THE TRANSVERSE FREE VIBRATION OF A BEAM OF ANNULAR CROSS-SECTION MADE OF FUNCTIONALLY GRADED MATERIALS

For a beam of annular cross-section made of functionally graded materials (FGM), we assume that the physical parameters of the materials along the direction of the wall thickness varying in a simple power law. Based on the Lagrange's function and the Hamilton's principle, the Hamilton's canonical equations for the transverse free vibration of the beam are established. A symplectic eigenvalue problem of the Hamilton matrix is solved by using the symplectic method. Then, the natural frequencies and the vibration mode functions of the beam are obtained, with conditions on the two ends as the simply supported, the fixed, the cantilever and the fixed-simply supported. Numerical examples are given for the first eight order dimensionless natural frequencies against the material volume fraction, and the effect of the material volume fraction on the natural frequency is analyzed.