Vibration Analysis of Axially Functionally Graded Non-Prismatic Euler − Bernoulli Beams Using the Finite Difference Method

: This paper presents an approach to the vibration analysis of axially functionally graded (AFG) non-prismatic Euler − Bernoulli beams using the finite difference method (FDM). The characteristics (cross-sectional area, moment of inertia, elastic moduli, and mass density) of AFG beams vary along the longitudinal axis. 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 AFG non-prismatic Euler − Bernoulli beams 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 AFG non-prismatic Euler − Bernoulli beams, considering the damping. The results obtained in this paper 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 (FDM), used for the homogeneous Euler−Bernoulli beam, to the vibration analysis of axially functionally graded (AFG) non-prismatic Euler−Bernoulli beam. Various studies have focused on the vibration analysis of AFG Euler−Bernoulli beams, most of which using numerical methods. Chen [2] investigated the bending behavior of a non-uniform AFG Euler−Bernoulli beam based on the Chebyshev collocation method; the Chebyshev differentiation matrices were used to reduce the ordinary differential equations into a set of algebraic equations to form the eigenvalue problem associated with the free vibration. Soltani et al. [3] 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. [4] 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. Liu et al. [5] developed a model for the free transverse vibration of AFG tapered Euler−Bernoulli beam through the spline finite point method; the beam was discretized with a set of uniformly scattered spline nodes along the beam axis, and the displacement field was approximated by the particularly constructed cubic B-spline interpolation functions. Kukla et al. [6] proposed an approach to free vibration analysis of functionally graded beams by approximating the beam by an equivalent beam with piece-wise exponentially varying material and geometrical properties. Cao et al. [7] studied the free vibration of AFG beam using analytical method based on the asymptotic perturbation method and Meijer-Function, respectively.
Classical vibration analysis of Euler−Bernoulli beams 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. Assuming a harmonic vibration, the equations of dynamic equilibrium on an infinitesimal beam element are as follows: where (x) is the beam's mass per unit volume, A(x) is the cross-sectional area, k(x) is the stiffness of the elastic Winkler foundation, and  is the circular frequency of the beam. The transverse force T(x) is related to the shear force V(x), as follows: . ( Let us consider an external distributed axial load n(x) positive along the + x axis . ( Substituting Equations (2) and (3) into Equations (1a-b) yields (4a) . (4b) The bending moment, the rotation of the cross section (x) (positive in clockwise), and the deflection are related together as follows: where at a position x, E(x) and I(x) denote the elastic modulus and the moment of inertia, respectively. Combining Equations After some manipulations, Equation (7) yields .

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 bending moment curves w(x) and M(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.
(9a) (9b) Equation (8) has a fourth-order derivative, and the deflection curve is consequently approximated around the point of interest i as a fourth-degree polynomial. Thus, a five-point stencil is used to write finite difference approximations to derivatives at grid points. The derivatives at i are expressed with deflection values at points i-2, i-1, i, i+1, and i+2.
Two FDM approximations were considered for the analysis of AFG beams: the M−W and the W FDM approximation.

W FDM approximation
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, and the elastic modulus are denoted by  r , A r , I r , and E 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 . We set The parameters EI(x) and EI(x) are related to the first and second derivative of E(x)I(x) with respect to x, respectively, as follows: The vibration frequency  is defined as follows: The bending moment, the shear force, and the rotation of the cross section are calculated using Equations (6b), (10c), The shear force and the rotation of the cross section are calculated using Equations (4b) and (9b), and Equations (5b), (9b), and (12b), respectively, as follows: Combining Equations (2), (9b), and (20a) yields the transverse force as follows: (21)    Effect of a spring−mass system: The dynamic behavior of a beam carrying a spring−mass system was analyzed.

Analysis at beam's ends and at positions of discontinuity
The deflection of the mass is denoted by w iM . The spring stiffness K p is defined as follows:

Free vibration analysis of AFG tapered Euler−Bernoulli beams
The natural frequencies (coefficients ) of AFG non-prismatic Euler−Bernoulli beams were determined. Fixed−free, pinned− pinned, and fixed−fixed beams were considered. The geometric and material properties of the beams were represented as follows: , , where Cb and Ch denote the width and height taper ratios, respectively, and E 0 , I 0 ,  0 , and A 0 denote the elastic modulus, the moment of inertia, the beam's mass per unit volume, and the cross-sectional area, respectively, at x = 0.

Pinned−pinned beam 4 Conclusion
The FDM-based model developed in this paper enabled, with relative easiness, vibration analysis of axially functionally graded non-prismatic Euler−Bernoulli 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 Euler−Bernoulli beams resting on Pasternak foundations ✓ Elastically connected multiple-beam system ✓ First-and second-order dynamic stiffness matrices