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Coupled Bending and Axial Vibrations of Axially Functionally Graded Euler–Bernoulli Beams

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Abstract

Purpose

This study aims to obtain and present the closed-form solution of coupled axial-bending vibration problem for the general case of axially functionally graded (AFG) beams, to create the frequency equation and to propose numerical method for computing natural frequencies.

Methods

The general model of a beam is introduced using two cylindrical springs, one rotational spring, and a rigid body at each beam end. Mass centres of bodies are placed eccentrically in axial and transverse direction with respect to end points of the beam. The general boundary conditions are modelled by linearized Newton–Euler differential equations and the general case of the in-plane axial-bending vibrations of AFG beams is covered. It is assumed that the beam is made of a functionally graded material whose material and cross-sectional characteristics change along its longitudinal axis without any restrictions. The Euler–Bernoulli constitutive theory is applied for modelling. Partial differential equations of motion are transformed into a system of ordinary differential equations with variable coefficients, the form suitable for the implementation of the symbolic-numeric methods of initial parameters. Natural frequencies of the beam are computed as numerical solutions of the exact frequency equation.

Results and Conclusions

The closed-form solution of coupled axial-bending vibrations is derived for general case of AFG beams. Orthogonality conditions of mode shapes are derived, and constants in time function. Also, the frequency equation is derived and solved numerically to obtain natural frequencies. Obtained results of natural frequencies and mode shapes are compared to those available in open literature.

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Data availability

No data were used in the paper.

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Acknowledgements

Support for this research was provided by the Ministry of Science, Technological Development, and Innovation of the Republic of Serbia under Grants No. 451-03-47/2023-01/200105 and No. 451-03-47/2023-01/200108. This support is gratefully acknowledged.

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120, Belgrade 35, Serbia

    Aleksandar Tomović, Aleksandar Obradović, Nemanja Zorić & Zoran Mitrović

  2. Faculty of Mechanical and Civil Engineering in Kraljevo, University of Kragujevac, Dositejeva 19, 36000, Kraljevo, Serbia

    Slaviša Šalinić

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  1. Aleksandar Tomović
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  2. Slaviša Šalinić
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Correspondence to Aleksandar Tomović.

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Tomović, A., Šalinić, S., Obradović, A. et al. Coupled Bending and Axial Vibrations of Axially Functionally Graded Euler–Bernoulli Beams. J. Vib. Eng. Technol. 12, 2987–3004 (2024). https://doi.org/10.1007/s42417-023-01027-y

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