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In this paper, the divergent instability and coupled flutter characteristics of axially moving beams made of functionally graded materials (FGM) are studied using the interpolation matrix method. The material property of the beam is designed to change smoothly and continuously along the thickness direction. In considering the Euler-Bernoulli beam theory, Hamilton’s principle is used to derive the differential equation of the transverse vibration kinematics of axially moving FGM beams. In addition, the calculation model for solving the complex frequency of the beam based on the interpolation matrix method has been established. The presented solutions are compared with those in the literature to illustrate the effectiveness of the interpolation matrix method. The results show that the divergence and flutter velocities of axially moving FGM beams tend to decrease with the increase of the material gradient index, and there is a very narrow stability region between the first static instability region (divergence) and the first dynamic instability region (first- and second-order coupled flutter).

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In this paper, the interpolation matrix method (IMM) is proposed to solve the buckling critical load of axially functionally graded (FG) Timoshenko beams. Based on Timoshenko beam theory, a set of governing equations coupled by the deflection function and rotation function of the beam are obtained. Then, the deflection function and rotation function are decoupled and transformed into an eigenvalue problem of a variable coefficient fourth-order ordinary differential equation with unknown deflection function. According to the theory of interpolation matrix method, the eigenvalue problem of the variable coefficient fourth-order ordinary differential equation is transformed into an eigenvalue problem of a set of linear algebraic equations, and the critical buckling load and the corresponding deflection function of the axially functionally graded Timoshenko beam can be calculated by the orthogonal triangular (QR) decomposition method, which is the most effective and widely used method for finding all eigenvalues of a matrix. The numerical results are in good agreement with the existing results, which shows the effectiveness and accuracy of the method.