Document Type : Research Paper

Authors

1 Civil Engineering Dept., Enugu State University of Science and Technology, Agbani, Enugu State, Nigeria.

2 Mathematics Dept., Faculty of Sciences and Arts, Al-Kamil, University of Jeddah, Saudi Arabia.

Abstract

Free vibration analysis of thin beams under dynamic excitation is important in design to prevent resonance failures that occur when the excitation frequency coincides with the natural vibration frequency. In this paper, the Elzaki transform method (ETM) is used, for the first time, to solve the free vibration problem of thin beams. The beam is assumed to be homogeneous and prismatic, and the vibration is assumed to be harmonic. As a result, the field equation becomes a fourth-order homogeneous ordinary differential equation (ODE). The Elzaki transform method is chosen in this study due to its proven ability to solve ODEs, systems of ODEs, integro-differential equations, integral equations, and fractional differential equations. The Elzaki transformation simplifies the field equation into an algebraic equation in terms of the unknown deflection in the Elzaki space. By inverting the transformation, the general solution for the deflection is obtained in the physical domain, considering the initial conditions. The enforcement of boundary conditions for each case of end supports is utilized to determine the eigenequations, which are then solved for their roots using Symbolic Algebra methods. The eigenvalues are used to determine the exact natural frequencies of flexural vibration for each considered classical boundary condition. The eigenequations obtained are exact and identical to the ones previously derived by other scholars.

Graphical Abstract

Highlights

• Natural frequencies of vibrating thin beams with different supports are found using the Elzaki transform method.
• The Elzaki transform converts the governing equation into an integral equation, simplifying the solution process.
• By considering boundary conditions, the transform space parameter facilitates a simplified solution approach.

Keywords

Main Subjects

###### References
1. C Ike, Point Collocation Method for the Analysis of Euler-Bernoulli Beam on Winkler Foundation, Int. J. Darshan Inst. Eng. Res. Emerging Technol., 7 (2018) 1 – 7.
2. C. Ike, Fourier Sine Transform Method for the Free Vibration of Euler-Bernoulli Beam Resting on Winkler Foundation, Int. J. Darshan Inst. Eng. Res. Emerging Technol., 7 (2018) 1 – 6.
3. C. Ike , E.U. Ikwueze, Fifth Degree Hermittian Polynomial Shape Functions for the Finite Element Analysis of Clamped-Simply Supported Euler-Bernoulli Beam, Am. J. Eng. Res., 7 (2018) 97 – 105.
4. C. Ike , E.U. Ikwueze, Ritz Method for the Analysis of Statically Indeterminate Euler-Bernoulli Beams, Saudi J. Eng. Technol., 3 (2018) 133 – 140.
5. O Mama, C.C. Ike, C.U. Nwoji, H.N. Onah, Analysis of Infinitely Long Euler-Bernoulli Beam on Two-Parameter Elastic Foundation: Case of Point Load, Electron. J. Geotech. Eng., 22 (2017) 4929 – 4944.
6. Avcar, Free Vibration Analysis of Beams Considering Different Geometric Characteristics and Boundary Conditions, Int. J. Mech. Appl., 4 (2014) 94 – 100. https://doi.org/10.5923/j.mechanics.20140403.03
7. C. Ike, C.U. Nwoji, B.O. Mama, H.N. Onah, M.E. Onyia, Laplace Transform Method for the Elastic Buckling Analysis of Moderately Thick Beams, Int. J. Eng. Res. Technol., 12 (2019) 1626 – 1638.
8. C. Ike, C.U. Nwoji, H.N. Onah, B.O. Mama, M.E. Onyia, Modified Single Finite Fourier Cosine Integral Transform Method for Finding the Critical Elastic Buckling Loads of First Order Shear Deformable Beams with Fixed Ends, Rev. Compos. Mater. Av., – J. Adv. Compos. Mater., 29 (2019) 357 – 362. https://doi.org/10.18280/rcma.290603
9. N. Onah, C.U. Nwoji, M.E. Onyia, B.O. Mama, C.C. Ike, Exact Solutions for the Elastic Buckling Loads of Moderately Thick Beams, Rev. Compos. Mater. Av.,–J. Adv. Compos. Mater., 30 (2020) 83 – 93. https://doi.org/10.18280/rcma.300205
10. C. Ike, Ritz Variational Method for the Flexural Analysis of Third Order Shear Deformable Beams, Proceedings on Engineering Research Technology Innovation and Practice (CERTIP), 2020. http://dx.doi.org/10.13140/RG.2.2.11242.95688
11. O. Mama, O.A. Oguaghamba, C.C. Ike, Finite Fourier Sine Transform Method for Finding Natural Frequencies of Flexural Vibration of Moderately Thick Beams, Proc. Sustain. Eng. Ind. Technol. Conf., 2020, A5-1- 5.
12. Hurty, W.C. and Rubinstein, M.F. Dynamics of Structures, Prentice Hall, New Delhi, 1967.
13. R. Chun, Free Vibration of a Beam with One End Spring-Hinged and the Other Free, J. Appl. Mech., 39 (1972) 1154 – 1155. https://doi.org/10.1115/1.3422854
14. Kanber, O.M. Tufik, Vibration of Thin Beams by PIM and RPIM Methods, Int. J. Aerosp. Lightweight Struct., 3 (2013) 1 – 8. https://doi.org/10.3850/s2010428613000020
15. T.S. Wang , C.C. Lin, Dynamic Analysis of Generally Supported Beams Using Fourier Series, J. Sound Vib., 196 (1996) 285 – 293.https://doi.org/10.1006/jsvi.1996.0484
16. K. Kim , M.S. Kim, Vibration of Beams with Generally Restrained Boundary Conditions Using Fourier Series, J. Sound Vib., 245 (2001) 771 – 784. https://doi.org/10.1006/jsvi.2001.3615
17. L. Li, Free Vibrations of Beams with General Boundary Conditions, J. Sound Vib., 237 (2000) 709-725. https://doi.org/10.1006/jsvi.2000.3150
18. Achawakorn, T. Jearsiripongkul, Vibration Analysis of Exponential Cross-Section Beam using Galerkin’s Method, Int. J. Appl. Sci. Technol., 2 (2012) 7 – 13.
19. Liu , C.S. Gurran, The Use of He’s Variational Iteration Method for Obtaining the Free Vibration of an Euler – Bernoulli Beam, Math. Comput. Model., 50 (2009) 1545 – 1552. https://doi.org/10.1016/j.mcm.2009.09.005
20. O. Agboola, J.A. Gbadeyan, A.A. Opanuga, M.C. Agarana, S.A. Bishop and I.G. Oghonyon, Variational Iteration Method for Natural Frequencies of a Cantilever Beam with Special Attention to the Higher Modes, Proceedings of the World Congress on Engineering, WCE 2017 Lecture Notes in Engineering and Computer Science, London UK,1, 2017,148 – 151.
21. E. Sakman , A. Mutlu, The Natural Frequencies and Mode Shapes of an Euler-Bernoulli Beam with a Rectangular Cross-Section which has a Surface Crack, Int. Sci. J. Sci. Tech. Union Mech.Eng., 5 (2017) 6-9
22. Hong, J. Dodson, S. Laflamme, A. Downey, Transverse Vibration of Clamped-Pinned-Free Beam with Mass at Free End, Appl. Sci., 9 (2019) 1 – 16. https://doi.org/10.3390/app9152996
23. O. Agboola ,J.A. Gbadeyan, Free vibration analysis of Euler-Bernoulli beam using differential transformation method, Nigerian Defence Academy J.Sci.Technol., 7 (2013) 77 – 88.
24. Gritsenko, J. Xu, R. Paolo, Transverse Vibrations of Cantilever Beams: Analytical Solutions with General Steady-State Forcing, Appl. Eng. Sci., 3 (2020) 100017. https://doi.org/10.1016/j.apples.2020.100017
25. B. Coskun, M.T. Atay, B. Özturk, Transverse Vibration Analysis of Euler-Bernoulli Beams Using Analytical Approximate Techniques, Adv. Vib. Anal. Res.,(2011) 1 – 22.https://doi.org/10.5772/15891
26. Adair , M. Jaeger, Simulation of the Vibrations of a Non-Uniform Beam Loaded with Both a Transversely and Axially Eccentric Tip Mass, Int. J. Comput. Methods Exp. Meas., 6 (2018) 679 – 690. https://doi.org/10.2495/CMEM-V6-N4-679-690
27. Torabi, D. Sharifi, M. Ghassabi, A. Mohebbi, Nonlinear Free Transverse Vibration Analysis of Beams Using Variational Iteration Method, AUT J. Mech. Eng., 2 (2018) 233 – 242.https://doi.org/10.22060/mej.2017.12332.5315
28. Chalah-Rezgui, F. Chalah, K. Falek, A. Bali, A. Nechnech, Transverse Vibration Analysis of Uniform Beams under Various Ends Restraints, APCBEE Procedia, 9 (2014) 328 – 333. https://doi.org/10.1016/j.apcbee.2014.01.058
29. H. Flaieh, A.A. Dwech, M.R. Mosheer, Modal Analysis of Fixed-Free Beam Considering Different Geometric Parameters and Materials, IOP Conf. Ser.: Mater. Sci. Eng., 1094,2021,012118. https://doi.org/10.1088/1757-899X/1094/1/012118
30. H. Al-Raheimy, Free Vibration of Simply Supported Beam Subjected to Axial Force, J. Babylon Univ. Eng. Sci., 20 (2012) 301 – 315.
31. Rama, Application of Elzaki Transform to Vibrations in Mechanical System, Adv. Math.: Sci. J., 10 (2021) 1987 – 1996.https://doi.org/10.37418/amsj.10.4.13
32. R. Ramachandruni, Y. Namala, Analytical Solutions of Some Mechanics Problems by Elzaki Transform, Indian J. Sci. Technol., 13 (2020) 4606 – 4618. https://doi.org/10.17485/IJST/v13i47.1971
33. M. Elzaki, S.M. Elzaki, E. Elnour , Application of New Transform “Elzaki Transform” to Mechanics, Electrical Circuits and Beams Problems, Global J. Math. Sci., 4 (2012) 25 – 34.
34. W. Lim and Z. Chen, A New Static Analysis Approach for Free Vibration of Beams, Int. J. Appl. Mech.,10 (2018) 1850004. https://doi.org/10.1142/s1758825118500047
35. C. Ike, Sumudu Transform Method for Finding the Transverse Natural Harmonic Vibration Frequencies of Euler-Bernoulli Beams, ARPN J. Eng. Appl. Sci., 16 (2021) 903 – 911.