Vibration Analysis of Axially Functionally Graded Non-Prismatic Timoshenko Beams Using the Finite Difference Method

: This paper presents an approach to the vibration analysis of axially functionally graded non-prismatic Timoshenko beams (AFGNPTB) using the finite difference method (FDM). The characteristics (cross-sectional area, moment of inertia, elastic moduli, shear moduli, and mass density) of axially functionally graded beams vary along the longitudinal axis. The Timoshenko beam theory covers cases associated with small deflections based on shear deformation and rotary inertia considerations. The FDM is an approximate method for solving problems described with differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, differential equations were formulated with finite differences, and additional points were introduced at the beam’s ends and at positions of discontinuity (supports, hinges, springs, concentrated mass, spring − mass system, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. Vibration analysis of AFGNPTB was conducted with this model, and natural frequencies were determined. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of AFGNPTB, considering the damping. The results obtained in this study showed good agreement with those of other studies, and the accuracy was always increased through a grid refinement.


Introduction
This paper describes the application of Fogang's model [1] based on the finite difference method, used for the Euler−Bernoulli beam, to the vibration analysis of axially functionally graded non-prismatic Timoshenko beams (AFGNPTB). Various studies have focused on the vibration analysis of AFGNPTB. Kindelan et al. [2] presented a method of obtaining optimal finite difference formulas that maximize their frequency range of validity. Both conventional and staggered equispaced stencils for first and second derivatives were considered. Mwabora et al. [3] considered numerical solutions for static and dynamic stability parameters of an axially loaded uniform beam resting on VIBRATION ANALYSIS OF AXIALLY FUNCTIONALLY GRADED TIMOSHENKO BEAMS simply supported foundations using the finite difference method (FDM), where a central difference scheme was developed. Soltani et al. [4] applied the FDM to evaluate natural frequencies of non-prismatic beams with different boundary conditions and resting on variable one-or two-parameter elastic foundations. Torabi et al. [5] presented an exact closed-form solution for free vibration analysis of Euler−Bernoulli conical and tapered beams carrying any desired number of attached masses; the concentrated masses were modeled by Dirac's delta functions. Soltani [6] developed a semi-analytical technique to investigate the free bending vibration behavior of an axially functionally graded nonprismatic Timoshenko beam subjected to a point force at both ends, based on the power series expansion.
Classical analysis of the Timoshenko beam involves solving the governing equations that are expressed via means of differential equations, and considering boundary and continuity conditions. However, solving differential equations may be difficult in the presence of an axial force (or external distributed axial forces), an elastic Winkler foundation, a Pasternak foundation, or damping. Generally, the FDM does not consider points outside the beam; the boundary conditions are applied at the beam's ends, not the governing equations. Consequently, the non-application of governing equations at the beam's ends leads to inaccurate results, making the FDM less useful compared with other numerical methods, such as the finite element method. This paper presented a model based on FDM; this model consisted of formulating differential equations with finite differences and introducing additional points at the beam's ends and at positions of discontinuity (supports, hinges, springs, concentrated mass, spring−mass system, etc.). The introduction of additional points allowed us to apply the governing equations at the beam's ends and to satisfy the boundary and continuity conditions. Vibration analysis of structures was conducted using the model, and natural frequencies were determined. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of Euler−Bernoulli beams, considering the damping.

2.1
Free vibration analysis

Governing equations of the free vibration
The focus here is to determine the eigenfrequencies of the beam. A second-order analysis is conducted; and the first-order analysis can easily be deduced. The sign convention adopted for loads, bending moments, transverse forces, and displacements is illustrated in Figure 1. The transverse force T(x) is related to the shear force V(x), as follows: (2) Let us consider an external distributed axial load n(x) positive along the + x axis Substituting Equations (2) and (3) into According to the Timoshenko beam theory, the bending moment and shear force are related to the deflection and rotation of the cross section (x), as follows: Where at a position x, E(x) is the elastic modulus, G(x) is the shear modulus, and  is the shear correction factor.

Fundamentals of the FDM
Let us consider a segment k of the beam having equidistant grid points with spacing h k .
Equations (6a-b) have a second-order derivative; consequently, the deflection and rotation curves w(x), and (x), respectively, are approximated around the point of interest i as second-degree polynomials.
Thus, a three-point stencil is used to write finite difference approximations to derivatives at grid points. The derivatives (S(x) representing w(x) or (x)) at i are expressed with deflection values at points i-1, i, and i+1.

FDM Formulation of equations and efforts
Since the characteristics of the beam vary throughout the longitudinal axis, reference values are defined. The reference values of the beam's mass per unit volume, the cross-sectional area, the moment of inertia, the elastic modulus, and the shear modulus are denoted by  r , A r , I r , E r , and G r , respectively. At a position x the beam's mass per unit volume, the cross-sectional area, the moment of inertia, the elastic modulus, and the shear modulus are related to the reference values as follows: The reference length is denoted by l r . The bending shear factor  r and other parameters are defined as follows: The following parameters describing the rate of change of stiffnesses E(x)I(x) and G(x)A(x) are defined as follows: The reference coefficient of rotary inertia k RIr and the vibration frequency  are defined as follows: (11a) (11b)  (2) and (5a-b) yields the bending moment and transverse force, as follows: For the special case of an AFGNPTB without an axial force or a Winkler foundation, Equation (12a) becomes The shear force is then calculated as follows: VIBRATION ANALYSIS OF AXIALLY FUNCTIONALLY GRADED TIMOSHENKO BEAMS

Analysis at beam's ends and at positions of discontinuity
Positions of discontinuity are positions of concentrated mass, spring−mass system, supports, hinges, springs, abrupt change in cross section, positions where E(x)I(x) and G(x)A(x) are not differentiable, and change in grid spacing.
The model used in this paper (developed in Fogang [1]) consists of realizing an opening of the beam at the position of discontinuity and introducing additional points (imaginary points ia and id) in the opening, as represented in Figure 2.
Imaginary points are also introduced at beam's ends (points 0 and n+2), so governing equations are applied at the beam's ends, as well as boundary conditions.  Effect of a spring−mass system: The dynamic behavior of a beam carrying a spring−mass system was analyzed.
The deflection of the mass is denoted by w iM . The spring stiffness K p is defined as follows: (20) Applying

Free vibration analysis of AFG tapered Timoshenko beams
The natural frequencies (coefficients ) of axially functionally graded non-prismatic (AFGNP) Timoshenko beams were determined. Pinned−pinned, fixed−free, and fixed−fixed beams were considered. Soltani [6] presented results obtained with the power series method (PSM) and those obtained by Shahba et al. [7] with the finite element method (FEM in terms of ABAQUS software). Case B of [6], described as follows, was considered: the height and breadth of the beam were assumed to vary linearly in longitudinal axis with the same tapering ratio  = 1 -h0/h1 = 1 -b0/b1. The heights and breadths at the left and the right beam's end, respectively, were denoted by h0, b0, h1, and b1. The results of this study are compared to those of Soltani [6] and Shahba et al. [7] in Table 1. The results of this study are in good agreement with those of Soltani [6] and Shahba et al. [7] .
The FDM-based model developed in this paper enabled, with relative easiness, vibration analysis of axially functionally graded non-prismatic Timoshenko beams. The results showed that the calculations conducted as described in this paper were accurate.
The following aspects were not addressed in this study but could be analyzed with the model in the future: ✓ Axially functionally graded Timoshenko beams resting on Pasternak foundations ✓ Elastically connected multiple-beam system Supplementary Materials: The following file was uploaded during submission: • "Vibration analysis of AFG tapered Timoshenko beams"