Nonlinear vibration of a double-walled carbon nanotube embedded in a polymer matrix

In the current work, the nonlinear vibration of an embedded double-walled carbon nanotube (DWCNT) aroused by nonlinear van der Waals (vdW) interaction forces from both surrounding medium and adjacent tubes is studied. Using both Euler–Bernoulli and Timoshenko beam models, the relation between deflection amplitudes and resonant frequencies of the DWCNT is derived through harmonic balance method. It is found that the nonlinear vdW forces from the surrounding medium result in noncoaxial vibration of the embedded DWCNT. The noncoaxial vibration includes both uni-directional and bi-directional vibration modes. It is found that the surrounding matrix has more prominent effect on the uni-directional vibration in comparison to the bi-directional vibration. The axial load effect on the vibrational behavior of the embedded DWCNT is also discussed. Due to the influence of the surrounding polymer, the prediction on the resonant frequencies of embedded CNTs is quite different from that for free-standing CNTs. A softening behavior for the deflection amplitude-resonant frequency relation is observed for the first time in the bi-directional vibration of the embedded DWCNT, which can only be obtained using the Timoshenko beam theory.