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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

The Nonlinear Thermo-Mechanical Vibration of an FGM Beam on Winkler-Pasternak Foundation via Perturbation Analysis Method

Journal of Applied Nonlinear Dynamics 13(2) (2024) 337--350 | DOI:10.5890/JAND.2024.06.011

Nader Mohammadi$^{1}$, Mehrdad Nasirshoaibi$^{2}$

$^{1}$ Department of Mechanical Engineering, Islamic azad University Parand Branch, Parand, Tehran, Iran

$^{2}$ Department of Precision and Microsystems Engineering, Faculty of Mechanical, Maritime and Materials Engineering, Delft University of Technology, 2628 CD, Delft, the Netherlands

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Abstract

In this work, the perturbation analytical method is used to analyze the nonlinear free vibration of a functionally graded beam on the Winkler-Pasternak elastic basis exposed to mechanical and thermal loadings. The partial differential equation describing the beam's nonlinear motion is first developed under the assumption of the Euler-Bernoulli theory and with the aid of Von-strain-displacement Karman's relation. The Galerkin technique is then used to change the partial differential equation into a nonlinear ordinary differential equation. The ordinary differential equation is resolved using the analytical perturbation method. The effect of boundary conditions, the elastic foundation coefficients (including linear and nonlinear Winkler foundation moduli and elastic shear modulus foundation), axial force, and temperature on the nonlinear natural frequency of FG beams is studied. Results reveal that the presented analytical solution provides accurate predictions with minimum computational effort, as the first-order solution is enough to obtain results with the desired accuracy.

References

  1. [1]  Sankar, B.V. (2001), An elasticity solution for functionally graded beams, Composites Science and Technology, 61(5), 689-696.
  2. [2]  Zhong, Z. and Yu, T. (2007), Analytical solution of a cantilever functionally graded beam, Composites Science and Technology, 67(3-4), 481-488.
  3. [3]  Lu, C.F., Chen, W.Q., Xu, R.Q., and Lim, C.W. (2008), Semi-analytical elasticity solutions for bi-directional functionally graded beams, International Journal of Solids and Structures, 45(1), 258-275.
  4. [4]  Au, F.T.K., Zheng, D.Y., and Cheung, Y.K. (1999), Vibration and stability of non-uniform beams with abrupt changes of cross-section by using C1 modified beam vibration functions, Applied Mathematical Modelling, 23(1), 19-34.
  5. [5]  Ying, J., Lu, C.F., and Chen, W.Q. (2008), Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations, Composite Structures, 84(3), 209-219.
  6. [6]  Ke, L.L., Yang, J., and Kitipornchal, S. (2010), An analytical study on the nonlinear vibration of functionally graded beams, Meccanica, 45(6), 743-752.
  7. [7]  Simsek, M. (2010), Non-linear vibration analysis of a functionally graded Timoshenko beam under action of a moving harmonic load, Composite Structures, 92(10), 2532-2546.
  8. [8]  Fallah, A. and Aghdam, M.M. (2011), Nonlinear free vibration and post buckling analysis of functionally graded beams on nonlinear elastic foundation, European Journal of Mechanics - A/Solids, 30(4), 571-583.
  9. [9]  Librescu, L., Oh, S.Y., and Song, O. (2005), Thin-walled beams made of functionally graded materials and operating in a high temperature environment: vibration and stability, Journal of Thermal Stresses, 28(6-7), 649-712.
  10. [10]  Bian, Z.G., Lim, C.W., and Chen, W.Q. (2006), On functionally graded beams with integrated surface piezoelectric layers, Composite Structures, 72(3), 339-351.
  11. [11]  Bhangale, R.K. and Ganesan, N. (2006), Thermoelastic buckling and vibration behavior of a functionally graded sandwich beam with constrained viscoelastic core, Journal of Sound and Vibration, 295(1-2), 294-316.
  12. [12]  Bangera, K.M. and Chandrashekhara, K. (1991), Nonlinear vibration of moderately thick laminated beams using finite element method, Finite Elements in Analysis and Design, 9(4), 321-333.
  13. [13]  Malekzadeh, P. and Vosoughi, A.R. (2009), DQM large amplitude vibration of composite beams on nonlinear elastic foundations with restrained edges, Communications in Nonlinear Science and Numerical Simulation, 14(3), 906-915.
  14. [14]  He, J.H. (2007), Variational iteration method: Some recent results and new interpretations, Journal of Computational and Applied Mathematics, 207(1), 3-17.
  15. [15]  Younesian, D., Askari, H., Saadatnia, Z., and KalamiYazdi, M. (2010), Frequency analysis of strongly nonlinear generalized Duffing oscillators using He's frequency amplitude formulation and He's energy balance method, Computers $\&$ Mathematics with Applications, 59(9), 3222-3228.
  16. [16]  Pirbodaghi, T., Ahmadian, M.T., and Fesanghar, M. (2009), On the homotopy analysis method for non-linear vibration of beams, Mechanics Research Communications, 36(2), 143-148.
  17. [17]  Baghani, M., Jafari-Talookolaei, R.A., and Salarieh, H. (2011), Large amplitudes free vibrations and post-buckling analysis of un-symmetrically laminated composite beams on nonlinear elastic foundation, Applied Mathematical Modelling, 35(1), 130-138.
  18. [18]  Ding, H.J., Huang, D.J., and Chen, W.Q. (2007), Elasticity solutions for plane anisotropic functionally graded beams, International Journal of Solids and Structures, 44(1), 176-196.
  19. [19]  Hoang, V.N.V., Minh, V.T., Ninh, D.G., Nguyen, C.T., and Huy, V.L. (2020), Effects of non-uniform elastic foundation on the nonlinear vibration of nanocomposite plates in thermal environment using Selvadurai methodology, Composite Structures, 253, 112812.
  20. [20]  Ninh, D.G., Eslami, H., and Hoang, V.N.V. (2021), Dynamical behaviors of conveying-fluid nanocomposite toroidal shell segments with piezoelectric layer in thermal environment using the Reddy's third-order shear deformation shell theory, Thin-Walled Structures, 159, 107204.
  21. [21]  Hoang, V.N.V., Ninh, D.G., and Huy, V.L. (2021), A new structure study: Vibrational analyses of fgm convex-concave shells subjected to electro-thermal-mechanical loads surrounded by pasternak foundation, European Journal of Mechanics - A/Solids, 86, 104168.
  22. [22]  Hoang, V.N.V., Ninh, D.G., Truong, D.V., Bao, H.V., and Huy, V.L. (2021), behaviors of dynamics and stability standard of graphene nanoplatelet reinforced polymer corrugated plates resting on the nonlinear elastic foundations, Composite Structures, 260, 113253.
  23. [23]  Ninh, D.G., Quan, N.M., and Hoang, V.N.V. (2021), Thermally vibrational analyses of functionally graded graphene nanoplatelets reinforced funnel shells with different complex shapes surrounded by elastic foundation, Composite Structures, 29(26), 4654-4676.
  24. [24]  Emam, S.A. and Nayfeh, A.H. (2009), Postbuckling and free vibrations of composite beams, Composite Structures, 88, 636-642.
  25. [25]  Oh, S.Y., Librescu, L., and Song, O. (2003), Thermoelastic modeling and vibration of functionally graded thin-walled rotating blades, AIAA Journal, 41(10), 2051-2061.
  26. [26]  Rao, S.S. (2004), Mechanical Vibrations, fourth ed. Pearson Prentice Hall.
  27. [27]  He, J.H. (1999), Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 257-262.
  28. [28]  He, J.H. (2000), A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Non-Linear Mechanics, 35(1), 37-43.
  29. [29]  El-Shahed, M. (2005), Application of He's homotopy perturbation method to Volterra's integro-differential equation, International Journal of Nonlinear Sciences and Numerical Simulation, 6(2), 163-168.
  30. [30]  He, J.H. (2004), The homtopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathematics and Computation, 151(1), 287-292.
  31. [31]  He, J.H. (2005), Homotopy perturbation method for bifurcation of nonlinear problems, International Journal of Nonlinear Sciences and Numerical Simulation, 6(2), 207-208.
  32. [32]  He, J.H. (2005), Periodic solutions and bifurcations of delay-differential equations, Physics Letters A, 347(4-6), 228-230.
  33. [33]  He, J.H. (2005), Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals, 26(3), 695-700.
  34. [34]  He, J.H. (2006), Homotopy perturbation method for solving boundary value problems, Physics Letters A, 30(1-2), 87-88.
  35. [35]  Odibat, Z. and Momani, S. (2008), Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos, Solitons $\&$ Fractals, 36, 167-174.
  36. [36]  Azrar, L., Benamar, R., and White, R.G. (1999), Semi-analytical approach to the nonlinear 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, 224(2), 183-207.
  37. [37]  Qaisi, M.I. (1993), Application of the harmonic balance principle to the nonlinear free vibration of beams, Applied Acoustics, 40(2), 141-151.
  38. [38]  Lai, S.K., Lam, C.W., Wu, B.S., Wang, C., Zeng, Q.C., and He, X.F. (2008), Newton-harmonic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators, Applied Mathematical Modelling, 33(2), 852-866.

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