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<p>The S-shaped test piece is an effective workpiece for evaluating the machining accuracy and dynamic performance of five-axis CNC machine tools. Addressing the lack of precise mathematical models and systematic geometric analyses for the third-version S-shaped test piece specified in ISO 10791-7:2020, this study constructs a set of cubic quasi-uniform rational B-spline (QURBS) basis functions via the de Boor-Cox recurrence formula for the first time. A parametric ruled surface model integrated with control vertex coordinates is established. Model analysis reveals four curvature peaks on the upper and lower directrices of both ruled surfaces A and B, with a maximum curvature of 0.032&#xa0;mm⁻¹ for surface A and 0.042&#xa0;mm⁻¹ for surface B. The opening/closing angles exhibit two dynamic transitions within the 300–400&#xa0;mm arc length interval. A novel twist angle calculation formula based on normal vector projection angles is proposed, revealing maximum twist angles at the Z = 0&#xa0;mm directrix: 3.794° for ruled surface A and 3.288° for ruled surface B. By introducing a concavity/convexity criterion for the lower directrix, the machining principle error computation is optimized, confirming a positive correlation between tool-workpiece contact length and machining principle error.</p>

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Research on mathematical modeling and geometric characteristics of the S-shaped fillet

  • 期刊论文

  • 发表日期：

    2026年1月3日

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  • Xiaoxiang Wang,
  • Dan Zhang,
  • Yuli Sun,
  • San Li,
  • Qiang Guan,
  • Jijian Wei,
  • Yuancan Zhang,
  • Dunwen Zuo

摘要

The S-shaped test piece is an effective workpiece for evaluating the machining accuracy and dynamic performance of five-axis CNC machine tools. Addressing the lack of precise mathematical models and systematic geometric analyses for the third-version S-shaped test piece specified in ISO 10791-7:2020, this study constructs a set of cubic quasi-uniform rational B-spline (QURBS) basis functions via the de Boor-Cox recurrence formula for the first time. A parametric ruled surface model integrated with control vertex coordinates is established. Model analysis reveals four curvature peaks on the upper and lower directrices of both ruled surfaces A and B, with a maximum curvature of 0.032 mm⁻¹ for surface A and 0.042 mm⁻¹ for surface B. The opening/closing angles exhibit two dynamic transitions within the 300–400 mm arc length interval. A novel twist angle calculation formula based on normal vector projection angles is proposed, revealing maximum twist angles at the Z = 0 mm directrix: 3.794° for ruled surface A and 3.288° for ruled surface B. By introducing a concavity/convexity criterion for the lower directrix, the machining principle error computation is optimized, confirming a positive correlation between tool-workpiece contact length and machining principle error.