This paper investigates the nonlinear forced vibration of doubly curved sandwich nanoshells with auxetic honeycomb core having negative Poisson's ratio and nanocomposite-reinforced coatings. It is assumed that the nanoshell structure rests on Winkler-Pasternak foundation. The dynamic response is analyzed under various periodic and impulsive pressure excitations. The basic governing, compatibility, and constitutive equations are derived in the context of the non-classical nonlocal strain gradient elasticity theory to rigorously account for the size-dependent effects in nonlinear dynamic response at nanoscale. Utilizing the first-order shear deformation theory, representing the strain components in terms of the deformation field and its derivatives, the derived governing equations can be expressed in a system of three-dimensional nonlinear partial differential equations. To achieve a closed-form solution avoiding the complexities of the numerical iterative methods in presence of geometrical nonlinearities, the considered shallow nanoshell panels are taken as simply supported at their different fixed or moveable states. Further, assuming an appropriate approximation for the deformation field, utilizing the Galerkin method, the governing partial differential equations are reduced to an explicit formulation of the corresponding ordinary differential equation of motion which is numerically solved by the Runge-Kutta (RK) method. In numerical investigations, the accuracy and reliability of the proposed analytical-numerical approach are first validated by comparing the obtained results with benchmark solutions available in the literature. Subsequently, the influence of various parameters, including mechanical and geometrical properties, boundary conditions, and different periodic and impulsive external pressure excitations, on the geometrically nonlinear forced vibration behavior of the nanoshell panels is systematically analyzed using the developed solution methodology.
