CATIA齿轮

Designing parametric bevel gears with Catia V5 Published at http://gtrebaol.free.fr/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 \cis 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 \canismes des machines y compris les automobiles\1930 edition.

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