Natural Vibration Analysis of Axially Graded Tapered Rayleigh Beams on a Quadratic Bi-Parametric Elastic Foundation
DOI:
https://doi.org/10.4314/njtd.v22i1.3341Keywords:
Natural vibration, functionally graded beam, differential transform method, dimensionless frequency, Rayleigh beam theoryAbstract
Understanding the dynamic behaviour of non-homogeneous beams on elastic foundations is vital for ensuring the stability and safety of engineering structures such as bridges, buildings, and mechanical systems. This study aims to address this problem by investigating the natural vibration characteristics of such beams. The mechanical properties of the beam and the foundation stiffness are assumed to vary along the longitudinal direction. Two boundary conditions-clamped-free and clamped-clamped are considered. The differential transform method (DTM) is employed to solve the governing equation of motion, providing a numerical approximation of the beam's dynamic behaviour. The results, presented in both tabular and graphical formats, highlight the influence of key parameters such as the inverse of the slenderness ratio, foundation stiffness, and boundary conditions on the free vibration response. The findings indicate that increasing foundation stiffness leads to higher natural frequencies, with the Pasternak foundation yielding the highest values. Additionally, an axially functionally graded Rayleigh beam on a Pasternak foundation exhibits superior natural frequencies compared to a similar beam on a Winkler foundation. The effectiveness and accuracy of the DTM are further validated through two illustrative dynamic response problems, with results demonstrating excellent agreement with existing literature.
References
Adair, D., Ismailov, K., and Jaeger, M. (2018). Vibration of a Beam on an Elastic Foundation Using the Variational Iteration Method. International Journal of Aerospace and Mechanical Engineering, 12(9), 914-919.
Alshorbagy, A. E., Eltaher, M. A., and Mahmoud, F. F. (2011). Free vibration characteristics of functionally graded beams using finite element approach. Journal of Theoretical and Applied Mechanics, 49(2), 321-339.
Avcar, M., and Alsaid, S. S. (2017). Free vibration analysis of Rayleigh beam composed of functionally graded materials considering simply-supported boundary conditions. Journal of Vibration and Control, 23(12), 2026-2044.
Aydogdu, M., and Taskin, M. (2007). Free vibration analysis of functionally graded beams using Navier solution method. Composite Structures, 80(1), 45-53.
Catal, S. (2008). Solution of free vibration equations of beam on elastic soil by using differential transform method. Applied Mathematical Modeling, 32(9), 1744-1757.
Chen, Y. F., Zang, Z. P., Dong, S. H., Ao, C., Liu, H., Ma, S., ... and Feng, W. (2021). Flapwise bending vibration analysis of rotating tapered Rayleigh beams for the application of offshore wind turbine blades. China Ocean Engineering, 35(4), 544-553.
Coskun, I., Ozturk, M., and Mutman, U. (2014). Vibration analysis of non-uniform Euler-Bernoulli beam on elastic foundations using Adomain Decomposition Method. Journal of Sound and Vibration, 333(5), 1490-1506.
De Rosa, M. A., and Maurizi, M. J. (1998). The influence of concentrated masses and Pasternak soil on the free vibrations of Euler beam - Exact solution. Journal of Sound and Vibration, 212(4), 573-581.
Ebrahimi-Mamaghani, A., Sarparast, M., and Rezaei, M. (2020). Free and forced vibration analysis of axially functionally graded Rayleigh and Euler-Bernoulli beams moving under load. Journal of Solid Mechanics, 12(2), 429-447.
Kumar, S. (2022). Natural frequencies of beams with axial material gradation resting on two parameter elastic foundation. Trends in Sciences, 19(6), 3048-3058.
Lamprea-Pineda, A. C., Connolly, D. P., and Hussein, M. F. (2022). Beams on elastic foundations–A review of railway applications and solutions. Transportation Geotechnics, 33, 1-63.
Lazreg, H. and Fabrice, B. (2020). Bending and free vibration analysis of functionally graded beams on elastic foundations with analytical validation. Advances in Materials Research, 9(1) 63-98.
Ma, J., Wang, J., Wang, C., and Guo Y., (2024). Vibration response of beams-supported by finite-thickness elastic foundation under a concentrated force. Journal of Mechanical Science and Technology, 38, 595-604.
Ma J., Li, D., Yang, H., Wang, J., Wang, C., and Guo, Y., (2024). Analytical investigation on the dynamic behavior of multi-span continuous beams supported on soil with finite depth. Coatings, 14, 1-16, https://doi.org/10.3390/coatings14070864.
Malik, M., and Dang, V. (1998). Structural dynamics: differential transform method and its applications. Computer Modeling in Engineering & Sciences, 1(1), 73-88.
Mei, C. (2008). Free lateral vibrations of centrifugally stiffened rotating Euler-Bernoulli beam using the differential transform method. Journal of Sound and Vibration, 318(1-2), 397-412.
Olotu, O. T., Gbadeyan, J. A., and Agboola, O. O. (2023). Free Vibration Analysis of Tapered Rayleigh Beams resting on Variable Two-Parameter Elastic Foundation. Forces in Mechanics, 12, 1-12.
Shahba, A., Attarnejad, R., Marvi, A., and Hajilar, S. (2011). Finite element analysis of free vibration and stability of functionally graded Timoshenko beams. European Journal of Mechanics-A Solids, 30(2), 274-283.
Simsek, M. (2010). Fundamental frequency analysis of functionally graded beams using higher-order beam theories. Composite Structures, 92(5), 1220-1230.
Skalba, P., and Liskiewicz, A. (2018). Natural frequencies of beams with axial material gradation resting on two parameter elastic foundation. Trends in Sciences, 11(1), 107-114.
Sulaiman M.A., Usman M.A., Mustapha R.A., Hammed F.A. and Raheem T.L., (2024). Effect of coefficient of viscous damping on dynamic analysis of Euler-Bernoulli beam resting on elastic foundation using Integral Numerical Method. J. Appl. Sci. Environ. Manage, 28(3), 883-887, https://www.ajol.info/index.php/jasem.
Lohar, H., Mitra, A., and Sahoo, S., (2016). Free vibration analysis of axially functionally graded linearly taper beam on elastic foundation. In IOP Conference Series: Materials Science and Engineering, 149(1), 3-8.
Wattanasakulpong, N. (2012). Free vibration of functionally graded beams with general elastically end constraints using differential transform method. Journal of Mechanical Science and Technology, 26(12), 3951-59.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Nigerian Journal of Technological Development
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
In accordance with the Copyright Act of 1976, which became effective January 1, 1978, the following statement signed by each author must accompany the manuscript submitted: "I, the undersigned author, transfer all copyright ownership of the manuscript referenced above to the Nigerian Journal of Technological Development, in the event the work is published. I warrant that the article is original, does not infringe upon any copyright or other proprietary right of any third party, is not under consideration by another journal, and has not been published previously. I have reviewed and approve the submitted version of the manuscript and agree to its publication in the Nigerian Journal of Technological Development." A copyright transfer form should be downloaded from the NJTD Website ( http://njtd.com.ng/index.php/njtd). Author(s) will be consulted, whenever possible, regarding republication of material. All authors must have access to the data presented and the authors and sponsor (if applicable) must agree to share original data with the editor if requested.
