Calculating the free vibrations of functionally graded plates using the unlimited integral
DOI:
https://doi.org/10.34115/basrv9n1-013Keywords:
analytical solutions, vibrations, unlimited integral, Hamilton's principleAbstract
This paper discusses the use of the infinite integral to calculate free vibrations of composite structures. This technique allows us to reduce the number of unknowns to four, resulting in simpler governing equations. This model is meant to look at the uncontrolled shaking of functionally graded plates (FGM), where the transverse shear stresses are spread out in a hyperbolic way and meet the requirements of free tension on both the top and bottom surfaces of the plate. The motion equations are based on Hamilton's principle and Navier-type analytical solutions for simply supported plates. The discovered solutions are compared to existing solutions to ensure the model's validity. The material's characteristics vary continuously over the plate's thickness according to a power law and an exponential function. We investigate the effects of the power law index and geometric ratio on natural frequencies using numerical data.
References
AKAVCI, S. S. An efficient shear deformation theory for free vibration of functionally graded thick rectangular plates on elastic foundation. Composite Structures, [s. l.], v. 108, n. 1, p. 667–676, 2014. Disponível em: http://dx.doi.org/10.1016/j.compstruct.2013.10.019.
BELARBI, M.-O. et al. Bending, free vibration and buckling finite element analysis of porous functionally graded plates with various porosity distributions using an improved FSDT. Mechanics Based Design of Structures and Machines, [s. l.], v. 53, n. 1, p. 401–445, 2025. Disponível em: https://www.tandfonline.com/doi/full/10.1080/15397734.2024.2366530.
BELKHODJA, Y. et al. An exponential-trigonometric higher order shear deformation theory (HSDT) for bending, free vibration, and buckling analysis of functionally graded materials (FGMs) plates. Advanced Composites Letters, [s. l.], v. 29, 2020.
BENACHOUR, A. et al. A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient. Composites Part B: Engineering, [s. l.], v. 42, n. 6, p. 1386–1394, 2011. Disponível em: https://linkinghub.elsevier.com/retrieve/pii/S1359836811002502.
CHEN, C. S.; HSU, C. Y.; TZOU, G. J. Vibration and stability of functionally graded plates based on a higher-order deformation theory. Journal of Reinforced Plastics and Composites, [s. l.], v. 28, n. 10, p. 1215–1234, 2009.
FERREIRA, A. J. M. et al. Natural frequencies of functionally graded plates by a meshless method. Composite Structures, [s. l.], v. 75, n. 1–4, p. 593–600, 2006.
GUPTA, B. Few Studies on Biomedical Applications of Functionally Graded Material. International Journal of Engineering Technology Science and Research, 4 (3), 39–43. [S. l.: s. n.], 2017.
HOSSEINI-HASHEMI, S. et al. On the exact in-plane and out-of-plane free vibration analysis of thick functionally graded rectangular plates: Explicit 3-D elasticity solutions. Composites Part B: Engineering, [s. l.], v. 46, p. 108–115, 2013.
HOSSEINI-HASHEMI, S.; FADAEE, M.; ES’ HAGHI, M. A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates. International Journal of Mechanical Sciences, [s. l.], v. 52, n. 8, p. 1025–1035, 2010.
HUANG, X.-L.; SHEN, H.-S. Nonlinear vibration and dynamic response of functionally graded plates in thermal environments. International Journal of Solids and Structures, [s. l.], v. 41, n. 9–10, p. 2403–2427, 2004.
KARAMA, M.; AFAQ, K. S.; MISTOU, S. Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. International Journal of Solids and Structures, [s. l.], v. 40, n. 6, p. 1525–1546, 2003. Disponível em: https://linkinghub.elsevier.com/retrieve/pii/S0020768302006479. Acesso em: 29 out. 2021.
KAREEM, M. G. et al. Analysis the free vibration of functionally graded material plate by using new displacement function. Results in Engineering, [s. l.], v. 25, p. 103756, 2025. Disponível em: https://linkinghub.elsevier.com/retrieve/pii/S2590123024019996.
LI, M.; guedes soares, C.; YAN, R. Free vibration analysis of FGM plates on Winkler/Pasternak/Kerr foundation by using a simple quasi-3D HSDT. Composite Structures, [s. l.], v. 264, n. January, p. 113643, 2021. Disponível em: https://doi.org/10.1016/j.compstruct.2021.113643.
LI, Q.; IU, V. P.; KOU, K. P. Three-dimensional vibration analysis of functionally graded material plates in thermal environment. Journal of Sound and Vibration, [s. l.], v. 324, n. 3–5, p. 733–750, 2009.
LIU, D. Y.; WANG, C. Y.; CHEN, W. Q. Free vibration of FGM plates with in-plane material inhomogeneity. Composite Structures, [s. l.], v. 92, n. 5, p. 1047–1051, 2010.
LÜ, C. F.; LIM, C. W.; CHEN, W. Q. Exact solutions for free vibrations of functionally graded thick plates on elastic foundations. Mechanics of Advanced Materials and Structures, [s. l.], v. 16, n. 8, p. 576–584, 2009. Disponível em: https://www.tandfonline.com/doi/abs/10.1080/15376490903138888. Acesso em: 29 out. 2021.
MATSUNAGA, H. Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory. Composite Structures, [s. l.], v. 82, n. 4, p. 499–512, 2008.
MEFTAH, A. et al. A non-polynomial four variable refined plate theory for free vibration of functionally graded thick rectangular plates on elastic foundation. Steel and Composite Structures, [s. l.], v. 23, n. 3, p. 317–330, 2017. Disponível em: http://koreascience.or.kr/journal/view.jsp?kj=KJKHEW&py=2017&vnc=v23n3&sp=317.
MEFTAH, A. Bending and Buckling Analysis of Functionally Graded Plates using a New Shear Strain Function with Reduced Unknowns. Journal of Mechanical Engineering, [s. l.], v. 20, n. 3, p. 105–129, 2023. Disponível em: https://ir.uitm.edu.my/id/eprint/84114/1/84114.pdf.
MOUAS, Z. et al. Free vibration analysis of functionally graded composite plates reinforced with linearly and nonlinearly distributed carbon nanotubes in hygrothermal environments. Mechanics Based Design of Structures and Machines, [s. l.], v. 53, n. 2, p. 1516–1543, 2025. Disponível em: https://www.tandfonline.com/doi/full/10.1080/15397734.2024.2389112.
NGUYEN, T.-K. K. A higher-order hyperbolic shear deformation plate model for analysis of functionally graded materials. International Journal of Mechanics and Materials in Design, [s. l.], v. 11, n. 2, p. 203–219, 2014. Disponível em: http://dx.doi.org/10.1007/s10999-014-9260-3. Acesso em: 29 out. 2021.
NIE, G. J.; ZHONG, Z. Vibration analysis of functionally graded annular sectorial plates with simply supported radial edges. Composite Structures, [s. l.], v. 84, n. 2, p. 167–176, 2008.
UYMAZ, B.; AYDOGDU, M. Three-dimensional vibration analyses of functionally graded plates under various boundary conditions. Journal of Reinforced Plastics and Composites, [s. l.], v. 26, n. 18, p. 1847–1863, 2007. Disponível em: https://journals.sagepub.com/doi/abs/10.1177/0731684407081351. Acesso em: 29 out. 2021.
VEL, S. S.; BATRA, R. C. Three-dimensional exact solution for the vibration of functionally graded rectangular plates. Journal of Sound and Vibration, [s. l.], v. 272, n. 3–5, p. 703–730, 2004.
WOO, J.; MEGUID, S. A.; ONG, L. S. Nonlinear free vibration behavior of functionally graded plates. Journal of sound and vibration, [s. l.], v. 289, n. 3, p. 595–611, 2006.
ZAOUI, F. Z.; OUINAS, D.; TOUNSI, A. New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations. Composites Part B: Engineering, [s. l.], v. 159, p. 231–247, 2019. Disponível em: https://doi.org/10.1016/j.compositesb.2018.09.051.
ZENKOUR, A. M. On vibration of functionally graded plates according to a refined trigonometric plate theory. International Journal of Structural Stability and Dynamics, [s. l.], v. 5, n. 2, p. 279–297, 2005.
ZHAO, X.; LEE, Y. Y.; LIEW, K. M. Free vibration analysis of functionally graded plates using the element-free kp-Ritz method. Journal of Sound and Vibration, [s. l.], v. 319, n. 3–5, p. 918–939, 2009.
ZHOU, D. et al. Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method. International Journal of Solids and Structures, [s. l.], v. 39, n. 26, p. 6339–6353, 2002.
