Authors: A.F. Denisenko, R.V. Ladyagin

Title of the article: Determination of natural frequencies of a metal-cutting machine spindle using mass reduction and influence coefficients

Year: 2026, Issue: 1, Pages: 15-25

Branch of knowledge: 2.5.5. Technology and equipment for mechanical and physical-technical processing

Index UDK: 621.9.06-229.33.001.24

DOI: 10.26730/1999-4125-2026-1-15-25

Abstract: This article considers the problem of determining the natural frequencies of a machine tool spindle, a pressing issue due to the increase in vibration during high-speed machining. Existing approaches, such as finite element modeling, offer high accuracy but are labor-intensive at the preliminary design stage due to the uncertainty of geometric parameters. The aim of this work is to develop adapted analytical methods for quickly assessing the dynamic characteristics of spindle assemblies, taking into account their key design features: asymmetry, the presence of a massive cantilever section, and the difference in the stiffness of the front and rear supports. The proposed approach is based on a combination of the influence coefficient method and mass reduction, taking into account both the distributed and concentrated masses of the spindle. Particular attention is paid to modeling the cantilever section: two schemes for reducing its mass—to the middle and to the end of the cantilever—are compared. The effect of support elasticity on natural frequencies is also studied. The methodology was verified by comparing the calculated results with modal analysis data in ANSYS Workbench for a real spindle with various bearing configurations. It was found that the model with the console mass reduced to its end provides the best accuracy: the discrepancy with finite element modeling is only 1.2–1.6%. Accounting for support elasticity is shown to reduce natural frequencies by almost half compared to the model with rigid supports. The spindle's design feature of mass concentration on a short console results in close values ​​of the first and second natural frequencies, with a difference of 6.3–13.8%. The developed analytical relationships represent an effective tool for engineering analysis and dynamic design of spindle systems at early stages, providing a compromise between sufficient accuracy and computational simplicity.

Key words: machine spindle natural frequencies influence coefficients mass reduction console part elastic supports finite element modeling

Receiving date: 15.10.2025

Approval date: 15.01.2026

Publication date: 19.03.2026

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