Nonlinear free vibration of carbon nanotube reinforced composite beams based on the third order beam theory

This paper investigates the linear and nonlinear free vibration of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) aligned in the beam axial direction. Two types of CNT distributions over the beam cross section are considered, i.e. functionally graded type Λ and V distributions. The third order shear deformable beam theory is employed and the geometric nonlinearity of von Kármán sense is included in this study. The energy functional of CNT composite beams is obtained using the classical variational method of Hamilton's principle and then is solved by the p-Ritz method. Nonlinear vibration characteristics of the CNT beams are presented and the effects of CNT filler volume fraction, distribution, beam length to depth ratio and end support conditions on nonlinear to linear frequency ratios are discussed. It is found that there are significant differences between the results using the first order beam theory and the results using the third order beam theory when both ends of the beam are soft-clamped.