<p>This study proposes a method for quantitatively evaluating the characteristics of a gear shape deviation network by analyzing the eigenvalues and eigenmodes of the Graph Laplacian constructed from the network. Previous research represented each tooth of a gear as a node and the correlation coefficients between measured tooth helix deviations as edges, forming a tooth helix deviation network. By generating graph of the adjacency matrix in this network, it was possible to visually confirm that, for example, the gate arrangement of the injection molding mold has a significant effect on the accuracy of injection-molded plastic gears, However, a quantitative evaluation was not conducted. eigen Analysis of matrices, commonly used in linear algebra, provides a coordinate-independent representation of matrix properties and is applied in mechanical engineering for tasks such as principal stress derivation and vibration analysis. This study applies eigen Analysis of the Graph Laplacian derived from the tooth shape deviation network to quantitatively assess the characteristics of the network. To facilitate the interpretation of the eigen Analysis results of the tooth shape deviation network, a shift and scale operation is performed so that the correlation coefficients between tooth helix deviation curves, originally ranging from <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\pm 1\)</EquationSource> </InlineEquation>, are transformed into a range from 0 to 1. The Graph Laplacian matrix is then constructed from the adjacency matrix of this network, and eigen Analysis is conducted. This operation enables the evaluation of network characteristics by making the problem analogous to the eigenvalue problem of a one-dimensional mechanical vibration system, where the number of mass points is equal to the number of teeth, and each mass point is connected by springs with stiffness proportional to the correlation coefficients. Furthermore, by normalizing the Graph Laplacian, the issue of increasing maximum eigenvalues in proportion to the number of teeth is mitigated. Additionally, structural simplification of the network is performed to address the issue of complex higher-order eigenvector shapes. In this study, the proposed method is applied to the tooth helix deviation network of hobbed gears to investigate the impact of workpiece mounting errors on network characteristics. The results reveal that tooth helix slope deviations caused by workpiece errors lead to reduced modified correlation coefficients between opposing tooth on the base circle. By interpreting these correlation coefficients, it was found that the resulting vibration system exhibits changes in natural frequencies, and the curvature of the eigenvectors is altered at locations with low spring constants. These findings confirm the effectiveness of the proposed method.</p>

Eigen analysis of graph laplacian derived from gear shape deviation networks

期刊论文
发表日期：

2025年8月26日

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Shu Takata,
Yuichiro Seo,
Daisuke Iba,
Jing Chong Low,
Shunta Takahashi,
Naoki Yamashita,
Junichi Hongu

摘要

This study proposes a method for quantitatively evaluating the characteristics of a gear shape deviation network by analyzing the eigenvalues and eigenmodes of the Graph Laplacian constructed from the network. Previous research represented each tooth of a gear as a node and the correlation coefficients between measured tooth helix deviations as edges, forming a tooth helix deviation network. By generating graph of the adjacency matrix in this network, it was possible to visually confirm that, for example, the gate arrangement of the injection molding mold has a significant effect on the accuracy of injection-molded plastic gears, However, a quantitative evaluation was not conducted. eigen Analysis of matrices, commonly used in linear algebra, provides a coordinate-independent representation of matrix properties and is applied in mechanical engineering for tasks such as principal stress derivation and vibration analysis. This study applies eigen Analysis of the Graph Laplacian derived from the tooth shape deviation network to quantitatively assess the characteristics of the network. To facilitate the interpretation of the eigen Analysis results of the tooth shape deviation network, a shift and scale operation is performed so that the correlation coefficients between tooth helix deviation curves, originally ranging from \(\pm 1\) , are transformed into a range from 0 to 1. The Graph Laplacian matrix is then constructed from the adjacency matrix of this network, and eigen Analysis is conducted. This operation enables the evaluation of network characteristics by making the problem analogous to the eigenvalue problem of a one-dimensional mechanical vibration system, where the number of mass points is equal to the number of teeth, and each mass point is connected by springs with stiffness proportional to the correlation coefficients. Furthermore, by normalizing the Graph Laplacian, the issue of increasing maximum eigenvalues in proportion to the number of teeth is mitigated. Additionally, structural simplification of the network is performed to address the issue of complex higher-order eigenvector shapes. In this study, the proposed method is applied to the tooth helix deviation network of hobbed gears to investigate the impact of workpiece mounting errors on network characteristics. The results reveal that tooth helix slope deviations caused by workpiece errors lead to reduced modified correlation coefficients between opposing tooth on the base circle. By interpreting these correlation coefficients, it was found that the resulting vibration system exhibits changes in natural frequencies, and the curvature of the eigenvectors is altered at locations with low spring constants. These findings confirm the effectiveness of the proposed method.