A homogenization procedure and a physical discrete model for geometrically nonlinear transverse vibrations of a clamped beam made of a functionally graded material
Treść / Zawartość
Functionally graded materials are used in aircrafts, space vehicles and defence industries because of their good thermal resistance. Geometrically nonlinear free vibration of a functionally graded beam with clamped ends (FGCB) is modeled here by an N-dof discrete system presenting an equivalent isotropic beam, with effective bending and axial stiffness parameters obtained via a homogenization procedure. The discrete model is made of N masses placed at the ends of solid bars connected by rotational springs, presenting the flexural rigidity. Transverse displacements of the bar ends induce a variation in their lengths giving rise to axial forces modeled by longitudinal springs. The nonlinear semi-analytical model previously developed is used to reduce the vibration problem, via application of Hamilton’s principle and spectral analysis, to a nonlinear algebraic system involving the mass and rigidity tensors mij and kij and the nonlinearity tensor bijkl. The material properties of the (FGCB) examined is assumed to be graded according to a power rule of mixture in the thickness direction. The fundamental nonlinear frequency parameters found for the (FGCB) are in a good agreement with previously published results showing the validity of the present equivalent discrete model and its availability for further applications to non-uniform beam.
Bibliogr. 17 poz., rys., tab., wykr.
- LMPGI (Laboratoire de Mécanique Productique et Génie Industriel) Université Hassan II Ain Chock in Casablanca, Ecole Supérieure de Technologie, KM 7 Route El Jadida, B.P 8012 Oasis, Casablanca, Morocco
- ERSIM (Equipe d’Etudes et Recherches en Simulation, Instrumentation et Mesures) Mohammed V University in Rabat - Ecole Mohammadia des Ingénieurs, Avenue Ibn Sina, Agdal, Rabat, Morocco
- 1. Khor KA, Gu YW, Dong ZL, Srivatsan TS. et al. (Eds.), Composites and Functionally Graded Materials, 1997; 80:89-105.
- 2. Yung YY, Munz D, in: Shiota T, Miyamoto MY. (Eds.), Functionally Graded Material, 1996; 41-46.
- 3. Zhang Da-Guang, Zhou You-He, Computational Materials Science, in press, doi:10.1016/j.commatsci.2008.05.016.
- 4. Joshi S, Mukherjee A, Schmauder S. Computational Materials Science 2003; 28: 548-555.
- 5. Bao G, Wang L, Multiple cracking in functionally graded ceramic/metal coatings. International Journal of Solids and Structure 1995; 32: 2853-2871.
- 6. Mahamood RM, Akinlabi ET, Shukla M, Pityana S. Functionally graded material: An overview. 2012; http://researchspace.csir.co.za/dspace/handle/10204/65 48.
- 7. Ziane N, Meftah SA, Belhadj HA, Tounsi A, Bedia EAA. Free vibration analysis of thin and thick-walled FGM box beams. International Journal of Mechanical Sciences, 2013a; 66: 273-282. doi:10.1016/j.ijmecsci.2012.12.001.
- 8. Ziane N, Meftah SA, Belhadj HA, Tounsi A, Bedia EAA. Free vibration analysis of thin and thick-walled FGM box beams. International Journal of Mechanical Sciences 2013b; 66: 273-282. doi:10.1016/j.ijmecsci.2012.12.001.
- 9. Zerkane A, El Bikri K, Benamar R. A homogenization procedure for the free vibration analysis of functionally graded beams at large vibration amplitudes. International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering 2013; 17344: 2198-2201.
- 10. Rahmouni A, Beidouri Z, Benamar R. A discrete model for the natural frequencies and mode shapes of constrained vibrations of beams with various boundary conditions web of conferences MATEC http://dx.doi.org/10.1051/matecconf/20120100001.
- 11. Rahmouni A, Beidouri Z, Benamar R. A discrete model for geometrically nonlinear transverse free constrained vibrations of beams with various end conditions. Journal of Sound and Vibration 2013; 332(20): 5115-5134. http:// 10.1016/j.jsv.2013.04.011.
- 12. Rahmouni A and Benamar R. A discrete model for geometrically non-linear transverse free constrained vibrations of beams carrying a concentrated mass at various locations Eurodyn 2014.
- 13. Rahmouni A, Benamar R. A Non-linear transverse vibrations of clamped beams carrying two or three concentrated masses at various locations MATEC Web of Conferences 477, 83, DOI: 10.1051/05009 (2016) matecconf/ 2016 68305009 CSNDD 2016
- 14. Rahmouni A, Benamar R. A Examining non-linear transverse vibrations of clamped beams carrying n concentrated masses at various locations using discrete model International Journal of Computational Engineering Research (IJCER),2016; 6(12): 23-27.
- 15. Benamar R, Bennouna MMK, White RG. The effects of large vibration amplitudes on the fundamental mode shape of thin elastic structures, part I: Simply supported and clamped-clamped beams, Journal of Sound and Vibration 1991; 149: 179-195.
- 16. Kadiri ME, Benamar R, White RG. Improvement of the semi-analytical method for determining the geometrically nonlinear response of thin straight structures, part I: application to clamped-clamped and simply supported–clamped beams, Journal of Sound and Vibration 2002; 249 (2): 263-305.
- 17. Fallah A, Aghdam MM. Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation. European Journal of Mechanics - A/Solids 2011; 30: 571-583.