Identification of Tool and Machine Settings for Hypoid Gear Based on Non-Uniform Discretization

Identifying the tool and machine settings of tooth surfaces in hypoid gears is challenging, considering the highly model nonlinearities and the ill-conditioned Jacobian matrix. To tackle these problems, we propose a novel identification model based on non-uniform discretization for hypoid gear, with the goal of efficiently obtaining accurate design parameters. The model employs a non-uniform discretization scheme for the tooth surface, approximating the quadrature of the surface variation using the Gaussian rule. This scheme is based on the Chebyshev node, which better captures gradient variation of surface variation and provides more accurate quadrature results than a uniform grid of the same size. The fundamental analysis of the problem characteristics is performed through the condition number of the Jacobian matrix, and numerical stability is guaranteed using the non-uniform discretization and fixing non-influential variables. Finally, a numerical example is presented, and the simulations in variations scenarios are conducted to validate the proposed model. The results demonstrate that the model guarantees both identification accuracy and efficiency, with outcomes aligning with the expectations based on condition number analysis.