OICC Press Journal of Solid Mechanics 2008-7683 12 4 2020 09 30 Investigation of Strain Gradient Theory for the Analysis of Free Linear Vibration of Nano Truncated Conical Shell 632 648 EN A.R Sheykhi Department of Mechanical Engineering , Science and Research Branch, Islamic Azad University, Tehran, Iran Sh Hosseini Hashemi School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran A Maghsoudpour Department of Mechanical Engineering , Science and Research Branch, Islamic Azad University, Tehran, Iran Sh Etemadi Haghighi Department of Mechanical Engineering , Science and Research Branch, Islamic Azad University, Tehran, Iran Journal Article 2020 09 30 In this paper the nano conical shell model is developed based on modified strain gradient theory. The governing equations of the nano truncated conical shell are derived using the FSDT, and the size parameters through modified strain gradient theory have been taken into account. Hamilton’s principle is used to obtain the governing equations, and the shell’s equations of motion are derived with partial differentials along with the classical and non-classical boundary conditions. Galerkin’s method and the Generalized Differential Quadrature (GDQ) approach are applied to obtain the linear free vibrations of the carbon nano cone (CNC). The CNC is studied with simply supported boundary condition. The results of the new model are compared with those of the classical and couple stress theories, which point to the conclusion that the classical and couple stress models are special cases of modified strain gradient theory. Results also reveal that rigidity of the nano truncated conical shell in the strain gradient theory is greater than that in the modified couple stress and classical theories respectively, which leads to an increase in dimensionless natural frequency ratio. Moreover, the study investigates the effect of the size parameters on nano shell vibration for different lengths and vertex angles.