Free vibration analysis of functionally graded material beams based on Levinson beam theory

• Published in: Applied mathematics and mechanics Vol. 37; no. 7; pp. 861 - 878
• Main Authors: Wang, Xuan, Li, Shirong
• Format: Journal Article
• Language: English
• Published: Shanghai Shanghai University 01.07.2016; School of Hydraulic, Energy and Power Engineering, Yangzhou University, Yangzhou 225127, Jiangsu Province, China%School of Hydraulic, Energy and Power Engineering, Yangzhou University, Yangzhou 225127, Jiangsu Province, China; School of Civil Science and Engineering, Yangzhou University, Yangzhou 225127, Jiangsu Province, China
• Subjects: Applications of Mathematics; Classical Mechanics; Fluid- and Aerodynamics; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Partial Differential Equations; O343.7; 74E05; 74K10; functionally graded material (FGM) beam, Levinson beam theory (LBT), free vibration, shooting method, natural frequency; O343.1; shooting method; Levinson beam theory (LBT); functionally graded material (FGM) beam; free vibration; natural frequency
• ISSN: 0253-4827; 1573-2754
• DOI: 10.1007/s10483-016-2094-9

Free vibration response of functionally graded material （FGM） beams is studied based on the Levinson beam theory （LBT）. Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.