Nonlinear and chaotic vibrations of FG double curved sandwich shallow shells resting on visco-elastic nonlinear Hetenyi foundation under combined resonances

Abstract

In this study, the nonlinear and chaotic instability of functionally graded (FG) double curved shallow sandwich shells resting on a viscoelastic Hetenyi foundation, under simultaneous effect of in-plane and transverse excitations is studied. Employing the third-order Reddy theory and von-Karman relations and the Hamilton principle, the partial differential equations of motion under movable SS boundary conditions are derived. Introducing the trigonometric Airy stress function and applying the Galerkin’s method, the equations are reduced to a set of nonlinear ODEs with time. Then, under forcing resonance conditions and using the perturbation method, the modulation equations at the stationary conditions are derived and solved numerically. Then stability of non-trivial solutions for the resonance amplitude corresponding to the presence of limit cycle oscillation is investigated. Then by extracting the characteristic resonance amplitude curves, the effect of various parameters, including: frequency detuning parameter, in-plane excitation amplitude, linear, shear and nonlinear stiffness and damping parameters of the foundation on nonlinear response are analyzed. Finally, by extracting two degrees of freedom system time response, bifurcation and chaotic characteristic curves of the problem, the conditions for occurrence the periodic, double periodic, multi-periodic and chaotic behaviors under the simultaneous effect of parametric and internal resonances are studied comprehensively.