Extremely large oscillation and nonlinear frequency of a multi-scale hybrid disk resting on nonlinear elastic foundation

This is a fundamental study on the nonlinear vibrations considering large amplitude in multi-sized hybrid Nano-composites (MHC) disk (MHCD) relying on nonlinear elastic media and located in an environment with gradually changed temperature feature. Carbon fibers (CF) or carbon nanotubes (CNTs) in the macro or nano sizes respectively are responsible for reinforcing the matrix. For prediction of the efficiency of the properties MHCD's modified Halpin-Tsai theory has been presented. The strain-displacement relation in multi-sized laminated disk's nonlinear dynamics through applying Von Karman nonlinear shell-theory and using third-order-shear-deformation-theory (TSDT) is determined. The energy methods called Hamilton's principle is applied for deriving the motion equations along with Boundary Conditions (BCs), which has ultimately been solved using the perturbation approach (PA) and generalized differential quadrature method (GDQM). At the final stage, the outcomes illustrate that patterns of FG, fibers' various directions, the WCNT and VF factors, top surface's applied temperature and temperature gradient have considerable impact on the MHCD's nonlinear dynamics. Another important consequence is that, the influences of the θ, WCNT and VF amounts on the disk's nonlinear vibrations may be taken into account at the larger amounts of the high deflection element and the negative axial load's impact on the structure's nonlinear dynamics is more extreme. A more general conclusion of this study is that for designing the MHCD should be more attention to the nonlinearity parameter.