A simple tool to forecast the natural frequencies of thin-walled cylinders

A simple tool to forecast the natural frequencies of thin-walled cylinders

Marco Cammalleri, Antonella Castellano, Marco Abella

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Abstract. This paper presents an approximate method to predict the natural frequencies of thin-walled cylinders. By taking inspiration from a previous work of one of the authors, the starting point of the proposed approach is a proper construction of reasonable eigenfunctions. However, a new simple tool based on the principle of virtual work has been developed to estimate the natural frequencies and the amplitude of vibration without complex numerical resolution. Moreover, the applicability of the model is extended to all the most common constraint conditions. The identification of the natural frequencies of a continuous cylinder is reduced to an eigenvalue problem based on a matrix whose elements depend only on the geometric characteristics of the cylinder, the mechanical properties of the material and known numerical parameters. The latter are precalculated for given boundary conditions, covering clamped or pinned end constraints. Although the proposed formulation can address any constraints combination, only a pinned-pinned cylinder is analyzed here for brevity. The reliability of the model was tested against FEM analysis results. These comparisons showed that the maximum error versus the exact solutions for the lowest natural frequency is around 2% for all the mode shapes of the pinned-pinned case, offering an excellent trade-off between accuracy and ease of use.

Circular Cylindrical Shell, Natural Frequencies, Eigenvalue Problem

Published online 3/17/2022, 6 pages
Copyright © 2023 by the author(s)
Published under license by Materials Research Forum LLC., Millersville PA, USA

Citation: Marco Cammalleri, Antonella Castellano, Marco Abella, A simple tool to forecast the natural frequencies of thin-walled cylinders, Materials Research Proceedings, Vol. 26, pp 635-640, 2023

DOI: https://doi.org/10.21741/9781644902431-102

The article was published as article 102 of the book Theoretical and Applied Mechanics

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

[1] Y. Xing, B. Liu, T. Xu, Exact solutions for free vibration of circular cylindrical shells with classical boundary conditions, Int. J. Mech. Sci. 75 (2013) 178–188. https://doi.org/10.1016/j.ijmecsci.2013.06.005
[2] Y. Dubyk, I. Orynyak, O. Ishchenko, An exact series solution for free vibration of cylindrical shell with arbitrary boundary conditions, Sci. J. Ternopil Natl. Tech. Univ. 89 (2018) 79–88. https://doi.org/10.33108/visnyk_tntu2018.01.079
[3] Cammarata, A., Sinatra, R., & Maddio, P. D. (2019, July). An extended Craig-Bampton method for the modal analysis of mechanisms. In IFToMM World Congress on Mechanism and Machine Science (pp. 3329-3338). Springer, Cham. https://doi.org/10.1007/978-3-030-20131-9_329
[4] Cammarata, A., Sinatra, R., & Maddìo, P. D. (2019). Static condensation method for the reduced dynamic modeling of mechanisms and structures. Archive of Applied Mechanics, 89(10), 2033-2051. https://doi.org/10.1007/s00419-019-01560-x
[5] M. Cammalleri, E. Pipitone, S. Beccari, G. Genchi, A Mathematical Model for the Prediction of the Injected Mass Diagram of a S.I. Engine Gas Injector, Journal of Mechanical Science and Technology, Springer vol. 11 (2013) pagg. 3253-3265. https://doi.org/10.1007/s12206-013-0848-6
[6] Cammalleri M., Castellano A., Analysis of hybrid vehicle transmissions with any number of modes and planetary gearing: kinematics, power flows, mechanical power losses. Mechanism and Machine Theory 162 (2021) 104350. https://doi.org/10.1016/j.mechmachtheory.2021.104350
[7] S.M.R. Khalili, A. Davar, K. Malekzadeh Fard, Free vibration analysis of homogeneous isotropic circular cylindrical shells based on a new three-dimensional refined higher-order theory, Int. J. Mech. Sci. 56 (2012) 1–25. https://doi.org/10.1016/j.ijmecsci.2011.11.002
[8] C. Wang, J.C.S. Lai, Prediction of natural frequencies of finite length circular cylindrical shells, Appl. Acoust. 59 (2000) 385–400. https://doi.org/10.1016/S0003-682X(99)00039-0
[9] M. Cammalleri, A. Costanza, A closed-form solution for natural frequencies of thin-walled cylinders with clamped edges, Int. J. Mech. Sci. 110 (2016) 116–126. https://doi.org/10.1016/j.ijmecsci.2016.03.005
[10] F. Sorge, M. Cammalleri, Control of hysteretic instability in rotating machinery by elastic suspension systems subject to dry and viscous friction, J. Sound Vib. 329 (2010) 1686. https://doi.org/10.1016/j.jsv.2009.12.007
[11] F. Sorge, M. Cammalleri, On the beneficial effect of rotor suspension anisotropy on viscous-dry hysteretic instability, Meccanica. 47 (2012) 1705–1722. https://doi.org/10.1007/s11012-012-9549-y
[12] A. Kumar, S.L. Das, P. Wahi, Effect of radial loads on the natural frequencies of thin-walled circular cylindrical shells, Int. J. Mech. Sci. 122 (2017) 37–52. https://doi.org/10.1016/j.ijmecsci.2016.12.024
[9] A.E.H. Love, XVI. The small free vibrations and deformation of a thin elastic shell, Philos. Trans. R. Soc. London. 179 (1888) 491–546. https://doi.org/10.1098/rsta.1888.0016
[10] E. Reissner, A New Derivation of the Equations for the Deformation of Elastic Shells, Am. J. Math. 63 (1941) 177. https://doi.org/10.2307/2371288
[11] L.H. Donnell, Beams, plates and shells, McGraw-Hill, New York, 1976.