Nonlinear free vibration of double walled carbon nanotubes by using describing function method with multiple trial functions

In this paper, nonlinear free vibration of double walled carbon nanotubes (DWCNTs) embedded in an elastic medium with geometric nonlinearity and interlayer van der Waals force nonlinearity are studied. The motion of the DWCNT is represented by multiple eigenfunctions of the linear system which are referred as trial functions. Describing function method (DFM) is employed in order to represent the nonlinear forces as a multiplication of a nonlinear stiffness matrix and a displacement vector, which made it possible to identify when it is necessary to consider multiple trial functions. The results revealed that it is important to consider multiple trial functions, especially in case of out-of-phase vibrations of DWCNTs with interlayer van der Waals force nonlinearity. The effects of number of trial functions and medium stiffness on the free vibration of DWCNTs are investigated. It is observed that utilizing multiple trial functions made it possible to detect different vibration modes occurring at a single vibration frequency.

Graphical abstractDescribing function method (DFM) with multiple trial functions are used to determine nonlinear free vibration frequencies of DWCNTs with geometric and vdW force nonlinearities.Figure optionsDownload full-size imageDownload as PowerPoint slideHighlights► Describing function method (DFM) is used for nonlinear free vibration of CNTs. ► Multiple trial functions are used to determine nonlinear free vibration frequencies. ► Use of DFM determines whether multiple trial functions are necessary or not. ► Single trial function is sufficient to consider geometric nonlinearity. ► Multiple trial functions are required for van der Waals force nonlinearity.