This paper studies free vibration of axially functionally graded beams with non-uniform cross-section. A novel and simple approach is presented to solve natural frequencies of free vibration of beams with variable flexural rigidity and mass density. For various end supports including simply supported, clamped, and free ends, we transform the governing equation with varying coefficients to Fredholm integral equations. Natural frequencies can be determined by requiring that the resulting Fredholm integral equation has a non-trivial solution. Our method has fast convergence and obtained numerical results have high accuracy. The effectiveness of the method is confirmed by comparing numerical results with those available for tapered beams of linearly variable width or depth and graded beams of special polynomial non-homogeneity. Moreover, fundamental frequencies of a graded beam combined of aluminum and zirconia as two constituent phases under typical end supports are evaluated for axially varying material properties. The effects of the geometrical and gradient parameters are elucidated. The present results are of benefit to optimum design of non-homogeneous tapered beam structures.
"Furthermore, Wang and Wang  analyzed a vibration problem for a class of nonuniform Euler–Bernoulli beams with special geometry. Huang and Li   proposed an integral equation method to determine the natural frequencies and buckling loads of axially functionally graded beams with variable cross-section. Rajasekaran  used the differential transform and differential quadrature methods to analyze centrifugally stiffened axially functionally graded tapered beams. "
[Show abstract][Hide abstract] ABSTRACT: Free vibration of non-uniform functionally graded beams is analyzed via the Timoshenko beam theory. Bending stiffness and distributed mass density are assumed to obey a unified exponential law. For various boundary conditions, exact frequency equations are derived in closed form. These frequency equations can reduce to those for classical Timoshenko beams if the gradient index disappears. Moreover, the frequency equations of exponentially graded Rayleigh, shear, and Euler–Bernoulli beams can be obtained as special cases of the present. The gradient index has a strong influence on the natural frequencies. For Timoshenko beams, there exist two critical frequencies depending on the gradient index. Harmonic vibration cannot be excited for frequencies less than the lower critical frequency. The obtained results can serve as a benchmark for examining the accuracy of numerical frequencies based on other approaches for analyzing transverse vibration of non-uniform axially graded Timoshenko beams. The results also apply to bending vibration of rectangular Timoshenko beams with constant thickness and exponentially decaying/amplifying width.
International Journal of Mechanical Sciences 12/2014; 89:1–11. DOI:10.1016/j.ijmecsci.2014.08.017 · 2.03 Impact Factor
"Lü et al.  predicted the thermal deformation and bending characteristics of bi-directional FG beams using state space-based differential quadrature method (DQM). Huang and Li  studied the free vibration of nonuniform axially FG Euler-Bernoulli beams by transforming the governing equation with varying coefficients to Fredholm integral equations. Shahba et al.  used the finite element method to examine the free vibration and buckling of tapered axially FG Timoshenko beams. "
[Show abstract][Hide abstract] ABSTRACT: Nonlinear free vibration of functionally graded (FG) plates with in-plane material inhomogeneity subjected to different boundary conditions is presented. The nonlinear equations of motion and the related boundary conditions are extracted based on the classical plate theory. Green's strain tensor together with von Kármán assumptions is employed to model the geometrical nonlinearity. The differential quadrature method as an efficient and accurate numerical tool is employed to discretize the governing equations in spatial domain. After validating the presented approach, parametric studies are performed to clarify the effects of different parameters on the nonlinear frequency parameters of the in-plane FG plates.
Mechanics of Advanced Materials and Structures 11/2014; 22(8):00-00. DOI:10.1080/15376494.2013.828818 · 0.77 Impact Factor
"The linear and nonlinear dynamics of FGM beams were studied in Refs.        . Over the last decade, the dynamic modeling and analysis of rotating blades made of FGMs has become a topic of considerable research. "
[Show abstract][Hide abstract] ABSTRACT: A comprehensive dynamic model of a rotating hub–functionally graded material (FGM) beam system is developed based on a rigid–flexible coupled dynamics theory to study its free vibration characteristics. The rigid–flexible coupled dynamic equations of the system are derived using the method of assumed modes and Lagrange's equations of the second kind. The dynamic stiffening effect of the rotating hub–FGM beam system is captured by a second-order coupling term that represents longitudinal shrinking of the beam caused by the transverse displacement. The natural frequencies and mode shapes of the system with the chordwise bending and stretching (B–S) coupling effect are calculated and compared with those with the coupling effect neglected. When the B–S coupling effect is included, interesting frequency veering and mode shift phenomena are observed. A two-mode model is introduced to accurately predict the most obvious frequency veering behavior between two adjacent modes associated with a chordwise bending and a stretching mode. The critical veering angular velocities of the FGM beam that are analytically determined from the two-mode model are in excellent agreement with those from the comprehensive dynamic model. The effects of material inhomogeneity and graded properties of FGM beams on their dynamic characteristics are investigated. The comprehensive dynamic model developed here can be used in graded material design of FGM beams for achieving specified dynamic characteristics.
Journal of Sound and Vibration 02/2014; 333(5):1526–1541. DOI:10.1016/j.jsv.2013.11.001 · 1.81 Impact Factor
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