Free vibration of functionally graded beams based on both classical and first-order shear deformation beam theories

• Published in: Applied mathematics and mechanics Vol. 35; no. 5; pp. 591 - 606
• Main Author: 李世荣 万泽青 张静华
• Format: Journal Article
• Language: English
• Published: Heidelberg Shanghai University 01.05.2014
• Subjects: Applications of Mathematics; Beams (structural); Boundary conditions; Classical Mechanics; Fluid- and Aerodynamics; Free vibration; Functionally gradient materials; Joining; Mathematical Modeling and Industrial Mathematics; Mathematical models; Mathematics; Mathematics and Statistics; Partial Differential Equations; Resonant frequency; Shear deformation; Timoshenko beams; shooting method; 74E05; 74K10; O343.1; Timoshenko beam; O343.7; free vibration; analogous transformation; functionally graded material (FGM)
• ISSN: 0253-4827; 1573-2754
• DOI: 10.1007/s10483-014-1815-6

The free vibration of functionally graded material （FGM） beams is studied based on both the classical and the first-order shear deformation beam theories. The equations of motion for the FGM beams are derived by considering the shear deforma- tion and the axial, transversal, rotational, and axial-rotational coupling inertia forces on the assumption that the material properties vary arbitrarily in the thickness direction. By using the numerical shooting method to solve the eigenvalue problem of the coupled ordinary differential equations with different boundary conditions, the natural frequen- cies of the FGM Timoshenko beams are obtained numerically. In a special case of the classical beam theory, a proportional transformation between the natural frequencies of the FGM and the reference homogenous beams is obtained by using the mathematical similarity between the mathematical formulations. This formula provides a simple and useful approach to evaluate the natural frequencies of the FGM beams without dealing with the tension-bending coupling problem. Approximately, this analogous transition can also be extended to predict the frequencies of the FGM Timoshenko beams. The numerical results obtained by the shooting method and those obtained by the analogous transformation are presented to show the effects of the material gradient, the slenderness ratio, and the boundary conditions on the natural frequencies in detail.