Analysis of the associated stress distributions to the nonlinear forced vibrations of functionally graded multi-cracked beams

• Autorzy: Chajdi Mohcine , Adri Ahmed , El Bikri Khalid , Benamar Rhali

Identyfikatory

DOI

10.29354/diag/133702

• Języki publikacji: EN

Abstrakty

EN

Geometrically non-linear vibrations of functionally graded Euler-Bernoulli beams with multi-cracks, subjected to a harmonic distributed force, are examined in this paper using a theoretical model based on Hamilton's principle and spectral analysis. The homogenisation procedure is performed, based on the neutral surface approach, and reduces the FG beams analysis to that of an equivalent homogeneous multi-cracked beam. The so-called multidimensional Duffing equation obtained and solved using a simplified method (second formulation) previously applied to various non-linear structural vibration problems. The curvature distributions associated to the multi-cracked beam forced deflection shapes are obtained for each value of the excitation level and frequency. The parametric study performed in the case of a beam and the detailed numerical results are given in hand to demonstrate the effectiveness of the proposed procedure, and in the other hand conducted to analyse many effects such as the beam material property, the presence of crack, the vibration amplitudes and the applied harmonic force on the non-linear dynamic behaviour of FG beams.

Słowa kluczowe

EN

nonlinear vibration; multi-crack; functionally graded beams; forced vibrations

PL

drgania nieliniowe; belki gradientowe; rozkład naprężeń; drgania wymuszone

• Wydawca: Polskie Towarzystwo Diagnostyki Technicznej
• Czasopismo: Diagnostyka
• Rocznik: 2021
• Tom: Vol. 22, No. 1
• Strony: 101--112
• Opis fizyczny: Bibliogr. 34 poz., rys., tab.

Twórcy

autor

Chajdi Mohcine

• Mohammed V University in Rabat, ENSAM - Rabat, MSSM, B.P.6207, Rabat Instituts, Rabat, Morocco

autor

Adri Ahmed

• Hassan II University of Casablanca, EST - Casablanca, LMPGI, B.P.8012, Oasis Casablanca, Morocco

autor

El Bikri Khalid

• Mohammed V University in Rabat, ENSAM - Rabat, MSSM, B.P.6207, Rabat Instituts, Rabat, Morocco

autor

Benamar Rhali

• Mohammed V University in Rabat, EMI - Rabat, LERSIM, B.P.765, Agdal, Rabat, Morocco

Bibliografia

• 1. Sridhar R, Chakraborty A, Gopalakrishnan S. Wave propagation analysis in anisotropic and inhomogeneous uncracked and cracked structures using pseudospectral finite element method. International Journal of Solids and Structures. 2006; 43(16):4997-5031. https://doi.org/10.1016/j.ijsolstr.2005.10.005.
• 2. Yang J, Chen Y. Free vibration and buckling analyses of functionally graded beams with edge cracks. Composite Structures. 2008; 83(1):48-60.
• 3. Ke L-L, Yang J, Kitipornchai S, Xiang Y. Flexural vibration and elastic buckling of a cracked timoshenko beam made of functionally graded materials. Mechanics of Advanced Materials and Structures. 2009;16(6):488-502. https://doi.org/10.1080/15376490902781175.
• 4. Matbuly MS, Ragb O, Nassar M. Natural frequencies of a functionally graded cracked beam using the differential quadrature method. Applied Mathematics and Computation. 2009; 215(6):2307-2316. https://doi.org/10.1016/j.amc.2009.08.026.
• 5. Yu Z, Chu F. Identification of crack in functionally graded material beams using the p-version of finite element method. Journal of Sound and Vibration. 2009;325(1):69-84. https://doi.org/10.1016/j.jsv.2009.03.010.
• 6. Ferezqi HZ, Tahani M, Toussi HE. Analytical approach to free vibrations of cracked Timoshenko beams made of functionally graded materials. Mechanics of Advanced Materials and Structures. 2010;17(5):353-365. https://doi.org/10.1080/15376494.2010.488608.
• 7. Van Lien T, Duc NT, Khiem NT. A new form of frequency equation for functionally graded Timoshenko beams with arbitrary number of open transverse cracks. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering. 2019; 43(1):235-250. https://doi.org/10.1007/s40997-018-0152-2.
• 8. Shabani S, Cunedioglu Y. Free vibration analysis of functionally graded beams with cracks. Journal of Applied and Computational Mechanics. 2019. https://doi.org/10.22055/jacm.2019.30065.1672.
• 9. Kou KP, Yang Y. A meshfree boundary-domain integral equation method for free vibration analysis of the functionally graded beams with open edged cracks. Composites Part B: Engineering. 2019; 156:303-309. https://doi.org/10.1016/j.compositesb.2018.08.089.
• 10. Birman V, Byrd LW. Vibrations of damaged cantilever beams manufactured from functionally graded materials. AIAA Journal. 2007; 45(11):2747-2757. https://doi.org/10.2514/1.30076.
• 11. Wei D, Liu Y, Xiang Z. An analytical method for free vibration analysis of functionally graded beams with edge cracks. Journal of Sound and Vibration. 2012; 331(7):1686-1700. https://doi.org/10.1016/j.jsv.2011.11.020.
• 12. Yan T, Kitipornchai S, Yang J, He XQ. Dynamic behaviour of edge-cracked shear deformable functionally graded beams on an elastic foundation under a moving load. Composite Structures. 2011; 93(11):2992-3001. https://doi.org/10.1016/j.compstruct.2011.05.003.
• 13. Yang J, Chen Y, Xiang Y, Jia XL. Free and forced vibration of cracked inhomogeneous beams under an axial force and a moving load. Journal of Sound and Vibration. 2008;312(1):166-181. https://doi.org/10.1016/j.jsv.2007.10.034.
• 14. Yan T, Yang J. Forced vibration of edge-cracked functionally graded beams due to a transverse moving load. Procedia Engineering. 2011; 14:3293-3300. https://doi.org/10.1016/j.proeng.2011.07.416.
• 15. Yan T, Kitipornchai S, Yang J. Parametric instability of functionally graded beams with an open edge crack under axial pulsating excitation. Composite Structures. 2011;93(7):1801-1808. https://doi.org/10.1016/j.compstruct.2011.01.019.
• 16. Lien TV, Đuc NT, Khiem NT, Lien TV, Đuc NT, Khiem NT. Free and forced vibration analysis of multiple cracked FGM multi span continuous beams using dynamic stiffness method. Latin American Journal of Solids and Structures. 2019; 16(2). https://doi.org/10.1590/1679-78255242.
• 17. Zhu L-F, Ke L-L, Xiang Y, Zhu X-Q, Wang Y-S. Vibrational power flow analysis of cracked functionally graded beams. Thin-Walled Structures. 2020;150:106626. https://doi.org/10.1016/j.tws.2020.106626.
• 18. Kitipornchai S, Ke LL, Yang J, Xiang Y. Nonlinear vibration of edge cracked functionally graded Timoshenko beams. Journal of Sound and Vibration. 2009;324(3):962-982. https://doi.org/10.1016/j.jsv.2009.02.023.
• 19. Akbaş ŞD. Geometrically nonlinear static analysis of edge cracked Timoshenko beams composed of functionally graded material. Mathematical Problems in Engineering. 2013. https://www.hindawi.com/journals/mpe/2013/871815/abs/. Accessed December 31, 2018.
• 20. Panigrahi B, Pohit G. Nonlinear modelling and dynamic analysis of cracked Timoshenko functionally graded beams based on neutral surface approach. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 2016;230(9):1486-1497. https://doi.org/10.1177/0954406215576560.
• 21. Yan T, Yang J, Kitipornchai S. Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation. Nonlinear Dynamics. 2012; 67(1):527-540. https://doi.org/10.1007/s11071-011-0003-9.
• 22. Panigrahi B, Pohit G. Study of non-linear dynamic behavior of open cracked functionally graded Timoshenko beam under forced excitation using harmonic balance method in conjunction with an iterative technique. Applied Mathematical Modelling. 2018;57:248-267. https://doi.org/10.1016/j.apm.2018.01.022.
• 23. Gayen D, Tiwari R, Chakraborty D. Static and dynamic analyses of cracked functionally graded structural components: A review. Composites Part B: Engineering. 2019; 173:106982. https://doi.org/10.1016/j.compositesb.2019.106982.
• 24. Chajdi M, Adri A, El bikri K, Benamar R. Geometrically nonlinear free and forced vibrations analysis of clamped-clamped functionally graded beams with multicracks. MATEC Web of Conferences. 2018;211:02002. https://doi.org/10.1051/matecconf/201821102002.
• 25. El Kadiri M, Benamar R, White RG. Improvement of the semi-analytical method, for determining the geometrically non-linear response of thin straight structures. Part i: application to clamped-clamped and simply supported–clamped beams. Journal of Sound and Vibration. 2002; 249(2):263-305. https://doi.org/10.1006/jsvi.2001.3808.
• 26. Chajdi M, Fakhreddine H, Adri A, Bikri KE, Benamar R. Geometrically non-linear forced vibrations of fully clamped functionally graded beams with multi-cracks resting on intermediate simple supports. Journal of Physics: Conference Series. 2019; 1264:012023. https://doi.org/10.1088/1742-6596/1264/1/012023.
• 27. Chajdi M, Merrimi EB, El Bikri K. Geometrically nonlinear free vibration of composite materials: clamped-clamped functionally graded beam with an edge crack using homogenisation method. Key Engineering Materials. 2017; 730:521-526. https://doi.org/10.4028/www.scientific.net/KEM.730 .521.
• 28. Adri A, Benamar R. Linear and geometrically nonlinear frequencies and mode shapes of beams carrying a point mass at various locations. An analytical approch and a parametric study. Diagnostyka 2017; 18(2): 13-21.
• 29. Chajdi M, Adri A, Bikri KE, Benamar R. Linear and geometrically nonlinear free and forced vibrations of fully clamped multi-cracked beams. Diagnostyka. 2019;20(1):111-125. https://doi.org/10.29354/diag/103125.
• 30. Fakhreddine H, Adri A, Rifai S, Benamar R. A multimode approach to geometrically non-linear forced vibrations of Euler–Bernoulli multispan beams. Journal of Vibration Engineering & Technologies. 2020;8(2):319-326. https://doi.org/10.1007/s42417-019-00139-8.
• 31. Aldlemy MS, Al-jumaili S a. K, Al-Mamoori R a. M, Ya T, Alebrahim R. Composite patch reinforcement of a cracked simply-supported beam traversed by moving mass. Journal of Mechanical Engineering and Sciences. 2020;14(1):6403-6415. https://doi.org/10.15282/jmes.14.1.2020.16.0501.
• 32. El Bikri K, Benamar R, Bennouna MM. Geometrically non-linear free vibrations of clamped-clamped beams with an edge crack. Computers & Structures. 2006;84(7):485-502. https://doi.org/10.1016/j.compstruc.2005.09.030.
• 33. Azrar L, Benamar R, White RG. A semi-analytical approach to the non-linear dynamic response problem of beams at large vibration amplitudes, part ii: multimode approach to the steady state forced periodic response. Journal of Sound and Vibration. 2002;255(1):1-41. https://doi.org/10.1006/jsvi.2000.3595.
• 34. Azrar L, Benamar R, White RG. Semi-analytical approach to the non-linear dynamic response problem of s-s and c-c beams at large vibration amplitudes part I: general theory and application to the single mode approach to free and forced vibration analysis. Journal of Sound and Vibration. 1999; 224(2):183-207. https://doi.org/10.1006/jsvi.1998.1893.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).