Abstract
This paper aims to obtain the analytical free vibration solution of an orthotropic rectangular thin plate using the finite integral transformation. Due to the mathematical difficulty of complex boundary value problems, it is very hard to solve the title problem with common analytical methods. By imposing the integral transformation, the high-order partial differential equation with specified boundary conditions is converted into linear algebraic equation and the exact solutions with high precision and fast convergence are obtained elegantly. The main advantage of the proposed method is that it is simple and general, and it does not need to pre-determine the deflection function, which makes the solution procedure more reasonable. The method has a wide range of applications and can deal with other elastic plate problems, such as buckling and bending. The present results are validated by comparison with the existing analytical solutions that showed satisfactory agreement. The new analytical solution obtained can be used as a benchmark for validating other numerical and approximate methods.
Similar content being viewed by others
References
Biancolini, M.E., Brutti, C., Reccia, L.: Approximate solution for free vibrations of thin orthotropic rectangular plates. J. Sound Vib. 288, 321–344 (2005). https://doi.org/10.1016/j.jsv.2005.01.005
Rossi, R.E., Bambill, D.V., Laura, P.A.A.: Vibrations of a rectangular orthotropic plate with a free edge: a comparison of analytical and numerical results. Ocean Eng. 25, 521–527 (1998). https://doi.org/10.1016/S0029-8018(97)00022-X
Chen, W.Q., Lü, C.F.: 3D free vibration analysis of cross-ply laminated plates with one pair of opposite edges simply supported. Compos. Struct. 69, 77–87 (2005). https://doi.org/10.1016/j.compstruct.2004.05.015
Richard, J.: Sylvester: buckling of sandwich cylinders under axial load. J. Aerosp. Sci. 29, 863–872 (1962). https://doi.org/10.2514/8.9620
Bhat, R.B.: Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh–Ritz method. J. Sound Vib. 102, 493–499 (1985)
Dickinson, S.M., Di Blasio, A.: On the use of orthogonal polynomials in the Rayleigh–Ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates. J. Sound Vib. 108, 51–62 (1986)
Eftekhari, S.A., Jafari, A.A.: Accurate variational approach for free vibration of simply supported anisotropic rectangular plates. Arch. Appl. Mech. 84, 607–614 (2014). https://doi.org/10.1007/s00419-013-0812-z
Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C., Jorge, R.M.N., Soares, C.M.: Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique. Compos. B Eng. 44, 657–674 (2013)
Zhou, D., Cheung, Y.K., Kong, J.: Free vibration of thick, layered rectangular plates with point supports by finite layer method. Int. J. Solids Struct. 37, 1483–1499 (2000)
Bodaghi, M., Saidi, A.R.: Thermoelastic buckling behavior of thick functionally graded rectangular plates. Arch. Appl. Mech. 81, 1555–1572 (2011)
Mercan, K., Civalek, Ö.: Buckling analysis of Silicon carbide nanotubes (SiCNTs) with surface effect and nonlocal elasticity using the method of HDQ. Compos. B Eng. 114, 34–45 (2017). https://doi.org/10.1016/j.compositesb.2017.01.067
Demir, Ç., Civalek, Ö.: A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix. Compos. Struct. 168, 872–884 (2017). https://doi.org/10.1016/j.compstruct.2017.02.091
Civalek, Ö.: Numerical solutions to the free vibration problem of Mindlin sector plates using the discrete singular convolution method. Int. J. Struct. Stab. Dyn. 9, 267–284 (2009)
Civalek, Ö., Ozturk, B.: Vibration analysis of plates with curvilinear quadrilateral domains by discrete singular convolution method. Struct. Eng. Mech. 36, 279–299 (2010)
Mercan, K., Civalek, Ö.: DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix. Compos. Struct. 143, 300–309 (2016). https://doi.org/10.1016/j.compstruct.2016.02.040
Demir, Ç., Mercan, K., Civalek, Ö.: Determination of critical buckling loads of isotropic, FGM and laminated truncated conical panel. Compos. B Eng. 94, 1–10 (2016). https://doi.org/10.1016/j.compositesb.2016.03.031
Civalek, Ö.: Free vibration of carbon nanotubes reinforced (CNTR) and functionally graded shells and plates based on FSDT via discrete singular convolution method. Compos. B Eng. 111, 45–59 (2017). https://doi.org/10.1016/j.compositesb.2016.11.030
Akgoz, B., Civalek, O.: Nonlinear vibration analysis of laminated plates resting on nonlinear two-parameters elastic foundations. Steel Compos. Struct. 11(5), 403–421 (2011)
Wu, Y., Xing, Y., Liu, B.: Analysis of isotropic and composite laminated plates and shells using a differential quadrature hierarchical finite element method. Compos. Struct. 205, 11–25 (2018). https://doi.org/10.1016/j.compstruct.2018.08.095
Thai, H.-T., Kim, S.-E.: A review of theories for the modeling and analysis of functionally graded plates and shells. Compos. Struct. 128, 70–86 (2015). https://doi.org/10.1016/j.compstruct.2015.03.010
Liew, K.M., Zhao, X., Ferreira, A.J.M.: A review of meshless methods for laminated and functionally graded plates and shells. Compos. Struct. 93, 2031–2041 (2011). https://doi.org/10.1016/j.compstruct.2011.02.018
Bui, T.Q., Nguyen, M.N., Zhang, C.: Buckling analysis of Reissner–Mindlin plates subjected to in-plane edge loads using a shear-locking-free and meshfree method. Eng. Anal. Boundary Elem. 35, 1038–1053 (2011). https://doi.org/10.1016/j.enganabound.2011.04.001
Zhao, X., Liew, K.M.: Free vibration analysis of functionally graded conical shell panels by a meshless method. Compos. Struct. 93, 649–664 (2011)
Fallah, A., Aghdam, M.M., Kargarnovin, M.H.: Free vibration analysis of moderately thick functionally graded plates on elastic foundation using the extended Kantorovich method. Arch. Appl. Mech. 83, 177–191 (2013). https://doi.org/10.1007/s00419-012-0645-1
Sakata, T., Takahashi, K., Bhat, R.B.: Natural frequencies of orthotropic rectangular plates obtained by iterative reduction of the partial differential equation. J. Sound Vib. 189, 89–101 (1996). https://doi.org/10.1006/jsvi.1996.9999
Singhatanadgid, P., Taranajetsada, P.: Vibration analysis of stepped rectangular plates using the extended Kantorovich method. Mech. Adv. Mater. Struct. 23, 201–215 (2016). https://doi.org/10.1080/15376494.2014.949922
Park, S.-J.: Three-dimensional free vibration analysis of orthotropic plates. J. Korean Soc. Disaster Inf. 10, 1–14 (2014)
Ganesh, S., Kumar, K.S., Mahato, P.K.: Free vibration analysis of delaminated composite plates using finite element method. Procedia Eng. 144, 1067–1075 (2016). https://doi.org/10.1016/j.proeng.2016.05.061
Hu, N., Fukunaga, H., Kameyama, M., Aramaki, Y., Chang, F.K.: Vibration analysis of delaminated composite beams and plates using a higher-order finite element. Int. J. Mech. Sci. 44, 1479–1503 (2002)
Ju, F., Lee, H.P., Lee, K.H.: Finite element analysis of free vibration of delaminated composite plates. Compos. Eng. 5, 195–209 (1995). https://doi.org/10.1016/0961-9526(95)90713-L
Rock, T.A., Hinton, E.: A finite element method for the free vibration of plates allowing for transverse shear deformation. Comput. Struct. 6, 37–44 (1976)
Bahmyari, E., Rahbar-Ranji, A.: Free vibration analysis of orthotropic plates with variable thickness resting on non-uniform elastic foundation by element free Galerkin method. J. Mech. Sci. Technol. 26, 2685–2694 (2012)
Talebitooti, R., Zarastvand, M., Rouhani, A.S., Talebitooti, R., Zarastvand, M., Rouhani, A.S.: Investigating Hyperbolic Shear Deformation Theory on vibroacoustic behavior of the infinite Functionally Graded thick plate. Latin Am. J. Solids Struct. (2019). https://doi.org/10.1590/1679-78254883
Talebitooti, R., Gohari, H.D., Zarastvand, M.R.: Multi objective optimization of sound transmission across laminated composite cylindrical shell lined with porous core investigating non-dominated sorting genetic algorithm. Aerosp. Sci. Technol. 69, 269–280 (2017). https://doi.org/10.1016/j.ast.2017.06.008
Talebitooti, R., Zarastvand, M.R.: The effect of nature of porous material on diffuse field acoustic transmission of the sandwich aerospace composite doubly curved shell. Aerosp. Sci. Technol. 78, 157–170 (2018). https://doi.org/10.1016/j.ast.2018.03.010
Talebitooti, R., Zarastvand, M.R., Gohari, H.D.: The influence of boundaries on sound insulation of the multilayered aerospace poroelastic composite structure. Aerosp. Sci. Technol. 80, 452–471 (2018). https://doi.org/10.1016/j.ast.2018.07.030
Talebitooti, R., Zarastvand, M.R.: Vibroacoustic behavior of orthotropic aerospace composite structure in the subsonic flow considering the Third order Shear Deformation Theory. Aerosp. Sci. Technol. 75, 227–236 (2018). https://doi.org/10.1016/j.ast.2018.01.011
Talebitooti, R., Zarastvand, M., Darvishgohari, H.: Multi-objective optimization approach on diffuse sound transmission through poroelastic composite sandwich structure. J Sandw. Struct. Mater. (2019). https://doi.org/10.1177/1099636219854748
Ghassabi, M., Talebitooti, R., Zarastvand, M.R.: State vector computational technique for three-dimensional acoustic sound propagation through doubly curved thick structure. Comput. Methods Appl. Mech. Eng. 352, 324–344 (2019). https://doi.org/10.1016/j.cma.2019.04.011
Talebitooti, R., Johari, V., Zarastvand, M., Talebitooti, R., Johari, V., Zarastvand, M.: Wave transmission across laminated composite plate in the subsonic flow Investigating Two-variable Refined Plate Theory. Latin Am. J. Solids Struct. (2018). https://doi.org/10.1590/1679-78254352
Thai, H.-T., Kim, S.-E.: Levy-type solution for free vibration analysis of orthotropic plates based on two variable refined plate theory. Appl. Math. Model. 36, 3870–3882 (2012). https://doi.org/10.1016/j.apm.2011.11.003
Hosseini Hashemi, S., Atashipour, S.R., Fadaee, M.: An exact analytical approach for in-plane and out-of-plane free vibration analysis of thick laminated transversely isotropic plates. Arch Appl Mech. 82, 677–698 (2012). https://doi.org/10.1007/s00419-011-0583-3
Bodaghi, M., Saidi, A.R.: Stability analysis of functionally graded rectangular plates under nonlinearly varying in-plane loading resting on elastic foundation. Arch. Appl. Mech. 81, 765–780 (2011). https://doi.org/10.1007/s00419-010-0449-0
Gorman, D.J.: Free vibration analysis of cantilever plates by the method of superposition. J. Sound Vib. 49, 453–467 (1976). https://doi.org/10.1016/0022-460x(76)90828-2
Xing, Y.F., Liu, B.: Exact solutions for the free in-plane vibrations of rectangular plates. Int. J. Mech. Sci. 51, 246–255 (2009). https://doi.org/10.1016/j.ijmecsci.2008.12.009
Liu, B., Xing, Y.: Exact solutions for free vibrations of orthotropic rectangular Mindlin plates. Compos. Struct. 93, 1664–1672 (2011). https://doi.org/10.1016/j.compstruct.2011.01.014
Li, R., Tian, Y., Wang, P., Shi, Y., Wang, B.: New analytic free vibration solutions of rectangular thin plates resting on multiple point supports. Int. J. Mech. Sci. 110, 53–61 (2016)
Li, R., Wang, B., Li, G., Du, J., An, X.: Analytic free vibration solutions of rectangular thin plates point-supported at a corner. Int. J. Mech. Sci. 96, 199–205 (2015)
Li, R., Wang, P., Xue, R., Guo, X.: New analytic solutions for free vibration of rectangular thick plates with an edge free. Int. J. Mech. Sci. 131, 179–190 (2017)
Timoshenko, S.P.: Theory of Elastic Stability, 2nd edn. McGraw-Hill Book Company, New York (1961)
Yao, W., Zhong, W., Lim, C.W.: Symplectic Elasticity. World Scientific, Singapore (2009)
Lim, C.W., Xu, X.S.: Symplectic elasticity: theory and applications. Appl. Mech. Rev. 63, 050802 (2010). https://doi.org/10.1115/1.4003700
Li, R., Zheng, X., Yang, Y., Huang, M., Huang, X.: Hamiltonian system-based new analytic free vibration solutions of cylindrical shell panels. Appl. Math. Model. 76, 900–917 (2019). https://doi.org/10.1016/j.apm.2019.07.020
Zheng, X., Sun, Y., Huang, M., An, D., Li, P., Wang, B., Li, R.: Symplectic superposition method-based new analytic bending solutions of cylindrical shell panels. Int. J. Mech. Sci. 152, 432–442 (2019). https://doi.org/10.1016/j.ijmecsci.2019.01.012
Xing, Y.F., Xu, T.F.: Solution methods of exact solutions for free vibration of rectangular orthotropic thin plates with classical boundary conditions. Compos. Struct. 104, 187–195 (2013)
Liu, B., Xing, Y.: Exact solutions for free vibrations of orthotropic rectangular Mindlin plates. Compos. Struct. 93, 1664–1672 (2011)
Liu, B., Xing, Y.F., Reddy, J.N.: Exact compact characteristic equations and new results for free vibrations of orthotropic rectangular Mindlin plates. Compos. Struct. 118, 316–321 (2014)
Xing, Y.F., Liu, B.: New exact solutions for free vibrations of thin orthotropic rectangular plates. Compos. Struct. 89, 567–574 (2009). https://doi.org/10.1016/j.compstruct.2008.11.010
Li, R., Zhong, Y., Tian, B., Liu, Y.: On the finite integral transform method for exact bending solutions of fully clamped orthotropic rectangular thin plates. Appl. Math. Lett. 22, 1821–1827 (2009). https://doi.org/10.1016/j.aml.2009.07.003
Tian, B., Li, R., Zhong, Y.: Integral transform solutions to the bending problems of moderately thick rectangular plates with all edges free resting on elastic foundations. Appl. Math. Model. 39, 128–136 (2015)
Zhong, Y., Zhao, X.-F., Li, R.: Free vibration analysis of rectangular cantilever plates by finite integral transform method. Int. J. Comput. Methods Eng. Sci. Mech. 14, 221–226 (2013). https://doi.org/10.1080/15502287.2012.711424
Zhong, Y., Zhao, X., Liu, H.: Vibration of plate on foundation with four edges free by finite cosine integral transform method. Latin Am. J. Solids Struct. 11, 854–863 (2014). https://doi.org/10.1590/S1679-78252014000500008
Bidgoli, A.M.M., Daneshmehr, A.R., Kolahchi, R.: Analytical bending solution of fully clamped orthotropic rectangular plates resting on elastic foundations by the finite integral transform method. J. Appl. Comput, Mech (2015). https://doi.org/10.22055/jacm.2014.10742
Ike, C.: Flexural analysis of Kirchhoff plates on Winkler foundations using finite Fourier sine integral transform method. Math. Model. Eng. Problems 4, 145–154 (2017). https://doi.org/10.18280/mmep.040402
Nwoji, C.U., Mama, B.O., Onah, H.N., Ike, C.C.: Flexural analysis of simply supported rectangular Mindlin plates under bisinusoidal transverse load. APRN J. Eng. Appl. Sci. 13, 4480–4488 (2018)
Simulia, D.S.: Abaqus 6.13 Analysis User’s Guide. Dassault Systems, Providence (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhang, J., Ullah, S. & Zhong, Y. Accurate free vibration solutions of orthotropic rectangular thin plates by straightforward finite integral transform method. Arch Appl Mech 90, 353–368 (2020). https://doi.org/10.1007/s00419-019-01613-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-019-01613-1