Nonlinear Vibration and Stability Analysis of Functionally Graded Nanobeam Subjected to External Parametric Excitation and Thermal Load

Analysis of vibration stability of simply supported Euler-Bernoulli functionally graded (FG) nanobeam embedded in viscous elastic medium with thermal effect under external parametric excitation is presented in this work. An attempt has been made for the first time is investigating the effect of thermal load on dynamic behavior, amplitude response, instability region and bifurcation points of functionally graded nanobeam. Thermal loads are supposed to be uniform, linear or nonlinear distribution along the thickness direction. Nonlocal continuum theory and the principle of the minimum total potential energy are applied to derive the governing equations. The partial differential equations (PDE) are transported to the ordinary differential equations (ODE) by using the Petrov-Galerkin method and the multiple time scales method are manipulated to solve the motion equation. To study the effect of external parametric excitation and thermal effect, different temperature distributions along the thickness such as uniform, linear, and nonlinear distribution are considered. Moreover, stable and unstable regions and bifurcation points are determined. It is obtained that the thermal load can affect the amplitude response of FG nanobeam. Also, it is observed that the instability of the system is affected by the detuning parameter and the parametric excitation amplitude plays great role in the instability of system. Nanobeams are used in many devices like nanoresonators, nanosensors and nanoswitches. This paper is helpful for designing and manufacturing nanoscale structures specially nanoresonators under different thermal loads.

Nonlinear Vibration of a Functionally Graded Nanobeam Based on the Nonlocal Strain Gradient Theory considering Thickness Effect

In this work, a nonlocal strain gradient beam model considering the thickness effect is developed to study the nonlinear vibration response of a functionally graded nanobeam. The governing equation of the functionally graded nanobeam is derived by using the Euler–Bernoulli beam theory with von Kármán’s nonlinear strain-gradient relationship and the Hamilton principle. The expression of the nonlinear frequency for the functionally graded nanobeam with pinned-pinned boundary conditions is obtained with the help of Galerkin technique and the Hamiltonian approach. The obtained results show that the effect of thickness is very important for the size-dependent vibration response of the functionally graded nanobeam; the nonlinear vibration response of the nanobeam depends not only on the material length scale parameter and nonlocal parameter but also on the slenderness ratio. Effects of the slenderness ratio and the power-law index on the vibration response of the functionally graded nanobeam are also investigated and discussed. The numerical results show that the nonlocal parameter reduces the nonlinear frequency of the functionally graded nanobeam, while the material length scale parameter increases the nonlinear frequency of the functionally graded nanobeam. The slenderness ratio leads to an increase in the nonlinear frequency of the functionally graded nanobeam, while the power-law index leads to a decrease in the nonlinear frequency of the functionally graded nanobeam.