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Numerical study on the free vibration of carbon nanocones resting on elastic foundation using nonlocal shell model

ABSTRACT

Abstract Employing the variational differential quadrature method, the free vibration of carbon nanocones (CNCs) embedded in an elastic foundation, is studied based on nonlocal elasticity theory. On the basis of the first-order shear deformation theory, the energy functional of the CNC is presented and then discretized by employing the generalized differential quadrature method in the axial direction and periodic differential operators in the circumferential direction. According to Hamilton’s principle and using matrix relations, the reduced forms of mass and stiffness matrices are readily obtained. The results of present study are compared to those obtained by molecular mechanics to verify the proposed approach. In addition, the effects of nonlocal parameter, boundary conditions, semi-apex angle and both Winkler and Pasternak coefficients of elastic foundation are examined on the vibrational behavior of CNCs. The results indicate that the increase in nonlocal parameter and elastic foundation coefficients decreases and increases the fundamental frequency of CNCs, respectively.AbstractEmploying the variational differential quadrature method, the free vibration of carbon nanocones (CNCs) embedded in an elastic foundation, is studied based on nonlocal elasticity theory. On the basis of the first-order shear deformation theory, the energy functional of the CNC is presented and then discretized by employing the generalized differential quadrature method in the axial direction and periodic differential operators in the circumferential direction. According to Hamilton’s principle and using matrix relations, the reduced forms of mass and stiffness matrices are readily obtained. The results of present study are compared to those obtained by molecular mechanics to verify the proposed approach. In addition, the effects of nonlocal parameter, boundary conditions, semi-apex angle and both Winkler and Pasternak coefficients of elastic foundation are examined on the vibrational behavior of CNCs. The results indicate that the increase in nonlocal parameter and elastic foundation coefficients decreases and increases the fundamental frequency of CNCs, respectively.