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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