Abstract
This paper deals with the 3/2 superharmonic resonance and 1/2 subharmonic resonance of functionally graded carbon nanotube reinforced composite (FG-CNTRC) beams. In the framework of Timoshenko beam theory and von Kármán type of geometric nonlinearity, the nonlinear equations of motion for FG-CNTRC beams are derived by Hamilton’s principle and discretized by the Galerkin method. The incremental harmonic balance (IHB) method is used to calculate the dynamic responses of FG-CNTRC beams subjected to transverse harmonic excitation. The stability of steady state solutions obtained from the IHB method is evaluated by Floquet theory. It is found that when the period-1 solutions of FG-CNTRC beams with asymmetric carbon nanotube distribution lose stabilities and the period doubling bifurcation occurs, the nonlinear system will generate 3/2 superharmonic resonance and 1/2 subharmonic resonance respectively in two specified frequency ratio intervals. The frequency response curves of 3/2 superharmonic resonance and 1/2 subharmonic resonance are constructed by the IHB method. The numerical results reveal the effects of material, geometry and excitation parameters on the 3/2 superharmonic and 1/2 subharmonic resonant responses of the system. Additionally, the two unstable regions where the 3/2 superharmonic and 1/2 subharmonic resonances occur are determined under different parameters.
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