Tooth surface error and sensitivity matrix of hypoid gear

According to the hypoid gear model, the sensitivity of tooth surface errors at different positions to various machining parameters is analyzed, and the sensitivity function of each position of tooth surface relative to different machining parameters is accurately calculated and fitted, and then the accurate nonlinear sensitivity function matrix of tooth surface error is obtained, so as to take into account the efficiency and accuracy of reverse processing parameter adjustment during tooth surface modification.

Tooth surface error refers to that the actual tooth surface deviates from the theoretical design tooth surface (standard tooth surface) due to processing, deformation and other factors. It is generally expressed by the normal deviation of each point on the tooth surface, as shown in the figure.

The figure shows the tooth surface error in normal direction of a point on the theoretical tooth surface of complex curved surface gear. Point m and point m ‘represent the corresponding two points on the theoretical tooth surface and the actual tooth surface respectively, then points R and R’ are the position vectors of M and point m ‘, and are the normal vectors of M points on the theoretical tooth surface. The tooth surface error of this point can be expressed by the projection of the vector difference between two points of the corresponding position on the normal vector n, which is denoted as δ

The tooth surface error can be measured by selecting a finite number of points uniformly on the theoretical tooth surface. The degree of deviation of the actual tooth surface from the theoretical tooth surface can be shown by calculating the tooth surface error of the finite points. In order to discuss the sensitivity of tooth surface error to gear machining parameters, suppose that there are a total of P gear processing parameters, and the variation of the j th gear parameter is Δ RJ. A total of Q tooth surface points are selected to analyze the tooth surface error. The tooth surface error value is recorded as δ I (I = 1,2,…, q). It is assumed that the tooth surface error can be expressed in matrix form:

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