Semi-inverse method for axially functionally graded beams with an anti-symmetric vibration mode

By Lei Wu, Qi-shen Wang, Isaac E. Elishakoff. 2005

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Geometrically nonlinear free vibration analysis of axially functionally graded taper beams (2015, methodology): “[46] investigated the free vibrations of AFG tapered beams using the semi inverse method.”
Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section (2013, methodology): “For example, Elishakoff and co-workers used the inverse method to find the exact solutions of natural frequency for non-homogeneous beams by choosing the transverse displacement, the stiffness and material density of the axially direction as some certain polynomials [3–5].”
Free and forced vibration analysis of FG beam considering temperature dependency of material properties (2011, background): “[16] obtained closed-form natural frequencies of a simply supported FG beam with properties as polynomial functions of the axial coordinates.”
Nonlinear forced vibration analysis of clamped functionally graded beams (2011, background): “Wu et al. [ 13 ] obtained the closed-form natural frequencies of a simply supported FG beam with mass density and Young’s modulus being polynomial functions of the axial coordinate.”