# Calculation of initial clearance of spiral bevel gear tooth surface

Spiral bevel gear is point contact in theory. After bearing, the contact area becomes surface contact due to elastic deformation. The elastic contact finite element method is to calculate the size and shape of the contact area. Therefore, we must first calculate the theoretical meshing point, the initial clearance of the tooth surface in the field and the unit normal vector direction of each point. The initial tooth surface clearance is the distance between the corresponding pairs of possible contact points on the meshing tooth surface. When using the numerical calculation method to solve the contact problem, whether the finite element method, boundary element method or other methods, the initial gap and element division of the contact area and nearby point pairs of two contact bodies are relatively strict, because this is the key to the convergence of the iterative solution of the contact boundary.

The previous methods to solve the initial clearance between the contact point and the tooth surface are to use the node on the tooth surface Ω 1 as the parallel line of the unit normal vector of the theoretical meshing point and intersect with the tooth surface Ω 2, that is, the corresponding node of the contact point pair on the tooth surface Ω 2. The contact problem of involute gear with convex tooth surface along the tooth height direction can be used as an approximate calculation method. In order to improve the calculation accuracy of the initial clearance of the tooth surface, it is proposed to divide the elements on the tooth surface Ω 1 and calculate the coordinate values of each node. Through the tooth surface Ω 1 node, make the intersection with the tooth surface Ω 2 along the direction of the external normal vector of the current node. The intersection point is the corresponding contact point to node on the tooth surface Ω 2, that is, the current external normal vector method. At the same time, the initial clearance of the corresponding point pair can be obtained.

According to this method, firstly, the position vector of any node i (1) of tooth surface Ω 1 is R1, the outer normal vector is N1, and the initial clearance is δ, The position vector R2 of the possible contact point I (2) on the tooth surface Ω 2 is the quantity to be calculated. As shown in Fig. 1, the corresponding relationship between nodes i (1) and I (2) can be established as follows:

This is a set of three-dimensional nonlinear equations, which can be solved by quasi Newton method to obtain the initial clearance of the possible contact points in and near the tooth surface contact area of spiral bevel gear, and the initial clearance of the possible contact points when the gear pair meshes at the pitch cone. The results are shown in figure 2. 