Free Vibration Analysis of Curvilinearly Tapered Axially Functionally Graded Material Beams

Abstract

The study focuses on the free vibration analysis of beams made of axially functionally graded materials (AFGM) with curvilinear variable cross-sections along their length. The beams encompass various shapes, including concave and convex conic sections, with axial material properties varying according to polynomial and exponential laws. The equations of motion are derived using Hamilton’s principle within the framework of Timoshenko beam theory. These governing equations, subjected to various boundary conditions, are solved using the differential transform method (DTM). The proposed solution technique is validated by comparing computed natural frequencies with the existing literature and results obtained using three-dimensional finite element analysis in ABAQUS. The incorporation of material gradients into the beam finite element models was achieved using the user-defined material subroutine (UMAT). Additionally, a comprehensive study is conducted to examine the influence of various factors on the natural frequencies of functionally graded beams. These factors include parameters of material laws, types of variable beam shapes, slenderness ratio, and specific boundary conditions. This study provides a thorough understanding of the modal dynamics of the considered beams, offering valuable insights into the behavior of FGM structures.

1. Introduction

Incorporating variable cross-sections in structural elements facilitates the achievement of an optimal balance between weight and strength, especially in critical applications like helicopter blades, airplane propellers, turbine blades, and wind turbines. Given their significance, there has been extensive research on the vibrations of these structures, often modeled as beams with variable cross-sections [1,2,3].

The increasing use of functionally graded materials (FGMs) in beam-like structures underscores the need to account for material variations. FGMs enhance performance and provide a continuous stress distribution, unlike the discontinuities found in laminated and sandwich composites [4,5,6].

While initial research focused on material gradients in the thickness direction, there is significant interest in longitudinally graded materials, known as axially functionally graded material (AFGM) structures. Analyzing the free vibration of non-uniform AFGM beams involves recognizing that the coefficients in the equations of motion, derived from any beam theory, vary with changes in cross-sections and material properties along the beam’s length. Consequently, solving these differential equations is not straightforward and often necessitates numerical procedures.

In the context of one-dimensional continuous models with inhomogeneous material parameters, Elishakoff and co-workers have presented exact solutions for the fundamental frequency of diverse AFG beams with different end supports [7]. Yuan et al. (2016) [8] applied the confluent hypergeometric function method to obtain exact solutions for Timoshenko beams with varying bending stiffness. Zhao et al. (2017) [9] employed the Chebyshev polynomials theory to analyze the free vibration of both AFGM Euler–Bernoulli and Timoshenko beams with non-uniform cross-sections. Xie et al. (2017) [10] exploited the spectral collocation method to examine the statics and free vibrations of Euler–Bernoulli beams with axially variable cross-sections and material parameters, while Chen (2021) [11] extended this technique to non-uniform AFG Timoshenko beams. Additionally, Zhang et al. (2019) [12] utilized a Jacobi polynomial-based approximation, and Cao et al. (2019) [13] used the asymptotic perturbation method for vibrational analysis of non-uniform AFGM beams.

Other researchers have creatively adapted methods known from studying homogeneous beams with variable cross-sections to address solutions for non-uniform AFGM beams. For instance, Ghazaryan et al. (2018) [14] employed the differential transform method to study the free vibration characteristics of AFGM Euler–Bernoulli beams, while Rajasekaran and Tochaei (2014) [15] used both the differential transform element and differential quadrature element methods to study AFGM Timoshenko beams, both with varying cross-sectional profiles and material properties along the beam axis. Keshmiri et al. (2018) [16] determined natural frequencies of cantilever beams with different variable cross-sections and material gradients using the Adomian decomposition method (ADM), and Lin et al. (2022) [17] investigated the vibration of rotating non-uniform AFG Euler–Bernoulli beams using the Laplace ADM. In turn, Wang et al. (2022) [18] combined ADM with an iterative process for reliability analysis of composite beams with varying cross-section rigidities and mass distributions along their length. Mahmoud (2019) [19] presented a general solution for the free transverse vibration of cantilever non-uniform AFGM Euler–Bernoulli beams, loaded at the tips with point masses, using the Myklestad method. Chen et al. (2021) [20] employed the variational iteration method to determine the modal characteristics of tapered AFGM Euler–Bernoulli and Timoshenko beams. Liu et al. (2022) [21] proposed a closed-form dynamic stiffness formulation for free vibration analysis of tapered and/or FGM Euler–Bernoulli beams. Adelkhani and Ghanbari (2022) [22] analyzed the vibrations of tapered AFGM beams with nonlinear profiles using the point collocation method.

To investigate the dynamics of AFGM tapered beams under large deflection scenarios, Kumar et al. (2015) [23] employed the Rayleigh–Ritz method with start functions derived from nonlinear static analysis within the Euler–Bernoulli beam theory with von Kármán geometric nonlinearity, while Ghayesh (2018) [24] utilized the Galerkin method and the third-order shear deformation beam theory. Soltani and Asgarian (2019) [25] developed a hybrid approach combining the power series expansions and the Rayleigh–Ritz method for stability and free vibration analyses of non-uniform AFGM beams resting on Winkler–Pasternak elastic foundation. Singh and Sharma (2022) [26] carried out a vibration analysis of an AFGM non-prismatic Timoshenko beam under axial thermal variation in a humid environment using the harmonic differential quadrature method.

Finite element method-based approaches, tailored for precise and efficient study of this task, have also been proposed. Özdemir (2022) [27] studied free vibration and buckling characteristics of AFGM tapered rotating Euler–Bernoulli and Timoshenko beams with different end conditions using a two-noded beam element. Bazoune (2024) [28] proposed a Fourier p-element model for accurate predictions of Timoshenko–Ehrenfest beam frequencies with high precision compared to other existing models. Chen et al. (2019) [29] presented isogeometric analysis (IGA) in conjunction with elasticity theory for the three-dimensional vibration problem of AFGM beams with variable thickness. Murillo et al. [30] used the seven-parameter spectral finite element formulation to perform the analysis of FG shells of either uniform or non-uniform thickness. Burlayenko et al. (2024) [31] conducted simulations on the free vibrations of AFGM beams with non-uniform cross-sections, utilizing both one-dimensional and three-dimensional FGM finite element models developed in ABAQUS via user-defined subroutine UMAT [32,33,34]. Chaker et al. (2020) [35] used the user element (UEL) interface as an alternative method for implementing graded solid shell elements in ABAQUS.

Most of the examined non-uniform beams had rectangular cross-sections with linear taper. There were also studies on beams with parabolic and exponential thickness and constant or linear varying width, but these were less common. In some cases, vibrations of beams were investigated in scenarios such as truncated beams, beams with one sharp end, and beams with both ends sharp, as discussed in [1,3]. However, research on AFGM beams having arbitrary cross-section variations remains limited. Recently, Lee and Lee (2022) [36] studied the coupled flexural-torsional free vibration of circular horizontally curved beams with rectangular and elliptical cross-sections made of AFGMs with quadratic functions of Young’s modulus and the mass density, using the trial eigenvalue method and numerical integration within the Timoshenko and Saint-Venant beam theories. Rezaiee-Pajand et al. (2022) [37] proposed a closed-form solution for the lateral-torsional buckling moment of a bidirectional exponentially functionally graded monosymmetric C-shaped Euler–Bernoulli beam. Liu et al. (2024) [38] presented a nonlinear model to obtain the mode shape and frequency of FGM Euler–Bernoulli thin-walled beams with varying cross-sections, employing the perturbation approach and the Galerkin method. Other researchers have explored the concept of transforming the equations of motion of non-uniform FGM beams into those of equivalent uniform beams. For instance, Chen et al. (2017) [39] applied this idea to a specific class of non-uniform FGM beams. Additionally, Martin and Salehian (2020) [40] developed a method based on metric minimization, a modal participation factor, and the proper orthogonal decomposition to accurately approximate a Euler–Bernoulli motion equation with spatially varying coefficients with an equivalent constant coefficient model.

A literature search indicates that studying the dynamics of tapered AFGM beams involves complexities. While various techniques have been reported, closed-form solutions are limited to specific cross-sections, material variations, and defined boundary conditions. Numerical methods like FEM are proficient but make it challenging to derive general conclusions about system behavior [41]. In this respect, accurate semi-analytical approaches can be a good alternative. Despite extensive research, the vibration characteristics of AFGM beams with arbitrary continuously variable cross-sections are not fully addressed. This study aims to use the differential transform method (DTM) to obtain accurate semi-analytical solutions for the free vibration of curvilinearly tapered AFGM beams under various boundary conditions, determining the effects of variable cross-sections on natural frequencies and mode shapes. This is an innovation of the present contribution.

2. AFGM Beams with Variable Cross-Section

2.1. Geometry of the Beams

Focusing on the closed-form solution of the free vibration problem, we aim to describe various non-uniform beam geometries in a unified analytical manner. One effective way to achieve this is by using rational Bézier curves [42].

We consider planar beams of length of $L$ that are symmetrical about their midlines, with the left ends positioned at the origin of the Cartesian coordinate system as shown in Figure 1. The top curve of these beams is a conic arc, which can be represented by an algebraic quadratic equation in ${ℝ}^2$ as follows:

$$A{x}^2+Bxy+C{x}^2+Dx+Ey+F=0.$$

(1)

The coefficients of (1) can be derived from the quadratic rational Bézier curve, which involves the control points $P_0\left(x_0;y_0\right)$, $P_1\left(x_1;y_1\right)$, and $P_2\left(x_2;y_2\right)$ and the weight $w_1$, utilizing the Gröbner basis as presented in [43]:

$$\begin{gathered} \begin{array}{l}A=y_0^2-4w_1^2{y}_0{y}_1+4w_1^2{y}_1^2-2y_0{y}_2+4w_1^2{y}_0{y}_2-4w_1^2{y}_1{y}_2+y_2^2,\\ B=-2x_0{y}_0+4w_1^2{x}_1{y}_0+2x_2{y}_0-4w_1^2{x}_2{y}_0+4w_1^2{x}_0{y}_1-8w_1^2{x}_1{y}_1+4w_1^2{x}_2{y}_1+2x_0{y}_2-4w_1^2{x}_0{y}_2 \\ +4w_1^2{x}_1{y}_2-2x_2{y}_2,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\\ C=x_0^2-4w_1^2{x}_0{x}_1+4w_1^2{x}_1^2-2x_0{x}_2+4w_1^2{x}_0{x}_2-4w_1^2{x}_1{x}_2+x_2^2,\\ D=-2x_2{y}_0^2+4w_1^2{x}_1{y}_0{y}_1+4w_1^2{x}_2{y}_0{y}_1-4w_1^2{x}_0{y}_1^2-4w_1^2{x}_2{y}_1^2+2x_0{y}_0{y}_2-8w_1^2{x}_1{y}_0{y}_2+2x_2{y}_0{y}_2 \\ +4w_1^2{x}_0{y}_1{y}_2+4w_1^2{x}_1{y}_1{y}_2-2x_0{y}_2^2,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\\ E=-4w_1^2{x}_1^2{y}_0+2x_0{x}_2{y}_0+4w_1^2{x}_1{x}_2{y}_0-2x_2^2{y}_0+4w_1^2{x}_0{x}_1{y}_1-8w_1^2{x}_0{x}_2{y}_1+4w_1^2{x}_1{x}_2{y}_1-2x_0^2{y}_2 \\ +4w_1^2{x}_0{x}_1{y}_2-4w_1^2{x}_1^2{y}_2+2x_0{x}_2{y}_2,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\\ F=x_2^2{y}_0^2-4w_1^2{x}_1{x}_2{y}_0{y}_1+4w_1^2{x}_0{x}_2{y}_1^2+4w_1^2{x}_1^2{y}_0{y}_2-2x_0{x}_2{y}_0{y}_2-4w_1^2{x}_0{x}_1{y}_1{y}_2+x_0^2{y}_2^2.\end{array} \end{gathered}$$

(2)

For convenience, let $l_f$ and $l_e$ denote the length of lines ${\mathit{P}}_1{\mathit{P}}_m$ and ${\mathit{P}}_s{\mathit{P}}_m$, respectively, where ${\mathit{P}}_S=\left(1-s\right){\mathit{P}}_m+s{\mathit{P}}_1$ is the shoulder point, ${\mathit{P}}_m$ is the midpoint of line ${\mathit{P}}_0{\mathit{P}}_2$, and $s={\displaystyle \frac{l_e}{l_f}}$ is a shape parameter related to $w_1={\displaystyle \frac{s}{1-s}}$. By specifying the coordinates of the points ${\mathit{P}}_0$, ${\mathit{P}}_1$, and ${\mathit{P}}_2$ and the parameter $s$, the implicit equation of a conic curve (1) can be constructed. The curve will have a convex upward shape if the vertex ${\mathit{P}}_1$ is above the line ${\mathit{P}}_0{\mathit{P}}_2$ and a concave downward shape if ${\mathit{P}}_1$ below the line ${\mathit{P}}_0{\mathit{P}}_2$ as illustrated in Figure 1a,b, respectively.

2.2. Graded Materials of the Beams

We suppose that the materials of the beams with variable thickness are homogeneous in the thickness direction but graded along the axial direction. Consequently, the material properties, such as mass density and Young’s modulus, vary with the position along the $x$-axis. Specifically, we analyze the following types of material distributions:

Type 1. The effective mass density and Young’s modulus are power law functions of $x$ written as follows:

$$\rho \left(x\right)={\rho }_0{\displaystyle \sum }_{i=0}^m{a}_i{\left({\displaystyle \frac{x}{L}}\right)}^i, E\left(x\right)=E_0{\displaystyle \sum }_{i=0}^n{b}_i{\left({\displaystyle \frac{x}{L}}\right)}^i,$$

(3)

where ${\rho }_0$ and $E_0$ are the mass density and Young’s modulus at $x=0$, respectively; $m$ and $n$ are power coefficients that account for arbitrary variations of the material parameters along the beam’s length; and $a_i$ and $b_i$ constant coefficients.

Type 2. The effective density and Young’s modulus follow exponential law functions of $x$, given as follows:

$$\rho \left(x\right)={\rho }_0{V}_e,E\left(x\right)=E_0{V}_e,$$

(4)

where the exponential function $V_e=e^{{\gamma }_e\left(1-a\left(1-{\displaystyle \frac{x}{L}}\right)\right)}$ is determined by the parameter $a$ and gradation index ${\gamma }_e$, [29].

Type 3. The beams with variable thickness are composed of a metal–ceramic mixture. The effective material properties are calculated based on the rule of mixtures in the following form:

$$\rho \left(x\right)={\rho }_c+\left({\rho }_m-{\rho }_c\right)V_c,E\left(x\right)=E_c+\left(E_m-E_c\right)V_c,$$

(5)

where the subscripts ‘$m$’ and ‘$c$’ refer to the properties of metal and ceramic constituents, respectively; $V_c$ denotes the ceramic volume fraction, expressed as $V_c={\left(1-b\left(1-{\displaystyle \frac{x}{L}}\right)\right)}^{{\gamma }_c}$ with the parameter $b$ and gradation index ${\gamma }_c$, [29].

Poisson’s ratio $\nu$ is assumed to be constant throughout this study.

2.3. Governing Equations of the Beams

We consider the transverse vibrations of an AFGM beam in the $xz$-plane using the Timoshenko beam theory. The equations of motion governing such beams with generally variable cross-sections in terms of the transverse displacement $w\left(x,t\right)$ of the midline and the cross-sectional rotation angle $\phi \left(x,t\right)$ at time $t$ are written as follows:

$$\begin{gathered} -{\displaystyle \frac{\partial }{\partial x}}\left(D_{11}\left(x\right){\displaystyle \frac{\partial \phi }{\partial x}}\right)-A_{55}\left(x\right)\left({\displaystyle \frac{\partial w}{\partial x}}-\phi \right)+I_2\left(x\right){\displaystyle \frac{{\partial }^2\phi }{\partial t^2}}=0, \\ -{\displaystyle \frac{\partial }{\partial x}}\left(A_{55}\left(x\right)\left({\displaystyle \frac{\partial w}{\partial x}}-\phi \right)\right)+I_0\left(x\right){\displaystyle \frac{{\partial }^{2}w}{\partial t^2}}=0. \end{gathered}$$

(6)

The rigidities and the inertia coefficients are expressed as integrals over the cross-sectional domain in the following form:

$$\begin{gathered} D_{11}\left(x\right)={{\displaystyle \int }}_{A\left(x\right)}{z}^{2}E\left(x\right)dA,A_{55}\left(x\right)=\kappa {{\displaystyle \int }}_{A\left(x\right)}G\left(x\right)dA, \\ \left(I_0\left(x\right),I_2\left(x\right)\right)={{\displaystyle \int }}_{A\left(x\right)}\left(1,z^2\right)\rho \left(x\right)dA, \end{gathered}$$

(7)

where $\rho \left(x\right)$ is the mass density, $G\left(x\right)={\displaystyle \frac{E\left(x\right)}{2\left(1+\nu \right)}}$ is the shear modulus, and $\kappa$ is the shear correction factor. Throughout this study, the shear correction factor is accepted as 5/6 unless specified otherwise.

Assuming simple harmonic oscillations, the deflection and the rotation angle can be expressed as follows:

$$w\left(x,t\right)=\overline{w}\left(x\right)e^{i\omega t},\phi \left(x,t\right)=\overline{\phi }\left(x\right)e^{i\omega t},$$

(8)

where $\overline{w}\left(x\right)$ and $\overline{\phi }\left(x\right)$ are the amplitudes of the displacement and rotation angle, respectively; $\omega$ is the angular frequency; and $i^2=-1$ represents the imaginary unit.

Substituting (8) into (6) yields a system of coupled ordinary differential equations that describe the free transverse vibrations of a Timoshenko beam with a non-uniform cross-section as follows:

$$\begin{gathered} {\displaystyle \frac{d}{dx}}\left(D_{11}\left(x\right){\displaystyle \frac{d\overline{\phi }}{dx}}\right)+A_{55}\left(x\right)\left({\displaystyle \frac{d\overline{w}}{dx}}-\overline{\phi }\right)+I_2\left(x\right){\omega }^2\overline{\phi }=0, \\ {\displaystyle \frac{d}{dx}}\left(A_{55}\left(x\right)\left({\displaystyle \frac{d\overline{w}}{dx}}-\overline{\phi }\right)\right)+I_0\left(x\right){\omega }^2\overline{w}=0. \end{gathered}$$

(9)

Here, the rigidities and inertia coefficients (7) can be conveniently presented using the following expressions:

$$\begin{gathered} D_{11}\left(x\right)=E\left(x\right)I\left(x\right),A_{55}\left(x\right)=\kappa G\left(x\right)A\left(x\right), \\ {I}_0\left(x\right)=\rho \left(x\right)A\left(x\right),I_2\left(x\right)=\rho \left(x\right)J\left(x\right), \end{gathered}$$

(10)

where $A\left(x\right)$ and $J\left(x\right)$ are the area and second inertia moment of the cross-section, respectively.

3. Solution Methodology

3.1. Differential Transform Method

Due to the variable coefficients, reflecting the dependence of material properties and cross-sectional geometry on the $x$-coordinate, the coupled nature of the differential equations in (9) presents mathematical challenges for finding a closed-form solution. To address this, we employ the differential transform method (DTM) for its versatility and simplicity. This method converts the differential equations into a set of recurrent algebraic equations, enabling the solution to be represented as an infinite power series. The fundamental aspects of the DTM can be found in G.E. Pukhov’s original work [44].

We reformulate system (9) into a more suitable form for applying the DTM as follows:

$$\begin{gathered} {\displaystyle \frac{d^2\overline{\phi }}{d{x}^2}}+{\overline{D}}_1\left(x\right){\displaystyle \frac{d\overline{\phi }}{dx}}+{\overline{D}}_2\left(x\right){\displaystyle \frac{d\overline{w}}{dx}}-{\overline{D}}_2\left(x\right)\overline{\phi }+{\omega }^2{\overline{D}}_3\left(x\right)\overline{\phi }=0, \\ {\displaystyle \frac{d^2\overline{w}}{d{x}^2}}+{\overline{G}}_1\left(x\right){\displaystyle \frac{d\overline{w}}{dx}}-{\displaystyle \frac{d\overline{\phi }}{dx}}-{\overline{G}}_1\left(x\right)\overline{\phi }+{\omega }^2{\overline{G}}_2\left(x\right)\overline{w}=0, \end{gathered}$$

(11)

where the variable coefficients are defined as ${\overline{D}}_1\left(x\right)={\displaystyle \frac{D_{11}^{\prime }\left(x\right)}{D_{11}\left(x\right)}}$, ${\overline{D}}_2\left(x\right)={\displaystyle \frac{A_{55}\left(x\right)}{D_{11}\left(x\right)}}$, ${\overline{D}}_3\left(x\right)={\displaystyle \frac{I_2\left(x\right)}{D_{11}\left(x\right)}}$, ${\overline{G}}_1\left(x\right)={\displaystyle \frac{A_{55}^{\prime }\left(x\right)}{A_{55}\left(x\right)}}$, and ${\overline{G}}_2\left(x\right)={\displaystyle \frac{I_0\left(x\right)}{A_{55}\left(x\right)}}$. Here, and in what follows, the prime denotes differentiation with respect to the $x$-coordinate.

Hence, the solution of (11) at a specific point $x_0$ within the interval $0\le x\le L$ in the domain of differential transformation is expressed by the recurrent algebraic equations:

$$\begin{array}{l}\Phi \left(k+2\right)=k_{*}\{-{\displaystyle {\displaystyle \sum }_{p=0}^k}\left(p+1\right)\Phi \left(p+1\right)D_1\left(k-p\right)+{\displaystyle {\displaystyle \sum }_{p=0}^k}\Phi \left(p\right)D_2\left(k-p\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{\displaystyle {\displaystyle \sum }_{p=0}^k}\left(p+1\right)W\left(p+1\right)D_2\left(k-p\right)-{\omega }^2{\displaystyle {\displaystyle \sum }_{p=0}^k}\Phi \left(p\right)D_3\left(k-p\right)\}\\ W\left(k+2\right)=k_{*}\{-{\displaystyle {\displaystyle \sum }_{p=0}^k}\left(p+1\right)W\left(p+1\right)G_1\left(k-p\right)+{\displaystyle {\displaystyle \sum }_{p=0}^k}\Phi \left(p\right)G_1\left(k-p\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle {\displaystyle \sum }_{p=0}^k}\left(p+1\right)\Phi \left(p+1\right)L\left(k-p\right)-{\omega }^2{\displaystyle {\displaystyle \sum }_{p=0}^k}W\left(p\right)G_2\left(k-p\right)\}\end{array}$$

(12)

where $k_{*}={\displaystyle \frac{k!}{\left(k+2\right)!}}$, $L\left(k\right)$ is a unity image, and $W\left(k\right)$, $\Phi \left(k\right)$, $D_1\left(k\right)$, $D_2\left(k\right)$, $D_3\left(k\right)$, $G_1\left(k\right)$, and $G_2\left(k\right)$ are the images of the unknown amplitudes $\overline{w}\left(x\right)$ and $\overline{\phi }\left(x\right)$ and the corresponding functional coefficients in (11).

By incrementing the index $k$ sequentially, we can generate all images from (12) except for $W\left(0\right)$, $W\left(1\right)$, $\Phi \left(0\right)$, and $\Phi \left(1\right)$. Consequently, the system (12) can be simplified to the following form:

$$\begin{gathered} \Phi \left(k+2\right)=B_{\phi }\left(k\right)W\left(0\right)+C_{\phi }\left(k\right)\Phi \left(0\right)+G_{\phi }\left(k\right)W\left(1\right)+H_{\phi }\left(k\right)\Phi \left(1\right), \\ W\left(k+2\right)=B_w\left(k\right)W\left(0\right)+C_w\left(k\right)\Phi \left(0\right)+G_w\left(k\right)W\left(1\right)+H_w\left(k\right)\Phi \left(1\right) \end{gathered}$$

(13)

where the explicit forms of the recurrent coefficients in (13) are presented in [45] (see Appendix B in [45]).

Given that the images $W\left(k\right)$ and $\Phi \left(k\right)$ at $k\ge 2$ depend on $W\left(0\right)$, $W\left(1\right)$, $\Phi \left(0\right)$, and $\Phi \left(1\right)$, the original functions $\overline{w}\left(x\right)$ and $\overline{\phi }\left(x\right)$ can be reconstructed for a selected number of images $N$ as follows:

$$\begin{array}{l}\overline{w}\left(x,\omega \right)=\left\{1+{\displaystyle {\displaystyle \sum }_{p=0}^{N-2}}{B}_w\left(p\right){\left(x-x_0\right)}^{p+2}\right\}W\left(0\right)+\left\{{\displaystyle {\displaystyle \sum }_{p=0}^{N-2}}{C}_w\left(p\right){\left(x-x_0\right)}^{p+2}\right\}\Phi \left(0\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left\{\left(x-x_0\right)+{\displaystyle {\displaystyle \sum }_{p=0}^{N-2}}{G}_w\left(p\right){\left(x-x_0\right)}^{p+2}\right\}W\left(1\right)+\left\{{\displaystyle {\displaystyle \sum }_{p=0}^{N-2}}{H}_w\left(p\right){\left(x-x_0\right)}^{p+2}\right\}\Phi \left(1\right),\\ \overline{\phi }\left(x,\omega \right)=\left\{{\displaystyle {\displaystyle \sum }_{p=0}^{N-2}}{B}_{\phi }\left(p\right){\left(x-x_0\right)}^{p+2}\right\}W\left(0\right)+\left\{1+{\displaystyle {\displaystyle \sum }_{p=0}^{N-2}}{C}_{\phi }\left(p\right){\left(x-x_0\right)}^{p+2}\right\}\Phi \left(0\right)+\left\{{\displaystyle {\displaystyle \sum }_{p=0}^{N-2}}{G}_{\phi }\left(p\right){\left(x-x_0\right)}^{p+2}\right\}W\left(1\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left\{\left(x-x_0\right)+{\displaystyle {\displaystyle \sum }_{p=0}^{N-2}}{H}_{\phi }\left(p\right){\left(x-x_0\right)}^{p+2}\right\}\Phi \left(1\right).\end{array}$$

(14)

The boundary conditions at the beam ends can include transverse displacement $\overline{w}$, rotation angle $\overline{\phi }$, bending moment $\overline{M}$, and shear force $\overline{Q}$, which are related by the following equations

$$\overline{M}=D_{11}\left(x\right){\displaystyle \frac{d\overline{\phi }}{dx}},\overline{Q}=A_{55}\left(x\right)\left({\displaystyle \frac{d\overline{\overline{w}}}{dx}}-\overline{\phi }\right)$$

(15)

Taking into account (14) and (15), we can formulate the eigenvalue problem in a form that allows for the incorporation of any constraints as follows:

$$\left[\begin{array}{cccc}{A}_{11}\left(\omega \right)& A_{12}\left(\omega \right)& A_{13}\left(\omega \right)& A_{14}\left(\omega \right)\\ A_{21}\left(\omega \right)& A_{22}\left(\omega \right)& A_{23}\left(\omega \right)& A_{24}(\omega \\ A_{31}\left(\omega \right)& A_{32}\left(\omega \right)& A_{33}\left(\omega \right)& A_{34}(\omega \\ A_{41}\left(\omega \right)& A_{42}\left(\omega \right)& A_{43}\left(\omega \right)& A_{44}(\omega \end{array}\right]\left(\begin{array}{c}W\left(0\right)\\ \Phi \left(0\right)\\ W\left(1\right)\\ \Phi \left(1\right)\end{array}\right)=0.$$

(16)

Here, the functions $A_{ij}\left(\omega \right)$ are appropriate polynomials of the unknown angular frequency $\omega$, which involve the recurrent coefficients and the power terms of $\left(x – x_0\right)$.

Finally, the eigenvalues are computed by solving the root-finding problem resulting from Equation [14]:

$$\det \left|A_{ij}\left(\omega \right)\right|=0.$$

(17)

A computational program has been developed in the Matlab R2016b environment to apply the DTM for analyzing the free vibration of curvilinear tapered inhomogeneous Timoshenko beams. The eigenvalue problem is addressed using standard algorithms provided in the Matlab R2016b package [46].

3.2. Finite Element Modeling

To ensure the accuracy and validation of the proposed semi-analytical methodology for analyzing the free vibration parameters of curvilinearly tapered AFGM beams, a finite element model has been developed and utilized for comparison purposes. It is well-known that the optimal modeling strategy involves using graded finite elements (GFEs), which allow for the variation of material properties at the element level. Additionally, advanced three-dimensional models are essential to precisely capture the non-uniform geometry of axially inhomogeneous beams [31].

The free vibration analysis of AFGM beams is conducted using ABAQUS [47], incorporating a user-defined material subroutine (UMAT) to implement material gradients into a convenient three-dimensional element, C3D20, as detailed in [31,33] and illustrated in Figure 2b. The material gradient is assigned within the element volume during the computation of the element stiffness matrix, ${\mathit{K}}^e$ using Gauss quadrature, as follows:

$${\mathit{K}}^e={\displaystyle \sum }_{p=1}^{n_p}{\displaystyle \sum }_{q=1}^{n_q}{\displaystyle \sum }_{r=1}^{n_r}{\left({\mathit{B}}^T\mathit{D}\left(x\right)\mathit{B}\left|\mathit{J}\right|\right)}_{p,q,r}{w}_p{w}_q{w}_r,$$

(18)

where the indices $p$, $q$, and $r$ refer to the integration points over the element volume; $\left|\mathit{J}\right|$ is the determinant of the geometric Jacobian matrix; $w_p$, $w_q$, and $w_r$ are the Gaussian weights; $\mathit{B}$ stands for a matrix of gradients of the shape functions for the 3-D element; and the material matrix $\mathit{D}\left(x\right)$ of the 3-D GFE relevant to this study can be found in [33].

ABAQUS currently does not support calculating the mass matrix based on a user-defined mass density in frequency analysis [47]. The elements available in ABAQUS use a constant mass density, such as that noted in the following equation:

$${\mathit{M}}^e=\underset{V^e}{{\displaystyle \int }}{\rho }_0{\mathit{N}}^T·\mathit{N}d{V}^e,$$

(19)

where $\mathit{N}$ is an element interpolation matrix.

To address this limitation, we assigned a constant effective mass density within the GFE by using an average value of the actual mass density distribution over the beam’s volume in the following form:

$${\rho }_{avr}={\displaystyle \frac{1}{V}}\underset{V}{{\displaystyle \int }}\rho \left(x\right)dx$$

(20)

In the case of the free vibration analysis without damping, the global discrete system of governing equations of motion is reduced to the problem of eigenvalues and associated eigenvectors, which is written as follows:

$$\left(\mathit{K}-{\omega }^2\mathit{M}\right)\mathbf{X}=0,$$

(21)

where $\mathit{M}$ and $\mathit{K}$ are global mass and stiffness matrices, respectively, obtained by assembling the formulated element mass and stiffness matrices; $\omega$ is the undamped circular frequency; and $\mathbf{X}$ is the vector of nodal displacements associated with oscillations at each $\omega$. The free vibration analysis is carried out using linear perturbation load step, obtaining the eigenvalues by means of Lanczos or subspace iteration methods [47]. Figure 2a illustrates a three-dimensional finite element beam model created using the ABAQUS pre-processor, utilizing the ‘loft’ feature. To provide accurate results, the element aspect ratio in the mesh was consistently maintained close to one throughout all subsequent calculations.

4. Results

We will now examine beams with various curvilinear tapers in rectangular cross-sections, incorporating three distinct material variations along the beam axis. This analysis aims to demonstrate the effectiveness of the proposed computational technique by comparing the results with those obtained from other methods. First, we validate the accuracy of the DTM approach by benchmarking our findings against known solutions for uniform homogeneous and inhomogeneous Timoshenko beams under various boundary conditions. Then, we present a selection of examples for curvilinear taper beams, including both known and previously unexplored cases, to further illustrate the method’s applicability and provide new insights that may be valuable to other researchers in this field.

4.1. Homogeneous Timoshenko Beams with Uniform Cross-Sections

The accuracy and effectiveness of the proposed approach are first benchmarked by computing the frequencies of homogeneous beams with uniform cross-sections.

We consider geometrically uniform ($w_1=0$) and homogeneous ($m=n=0$ with unit coefficients in (3)) beams with clamped-clamped (C-C), clamped-free (C-F), and simply supported (S-S) boundary conditions. The material and geometric parameters of the beams are listed in

Table 1. The parameters of uniform beams.

L, m	$\mathit{b}$, m	$\mathit{h}$, m	$\mathit{E}$, Pa	$\mathit{G}$, Pa	$\mathit{\nu }$	$\mathit{\kappa }$	$\mathit{\rho }$, kg/m3
1	0.02	0.1	2.1 × 1011	0.875 × 1011	0.2	0.845	7860

.

The first seven natural frequencies of the S-S, C-C, and C-F beams compared with those computed using the dynamic stiffness matrix method (DSM) in conjunction with the Timoshenko beam theory in [48] are presented in

Table 2. The natural frequencies of uniform S-S, C-C, and C-F homogeneous beams based on different methods (Hz).

Mode Number	S-S			C-C			C-F
	DTM	[48]	Δ *, %	DTM	[48]	Δ, %	DTM	[48]	Δ, %
1	230.7888	230.7890	6.9 × 10−5	500.5870	500.5870	3.6 × 10−6	82.88577	82.88600	2.8 × 10−5
2	884.4691	884.4690	1.3 × 10−5	1288.763	1288.763	1.7 × 10−5	498.1911	498.1910	2.9 × 10−5
3	1870.358	1870.358	1.4 × 10−5	2339.160	2339.160	7.0 × 10−6	1315.041	1315.041	1.4 × 10−5
4	3091.056	3091.056	4.4 × 10−6	3565.013	3565.013	9.4 × 10−6	2394.776	2394.776	7.0 × 10−6
5	4468.225	4468.225	9.5 × 10−6	4910.439	4910.438	2.3 × 10−5	3658.662	3658.662	5.9 × 10−6
6	5946.852	5946.679	2.9 × 10−3	6338.321	6337.691	9.9 × 10−3	5044.291	5044.277	2.9 × 10−3
7	7472.753	7489.650	2.3 × 10−1	7822.471	7821.911	7.2 × 10−3	6508.492	6509.954	2.3 × 10−2

* $\mathsf{\Delta }={\displaystyle \frac{\left|{\omega }_{DSM}-{\omega }_{DTM}\right|}{{\omega }_{DSM}}}\times 100\%$.

.

As shown in Table 2, there is excellent agreement between the solutions obtained using the DTM and DSM methods for all vibrational modes.

4.2. Inhomogeneous Timoshenko Beams with Uniform Cross-Sections

Next, the ability of the proposed approach to accurately predict the frequencies of beams with uniform cross-sections made of inhomogeneous materials is evaluated.

We assume that the beam’s mass density and modulus of elasticity follow power law distributions of Type 1 as described in (3). Since natural frequencies for Timoshenko beams under these conditions are not documented in the literature, we will compare our results with those obtained using the Euler–Bernoulli beam theory in [14] for AFGM beams with a slenderness ratio of $L/h=50$. The geometrical and material parameters are consistent with those detailed in [14] (see Tables 4–6 and Appendix B in [14]).

The comparison of the first five dimensionless frequencies $\overline{\omega }=\omega L^2\sqrt{{\displaystyle \frac{{\rho }_{0}A}{E_{0}J}}}$ for the cantilever AFGM beam is presented in

Table 3. Dimensionless natural frequencies of uniform cantilever AFGM beams.

$\begin{array}{c}\\ \mathit{n}\end{array}\backslash \begin{array}{c}\mathit{m}\\ \end{array}$	0				1				2
	[14]	DTM	$\mathbf{\Delta }{ }^{**},\mathbf{\%}$	DTM *	[14]	DTM	$\mathbf{\Delta },\mathbf{\%}$	DTM *	[14]	DTM	$\mathbf{\Delta },\mathbf{\%}$	DTM *
0	17.92823	17.9268	0.01	17.78719	21.05590	21.0543	0.01	20.89837	24.29448	24.29265	0.01	24.11362
	112.3543	112.2918	0.06	106.6048	140.9285	140.8521	0.05	133.8822	162.7971	162.7083	0.05	154.6099
	314.5953	314.1788	0.13	280.3868	403.4795	402.9577	0.13	360.512	462.7162	462.1126	0.13	413.0607
	616.4812	614.9754	0.25	508.6127	795.2794	793.3707	0.24	658.1122	909.6774	907.4782	0.24	751.8375
	1019.088	1015.124	0.39	774.2471	1317.894	1312.838	0.39	1004.356	1505.584	1499.772	0.39	1145.767
1	18.89966	18.89824	0.01	18.75921	22.29867	22.29709	0.01	22.14241	25.73713	25.73531	0.01	25.55769
	124.9620	124.8901	0.06	118.3494	158.3713	158.2830	0.06	150.2293	182.9067	182.8039	0.06	173.4348
	355.0148	354.5324	0.14	315.5107	460.6318	460.0258	0.13	410.8001	527.8895	527.1870	0.13	470.1954
	698.2161	696.4732	0.25	573.9377	911.2419	909.0284	0.25	752.4501	1041.633	1039.076	0.25	858.5914
	1155.920	1151.340	0.40	874.6708	1512.257	1506.403	0.39	1150.092	1726.572	1719.828	0.39	1310.355
2	18.99432	18.99291	0.01	18.85473	22.45531	22.45374	0.01	22.3005	25.92123	25.91943	0.01	25.74348
	127.1780	127.1041	0.06	120.3839	162.3684	162.2768	0.06	153.9285	187.5542	187.4475	0.06	177.7289
	363.0379	362.5411	0.14	322.3891	475.3808	474.7514	0.13	423.6509	544.8226	544.0922	0.13	484.8789
	714.8085	713.0133	0.25	586.9809	941.7464	939.4477	0.25	776.9921	1076.597	1073.939	0.25	886.5256
	1183.930	1179.213	0.40	894.8416	1563.728	1557.652	0.39	1188.231	1785.535	1778.528	0.39	1353.679
3	18.97105	18.96963	0.01	18.83099	22.45202	22.45045	0.01	22.29715	25.91884	25.91704	0.01	25.74104
	126.3443	126.2712	0.06	119.6185	162.2392	162.1477	0.06	153.8091	187.4615	187.3548	0.06	177.6432
	359.5311	359.0406	0.14	319.3864	474.8301	474.2016	0.13	423.1715	544.4325	543.7027	0.13	484.5410
	707.2811	705.5100	0.25	581.0888	940.5708	938.2755	0.25	776.0493	1075.765	1073.110	0.25	885.8633
	1171.009	1166.357	0.40	885.5952	1561.717	1555.650	0.39	1186.747	1784.113	1777.111	0.39	1352.638
4	18.97367	18.97225	0.01	18.83369	22.44933	22.44776	0.01	22.29437	25.92410	25.9223	0.01	25.74647
	126.4718	126.3986	0.06	119.7366	162.0943	162.0029	0.06	153.6739	187.7400	187.6332	0.06	177.9022
	360.1566	359.6650	0.14	319.9237	474.1105	473.4831	0.13	422.5427	545.7931	545.0612	0.13	485.7226
	708.6890	706.9135	0.25	582.1911	938.9675	936.6766	0.25	774.7618	1078.791	1076.127	0.25	888.2724
	1173.481	1168.816	0.40	887.3617	1558.918	1552.864	0.39	1184.683	1789.389	1782.364	0.39	1356.492
5	−	−	−	−	22.44994	22.44838	0.01	22.29501	25.92274	25.92094	0.01	25.74504
	−	−	−	−	162.1362	162.0448	0.06	153.7134	187.6486	187.5419	0.06	177.8163
	−	−	−	−	474.3503	473.7225	0.13	422.7534	545.2794	544.5482	0.13	485.2744
	−	−	−	−	939.5286	937.2362	0.25	775.2134	1077.591	1074.931	0.25	887.3160
	−	−	−	−	1559.921	1553.862	0.39	1185.423	1787.249	1780.233	0.39	1354.929
6	−	−	−	−	−	−	−	−	25.92296	25.92116	0.01	25.74528
	−	−	−	−	−	−	−	−	187.6670	187.5602	0.06	177.8338
	−	−	−	−	−	−	−	−	545.3964	544.6650	0.13	485.3772
	−	−	−	−	−	−	−	−	1077.880	1075.218	0.25	887.5466
	−	−	−	−	−	−	−	−	1787.776	1780.758	0.39	1355.315

* Dimensionless frequencies of the cantilever AFGM beam with $L/h=10$; ** $\mathsf{\Delta }={\displaystyle \frac{\left|{\omega }_{Ref}-{\omega }_{DTM}\right|}{{\omega }_{Ref}}}\times 100\%$.

. Furthermore, to extend this study to beams, where shear deformation plays a significant role, in Table 3, we also provide the natural frequencies computed for beams with a slenderness ratio of 10.

Upon reviewing Table 3, it becomes evident that both Euler–Bernoulli and Timoshenko beams with slenderness ratio of 50 demonstrate excellent agreement for the fundamental frequency and show slight deviations for higher frequencies. This is because the Euler–Bernoulli beam theory overestimates especially higher frequencies due to its neglect of shear effect. Specifically, when the slenderness ratio is 10, the fundamental frequencies of the beams show only satisfactory matching, while significant discrepancies are observed in higher-order frequencies, consistent with theoretical expectations.

In the second example, we evaluate the accuracy of predicted natural frequencies for exponential AFGM beams. The current results are compared with those obtained in [11] using the Chebyshev collocation method. The beams are composed of zirconia and aluminum with axial gradation according to law (5), where the ceramic volume fraction is an exponential function: $V_c={\displaystyle \frac{e^{\alpha x/L}-1}{e^{\alpha }-1}}$. The material and geometrical properties of the beams are described in [11] and detailed in

Table 4. The parameters of uniform exponential law AFGM beams.

L, m	$\mathit{b}$, m	$\mathit{h}$, m	$\mathit{E}$, GPa		$\mathit{\rho }$, kg/m3		ν
			ZrO2	Al	ZrO2	Al
1	0.01	0.03	200	70	3800	2702	0.3

as follows:

The computed first four dimensionless natural frequencies $\overline{\omega }=\omega L^2\sqrt{{\displaystyle \frac{{\rho }_{Al}A}{E_{Al}J}}}$ of the exponential law AFGM beams under S-S and C-C boundary conditions for various exponents $\alpha$ compared with the results in [11] are given in

Table 5. Dimensionless natural frequencies of uniform exponential law AFGM beams.

BCs	$\mathit{\alpha }$	−10			−3			0			3			10
	Mode	[11]	DTM	$\mathbf{\Delta }{ }^{*}, \mathbf{\%}$	[11]	DTM	$\mathbf{\Delta }, \mathbf{\%}$	[11]	DTM	$\mathbf{\Delta }, \mathbf{\%}$	[11]	DTM	$\mathbf{\Delta }, \mathbf{\%}$	[11]	DTM	$\mathbf{\Delta }, \mathbf{\%}$
S-S	1	9.9225	9.8357	0.88	10.351	10.351	4.8 × 10−4	10.849	10.849	3.7 × 10−4	11.225	11.225	3.7 × 10−4	11.437	11.447	3.7 × 10−4
	2	39.834	39.363	1.18	41.716	41.716	2.4 × 10−5	43.396	43.396	6.9 × 10−5	44.571	44.571	6.9 × 10−5	45.394	45.446	6.9 × 10−5
	3	89.325	89.576	0.28	93.239	93.239	2.2 × 10−5	96.793	96.793	3.1 × 10−5	99.431	99.431	3.1 × 10−5	101.21	101.31	3.1 × 10−5
	4	157.50	156.01	0.94	164.07	164.07	6.1 × 10−6	170.22	170.22	2.4 × 10−5	174.87	174.87	2.4 × 10−5	177.98	178.17	2.4 × 10−5
C-C	1	24.646	24.613	0.14	24.780	24.780	1.6 × 10−4	24.223	24.223	1.2 × 10−4	23.794	23.794	1.2 × 10−4	23.907	23.925	1.2 × 10−4
	2	65.291	65.232	0.09	66.155	66.155	1.5 × 10−5	66.621	66.621	4.5 × 10−5	66.780	66.781	4.5 × 10−5	67.120	67.176	4.5 × 10−5
	3	124.53	123.97	0.44	127.04	127.04	1.6 × 10−5	129.55	129.55	7.7 × 10−6	131.09	131.09	7.7 × 10−6	131.94	132.03	7.7 × 10−6
	4	201.23	201.32	0.05	206.29	206.29	1.6 × 10−4	211.61	211.61	1.9 × 10−5	215.14	215.15	1.9 × 10−5	216.92	217.10	1.9 × 10−5

* $\mathsf{\Delta }={\displaystyle \frac{\left|{\omega }_{Ref}-{\omega }_{DTM}\right|}{{\omega }_{Ref}}}\times 100\%$.

.

From Table 5, it is obvious that the present results for all first four frequencies closely align with those obtained in the referenced study.

Finally, uniform aluminum-zirconia Timoshenko beams with axial variation of the material characteristics according to the power law function $V_c={\left({\displaystyle \frac{x}{L}}\right)}^{\beta }$ of the ceramic volume fraction in the rule of mixtures (5) are tested. The natural frequencies of these beams are benchmarked against the findings in [15]. The parameters of the AFGM beams are the same as those in [15] and are listed in Table 4.

The first four dimensionless natural frequencies $\overline{\omega }=\omega L^2\sqrt{{\displaystyle \frac{{\rho }_{Zr}A}{E_{Zr}J}}}$ of the AFGM beams under C-F, S-S, and C-C boundary conditions for various indices $\beta$, compared with the solutions obtained using the differential quadrature element method in [15], are shown in

Table 6. Dimensionless natural frequencies of uniform power law AFGM beams.

BCs	$\mathit{\beta }$	1			2			3
	Mode	[15]	DTM	$\mathbf{\Delta }{ }^{*}, \mathbf{\%}$	[15]	DTM	$\mathbf{\Delta }, \mathbf{\%}$	[15]	DTM	$\mathbf{\Delta }, \mathbf{\%}$
C-F	1	3.89630	3.896310	2.6 × 10−4	3.88280	3.883120	8.2 × 10−3	3.79570	3.797640	5.4 × 10−2
	2	15.0505	15.05049	6.0 × 10−4	15.2590	15.25982	6.0 × 10−3	15.2936	15.30040	4.5 × 10−2
	3	30.9409	30.94087	1.2 × 10−3	31.6180	31.61871	3.2 × 10−3	31.9962	32.00717	3.6 × 10−2
	4	46.3782	46.37814	1.2 × 10−3	47.6349	47.63506	1.6 × 10−3	48.3335	48.34427	2.4 × 10−2
S-S	1	7.84590	7.845920	2.6 × 10−4	7.98730	7.988010	1.0 × 10−2	8.06450	8.069630	6.5 × 10−2
	2	23.9406	23.94060	8.4 × 10−4	24.2521	24.25241	2.1 × 10−3	24.3532	24.35745	1.9 × 10−2
	3	41.6789	41.67887	1.4 × 10−3	42.3024	42.30265	2.0 × 10−3	42.5010	42.50566	1.2 × 10−2
	4	53.7111	53.71108	2.0 × 10−3	54.8771	54.87722	2.2 × 10−3	55.3881	55.38793	1.7 × 10−3
C-C	1	12.9065	12.90653	1.0 × 10−3	12.6873	12.68758	3.8 × 10−3	12.6030	12.60572	2.3 × 10−2
	2	26.7597	26.75967	1.4 × 10−3	26.6416	26.64177	1.8 × 10−3	26.5809	26.58381	1.2 × 10−2
	3	43.0417	43.04166	1.5 × 10−3	43.3694	43.36938	1.3 × 10−3	43.4958	43.49723	4.7 × 10−3
	4	58.5205	58.52006	7.9 × 10−4	59.2779	59.27826	2.0 × 10−3	59.5726	59.57154	2.7 × 10−4

* $\mathsf{\Delta }={\displaystyle \frac{\left|{\omega }_{Ref}-{\omega }_{DTM}\right|}{{\omega }_{Ref}}}\times 100\%$.

.

Inspecting Table 6 reveals excellent agreement between the first four frequencies obtained in this study and those reported in the referenced research across all boundary conditions and power indices.

It should be noted that throughout this series of examples, the proposed approach demonstrates high accuracy, efficiency, and rapid convergence. Only thirty terms in the series expansions (14) are sufficient to yield accurate results for the first several natural frequencies.

4.3. Homogeneous Timoshenko Beams with Curvilinearly Tapered Cross-Sections

In order to demonstrate the applicability and effectiveness of the DTM in solving the free vibration problem for axially curvilinear cross-sectional beams, we now consider various specific cases. Whenever possible, we compare the results obtained using DTM with those already documented in the literature. In cases, where comparative data are not available, we perform the frequency analysis using ABAQUS 6.14 software.

The first problem addressed is a homogeneous beam with a double-parabolic taper, truncated at the left end. The shape of the parabolic taper beam is analytically defined by the weight $w_1=1$, and the parameter $l_e$ is equal to the truncation factor ${\delta }_0$ and the triangle ${\mathit{P}}_0{\mathit{P}}_1{\mathit{P}}_2$ specified as shown in Figure 3a.

Due to the lack of data for such Timoshenko beams in the published literature, the results are compared with those obtained using the Rayleigh–Ritz method within the Euler–Bernoulli beam theory in [2].

Table 7. Dimensionless natural frequencies of homogeneous beams with double parabolic taper.

${\mathit{\delta }}_0$	BCs	C-C			S-S			F-C
	Mode	[2]	DTM	$\mathbf{\Delta }{ }^{*}, \mathbf{\%}$	[2]	DTM	$\mathbf{\Delta }, \mathbf{\%}$	[2]	DTM	$\mathbf{\Delta }, \mathbf{\%}$
0.1	1	14.368	14.35795	0.07	6.3778	6.379480	0.03	5.3883	5.371650	0.31
	2	40.479	40.36953	0.27	27.310	27.28302	0.10	20.699	20.55328	0.70
	3	80.211	79.77238	0.55	61.023	60.84181	0.30	48.471	47.93399	1.11
0.2	1	15.972	15.9348	0.23	7.0594	7.054580	0.07	4.8654	4.862540	0.06
	2	44.578	44.3374	0.54	29.395	29.32546	0.24	20.372	20.32497	0.23
	3	87.897	87.0621	0.95	65.800	65.45681	0.52	49.870	49.61366	0.51
0.3	1	17.176	17.11698	0.34	7.5823	7.575210	0.09	4.5179	4.514900	0.07
	2	47.701	47.32308	0.79	31.082	30.97545	0.34	20.420	20.35704	0.31
	3	93.828	92.52728	1.39	69.692	69.16329	0.76	51.543	51.17858	0.71
0.4	1	18.175	18.08114	0.52	8.0198	8.008940	0.14	4.2660	4.262320	0.09
	2	50.325	49.73045	1.18	32.561	32.39500	0.51	20.592	20.50043	0.44
	3	98.855	96.82087	2.06	73.098	72.27499	1.13	53.199	52.65124	1.03
0.5	1	19.044	18.89066	0.81	8.4018	8.384520	0.21	4.0729	4.067820	0.12
	2	52.633	51.66934	1.83	33.904	33.63600	0.79	20.812	20.67111	0.68
	3	103.30	100.0365	3.16	76.179	74.85468	1.74	54.785	53.92227	1.57
0.6	1	19.852	19.55547	1.49	8.7443	8.71463	0.34	3.9188	3.9112	0.19
	2	54.718	53.06449	3.02	35.148	34.68662	1.31	21.052	20.81738	1.11
	3	107.34	101.82207	5.14	79.021	76.76144	2.86	56.296	54.84588	2.58
0.7	1	20.531	20.01665	2.51	9.0569	8.99939	0.63	3.7922	3.77915	0.34
	2	56.634	53.51562	5.51	36.315	35.43022	2.44	21.299	20.85956	2.06
	3	111.06	96.84761	12.80	81.676	77.40895	5.22	57.736	55.03705	4.67
0.8	1	21.186	19.98305	5.68	9.3459	9.207100	1.49	3.6859	3.493240	5.23
	2	58.417	43.48449	25.6	37.419	35.37001	5.48	21.547	19.53236	9.35
	3	114.54	69.00506	39.8	84.180	51.02468	39.4	59.112	63.26979	7.03

* $\mathsf{\Delta }={\displaystyle \frac{\left|{\omega }_{Ref}-{\omega }_{DTM}\right|}{{\omega }_{Ref}}}\times 100\%$.

presents the first three dimensionless frequencies $\overline{\omega }=\omega {\left(L-{\delta }_0\right)}^2\sqrt{{\displaystyle \frac{\rho A_0}{E{J}_0}}}$ for both the reference Euler–Bernoulli and present Timoshenko beams with a slenderness ratio $L/h_0=50$ under three different boundary conditions and different truncation factors.

One can observe from Table 7 that the eigenfrequencies closely match for both types of beams when the truncation factor is small. However, as the truncation factor increases, discrepancies between them become more pronounced, especially for the third mode. This difference is anticipated because shorter beams necessitate accounting for shear deformation, which the simpler Euler–Bernoulli beam model does not address.

The second benchmark example involves a beam with a parabolic-tapered height and constant width, truncated at the left end and tapered to a point at the right end (${\delta }_1=0$), as illustrated in Figure 3b. A detailed specification of this beam configuration is provided in [3]. The analytical presentation of this shape of the parabolic beam with Bézier curves is discussed in Section 2.1.

Once again, due to limited availability of data for Timoshenko beams in the existing literature, comparisons are made with results obtained using hypergeometric functions within the framework of the Euler–Bernoulli beam theory, as described in [3].

Table 8. Dimensionless natural frequencies of cantilever homogeneous beams with parabolic taper in height and constant width.

${\mathit{\delta }}_0$	0.15			0.2			0.25			0.3			0.35
Mode	[3]	DTM	$\mathbf{\Delta }{ }^{*}, \mathbf{\%}$	[3]	DTM	$\mathbf{\Delta }, \mathbf{\%}$	[3]	DTM	$\mathbf{\Delta }, \mathbf{\%}$	[3]	DTM	$\mathbf{\Delta }, \mathbf{\%}$	[3]	DTM	$\mathbf{\Delta }, \mathbf{\%}$
1	1.050	1.05057	0.05	1.469	1.46748	0.10	1.936	1.93539	0.03	2.465	2.46322	0.07	3.070	3.06690	0.10
2	7.606	7.58803	0.24	8.958	8.93699	0.23	10.42	10.3849	0.34	12.02	11.9846	0.29	13.83	13.7831	0.34
3	18.25	18.2256	0.13	20.91	20.8092	0.48	23.79	23.6579	0.56	26.96	26.7977	0.60	30.54	30.3304	0.69
4	32.54	31.9653	1.77	36.96	36.6841	0.75	41.73	41.3659	0.87	46.99	46.5406	0.96	52.92	52.3541	1.07
$\delta$	0.4			0.45			0.5			0.55			0.6
1	3.770	3.76591	0.11	4.593	4.58701	0.13	5.576	5.56729	0.16	6.772	6.76070	0.17	8.263	8.24726	0.19
2	15.91	15.8416	0.43	18.32	18.2384	0.45	21.18	21.0814	0.47	24.65	24.5248	0.51	28.96	28.7980	0.56
3	34.63	34.3716	0.75	39.40	39.0763	0.82	45.05	44.6586	0.87	51.9	51.4115	0.94	60.41	59.7941	1.02
4	59.70	59.0060	1.16	67.61	66.7478	1.28	76.98	75.9204	1.38	88.34	87.0392	1.47	102.4	100.825	1.54
$\delta$	0.65			0.7			0.75			0.8			0.85
1	10.17	10.1532	0.16	12.72	12.6888	0.25	16.27	16.2322	0.23	21.60	21.5566	0.20	30.47	30.3768	0.31
2	34.47	34.2610	0.61	41.78	41.5124	0.64	51.99	51.6294	0.69	67.26	66.6661	0.88	92.66	91.9427	0.77
3	71.29	70.5089	1.10	85.72	84.7297	1.16	105.9	104.568	1.26	136.0	134.452	1.14	186.2	183.606	1.39
4	120.5	118.443	1.71	144.4	141.823	1.78	177.8	174.436	1.89	227.8	222.612	2.28	310.9	304.354	2.11

* $\mathsf{\Delta }={\displaystyle \frac{\left|{\omega }_{Ref}-{\omega }_{DTM}\right|}{{\omega }_{Ref}}}\times 100\%$.

displays the first four dimensionless frequencies $\overline{\omega }=\omega {\left(L/2\right)}^2\sqrt{{\displaystyle \frac{\rho A_0}{E{J}_0}}}$ of cantilever Euler–Bernoulli and Timoshenko beams with a slenderness ratio $L/h_0=50$ for various truncation factors.

From Table 8, one can see that the compared natural frequencies exhibit close agreement. Similar to the previous example, as the beam length decreases, discrepancies between the present and benchmarked results increase, especially for higher modes, due to the greater influence of shear deformation, as discussed earlier.

Therefore, these examples thoroughly confirm the reliability of the proposed computational method for analyzing curvilinear tapered beams.

4.4. AFGM Timoshenko Beams with Curvilinearly Tapered Cross-Sections

This subsection presents the performance of the DTM for frequency analysis of inhomogeneous curvilinearly tapered beams of convex-up and concave-down shapes, referred to as plump and slender models in [29] and illustrated in Figure 1a,b, respectively. The material properties of the beams vary axially according to the laws specified in (4) for Type 2 and (5) for Type 3 distributions. Detailed geometric and material properties of the beams can be found in [29] and are summarized in

Table 9. The parameters of curvilinearly non-uniform AFGM beams.

Material	$\mathit{E}$, GPa	$\mathit{\rho }$, kg/m3	$\mathit{\nu }$	L, m	${\mathit{b}}_0$, m	${\mathit{h}}_0$, m	${\mathit{l}}_{\mathit{e}}$, m	${\mathit{l}}_{\mathit{f}}$, m
Aluminum	70	2700	0.3	1	0.2	0.2	0.04	0.1
Steel	348.43	2370	0.24
Si3N4	201.04	8166	0.33

.

First, we analyze beams with varying thickness and constant width for both Type 2 and Type 3 material laws. The exponential law (4) with parameter $a=1$ and gradient indices ${\gamma }_e=0, 1, 5$ is applied to beams made of aluminum, whereas the power law (5) with parameter $b=1$ and gradient indices ${\gamma }_c=0, 1, 5$ is utilized for beams composed of a combination of steel (SUS304) and ceramic (Si₃N₄) phases, as noted in [29].

Table 10. Natural frequencies, Hz, of the C-C AFGM beams with curvilinear taper in height and constant width.

AFGM	Model	${\mathit{\gamma }}_{\mathit{e}}/{\mathit{\gamma }}_{\mathit{c}}$	0			1			5
		Mode	[29]	DTM	$\mathbf{\Delta }{ }^{*}, \mathbf{\%}$	[29]	DTM	$\mathbf{\Delta }, \mathbf{\%}$	[29]	DTM	$\mathbf{\Delta }, \mathbf{\%}$
Type 2	Plump	1	889.39	887.304	0.23	891.48	889.589	0.21	997.39	1002.94	0.56
		2	1889.1	2034.36	7.69	1894.4	2041.25	7.75	2025.3	2203.40	8.79
	Slender	1	800.93	779.141	2.72	805.63	783.893	2.70	920.25	899.469	2.26
		2	1774.6	1728.28	2.61	1778.8	1732.06	2.63	1882.4	1838.13	2.35
Type 3	Plump	1	2128.3	2133.71	0.25	1229.4	1226.23	0.26	947.74	947.074	0.07
		2	4532.2	4911.19	8.36	2632.4	2837.54	7.79	2036.2	2199.87	8.04
	Slender	1	1906.2	1865.29	2.15	1108.2	1078.24	2.70	851.24	828.860	2.63
		2	4239.8	4151.87	2.07	2471.8	2406.75	2.63	1902.5	1856.66	2.41

* $\mathsf{\Delta }={\displaystyle \frac{\left|{\omega }_{Ref}-{\omega }_{DTM}\right|}{{\omega }_{Ref}}}\times 100\%$.

presents the first two frequencies associated with the bending modes of the curvilinear taper beams with C-C boundary conditions. These frequencies are compared with those computed using NURBS in the 3-D finite element analysis conducted in [29].

The current results align satisfactory with the reference data, demonstrating differences of less than 0.6% for the fundamental frequency and less than 9% for the second frequency in the plump beams. For slender beams, the discrepancies are approximately 3% for both frequencies. These deviations arise from differences in modeling approaches. The current model uses a line beam structure with a first-order shear theory, whereas the IGA analysis in [29] employs a more detailed 3-D beam modeling approach.

Next, we analyze the parabolic convex beams depicted in Figure 3b. These beams feature a parabolic taper in height and maintain a constant width. They are truncated at the left end with a fixed factor ${\delta }_0=0.5$, while the right end is characterized by a variable truncation factor ${\delta }_1$. The beam ends are subjected to C-C and C-F constraints. The material constants, given in Table 1, vary along the beam length according to the Type 1 material law with a linear distribution for Young’s modulus ($n=1$) and a parabolic distribution for mass density ($m=2$) as described in (3). Detailed specifications of these beam configurations are provided in [23].

While benchmark results for the fundamental frequencies of these beams are available in [23], higher-order frequencies are absent in the existing literature. Therefore, the results obtained using the DTM are validated against finite element analysis for these beams, as discussed in Section 3.2. The comparison of first five dimensionless bending frequencies $\overline{\omega }=\omega L^2\sqrt{{\displaystyle \frac{{\rho }_0{A}_0}{E_0{J}_0}}}$, depending on the truncation factor ${\delta }_1$ for the two types of boundary conditions is shown in

Table 11. Dimensionless natural frequencies of truncated AFGM beams with parabolic taper in height and constant width.

BCs	${\mathit{\delta }}_1$	0.2				0.4				0.6
	Mode	[23]	FEM	DTM	$\mathbf{\Delta }{ }^{*}, \mathbf{\%}$	[23]	FEM	DTM	${\mathbf{\Delta }}^{ *}, \mathbf{\%}$	[23]	FEM	DTM	$\mathbf{\Delta }{ }^{*}, \mathbf{\%}$
C-C	1	18.4750	18.8461	18.5033	0.15	16.423	16.7328	16.4583	0.21	14.1591	14.7828	14.2013	0.30
	2	-	53.7872	51.5889	4.09	-	47.1474	46.5676	1.23	-	42.3017	40.8931	3.33
	3	-	105.999	101.332	4.40	-	93.1729	92.1155	1.13	-	84.2618	81.5879	3.17
	4	-	175.772	167.090	4.94	-	154.114	152.461	1.07	-	139.905	135.640	3.05
	5	-	261.850	248.400	5.14	-	229.652	227.231	1.05	-	208.997	202.736	3.00
C-F	1	2.5715	2.71073	2.57115	0.01	2.7594	2.94885	2.75933	0.00	3.0151	3.17381	3.0146	0.02
	2	-	18.8484	18.0533	4.22	-	18.3961	17.5276	4.72	-	18.0266	17.0113	5.63
	3	-	54.3519	51.7463	4.79	-	49.5819	48.3800	2.42	-	46.8092	44.6929	4.52
	4	-	107.161	101.628	5.16	-	95.5807	94.1178	1.53	-	88.8564	85.6122	3.65
	5	-	177.343	167.573	5.51	-	156.448	154.637	1.16	-	142.196	139.622	1.81

* $\mathsf{\Delta }={\displaystyle \frac{\left|{\omega }_{Ref}-{\omega }_{DTM}\right|}{{\omega }_{Ref}}}\times 100\%$.

.

It is observed from Table 11 that the differences between dimensionless fundamental frequencies obtained from the present analysis, published results, and FEM predictions are quite small. However, the higher frequencies show more significant deviations between the DTM and FEM solutions. Likely, these discrepancies between the two sets of data are attributed to methodological differences in formulation and solution approach. Overall, these findings confirm the reliability of the DTM approach within acceptable limits.

As mode shapes play a crucial role in evaluating the accuracy of structural dynamics modeling, the first five bending mode shapes associated with the appropriate frequencies of AFGM beams with ${\delta }_1=0.2$ for C-C and C-F boundary conditions, as listed in Table 11 and obtained using the DTM, are plotted in Figure 4a,b, respectively.

Analogously, Figure 5a,b illustrates the bending mode shapes of the same truncated AFGM beams, constructed based on the FEM results, which are given in Table 11.

It is worth noticing that the geometric non-uniformity and the variation of material parameters along the beam length significantly affect the mode shapes. The figures clearly show that the positions of the nodal points shift along the beam length, and this shift is more pronounced as the truncation factor ${\delta }_1$ becomes smaller.

4.5. AFGM Timoshenko Beams with Curvilinearly Double-Tapered Cross-Sections

To the best of authors’ knowledge, benchmark results for curvilinearly double-tapered AFGM beams are not available in the existing literature. Hence, the results of the present analysis are validated against axial plumb and slender beams with Type 2 and Type 3 material distributions, as considered in [29], which feature variable thickness and constant width. Further, the analysis will be extended to include beams with double-tapered cross-sectional variations. The geometrical and material parameters are identical to those used in the previous examples of curvilinear taper beams with exponential and power law gradations as presented in [29].

The first two dimensionless frequencies $\overline{\omega }=\omega L\sqrt{{\displaystyle \frac{{\rho }_{Al}}{E_{Al}}}}$ of axially graded Type 2 plump and slender C-C beams with variable thickness and constant width, as well beams with simultaneously variable thickness and width, depending on the exponential law parameter $a$ and gradient index ${\gamma }_e$ are provided in

Table 12. Dimensionless natural frequencies of curvilinear taper C-C AFGM beams with various exponential law (4) parameters.

Model	a	0.2			0.5			0.8			1.0
	${\mathit{\gamma }}_{\mathit{e}}$	[29]	DTM	DTM *	[29]	DTM	DTM *	[29]	DTM	DTM *	[29]	DTM	DTM *
Plump	0	1.0975	1.09493	1.03005	1.0975	1.09493	1.03005	1.0975	1.09493	1.03005	1.0975	1.09493	1.03005
		2.3311	2.51039	2.41452	2.3311	2.51039	2.41452	2.3311	2.51039	2.41452	2.3311	2.51039	2.41452
	0.5	1.0975	1.09495	1.03006	1.0977	1.09509	1.03012	1.0979	1.09669	1.03025	1.0981	1.09560	1.03037
		2.3312	2.51047	2.41461	2.3316	2.51092	2.41510	2.3322	2.51583	2.41601	2.3328	2.51252	2.41685
	1	1.0976	1.09503	1.03010	1.0981	1.09560	1.03037	1.0991	1.09923	1.03093	1.1001	1.09774	1.03149
		2.3314	2.51073	2.41489	2.3328	2.51252	2.41685	2.3353	2.52559	2.42049	2.3377	2.51889	2.42384
	2	1.0979	1.09536	1.03025	1.1001	1.09774	1.03149	1.1047	1.10274	1.03446	1.1095	1.10799	1.03796
		2.3322	2.51175	2.41601	2.3377	2.51889	2.42384	2.348	2.53212	2.43829	2.3575	2.54429	2.45154
	5	1.1001	1.09774	1.03149	1.1181	1.11733	1.04478	1.1689	1.17175	1.09083	1.2308	1.23763	1.15183
		2.3377	2.51889	2.42384	2.3725	2.56324	2.47209	2.4379	2.6446	2.55945	2.4992	2.71898	2.63835
Slender	0	0.9883	0.96145	1.06330	0.9883	0.96145	1.06330	0.9883	0.96145	1.06330	0.9883	0.96145	1.06330
		2.1899	2.13269	2.27063	2.1899	2.13269	2.27063	2.1899	2.13269	2.27063	2.1899	2.13269	2.27063
	0.5	0.9884	0.96151	1.06337	0.9887	0.96292	1.06368	0.9893	0.96239	1.06427	0.9898	0.96292	1.06482
		2.1899	2.13273	2.27063	2.1902	2.13385	2.27066	2.1907	2.13343	2.27071	2.1912	2.13385	2.27076
	1	0.9886	0.96169	1.06355	0.9898	0.96292	1.06482	0.9921	0.96521	1.06718	0.9941	0.96732	1.06936
		2.1901	2.13287	2.27065	2.1912	2.13385	2.27076	2.1932	2.13567	2.27101	2.1951	2.13735	2.27128
	2	0.9893	0.96239	1.06427	0.9941	0.96732	1.06936	1.0032	0.97648	1.07883	1.0116	0.98496	1.08759
		2.1907	2.13343	2.27071	2.1951	2.13735	2.27128	2.2032	2.14468	2.27284	2.2108	2.15152	2.27483
	5	0.9941	0.96732	1.06936	1.0248	0.99839	1.10130	1.0821	1.05618	1.16083	1.1356	1.10994	1.21554
		2.1951	2.13735	2.27128	2.2226	2.16369	2.27892	2.2800	2.21822	2.30846	2.3998	2.26823	2.35005

* Curvilinear double-tapered beam.

. Satisfactory agreement between the reference and present results for plump and slender AFGM beams with variable thickness is observed. The errors between the frequencies fall within the range observed for frequencies of similar beams with $a=1$ and ${\gamma }_e=1$ in Table 10. In particular, minor deviations are noted for the first frequency, while higher discrepancies are observed for the second frequency.

Upon reviewing Table 12, it becomes evident that introducing variations in width influences the frequencies of the beams. Specifically, plump double-tapered beams show slightly lower frequencies compared to those with only variable thickness, whereas slender double-tapered beams exhibit a slight increase. However, the trends in frequency changes with increasing gradient index ${\gamma }_e$ and parameter $a$ remain consistent for both beams with only variable thickness and double-tapered beams. The frequencies increase with higher ${\gamma }_e$ and $a$. This increase is more pronounced as ${\gamma }_e$ or $a$ become larger.

Analogously, axially graded Type 3 plump and slender C-C beams with variable thickness and constant width and with simultaneously variable thickness and width are examined.

Table 13. Dimensionless natural frequencies of curvilinear taper C-C AFGM beams with various power law (5) parameters.

Model	b	0.2			0.5			0.8			1.0
	${\mathit{\gamma }}_{\mathit{c}}$	[29]	DTM	DTM *	[29]	DTM	DTM *	[29]	DTM	DTM *	[29]	DTM	DTM *
Plump	0	1.1029	1.10568	1.04031	1.1029	1.10568	1.04031	1.1029	1.10569	1.04031	1.1029	1.10569	1.04031
		2.3486	2.54497	2.44810	2.3486	2.54497	2.44810	2.3486	2.54498	2.44813	2.3486	2.54497	2.44813
	0.5	1.0270	1.02926	0.96840	0.9236	0.92519	0.87025	0.8254	0.82580	0.77672	0.7529	0.75141	0.70703
		2.1870	2.36872	2.27858	1.9669	2.12879	2.04736	1.7563	1.89754	1.82512	1.5969	1.72072	1.65589
	1	0.9653	0.96706	0.90989	0.8139	0.81428	0.76616	0.7002	0.69935	0.65802	0.6371	0.63543	0.59782
		2.0571	2.22708	2.14243	1.7383	1.87906	1.80811	1.4981	1.61675	1.55635	1.3641	1.47041	1.41596
	2	0.8708	0.87183	0.82031	0.6878	0.68729	0.64668	0.5897	0.58870	0.55389	0.5512	0.55024	0.51771
		1.8597	2.01193	1.93575	1.4761	1.59425	1.53483	1.2703	1.37125	1.32095	1.1901	1.28514	1.23845
	5	0.7083	0.70814	0.66630	0.5512	0.55045	0.51792	0.5044	0.50397	0.47425	0.4911	0.49077	0.46188
		1.5204	1.64301	1.58161	1.1883	1.28345	1.23660	1.0864	1.17376	1.13122	1.0551	1.13997	1.09864
Slender	0	0.9878	0.96659	1.06906	0.9878	0.96659	1.06906	0.9878	0.96659	1.06906	0.9878	0.96659	1.06906
		2.1970	2.15149	2.29122	2.1970	2.15149	2.29122	2.1970	2.15149	2.29122	2.1970	2.15149	2.29122
	0.5	0.9203	0.90015	0.99556	0.8285	0.81359	0.89622	0.7420	0.72427	0.80065	0.6790	0.66130	0.73109
		2.0466	2.00327	2.13265	1.8421	1.74851	1.92080	1.6476	1.60620	1.70524	1.5028	1.46410	1.54951
	1	0.8654	0.84606	0.93568	0.7309	0.71349	0.78886	0.6302	0.61411	0.67858	0.5743	0.55882	0.61731
		1.9255	1.88386	2.00444	1.6286	1.59073	1.68759	1.4054	1.37004	1.44874	1.2809	1.24838	1.31629
	2	0.7812	0.76321	0.84391	0.6183	0.60271	0.66606	0.5307	0.51674	0.57062	0.4959	0.48271	0.53300
		1.7411	1.70209	1.80889	1.3828	1.34864	1.42756	1.1895	1.15997	1.22235	1.1133	1.08328	1.14099
	5	0.6363	0.62055	0.68581	0.4955	0.48246	0.53250	0.4532	0.44121	0.48651	0.4411	0.42951	0.47381
		1.4235	1.38932	1.47301	1.1113	1.08346	1.15111	1.015	0.99219	1.06980	0.9859	0.96212	1.03009

* Curvilinear double-tapered beam.

presents the first two dimensionless frequencies $\overline{\omega }=\omega L\sqrt{{\displaystyle \frac{{\rho }_{sus}}{E_{sus}}}}$ of these beams for different values of power law parameter $b$ and gradient index ${\gamma }_c$.

Table 13 shows that the agreement between the results obtained from the IGA analysis in [29] and the present study is satisfactory, with errors comparable to those observed for Type 3 plump and slender C-C beams with $b=1$ and ${\gamma }_c=1$ in Table 11. Additionally, Table 13 demonstrates that introducing variable width along with variable height has similar effects on the frequencies of both beams with variable thickness only and curvilinearly double-tapered beams, similar to what was observed for Type 2 plump and slender C-C beams, as mentioned earlier.

On the other hand, the trends of the frequency changes with increasing $b$ and ${\gamma }_c$ are found to be opposite to those in Type 2 plump and slender C-C beams. In Type 3 plump and slender beams, the frequencies decrease with higher values of ${\gamma }_c$ and $b$. This decrease begins more rapidly and exhibits a faster decay as ${\gamma }_c$ or $b$ increase.

Table 14. Dimensionless natural frequencies of curvilinear taper AFGM beams with various boundary conditions.

AFGM	Model	BCs	C-C			C-S			C-F			S-S
		Mode	[29]	DTM	DTM *	[29]	DTM	DTM *	[29]	DTM	DTM *	[29]	DTM	DTM *
Type 2	Plump	1	1.1001	1.09774	1.03149	0.8392	0.83336	0.80370	0.1669	0.16736	0.16262	0.6856	0.67765	0.66811
		2	2.3377	2.51889	2.42384	1.5533	2.31624	2.25450	0.9638	1.13797	1.12997	1.8524	2.16694	2.13405
	Slender	1	0.9941	0.98513	1.06936	0.5764	0.56832	0.58619	0.1070	0.10628	0.09795	0.3468	0.34744	0.33893
		2	2.1951	2.15107	2.27128	1.7610	1.74672	1.89129	0.7314	0.7207	0.72082	1.3943	1.4032	1.43892
Type 3	Plump	1	0.6371	0.63543	0.59782	0.5215	0.51757	0.49565	0.1460	0.14617	0.13912	0.4070	0.40223	0.39643
		2	0.8972	1.47041	1.41596	0.9537	1.36093	1.32281	0.6423	0.74588	0.73694	1.0946	1.27727	1.25659
	Slender	1	0.5743	0.55882	0.61731	0.3808	0.37381	0.39302	0.1008	0.09998	0.09508	0.2065	0.20703	0.2028
		2	1.2809	1.24838	1.31629	1.0547	1.0332	1.06555	0.4992	0.48937	0.49201	0.8293	0.82974	0.82545

* Curvilinear double-tapered beam.

presents the dimensionless natural frequencies of beams with variable thickness only and curvilinearly double-tapered beams under different boundary conditions. The calculation errors for the first and second frequencies are within 3%. Notably, the second bending frequency is not available from the IGA analysis in [29] in some cases. Analyzing the data in Table 14, it is evident that the frequencies of curvilinearly double-tapered beams behave similarly to those of beams with variable thickness only, depending on the boundary conditions. In particular, the highest frequencies correspond to C-C restraints, while the lowest frequencies are observed under C-F boundary conditions.

5. Conclusions

Free vibration analysis of axially functionally graded beams with curvilinear tapered cross-sections and various material distributions is performed using the semi-analytical differential transform method. Unlike the standard DTM procedure, which typically imposes boundary conditions on transformed images of unknown functions, the proposed DTM approach combines various boundary conditions subjected directly to the original continuous problem formulation. The non-uniform beams shaped by conic sections, are analytically represented using rational Bézier curves within the Gröbner basis.

The equations of motion for free vibration are derived using the principle of virtual work under the assumptions of Timoshenko beam theory. It is assumed that both the material properties of the beam and its geometric features vary continuously along the beam’s axial direction.

Comparisons are presented to validate the accuracy and assess the efficiency of the proposed computational technique. The computed examples encompass both documented cases and previously unexamined scenarios, ranging from uniform homogeneous and inhomogeneous beams to curvilinearly double-tapered axially functionally graded beams. In cases where benchmark results are unavailable in the literature, the DTM solutions are compared with finite element simulations using ABAQUS, incorporating material gradients into the 3-D C3D20 elements via a user-defined subroutine UMAT.

Based on these results, it can be concluded that the proposed DTM formulation excels in modeling the free vibrations of beams with complex geometries and material distributions. The frequency analysis of Timoshenko beams, whether homogeneous or inhomogeneous with exponential and power law material distributions and with uniform cross-sections, reveals that the method achieves high accuracy, efficiency, and rapid convergence. Accurate results for the first few natural frequencies are achieved with only thirty terms in the series solution.

For homogeneous truncated beams with a parabolic taper in height or double-parabolic taper, where the published literature lacks frequency data within Timoshenko beam theory, comparisons with Euler–Bernoulli beams indicate close agreement when the truncation factor is small. However, as the truncation factor increases, discrepancies become more pronounced, especially for higher modes. Consequently, the present results offer valuable reference data for researchers using Timoshenko beam theory.

Benchmarking AFGM beams with curvilinear taper in height for both plump and slender geometries and for exponential and power material distributions shows satisfactory matching between the present and reference results. For AFGM beams with a parabolic taper in height that are truncated at both ends with a power material law, comparisons of fundamental frequencies between the DTM and reference solutions show excellent agreement. However, essential discrepancies occur between the DTM and FEM predictions for higher frequencies.

Additionally, this study provides new results from frequency analysis of curvilinearly double-tapered AFGM beams. It was found that simultaneous variations in height and width of the beam cross-section influence the beam’s frequencies. While these changes may not be substantial, conducting dynamic structural modeling remains crucial. Therefore, these findings can be useful in the design process of axially functionally graded beams and serve as benchmarks for researchers interested in exploring similar problems using alternative methods.