Haar wavelet technique applied on the functionally graded carbon nanotube reinforced conical shells to study free vibration and buckling behaviors in thermal environments

This article is provided to investigate the free vibrational behaviors accompanied by buckling characteristics of carbon nanotube-reinforced conical shells via the Haar wavelet technique. For a better performance in the thermal environments, materials used in the structure obey the temperature-dependent relations. On the other hand, carbon nanotubes are embedded across the thickness uniformly or functionally. Using Hamilton’s principle through the first-order shear deformation theory yields governing equations. Both dynamic and stability states are evaluated in the vicinity of equilibrium state and are also derived with regard to initial thermal stresses that are obtained via linear membrane technique in static analysis. The obtained partial differential equations are converted into ordinary ones with the aid of separating variable technique. Subsequently, the Haar wavelet approach is used to discretize the equations meridionally and transform them into new algebraic ones. Capability of this approach in calculation of frequencies with only a few numbers of the collocation points is demonstrated. Finally, to verify the integrity and precision of the proposed approach, some comparison studies are made with those of relevant results in the literature primarily. Thereafter, some effective parameters such as the geometry of nanotube, different boundary conditions, temperatures, and material properties are studied.