Dynamic Stability Analysis of FGM Beams Based on the Nonlinear Timoshenko Model

Various types of dynamic instabilities in mechanical systems are one of the most important disruptive factors in such structures. Therefore, an accurate study of dynamic instability in beams, as one of the fundamental engineering structures, is of great importance. In this paper, dynamic instability problem of beams made of Functionally Graded Materials (FGM) is investigated. For this purpose, the first-order shear deformation (or the Timoshenko) beam theory with the effects of geometric nonlinearity is considered. Thus, the proposed model has the ability to determine mechanical behavior of thin and thick beams. By considering the energy functions of the system, and implementing the Hamilton’s principle, the governing equations are obtained along with different types of common boundary conditions. The Differential Quadrature Method (DQM), as one of the best-known numerical methods, is used. The nonlinear partial differential equations are written in the form of equivalent ordinary differential equations. Then, considering the harmonic responses for the system, the differential equations are converted to a set of nonlinear algebraic equations. Finally, in order to study the important parameters, various numerical examples are provided. The obtained numerical results are compared with the literature and thus, the validity of the presented formulation and solution methodology is revealed. Also, a comparative study between linear and nonlinear kinematic models shows that the importance of geometric nonlinearity of the model is quite significant.