Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams

This research deals with the forced vibration behavior of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on the Timoshenko beam theory along with von Kármán geometric nonlinearity. For the carbon-nanotube reinforced composite (CNTRC) beams, uniform distribution (UD) and three types of functionally graded (FG) distribution patterns of SWCNT reinforcements are considered. It is assumed that the material properties of FG-CNTRC beams are graded in the thickness direction and estimated through the rule of mixture. The nonlinear governing equations and corresponding boundary conditions are derived based on the Hamilton principle and discretized by means of the generalized differential quadrature (GDQ) method. After that, a Galerkin-based numerical technique is employed to reduce the set of nonlinear governing equations into a time-varying set of ordinary differential equations of Duffing type. Since the nanobeam responds periodically to harmonic excitations, a set of periodic differential matrix operators is introduced to discretize the Duffing equations on the time domain using the derivatives of a periodic base function. The vectorized form of final nonlinear parameterized equations is then solved through the use of pseudo-arc length continuum technique. Numerical results are presented to examine the effects of different parameters such as nanotube volume fraction, slenderness ratio, dimensionless damping parameter, dimensionless transverse force, CNT distributions and boundary conditions on the natural frequencies and frequency responses of FG-CNTRC beams.