Formative or equal no. Of teeth for bevel gears? Tredgold’s Approximation?

Introduction:

The involute teeth for a spur gear may be generated by the edge of a plane as it rolls on a base cylinder.

Tredgold’s Approximation:

• A true section of the resulting involute lies on the surface of a sphere. But it is not possible to represent on a plane surface the exact profile of a bevel gear tooth lying on the surface of a sphere.

• Therefore, it is important to approximate the bevel gear tooth profiles as accurately as possible.

• The approximation (known as Tredgold’s approximation) is based upon the fact that a cone tangent to the sphere at the pitch point will closely approximate the surface of the sphere for a short distance either side of the pitch point, as shown in Fig. (a).

• The cone (known as back cone) may be developed as a plane surface and spur gear teeth corresponding to the pitch and pressure angle of the bevel gear and the radius of the developed cone can be drawn.

• This procedure is shown in Fig. (b).

Let θ_P = Pitch angle or half of the cone angle,
R = Pitch circle radius of the bevel pinion or gear, and
R_B = Back cone distance or equivalent pitch circle radius of spur pinion or gear.
Now from Fig.(b), we find that
 

R_B = R sec θ_P

We know that the equivalent (or formative) number of teeth,

where T = Actual number of teeth on the gear

Notes :

1. The action of bevel gears will be same as that of equivalent spur gears.
2. Since the equivalent number of teeth is always greater than the actual number of teeth, therefore a given pair of bevel gears will have a larger contact ratio. Thus, they will run more smoothly than a pair of spur gears with the same number of teeth.