# Establishment of mathematical model of real tooth surface of logarithmic spiral bevel gear

There are many ways to express the tooth surface equation of logarithmic spiral bevel gear, including explicit, implicit, parameter and vector. According to the collected discrete data point coordinates, the most feasible method to construct the logarithmic spiral bevel gear is to use the parameter equation. Because the parameter equation has more advantages than explicit and implicit, NURBS interpolation theory is used to study when establishing the real tooth surface mathematical model of logarithmic spiral bevel gear.

For the discrete data points of tooth surface collected by 3D laser scanner, the parametric change law of discrete data points of tooth surface must be determined first. To determine the interpolation curve of tooth surface, we must first give the data points and give the corresponding parameter values to make it a strict increasing sequence, that is, the segmentation of parameters. The specific curve parameter equation can be determined by a specific single parameter vector function, which not only determines the one-to-one correspondence between the points on the curve and the parameter values, but also has greater degrees of freedom to control the shape of the curve. Similarly, for the parameterization of the tooth surface of logarithmic spiral bevel gear, it not only determines the one-to-one correspondence between the points on the tooth surface and the values in the parameter domain, but also controls the shape of the tooth surface. However, the process of tooth surface parameterization is not unique.

According to the obtained control point coordinates of the interpolation surface, the spline basis function Ni, K (U) of B-spline surface can be obtained by substituting into the formula. According to the collected discrete data points, the data points can be obtained by NURBS interpolation theory and MATLAB programming. The tooth surface mathematical model of cubic B-spline interpolation of logarithmic spiral bevel gear is as follows:

Of which:

u. The parameter V of the spiral surface of the gear is expressed by the logarithm of the spiral surface.

The tooth surface reconstructed by Matlab is shown in the figure:

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