Nonlinear vibration of carbon nanotube embedded in viscous elastic matrix under parametric excitation by nonlocal continuum theory

• The amplitude–frequency response is presented by the multiple scale method.
• The gap between negative and positive bifurcation points can be enhanced by parametric load.
• The nonlocal continuum theory can present a more proper model.

In the present work, the nonlinear vibration of a carbon nanotube which is subjected to the external parametric excitation is studied. By the nonlocal continuum theory and nonlinear von Kármán beam theory, the governing equation of the carbon nanotube is derived with the consideration of the large deformation. The principle parametric resonance of the nanotube is discussed and the approximation explicit solution is presented by the multiple scale method. Numerical calculations are performed. It can be observed that when the mode number is 1, the stable region can be significantly changed by the parametric excitation, length-to-diameter ratio and matrix stiffness. This phenomenon becomes different to appear if the mode number increases. Moreover, the small scale effects have great influences on the positive bifurcation point for the short carbon nanotube, and the nonlocal continuum theory can present the proper model.

It can be observed that the gap between two bifurcation points becomes wider with the axial load increasing, which means the parametric excitation can enhance the stable region.Relation between the detuning parameter and response of principle parametric resonance for nanotube with the effects of parametric excitation. (a) F=0.0005 EAc, (b) F=0.001 EAc and (c) F=0.002 EAc.Figure optionsDownload as PowerPoint slide