Designing parametric bevel gears with Catia V...

mrhg 2010-12-18

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Designing parametric
bevel gears with Catia V5

Published at http://gtrebaol./doc/catia/bevel_gear.html
Written by Gildas Trébaol on June 25, 2005.
Zipped parts: bevel_gear.zip (340 KB).
VRML97 model: bevel_gear.wrl (58 KB).

The knowledge used for designing spur gears can be reused for making bevel gears

This tutorial shows how to make a basic bevel gear that you can freely re-use in your assemblies.

1 Sources, credits and links

The conventional formulas and their names in French come from the page 100 of the book
"Précis de construction mécanique" by R. Quatremer and J.P. Trotignon, Nathan publisher, 1983 edition.
I found a clear explanation of bevel gears in the pages 258 to 280 of the book
"Les mécanismes des machines y compris les automobiles" by H. Leblanc, Garnier publisher, 1930 edition.
For an exhaustive analysis, we could also use the famous old book "Les engrenages" written by Mr Henriot.

The principle for designing a bevel gear consists in drawing two primitive conical surfaces:

The front cone, parallel to the edges of the teeth.
The rear cone, used for designing the profile of a tooth.

The half angle delta of the front cone depends on:

The module m.
The number of teeth of the gear Z1.
The number of teeth of the other gear Z2.
The angle between the axis of the two gears.

In most applications using bevel gears, the angle between the axis of the two gears is equal to π/2.
In that case, the half angle delta of the front cone is defined by the formula:

delta = atan( Z1 / Z2 )

2 Table of gear parameters and formulas

The following table contains:

The parameters and formulas used for standard spur gears.
The specific parameters and formulas added for bevel gears (in the cells colored in pink).

#	Parameter	Type or unit	Formula	Description	Name in French
1	a	angular degree	20deg	Pressure angle: technologic constant (10deg ≤ a ≤ 20deg)	Angle de pression.
2	m	millimeter	—	Modulus.	Module.
3	Z1	integer	—	Number of teeth (11 ≤ Z1 ≤ 200).	Nombre de dents.
4	Z2	integer	—	Number of teeth of the complementary bevel gear.	Nombre de dents de la roue conique complémentaire.
5	delta	angular degree	atan( Z1 / Z2 )	Half angle of the front primitive cone.	Demi angle au sommet du cône primitif avant
6	ld	millimeter	—	Length of the teeth on the front primitive cone.	Longueur des dents sur le cone primitif avant.
7	ratio		1 - ld / ( lc * cos( delta ) )		pour calculer les homothéties du flanc intérieur
8	dZ	millimeter	0mm	Translation offset of the generative geometry on the Z axis.	Décalage des constructions géométriques suivant l'axe Z.
9	p	millimeter	m * π	Pitch of the teeth on a straight generative rack.	Pas de la denture sur une crémaillère génératrice rectiligne.
10	e	millimeter	p / 2	Circular tooth thickness, measured on the pitch circle.	Epaisseur d'une dent mesurée sur le cercle primitif.
11	ha	millimeter	m	Addendum = height of a tooth above the pitch circle.	Saillie d'une dent.
12	hf	millimeter	m * 1.25	Dedendum = depth of a tooth below the pitch circle.	Creux d'une dent.
13	rp	millimeter	m * Z / 2	Radius of the pitch circle.	Rayon du cercle primitif.
14	rc	millimeter	rp / cos( delta )		Rayon du cône primitif arrière
15	ra	millimeter	rp + ha	Radius of the outer circle.	Rayon du cercle de tête.
16	rf	millimeter	rp - hf	Radius of the root circle.	Rayon du cercle de fond.
17	rb	millimeter	rc * cos( a )	Radius of the base circle.	Rayon du cercle de base.
18	rr	millimeter	m * 0.38 = "arc cercle fond" * 0.7763	Radius of the root concave corner. (m * 0.38) is a normative formula.	Congé de raccordement à la racine d'une dent. (m * 0.38) vient de la norme.
19	t	floating point number	0 ≤ t ≤ 1	Sweep parameter of the involute curve.	Paramètre de balayage de la courbe en développante.
20	tc	angular degree	-atan( yd( a / 180deg ) / xd( a / 180deg ) )	Trim angle used to put the contact point in the ZX plane.	Angle d'ajustement pour placer le point de contact dans le plan ZX.
21	xd	millimeter	rb * ( cos(t * π) + sin(t * π) * t * π )	X coordinate of the involute tooth profile, generated by the t parameter.	Coordonnée X du profil de dent en développante de cercle, généré par le paramètre t.
22	yd	millimeter	rb * ( sin(t * π) - cos(t * π) * t * π )	Y coordinate of the involute tooth profile.	Coordonnée Y du profil de dent en développante de cercle.

1 First attempt: a simple projection on the rear primitive cone

This view shows that the whole geometry must be rebuilt, because the simple projection on a cone implies interferences between the root circles:

2 Projection of the involute on the rear primitive cone

Now, the tooth is actually designed on a cone:

The involute is still designed on the XY plane.
Then it is projected on the rear primitive cone.
The root circle and outer circle are defined in planes orthogonal to the axis of the cone.
The tooth profile is made with "cut and assemble" operations on the root circle,
the projection of the involute curve on the cone, and the outer circle.
The whole profile is a circular repetition around axis of the cone.
The profile is good, but it has a major drawback: the axis of the cone (in red) is not parallel to X, Y or Z (in green).

3 Designing the involute curve on an inclined plane

In order to make the gear aligned with the Z axis (shown in green), the involute curves is designed on an inclined plane (shown in red):

4 Making the tooth profile

The inner tooth profile is generated by a scale operation on the outer tooth profile.
The scale factor is computed by the ratio between the length of the front cone and the length of the teeth:
ratio = 1 - teeth_length / front_cone_length.
The tooth is generated by a multi-section surface, guided by 2 line segments
connected to the end points of the outer tooth profile and innner tooth profile.
The whole profile is a circular repetition of the tooth profile around the Z axis.
Now the teeth surface is ready, but the generation parameters are not well defined yet.

5 Making the outer and inner side cones

On most bevel gears, the teeth are delimited by an exterior cone and an interior cone. In order to build these cones:

The tooth profile is duplicated on the whole circle.
That profile is then used for cutting the rear cone.
The remaining part of the rear cone makes the outer side of the teeth.
The inner side is made by a scale-down operation on the outer side surface.
Then we can merge the inner side cone, the teeth surfcaces and the outer side cone.
The resulting surface can be converted to a solid body in the Mechanical Part workshop.

6 Checking and improving the robustness of the theet surface

The parametric gear show in the previous section fails when the delta angle is greater than 70degrees.
After hacking some parameters, the following image shows an improved extreme geometry:

Minimal number of teeth Z1 = 11.
Maximal delta angle = 79degrees.

7 Checking the generation of the side surface

We do the same work on the generation of the side surfaces.
Of course, this geometry should never be used in a real mechanism.

8 Putting the primitive cones in a separate group of surfaces

Now that the gear design is completed, we can put the fundamental geometric elements in a separate group and display them in green.

The following image show the rotation axis, the primitive circle, the front and rear primitive cones.
These elements can be are useful for checking the position of the bevel gears in the mechanical assembly workshop.

9 Flat gear

This figure shows a gear generated with the widest front cone:

10 Normal gear

On the opposite, we can check that we go back to the ordinary spur gear when the delta angle tends to zero:

11 Check if the curved surfaces could be simplified

The final bevel gear file is large: 950 KB for 13 teeth.
So we can wonder if the file could be smaller with simpler surfaces.
In order to check that, I replace all the surfaces generated by circles, arcs or involute curves with surfaces generated by straight lines.
The file size only decreased to 890KB, so the curved surfaces of the bevel gear are not worth being simplified.