Free vibration analysis of functionally graded beams

Abstract

Abstract. In this paper free vibration of axially functionally graded (FG) beams consisting of two segments is studied. The FG beams under consideration are characterized by axially varying cross-section areas and/or functionally grading material properties. Numerical example for a beam that is clamped at both ends is presented.

Free Vibration Analysis of Axially Functionally Graded (AFG) Cantilever Columns of a Regular Polygon Cross-Section with Constant Volume

“…There are various kinds of functions for Young's modulus E in the literature: linear [2,3,9,12,14,17], trigonometric [4], polynomial [5,[14][15][16], piecewise [6], exponential [7,10], and periodic [14] functions, etc. The linear function is selected in this study, and then the function of E at the coordinate x is expressed as [17]…”

“…For the variable function of the mass density ρ, it is usual that ρ is equal to E [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], and then the density ratio is the same of modular ratio m (see Figure 1c). Therefore, the mass density ratio m defined as a ratio of ρ f to ρ c and the linear function of ρ at the coordinate x can be written as…”

“…The following references and their citations include the mathematical models and historical reviews of this subject. The typical studies on free vibration of AFG columns are briefly reviewed here in chronological order: Akgoz and Civalek [3] studied natural frequencies of a linearly tapered microbeam based on the modified couple stress theory; Calio and Elishakoff [4] investigated the closed-form solutions of natural frequency for a simply supported beam-column with an elastically guided end condition at one end, where the trigonometric functions of the Young's modulus in the mathematical formulation were considered; Li [5] studied the static and dynamic behaviors of a prismatic beam, including the rotatory inertia and shear deformation; Singh and Li [6] studied critical buckling loads of a cantilever column with a piecewise element, restrained by the elastic foundation; Huang and Li [7] researched a new approach for the free vibration of tapered beams with simply supported, both clamped, clamped-pinned, and cantilevered end conditions, respectively, where the Fredholm integral equations were used in the mathematical formulations; Shahba et al [8] investigated the free vibration and stability of a non-prismatic column with classical and non-classical boundary conditions; Shahba and Rajasekaran [9] studied the free vibration and stability of tapered beams, in which governing equations for the free vibration and buckling were solved by the differential quadrature element method (DQEM); Kukla and Rychlewska [10] dealt with beams with both ends fixed, which were fabricated from two different FGMs for analyzing free vibration; Yilmaz et al [11] investigated the buckling loads of non-prismatic columns restrained by the elastic foundation using the differential quadrature method (DQM); Chandran and Rajendran [12] studied closed-form solutions of the buckling load of a prismatic cantilever column using the principle of conservation energy; Shafiei et al [13] studied non-linear vibrations of a linearly tapered microbeam with a square cross-section, where the governing equations were solved by the differential quadrature method (DQM); Ranganathan et al [14] studied the buckling of slender columns that determined the buckling loads by the linear perturbation method together with the Rayleigh-Ritz method and investigated the maximum buckling loads under the same average mean Young's modulus; Elishakoff et al [15] studied the buckling and vibrations of a column sharing Duncan's mode shapes and assuming a fifth order polynomial; Rezaiee and Masoodi [16] investigated closed-form solutions of the natural frequencies and buckling loads of tapered beam-columns supported by semi-rigid connections; and Lee and Lee [17] studied the free vibration and buckling of tapered cantilever columns with square and circular cross-sections. As discussed above, FG materials developed in 1984 are of particular interest in dealing with the free vibration of FG columns.…”

Optimal design of the transversely vibrating Euler–Bernoulli beams segmented in the longitudinal direction

“…As far as fundamental vibration frequency is concerned, from the parametric studies, they discerned that in the former case, a column with piecewise constant cross-section is less efficient than a uniform column, whereas, in the latter case, the column with piecewise constant cross-section is more efficient than the uniform column, based on either fundamental vibration frequency or buckling load. Kukla and Rychlewska [5] conducted free vibration analysis of the axially functionally graded clamped-clamped two-segmented beams. This analysis is based on the approximation of the functionally graded beam by piecewise exponentially varying geometrical and material properties.…”

Free vibration of axially functionally graded Euler-Bernoulli beams

“…The exact solution to free vibration of axially exponentially graded beams is presented by Li et al in reference [5]. Free vibration of axially FG beams consisting of two segments was studied in paper [6]. The analytical solution to the problem was obtained by assumption that the changes of the cross-sectional area and material properties in the beam segments have an exponential form.…”