Relative differential method of spiral bevel gear meshing

Let s (T) be a surface moving continuously in space, u and V are its surface parameters. When expressed by vector function R, it is not only a function of u and V, but also a function of time t. If we establish a coordinate system that is fixed to the surface s (T) and moves with it σ( t)={o(t); I (T), J (T), K (T)}, then the coordinate system origin o (T), coordinate vectors I (T), J (T), K (T) are all functions of time t. Let the surface be opposite σ( t) The coordinates of are (x (U, V), y (U, V), Z (U, V)), then the equation of moving surface s (T) can be written as:

Find the total differential of the formula by differential method, and make:

Then there are:

Let the angular velocity of the moving surface s (T), that is, the moving coordinate system σ( t) The angular velocity of is ω( t) , then we can see that the motion speed of the endpoint of the coordinate vector is:

Bring them into the formula and note:

Then the formula can be written as:

We call D 1R a vector r with respect to the moving coordinate system σ( t) The relative differential of, that is, the assumed surface s (T) and the motion coordinate system σ( t) Differential at rest. The formula gives the relationship between absolute differential DR and relative differential D1R. We call the formula relative differential formula.

The formula holds not only for the surface equation, but also for the surface normal vector. set up

The normal vector of moving surface s (T) is:

This differential method of vector function is called relative differential method. It is a powerful tool to study gear meshing problems. All the formulas of surface theory in differential geometry can be used in gear meshing only by relative differential representation.