# Free Vibration Analysis of Curvilinearly Tapered Axially Functionally Graded Material Beams
###### Abstract
[PERSON] and [PERSON] (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, [PERSON] 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 Karman geometric nonlinearity, while [PERSON] (2018) [24] utilized the Galerkin method and the third-order shear deformation beam theory. [PERSON] and [PERSON] (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. [PERSON] and [PERSON] (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. [PERSON] (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. [PERSON] (2024) [28] proposed a Fourier p-element model for accurate predictions of Timoshenko-Ehrenfest beam frequencies with high precision compared to other existing models. [PERSON] 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. [PERSON] 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. [PERSON] 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]. [PERSON] 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, [PERSON] and [PERSON] (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. [PERSON] 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. [PERSON] 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, [PERSON] et al. (2017) [39] applied this idea to a specific class of non-uniform FGM beams.