Nonlinear vibration analysis of FG-CNTRC sandwich Timoshenko beam based on modified couple stress theory subjected to longitudinal magnetic field using generalized differential quadrature method

In this paper, the magneto-mechanical nonlinear vibration behavior of rectangular functionally graded carbon nanotube reinforced composite (FG-CNTRC) sandwich Timoshenko beam based on modified couple stress theory (MCST) is investigated by the generalized differential quadrature method. The FG-CNTRC sandwich beam consists of two FG-CNTRC face sheets and homogenous core subjected to longitudinal magnetic field. In the FG-CNTRC face sheets, carbon nanotubes are disseminated in different patterns as uniform distribution, FG-X, FG-V, and FG-O. Based on von Kármán geometric nonlinearity, the nonlinear governing equations are derived by the Hamilton’s principle. Accuracy and convergence of this study are validated by comparing the numerical results with those found in literature. Various parameters effects are examined on the nonlinear 1st frequency of the sandwich beam. The results reveal that composition of the CNTRC face sheets and homogenous core in sandwich beam can be achieved remarkable stiffness in comparison to only CNTRC beam or only homogenous beam. For achieving the highest stiffness of FG-CNTRC sandwich beam, the value of thickness ratio is obtained about 0.34 and 0.27 in the presence and absence of Pasternak foundation, respectively. Moreover, the linear and nonlinear 1st frequencies increase with an increase in the magnetic field for all of CNT distribution types of sandwich beam and different boundary conditions whereas the frequency ratio decreases. Also the highest and lowest nonlinear 1st frequencies are corresponding to FG-A and FG-V distributions of CNT in face sheets, respectively.