Hello friends, in this post I want to share: Calculation of the Metric Helical Bevel Gear.
This is a topic that has been asked frequently on my YouTube channel and despite not having had the opportunity to manufacture a gear with these characteristics, I decided to task of consulting, understanding, and explaining this topic at least the theoretical part, and in this post, I want to convey it to you. So without further ado, let’s start!
Known Data:
m: Metric Gear Module.
z: Number of Teeth.
α: Pitch Angle.
L: Face Width.
Ω: Gear Helix Angle.
Cl: Chordal thickness of the cutter on the pitch circle, at the larger end of the gear.
Cs: Chordal thickness of the cutter tooth on the pitch circumference, at the smaller end of the gear.
Formulas for Bevel Gear
These are the machining formulas for the Bevel Gear:
Apparent module (am) am = m / cos Ω
Pitch diameter (pd) pd = am * z
Larger outside diameter (Lod) Lod = pd + (2 * m * cos α)
Cone distance (E) E = pd / (2 * sin α )
Larger inside diameter (Lid) Lid = pd – (1.157 * 2 * m * Cos α)
Smaller outside diameter (sod) sod = Lod (E – L) / E
Tooth height (h) h = 2.167 * m
Angle corresponding to the module (β) tan β = m / E
Angle corresponding to the bottom of the toothing clearance (β’) tan β’ = 0,157 (m / E)
Half angle of the face cone (Δ) Δ = α + β
Dividing Head Tilt Angle (ϗ) ϗ = α – (β + β’)
Lead (L) L = π * pd / tan Ω
EXAMPLE:
Make the Calculation of the Metric Helical Bevel Gear with the following data:
Metric Module (m) = 4
Number of teeth (z) = 16
The angle at the center of the gear (α) = 37° 20′
Face Width (L) = 19 millimeters
Gear helix angle (Ω) = 30°
The chordal thickness of the cutter on the pitch circle, at the larger end of the gear (Cl) = 5.8 millimeters
The chordal thickness of the cutter tooth on the pitch circumference, at the smaller end of the gear (Cs) = 4.3 millimeters
These last two values: Cl and Cs are taken from the respective cutters. I will talk about this topic later.
Now let’s go with the calculations:
In the first instance, we will calculate everything related to the geometry of the bevel gear, which does not differ much from the metric bevel gear with straight teeth.
Apparent module (am) am = m / cos Ω am = 4 / cos 30° am = 4 / 0.86603 am = 4.618
Pitch diameter (pd) pd = am * z pd = 4.618 * 16 pd = 73.90 millimeters.
Lead (L) L = π * pd / tan Ω L = (3.1416 * 73.9) / tan 30° L = 232.164 / 0.57735 L = 402.119 millimeters
With these data, it is possible to carry out the turning part, the inclination of the dividing head, and search for the series of gears in the banjo to mount on the universal milling machine.
As a second step, we go with the calculation of the angle of rotation of the dividing head and the deviation for the correct cutting of the teeth of the helical bevel gear.
Number of teeth of the equivalent helical gear (eZ) eZ = Pd / (m * Cos² Ω) eZ = 92.9 / 4 * Cos ² 30° eZ = 92.9 / 4 * 0.86603 ² eZ = 92.9 / 4 * 0.75 eZ = 92.9 / 3 eZ = 30.96 For the Number of teeth of the equivalent helical gear Zf, cutter number 5 corresponds
The apparent pitch at the larger end of the gear (ap) ap = π * m / Cos Ω ap = 3.1416 * 4 / cos 30° ap = 3.1416 * 4 / 0.86603 ap = 12 .5664 / 0.86603 ap = 14.5103
Pitch taper radius at the larger end of the gear (TR) TR = pd / (2 * Sin α) TR = 73.90 /(2 * Sin 37° 20′) TR = 73.90 /(2 * 0.60645) TR = 73.9 / 1.2129 TR = 60.928
Here is a small parenthesis. The value “Cl” cordal thickness of the cutter on the pitch circle, at the larger end of the gear, is taken in the milling cutter with a special gauge, or it can be done approximately with a conventional vernier, and in this case, it is done in the milling cutter module 4, and the value is 5.8 millimeters,
In the same way, for the chordal thickness of the cutter on the pitch circle, at the smaller end of the gear.: “Cs” is made in the cutter module 2.75 that We calculated in the point: “cutter Module and that gave a value: of 2.75 and whose value is: 4.3 millimeters.
The rotation angle of the gear (ra) ra = (57.3 / pd)*((ap/2) – (TR / (L * Cos Ω) * (Cl – Cs))) ra = 57.3 / 73.9 ((14.51/2) – (60.92/19 * Cos 30°)(5.8 – 4.3)) ra = 0.775 (7.25 – (60.92/19 * 0.86603) * 1.5) ra = 0.775 (7.25 – (60.92/16.45) * 1.5) ra = 0.775 (7.25 – (3.703 * 1.5)) ra = 0.775 (7.25 – 5.55) ra = 0.775 (1.7) ra = 1.31
Divisions in the dividing head of the milling machine: (Dd) Dd = 40 / z Dd = 40/16 Dd = 2 turns, 1 hole in the plate of 2 holes. Now amplifying the fraction by 14 times Dd = 2 turns, 14 holes in the plate of 28 holes.
Number of degrees for a full rotation of the dividing head (Nd) Nd = 9°
Dividing head holes (dhh) dhh = q * (ra / Nd) dhh = 28 * (1.31 / 9°) dhh = 28 * 0.145 dhh = 4.06 holes or 4 holes of the dividing head plate.
With this information, I hope to have explained the best way the features, concepts, and information about the Calculation of the Metric Helical Bevel Gear
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