  Citation: Alshabatat, N.T. Natural Frequencies Optimization of Thin-Walled Circular Cylindrical Shells Using Axially Functionally Graded Materials. Materials 2022, 15, 698. https://doi.org/10.3390/ ma15030698 Academic Editor: Martin Vašina Received: 29 December 2021 Accepted: 15 January 2022 Published: 18 January 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). materials Article Natural Frequencies Optimization of Thin-Walled Circular Cylindrical Shells Using Axially Functionally Graded Materials Nabeel Taiseer Alshabatat Mechanical Engineering Department, Tafila Technical University, Tafila 66110, Jordan; nabeel@ttu.edu.jo Abstract: One method to avoid vibration resonance is shifting natural frequencies far away from excitation frequencies. This study investigates optimizing the natural frequencies of circular cylin- drical shells using axially functionally graded materials. The constituents of functionally graded materials (FGMs) vary continuously in the longitudinal direction based on a trigonometric law or using interpolation of volume fractions at control points. The spatial change of material properties alters structural stiffness and mass, which then affects the structure’s natural frequencies. The local material properties at any place in the structure are obtained using Voigt model. First-order shear deformation theory and finite element method are used for estimating natural frequencies, and a genetic algorithm is used for optimizing material volume fractions. To demonstrate the proposed method, two optimization problems are presented. The goal of the first one is to maximize the fundamental frequency of an FGM cylindrical shell by optimizing the material volume fractions. In the second problem, we attempt to find the optimal material distribution that maximizes the distance between two adjoining natural frequencies. The optimization examples show that building cylindrical shells using axially FGM is a useful technique for optimizing their natural frequencies. Keywords: cylindrical shell; functionally graded material; natural frequency; optimization 1. Introduction Cylindrical shells are extensively employed in various applications such as aerospace, automobile, marine, and construction industries. They are usually exposed to dynamic excitations, which cause vibrations and make noise. Reducing vibrations and noise radi- ation from these structures are important issues in the early stages of structural design process. A common technique to decrease vibrations is shifting the natural frequencies of a structure away from the excitation frequencies to prevent resonance. Different passive methods are employed for optimizing the natural frequencies of cylindrical shells, and these include adjusting the thickness, adding stiffeners, and tailoring material. For example, Alzahabi [1] shifted the natural frequencies of a submarine hull by controlling the local thickness. The length of the hull was divided into equal segments, and then the thickness of each segment was optimized to achieve the sought natural frequency. Nasrekani et al. [2] studied the effect of changing the thickness, continuously along a cylindrical shell, on its first axisymmetric natural frequency. Bagheri et al. [3] investigated a multi-objective problem for simultaneously maximizing the fundamental frequency and minimizing the weight of cylindrical shell using ring stiffeners. Mehrabami et al. [4] maximized the funda- mental frequency-to-weight ratio of cylindrical shell using rings and strings. Akl et al. [5] minimized the vibration and/or the sound radiation from under water cylindrical shells using stiffeners. In addition to optimizing the natural frequencies, the literature about the optimization of circular cylindrical shells with other objective functions is very wide. For example, Biglar et al. [6] optimized the locations and orientations of piezoelectric patches on cylindrical shells to reduce the vibration. Sadeghifar et al. [7] studied maximizing the critical buckling load and minimizing the weight of orthogonally stiffened cylindrical shells. Belradi et al. [8] utilized anisogrid composite lattice structures in building circular cylindri- cal shells to optimize the critical buckling load with strength and stiffness constraints. Materials 2022, 15, 698. https://doi.org/10.3390/ma15030698 https://www.mdpi.com/journal/materials