Mathematical Modeling and Generation Principle of Skew Bevel Gears

There is assumed a sphere which is setting radius dimension as R and a base cone which

culmination and the sphere center coincide, as shown in figure1. The base cone and the great circle

of a sphere-a plane Q are tangent to a line. When the base cone is fixed, the plane Q rotate around

the base cone and they keep tangency all along. Under the condition of the above expression, the

movement of the plane Q relative to the base cone is what is needed pure roll for generating

spherical involute. If there is a straight line ML on the plane Q. In the expansion, the straight line ML

can spread a curved surface, while the curved surface is then involute helicoids. As shown in figure.

2. Base cone angle δb generatrix of small end about base cone Rb base cone circle spreading angle φ

plane Q spreading angle γ. Fixed coordinate systems S(o-x,y,z) are rigidly connected to the gear

blanks. In this systems, origin o1 and x1-y1 plane coincide with center of cone and bottom of base

cone respectively; At the same time, Matrix M1t provides the coordinate transformations from

system S1 into the top coordinate system St(ot-xt,yt,zt) by offsetting A along the axis z1; And matrix

Mts defines the relation between the stationary coordinate systems St and Ss(os-xs,ys,zs), in which

coordinate systems Ss rotate φ on the axis zt ; Afterward Matrix Msb performs the coordinate

transformations from system Ss into the top coordinate system Sb(ob-xb,yb,zb) by moving π/2-δb

around the axis ys, at the same time axis xb and origin ob coincide with the line of base cone and the

plane Q respectively Finally, Mbq denote the transformations from Sb into Sq(oq-xq,yq,zq) by

revolving γ around zb connected to the plane Q.

In the spreading process, a cluster of tracing line that are formed by generation line ML make up

involute helicoids. In the coordinate systems S, the value of the generation line ML can be acquired

by Coordinate Transform from coordinate systems. When parameter φ continuous change, In the

coordinates system S (o-x,y,z) the mathematical model of a point on the Tooth-surface can be

performed by applying the following equation:

When big circle (plane Q) pure roll, the circular arc that base cone bottom circle is bypassed by big

circle is same length. In the Cartesian coordinate system, the vector equation of tooth surface about

circle spreading angle φ and polar Radius ρ is shown :

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