'Circle (0,0), Rr, 6

' draw all reference circles
For t As Double = 0 To 2 * Pi Step Pi * .01 / Ra
Dim As Double ux = Cos(t)
Dim As Double uy = Sin(t)
Pset ( Rr*ux, Rr*uy), 7
Pset ( Rp*ux, Rp*uy), 7
Pset ( Rb*ux, Rb*uy), 7
Pset ( Ra*ux, Ra*uy), 7
Next t

'-----------------------------------------------------------------------
Draw String (Rr, 0.2), " Root" , 7
Draw String (Rb, -.1), " Base" , 7
Draw String (Rp, 0.2), " Pitch", 7
Draw String (Ra, -.1), " Tip"  , 7

'=======================================================================
' draw the tooth profile
' x = Rb * Cos(a) + a * Sin(a)
' y = Rb * Sin(a) - a * Cos(a)
Dim As Integer k = 14
Dim As Double Ta = Sqr(Ra*Ra - Rb*Rb) / Rb  ' T at involute crossing addendum
Dim As Double Tp = Sqr(Rp*Rp - Rb*Rb) / Rb  ' T at pitch point on involute
For t As Double = 0 To Ta Step .001  ' t is the parametric variable
Dim As Double ux = Cos(t)
Dim As Double uy = Sin(t)
Dim As Double px = Rb * ( ux + t * uy )
Dim As Double py = Rb * ( uy - t * ux )

If t > Tp Then k = 13
Pset(px, py), k

Next t

'-----------------------------------------------------------------------
pset (Rb, 0), 0
for r as double = Rb to Ra + 1e-6 step (Ra - Rb) / 15  ' n+1 points
Dim As Double t = Sqr(r*r - Rb*Rb) / Rb
Dim As Double ux = Cos(t)
Dim As Double uy = Sin(t)
Dim As Double px = Rb * ( ux + t * uy )
Dim As Double py = Rb * ( uy - t * ux )
line -(px, py), 15
Circle(px, py), .07, 13,,,,F
next r

'=======================================================================
Sleep
'=======================================================================

4. Mar 1, 2015

### Jarfi

Okay thank you for that info, that's a lot about involute profiles. But say I have a bevel and spur gear with the same teeth ratio, teeth numbers and gear width etc , will I be using the same profile/involute equation for bevel gears and spur gears? that is can this be applied to bevel gear teeth

5. Mar 1, 2015

### Baluncore

Yes. Spur gears are cylindrical, bevel gears are conical. The projected apex of two meshed conical gears coincide. Lines drawn from the common apex point to a spur gear involute profile define the surface of a bevel gear.

6. Mar 2, 2015

7. Mar 2, 2015

### Baluncore

@ Jarfi.
Are your bevel gears cut straight, spiral, skew or hypoid?
What is the angle between the two bevel gear shafts?
What is the gear tooth count ratio?

8. Mar 2, 2015

### Jarfi

It's a straight bevel gear, 90° angle. Gear tooth ratio is large, maybe 1:10. Pinion is 1.6cm and bevel 16cm diameter with a standard 20° Pressure angle.

9. Mar 2, 2015

### Baluncore

Attached are four pages from "Dudley's Gear Handbook", 2'nd Edn, by D.P.Townsend.

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10. Mar 2, 2015

### Jarfi

That's a lot of information, saved if I'm getting serious

Just did some research anyways and it seems that Autocad has a gear development tool, which will create the teeth curves for bevel gears and calculate everything from max power to max RPM. Managed to make meshing internal spur gears with it which was nice, you can even choose the backlash in mm, insane.