Dynamic modeling and analysis of rotating FG beams for capturing steady bending deformation

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A dynamic analysis of rotating functionally gradient (FG) beams is presented for capturing the steady bending deformation by using a novel floating frame reference (FFR) formulation. Usually, the cross section of bending beams should rotate round the point at the neutral axis while centrifugal inertial forces are supposed to act on centroid axis. Due to material inhomogeneity of FG beams, centroid and neutral axes may be in different positions, which leads to the eccentricity of centrifugal forces. Thus, centrifugal forces can be divided into three componets: transverse component, axial component and force moment acting on the points of the neutral axis, in which transverse component and force moment can make the beam produce the steady bending deformation. However, this speculation has not been presented and discussed in existing literatures. To this end, a novel FFR formulation of rotating FG beams is especially developed considering centroid and neutral axes. The FFR and its nodal coordinates are used to determine the displacement field, in which kinetic and elastic energies can be accurately formulated according to centroid and neutral axes, respectively. By using the Lagrange's equations of the second kind, the nonlinear dynamic equations are derived for transient dynamics problems of rotating FG beams. Simplifying the nonlinear dynamic equations obtains the equilibrium equations about inertial and elastic forces. The equilibrium equations can be solved to capture the steady bending deformation. Based on the steady bending state, the nonlinear dynamic equations are linearized to obtain eigen-frequency equations. Transient responses obtained from the nonlinear dynamic equations and frequencies obtained from the eigen-frequency equations are compared with available results in existing literatures. Finally, effects of material gradient index and angular speed on the steady bending deformation and vibration characteristics are investigated in detail.