Free vibrations of stepped axially functionally graded Timoshenko beams

Abstract

This paper provides an analytical solution for free transverse vibrations of axially functionally graded beams with step changes in geometry and in material properties. The differential quadrature method using domain decomposition technique is used. Based on Timoshenko beam theory, the equations of motion are derived using Hamilton's principle. Material properties are assumed to vary along the beam in a continuous or an abrupt fashion. The combinations of classical boundary conditions (Free, Simply Supported and Clamped) are considered to determine the natural frequencies of many numerical examples. The results for different step locations with different axially functionally graded materials are presented. The phenomenon of dynamic stiffness of beams can be observed in various situations. As there are no available previous results of axially functionally graded beams with step changes, only the results for beams with no abrupt discontinuities are compared with published results. The developed differential quadrature solution has proved its simplicity and robustness to solve the problem presented in the title.

Free vibration of functionally graded beams and frameworks using the dynamic stiffness method

“…The algorithm ensures that no natural frequency of the structure is missed, and it has featured in literally hundreds of papers. It is worth noting that earlier investigations on the free vibration of FGBs were focused on individual FGBs except for a few isolated cases where stepped FGBs with collinear axes were reported [40,41]. Accordingly, the literature on the free vibration of frameworks containing FGBs is virtually non-existent.…”

A Unified Approach for Out-of-Plane Forced Vibration of Axially Functionally Graded Circular Rods

“…One type of time depended point load, shown in figure 1.b, with the amplitude p0 = 1 kgf is implemented to the crown point of the rod. The equations (13)(14)(15)(16)(17)(18) given in canonical form are solved numerically in the Laplace domain by the CFM. Here the torsional, flexural rigidity and cross-sectional area of the rod are; Shear correction factor α b = 1.11, radius of the circular rod r = 100 cm and ϕ 0 = π/4.…”

Free vibration analysis of a single edge cracked symmetric functionally graded stepped beams

“…Furthermore, the authors extended the studies also on intact stepped beams (Bambill et al, 2015; Su et al, 2018; Suddoung et al, 2014; Wattanasakulpong and Charoensuk, 2015), cracked homogeneous isotropic stepped beams (Al-Said, 2008; Attar, 2012; Kisa and Arif Gurel, 2007; Mao and Pietrzko, 2010; Naguleswaran, 2002; Nandwana and Maiti, 1997) and only one study on cracked FG stepped beams (Khiem et al, 2019).…”