Influence of the boundary relaxation on free vibration of functionally graded carbon nanotube-reinforced composite beams with geometric imperfections

In this paper, the influence of boundary relaxation on the free vibration characteristics of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) imperfect beams is studied based on the first-order shear theory. An analysis model of the imperfect FG-CNTRC beams with arbitrary boundary conditions is presented using the boundary spring technique. The relaxation degree of the boundary is evaluated by introducing relaxation parameters, which are simulated by adjusting the stiffness of springs. The governing equations are derived using the Rayleigh-Ritz method and solved to obtain the frequencies of the beams with geometric imperfections and relaxed boundaries. The results reveal that boundary relaxation and geometric imperfections have a coupling influence on the vibration behavior of FG-CNTRC beams. The influence of boundary relaxation on frequency is highly dependent on the amplitude and modes of the geometric imperfection, but not sensitive to the geometric imperfection location. Boundary restraint enhancement reduces the influence of relaxation on the beam vibration. The influences of CNTs distribution pattern and volume fraction of on the vibration characteristics of the beams with relaxed boundaries are also discussed.