Comparative analysis of beam and plate models in assessing the compliance of a thin-walled cantilever plate during milling
DOI:
https://doi.org/10.15276/opu.1.73.2026.02Keywords:
thin plate, compliance, deflection, workpiece stiffness, milling, vibrations, two-dimensional model, machining trajectoryAbstract
In this paper, the problem of evaluating the local compliance of a thin, cantilever-fixed steel plate during milling is considered, accounting for the displacement of the force application point along the machining trajectory. The relevance of the study stems from the widespread use of simplified one-dimensional models in the design of milling processes for low-rigidity parts, which systematically lead to errors in assessing local stiffness and, consequently, to excessively conservative cutting conditions. A comparison of one-dimensional (beam) and two-dimensional (plate) deformation models is performed: in the beam model, the plate is treated as a cantilever beam of constant cross-section, the deflection of which is determined analytically based on classical strength-of-materials equations; in the plate model, the Kirchhoff-Love theory is used, accounting for the two-dimensional distribution of deformations, the interaction of bending in two coordinate directions, and the actual boundary conditions of the fixation. For each position of the tool along the machining trajectory, the local compliance was determined within both models, enabling their direct quantitative comparison. It is shown that the one-dimensional model systematically overestimates deflection and underestimates local stiffness, particularly along the plate’s free edge, leading to an underestimation of the calculated limiting feed rate and overly conservative process parameters. The two-dimensional model correctly accounts for the redistribution of deformations over the plate surface, produces a spatial compliance map as a function of two coordinates, and reflects the actual stiffness characteristics with significantly greater accuracy. The comparative analysis reveals fundamental differences in the assessment of vibration-prone zones, confirms the need to apply plate models when calculating adaptive milling conditions for thin-walled cantilever elements on CNC machines, and opens the possibility of increasing machining productivity without increasing the risk of chatter.
References
1. How two-dimensional bending can extraordinarily stiffen thin sheets / V. Pini, J. J. Ruz, P. M. Kosaka,
O. Malvar, M. Calleja, J. Tamayo. Scientific Reports. 2016. Vol. 6. Art. 29627. DOI:
https://doi.org/10.1038/srep29627.
2. Domagalski Ł., Kowalczyk I. Optimization and analysis of plates with a variable stiffness distribution in terms of dynamic properties. Materials. 2025. Vol. 18, no. 9. Art. 2150. DOI: https://doi.org/10.3390/ma18092150.
3. Rodriguez C. A midsurface elasticity model for a thin, nonlinear, gradient elastic plate. International Journal of Engineering Science. 2024. Vol. 197. Art. 104026. DOI: https://doi.org/10.1016/j.ijengsci.2024.104026.
4. Semi-analytical period-doubling chatter analysis in thin wall milling / M. Sanz-Calle, J. Munoa, A. Iglesias, L. N. Lopez de Lacalle, Z. Dombovari. MM Science Journal. 2021. Vol. 2021, no. 5. P. 5126–5133. DOI: https://doi.org/10.17973/MMSJ.2021_11_2021167.
5. Dynamic equations of motion for inextensible beams and plates / M. Deliyianni, K. McHugh, J. T. Webster, E. Dowell. Archive of Applied Mechanics. 2022. Vol. 92, no. 6. P. 1929–1952. DOI: https://doi.org/10.1007/s00419-022-02157-7.
6. Evaluation of the Vibration Signal during Milling Vertical Thin-Walled Structures from Aerospace Materials / S. Kurpiel, K. Zagórski, J. Cieślik, K. Skrzypkowski, W. Brostow. Sensors. 2023. Vol. 23, no. 14. Art. 6398. DOI: https://doi.org/10.3390/s23146398.
7. Deflection error modeling during thin-wall machining / I. Llanos, A. Robles, J. Condón, M. Arizmendi, A. Beristain. Procedia CIRP. 2023. Vol. 117. P. 169–174. DOI: https://doi.org/10.1016/j.procir.2023.03.030.
8. Dynamic response analysis of a thin plate with partially constrained layer damping optimization under moving loads for various boundary conditions / Y. Qin, Q. Song, Z. Liu, J. Shi. Applied Sciences. 2021. Vol. 11, no. 7. Art. 3282. DOI: https://doi.org/10.3390/app11073282.
9. Ghaemifard S., Ghannadiasl A. A review of moving-load dynamic problems: Specialized modeling for beam and plate vibrancy. Journal of Vibration Engineering & Technologies. 2025. Vol. 13, no. 5. P. 280–309. DOI: https://doi.org/10.1007/s42417-025-01837-2.
10. Wang X., Song Q., Liu Z. Dynamic model and stability prediction of thin-walled component milling with multi-modes coupling effect. Journal of Materials Processing Technology. 2021. Vol. 2(88). Art. 116869. DOI: https://doi.org/10.1016/j.jmatprotec.2020.116869.
11. Zelinskyi S. A., Morozov Yu. A., Serebriy Yu. A. Mathematical model of process of contour milling of nonrigid details. Proceedings of Odessa Polytechnic University. 2015. Vol. 1(45). P. 28–33. DOI: https://doi.org/10.15276/opu.1.45.2015.06.
12. Control of the milling mode of a thin-walled console plate by indirect indicators / A. Tkach, I. Sydorenko, I. Prokopovych, V. Kurhan, A. Toropenko. Proceedings of Odessa Polytechnic University. 2025. Vol. 1(71). P. 39–45. DOI: https://doi.org/10.15276/opu.1.71.2025.04.
13. Chatter suppression with productivity improvement by scheduling a C3 continuous feedrate to match spindle speed variation / X. Qin, M. Wan, W.-H. Zhang, Y. Yang. Mechanical Systems and Signal Processing. 2023. Vol. 1(88). Art. 110021. DOI: 10.1016/j.ymssp.2022.110021.
14. Methods for implementing vibration suppression in machining parts on CNC machines / V. M. Tonkonohyi, S. A. Zelinskyi, V. A. Vodichev, V. V. Natalchyshyn, A. Z. Tkach. Proceedings of Odessa Polytechnic University. 2017. Vol. 1(51). P. 34–39. DOI: https://doi.org/10.15276/opu.1.51.2017.07.
