An analytical approach to large amplitude vibration and post-buckling of functionally graded beams rest on non-linear elastic foundation

• Autorzy: Yaghoobi H. , Torabi M.

Warianty tytułu

PL

Analityczne badania drgań o dużej amplitudzie i ugięcia po wyboczeniu belek z materiału gradientowego spoczywających na nieliniowo sprężystym podłożu

• Języki publikacji: EN

Abstrakty

EN

In this paper, non-linear vibration and post-buckling analysis of beams made of functionally graded materials (FGMs) rest on a non-linear elastic foundation subjected to an axial force are studied. Based on Euler-Bernoulli beam theory and von-Karman geometric non-linearity, the partial differential equation (PDE) of motion is derived.Then, this PDE problem is simplified into an ordinary differential equation problem by using the Galerkin method. Finally, the governing equation is analytically solved using the variational iteration metod (VIM). The results from the VIM solution are compared and shown to be in excellent agreement with the available solutions from the open literature. Some new results for the non-linear natural frequencies and buckling load of functionally graded (FG) beams, such as effects of vibration amplitude, elastic coefficients of foundation, axial force, end supports and material inhomogenity are presented for future references.

PL

W pracy przedstawiono analizę drgań nieliniowych i zjawisk następujących po wyboczeniu w belkach wykonanych z funkcjonalnych materiałów gradientowych (FGMs), spoczywających na nieliniowo sprężystym podłożu i jednocześnie poddanych osiowemu ściskaniu. Na podstawie teorii Eulera-Bernoulliego oraz przy uwzględnieniu geometrycznej nieliniowości von Karmana wyprowadzono cząstkowe równanie różniczkowe ruchu takich układów. Równanie to sprowadzono do postaci różniczkowej zwyczajnej za pomocą metody Galerkina. Na koniec, rozwiązano je analitycznie poprzez zastosowanie iteracyjnej metody wariacyjnej (VIM), a uzyskane rozwiązanie porównano z innymi, już istniejącymi i znanymi w literaturze, stwierdzając doskonałą zgodność. Otrzymano również nowe rezultaty w postaci określenia wpływu amplitudy drgań, sprężystości podłoża, wartości siły osiowej, rodzaju podparcia brzegów oraz niejednorodności materiału na częstości własne i obciążenie krytyczne belek gradientowych.

Słowa kluczowe

EN

functionally graded beams; non-linear vibration; post-buckling; Galerkin method; variational iteration method (VIM)

• Wydawca: Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej
• Czasopismo: Journal of Theoretical and Applied Mechanics
• Rocznik: 2013
• Tom: Vol. 51 nr 1
• Strony: 39--52
• Opis fizyczny: Bibliogr.35 poz., rys., tab.

Twórcy

autor

Yaghoobi H.

• Young Researchers Club, Central Tehran Branch, Islamic Azad University, Tehran, Iran

autor

Torabi M.

• Young Researchers Club, Central Tehran Branch, Islamic Azad University, Tehran, Iran

Bibliografia

• 1. Aydogdu M., 2007, Thermal buckling analysis of cross-ply laminated composite beams with general boundary conditions, Composites Science and Technology, 67, 1096-1104
• 2. Azrar L., Benamar R., White R.G., 1999, A 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, 183-207
• 3. Behdinan K., Stylianou M.C., Tabarrok B., 1997a, Dynamics of flexible sliding beams – nonlinear analysis. Part I: Formulation, Journal of Sound and Vibration, 208, 4, 517-539
• 4. Behdinan K., Tabarrok B., 1997b, Dynamics of flexible sliding beams – nonlinear analysis. Part II: Transient response, Journal of Sound and Vibration, 208, 4, 541-565
• 5. Bhashyam G.R., Prathap G., 1980, Galerkin finite element method for nonlinear beam vibrations, Journal of Sound and Vibration, 72, 91-203
• 6. Emam S.A., Nayfeh A.H., 2009, Postbuckling and free vibrations of composite beams, Composite Structures, 88, 636-642
• 7. Fallah A., 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
• 8. Fertis D.G., Afonta A., 1992, Free vibration of variable stiffness flexible bars, Computers and Structures, 43, 3, 445-450
• 9. Ganapathi M., Patel B.P., Saravanan J., Touratier M., 1998, Application of spline element for large-amplitude free vibrations, Composite Part B, 29B, 1-8
• 10. Gunda J.B., Gupta R.K., Janardhan G.R., Rao G.V., 2010, Large amplitude vibration analysis of composite beams: simple closed-form solutions, Composite Structures, 93, 870-879
• 11. Guo Q., Zhong H., 2004, Nonlinear vibration analysis of beams by a spline-based differentia quadrature method, Journal of Sound and Vibration, 269, 413-420
• 12. Gupta R.K., Jagadish Babu Gunda, Ranga Janardhan G., Venkateswara Rao G., 2009, Relatively simple finite element formulation for the large amplitude free vibrations of uniform beams, Finite Elements in Analysis and Design, 45, 624-631
• 13. Gupta R.K., Jagadish Babu Gunda, Ranga Janardhan G., Venkateswara Rao G., 2010a, Post-buckling analysis of composite beams: simple and accurate closed-form expressions, Composite Structures, 92, 8, 1947-1956
• 14. Gupta R.K., Jagadish Babu Gunda, Ranga Janardhan G., Venkateswara.Rao G., 2010b, Thermal post-buckling analysis of slender columns using the concept of coupled displacement field, International Journal of Mechanical Sciences, 52, 4, 590-594
• 15. Hatsunaga H., 2001, Vibration and buckling of multilayered composite beams according to high er order deformation theories, Journal of Sound and Vibration, 246, 47-62
• 16. He J.H., 1999, Variational iteration method - a kind of non-linear analytical technique: Some examples, International Journal of Non-Linear Mechanics, 34, 699-708
• 17. Jun L., Hongxing H., Rongying S., 2008, Dynamic stiffness analysis for free vibrations of axially loaded laminated composite beams, Computers and Structures, 84, 87-98
• 18. Kapania R.K., Raciti S., 1989, Nonlinear vibrations of unsymmetrically laminated beams, AIAA J., 27, 201-210
• 19. Karlik B., ¨ Ozkaya E., Aydin S., Pakdemirli M., 1998, Vibrations of a beam-mass system using artificial neural networks, Computers and Structures, 69, 3, 339-347
• 20. Ke L.L, Yang J., Kitipornchai S., 2010, An analytical study on the nonlinear vibration of functionally graded beams, Meccanica, 45, 743752, doi:10.1007/s11012-009-9276-1
• 21. Ma L.S., Lee D.W., 2011, A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading, Composite Structures, 93, 2, 831-842
• 22. Malekzadeh P., 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, 906-915
• 23. Mazzilli C.E.N., Soares M.E.S., Baracho Neto O.G.P., 2004, Non-linear normal modes of a simply supported beam: continuous system and finite element models, Computers and Structures 82, 2683-2691
• 24. Nayfeh A.H., Emam S.A., 2008, Exact solutions and stability of the postbuckling configurations of beams, Nonlinear Dynamics, Published on line on 23 February
• 25. Nayfeh A.H., Nayfeh S.A., 1995, Nonlinear normal modes of a continuous system with quadratic nonlinearities, Journal of Vibration and Acoustics – Trans. ASME, 117, 2, 199-205
• 26. Noda N., 1991, Thermal stresses in materials with temperature-dependent properties, Applied Mechanics Reviews, 44, 383-397
• 27. Patel B.P., Ganapathi M., Touratier M., 1999, Nonlinear free flexural vibrations/postbuckling analysis of laminated orthotropic beams/columns on a two parameter elastic foundation, Composite Structures, 46, 189-196
• 28. Pirbodaghi T., Ahmadian M.T., Fesanghary M., 2009, On the homotopy analysis metod for nonlinear vibration of beams, Mechanics Research Communications, 36, 2, 143-148
• 29. Qaisi M.I., 1993, Application of the harmonic balance principle to the nonlinear free vibration of beams, Applied Acoustics, 40, 141-151
• 30. Sapountzakis E.J., Tsiatas G.C., 2007, Elastic flexural buckling analysis of composite beams of variable cross-section by BEM, Engineering Structures, 29, 675-681
• 31. Simsek M., 2010, Non-linear vibration analysis of a functionally graded Timoshenko beam under action of a moving harmonic load, Composite Structures, 92, 2532-2546
• 32. Singh G., Venkateswara, Rao G., Iyengar N.G.R., 1991, Analysis of the nonlinear vibrations of unsymmetrically laminated composite beams, AIAA J., 29, 10, 1727-1735
• 33. Tanigawa Y., 1995, Some basic thermoelastic problems for nonhomogeneous structural materials, Applied Mechanics Reviews, 48, 287-300
• 34. Xie W.C., Lee H.P., Lin S.P., 2002, Normal modes of a nonlinear clamped-clamped beam, Journal of Sound and Vibration, 250, 2, 339-349
• 35. Younesian D., Askari H., Saadatnia Z., Kalami Yazdi M., 2010, Frequency analysis of stronglynonlineargeneralizedDuffingoscillators usingHe’s frequency-amplitude formulation and He’s energy balance method, Computers and Mathematics with Applications, 59, 3222-3228.