Vibration Analysis of a Tetra-Layered FGM Cylindrical Shell Using Ring Support

Abstract

In the present study, the vibration characteristics of a cylindrical shell (CS) made up of four layers are investigated. The ring is placed in the axial direction of a four-layered functionally graded material (FGM) cylindrical shell. The layers are made of functionally graded material (FGM). The materials used are stainless steel, aluminum, zirconia, and nickel. The frequency equations are derived by employing Sander’s shell theory and the Rayleigh–Ritz (RR) mathematical technique. Vibration characteristics of functionally graded materials have been investigated using polynomial volume fraction law for all FGM layers. The characteristic beam functions have been used to determine the axial model dependency. The natural frequencies are obtained with simply supported boundary conditions by using MATLAB software. Several analogical assessments of shell frequencies have also been conducted to confirm the accuracy and dependability of the current technique.

1. Introduction

Research on the vibration of FGM cylindrical shells is extensive in both theoretical and applied mechanics. Cylindrical shells are used in a variety of technical fields, including pipelines and conduits, automotive bodywork, space shuttles, airplane fuselages, submarines, pressure vessels, and building construction. The materials used to make the shells are diverse. Shells come in a variety of forms, but CS is the most significant. Functionally graded material is a combination of more than one material.

Functionally graded materials (FGMs) possess varied compositions and structures that can be optimized for real-life conditions, making them valuable for many industrial applications, including aerospace. Initially developed in 1983 for the Japanese space shuttle project, FGMs were designed to reduce thermal stress at the metal-ceramic interface caused by high temperatures. Their enhanced durability, heat resistance, and corrosion resistance provide a reliable foundation for advancing the future of the nuclear industry.

Numerous studies [1,2,3,4,5] have been conducted on the vibrational performance of functionally graded (FG) cylindrical shells (CSs) having different boundary conditions. Naeem and Sharma [6] examined the properties of thin circular CS during free vibrations. Ritz polynomials modeled the axial mode dependency. The impact of several boundary conditions, C-C, SS-SS, and CF, as well as the impact of changing shell frequencies, were investigated. Pradhan and Loy [7] concluded that NFs of the shells were demonstrated to rely on boundary conditions and constituent volume fractions and also discovered that the FGM shell’s frequency characteristics resemble those of isotropic cylindrical shells. Xiang and Ma [8] provided novel exact solutions, formed on the Goldenveizer–Novozhilov shell theory (theory of thin shells; the theory of thin elastic shells), for the vibration of thin circular cylindrical shells with intermediate ring supports. The suggested method generates the homogeneous differential equation system for a shell segment using the state-space technique and then develops a domain decomposition methodology to handle the continuity requirements between shell segments. Precise frequency characteristics are displayed in tables and design charts for circular cylindrical shells with different combinations of end support conditions and several intermediate ring supports. Researchers can use these precise vibration frequencies as critical benchmarks to check the accuracy of their numerical solutions for circular cylindrical shell problems.

Chen et al. [9] constructed the state equation in a unified matrix form from a three-dimensional fundamental equation of anisotropic elasticity and studied the free vibration of fluid-filled simply-supported cylindrical orthotropic FGM shells with arbitrary thickness. Zhang et al. [10] examined the shell-free vibrations utilizing a differential quadrature-type approach for several edge circumstances. Arshad et al. [11] used Love’s first-order shell theory to analyze the vibration frequency of an FG shell in relation to various fraction laws. Iqbal et al. [12] looked into accurate programs by the Goldenviezer–Novozhilov shell principle for the circular CS’s motions with intermediate ring support and enquired about the vibration of CSs and NFs under various boundary conditions. The ring support cylinder’s vibrational behavior was examined using the analytical method.

Naeem et al. [13] utilized a generalized DQM to examine the vibration behavior of the cylindrical shells made of functionally graded material. Rahimi et al. [14] looked at the behavior of the vibrational characteristics of the CS with the ring supports positioned in the center of the shell. They considered FGM CS made of ceramic and metal combination and used the energy functional to obtain the equation of motion. Arshad and Naeem [15] investigated FG three-layered CS for free vibration. Their research focused on the impact of ring supports positioned at various edge circumstances. The analysis was predicated on Love’s idea of a thin shell.

Vibration frequency analysis for a three-layered CS, with a FGM middle layer was investigated by Ghamkhar et al. [16]. They looked at the impact on shell vibrations for various central layer thicknesses. The Ritz mathematical technique and Sanders’ shell theory served as the foundation for the analysis. The volume fraction law using trigonometric functions was used to manage the distribution of FGM.

The frequency of vibration of a tri-layered FGM cylindrical shell consisting of an interior layer made of homogeneous material and a central layer made of FGM was investigated by Ghamkhar et al. [17]. The connections between stress and strain were derived from Sanders’ shell theory. The result for shell frequency was obtained by using the RRM. They also assessed the impact of NFs for varying central-layer thicknesses. The axial modal dependency was estimated using the features’ beam functions. Through this research, the cylindrical shell’s basic natural frequency was reported and explored against thickness-to-radius percentages across a broad range.

Vibration analysis of a CS with three layers, with the central layer made of FGM using three different volume fraction laws, was studied by Ghamkhar et al. [18] and also determined that the material distribution governed by the VFL has little effect (<1%). The Natural Frequency becomes maximal with the increase in thickness-to-radius ratios. Ghamkhar et al. [19] investigated four-layered CS composed of FGM and isotropic materials using simple and clamped boundary conditions and also concluded that varying ring support position shows an increase in the natural frequencies along the longitudinal path.

The present study examines the vibrational analysis of tetra-layered FGM CS across the axial direction, supported by ring support. The shell layers comprise stainless steel, aluminum, nickel, and zirconia. The novelty of this study is that all FGM layers have been investigated using polynomial VFL. Based on Sander’s shell theory, the Rayleigh–Ritz method is used to obtain the shell frequency equation. It derives the governing equations in an appropriate form using the strain-displacement curvature relations. To find NFs and mode shapes, as an eigenvalue problem, the resultant system of algebraic equations is resolved. The flow chart of this work is given as Figure 1.

Figure 1. Flow Chart showing research methodology.

2. Materials and Methods

Consider a CS whose radius is $R$, length is $L$, and thickness is H. The central surface is arranged to a symmetrical coordinate plot $(x,\psi ,z)$ of a cylindrical shell where $x,\psi$, and $z$ lye in the axial, circumferential, and radial directions of the shell and the shell displacements in the x, $\psi$, and $z$ positions are $u_1,u_2$, and $u_3$ accordingly, where the deformation of the displacement function is indicated as $u_1(x,\psi ,z),$ in the longitudinal, $u_2(x,\psi ,z),$ in tangential, and $u_3(x,\psi ,z),$ in transverse directions correspondingly (Figure 2).

Figure 2. CS structure of a four-layered FGM with ring support.

The relation of strain energy is defined by

$$\wp ={\displaystyle \frac{1}{2}}{\displaystyle \underset{0}{\overset{L}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\left\{J\right\}}^T[P]\left\{J\right\}}Rd\psi dx},$$

(1)

where

$${\left\{J\right\}}^T=\left\{\begin{array}{cccccc}{\xi }_1& {\xi }_2& {\xi }_{12}& {\kappa }_1& {\kappa }_2& 2\tau \end{array}\right\}$$

(2)

using different shell theories [5], the stiffness matrix [P] is defined as

$$\left[P\right]=\left[\begin{array}{cccccc}{H}_{11}& H_{12}& 0& I_{11}& I_{12}& 0\\ H_{12}& H_{22}& 0& I_{12}& I_{22}& 0\\ 0& 0& H_{66}& 0& 0& I_{66}\\ I_{11}& I_{12}& 0& J_{11}& J_{12}& 0\\ I_{12}& I_{22}& 0& J_{12}& J_{22}& 0\\ 0& 0& I_{66}& 0& 0& J_{66}\end{array}\right]$$

(3)

where ${\xi }_1,{\xi }_2,{\xi }_{12}$ are the strains containing the reference surface, ${\kappa }_1,{\kappa }_2,2\tau$ represent the curvatures, and $H_{ij},I_{ij}$, and $J_{ij}$ (i, j = 1, 2 and 6) expresses the stiffness in term of extension, coupling, and bending, where

$$\left\{H_{ij},I_{ij},J_{ij}\right\}={\displaystyle \underset{-H/2}{\overset{H/2}{\int }}{Q}_{ij}\left\{1,z,z^2\right\}dz.}$$

(4)

[5] Reduced stiffness, $Q_{ij}$, for FGM materials is defined as

$$\begin{gathered} Q_{11}={\displaystyle \frac{E}{(1-{\nu }^2)}}=Q_{22} \\ {Q}_{12}={\displaystyle \frac{\nu E}{1-{\nu }^2}} \end{gathered}$$

and

$$Q_{66}={\displaystyle \frac{E}{2\left(1+\nu \right)}}$$

(5)

where E is young’s modulus and v is Poisson’s ratio. The matrix $I_{ij}\ne 0$ is for FGM CS, taking into account its composition and the constituent characteristics, using (2) and (3) in (1), ${\wp }_{strain}$, and the strain energy is written as

$$\begin{array}{c}{\wp }_{strain}={\displaystyle \frac{R}{2}}{\displaystyle \underset{0}{\overset{L}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\left(H_{11}{{\xi }_1}^2+2H_{12}{\xi }_1{\xi }_2+H_{22}{{\xi }_2}^2+H_{66}{{\xi }_{12}}^2+2I_{11}{\xi }_1{\kappa }_1+2I_{21}{\xi }_1{\kappa }_2+2I_{12}{\xi }_2{\kappa }_1\right.}}+2I_{22}{\kappa }_2{\xi }_2\\ \left.+4I_{66}{\xi }_{12}\tau +J_{11}{{\kappa }_1}^2+2J_{12}{\kappa }_1{\kappa }_2+J_{22}{{\kappa }_2}^2+4J_{66}{\tau }^2\right)Rd\psi dx,\end{array}$$

(6)

The following values are taken from [17] and expressed as follows:

$$\left\{{\xi }_1, {\xi }_2, {\xi }_{12}\right\}=\left\{{\displaystyle \frac{\partial u_1}{\partial x}},{\displaystyle \frac{1}{R}}\left({\displaystyle \frac{\partial u_2}{\partial \psi }}+u_3\right),\left({\displaystyle \frac{\partial u_2}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial u_1}{\partial \psi }}\right)\right\}$$

(7)

$$\left\{ {\kappa }_1, {\kappa }_2, 2\tau \right\}=\left\{-{\displaystyle \frac{{\partial }^2{u}_3}{\partial x^2}},-{\displaystyle \frac{1}{R^2}}\left({\displaystyle \frac{{\partial }^2{u}_3}{\partial {\psi }^2}}-{\displaystyle \frac{\partial u_2}{\partial \psi }}\right),-{\displaystyle \frac{1}{R}}\left({\displaystyle \frac{{\partial }^2{u}_3}{\partial x\partial \psi }}-{\displaystyle \frac{3}{4}}{\displaystyle \frac{\partial u_2}{\partial x}}+{\displaystyle \frac{1}{4R}}{\displaystyle \frac{\partial u_1}{\partial \psi }}\right)\right\}$$

(8)

Equation (9) is obtained by substituting expressions (7) and (8) into (6), as follows:

$$\begin{array}{ll}{\wp }_{strain}=& {\displaystyle \frac{1}{2}} {{\displaystyle \int }}_0^L{{\displaystyle \int }}_0^{2\pi }[H_{11}{\left({\displaystyle \frac{\partial u_1}{\partial x}}\right)}^2+{\displaystyle \frac{H_{22}}{R^2}}{\left({\displaystyle \frac{\partial u_2}{\partial \psi }}+u_3\right)}^2+ 2{\displaystyle \frac{H_{12}}{R}}\left({\displaystyle \frac{\partial u_1}{\partial x}}\right)\left({\displaystyle \frac{\partial u_2}{\partial \psi }}+u_3\right)+H_{66}{\left({\displaystyle \frac{\partial u_2}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial u_1}{\partial \psi }}\right)}^2\\ & -2I_{11}\left({\displaystyle \frac{\partial u_1}{\partial x}}\right)\left({\displaystyle \frac{{\partial }^2{u}_3}{\partial x^2}}\right)-2{\displaystyle \frac{I_{21}}{R^2}}\left({\displaystyle \frac{\partial u_1}{\partial x}}\right)\left({\displaystyle \frac{{\partial }^2{u}_3}{\partial {\psi }^2}}-{\displaystyle \frac{\partial u_2}{\partial \psi }}\right)-2{\displaystyle \frac{I_{12}}{R}}\left({\displaystyle \frac{\partial u_2}{\partial \psi }}+u_3\right)\left({\displaystyle \frac{{\partial }^2{u}_3}{\partial x^2}}\right)\\ & -{\displaystyle \frac{2I_{22}}{R^3}}\left({\displaystyle \frac{{\partial }^2{u}_3}{\partial {\psi }^2}}-{\displaystyle \frac{\partial u_2}{\partial \psi }}\right)\left({\displaystyle \frac{\partial u_2}{\partial \psi }}+u_3\right)-{\displaystyle \frac{4I_{66}}{R}}\left({\displaystyle \frac{\partial u_2}{\partial x}}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial u_1}{\partial \psi }}\right)\left({\displaystyle \frac{{\partial }^2{u}_3}{\partial x\partial \psi }}-{\displaystyle \frac{3}{4}}{\displaystyle \frac{\partial u_2}{\partial x}}+{\displaystyle \frac{1}{4R}}{\displaystyle \frac{\partial u_1}{\partial \psi }}\right)\\ & +J_{11}{\left(-{\displaystyle \frac{{\partial }^2{u}_3}{\partial x^2}}\right)}^2+{\displaystyle \frac{J_{22}}{R^2}}{\left({\displaystyle \frac{{\partial }^2{u}_3}{\partial {\psi }^2}}-{\displaystyle \frac{\partial u_2}{\partial \psi }}\right)}^2+{\displaystyle \frac{2J_{12}}{R^2}}\left({\displaystyle \frac{{\partial }^2{u}_3}{\partial x^2}}\right)\left({\displaystyle \frac{{\partial }^2{u}_3}{\partial {\psi }^2}}-{\displaystyle \frac{\partial u_2}{\partial \psi }}\right)\\ & +{\displaystyle \frac{4J_{66}}{R^2}}{\left({\displaystyle \frac{{\partial }^2{u}_3}{\partial x_1\partial \psi }}-{\displaystyle \frac{3}{4}}{\displaystyle \frac{\partial u_2}{\partial x}}+{\displaystyle \frac{1}{4R}}{\displaystyle \frac{\partial u_1}{\partial \psi }}\right)}^2]d\psi dx.\end{array}$$

(9)

The kinetic energy of the shell is given as

$$\eta ={\displaystyle \frac{1}{2}}{\displaystyle \underset{0}{\overset{L}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\rho }_T\left[{\left(\frac{\partial u_1}{\partial t}\right)}^2+{\left(\frac{\partial u_2}{\partial t}\right)}^2+{\left(\frac{\partial u_3}{\partial t}\right)}^2\right]Rd\psi dx,}}$$

(10)

where

$${\rho }_t=\underset{-{\displaystyle \frac{H}{2}}}{\overset{{\displaystyle \frac{H}{2}}}{{\displaystyle \int }}}\rho dz$$

(11)

and the mass density is represented by $\rho$.

The difference between the kinetic energy and strain energy for a CS is represented by the Lagrange energy functional $\vartheta$, as follows:

$$\vartheta =\eta -{\wp }_{strain}$$

(12)

Numerical Procedure

The natural frequencies of the shell are obtained by using the Rayleigh–Ritz method. The deformation displacement fields in the $u_1$, $u_2$, $u_3$, i.e., longitudinal, transverse, and angular directions are considered for a CS with ring support. Relationships are now taken for granted in the displacement fields, as follows:

$$\begin{array}{l}{u}_1(x,\psi ,t)=X_m{U}_1(x)cos(\mathrm{n}\psi )sin(\omega t),\\ u_2(x,\psi ,t)=Y_m{U}_2(x)sin(\mathrm{n}\psi )cos(\omega t),\\ u_3(x,\psi ,t)=Z_m{U}_3(x)cos(\mathrm{n}\psi )sin(\omega t).\end{array}$$

(13)

where $X_m,Y_m$, and $Z_m$ are the amplitudes of vibration in the $x_1,\psi$, and $t$ directions; m represents the axial and n is the circumferential wave numbers of the mode forms; ω represents the shell wave’s angular vibration frequency; and $U_1(x)={\displaystyle \frac{d\psi (x)}{dx}},U_2(x)=\psi (x)$, and $U_3(x)=\psi (x){\prod }_{i=}^{\upsilon }{(x-d_i)}^{l_i},$ which denotes the axial function meeting the boundary conditions for ring support, where d denotes the position of ring that varies between 0 and 1 only along the shell length.

There is $i^{th}$ ring support at $x=d_i$ in the longitudinal direction. Where $l_i=1$ and $l_i=0$ denote whether there is a ring support or not, correspondingly. $\psi (x)$ represents the axial function, is defined as follows, and satisfies the geometric edge condition requirement.

$$\psi (x)={\alpha }_1\cosh ({\gamma }_{m}x)+{\alpha }_2\cos ({\gamma }_{m}x)-{\epsilon }_m{\alpha }_{3}sinh({\gamma }_{m}x)+{\epsilon }_m{\alpha }_{4}sinh({\gamma }_{m}x).$$

(14)

${\gamma }_m$ represents several transcendental equation’s roots, ${\epsilon }_m$ parameters are dependent on ${\gamma }_m$, and ${\alpha }_i (i=\mathrm{1...4})$ alter in relation to the edge conditions.

SS-SS is mathematically expressible in terms of $\psi (x)$ as

SS-SS boundary conditions $\psi (x)=\psi {(x)}^{″}=0$.

This issue is generalized using the subsequent non-dimensional properties.

$$\begin{gathered} {\underset{⌢}{\mathrm{U}}}_1={\displaystyle \frac{U_1(x)}{H}},{\underset{⌢}{\mathrm{U}}}_2={\displaystyle \frac{U_2(x)}{H}},{\underset{⌢}{\mathrm{U}}}_3={\displaystyle \frac{U_3(x)}{R}}, \\ {\underset{\_}{H}}_{ij}={\displaystyle \frac{H_{ij}}{H}},{\underset{\_}{I}}_{ij}={\displaystyle \frac{I_{ij}}{H^2}},{\underset{\_}{J}}_{ij}={\displaystyle \frac{J_{ij}}{H^3}}, \\ a={\displaystyle \frac{R}{L}}, b={\displaystyle \frac{H}{L}}, X={\displaystyle \frac{x}{L}},{\underset{⌢}{\rho }}_t={\displaystyle \frac{{\rho }_t}{H}}. \end{gathered}$$

(15)

The following relations are taken for displacement fields.

$$\left.\begin{array}{l}{u}_1(x,\psi ,t)=H{X}_m{\underset{⌢}{\mathrm{U}}}_1\cos (\mathrm{n}\psi )\sin (\omega t)\\ u_2(x,\psi ,t)=H{Y}_m{\underset{⌢}{\mathrm{U}}}_2\sin (\mathrm{n}\psi )\cos (\omega t)\\ u_2(x,\psi ,t)=R{Z}_m{\underset{⌢}{\mathrm{U}}}_3\cos (\mathrm{n}\psi )\sin (\omega t)\end{array}\right\}$$

(16)

The maximum energy principle is utilized to alter the Lagrangian function, as follows:

$$\left.\begin{array}{ll}{\vartheta }_{max}=& {\displaystyle \frac{\pi hLR}{2}}\left[R^2{\omega }^2{\underset{⌢}{\rho }}_t{\displaystyle {\int }_0^1\left(b^2{\left(X_m{\underset{⌢}{\mathrm{U}}}_1\right)}^2+b^2{\left(Y_m{\underset{⌢}{\mathrm{U}}}_2\right)}^2+b^2{\left(Z_m{\underset{⌢}{\mathrm{U}}}_3\right)}^2\right)dX-}\right.\\ & {\displaystyle \underset{0}{\overset{1}{\int }}\left\{a^2{b}^2{\underset{\_}{H}}_{11}{\left(X_m\frac{d{\underset{⌢}{\mathrm{U}}}_1}{dX}\right)}^2\right.}+{\underset{\_}{H}}_{22}{\left(-nb{Y}_m{\underset{⌢}{\mathrm{U}}}_2+Z_m{\underset{⌢}{\mathrm{U}}}_3\right)}^2-2a^3{b}^2{\underset{\_}{I}}_{11}\left(X_m{\displaystyle \frac{d{\underset{⌢}{\mathrm{U}}}_1}{dX}}\right)\left(Z_m{\displaystyle \frac{d^2{\underset{⌢}{\mathrm{U}}}_3}{d{X}^2}}\right)\\ & +2ab{\underset{\_}{H}}_{12}\left(X_m{\displaystyle \frac{d{\underset{⌢}{\mathrm{U}}}_1}{dX}}\right)\left(-nb{Y}_m{\underset{⌢}{\mathrm{U}}}_2+Z_m{\underset{⌢}{\mathrm{U}}}_3\right)+{\underset{\_}{H}}_{66}{\left(ab{y}_n{\displaystyle \frac{d{\underset{⌢}{\mathrm{U}}}_2}{dX}}nb{X}_m{\underset{⌢}{\mathrm{U}}}_1\right)}^2\\ & -2a{b}^2{\underset{\_}{I}}_{12}\left(X_m{\displaystyle \frac{d{\underset{⌢}{\mathrm{U}}}_1}{dX}}\right)\left(-n^2{Z}_m{\underset{⌢}{\mathrm{U}}}_3+nb{y}_n{\underset{⌢}{\mathrm{U}}}_2\right)-2a^{2}b{\underset{\_}{I}}_{12}\left(-nb{Y}_m{\underset{⌢}{\mathrm{U}}}_2+Z_m{\underset{⌢}{\mathrm{U}}}_3\right)\left(Z_m^2{\displaystyle \frac{d^2{\underset{⌢}{\mathrm{U}}}_3}{d{X}^2}}\right)\\ & -2b{\underset{\_}{I}}_{22}\left(-nb{Y}_m{\underset{⌢}{\mathrm{U}}}_2+Z_m{\underset{⌢}{\mathrm{U}}}_3\right)\left(-n^2{Z}_m{\underset{⌢}{\mathrm{U}}}_3+nb{Y}_m{\underset{⌢}{\mathrm{U}}}_2\right)+a^4{b}^2{\underset{\_}{J}}_{11}{\left(Z_m^2{\displaystyle \frac{d^2{\underset{⌢}{\mathrm{U}}}_3}{d{X}^2}}\right)}^2\\ & -4b{\underset{\_}{I}}_{66}\left(ab{Y}_m{\displaystyle \frac{d{\underset{⌢}{\mathrm{U}}}_2}{dX}}+nb{X}_m{\underset{⌢}{\mathrm{U}}}_1\right)\left(na{Z}_m{\displaystyle \frac{d{\underset{⌢}{\mathrm{U}}}_3}{dX}}-{\displaystyle \frac{3ab{Y}_m}{4}}{\displaystyle \frac{d{\underset{⌢}{\mathrm{U}}}_2}{dX}}+{\displaystyle \frac{nb}{4}}{X}_m{\underset{⌢}{\mathrm{U}}}_1\right)\\ & +b^2{\underset{\_}{J}}_{22}{\left(-n^2{Z}_m{\underset{⌢}{\mathrm{U}}}_3+nb{Y}_m{\underset{⌢}{\mathrm{U}}}_2\right)}^2+2a^2{b}^2{\underset{\_}{J}}_{12}\left(Z_m^2{\displaystyle \frac{d^2{\underset{⌢}{\mathrm{U}}}_3}{d{X}^2}}\right)\left(-n^2{Z}_m{\underset{⌢}{\mathrm{U}}}_3+nb{Y}_m{\underset{⌢}{\mathrm{U}}}_2\right)\\ & +4{\underset{\_}{J}}_{66}\left.\left.{\left(na{Z}_m{\displaystyle \frac{d{\underset{⌢}{\mathrm{U}}}_3}{dX}}-{\displaystyle \frac{3ab{Y}_m}{4}}{\displaystyle \frac{d{\underset{⌢}{\mathrm{U}}}_2}{dX}}+{\displaystyle \frac{nb}{4}}{X}_m\underset{⌢}{U}\right)}^2\right\}\right]dX\end{array}\right\}$$

(17)

The Lagrangian energy functional ${\vartheta }_{max}$ is maximized w.r.t. to the vibrational amplitudes $X_m,Y_m$, and $Z_m,$ as shown below:

$${\displaystyle \frac{\partial {\vartheta }_{g(\max )}}{\partial X_m}}={\displaystyle \frac{\partial {\vartheta }_{g(\max )}}{\partial Y_m}}={\displaystyle \frac{\partial {\vartheta }_{g(\max )}}{\partial Z_m}}=0.$$

(18)

The resultant relation is defined as

$$\left\{\left[K\right]-{\mathrm{\Omega }}^2\left[V\right]\right\}\stackrel{⌢}{X}=0$$

(19)

where

$${\mathrm{\Omega }}^2=R^2{\omega }^2{\underset{⌢}{\rho }}_t,$$

(20)

$\left[K\right]$ and $\left[V\right]$ are the mass and stiffness matrices that match the CS.

3. Results

3.1. Classsification of Material

A CS consisting of four layers is considered in the present work. The materials used are nickel, zirconia, and stainless steel.

The polynomial volume fraction law (VFL) of the shell component for all FGM layers is defined by the following relation:

$$L=\left[{\left({\displaystyle \frac{8z+H}{2H}}\right)}^N\right],0\le N\le \infty .$$

where N denotes the power law exponent. It is believed that each layer has the same thickness. The following are the material parameters: $E_1,{\nu }_1, {\rho }_1$ for stainless steel; $E_2,{\nu }_2, {\rho }_2$ for aluminum; $E_3,{\nu }_3, {\rho }_3$ for nickel; $E_4,{\nu }_4, {\rho }_4$ for zirconia; and $E_5,{\nu }_5, {\rho }_5$ for aluminum and nickel. The exact material amounts are given as

$$\begin{gathered} E_{fgm1}=\left[E_1-E_2\right]{\left({\displaystyle \frac{8z+H}{2H}}\right)}^N+E_2, \\ {\nu }_{fgm1}=\left[{\nu }_1-{\nu }_2\right]{\left({\displaystyle \frac{8z+H}{2H}}\right)}^N+{\nu }_2, \\ {p}_{fgm1}=\left[p_1-p_2\right]{\left({\displaystyle \frac{8z+H}{2H}}\right)}^N+p_2. \\ {E}_{fgm2}=\left[E_4-E_5\right]{\left({\displaystyle \frac{8z+H}{2H}}\right)}^N+E_5, \\ {\nu }_{fgm2}=\left[{\nu }_3-{\nu }_4\right]{\left({\displaystyle \frac{8z+H}{2H}}\right)}^N+{\nu }_4, \\ {p}_{fgm2}=\left[p_3-p_4\right]{\left({\displaystyle \frac{8z+H}{2H}}\right)}^N+p_4. \end{gathered}$$

For the above equations at $z=-{\displaystyle \frac{H}{2}},{\displaystyle \frac{H}{2}}$

$E_{fgm1}=E_2,{\nu }_{fgm1}={\nu }_2,p_{fgm1}=p_2$ and

$E_{fgm2}=E_5,{\nu }_{fgm2}={\nu }_5,p_{fgm2}=p_5$, accordingly.

The stiffness moduli are altered as follows:

$$\begin{array}{l}{H}_{ij}=H_{ij}(FGM)+H_{ij}(FGM)+H_{ij}(FGM)+H_{ij}(FGM),\\ I_{ij}=I_{ij}(FGM)+I_{ij}(FGM)+I_{ij}(FGM)+I_{ij}(FGM), \\ J_{ij}=J_{ij}(FGM)+J_{ij}(FGM)+J_{ij}(FGM)+J_{ij}(FGM).\end{array}$$

where $i.j=1,2 \mathrm{a}\mathrm{n}\mathrm{d} 6;$ FGM represents the FGM layers.

Results and Discussion

FGM CS pertains to the present outcome for SS edge situations to explain the precision of the precise outcome. To obtain the results, the RR technique is used.

Table 1 shows how frequency parameters for FGM CS with SS conditions are compared with [4].

Table 1. Comparing frequency characteristics for SS-SS CS (m = 2, L/R = 22, H = 0.02 and d = 0.3).

n	[4]	This Study Analysis	Difference
1	0.016101	0.016000	0.62%
2	0.009382	0.009367	0.15%
3	0.022105	0.022103	0.009%
4	0.042095	0.042086	0.021%
5	0.068008	0.068005	0.004%

3.2. Figures, Tables, and Schemes

Configurations of shells:

Table 2 defines the four-layered CS’s material composition, which is constructed of FGM.

Table 2. Shell type configuration and material distribution.

Types of Shell	Layer 1 FGM	Layer 2 FGM	Layer 3 FGM	Layer 4 FGM
Type	Nickel, Zirconia	Stainless Steel, Zirconia	Stainless Steel, Nickel	Aluminum, Zirconia
Material	E (N/m2)		Poisson Ratio  $\nu$	Density  $p$  (kg/m3)
Nickel	2.05098 × 1011		0.31	8900
Aluminum	1.68063 × 1011		0.298	5700
Zirconia	2.07788 × 1011		0.311	8166
Stainless Steel	2.1 × 1011		0.28	7800

In Table 3, using SS-SS edge conditions, the variation in NFs for N is observed. Since n increases about 1 to 10, the value of NF gradually increases. With the power exponent law between N = 1 to N = 20, NF increases from 436.4223 to 861.0517.

Table 3. Natural frequency (NF) variation with varying power exponent laws in opposition to the circumferential wave number for a four-layered thin cylindrical shell, with SS-SS having (m = 1, L/R = 22 m, H = 0.002 m, d = 0.2).

n	N = 1	N = 2	N = 12	N = 15	N = 20
1	436.4223	503.2684	515.1651	828.1999	861.0517
2	550.0414	634.4965	650.3718	1350.1806	1403.7316
3	571.8594	659.6653	676.1785	1947.6793	2024.9268
4	579.4226	668.3880	685.1188	1517.9261	1575.8139
5	582.9021	672.3991	689.2283	1896.8769	1969.2163
6	584.7856	674.5685	691.4494	2275.9279	2362.7229
7	585.9188	675.8719	692.7822	2655.0295	2756.2819
8	586.6533	676.7147	693.6423	3034.1603	3149.8713
9	587.1564	677.2899	694.2277	3413.3096	3543.4798
10	587.5159	677.6989	694.6422	3792.4713	3937.1012

Table 4 compares the present analysis with Ghamkhar et al. [17] for SS-SS CS; NF against n with (m = 1, L/R = 50 m, H = 0.007 m, d = 0.5) have been compared. As n varies from 1 to 10 and for N = 1 and 2, it can be seen that the values of the present analysis are smaller than those in the previous study where NF increases with an increase in wave number.

Table 4. Comparison of the natural frequency (NF) variation with varying power exponent laws in opposition to circumferential wave number for four-layered thin CS, when (m = 1, L/R = 50 m, H = 0.007 m, d = 0.5).

	3 Layered CS		4 Layered CS
	Ghamkhar et al. [17]		Present Analysis
n	N = 1	N = 2	N = 1	N = 2
1	499.393	516.109	502.422	573.268
2	848.181	850.782	849.536	834.496
3	848.297	850.858	849.534	839.665
4	848.604	851.210	849.533	848.388
5	849.247	851.858	849.532	852.399
6	850.411	853.030	849.530	854.568
7	852.321	854.953	849.528	855.871
8	855.239	857.887	849.526	856.714
9	859.46	862.124	849.526	857.289
10	865.311	868.017	849.524	857.698

Table 5 shows the change in NFs for FGM CS with (m = 1, R = 1, N = 1, d = 0.3, L/R = 10 m), which decreases horizontally from 487.2966 to 487.2955 for n = 1 when the thickness H is increased from 0.001 to 0.05. When the circumferential wave number is increased vertically from 1 to 10, then NF increases vertically from 487.2966 to 697.1757.

Table 5. Natural frequency (NF) variation with varying power exponent laws in opposition to circumferential wave number for the four-layered thin cylindrical shell, when (m = 1, N = 1, d = 0.3, L/R = 10 m).

n	H = 0.001	H = 0.005	H = 0.007	H = 0.01	H = 0.05
1	487.2966	487.2966	487.2965	487.2965	487.2955
2	665.0011	665.0009	665.0007	665.0003	664.9841
3	684.8592	684.8588	684.8583	684.8574	684.8264
4	690.9708	690.9699	690.9691	690.9674	690.9322
5	693.6777	693.6764	693.6752	693.6725	693.6598
6	695.1178	695.1160	695.1141	695.1104	695.1680
7	695.9762	695.9736	695.9711	695.9662	696.1674
8	696.5293	696.5259	696.5227	696.5165	696.9650
9	696.9067	696.9025	696.8985	696.8909	697.7252
10	697.1757	697.1706	697.1658	697.1568	698.5548

Table 6 describes the change in NFs against n with m = 2, N = 3, d = 0.8, and H = 0.075 under SS-SS. The natural frequencies increase from 513.8298 to 603.1852 when the length is increased.

Table 6. Natural frequency (NF) variation with varying power exponent laws in opposition to circumferential wave number for the four-layered thin cylindrical shell, when (m = 2, N = 3, R = 1, d = 0.8 and H =0.075 m).

n	L/R = 10	L/R = 20	L/R = 30	L/R = 40	L/R = 50
1	513.8298	513.8310	514.0113	513.8298	513.8298
2	643.3578	643.3588	643.4328	643.3578	643.3578
3	667.8943	667.9943	668.1112	667.8943	667.8943
4	671.6022	671.6122	671.7532	672.1032	671.6022
5	668.3093	668.3193	668.3093	668.3093	668.3093
6	661.1119	661.1119	661.1119	661.1119	661.1119
7	650.9084	650.9184	650.9084	650.9084	650.9084
8	637.9234	637.9334	637.9234	637.9234	637.9234
9	622.0942	622.1112	622.3425	622.5622	623.0112
10	603.1852	603.2152	604.1861	604.2672	604.3225

Table 7 displays the change in NFs against n with parameters (m = 1, H = 0.006, N = 2, and R = 1) for different length values. The values of NFs decrease from 50.1 to 0.9438 for n = 1 to n = 10 for L/R = 10. For length L/R= 10 m to 50 m, there is also a decrease from 50.1 to 2.2471 for n = 1.

Table 7. Natural frequency (NF) variation with varying length in opposition to circumferential wave number for the four-layered thin CS without ring support, with SS-SS (m = 1, H = 0.006 m, N = 2, and R = 1).

n	L/R = 10	L/R = 20	L/R = 30	L/R = 40	L/R = 50
1	50.1	13.661	6.1824	3.5004	2.2471
2	17.3714	4.434	1.9784	1.1144	0.7138
3	8.3222	2.0991	0.935	0.5272	0.3388
4	4.8129	1.211	0.5428	0.3116	0.2077
5	3.1225	0.7915	0.3684	0.2305	0.1749
6	2.1892	0.5748	0.3018	0.2255	0.1998
7	1.6284	0.4723	0.3051	0.267	0.2557
8	1.2774	0.4484	0.3532	0.3346	0.3294
9	1.0614	0.4805	0.4281	0.4188	0.4162
10	0.9438	0.5498	0.5207	0.5158	0.5144

InTable 8, it is observed that NF varies from 485.9 to 677.4533. Compared with the previous table, it can be seen that natural frequencies have increased constantly due to ring support, whereas natural frequencies decrease without ring support.

Table 8. Natural frequency (NF) variation with varying length in opposition to the circumferential wave number for the four-layered thin CS with ring support, with SS-SS (m = 1, H = 0.006 m, N = 2, and R = 1).

n	L/R = 10	L/R = 20	L/R = 30	L/R = 40	L/R = 50
1	485.9	501.2853	508.6516	512.1725	514.0659
2	632.9204	634.3871	634.7180	634.8385	634.8951
3	659.3743	659.6283	659.6826	659.7022	659.7113
4	668.2699	668.3455	668.3615	668.3672	668.3699
5	672.3076	672.3375	672.3438	672.3460	672.3471
6	674.4669	674.4809	674.4838	674.4848	674.4853
7	675.7454	675.7527	675.7542	675.7547	675.7550
8	676.5547	676.5587	676.5596	676.5599	676.5600
9	677.0899	677.0922	677.0927	677.0929	677.0930
10	677.4533	677.4546	677.4549	677.4550	677.4551

Table 9 shows the comparison of vibrations in natural frequency against n for shell, both with and without ring support for SS-SS boundary conditions. It is noticed that for the same N, the frequency behavior is consistent. The values of NFs with ring support are higher than those without ring support.

Table 9. Comparison of variation in natural frequency (Hz) for a CS with simply supported boundary conditions with different power exponent laws against n, when (m = 2, d = 0.4, L/R = 20 m, and H = 0.02 m).

	Without Ring Support		With Ring Support
n	N = 1	N = 2	N = 1	N = 2
1	13.7000	15.7446	507.4000	584.2955
2	4.4386	5.1171	849.5325	979.8045
3	2.1897	2.5306	849.5273	979.5824
4	1.7101	1.9935	849.5204	979.2718
5	2.1114	2.4760	849.5121	978.8732

Table 10 shows the NFs against L/R for FGM CS with ring support (n = 1, m = 1, R = 1, d = 0.5, and H = 0.005 m) under SS-SS by changing the value of N. As N increases, the natural frequencies dropped by 21.11%.

Table 10. Natural frequency (Hz) against L/R for CS under SS-SS, with ring support (n = 1, m = 1, d = 0.5, and H = 0.005).

L/R	N = 1	N = 2	N = 3	N = 5	N = 10	N = 20	N = 30	N = 50
5	591.985	681.685	693.879	407.501	470.81	471.51	471.508	471.508
10	525.407	605.02	526.427	615.823	417.86	418.478	418.478	418.478
20	507.391	584.281	594.736	349.271	403.538	404.135	404.135	404.135

Figure 3 displays the behavior of natural frequency against n for four-layered FGM CS. The NFs are obtained under SS-SS. There is a significant decrease from n = 1 and 2 in NFs and gradually increases for m = 9, afterward exhibiting calm behavior for H = 0.05 m and 0.07 m.

Figure 3. Natural frequency (NF) variation with varying power exponent laws in opposition to circumferential wave number for four-layered CS without ring support, when (m = 3, R = 1, N = 3, d = 0.1, L/R = 24, and H = 0.05 m, 0.07 m).

Figure 4 shows that the NFs are observed to increase as n increases from 1 to 2 for power exponent law and then remain constant for N = 1 and N = 2.

Figure 4. Natural frequency (NF) variation with varying power exponent laws in opposition to circumferential wave number for the four-layered cylindrical shell, when (m = 2, d = 0.5, L/R = 20 m, and H = 0.02 m).

In Figure 5, it is noted that NFs become constant after a certain point from n = 2 to 10.

Figure 5. Natural frequency (NF) variation with varying power exponent laws in opposition to the circumferential wave number for four-layered CS (m = 1, H = 0.006 m, N = 2, R = 1, L/R = 10 m to 30 m and d = 0.2).

In Figure 6, the variation in NF for SS-SS CS with different positions of ring is displayed for different values of L/R ratios. The values obtained for L = 5 are 485.9432, 519.668, 569.507, 732.725, 867.99641, 732.719,569.500, 519.661, and 485.942 for d= 0.0 to 1.0. It can be observed that the behavior of NF increases from d = 0.0 to 0.5, obtains its extreme value at 0.5, and decreases for d = 0.5 to d = 1.0, creating a systematic curve.

Figure 6. Natural frequency (NF) variation with ring support ‘d’ at different L/R ratios for four-layered CS without ring support, when (m = 1, R = 1, N = 1, L/R = 30 m, and H = 0.04 m).

4. Discussion

This paper explores the complex vibrational behavior and NFs of CS made of FGM. This work attempts to provide an important new understanding of the dynamics of these kinds of composite systems by examining the vibrations of CS made of four layers supported by a ring throughout its length. The FGM cylindrical shell is composed of nickel, zirconia, stainless steel, and aluminum. These materials are smart and advanced in their physical properties.

The cylindrical shell, represented by the symbol H/4, has an identical thickness when all four of its layers are utilized. In Sander’s shell theory, the Rayleigh–Ritz method (RRM) is used to analyze the vibration behavior and natural frequencies of these positioned cylindrical shells. Additionally, the study looks into the shells’ inherent frequencies in the SS-SS scenario. The research intends to give a comprehensive learning of how various conditions affect the vibrational properties of the CSs by investigating these boundary conditions.

In conclusion, this study makes a substantial contribution to our knowledge of the natural frequencies and vibration behavior of positioned cylindrical structures made of functionally graded polynomial volume fraction law. Through the utilization of sophisticated analytical methods and investigation of diverse material compositions and boundary conditions, the research illuminates the intricate dynamics of these composite systems, potentially bearing on a broad spectrum of practical uses.

5. Conclusions

Since cylindrical shells are used in practical applications, theoretical vibration analysis of CSs is important in applied mathematics and mechanics. The purpose of the present work has been to examine the vibration properties of CS with different material compositions using FGM. The components of the CS are composed of zirconia, nickel, stainless steel, and aluminum. These three categories were examined by arranging the FG material layers differently. Deriving shell frequency equations was the main goal of using these methods. The shell frequency equations were solved using Sander’s shell theory, based on Kirchhoff’s hypotheses. By applying the Lagrangian energy function to derive the numerical form of the shell’s frequency equation, the Reighliegh–Ritz technique solved the vibration difficulties about the shell.

The study discovered that for all boundary conditions, the minimum frequency arises at particular wave numbers and is proportional to raising the values of N, L/R, n, d, and H. The NFs varied in N under various circumstances. The outcomes were all fairly similar to one another. This analysis can be expanded to explore various shell difficulties by varying factors like the ring’s position and the shell’s thickness or length. Due to the same edge constraints, the shell’s frequency under ring support at various places takes on symmetric shapes. Their dissimilar end-point circumstances cause them to be non-symmetrical around the center. When ring support is induced on a cylinder-shaped shell, the NFs are significantly affected in comparison to the shell frequencies when ring support is not present.

MATLAB 2019a software was used to obtain the presented CS results, with SS-SS boundary conditions. The NFs behaving as a function of n are illustrated by these data. The following are the noteworthy findings. As ‘N’ rises, so does the natural frequency. A rise in the shell’s length ‘L’ causes the natural frequency to rise too. Increasing the shell’s thickness ‘H’ leads to an increase in NFs. The ring support is placed along the axial direction, which reaches its extreme when the ring is at the core of the shell,

In comparison with the previous study, Table 4 compares the present analysis with Ghamkhar et al. [17] for SS-SS CS, and NF against n with (m = 1, L/R = 50, H = 0.007, d = 0.05). As n varies from 1 to 10 and for N = 1 and 2, it can be seen that the values of the present analysis are smaller than those in the previous study where NF increases with an increase in wave number. These results demonstrate the crucial role that geometric as well as material characteristics play on the CS’s vibrational behavior, offering valuable information for improving both the durability and efficiency of the FGM CS. Extensions can be made in the present study by increasing the number of layers or changing the material of layers used.