Structural Study of Four-Layered Cylindrical Shell Comprising Ring Support

Abstract

In this work, the vibration analysis of a layered, cylinder-shaped shell is undertaken. The structure of the shell layers is composed of functionally graded and isotropic materials. The vibrations of four-layered cylindrical shells with a ring support along the axial direction are investigated in this research. The two internal layers are composed of isotropic materials, and the external two layers are composed of functionally graded materials. The outer functionally graded material layers considered are stainless steel, zirconia, and nickel. The inner two isotropic layers considered are aluminum and stainless steel. The shell frequency equation is acquired by employing the Rayleigh–Ritz method under the shell theory of Sanders. The trigonometric volume fraction law is used to sort the functionally graded material composition of the FGM layers. The natural frequencies are attained under two boundary conditions, namely simply supported–simply supported and clamped–clamped.

1. Introduction

In the analysis of structures, a cylindrical shell is a major feature. Various mechanical facets of these shell structures are analyzed in addition to their potential implementation, one of which is shell vibration. In areas of technology such as liquid gases at high pressure, nuclear power stations, networks of pipes, and other marine and aircraft systems, the examination of such shells plays a crucial role. The biography for the past scientific work has been developed based on Love’s theory. Utilizing Sanders’ shell theory and the Ritz composition, many researchers [1,2,3,4,5,6,7,8] have analyzed the frequency of dimensionally stable cylindrical shells under ring support over many end situations. Zang et al. [9] described the study of a hypothesis in which, contrary to Love’s results, all strains disappear from any rigid body. It offered formulations for the stress effects and couples that satisfy the homogeneous equilibrium conditions based on three stress frictions. The effect of the boundary conditions on the free vibrations for a multilayered cylindrical shell was investigated using the concept based on Love’s first theorem and the beam functions utilized for the Ritz as inertial structural features. There were nine boundary conditions taken into account, four of which had the same finishing criterion as their starting criterion. The other five conditions had different end requirements. Zang et al. [10] created heat-resistant structures for space plane fusion reactors and airframes. Zhi et al. [11] compared the frequency parameters with a precise three-dimensional linear elasticity analysis to test the analysis. The vibrational analysis of the functionally graded materials (FGMs) also discussed the behavior of natural frequencies.

Iqbal et al. [12] discussed the vibrational analysis of cylindrical shells and natural frequencies (NFs) under different boundary conditions and investigated new precise programs for the motions of the circular cylindrical shell with intermediate ring support based on the Goldenviezer–Novozhilov shell principle. The analytical method investigated the vibrational behavior of the ring support cylinder, while the state-space technique was used to operate the homogeneous comparative methodology for the shell component, and the domain decomposition technique was investigated for the continuity criteria among shell parts in the sense of the previously described method. Iqbal et al. [13] suggested 3D anisotropic elasticity fundamental equations, from which time visibility with differential equations is constructed in a coherent matrix form, and examined numerous techniques such as power law, sigmoid, and exponential FGMs, and we will focus on the numerical solution that was predominantly derived from the theoretical formulation and calculated using the MARC system. With reference to the effects of the application of P-FGM to S-FGM and E-FGM, Li et al. [14] introduced four sets of in-plane boundary conditions for the strictly assisted FG cylindrical shell for which the free vibration analysis was conducted. Sofiyev et al. [15] investigated physical properties that were thought to be temperature-dependent and eventually modified in the stress distribution of the shell. The results of the temperature increase were studied by specifying an approximate elevated temperature on the cylinder’s outer surface and the average temperature on the cylinder’s inner surface. The properties were temperature-dependent and gradually changed as the intensity of the shell increased.

By evaluating the elevated temperature, Arshad et al. [16] also mentioned the rising temperature effect. The entire study examined the strength of cylindrical shells supported on Winkler–Pasternak frameworks and made of ceramic, FGM, and metal layers that were exposed to an axial load. The revised stability and durability equations of the Donnell form on the Pasternak base were gathered. Galerkin’s process analysis values were obtained for the critical axial load of three-layered cylindrical shells with and without an elastic foundation. Extensive parametric experiments were performed to investigate the effects of FGM sheet thickness differences on the radius-to-thickness ratio. Naeem et al. [17] studied the vibration properties of two-layered cylindrical shells made of isotropic material on a single layer and functionally graded material on another. A bi-layered cylindrical shell’s vibration frequency analysis consisting of two distinct functionally graded layers is proposed by Arshad et al. [18]. The thicknesses of the shell’s layers are considered to be constant and equivalent. It is anticipated that the components of the material of the bi-layered dynamically graded cylindrical shell will differ seamlessly.

Ghamkhar et al. [19] investigated the vibration analysis of a three-layered cylindrical shell, with the anterior and posterior layers made of mechanically graded materials and the buffer part made of isotropic material. Love’s thin shell theorem was applied to the relationships of strain vs. displacement and curvature vs. displacement. To manipulate the intensity formulas, the Rayleigh–Ritz method (RRM) was used to grow the computational Lagrangian to be expressed as an eigenvalue query. The frequency wavelengths for long, simple-assisted end states were calculated by altering the thickness-to-radius and length-to-radius ratios; two non-dimensional geometrical parameters; and small, dense, and thin CSs. Ghamkhar et al. [20] examined a three-layered FGM cylindrical shell’s vibration frequency with an FGM central layer and homogeneous material as the interior and posterior layers. The relationships between the strain and the stress were drawn from the shell theory of Sanders. By utilizing the RRM, the shell frequency result was derived. The effect of natural frequencies (NFs) was evaluated for different middle-layer thicknesses. The beam functions of the features were used to approximate the axial modal dependency. The fundamental natural frequency of a cylindrical shell against the time span and thickness-to-radius percentage in a broad range was reported and investigated through this research.

Liu et al. [21] focused on determining the axial buckling load of stiffened multilayer cylindrical shell panels made of functionally graded graphene-reinforced composites (FG-GPL RCs). They used rings and stringers as stiffening tools. The study utilized Halpin–Tsai relations for elastic properties and implemented the virtual work principle and finite element approach based on the first-order shear deformation theory and Lekhnitskii’s smeared stiffener approach to derive stability equations. It examined four nanofiller dispersion types (FG-X, FG-A, FG-O, and UD) and systematically investigated the effects of geometric and material parameters on buckling loads and mode shapes, providing valuable design insights. The present research investigates the vibrations of four-layered cylindrical shells with ring support along the axial direction. The shell layers consist of a combination of functionally graded and isotropic materials. Specifically, the two internal layers are made of isotropic materials, while the two external layers are composed of functionally graded materials. The outer layers are assumed to be made of stainless steel, zirconia, and nickel, whereas the inner isotropic layers are composed of aluminum and stainless steel. The shell frequency equation is derived using the Rayleigh–Ritz method based on Sander’s shell theory. It utilizes energy principles (strain energy and kinetic energy) to derive the governing equations in an approximate form. The resulting system of algebraic equations is solved as an eigenvalue problem to determine natural frequencies and mode shapes.

2. Materials and Methods

A coordinate mechanism of cylinders $\left(x,\theta ,t\right)$ is represented as the shell of the center reference surface with x, θ, and z as the axial, angular, and thickness coordinates, respectively, where $u_1\left(x,\theta ,t\right)$, $u_2\left(x,\theta ,t\right)$, and $u_3\left(x,\theta ,t\right)$ denote the deformations of the displacement functions in the longitudinal, tangential, and transverse instructions accordingly (see Figure 1).

The strain energy relation is given below

$$\mathcal{I}=\frac{1}{2} \underset{0}{\overset{L}{{\displaystyle \int }}}\underset{0}{\overset{2\pi }{{\displaystyle \int }}}{\left\{K\right\}}^{\prime }\left[S\right]\left\{K\right\}Rd\theta dx$$

(1)

where

$${\left\{K\right\}}^{\prime }=\{c_1,c_2,\mathcal{l},{\kappa }_1, {\kappa }_2, 2\sout{t}\}$$

(2)

The matrix [C] is represented as follows:

$$\left[C\right]=\left[\begin{array}{cc}\begin{array}{ccc}{\xi }_{11}& {\xi }_{12}& 0\\ {\xi }_{12}& {\xi }_{22}& 0\\ 0& 0& {\xi }_{66}\end{array}& \begin{array}{ccc}{\psi }_{11}& {\psi }_{12}& 0\\ {\psi }_{12}& {\psi }_{22}& 0\\ 0& 0& {\psi }_{66}\end{array}\\ \begin{array}{ccc}{\psi }_{11}& {\psi }_{12}& 0\\ {\psi }_{12}& {\psi }_{22}& 0\\ 0& 0& {\psi }_{66}\end{array}& \begin{array}{ccc}{\zeta }_{11}& {\zeta }_{12}& 0\\ {\zeta }_{12}& {\zeta }_{22}& 0\\ 0& 0& {\zeta }_{66}\end{array}\end{array}\right]$$

(3)

where $c_1,c_2, \mathcal{l}$ express the strains that consist of the reference surface, $\left\{{\kappa }_1,{\kappa }_2,\sout{t}\right\}$ indicate the curvatures, and ${\xi }_{ij}$,${\psi }_{ij}$, and ${\zeta }_{ij}$ (i, j = 1, 2, 6) represent the extensional, coupling, and bending stiffness [5] (Loy et al. 1999).

They are defined by the following formulas:

$$\{{\xi }_{ij},{\psi }_{ij},{\zeta }_{ij}\}={{\displaystyle \int }}_{-\frac{H}{2}}^{\frac{H}{2}}{Q}_{ij}\left\{1,z,z^2\right\}dz$$

(4)

The reduced stiffness $Q_{ij}$ for isotropic materials is stated as,

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

And

$$Q_{66}=\frac{E}{2\left(1-{\lambda }^2\right)}$$

(5)

The matrix ${\psi }_{ij}=0$ is for isotropic circular-shaped CS, and ${\psi }_{ij} \ne 0$ is for FG CS; considering its structure and the characteristics of its constituents, the expressions from (2) and (3) in (1), $E_{strain}$, are rewritten as:

$$\begin{array}{l}{E}_{strain}=\frac{1}{2} {{\displaystyle \int }}_0^L{{\displaystyle \int }}_0^{2\pi }\{{\xi }_{11}{c}_1^2+{\xi }_{22}{c}_2^2+2{\xi }_{11}{c}_1{c}_2+{\xi }_{66}{\mathcal{l}}^2+2{\psi }_{11}{c}_1{K}_1+2{\psi }_{12}{c}_1{K}_2+2{\psi }_{12}{c}_2{K}_1\\ +2{\psi }_{22}{c}_2{K}_2+4{\psi }_{66}\mathcal{l}\sout{t}+{\zeta }_{11}{K}_1^2+{\zeta }_{22}{K}_2^2+2{\zeta }_{12}{K}_1{K}_2+4{\zeta }_{66}{\mathcal{l}}^2\}Rd\theta dx\end{array}$$

(6)

The expressions that follow are extracted from [19,20] and written as follows:

$$\left\{c_1,c_2,\mathcal{l}\right\}=\left\{\frac{\partial u_1}{\partial x},\frac{1}{R}\left(\frac{\partial u_2}{\partial \theta }+u_3\right),\left(\frac{\partial u_2}{\partial x}+\frac{1}{R}\frac{\partial u_1}{\partial \theta }\right)\right\}$$

(7)

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

(8)

By substituting expressions (7) and (8) into Equation (6), expression (6) becomes as follows:

$$\begin{array}{l}{E}_{strain}=\frac{1}{2} {{\displaystyle \int }}_0^L{{\displaystyle \int }}_0^{2\pi }[{\xi }_{11}{\left(\frac{\partial u_1}{\partial x}\right)}^2+\frac{{\xi }_{22}}{R^2}{\left(\frac{\partial u_2}{\partial \theta }+u_3\right)}^2+\frac{2{\xi }_{12}}{R}\frac{\partial u_1}{\partial x}{\left(\frac{\partial u_2}{\partial \theta }+u_3\right)}^2+{\xi }_{66}{\left(\frac{\partial u_2}{\partial x}+\frac{1}{R}\frac{\partial u_1}{\partial \theta }\right)}^2\\ -{\psi }_{11}\left(\frac{\partial u_1}{\partial x}\right)\left(\frac{{\partial }^2{u}_3}{\partial x^2}\right)-\frac{2{\psi }_{12}}{R^2}\left(\frac{\partial u_1}{\partial x}\right)\left(\frac{{\partial }^2{u}_3}{\partial {\theta }^2}-\frac{\partial u_2}{\partial \theta }\right)-\frac{2{\psi }_{12}}{R}\left(\frac{\partial u_2}{\partial \theta }+u_3\right)\left(\frac{{\partial }^2{u}_3}{\partial x^2}\right)\\ -\frac{2{\psi }_{22}}{R^3}\left(\frac{\partial u_2}{\partial \theta }+u_3\right)\left(\frac{{\partial }^2{u}_3}{\partial {\theta }^2}-\frac{\partial u_2}{\partial \theta }\right)-\frac{4{\psi }_{66}}{R}\left(\frac{\partial u_2}{\partial x}+\frac{1}{R}\frac{\partial u_1}{\partial \theta }\right)\left(\frac{{\partial }^2{u}_3}{\partial x\partial \theta }-\frac{3}{4}\frac{\partial u_2}{\partial x}+\frac{1}{4R}\frac{\partial u_1}{\partial \theta }\right)\\ +{\zeta }_{11}{\left(\frac{{\partial }^2{u}_3}{\partial x^2}\right)}^2+\frac{{\zeta }_{22}}{R^4}{\left(\frac{{\partial }^2{u}_3}{\partial {\theta }^2}-\frac{\partial u_2}{\partial \theta }\right)}^2+\frac{2d{\zeta }_{12}}{R^2}\left(\frac{{\partial }^2{u}_3}{\partial x^2}\right)\left(\frac{{\partial }^2{u}_3}{\partial {\theta }^2}-\frac{\partial u_2}{\partial \theta }\right)\\ +\frac{4{\zeta }_{66}}{R^2}{\left(\frac{{\partial }^2{u}_3}{\partial x\partial \theta }-\frac{3}{4}\frac{\partial u_2}{\partial x}+\frac{1}{4R}\frac{\partial u_1}{\partial \theta }\right)}^2]dxd\theta \end{array}$$

(9)

Shell kinetic energy is stated as

$$I=\frac{1}{2}{{\displaystyle \int }}_0^L{{\displaystyle \int }}_0^{2\pi }{\rho }_t\left[{\left(\frac{\partial u_1}{\partial t}\right)}^2+{\left(\frac{\partial u_2}{\partial t}\right)}^2+{\left(\frac{\partial u_3}{\partial t}\right)}^2\right]Rd\theta dx$$

(10)

where

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

(11)

where $\rho$ is the mass density.

The Lagrange energy functional $\Gamma$ for a CS is described as follows as a function of the strain and kinetic energies:

$$\Gamma = I-E_{strain}$$

(12)

Numerical Procedure

The RR method controls the normal frequencies of CS. The distortion defects in the longitudinal $u_1$, transverse $u_2$, and angular directions $u_3$ are taken into account. The displacement domains are now assumed by the following relationships:

$$\begin{gathered} u_1\left(x,\theta ,t\right)=x_{n}D\left(x\right)\cos \left(m\theta \right)sin\left(\varpi t\right), \\ {u}_2\left(x,\theta ,t\right)=y_{n}E\left(x\right)\sin \left(m\theta \right)cos\left(\varpi t\right) \end{gathered}$$

And

$$u_3\left(x,\theta ,t\right)=z_{n}F\left(x\right)\cos \left(m\theta \right)\sin \left(\varpi t\right)$$

(13)

where $x_n,y_n$, and $z_n$ reflect the vibrational amplitudes in the x, θ, and t direction accordingly; the axial and circumferential wave numbers of mode shapes are indicated by n and m; ω symbolizes the angular vibration frequency of the shell wave; and $D(x)=\frac{d\phi (x)}{dx},E(x)=\phi (x)$, and $F(x)=\phi (x){\prod }_{i=}^{\upsilon }{(x-d_i)}^{l_i},$ which indicates the axial function that satisfies the end point conditions. In the longitudinal direction at $x=d_i$, $i^{th}$ ring support is present. Here $l_i=1$ and $l_i=0$ represent with and without a ring support, respectively. m is the circumferential wave number and ω is the angular vibration frequency. And n denotes the axial model dependence in the longitudinal, circumferential, and transverse directions, respectively.

• $\phi \left(x\right)$ symbolizes the axial function that meets the geometric edge condition requirement and is defined as

  $$\phi \left(x\right)= {\beta }_1\cosh \left({\mu }_{m}x\right)+{\beta }_2\cos \left({\mu }_{m}x\right)-{\sigma }_m{\beta }_3\sin \mathrm{h}\left({\mu }_{m}x\right)+{\sigma }_m{\beta }_4\sin \left({\mu }_{m}x\right)$$

  (14)

${\mu }_m$ represents the roots of some transcendental equations,${\sigma }_m$ parameters depend on the values of ${\mu }_m$, and ${\beta }_i (i=1,2,3,4)$ change with respect to the edge conditions.

The geometric boundary conditions, namely simply supported and clamped boundary conditions, can be expressed mathematically in terms of the characteristic beam function $\phi \left(x\right)$ as follows:

Simply supported boundary conditions $\phi \left(x\right)=\phi {\left(x\right)}^{″}=0$.

Clamped boundary conditions $\phi \left(x\right)=\phi {\left(x\right)}^{\prime }=0$.

The following non-dimensional parameters are used to expand this problem.

$$\begin{gathered} \underset{⌢}{\mathrm{U}}=\frac{D(x)}{H},\underset{⌢}{\mathrm{V}}=\frac{E(x)}{H},\underset{⌢}{\mathrm{W}}=\frac{F(x)}{R} \\ {a}_{ij}=\frac{{\xi }_{ij}}{H},b_{ij}=\frac{{\psi }_{ij}}{H^2},d_{ij}=\frac{{\zeta }_{ij}}{H^3} \\ a=R/L,b=H/R,X=x/L,\underset{⌢}{{\rho }_t}={\rho }_t/H \end{gathered}$$

(15)

Expression (13) alters the structure as follows:

$$\left.\begin{array}{l}{u}_1(x,\theta ,t)=H{x}_n\underset{⌢}{U}\cos (m\theta )\sin (\varpi t)\\ u_2(x,\theta ,t)=H{y}_n\underset{⌢}{V}\sin (m\theta )\cos (\varpi t)\\ u_3(x,\theta ,t)=R{z}_n\underset{⌢}{W}\cos (m\theta )\sin (\varpi t)\end{array}\right\}$$

(16)

The Lagrangian function ${\mathrm{\pounds }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is transformed by employing the principle of maximum energy.

$$\left.\begin{array}{l}{\Gamma }_{\max }=\frac{\pi HLR}{2}\left[R^2{\varpi }^2\underset{⌢}{{\rho }_t}{\displaystyle {\int }_0^1\left(b^2{\left(x_n\underset{⌢}{U}\right)}^2+b^2{\left(y_n\underset{⌢}{V}\right)}^2+b^2{\left(z_n\underset{⌢}{W}\right)}^2\right)dX-}\right.\\ {\displaystyle \underset{0}{\overset{1}{\int }}\left\{a^2{b}^2{\underset{⌢}{a}}_{11}{\left(x_n\frac{d\underset{⌢}{U}}{dX}\right)}^2\right.}+{\underset{⌢}{a}}_{22}{\left(-mb{y}_n\underset{⌢}{V}+z_n\underset{⌢}{W}\right)}^2-2a^3{b}^2{\underset{⌢}{b}}_{11}\left(x_n\frac{d\underset{⌢}{U}}{dX}\right)\left(z_n\frac{d^2\underset{⌢}{W}}{d{X}^2}\right)\\ +2ab{\underset{⌢}{a}}_{12}\left(x_n\frac{d\underset{⌢}{U}}{dX}\right)\left(-mb{y}_n\underset{⌢}{V}+z_n\underset{⌢}{W}\right)+{\underset{⌢}{a}}_{66}{\left(ab{y}_n\frac{d\underset{⌢}{V}}{dX}mb{x}_n\underset{⌢}{U}\right)}^2\\ -2a{b}^2{\underset{⌢}{b}}_{12}\left(x_n\frac{d\underset{⌢}{U}}{dX}\right)\left(-m^2{z}_n\underset{⌢}{W}+mb{y}_n\underset{⌢}{V}\right)-2a^{2}b{\underset{⌢}{b}}_{12}\left(-mb{y}_n\underset{⌢}{V}+z_n\underset{⌢}{W}\right)\left(z_n^2\frac{d^2\underset{⌢}{W}}{d{X}^2}\right)\\ -2b{\underset{⌢}{b}}_{22}\left(-mb{y}_n\underset{⌢}{V}+z_n\underset{⌢}{W}\right)\left(-m^2{z}_n\underset{⌢}{W}+mb{y}_n\underset{⌢}{V}\right)+a^4{b}^2{\underset{⌢}{d}}_{11}{\left(z_n^2\frac{d^2\underset{⌢}{W}}{d{X}^2}\right)}^2\\ -4b{\underset{⌢}{b}}_{66}\left(ab{y}_n\frac{d\underset{⌢}{V}}{dX}+mb{x}_n\underset{⌢}{U}\right)\left(na{z}_n\frac{d\underset{⌢}{W}}{dX}-\frac{3ab{y}_n}{4}\frac{d\underset{⌢}{V}}{dX}+\frac{mb}{4}{x}_n\underset{⌢}{U}\right)\\ +b^2{\underset{⌢}{d}}_{22}{\left(-m^2{z}_n\underset{⌢}{W}+mb{y}_n\underset{⌢}{V}\right)}^2+2a^2{b}^2{\underset{⌢}{d}}_{12}\left(z_n^2\frac{d^2\underset{⌢}{W}}{d{X}^2}\right)\left(-m^2{z}_n\underset{⌢}{W}+mb{y}_n\underset{⌢}{V}\right)\\ +4{\underset{⌢}{d}}_{66}\left.\left.{\left(ma{z}_n\frac{d\underset{⌢}{W}}{dX}-\frac{3ab{y}_n}{4}\frac{d\underset{⌢}{V}}{dX}+\frac{mb}{4}{x}_n\underset{⌢}{U}\right)}^2\right\}\right]dX\end{array}\right\}$$

(17)

Lagrangian energy functional ${\Gamma }_{\max }$ is extremized with respect to vibrational amplitudes $x_n,y_n$, and $z_n$, as shown below:

$$\frac{\partial {\Gamma }_{\max }}{\partial x_n}=\frac{\partial {\Gamma }_{\max }}{\partial y_n}=\frac{\partial {\Gamma }_{\max }}{\partial z_n}=0$$

(18)

The resulting relation is defined in the form of a matrix as

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

(19)

where

$${\Omega }^2=R^2{\omega }^2{\sout{\rho }}_t,$$

(20)

$\left[K\right]$ and $\left[M\right]$ are the stiffness and mass matrices corresponding to the cylindrical shell.

3. Classification of Material

In the current study, a CS is taken into consideration, which is made up of four layers: isotropic material is used to construct the internal layers, while nickel, zirconia, and stainless steel are used to construct the outermost layers(see Figure 2).

The two elements using the trigonometric volume fraction law (VFL) are as follows:

$$V_{f1}={\sin }^2\left[{\left\{\frac{z-H_1}{H_2-H_1}\right\}}^N\right],V_{f2}={\cos }^2\left[{\left\{\frac{z-H_1}{H_2-H_1}\right\}}^N\right],0\le N\le \infty$$

This relation fulfills the VFL, i.e., $V_{f1}+V_{f2}=1$, where $h$ is the shell’s thickness and $N$ indicates the power law exponent. Every layer is assumed to be the same thickness $h/4$. The material parameters are as follows: $E_1,{\lambda }_1,{\rho }_1$ for nickel; $E_2,{\lambda }_2,{\rho }_2$ for zirconia; $E_3,{\lambda }_3,{\rho }_3$ for stainless steel; $E_4,{\lambda }_4,{\rho }_4$ for aluminum; $E_5,{\lambda }_5,{\rho }_5$ for stainless steel and zirconia. Then, the effective material quantities are as follows:

$$\begin{gathered} E_{fgm1}=\left[E_1-E_2\right]{\sin }^2\left[{\left\{\frac{z-H_1}{H_2-H_1}\right\}}^N\right]+E_2 \\ {\lambda }_{fgm1}=\left[{\lambda }_1-{\lambda }_2\right]{\sin }^2\left[{\left\{\frac{z-H_1}{H_2-H_1}\right\}}^N\right]+{\lambda }_2 \\ {\rho }_{fgm1}=\left[{\rho }_1-{\rho }_2\right]{\sin }^2\left[{\left\{\frac{z-H_1}{H_2-H_1}\right\}}^N\right]+{\rho }_2 \\ {E}_{fgm2}=\left[E_4-E_5\right]{\sin }^2\left[{\left\{\frac{z-H_3}{H_4-H_3}\right\}}^N\right]+E_5 \\ {\lambda }_{fgm2}=\left[{\lambda }_4-{\lambda }_5\right]{\sin }^2\left[{\left\{\frac{z-H_3}{H_4-H_3}\right\}}^N\right]+{\lambda }_5 \\ {\rho }_{fgm2}=\left[{\rho }_4-{\rho }_5\right]{\sin }^2\left[{\left\{\frac{z-H_3}{H_4-H_3}\right\}}^N\right]+{\rho }_5 \end{gathered}$$

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

$E_{fgm1}=E_2,{\lambda }_{fgm1}={\lambda }_2,{\rho }_{fgm1}={\rho }_2$ and

$E_{fgm2}=E_5,{\lambda }_{fgm2}={\lambda }_5,{\rho }_{fgm2}={\rho }_5$, respectively.

The stiffness moduli are altered as follows:

$$\begin{gathered} {\xi }_{ij}={\xi }_{ij}\left(FGM\right)+{\xi }_{ij}\left(iso\right)+{\xi }_{ij}\left(iso\right)+{\xi }_{ij}\left(FGM\right), \\ {\psi }_{ij}={\psi }_{ij}\left(FGM\right)+{\psi }_{ij}\left(iso\right)+{\psi }_{ij}\left(iso\right)+{\psi }_{ij}\left(FGM\right), \\ {\zeta }_{ij}={\zeta }_{ij}\left(FGM\right)+{\zeta }_{ij}\left(iso\right)+{\zeta }_{ij}\left(iso\right)+{\zeta }_{ij}\left(FGM\right) \end{gathered}$$

where $i.j=1,2,6,$ and iso signify the two internal isotropic layers, and FGM symbolizes the two outer FGM layers.

4. Results and Discussion

To explain the accuracy and knowledge of the exact outcome, the current outcome is used by FGM and isotropic CS for SS edge conditions. RRM is used to achieve the current result.

Table 1. Comparison of frequency parameters for simply supported isotropic cylindrical shell (n = 1, L = 20, R = 1, H = 0.01, and λ = 0.3).

m	[4]	Present	Difference
1	0.016101	0.016100	0.006%
2	0.009382	0.009376	0.06%
3	0.022105	0.022101	0.02%
4	0.042095	0.042092	0.007%
5	0.068008	0.068006	0.009%

represents the comparison of frequency parameters for simply supported isotropic cylindrical shells (n = 1, L = 20, R = 1, H = 0.01, and λ = 0.3) with those in [4]. Three materials are used to make two types of FG materials known as FGM cylindrical layers: stainless steel, zirconia, and nickel. Two FGM variations for the form of substance are found.

Table 2. Configurations of shell types.

Types of Shell	1st Layer FGM	2nd Layer Isotropic	3rd Layer Isotropic	4th Layer FGM
Type 1	Nickel, Zirconia	Stainless Steel	Aluminum	Stainless Steel, Zirconia
Type 2	Zirconia, Nickel	Stainless Steel	Aluminum	Zirconia, Stainless Steel

represents the material distribution.

• Type 1:

In type 1 FGM CS, inner surfaces are made of nickel and zirconia; on the other hand, outer layers are composed of stainless steel and zirconia.

• Type 2:

In type 2 FGM CS, inner surfaces are made with zirconia and nickel, and the outer layers are composed of zirconia and stainless steel.

In

Table 3. Vibration of NFs (Hz) against n for FGM CS under SS-SS boundary conditions when (n = 1, L = 20, H = 0.02, R = 1, and d = 0.1).

m	N = 1	N = 2	N = 5	N = 7	N = 9
1	366.3000	364.0547	363.0547	363.0547	363.0547
2	456.1014	454.0393	452.3383	452.3383	452.3383
3	477.2445	475.0875	473.3106	473.3106	473.3106
4	485.2193	483.0266	481.2239	481.2239	481.2239
5	489.6059	487.3932	485.5785	485.5785	485.5785
6	493.1564	490.9264	489.1031	489.1031	489.1031
7	497.0930	494.8423	493.0085	493.0085	493.0085
8	502.1957	499.9165	498.0668	498.0668	498.0668
9	509.0963	506.7773	504.9033	504.9033	504.9033
10	518.3668	515.9934	514.0839	514.0839	514.0839

, the variation in natural frequencies (NFs) is presented under SS to SS edge conditions observed with (n = 1, R = 1, d = 0.01, H = 0.002, and L = 20). As the value of circumferential wave number m increases from 1 to 10, the value of NFs decreases. For power law exponent ranging from N = 1 to N = 9, the values of NFs also decrease, but slowly. For values ranging from N = 1 to 20, the values change from 366.3000 to 363.0547.

In

Table 4. Vibration of NFs (Hz) against n for FGM CS under SS-SS boundary conditions when (n = 2, L = 20, H = 0.02, R = 1, and d = 0.1).

m	N = 1	N = 3	N = 5	N = 7	N = 9	N = 11
1	362.6197	359.9578	359.2090	359.2090	359.2090	359.2090
2	455.8980	452.9469	452.1372	452.1372	452.1372	452.1372
3	477.2085	474.1213	473.2755	473.2755	473.2755	473.2755
4	485.2101	482.0728	481.2149	481.2149	481.2149	481.2149
5	489.6049	486.4406	485.5777	485.5777	485.5777	485.5777
6	493.1596	489.9726	489.1063	489.1063	489.1063	489.1063
7	497.0994	493.8851	493.0148	493.0148	493.0148	493.0148
8	502.2050	498.9528	498.0760	498.0760	498.0760	498.0760
9	509.1086	505.8025	504.9154	504.9154	504.9154	504.9154
10	518.3821	515.0016	514.0989	514.0989	514.0989	514.0989

, the variation in NFs for length-to-radius ratios is presented against the circumference wave number m for shell I with parameters (n = 2, R = 1, d = 0.01, H = 0.002, and L = 20) under the S-S Edge condition. As the value of m increases from 1 to 10, the values of NFs change, and for N = 1 to 11, the values of NFs change from 362.6197 to 359.2090.

In

Table 5. Behavior of NFs (Hz) against n for FGM CS under SS-SS boundary conditions when (n = 1, L = 20, N = 1, R = 1, and d = 0.4).

m	H = 0.001	H = 0.005	H = 0.007	H = 0.01	H = 0.05	H = 0.07
1	4493.442	4493.443	4493.444	4493.445	4493.46	4493.469
2	785.8470	785.7706	785.7388	785.6991	786.0900	786.9266
3	794.0083	793.8696	793.8343	793.8241	798.5644	804.2985
4	796.3821	796.2296	796.2626	796.4488	814.3950	833.7276
5	797.3917	797.3464	797.5922	798.2960	844.7577	891.3993
6	797.9048	798.1816	798.8778	800.6169	897.6645	988.5901
7	798.1968	799.1243	800.6216	804.1492	979.4344	1131.8350
8	798.3810	800.4223	803.2025	809.5400	979.4344	1131.8350
9	798.5141	802.2857	806.9778	817.4377	1242.7268	1560.6494
10	798.6292	804.9192	812.3117	828.5070	1425.3211	1842.6299

, the variation is the same as in Table 3 with parameters (n = 3, R = 1, d = 0.03, H = 0.002, and L = 90).

Table 6. Behavior of NFs (Hz) when (n = 1, L = 20, N = 5, R = 1, and d = 0.5).

m	H = 0.001	H = 0.005	H = 0.007	H = 0.01	H = 0.05	H = 0.07
1	4313.561	4314.657	4314.658	4314.659	4314.673	4314.682
2	779.3320	838.7948	838.7603	838.7167	839.0724	839.9035
3	787.4543	838.7012	838.6600	838.6395	843.0766	848.5493
4	789.8100	838.6410	838.6587	838.8150	855.6305	873.9499
5	790.8129	838.6853	838.8951	839.5266	883.1276	927.3678
6	791.3239	838.9250	839.5478	841.1367	932.4724	1019.1568
7	791.6158	839.4714	840.8337	844.0830	1009.7367	1156.0430
8	791.8012	840.4554	843.0066	848.8724	1009.7367	1156.0430
9	791.9362	842.0268	846.3550	856.0697	1262.0934	1571.3251
10	792.0536	844.3534	851.1977	866.2819	1438.9259	1846.7592

represents the variation in NFs for length-to-radius ratios against m for shell I with parameters (n = 4, R = 1, d = 0.04, H = 0.02, and L = 20) under the S-S Edge condition. NFs increase up to 10. The value of NFs decreases from 4755.773 to 4575.089.

Configurations of shells:

The composition of the four-layered cylindrical shell made by FGM and the isotropic type of material is defined below in Table 2.

Table 7. Behavior of NFs (Hz) against n for FGM cylindrical shell when (n = 4, H = 0.005, N = 1, R = 1, and d = 1).

m	L = 10	L = 20	L = 30	L = 40	L = 50	L = 70
1	461.9686	329.8816	306.7	302.7992	302.8946	304.7281
2	420.4381	387.3319	385.3606	385.4567	385.6814	385.9822
3	415.7960	406.0455	405.6847	405.7264	405.7787	405.8421
4	417.1420	413.4404	413.3143	413.3238	413.3382	413.3561
5	419.1609	417.3701	417.2904	417.2846	417.2859	417.2891
6	421.3530	420.2377	420.1592	420.1424	420.1365	420.1324
7	423.9559	423.0649	422.9713	422.9443	422.9328	422.9233
8	427.3108	426.4492	426.3323	426.2948	426.2780	426.2637
9	431.7691	430.8393	430.6941	430.6454	430.6232	430.6041
10	437.6788	436.6267	436.4497	436.3891	436.3612	436.3372

describes the variation in NFs for length-to-radius ratios against m for shell I with parameters (n = 1, d = 0.4, R = 1, N = 5, and L = 20) under the S-S edge condition, and it was observed that NFs change from 4493.442 to 798.6292 for m.

Table 8. Variation in NFs (Hz) against n for FGM cylindrical shell under the SS-SS boundary condition (n = 1, R = 1, H = 0.002, L = 20, and d = 0.1).

m	N = 1	N = 2	N = 3	N = 4	N = 5	N = 10
1	356.7561	353.1512	351.5667	350.1086	348.8782	348.4389
2	451.3025	447.1279	445.3168	443.6651	442.2836	441.7932
3	472.5032	468.1340	466.2384	464.5095	463.0634	462.5501
4	480.1950	475.7544	473.8275	472.0697	470.5992	470.0773
5	483.8605	479.3859	477.4437	475.6717	474.1890	473.6626
6	485.9757	481.4816	479.5303	477.7496	476.2593	475.7301
7	487.4275	482.9202	480.9625	479.1754	477.6793	477.1480
8	488.6148	484.0970	482.1339	480.3413	478.8402	478.3069
9	489.7555	485.2281	483.2599	481.4619	479.9556	479.4204
10	490.9919	486.4545	484.4807	482.6771	481.1654	480.6281

presents the variation in NFs for length-to-radius ratios against m for the shell I with parameters (n = 1, R = 1, N = 5, and L = 20). It was observed and noted that the values of NFs change from 4313.561 to 792.0536 for m = 1 to m = 10. And for exponential law H = 0.001 to H = 0.07, the values change from 4313.561 to 4314.682.

Table 9. Variation in NFs (Hz) against n for FGM cylindrical shell under SS-SS when (n = 2, R = 1, H = 0.002, L = 20, and d = 0.2).

m	N = 1	N = 2	N = 3	N = 4	N = 5	N = 10
1	419.8248	413.2517	410.2073	407.3157	404.8068	403.8956
2	542.1721	537.1515	534.9731	532.9863	531.3244	530.7345
3	563.7859	558.5723	556.3103	554.2473	552.5216	551.9091
4	571.2834	566.0004	563.7079	561.6167	559.8672	559.2462
5	574.7881	569.4725	567.1653	565.0602	563.2989	562.6735
6	576.7816	571.4475	569.1315	567.0180	565.2491	564.6210
7	578.1269	572.7805	570.4583	568.3385	566.5639	565.9337
8	579.2045	573.8485	571.5211	569.3958	567.6161	566.9839
9	580.2178	574.8530	572.5207	570.3902	568.6054	567.9712
10	581.2970	575.9233	573.5858	571.4497	569.6594	569.0231

shows the variation in NFs for length-to-radius ratios against m for the shell I with parameters (n = 1, R = 1, N = 10, d = 0.06, and L = 20) under the S-S Edge condition, and it is noted that NFs vary from 4280.537 to 4280.562.

Table 10. Variation in NFs (Hz) against n for FGM cylindrical shell under the SS-SS boundary condition (N = 1, R = 1, n = 1, H = 0.005, and d = 0.8).

m	L = 10	L = 20	L = 40	L = 50	L = 70
1	413.1000	419.8248	428.9463	431.2599	434.1054
2	540.9617	542.1822	542.5790	542.6293	542.6737
3	563.6582	563.8588	563.9190	563.9265	563.9331
4	571.4879	571.5395	571.5550	571.5570	571.5587
5	575.4332	575.4426	575.4459	575.4463	575.4467
6	578.1811	578.1709	578.1688	578.1686	578.1683
7	580.7581	580.7346	580.7289	580.7283	580.7277
8	583.7200	583.6847	583.6760	583.6750	583.6741
9	587.4637	587.4167	587.4051	587.4037	587.4025
10	592.3291	592.2701	592.2554	592.2536	592.2521

shows NFs for length-to-radius ratios against m for shell I with parameters (n = 1, R = 1, d = 0.7, N = 20, and L = 20). It is observed that its variation for m = 1 to m = 10 changes from 425.7616 to 425,7638. It has been noticed that a very small change occurs throughout the region.

Table 11. Variation in NFs (Hz) against n for FGM CS under SS-SS boundary condition (N = 10, R = 1, n = 1, H = 0.005, and d = 0.9).

m	L = 10	L = 20	L = 40	L = 50	L = 70
1	343.7526	348.8778	354.1291	355.0526	355.9365
2	441.4266	442.2936	442.6029	442.6428	442.6781
3	462.9772	463.1435	463.1979	463.2047	463.2108
4	470.8447	470.8886	470.9031	470.9049	470.9065
5	474.9309	474.9370	474.9399	474.9403	474.9406
6	477.8692	477.8566	477.8541	477.8538	477.8536
7	480.7127	480.6867	480.6805	480.6798	480.6791
8	484.0531	484.0148	484.0055	484.0043	484.0034
9	488.3237	488.2733	488.2608	488.2593	488.2580
10	493.8992	493.8362	493.8205	493.8186	493.8170

shows NFs for length-to-radius ratios against circumference wave number n for shell I with parameters (n = 1, R = 1, N = 1, H = 0.005, and d = 0.8) under the S-S Edge condition by changing the value of H. The value of NFs changes from 418.758 to 594.9993 as n increases from 1 to 10. As the thickness of the shell varies, NFs remain nearly the same for different thicknesses from 0.001 to 0.07. It is noticed that for small changes in thickness, the values of NFs remain the same. Another noticeable variation for L is that the value of NFs increases when the range is from L = 10 to L = 70, increasing from 418.758 to 439.2311, indicating an increase in the frequency value.

Figure 3 represents the variation in natural frequencies against circumferential wave number m for type 1 shell. The lower line shows the behavior of NFs for N = 1 and the upper line presents the behavior of NFs for N = 2. Natural frequencies increased swiftly from m = 1 to 2 and then increased slowly. Figure 4 represents the same for N = 1 and N = 3 under simply supported end point conditions for type 2 shell.

Figure 5 presents the behavior of NFs against m for type 1 four-layered cylinder-shaped shell. The natural frequencies are obtained under simply supported edge conditions. Natural frequencies rapidly decreased from m = 1 to 2 and then slowly increased for m = 3, after which they behaved smoothly. The results are acquired for thicknesses 0.001 and 0.005 of type 1 shell. In Figure 6, the same behavior of NFs is observed for thicknesses 0.001, 0.005, and 0.007 of type 2 shell.

Figure 7 represents the variation in NFs against m of type 2 SS-SS four-layer CS with n = 2, d = 0.9, N = 1, R = 1, and H = 0.005. It is noticed that NFs are very close for L = 10, 20, and 40. The results increased rapidly from m = 1 to 2 and then increased slowly with increasing m.

Figure 8 shows the vibrations of NFs (Hz) for a four-layered clamped–simply supported circular cylinder-shaped shell against the position of the ring support with different H/R ratios $(m=10,n=1,L/R=20,H/R=0.001,0.01,0.05)$.

Figure 8 describes the vibrations of NFs (Hz) for a four-layered free circular cylinder-shaped shell against the position of the ring support with different H/R ratios $(n=10,m=N=1,L=20,H/R=0.001,0.01,0.05)$. The extreme lower line is obtained for H/R = 0.001, the extreme upper line is obtained for H/R = 0.01, and the middle line represents H/R = 0.05. It is clearly seen that NFs behave symmetrically as they increase from d = 0 to 0.5, attain their maximum value at d = 0.5, and then decrease from d = 0.05 to d = 1.

5. Discussion

The study presented in this work delves into the intricate vibration behavior and natural frequencies of layered cylindrical structures composed of functionally graded and isotropic materials. By investigating the vibrations of four-layered cylindrical shells assisted by a ring over their length, this research aims to uncover valuable insights into the dynamics of such composite structures. One of the distinctive features of this study is the composition of the layered cylindrical shells. The internal two layers are constructed by using isotropic materials, specifically aluminum and stainless steel, while the external two layers consist of functionally graded materials (FGM), namely stainless steel, zirconia, and nickel. This combination offers a diverse range of mechanical properties, making it an intriguing subject for exploration. The research involves a cylindrical shell with each of its four layers having an equal thickness, denoted as h/4.

To effectively analyze the vibration behavior and natural frequencies of these layered cylindrical shells, the Rayleigh–Ritz method (RRM) is employed within the framework of Sander’s shell theory. This method allows for the derivation of the shell frequency equation, providing valuable insights into the system’s dynamic response. Furthermore, the research investigates the natural frequencies of the shells under two distinct boundary conditions: simply supported–simply supported (SS-SS) and clamped–clamped (C-C). By exploring these boundary conditions, the study aims to provide a thorough understanding of how different support configurations impact the vibrational characteristics of the cylindrical shells.

In summary, this research contributes significantly to the understanding of vibration behavior and natural frequencies in layered cylindrical structures composed of functionally graded and isotropic materials. By leveraging advanced analytical techniques and exploring various material compositions and boundary conditions, the study sheds light on the complex dynamics of such composite systems, with potential implications for a wide range of engineering applications.

6. Conclusions

The current study was conducted to check the vibration characteristics of thin circular CS formed from FGM. The FG cylindrical shell constituents are made of stainless steel, nickel, zirconia, and aluminum. These four types were studied by changing the configuration of the layers of FG material. These constituents in type 1 of the cylindrical shells vary across the shell’s thickness constantly, seamlessly, and steadily from the FG layer’s inner to its outermost layer. The basic objectives were to obtain shell frequency equations with the help of these theories and techniques. Sander’s shell theory was applied to solve the shell frequency equations based on Kirchhoff’s assumptions. RRM was used to solve the shell vibration problems by utilizing the Lagrangian energy relation to obtain the shell’s frequency equation in numerical form. Axial modal dependence was approximated by trigonometric functions for simply supported end conditions and characteristic beam functions for number-clamped boundary conditions.

The study found that the minimum frequency occurs at specific wave numbers for all boundary conditions and is influenced directly by increasing the value of N, L, n, d, H, and L/R. The NFs decreased by varying N under different conditions. All the results were very close to each other. This analysis may be extended for the study of different shell problems, such as changing parameters like the thickness or length of the shell and the position of the ring. The frequency curves of the shell with ring support at different positions obtain symmetric shapes because of the same edge conditions. They are not symmetrical around the center because of different end point conditions. The induction of ring support on a cylinder-shaped shell has a significant effect on the NFs compared to the shell frequencies without ring support.

For the presented cylindrical shell, the results were obtained using MATLAB software version 2019 under simply supported–simply supported and clamped–clamped boundary conditions. These results illustrate the natural frequency behavior as a function of the circumferential wave number. Significant findings are as follows: The natural frequency increases as the power law index N″ increases. An increase in the length L″ of the shell leads to an increase in the natural frequency. Increasing the shell thickness h″ results in higher natural frequencies. Varying the ring support position along the longitudinal path shows that the natural frequency increases, reaching its maximum when the ring is at the central position of the shell, and then decreases as the ring position moves beyond the center.

These findings highlight the critical influence of geometric and material parameters on the vibrational behavior of cylindrical shells, providing insights for optimizing design and performance.