Nonlinear primary and super-harmonic resonances of functionally graded carbon nanotube reinforced composite beams

Abstract

The nonlinear primary and super-harmonic resonances of functionally graded carbon nanotube reinforced composite (FG-CNTRC) beams under transverse harmonic excitation are studied. Taking into account four different distribution patterns of carbon nanotubes (CNTs), the material properties of FG-CNTRC beams are predicted through the extended rule of mixture. The equations of motion for the transverse, longitudinal and rotational motions are derived from Timoshenko beam theory and von Kármán type of geometric nonlinearity. These nonlinear partial differential equations are reduced into nonlinear ordinary differential equations by the Galerkin method. The discretized equations of motion are solved by using the incremental harmonic balance (IHB) method, resulting in the vibration responses of the nonlinear system. For all the three coupled motions, the frequency response curves of primary, 2 super-harmonic and 3 super-harmonic resonances are constructed. Comparison studies on the dimensionless natural frequencies, nonlinear to linear frequency ratios and numerical simulation of the frequency response are conducted to verify the correctness and accuracy of this study. The numerical results illustrate the influences of CNT distributions, CNT volume fraction, slenderness ratio, dimensionless transverse excitation and damping coefficient on the primary and super-harmonic resonant responses.