Nonlinear vibration of axially functionally graded tapered microbeams

Solving the nonlinear governing equations of a non-uniform micro- and nano-beam is a complicated challenge for researchers. For the first time, the nonlinear size-dependent vibration of a non-uniform axially functionally graded (AFG) microbeam is studied in this paper. The microbeam is modeled according to the Euler-Bernoulli beam theory and the modified couple stress theory with von-Kármán's geometric nonlinearity. The boundary conditions are considered as clamped, simply supported and clamped-simply supported. To derive the equations and boundary conditions, Hamilton's principle is utilized and then the governing equations are solved by the generalized differential quadrature method (GDQM) and direct iterative method. Finally, the effects of nonlinearity, small-scale parameter and rates of cross-section change on the fundamental and the second frequencies of the AFG, pure ceramic and pure metal microbeams are presented. It is shown that the effects of the rate of cross-section change of the microbeam along one direction depend on the nonlinearity and also the rate of cross-section change along the other direction. The results of this study can be used in designation of many microstructures such as micro electro mechanical systems (MEMS), micro-actuators, etc.