Nonlinear Free Vibration of Curved Double Walled Carbon Nanotubes Using Differential Quadrature Method

Abstract

Abstract In this study, nonlinear free vibration of a curved double walled carbon nanotube (DWCNT) is investigated, which are widely used in nano resonators, and sensors. Nonlinearities considered are due to large deflection of carbon nanotubes (geometric nonlinearity), initial curvature of tubes, and nonlinear interlayer van der Waals forces between inner and outer tubes. Assuming a harmonic solution in time domain, the differential quadrature method (DQM), is utilized to discretize the partial differential equations of motion in spatial domain, which resulted in a nonlinear set of algebraic equations of motion. The effect of nonlinearities, stiffness of the surrounding elastic medium, end conditions, and vibrational modes on the nonlinear free vibration of DWCNTs are investigated. The paper contribution can be divided into two main parts: in the first part, algebraic set of equation has been developed for the most general case considering different sources of nonlinearities and end conditions; hence, by selecting proper values for key parameters, the equation set can be converted into the equation of motion of desired application area. In the second part, the equations of motion are formulated using DQM in a matrix form which can be used in future modal and finite element analysis.