Nonlinear vibrations of functionally graded doubly curved shallow shells

• Published in: Journal of sound and vibration Vol. 330; no. 7; pp. 1432 - 1454
• Main Authors: Alijani, F., Amabili, M., Karagiozis, K., Bakhtiari-Nejad, F.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 28-03-2011; Elsevier
• Subjects: Bifurcations; Curved; Equations of motion; Exact sciences and technology; Functionally gradient materials; Fundamental areas of phenomenology (including applications); Galerkin methods; Nonlinearity; Physics; Shallow shells; Shells; Solid mechanics; Structural and continuum mechanics; Vibration; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Double curvature; Mode coupling; Poincaré map; Equation of motion; Volume fraction; Time frequency domain method; Forced vibration; Simple support; Modeling; Quadratic equation; Natural frequency; Primary resonance; Subharmonic; Shallow shell; Multiscale method; Functionnally graded material; Galerkin method; Non linear effect; Arc length; Structural analysis
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/j.jsv.2010.10.003

Nonlinear forced vibrations of FGM doubly curved shallow shells with a rectangular base are investigated. Donnell’s nonlinear shallow-shell theory is used and the shell is assumed to be simply supported with movable edges. The equations of motion are reduced using the Galerkin method to a system of infinite nonlinear ordinary differential equations with quadratic and cubic nonlinearities. Using the multiple scales method, primary and subharmonic resonance responses of FGM shells are fully discussed and the effect of volume fraction exponent on the internal resonance conditions, softening/hardening behavior and bifurcations of the shallow shell when the excitation frequency is (i) near the fundamental frequency and (ii) near two times the fundamental frequency is shown. Moreover, using a code based on arclength continuation method, a bifurcation analysis is carried out for a special case with two-to-one internal resonance between the first and second doubly symmetric modes with respect to the panel’s center (
ω
13≈2
ω
11). Bifurcation diagrams and Poincaré maps are obtained through direct time integration of the equations of motion and chaotic regions are shown by calculating Lyapunov exponents and Lyapunov dimension.