A new approach for free vibration of axially functionally graded beams with non-uniform cross-section
ABSTRACT 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.
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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
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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
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ABSTRACT: Free vibration of axially inhomogeneous beams is analyzed. For exponentially graded beams with various end conditions, characteristic equations are derived in closed form. These characteristic or frequency equations can analytically reduce to the classical forms of Euler–Bernoulli beams if the gradient index disappears. The gradient has a strong influence on the frequency spectrum, and the natural frequencies noticeably depend on the variation of the gradient parameter and end support conditions. For certain beams with exponential gradients, there exists a critical frequency depending on the gradient parameter. Vibration can be only excited by propagating waves with frequencies in excess of the critical frequency, and otherwise vibration is prohibited for pseudo-frequencies lower than the critical frequency. For some gradient index with small change, the natural frequencies have an abrupt jump when across its critical frequencies. Obtained results can serve as a benchmark for other numerical procedures for analyzing transverse vibration of axially functionally graded beams. The minimal natural frequency can be sought for certain gradient index, and this helps engineers to optimally design vibrating nonhomogeneous beam structures. Obtained results also apply to free vibration of nonuniform beams with constant thickness and exponentially decaying width.Applied Acoustics 03/2013; 74(3):413–420. DOI:10.1016/j.apacoust.2012.08.003