^{1}

^{*}

^{2}

^{2}

A novel CAD method for tooth profile based on the gearing feature has been proposed in this paper, and this method could be applied to the variable transmission ratio and three-dimension situation. The two pitch curves of the two gears can be generated due to the given transmission ratio. The tooth profile curves are formed in each coordinate system of its pitch curve by the gearing process. Finally, the tooth profile could be extracted and translated into a real dimension by CAD method. It is provided a simply thought for tooth profile design to avoid so much complicated mathematical reasoning.

Bevel gearings are used in the situation of driving between the unparallel axes and have a wide application field. Because the gearing motion is so complex, the mathematical theory is very deep and the calculations are multifarious [

At first, based on the gearing feature, a pair of pitch curves (from gear 1 and 2) are generated. Tooth profile 1 can be arbitrarily drawn around the pitch curve 1; tooth profile 2 is uniquely formed by the motion of the tooth profile 1 along the pitch curve 2. The motion of the tooth profile 1 divides the space into two regions (scanned by gear 1 and not scanned by gear 1) and their boundary line. The region not scanned by gear 1 can be the region where gear 2 exists; otherwise the tooth profile 2 should be the boundary line. The boundary line could be extracted by digital image processing [

The aim gears to be design are a pair of bevel gearings whose axes are vertical to each other as shown in

The transmission ratio:

where:

i is the transmission ratio;

c is adjusting constant;

The mathematical relationship between t_{1} and

The pitch curves are two fictitious closed curves on the two gears which have pure rolling behavior. Suppose that gear 2 (the axis is OY) lives in coordinate system {O} and don’t move, on the other hand, gear 1 lives in coordinate system {A}, and its rotating axis is OA. OA contrarotates based on OZ axis in XOY plane (see from

The {O} coordinate of a point K on the pitch curve in {A} is {x, y, z}. After gearing

where:

According to the meanings of pitch curve:

According to Equation (2):

where:

Because:

So, the equations have infinite solutions:

when

where transmission ratio

The intersecting line G2 between the bevel surface of Equation (8) and spherical surface

is the pitch curve of gear 2 in {O}. The equations of the intersecting line G2 are:

The equations of the pitch curve G1 of gear 1 in {A} can be also obtained:

The pitch curves G1 and G2 is shown in

When the transmission ratio is given, the two pitch curves are determined uniquely. However, the tooth profile curves of the two gears are not unique. When one is given, the other is determined. We can design a three- dimension curve based on G 1’ s pitch curve to be its tooth profile curve:

where:

So, G 1’ s tooth profile curve S1 is shown in

Suppose that on the spherical surface, the S1 rolls purely along the pitch curve G2 (G1 and G2 are tangent), then the region where S1 doesn’t sweep should be the region where gear 2 exist. The black region where is swept by S1 (see from

The black region (in

OA and OB are calibration lines to translate pixel size into real size; the color region is composed by civil points and the white region is composed by outside points.

We provide (this provision really makes sense) that these points are boundary points: if there is at least an “outside point” on the position “a, c, e, g” of point M as shown in

(a, b, c, d, e, f, g, h) of P1 (see in

The flow of the ordinal extraction of boundary line is shown in

The tooth profile S2 which has been extracted is shown in

where xp, yp are pixel coordinate; x, y are real coordinate;

Will the two gears run steadily? Will their profile curves leave each other? These are not in the scope of this paper

(which could be known from [

For example, as shown in

In the three-dimension situation, S2 is a space curve on the spherical surface (the “S 2” mentioned before is its projection on XOY). Given a radius “a” (from Equation (8’)), there will be a pair of gears S1 and S2, as shown in

In this paper, a novel design method of space bevel gearing based on gearing feature is proposed. The two pitch curves are generated based on the transmission ratio, and then a tooth profile could be designed based on its pitch curve independently. According to its motion, the second tooth profile could be found and translated into a real size by image processing technology. Comparing to complicated method [

This work was supported by China Post-Doctoral Foundation No. 2012M520572, Tianjin Municipal Education Commission Grant No. 20120401, and Tianjin Municipal Science and Technology Commission Key Grant No. 14JCZDJC39500.

None.

NanYang,DaweiZhang,YanlingTian, (2015) Tooth Surface Design for Variable Transmission Ratio Bevel Gearing. Applied Mathematics,06,1685-1695. doi: 10.4236/am.2015.610150