Abstract
Flexural vibration analysis of beams made of functionally graded materials (FGMs) with various boundary conditions is considered in this paper. Due to technical problems during FGM fabrication, porosities and micro-voids can be created inside FGM samples which may lead to the reduction in density and strength of materials. In this investigation, the FGM beams are assumed to have even and uneven distributions of porosities over the beam cross-section. The modified rule of mixture is used to approximate material properties of the FGM beams including the porosity volume fraction. In order to cover the effects of shear deformation, axial and rotary inertia, the Timoshenko beam theory is used to form the coupled equations of motion for describing dynamic behavior of the beams. To solve such a problem, Chebyshev collocation method is employed to find natural frequencies of the beams supported by different end conditions. Based on numerical results, it is revealed that FGM beams with even distribution of porosities have more significant impact on natural frequencies than FGM beams with uneven porosity distribution.
Similar content being viewed by others
References
Suresh S, Mortensen A (1998) Fundamental of functionally graded materials. Maney, London
Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG (1999) Functionally graded materials: design, processing and application. Kluwer Academic Publishers, London
Zhu J, Lai Z, Yin Z, Jeon J, Lee S (2001) Fabrication of ZrO2–NiCr functionally graded material by powder metallurgy. Mater Chem Phys 68:130–135
Wattanasakulpong N, Prusty BG, Kelly DW, Hoffman M (2012) Free vibration analysis of layered functionally graded beams with experimental validation. Mater Design 36:182–190
Sankar BV (2001) An elasticity solution for functionally graded beams. Compos Sci Technol 61:689–696
Zhong Z, Yu T (2007) Analytical solution of a cantilever functionally graded beam. Compos Sci Technol 67:481–488
Kang YA, Li XF (2009) Bending of functionally graded cantilever beams with power-law non-linearity subjected to an end force. Int J Non Linear Mech 44:696–703
Kapuria S, Bhattacharyya M, Kumar AN (2008) Bending and free vibration response of layered functionally graded beams: a theoretical model and its experimental validation. Compos Struct 82:390–402
Aydogdu M, Taskin V (2007) Free vibration analysis of functionally graded beams with simply-supported edges. Mater Design 36:182–190
Xiang HJ, Yang J (2008) Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction. Compos Part B Eng 39:292–303
Sina SA, Navazi HM, Haddadpour H (2009) An analytical method for free vibration analysis of functionally graded beams. Mater Design 30:741–747
Simsek M (2010) Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nuclear Eng Design 240:697–705
Thai HT, Vo TP (2012) Bending and free vibration of functionally graded beams using various higher-order shear deformation theories. Int Mech Sci 62:57–66
Wattanasakulpong N, Prusty BG, Kelly DW (2011) Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams. Int J Mech Sci 53:734–743
Giunta G, Crisafulli D, Belouettar S, Carrera E (2011) Hierarchical theories for the free vibration analysis of functionally graded beams. Compos Struct 94:68–74
Alshorbagy AE, Eltaher MA, Mahmoud FF (2011) Free vibration characteristics of a functionally graded beam by finite element method. Appl Math Model 35:412–425
Su H, Banerjee JR, Cheung CW (2013) Dynamic stiffness formulation and free vibration analysis of functionally graded beams. Compos Struct 106:854–862
Pradhan KK, Chakraverty S (2013) Free vibration and Timoshenko functionally graded beams by Rayleigh-Ritz method. Compos Part B 51:175–184
Suddoung K, Charoensuk J, Wattanasakulpong N (2014) Vibration response of stepped. Appl Acoust 77:20–28
Rajasekaran S (2013) Buckling and vibration of axially functionally graded nonuniform beams using differential transformation based dynamic stiffness approach. Meccanica 48:1053–1070
Vo TP, Thai HT, Nguyen TK, Inam F (2014) Static and vibration analysis of functionally graded beams using refined shear deformation theory. Meccanica 49:155–168
Rajasekaran S, Tochaei EN (2014) Free vibration analysis of axially functionally graded tapered Timoshenko beams using differential transformation element method and differential quadrature element method of lowest-order. Meccanica 49:995–1009
Wattanasakulpong N, Ungbhakorn V (2014) Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities. Aerosp Sci Technol 32:111–120
Fox L, Parker IB (1968) Chebyshev polynomials in numerical analysis. Oxford University Press, London
Elbarbary EME (2005) Chebyshev finite difference method for the solution of boundary-layer equations. Appl Math Comput 160:487–498
Celik I (2005) Approximate computation of eigenvalues with Chebyshev collocation method. Appl Math Comput 160:401–410
Biazar J, Ebrahimi H (2012) Chebyshev wavelets approach for nonlinear systems of Volterra integral equations. Comput Math Appl 63:608–616
Doha EH, Abd-Elhameed WM, Bassuony MA (2013) New algorithms for solving high even-order differential equations using third and fourth Chebyshev-Galerkin method. J Comput Phys 236:563–579
Lin CH, Jen MHR (2005) Analysis of a laminated anisotropic plate by Chebyshev collocation method. Compos B 36:155–169
Lee J, Schultz WW (2004) Eigenvalue analysis of Timoshenko beams and axisymmetric Mindlin plates by the pseudospectral method. J Sound Vib 269:609–621
Sari MS, Butcher EA (2010) Natural frequencies and critical loads of beams and columns with damaged boundaries using Chebyshev polynomials. Int J Eng Sci 48:862–873
Sari MS, Butcher EA (2012) Free vibration analysis of non-rotating and rotating Timoshenko beams with damaged boundaries using the Chebyshev collocation method. Int J Mech Sci 60:1–11
Delale F, Erdogan F (1983) The crack problem for a nonhomogeneous plane. ASME J Appl Mech 50:609–614
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wattanasakulpong, N., Chaikittiratana, A. Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method. Meccanica 50, 1331–1342 (2015). https://doi.org/10.1007/s11012-014-0094-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-014-0094-8