# 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 [PERSON]'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. free vibrations; axially functionally graded material; curvilinear tapered beam; differential transform method; ABAQUS +
Footnote †: journal: Medical  
frequency of diverse AFG beams with different end supports [7]. [PERSON] et al. (2016) [8] applied the confluent hypergeometric function method to obtain exact solutions for Timoshenko beams with varying bending stiffness. [PERSON] 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. [PERSON] 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 [PERSON] (2021) [11] extended this technique to non-uniform AFG Timoshenko beams. Additionally, [PERSON] et al. (2019) [12] utilized a Jacobi polynomial-based approximation, and [PERSON] 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, [PERSON] et al. (2018) [14] employed the differential transform method to study the free vibration characteristics of AFGM Euler-Bernoulli beams, while [PERSON] and [PERSON] (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. [PERSON] 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 [PERSON] et al. (2022) [17] investigated the vibration of rotating non-uniform AFG Euler-Bernoulli beams using the Laplace ADM. In turn, [PERSON] 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. [PERSON] (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 Myklesstad method. [PERSON] et al. (2021) [20] employed the variational iteration method to determine the modal characteristics of tapered AFGM Euler-Bernoulli and Timoshenko beams. [PERSON] et al. (2022) [21] proposed a closed-form dynamic stiffness formulation for free vibration analysis of tapered and/or FGM Euler-Bernoulli beams.