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Background <p>This paper presents a closed-form sinusoidal expansion (CFSE) modeling framework for analytically representing the cutting forces and time domain arbitrary DOF forced vibration solution of cylindrical milling tools under both conventional and oblique cutting mechanics.</p> Methods <p>The proposed CFSE model leverages the analyticity of local cutting forces, expressing them as sinusoidal expansions in which each term is the product of a unit sinusoid, a generalizable parametric grouping (GPG), and a window function. This structured form is preserved through integration over flute segments to produce the total cutting force. The conventional and oblique force models within the framework are validated experimentally using identification methods based on a time-domain linear regression method for the former and a reduced-order particle swarm optimization method for the latter.</p> Results <p>It is shown that the GPGs associated with the cutting force are the only offline quantities required for efficient substitution-based computation of cutting force within the proposed CFSE–GPG-Based Solution Framework, while the inclusion of modal matrices further enables the substitution-based computation of forced vibrations and surface location error (SLE) within the framework. For a given milling process, the analytical expressions of GPGs depend solely on the formulation of the local cutting force. In this work, three distinct formulations of the local force for cylindrical milling tools were shown to integrate seamlessly into the proposed framework. The corresponding numerical results exhibited strong agreement with published cutting force, vibration, and SLE data, thereby confirming both the accuracy and the broad applicability of the framework. The computational convergence of the proposed framework is demonstrated to be consistently superior to a benchmark numerical approach, and also demonstrated superior accuracy–cost trade-off (high accuracy at low cost) than the numerical approach.</p> Conclusions <p>The framework is established using the circular chip thickness model, while its extension to non-circular chip thickness formulations associated with micro-geometry effects is highlighted as promising directions for future research and development.</p>

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Novel Sinusoidal Expansion and Generalizable Parametric Groupings for Closed-Form Models of the Periodic Milling Dynamics

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月3日

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  • Chigbogu Ozoegwu

摘要

Background

This paper presents a closed-form sinusoidal expansion (CFSE) modeling framework for analytically representing the cutting forces and time domain arbitrary DOF forced vibration solution of cylindrical milling tools under both conventional and oblique cutting mechanics.

Methods

The proposed CFSE model leverages the analyticity of local cutting forces, expressing them as sinusoidal expansions in which each term is the product of a unit sinusoid, a generalizable parametric grouping (GPG), and a window function. This structured form is preserved through integration over flute segments to produce the total cutting force. The conventional and oblique force models within the framework are validated experimentally using identification methods based on a time-domain linear regression method for the former and a reduced-order particle swarm optimization method for the latter.

Results

It is shown that the GPGs associated with the cutting force are the only offline quantities required for efficient substitution-based computation of cutting force within the proposed CFSE–GPG-Based Solution Framework, while the inclusion of modal matrices further enables the substitution-based computation of forced vibrations and surface location error (SLE) within the framework. For a given milling process, the analytical expressions of GPGs depend solely on the formulation of the local cutting force. In this work, three distinct formulations of the local force for cylindrical milling tools were shown to integrate seamlessly into the proposed framework. The corresponding numerical results exhibited strong agreement with published cutting force, vibration, and SLE data, thereby confirming both the accuracy and the broad applicability of the framework. The computational convergence of the proposed framework is demonstrated to be consistently superior to a benchmark numerical approach, and also demonstrated superior accuracy–cost trade-off (high accuracy at low cost) than the numerical approach.

Conclusions

The framework is established using the circular chip thickness model, while its extension to non-circular chip thickness formulations associated with micro-geometry effects is highlighted as promising directions for future research and development.