Development of dynamic stiffness method for free vibration of functionally graded Timoshenko beams

Abstract

The free vibration of functionally graded Timoshenko beams is investigated by developing the dynamic stiffness method. Material properties of the beam are assumed to vary continuously in the thickness direction. The governing differential equations of motion are solved and expressions for axial force, shear force and bending moment are derived. The dynamic stiffness matrix is then formulated by relating the amplitudes of forces and displacements at the ends of the beam. The Wittrick-Williams algorithm is used as solution technique to yield the natural frequencies and mode shapes of some illustrative examples. The results are discussed and some conclusions are drawn.

Modal analysis of functionally graded Timoshenko beam

“…First, to check validity of the governing equations and computational programs, first three natural frequencies of homogeneous beam are computed and tabulated in Tab. 1 where the obtained herein results are compared with exact solution provided in [18] and with those obtained by Su and Banerjee in [6]. It can be seen excellent agreement of the compared results, especially in the case of large slenderness ratio L/h.…”

“…Comparison of present results with those given in [6] shows that the proposed above integrated method and the dynamic stiffness method developed by Su and Banerjee [6] are similar in predicting natural frequencies of flexural vibration modes. Nevertheless, they give different results in computing the axial vibration modes which are coupled with the flexural modes in FGM Timoshenko beam.…”

“…The results are also compared to those obtained by the dynamic stiffness method in [6] where neutral axis position is assumed to be centroid (h 0 = 0). Mode shapes related to the computed frequencies are also examined and shown in Figs.…”

“…Since the FGM is increasingly used nowadays in the practice of high-tech industries the dynamic behavior of FGM structures becomes a vital problem. Though the powerful methods such as finite element [3][4][5], dynamic stiffness [6] and spectral element [7] have been all developed for analysis of FGM structures, the analytical method shows to be most simply and accurately applied for beam-like FGM structures. Aydogdu and Taskin [8] have examined different high-order shear deformation theories by computing natural frequencies of simply supported functionally graded beam and shown that the classical beam theory gives higher results.…”

An analytical method for the vibration and buckling of functionally graded beams under mechanical and thermal loads

“…Due to the excellent properties in mechanical and thermal behaviours, a wide range of application for functionally graded (FG) structures can be found in different fields, leading to the intensive study in many types of FG structures in the last three decades. Chebyshev collocation method, finite element method and differential quadrature method [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. For analytical approaches, a Navier solution has been widely used to study various mechanical behaviours of simply supported beams [16][17][18][19][20].…”