Nonlinear free vibration of microbeams partially supported by foundation using a third-order finite element formulation - Ich, Le Cong; Dung, Trang Quang; Chinh, Nguyen Van; Dung, Lam Van; Kien, Nguyen Dinh (Vietnam Journal of Science and Technology, 2022)
Geometrically nonlinear free vibration of microbeams partially supported by a three-parameter nonlinear elastic foundation is studied in this paper. Equations of motion based on the modified couple stress theory (MCST) and a refined third-order shear deformation beam theory are derived using Hamilton’s principle, and they are solved by a finite element formulation. The validity of the derived formulation is verified by comparing the present results with the published data for the case of the microbeams fully resting on the foundation. Numerical investigation is carried out to show the effects of the length scale parameter, the aspect ratio, the nondimensional amplitude and the boundary conditions on the nonlinear free vibration behavior of the microbeams. The obtained numerical results reveal that the foundation supporting length plays an important role on the vibration of the microbeams, and the influence of the foundation supporting length on the frequency ratio is dependent on the boundary conditions. It is also shown that the frequency ratio is decreased by the increase of the length scale, regardless of the boundary condition and the initial deflection. The influence of the nonlinear foundation stiffness on the ratio of nonlinear frequency to linear frequency of the microbeams is also studied and discussed.