The free vibration analysis of ring-stiffened FG conical sandwich shells with auxetic and non-auxetic honeycomb cores

Abstract

In this paper, the natural frequencies of a ring-stiffened conical sandwich shell with a functionally graded (FG) honeycomb core are calculated. The core can be either a non-auxetic honeycomb (NAH) or an auxetic honeycomb (AH). The honeycomb core and face layers are fabricated from metal-ceramic functionally graded material (FGM) in which the volume fraction of the ceramic increases from zero at the inner surface of the shell to one at its outer surface based on either power-law function (P-FGM), sigmoid function (S-FGM), or exponential function (E-FGM). The sandwich shell is modeled based on Murakami’s zig-zag theory and the intermediate ring support is modeled as a rigid structure. Hamilton’s principle is employed to derive the governing equations, compatibility conditions, and boundary conditions. A semi-analytical solution is presented which includes an exact solution in the circumferential direction followed by an approximate numerical solution via the differential quadrature method (DQM) in the meridional direction. It is concluded that for each vibrational mode, optimal values can be found for the inclined angle, core-to-shell thickness ratio, and location of the ring support which bring about the highest natural frequency. The presented work is the first theoretical work associated with the free vibration analysis of a ring-stiffened FG conical sandwich shell with a honeycomb core which provides a reduction in the mass and improvement in the thermo-mechanical characteristics, simultaneously. Such properties make such a structure widely used in the aerospace industry. Considering the importance of the dynamic characteristics of aerospace structures, the results of this research can be used by these industries in the analysis and optimal design of the main body of airplanes and missiles.

Keywords

Free vibration conical shell sandwich shell auxetic honeycomb zig-zag theory

Introduction

A structure that benefits from low density and high capacity in bearing shear loading can be chosen as an excellent candidate to be utilized as a core in a three-layered sandwich structure. As one of the most popular and efficient ones, the honeycomb structure can play this role. These orthotropic artificial structures benefit from low density, remarkable indentation resistance, high fracture toughness, and extreme energy absorption capability. Due to these useful characteristics, sandwich structures with honeycomb structures have been utilized in various and different engineering fields including aerospace, mechanical, transportation, and civil engineering.

There are considerable types of honeycomb structures such as hexagonal, re-entrant, chiral, hexachiral, arrow-head, double-arrow-head, etc. These structures can be classified into two groups including those with positive Poisson’s ratios (non-auxetic) and those with negative Poisson’s ratios (auxetic). The most well-known types of non-auxetic and AH structures are hexagonal ones and re-entrant ones, respectively. Since honeycomb structures are orthotropic, their Poisson’s ratios are not limited to the well-known range 0 < ν < 0.5. Apart from the works which have proposed relations to calculate the density and elastic constants of honeycomb structures, a considerable number of papers have been presented in recent years to investigate the effects of utilizing honeycomb cores on the mechanical behavior of sandwich structures. Benjeddou and Guerich¹ examined the free vibration analysis of a sandwich panel with an NAH core. They compared the results predicted via FEM (finite element method)-based software with the experimental tests. Torabi et al.² inspected the aeroelastic stability and free vibration analyses of a cantilevered trapezoidal sandwich plate with an NAH core and two isotropic homogenous face layers. They inspected the effects of the geometric factors of the cells and the thickness of the core on the natural frequencies and the critical aerodynamic pressure. An experimental work was presented by Usta et al.³ to study the low-velocity impact response of sandwich plates with various types of honeycomb cores including re-entrant, hexagonal, arrowhead, and hexachiral. They presented a comparison between the impact resistance of the sandwich plates with honeycomb cores of the above-mentioned types to find the best one. Gupta and Pradyumna⁴ examined the nonlinear vibration analysis of cylindrical, spherical, and hyperboloid panels with AH cores and curvilinear fiber-reinforced face layers. They inspected the influences of the geometric factors of the honeycomb core on the dynamic response and natural frequencies of the panels. The stress analysis of a spinning cylindrical sandwich shell with an AH core and polymeric face layers enriched with graphene nanoplatelets (GNPs) in a hygrothermal environment was examined by Allam et al.⁵ They prepared a parametric examination to investigate the effects of the core-to-face thickness ratio and mass fraction of the GNPs on the bending response of the shell.

Li and Fu⁶ studied the free vibration analysis of a cylindrical sandwich shell with an AH core and two isotropic homogeneous face layers. They inspected the dependency of the natural frequencies on the geometric parameters of the AH core. An analytical exact solution was provided by Li et al.⁷ for the vibration analysis of a combined cylindrical-spherical sandwich shell with an NAH core and laminated composite face layers. They inspected the effects of the NAH core on the vibration suppression capability of the shell. An optimal design was presented by Nasrollahpoor Shirvani et al.⁸ for a sandwich panel with an NAH core and laminated composite face layers exposed to uniform out-of-plane pressure. They tried to minimize the mass of the panel by considering some constraints on the maximum deflection, the shear and buckling resistance of the core, and the yield of the face layers. Ha et al.⁹ inspected the nonlinear chaotic behavior of a cylindrical sandwich panel with an AH core in a thermal environment. They tried to investigate the effects of geometric factors of the AH core on the chaotic behavior of the panel. Chu et al.¹⁰ inspected the free vibration analysis of a cylindrical sandwich shell with an AH core and FGM face layers partially filled with liquid. They studied the influences of the geometric factors of the AH core on the natural frequencies of the shell. Phuong et al.¹¹ inspected the nonlinear buckling behavior of concave and convex toroidal sandwich shells with AH cores and GNP-reinforced face layers under radial loading. They investigated the dependencies of the post-buckling curves on the mass fraction of the GNPs and the geometric factors of the AH core. The dynamic response of a cylindrical sandwich shell with an AH core subjected to axial compression and transverse excitation loads was studied by Ebrahimi and Dadashi.¹² They studied the effects of several parameters on the dynamic response of the shell. Abasi et al.¹³ examined the aeroelastic stability analysis of a ring-stiffened conical sandwich shell with an FG AH core and FG face sheets. They examined the relationships between the critical aerodynamic pressure of the shell and several parameters such as the power-law index and location of the ring. Jafari and Azhari¹⁴ inspected the free vibration and static bending analyses of cylindrical sandwich panels of rectangular, skew, and trapezoidal planforms with an NAH core. The dependencies of the deflection and natural frequencies of the panel on the thickness of the NAH core were investigated by them.

Dang et al.¹⁵ inspected the nonlinear dynamic response of a sandwich plate with an AH core and FGM face layers. For the blast loading, they examined the dependency of the dynamic response of the plate on the geometric factors of the AH core. Amirabadi et al.¹⁶ studied the free vibration analysis of a conical sandwich shell with an FG AH core and isotropic homogenous face layers. They investigated the effects of the material gradation and the geometric factors of the AH core on the natural frequencies. Quan et al.¹⁷ inspected the free vibration analysis of a sandwich plate with an AH core and polymeric face layers enriched with carbon nanotubes (CNTs). They studied the effects of the mass fraction of the CNTs and the geometric parameters of the AH core on the natural frequencies. The low-velocity impact response of a sandwich plate with an AH core was inspected by Biglari et al.¹⁸ They tried to discover the effect of non-auxetic and auxetic cores in improving the low-velocity impact response of the plate and reducing the dynamic deflection created by the impact. Ghovehoud et al.¹⁹ examined the dynamic instability characteristics of a sandwich plate with an AH core and three-phase polymeric face layers enriched with fibers and GNPs. They tried to find out how the geometric factors of the AH core affect the dynamic instability behavior of the plate. Thang et al.²⁰ studied the free vibration behavior of a barrel-shaped sandwich shell with AH core and FGM face layers. The influences of material gradation and curvature radii on the natural frequencies were inspected by them. Amirabadi et al.²¹ studied the aeroelastic stability analysis of a cylindrical sandwich shell with an FG AH core covered with polymeric face layers enriched with either CNTs, GNPs, or graphene oxide powders (GOPs). The effects of the mass fraction of the nanofillers and material gradation and geometric parameters of the cells in the FG AH core on the critical aerodynamic pressure of the shell were examined by them.

Utilizing a honeycomb core significantly reduces the weight of sandwich structures. However, due to their low elastic and shear moduli, sandwich structures with honeycomb cores may be vulnerable to external loads, especially external pressure. To compensate for this weakness, the shell can be strengthened with an intermediate ring support. According to the best knowledge of the authors, there is no theoretical research associated with the vibration analysis of ring-stiffened conical sandwich shells fabricated from FG materials. To fill this research gap, the free vibration analysis of a ring-stiffened conical sandwich shell with an auxetic or non-auxetic FG honeycomb core and FG face layers is investigated in this paper for the first time. The influences of various factors on the natural frequencies are examined. The results of this work are useful in the design, analysis, and optimization of future aerospace structures.

Modeling

The mathematical modeling of the shell is presented in this section to attain the governing equations, compatibility conditions, and boundary conditions. In order to avoid mathematical complexities, the following assumptions and limitations are considered:

- The vibrational amplitude is relatively smaller than the dimensions of the shell. Thus, the linear strain-displacement relations can be utilized.

- There is no slip or separation between the layers of the sandwich shell.

- The intermediate ring support can be considered as a rigid structure compared to the shell.

Description of the investigated problem

As shown in Figure 1, a conical sandwich shell of semi-vertex angle λ, length L, thickness h, small radius R₁, and large radius R₂ = R₁ + Lsinλ is considered. The shell is reinforced with an intermediate ring of radius R_r = R₁ + L_rsinλ located at a distance L_r from the small radius. The sandwich shell has three layers including an FG honeycomb core of thickness h_c and two FG face layers of the same thickness h_f = (h−h_c)/2. The honeycomb core can be either a hexagonal one or a re-entrant one, and their geometric factors are described in Figure 2 including the inclined angle β, wall thickness t_c, and dimensions d_c and l_c. The whole shell is fabricated from FGM in which the volume fraction of the ceramic increases from zero at the inner surface of the shell to one at several patterns.

Figure 1.

A ring-stiffened FG conical sandwich shell with a honeycomb core.

Figure 2.

Geometric parameters of the cells.

Material properties

The properties of the material utilized to fabricate the honeycomb core and the material properties of the face layers change from a metal-rich surface (indicated by the subscript me) at the internal radius of the shell to a ceramic-rich surface (indicated by the subscript ce) at the outer radius of the shell according to either P-FGM, S-FGM, or E-FGM. The mechanical properties (P) such as the elastic modulus (E₀), the density (ρ₀), and the Poisson’s ratio (ν₀) of the non-homogeneous material used to fabricate the honeycomb core and the face sheets vary as follows^21–24:

\begin{array}{l} P - FGM : & P_{0} (z) = P_{m e} + (P_{c e} - P_{m e}) {(\frac{1}{2} + \frac{z}{h})}^{p}, \\ S - FGM : & P_{0} (z) = {\begin{cases} P_{m e} + \frac{P_{c e} - P_{m e}}{2} {(1 + \frac{2 z}{h})}^{p}, & - \frac{h}{2} \leq z \leq 0, \\ P_{c e} - \frac{P_{c e} - P_{m e}}{2} {(1 - \frac{2 z}{h})}^{p}, & 0 \leq z \leq \frac{h}{2}, \end{cases} \\ E - FGM : & P_{0} (z) = P_{m e} \exp [{(\frac{1}{2} + \frac{z}{h})}^{p} Ln (\frac{P_{c e}}{P_{m e}})], \end{array}

(1)

in which p is called the material index. Because the material used to fabricate the honeycomb core and the face sheets is an isotropic one, its shear modulus can be presented as follows:

G_{0} = \frac{E_{0}}{2 (1 + ν_{0})} .

(2)

For several values of the FG index, Figure 3 shows the variations of material properties through the thickness direction for P_ce/P_me = 2. As observed, in all cases, the inner surface of the shell (z = −h/2) is metal-rich and the outer one (z = h/2) is ceramic-rich.

Figure 3.

The variations of material properties through the thickness direction.

In order to perform the mathematical modeling of the intermediate ring support, it is considered as a rigid structure and the flexible shell is divided into two parts on two sides of the ring. The governing equations and boundary conditions for the kth part of the shell and the compatibility conditions between these two parts are attained.

The components of the stress (σ_ij) in the lth layer in the kth part of the shell are described in terms of normal and shear components (ε_ij, γ_ij) of the strain tensor as follows²⁵:

\begin{array}{l} {\begin{cases} σ_{x x}^{(l)} \\ σ_{θ θ}^{(l)} \\ σ_{θ z}^{(l)} \\ σ_{x z}^{(l)} \\ σ_{x θ}^{(l)} \end{cases}}^{(k)} = {[\begin{array}{c} C_{11}^{(l)} (z) & C_{12}^{(l)} (z) & 0 & 0 & 0 \\ C_{12}^{(l)} (z) & C_{22}^{(l)} (z) & 0 & 0 & 0 \\ 0 & 0 & C_{44}^{(l)} (z) & 0 & 0 \\ 0 & 0 & 0 & C_{55}^{(l)} (z) & 0 \\ 0 & 0 & 0 & 0 & C_{66}^{(l)} (z) \end{array}]}^{(k)} {\begin{cases} ε_{x x} \\ ε_{θ θ} \\ γ_{θ z} \\ γ_{x z} \\ γ_{x θ} \end{cases}}^{(k)}, & \begin{array}{l} k = 1, 2 \\ l = 1, 2, 3 \end{array} \end{array}

(3)

in which

\begin{array}{l} C_{11}^{(l)} = \frac{E_{11}}{1 - ν_{12} ν_{21}}, & C_{22}^{(l)} = \frac{E_{22}}{1 - ν_{12} ν_{21}}, & C_{12}^{(l)} = ν_{12} C_{22}^{(l)}, & C_{44}^{(l)} = G_{23}, & C_{55}^{(l)} = G_{13}, & C_{66}^{(l)} = G_{12}, \end{array}

(4)

where E_ij, G_ij, and ν_ij show elastic moduli, shear moduli, and Poisson’s ratio. For the FG face layers, the density and the elastic constants can be described as follows:

\begin{array}{l} E_{11} (z) = E_{22} (z) = E_{0} (z), & ν_{12} (z) = ν_{21} (z) = ν_{0} (z), \\ G_{12} (z) = G_{23} (z) = G_{13} (z) = G_{0} (z), & ρ (z) = ρ_{0} (z), \end{array}

(5)

For the FG honeycomb core, the density and the elastic constants can be presented as follows²⁶:

Hexagonal honeycomb core (non-auxetic)²:

\begin{array}{l} \frac{E_{11}}{E_{0}} = \frac{(1 - η_{2}^{2} \cot^{2} β) \cos β}{(η_{1} + \sin β) \sin^{2} β} η_{2}^{3}, & \frac{E_{22}}{E_{0}} = \frac{[1 - η_{2}^{2} (η_{1} \sec^{2} β + \tan^{2} β)] (η_{1} + \sin β)}{\cos^{3} β} η_{2}^{3}, \\ \frac{G_{12}}{E_{0}} = \frac{(η_{1} + \sin β) η_{2}^{3}}{(1 + 2 η_{1}) η_{1}^{2} \cos β}, & \frac{G_{13}}{G_{0}} = \frac{η_{2} \cos β}{η_{1} + \sin β}, & \frac{G_{23}}{G_{0}} = [\frac{η_{1} + \sin β}{1 + 2 η_{1}} + \frac{η_{1} + 2 \sin^{2} β}{2 (η_{1} + \sin β)}] \frac{η_{2}}{2 \cos β}, \\ ν_{21} = \frac{[1 - η_{2}^{2} (1 + η_{1}) \sec^{2} β] (η_{1} + \sin β) \sin β}{\cos^{2} β}, & ν_{12} = \frac{E_{11} ν_{21}}{E_{22}}, & \frac{ρ}{ρ_{0}} = \frac{(1 + 0.5 η_{1}) η_{2}}{(η_{1} + \sin β) \cos β}, \end{array}

(6)

Re-entrant honeycomb core (auxetic)²⁶:

\begin{array}{l} \frac{E_{11}}{E_{0}} = \frac{η_{2}^{3}}{(η_{1} - \sin β) (η_{2}^{2} + \tan^{2} β) \cos β}, & \frac{E_{22}}{E_{0}} = \frac{(η_{1} - \sin β) η_{2}^{3}}{[1 + η_{2}^{2} (\tan^{2} β + η_{1} \sec^{2} β)] \cos^{3} β}, \\ \frac{G_{12}}{E_{0}} = \frac{η_{2}^{3}}{η_{1} (1 + 2 η_{1}) \cos β}, & \frac{G_{13}}{G_{0}} = \frac{η_{2} \cos β}{η_{1} - \sin β}, & \frac{G_{23}}{G_{0}} = (\frac{η_{1} - \sin β}{1 + 2 η_{1}} + \frac{0.5 η_{1} + \sin^{2} β}{η_{1} - \sin β}) \frac{η_{2}}{2 \cos β}, \\ ν_{21} = - \frac{(η_{1} - \sin β) (1 - η_{2}^{2}) \sin β}{[1 + η_{2}^{2} (η_{1} \sec^{2} β + \tan^{2} β)] \cos^{2} β}, & ν_{12} = \frac{E_{11} ν_{21}}{E_{22}}, & \frac{ρ}{ρ_{0}} = \frac{(1 + 0.5 η_{1}) η_{2}}{(η_{1} - \sin β) \cos β}, \end{array}

(7)

in which η₁ = d_c/l_c and η₂ = t_c/l_c.

To make a fair comparison between these two types of honeycomb core, the aspect ratios (η₁) can be related to each other to attain the same density of the core. Thus, utilizing equations (6) and (7), one can find the relation below between the aspect ratio in the re-entrant honeycomb core ( $η_{1}^{R}$ ) and the aspect ratio in the hexagonal honeycomb core ( $η_{1}^{H}$ ):

η_{1}^{R} = \frac{(2 + \sin β) η_{1}^{H} + 4 \sin β}{2 - \sin β} .

(8)

It is noteworthy that due to the discontinuity in the material properties of the core and the face sheets, the components of the stress tensor vary discontinuously at the contact surface of layers (z = ±h_c/2). However, the components of the stress tensor vary continuously inside each layer.

Stress, strain, deformation

As the simplest theories associated with the analysis of shell structures, there are several classical shell theories like Love, Donnell, Flugge, Sanders, etc. In such theories, it is assumed that the shear deformation and rotational inertia can be neglected which limits the applicability of classical shell theories to thin ones. To improve the accuracy, especially for moderately thick shells, the first-order shear deformation theory (FSDT) was introduced.²⁷ In this theory, the shear deformation and rotational inertia are included by considering rotations created by the shear deformation. When the FSDT is used to model a sandwich structure it is assumed that all layers of the structure experience the same rotation which results in additional rigidity and reduces the accuracy of the model, especially for thick sandwich structures. To remove this shortcoming, the zig-zag theory was introduced by Murakami.²⁸ In the zig-zag theory, it is assumed that each layer of a sandwich structure experiences its own rotation. This assumption increases the degree of freedom in modeling which improves the accuracy of the model. The difference between the zig-zag theory and the FSDT is illustrated in Figure 4. As this figure shows, the zig-zag theory provides additional flexibility in comparison with the FSDT.

Figure 4.

Schematic of the displacement fields in the FSDT and zig-zag theories.

Based on Murakami’s zig-zag theory, the components of the displacement field in the kth part of the shell are presented as^28,29

{\begin{cases} u_{1}^{(k)} (t, x_{k}, θ, z) \\ u_{2}^{(k)} (t, x_{k}, θ, z) \\ u_{3}^{(k)} (t, x_{k}, θ, z) \end{cases}} = {\begin{cases} u^{(k)} (t, x_{k}, θ) \\ v^{(k)} (t, x_{k}, θ) \\ w^{(k)} (t, x_{k}, θ) \end{cases}} + z {\begin{array}{c} α_{x}^{(k)} (t, x_{k}, θ) \\ α_{θ}^{(k)} (t, x_{k}, θ) \\ 0 \end{array}} + f (z) {\begin{array}{c} Γ_{x}^{(k)} (t, x_{k}, θ) \\ Γ_{θ}^{(k)} (t, x_{k}, θ) \\ 0 \end{array}},

(9)

where

u_{1}^{(k)}

u_{2}^{(k)}

, and

u_{3}^{(k)}

show displacement along meridional (x_k), circumferential (θ), and transverse (z) directions, respectively; u^(k), v^(k), and w^(k) show the corresponding components of displacement at the middle surface (z = 0);

α_{x}^{(k)}

and

α_{θ}^{(k)}

are rotations about θ- and x_k-axes, sequentially;

Γ_{x}^{(k)}

and

Γ_{θ}^{(k)}

are used to include the zig-zag effect in the sandwich structure; and for a sandwich structure made of N_L individual layers (in this work N_L = 3), the function f(z) is defined as follows^28,29:

f (z) = \sum_{l = 1}^{N_{L}} \frac{2 {(- 1)}^{l}}{h_{l}} (z - \frac{z_{l} + z_{l + 1}}{2}) [H (z - z_{l}) - H (z - z_{l + 1})] .

(10)

where h_l is the thickness of the lth layer of the sandwich structure; H stands for the Heaviside function; and z_l and z_l+1 are the transverse position of the bottom and top of the lth layer, respectively. By removing f(z),

Γ_{x}^{(k)}

and

Γ_{θ}^{(k)}

in equation (9) and considering the shear correction factor (k_s = 5/6) for the transverse shear stresses (σ_xz and σ_θz), the FSDT can be obtained.

The following relations present the components of strain in the kth part of the shell³⁰:

\begin{array}{l} ε_{x x}^{(k)} = \frac{\partial u_{1}^{(k)}}{\partial x_{k}}, & γ_{x θ}^{(k)} = \frac{1}{r_{k}} \frac{\partial u_{1}^{(k)}}{\partial θ} - \frac{\sin λ}{r_{k}} u_{2}^{(k)} + \frac{\partial u_{2}^{(k)}}{\partial x_{k}}, \\ ε_{θ θ}^{(k)} = \frac{\sin λ}{r_{k}} u_{1}^{(k)} + \frac{1}{r_{k}} \frac{\partial u_{2}^{(k)}}{\partial θ} + \frac{\cos λ}{r_{k}} u_{3}^{(k)}, & γ_{x z}^{(k)} = \frac{\partial u_{1}^{(k)}}{\partial z} + \frac{\partial u_{3}^{(k)}}{\partial x_{k}}, \\ ε_{z z}^{(k)} = \frac{\partial u_{3}^{(k)}}{\partial z}, & γ_{θ z}^{(k)} = \frac{\partial u_{2}^{(k)}}{\partial z} - \frac{\cos λ}{r_{k}} u_{2}^{(k)} + \frac{1}{r_{k}} \frac{\partial u_{3}^{(k)}}{\partial θ}, \end{array}

(11)

where r₁ = R₁+x₁sinλ and r₂ = R_r+x₂sinλ represent the radius of the shell at the first and second parts of the shell, respectively. These radii are measured from the central line of the shell to the middle of the core.

By inserting equations (9) into (11) and considering the assumptions of shallow shells, the components of strain can be represented as follows:

\begin{array}{l} {\begin{cases} γ_{x z}^{(k)} \\ γ_{θ z}^{(k)} \end{cases}} = {\begin{array}{c} \frac{\partial w^{(k)}}{\partial x_{k}} + α_{x}^{(k)} \\ \frac{1}{r_{k}} (\frac{\partial w^{(k)}}{\partial θ} - v^{(k)} \cos λ) + α_{θ}^{(k)} \end{array}} + g (z) {\begin{cases} Γ_{x}^{(k)} \\ Γ_{θ}^{(k)} \end{cases}}, & ε_{z z}^{(k)} = 0, \\ {\begin{cases} ε_{x x}^{(k)} \\ ε_{θ θ}^{(k)} \\ γ_{x θ}^{(k)} \end{cases}} = {\begin{array}{c} \frac{\partial u^{(k)}}{\partial x_{k}} \\ \frac{1}{r_{k}} (u^{(k)} + \frac{\partial v^{(k)}}{\partial θ} + w^{(k)} \cos λ) \\ \frac{1}{r_{k}} (\frac{\partial u^{(k)}}{\partial θ} - v^{(k)} \sin λ) + \frac{\partial v^{(k)}}{\partial x_{k}} \end{array}} + z {\begin{array}{c} \frac{\partial α_{x}^{(k)}}{\partial x_{k}} \\ \frac{1}{r_{k}} (α_{x}^{(k)} \sin λ + \frac{\partial α_{θ}^{(k)}}{\partial θ}) \\ \frac{1}{r_{k}} (\frac{\partial α_{x}^{(k)}}{\partial θ} - α_{θ}^{(k)} \sin λ) + \frac{\partial α_{θ}^{(k)}}{\partial x_{k}} \end{array}} + f (z) {\begin{array}{c} \frac{\partial Γ_{x}^{(k)}}{\partial x_{k}} \\ \frac{1}{r_{k}} (Γ_{x}^{(k)} \sin λ + \frac{\partial Γ_{θ}^{(k)}}{\partial θ}) \\ \frac{1}{r_{k}} (\frac{\partial Γ_{x}^{(k)}}{\partial θ} - Γ_{θ}^{(k)} \sin λ) + \frac{\partial Γ_{θ}^{(k)}}{\partial x_{k}} \end{array}}, \end{array}

(12)

where

g (z) = \frac{d f (z)}{d z} = \sum_{l = 1}^{N_{L}} \frac{2 {(- 1)}^{l}}{h_{l}} [H (z - z_{l}) - H (z - z_{l + 1})] .

(13)

Hamilton’s principle

Known as Hamilton’s principle, the equation below can be employed to derive the governing equations, compatibility conditions, and boundary conditions associated with the vibrational behavior of the kth part of the shell^31,32:

\int_{t_{1}}^{t_{2}} (δ T_{s}^{(k)} - δ U_{s}^{(k)} + δ W_{n . c .}^{(k)}) d t = 0,

(14)

in which [t₁,t₂] is an arbitrary time interval, δ is the well-known variational operator, and

T_{s}^{(k)}

U_{s}^{(k)}

, and

W_{n . c .}^{(k)}

show the kinetic energy of the kth part of the shell, the strain energy of the kth part of the shell, and the work done by external non-conservative loads on the kth part of the shell, respectively.

The kinetic energy of the kth part of the shell can be described as follows:

T_{s}^{(k)} = \frac{1}{2} \underset{V_{k}}{∭} ρ [{(\frac{\partial u_{1}^{(k)}}{\partial t})}^{2} + {(\frac{\partial u_{2}^{(k)}}{\partial t})}^{2} + {(\frac{\partial u_{3}^{(k)}}{\partial t})}^{2}] d V_{k},

(15)

in which, as described in the equation below, V_k is the volume of the kth part of the shell:

\underset{V_{k}}{∭} () d V_{k} = \iint_{S_{k}} \int_{- \frac{h}{2}}^{\frac{h}{2}} () d z d S_{k},

(16)

where dS_k = r_kdx_kdθ is the surface area of the kth part of the shell at its middle surface.

Utilizing equations (9), (15), and (16) the following relation can be achieved:

\begin{array}{l} T_{s}^{(k)} = \frac{1}{2} \iint_{S_{k}} {I_{0} [{(\frac{\partial u^{(k)}}{\partial t})}^{2} + {(\frac{\partial v^{(k)}}{\partial t})}^{2} + {(\frac{\partial w^{(k)}}{\partial t})}^{2}] + 2 I_{1} (\frac{\partial u^{(k)}}{\partial t} \frac{\partial α_{x}^{(k)}}{\partial t} + \frac{\partial v^{(k)}}{\partial t} \frac{\partial α_{θ}^{(k)}}{\partial t}) \\ + I_{2} [{(\frac{\partial α_{x}^{(k)}}{\partial t})}^{2} + {(\frac{\partial α_{θ}^{(k)}}{\partial t})}^{2}] + 2 I_{3} (\frac{\partial u^{(k)}}{\partial t} \frac{\partial Γ_{x}^{(k)}}{\partial t} + \frac{\partial v^{(k)}}{\partial t} \frac{\partial Γ_{θ}^{(k)}}{\partial t}) \\ + 2 I_{4} (\frac{\partial α_{x}^{(k)}}{\partial t} \frac{\partial Γ_{x}^{(k)}}{\partial t} + \frac{\partial α_{θ}^{(k)}}{\partial t} \frac{\partial Γ_{θ}^{(k)}}{\partial t}) + I_{5} [{(\frac{\partial Γ_{x}^{(k)}}{\partial t})}^{2} + {(\frac{\partial Γ_{θ}^{(k)}}{\partial t})}^{2}]} d S_{k}, \end{array}

(17)

where the inertia terms of the shell are defined as follows:

{\begin{cases} I_{0} \\ I_{1} \\ I_{2} \\ I_{3} \\ I_{4} \\ I_{5} \end{cases}} = \int_{- \frac{h}{2}}^{\frac{h}{2}} ρ (z) {\begin{array}{c} 1 \\ z \\ z^{2} \\ f (z) \\ z f (z) \\ f^{2} (z) \end{array}} d z .

(18)

The variation of the kinetic energy of the kth part of the shell is presented as follows:

\begin{array}{l} δ T^{(k)} = \iint_{S_{k}} [I_{0} (\frac{\partial u^{(k)}}{\partial t} \frac{\partial δ u^{(k)}}{\partial t} + \frac{\partial v^{(k)}}{\partial t} \frac{\partial δ v^{(k)}}{\partial t} + \frac{\partial w^{(k)}}{\partial t} \frac{\partial δ w^{(k)}}{\partial t}) + I_{1} \\ (\frac{\partial u^{(k)}}{\partial t} \frac{\partial δ α_{x}^{(k)}}{\partial t} + \frac{\partial α_{x}^{(k)}}{\partial t} \frac{\partial δ u^{(k)}}{\partial t} + \frac{\partial v^{(k)}}{\partial t} \frac{\partial δ α_{θ}^{(k)}}{\partial t} + \frac{\partial α_{θ}^{(k)}}{\partial t} \frac{\partial δ v^{(k)}}{\partial t}) + I_{2} (\frac{\partial α_{x}^{(k)}}{\partial t} \frac{\partial δ α_{x}^{(k)}}{\partial t} + \frac{\partial α_{θ}^{(k)}}{\partial t} \frac{\partial δ α_{θ}^{(k)}}{\partial t}) + I_{3} (\frac{\partial u^{(k)}}{\partial t} \frac{\partial δ Γ_{x}^{(k)}}{\partial t} \\ + \frac{\partial Γ_{x}^{(k)}}{\partial t} \frac{\partial δ u^{(k)}}{\partial t} + \frac{\partial v^{(k)}}{\partial t} \frac{\partial δ Γ_{θ}^{(k)}}{\partial t} + \frac{\partial Γ_{θ}^{(k)}}{\partial t} \frac{\partial δ v^{(k)}}{\partial t}) + I_{4} (\frac{\partial α_{x}^{(k)}}{\partial t} \frac{\partial δ Γ_{x}^{(k)}}{\partial t} + \frac{\partial Γ_{x}^{(k)}}{\partial t} \frac{\partial δ α_{x}^{(k)}}{\partial t} \\ + \frac{\partial α_{θ}^{(k)}}{\partial t} \frac{\partial δ Γ_{θ}^{(k)}}{\partial t} + \frac{\partial Γ_{θ}^{(k)}}{\partial t} \frac{\partial δ α_{θ}^{(k)}}{\partial t}) + I_{5} (\frac{\partial Γ_{x}^{(k)}}{\partial t} \frac{\partial δ Γ_{x}^{(k)}}{\partial t} + \frac{\partial Γ_{θ}^{(k)}}{\partial t} \frac{\partial δ Γ_{θ}^{(k)}}{\partial t})] d S_{k}, \end{array}

(19)

The following relation describes the variation of the strain energy of the kth part of the shell³¹:

δ U_{s}^{(k)} = \underset{V_{k}}{∭} (σ_{x x}^{(k)} δ ε_{x x}^{(k)} + σ_{θ θ}^{(k)} δ ε_{θ θ}^{(k)} + σ_{x θ}^{(k)} δ γ_{x θ}^{(k)} + σ_{x z}^{(k)} δ γ_{x z}^{(k)} + σ_{θ z}^{(k)} δ γ_{θ z}^{(k)}) d V_{k},

(20)

which can be represented by utilizing equations (12) and (16) as

\begin{array}{l} δ U_{s}^{(k)} = \iint_{S_{k}} [N_{x x}^{(k)} \frac{\partial δ u^{(k)}}{\partial x_{k}} + M_{x x}^{(k)} \frac{\partial δ α_{x}^{(k)}}{\partial x_{k}} + P_{x x}^{(k)} \frac{\partial δ γ_{x}^{(k)}}{\partial x_{k}} + \frac{N_{θ θ}^{(k)} \sin λ}{r_{k}} δ u^{(k)} + \frac{N_{θ θ}^{(k)}}{r_{k}} \frac{\partial δ v^{(k)}}{\partial θ} \\ + \frac{N_{θ θ}^{(k)} \cos λ}{r_{k}} δ w^{(k)} + \frac{M_{θ θ}^{(k)}}{r_{k}} (\frac{\partial δ α_{θ}^{(k)}}{\partial θ} + δ α_{x}^{(k)} \sin λ) + \frac{P_{θ θ}^{(k)}}{r_{k}} (\frac{\partial δ Γ_{θ}^{(k)}}{\partial θ} + δ Γ_{x}^{(k)} \sin λ) \\ + \frac{N_{x θ}^{(k)}}{r_{k}} \frac{\partial δ u^{(k)}}{\partial θ} + N_{x θ}^{(k)} \frac{\partial δ v^{(k)}}{\partial x_{k}} - \frac{N_{x θ}^{(k)} \sin λ}{r_{k}} δ v^{(k)} + M_{x θ}^{(k)} (\frac{1}{r_{k}} \frac{\partial δ α_{x}^{(k)}}{\partial θ} + \frac{\partial δ α_{θ}^{(k)}}{\partial x_{k}} - \frac{\sin λ}{r_{k}} δ α_{θ}^{(k)}) \\ + P_{x θ}^{(k)} (\frac{1}{r_{k}} \frac{\partial δ Γ_{x}^{(k)}}{\partial θ} + \frac{\partial δ Γ_{θ}^{(k)}}{\partial x_{k}} - \frac{\sin λ}{r_{k}} δ Γ_{θ}^{(k)}) + Q_{x z}^{(k)} (\frac{\partial δ w^{(k)}}{\partial x_{k}} + δ α_{x}^{(k)}) + R_{x z}^{(k)} δ Γ_{x}^{(k)} \\ + Q_{θ z}^{(k)} (\frac{1}{r_{k}} \frac{\partial δ w^{(k)}}{\partial θ} - \frac{\cos λ}{r_{k}} δ v^{(k)} + δ α_{θ}^{(k)}) + R_{θ z}^{(k)} δ Γ_{θ}^{(k)}] d S_{k}, \end{array}

(21)

where the stress resultants are defined as follows:

\begin{array}{l} {\begin{cases} N_{i j}^{(k)} \\ M_{i j}^{(k)} \\ P_{i j}^{(k)} \end{cases}} = \int_{- \frac{h}{2}}^{\frac{h}{2}} σ_{i j}^{(k)} {\begin{array}{c} 1 \\ z \\ f (z) \end{array}} d z, & {\begin{cases} Q_{i z}^{(k)} \\ R_{i z}^{(k)} \end{cases}} = \int_{- \frac{h}{2}}^{\frac{h}{2}} σ_{i z}^{(k)} {\begin{array}{c} 1 \\ g (z) \end{array}} d z, & i, j = x, θ . \end{array}

(22)

By inserting equations (3) and (12) into equation (22), the following relation is attained:

{\begin{cases} N_{x x}^{(k)} \\ M_{x x}^{(k)} \\ P_{x x}^{(k)} \\ N_{θ θ}^{(k)} \\ M_{θ θ}^{(k)} \\ P_{θ θ}^{(k)} \end{cases}} = [\begin{array}{l} A_{11} & B_{11} & F_{11} & A_{12} & B_{12} & F_{12} \\ B_{11} & D_{11} & H_{11} & B_{12} & D_{12} & H_{12} \\ F_{11} & H_{11} & T_{11} & F_{12} & H_{12} & T_{12} \\ A_{12} & B_{12} & F_{12} & A_{22} & B_{22} & F_{22} \\ B_{12} & D_{12} & H_{12} & B_{22} & D_{22} & H_{22} \\ F_{12} & H_{12} & T_{12} & F_{22} & H_{22} & T_{22} \end{array}] {\begin{array}{c} \frac{\partial u^{(k)}}{\partial x_{k}} \\ \frac{\partial α_{x}^{(k)}}{\partial x_{k}} \\ \frac{\partial Γ_{x}^{(k)}}{\partial x_{k}} \\ \frac{1}{r_{k}} (u^{(k)} \sin λ + \frac{\partial v^{(k)}}{\partial θ} + w^{(k)} \cos λ) \\ \frac{1}{r_{k}} (α_{x}^{(k)} \sin λ + \frac{\partial α_{θ}^{(k)}}{\partial θ}) \\ \frac{1}{r_{k}} (Γ_{x}^{(k)} \sin λ + \frac{\partial Γ_{θ}^{(k)}}{\partial θ}) \end{array}}, {\begin{cases} N_{x θ}^{(k)} \\ M_{x θ}^{(k)} \\ P_{x θ}^{(k)} \end{cases}} = [\begin{array}{l} A_{66} & B_{66} & F_{66} \\ B_{66} & D_{66} & H_{66} \\ F_{66} & H_{66} & T_{66} \end{array}] {\begin{cases} \frac{\partial v^{(k)}}{\partial x_{k}} + \frac{1}{r_{k}} (\frac{\partial u^{(k)}}{\partial θ} - v^{(k)} \sin λ) \\ \frac{\partial α_{θ}^{(k)}}{\partial x_{k}} + \frac{1}{r_{k}} (\frac{\partial α_{x}^{(k)}}{\partial θ} - α_{θ}^{(k)} \sin λ) \\ \frac{\partial Γ_{θ}^{(k)}}{\partial x_{k}} + \frac{1}{r_{k}} (\frac{\partial Γ_{x}^{(k)}}{\partial θ} - Γ_{θ}^{(k)} \sin λ) \end{cases}}, {\begin{cases} Q_{θ z}^{(k)} \\ R_{θ z}^{(k)} \end{cases}} = [\begin{array}{l} A_{44} & Y_{44} \\ Y_{44} & Z_{44} \end{array}] {\begin{array}{c} \frac{1}{r_{k}} (\frac{\partial w^{(k)}}{\partial θ} - v^{(k)} \cos λ) + α_{θ}^{(k)} \\ Γ_{θ}^{(k)} \end{array}}, {\begin{cases} Q_{x z}^{(k)} \\ R_{x z}^{(k)} \end{cases}} = [\begin{array}{l} A_{55} & Y_{55} \\ Y_{55} & Z_{55} \end{array}] {\begin{array}{c} \frac{\partial w^{(k)}}{\partial x} + α_{x}^{(k)} \\ Γ_{x}^{(k)} \end{array}},

(23)

in which

\begin{array}{l} {\begin{cases} A_{i j} \\ B_{i j} \\ D_{i j} \\ F_{i j} \\ H_{i j} \\ T_{i j} \end{cases}} = \int_{- \frac{h}{2}}^{\frac{h}{2}} Q_{i j} {\begin{array}{c} 1 \\ z \\ z^{2} \\ f (z) \\ z f (z) \\ f^{2} (z) \end{array}} d z, & {\begin{cases} A_{q q} \\ Y_{q q} \\ Z_{q q} \end{cases}} = \int_{- \frac{h}{2}}^{\frac{h}{2}} Q_{q q} {\begin{array}{c} 1 \\ g (z) \\ g^{2} (z) \end{array}} d z, & \begin{array}{l} i, j = 1, 2, 6 . \\ q = 4, 5 . \end{array} \end{array}

(24)

There is no external load in the free vibrational analysis of a structure. Thus, the work done by non-conservative loads on the kth part of the shell should be set as zero ( $δ W_{n . c .}^{(k)} = 0$ ). By inserting equations (19) and (21) into equation (14), the governing equations for the free vibration analysis of the kth part of the shell can be derived as follows:

\begin{array}{l} - I_{0} \frac{\partial^{2} u^{(k)}}{\partial t^{2}} - I_{1} \frac{\partial^{2} α_{x}^{(k)}}{\partial t^{2}} - I_{3} \frac{\partial^{2} Γ_{x}^{(k)}}{\partial t^{2}} + \frac{\partial N_{x x}^{(k)}}{\partial x_{k}} + \frac{N_{x x}^{(k)} - N_{θ θ}^{(k)}}{r_{k}} \sin λ + \frac{1}{r_{k}} \frac{\partial N_{x θ}^{(k)}}{\partial θ} = 0, \\ - I_{0} \frac{\partial^{2} v^{(k)}}{\partial t^{2}} - I_{1} \frac{\partial^{2} α_{θ}^{(k)}}{\partial t^{2}} - I_{3} \frac{\partial^{2} Γ_{θ}^{(k)}}{\partial t^{2}} + \frac{1}{r_{k}} \frac{\partial N_{θ θ}^{(k)}}{\partial θ} + \frac{\partial N_{x θ}^{(k)}}{\partial x_{k}} + \frac{2 N_{x θ}^{(k)}}{r_{k}} \sin λ + \frac{Q_{θ z}^{(k)}}{r_{k}} \cos λ = 0, \\ - I_{0} \frac{\partial^{2} w^{(k)}}{\partial t^{2}} + \frac{\partial Q_{x z}^{(k)}}{\partial x_{k}} + \frac{Q_{x z}^{(k)}}{r_{k}} \sin λ + \frac{1}{r_{k}} \frac{\partial Q_{θ z}^{(k)}}{\partial θ} - \frac{N_{θ θ}^{(k)}}{r_{k}} \cos λ = 0, \\ - I_{1} \frac{\partial^{2} u^{(k)}}{\partial t^{2}} - I_{2} \frac{\partial^{2} α_{x}^{(k)}}{\partial t^{2}} - I_{4} \frac{\partial^{2} Γ_{x}^{(k)}}{\partial t^{2}} + \frac{\partial M_{x x}^{(k)}}{\partial x_{k}} + \frac{M_{x x}^{(k)} - M_{θ θ}^{(k)}}{r_{k}} \sin λ + \frac{1}{r_{k}} \frac{\partial M_{x θ}^{(n)}}{\partial θ} - Q_{x z}^{(k)} = 0, \\ - I_{1} \frac{\partial^{2} v^{(k)}}{\partial t^{2}} - I_{2} \frac{\partial^{2} α_{θ}^{(k)}}{\partial t^{2}} - I_{4} \frac{\partial^{2} Γ_{θ}^{(k)}}{\partial t^{2}} + \frac{1}{r_{k}} \frac{\partial M_{θ θ}^{(k)}}{\partial θ} + \frac{\partial M_{x θ}^{(k)}}{\partial x_{k}} + \frac{2 M_{x θ}^{(k)}}{r_{k}} \sin λ - Q_{θ z}^{(k)} = 0, \\ - I_{3} \frac{\partial^{2} u^{(k)}}{\partial t^{2}} - I_{4} \frac{\partial^{2} α_{x}^{(k)}}{\partial t^{2}} - I_{5} \frac{\partial^{2} Γ_{x}^{(k)}}{\partial t^{2}} + \frac{\partial P_{x x}^{(k)}}{\partial x_{k}} + \frac{P_{x x}^{(k)} - P_{θ θ}^{(k)}}{r_{k}} \sin λ + \frac{1}{r_{k}} \frac{\partial P_{x θ}^{(k)}}{\partial θ} - R_{x z}^{(k)} = 0, \\ - I_{3} \frac{\partial^{2} v^{(k)}}{\partial t^{2}} - I_{4} \frac{\partial^{2} α_{θ}^{(k)}}{\partial t^{2}} - I_{5} \frac{\partial^{2} Γ_{θ}^{(k)}}{\partial t^{2}} + \frac{1}{r_{k}} \frac{\partial P_{θ θ}^{(k)}}{\partial θ} + \frac{\partial P_{x θ}^{(k)}}{\partial x_{k}} + \frac{2 P_{x θ}^{(k)}}{r_{k}} \sin λ - R_{θ z}^{(k)} = 0, \end{array}

(25)

and the boundary conditions at x₁ = 0 and x₂ = L−L_r can be presented in the general form below:

\begin{array}{l} \begin{array}{l} N_{x x}^{(k)} δ u^{(k)} = 0, & N_{x θ}^{(k)} δ v^{(k)} = 0, & Q_{x z}^{(k)} δ w^{(k)} = 0, \end{array} \\ \begin{array}{l} M_{x x}^{(k)} δ α_{x}^{(k)} = 0, & M_{x θ}^{(k)} δ α_{θ}^{(k)} = 0, & P_{x x}^{(k)} δ Γ_{x}^{(k)} = 0, & P_{x θ}^{(k)} δ Γ_{θ}^{(k)} = 0, \end{array} \end{array}

(26)

In this work, three standard conditions are considered for each edge of the shell including clamped, free, and simply supported conditions. Based on equation (26), the mathematical modeling of these conditions can be described as follows:

Clamped (C):

\begin{array}{l} u^{(k)} = 0, & v^{(k)} = 0, & w^{(k)} = 0, & α_{x}^{(k)} = 0, & α_{θ}^{(k)} = 0, & Γ_{x}^{(k)} = 0, & Γ_{θ}^{(k)} = 0 . \end{array}

(27)

Simply supported (S):

\begin{array}{l} N_{x x}^{(k)} = 0, & v^{(k)} = 0, & w^{(k)} = 0, & M_{x x}^{(k)} = 0, & α_{θ}^{(k)} = 0, & P_{x x}^{(k)} = 0, & Γ_{θ}^{(k)} = 0 . \end{array}

(28)

Free (F):

\begin{array}{l} N_{x x}^{(k)} = 0, & N_{x θ}^{(k)} = 0, & Q_{x z}^{(k)} = 0, & M_{x x}^{(k)} = 0, & M_{x θ}^{(k)} = 0, & P_{x x}^{(k)} = 0, & P_{x θ}^{(k)} = 0 . \end{array}

(29)

The compatibility conditions at the location of the ring (x₁ = L_r and x₂ = 0) can be described as follows:

\begin{array}{l} u^{(1)} = u^{(2)}, & v^{(1)} = v^{(2)}, & w^{(1)} = 0, & w^{(2)} = 0, & α_{x}^{(1)} = α_{x}^{(2)}, & α_{θ}^{(1)} = α_{θ}^{(2)}, & Γ_{x}^{(1)} = Γ_{x}^{(2)}, & Γ_{θ}^{(1)} = Γ_{θ}^{(2)}, \\ N_{x x}^{(1)} = N_{x x}^{(2)} & N_{x θ}^{(1)} = N_{x θ}^{(2)}, & M_{x x}^{(1)} = M_{x x}^{(2)}, & M_{x θ}^{(1)} = M_{x θ}^{(2)}, & P_{x x}^{(1)} = P_{x x}^{(2)}, & P_{x θ}^{(1)} = P_{x θ}^{(2)} . \end{array}

(30)

Solution

A semi-analytical solution is presented in this section which consists of an exact solution in the circumferential direction followed by an approximate numerical one in the meridional direction.

Solution in the circumferential direction

Due to the continuity of all geometric and physical characteristics at θ = 0 and θ = 2π, the following solution can be considered in the circumferential direction:

\begin{array}{l} {\begin{cases} u^{(k)} (x_{k}, θ, t) \\ w^{(k)} (x_{k}, θ, t) \\ α_{x}^{(k)} (x_{k}, θ, t) \\ Γ_{x}^{(k)} (x_{k}, θ, t) \end{cases}} = {\begin{cases} U^{(k)} (x_{k}) \\ W^{(k)} (x_{k}) \\ X^{(k)} (x_{k}) \\ Ψ^{(k)} (x_{k}) \end{cases}} \exp (i ω t) \cos (m θ), & {\begin{cases} v^{(k)} (x_{k}, θ, t) \\ α_{θ}^{(k)} (x_{k}, θ, t) \\ Γ_{θ}^{(k)} (x_{k}, θ, t) \end{cases}} = {\begin{cases} V^{(k)} (x_{k}) \\ Θ^{(k)} (x_{k}) \\ Φ^{(k)} (x_{k}) \end{cases}} \exp (i ω t) \sin (m θ), \end{array}

(31)

where i² = −1, ω represents the natural frequency, and m = 0,1,2,3,… is known as the circumferential wave number.

By inserting equations (23) into (25) and applying the solution presented in equation (31), the following differential equations can be attained as the governing equations:

\begin{array}{l} A_{11} U_{, x x}^{(k)} + A_{11} r_{k}^{- 1} \sin λ U_{, x}^{(k)} - (A_{22} \sin^{2} λ + m^{2} A_{66}) r_{k}^{- 2} U^{(k)} + m (A_{12} + A_{66}) r_{k}^{- 1} V_{, x}^{(k)} \\ - m (A_{22} + A_{66}) r_{k}^{- 2} \sin λ V^{(k)} + A_{12} r_{k}^{- 1} \cos λ W_{, x}^{(k)} - 0.5 A_{22} r_{k}^{- 2} \sin 2 λ W^{(k)} \\ + B_{11} X_{, x x}^{(k)} + B_{11} r_{k}^{- 1} \sin λ X_{, x}^{(k)} - (B_{22} \sin^{2} λ + m^{2} B_{66}) r_{k}^{- 2} X^{(k)} + m (B_{12} + B_{66}) r_{k}^{- 1} Θ_{, x}^{(k)} \\ - m (B_{22} + B_{66}) r_{k}^{- 2} \sin λ Θ^{(k)} + F_{11} Ψ_{, x x}^{(k)} + F_{11} r_{k}^{- 1} \sin λ Ψ_{, x}^{(k)} - (F_{22} \sin^{2} λ + m^{2} F_{66}) r_{k}^{- 2} Ψ^{(k)} \\ + m (F_{12} + F_{66}) r_{k}^{- 1} Φ_{, x}^{(k)} - m (F_{22} + F_{66}) r_{k}^{- 2} \sin λ Φ^{(k)} + ω^{2} (I_{0} U^{(k)} + I_{1} X^{(k)} + I_{3} Ψ^{(k)}) = 0, \end{array}

\begin{array}{l} - m (A_{12} + A_{66}) r_{k}^{- 1} U_{, x}^{(k)} - m (A_{22} + A_{66}) r_{k}^{- 2} \sin λ U^{(k)} + A_{66} V_{, x x}^{(k)} + A_{66} r_{k}^{- 1} \sin λ V_{, x}^{(k)} \\ - (m^{2} A_{22} + A_{44} \cos^{2} λ + A_{66} \sin^{2} λ) r_{k}^{- 2} V^{(k)} - m (A_{22} + A_{44}) r_{k}^{- 2} \cos λ W^{(k)} - m (B_{12} + B_{66}) r_{k}^{- 1} X_{, x}^{(k)} \\ - m (B_{22} + B_{66}) r_{k}^{- 2} \sin λ X^{(k)} + B_{66} Θ_{, x x}^{(k)} + B_{66} r_{k}^{- 1} \sin λ Θ_{, x}^{(k)} + [A_{44} r_{k}^{- 1} \cos λ - (m^{2} B_{22} + B_{66} \sin^{2} λ) r_{k}^{- 2}] Θ^{(k)} \\ - m (F_{12} + F_{66}) r_{k}^{- 1} Ψ_{, x}^{(k)} - m (F_{22} + F_{66}) r_{k}^{- 2} \sin λ Ψ^{(k)} + F_{66} Φ_{, x x}^{(k)} + F_{66} r_{k}^{- 1} \sin λ Φ_{, x}^{(k)} \\ + [Y_{44} r_{k}^{- 1} \cos λ - (m^{2} F_{22} + F_{66} \sin^{2} λ) r_{k}^{- 2}] Φ^{(n)} + ω^{2} (I_{0} V^{(k)} + I_{1} Θ^{(k)} + I_{3} Φ^{(k)}) = 0, \end{array}

\begin{array}{l} - A_{12} r_{k}^{- 1} \cos λ U_{, x}^{(k)} - 0.5 A_{22} r_{k}^{- 2} \sin 2 λ U^{(k)} - m (A_{22} + A_{44}) r_{k}^{- 2} \cos λ V^{(k)} + A_{55} W_{, x x}^{(k)} \\ + A_{55} r_{k}^{- 1} \sin λ W_{, x}^{(k)} - (A_{22} \cos^{2} λ + m^{2} A_{44}) r_{k}^{- 2} W^{(k)} + (A_{55} - B_{12} r_{k}^{- 1} \cos λ) X_{, x}^{(k)} \\ + (A_{55} r_{k}^{- 1} \sin λ - 0.5 B_{22} r_{k}^{- 2} \sin 2 λ) X^{(k)} + m (A_{44} r_{k}^{- 1} - B_{22} r_{k}^{- 2} \cos λ) Θ^{(k)} + (Y_{55} - F_{12} r_{k}^{- 1} \cos λ) Ψ_{, x}^{(k)} \\ + (Y_{55} r_{k}^{- 1} \sin λ - 0.5 F_{22} r_{k}^{- 2} \sin 2 λ) Ψ^{(k)} + m (Y_{44} r_{k}^{- 1} - F_{22} r_{k}^{- 2} \cos λ) Φ^{(k)} + I_{0} ω^{2} W^{(k)} = 0, \end{array}

\begin{array}{l} B_{11} U_{, x x}^{(k)} + B_{11} r_{k}^{- 1} \sin λ U_{, x}^{(k)} - (B_{22} \sin^{2} λ + m^{2} B_{66}) r_{k}^{- 2} U^{(k)} + m (B_{12} + B_{66}) r_{k}^{- 1} V_{, x}^{(k)} \\ - m (B_{22} + B_{66}) r_{k}^{- 2} \sin λ V^{(k)} - (A_{55} - B_{12} r_{k}^{- 1} \cos λ) W_{, x}^{(k)} - 0.5 B_{22} r_{k}^{- 2} \sin 2 λ W^{(k)} + D_{11} X_{, x x}^{(k)} \\ + D_{11} r_{k}^{- 1} \sin λ X_{, x}^{(k)} - [A_{55} + (D_{22} \sin^{2} λ + m^{2} D_{66}) r_{k}^{- 2}] X^{(k)} + m (D_{12} + D_{66}) r_{k}^{- 1} Θ_{, x}^{(k)} \\ - m (D_{22} + D_{66}) r_{k}^{- 2} \sin λ Θ^{(k)} + H_{11} Ψ_{, x x}^{(k)} + H_{11} r_{k}^{- 1} \sin λ Ψ_{, x}^{(k)} - [Y_{55} + (H_{22} \sin^{2} λ + m^{2} H_{66}) r_{k}^{- 2}] Ψ^{(k)} \\ + m (H_{12} + H_{66}) r_{k}^{- 1} Φ_{, x}^{(k)} - m (H_{22} + H_{66}) r_{k}^{- 2} \sin λ Φ^{(k)} + ω^{2} (I_{1} U^{(k)} + I_{2} X^{(k)} + I_{4} Ψ^{(k)}) = 0, \end{array}

\begin{array}{l} - m (B_{12} + B_{66}) r_{k}^{- 1} U_{, x}^{(k)} - m (B_{22} + B_{66}) r_{k}^{- 2} \sin λ U^{(k)} + B_{66} V_{, x x}^{(k)} + B_{66} r_{k}^{- 1} \sin λ V_{, x}^{(k)} \\ + [A_{44} r_{k}^{- 1} \cos λ - (m^{2} B_{22} + B_{66} \sin^{2} λ) r_{k}^{- 2}] V^{(k)} + m (A_{44} r_{k}^{- 1} - B_{22} r_{k}^{- 2} \cos λ) W^{(k)} \\ - m (D_{12} + D_{66}) r_{k}^{- 1} X_{, x}^{(k)} - m (D_{22} + D_{66}) r_{k}^{- 2} \sin λ X^{(k)} + D_{66} Θ_{, x x}^{(k)} + D_{66} r_{k}^{- 1} \sin λ Θ_{, x}^{(k)} \\ - [A_{44} + (m^{2} D_{22} + D_{66} \sin^{2} λ) r_{k}^{- 2}] Θ^{(k)} - m (H_{12} + H_{66}) r_{k}^{- 1} Ψ_{, x}^{(k)} - m (H_{22} + H_{66}) r_{k}^{- 2} \sin λ Ψ^{(k)} \\ + H_{66} Φ_{, x x}^{(k)} + H_{66} r_{k}^{- 1} \sin λ Φ_{, x}^{(k)} - [Y_{44} + (m^{2} H_{22} + H_{66} \sin^{2} λ) r_{k}^{- 2}] Φ^{(k)} + ω^{2} (I_{1} V^{(k)} + I_{2} Θ^{(k)} + I_{4} Φ^{(k)}) = 0, \end{array}

\begin{array}{l} F_{11} U_{, x x}^{(k)} + F_{11} r_{k}^{- 1} \sin λ U_{, x}^{(k)} - (F_{22} \sin^{2} λ + m^{2} F_{66}) r_{k}^{- 2} U^{(k)} + m (F_{12} + F_{66}) r_{k}^{- 1} V_{, x}^{(k)} - m (F_{22} + F_{66}) r_{k}^{- 2} \sin λ V^{(k)} \\ - (Y_{55} - F_{12} r_{k}^{- 1} \cos λ) W_{, x}^{(k)} - 0.5 F_{22} r_{k}^{- 2} \sin 2 λ W^{(k)} + H_{11} X_{, x x}^{(k)} + H_{11} r_{k}^{- 1} \sin λ X_{, x}^{(k)} \\ - [Y_{55} + (H_{22} \sin^{2} λ + m^{2} H_{66}) r_{k}^{- 2}] X^{(k)} + m (H_{12} + H_{66}) r_{k}^{- 1} Θ_{, x}^{(k)} - m (H_{22} + H_{66}) r_{k}^{- 2} \sin λ Θ^{(k)} + T_{11} Ψ_{, x x}^{(k)} \\ + T_{11} r_{k}^{- 1} \sin λ Ψ_{, x}^{(k)} - [Z_{55} + (T_{22} \sin^{2} λ + m^{2} T_{66}) r_{k}^{- 2}] Ψ^{(k)} + m (T_{12} + T_{66}) r_{k}^{- 1} Φ_{, x}^{(k)} - m (T_{22} + T_{66}) r_{k}^{- 2} \sin λ Φ^{(k)} \\ + ω^{2} (I_{3} U^{(k)} + I_{4} X^{(k)} + I_{5} Ψ^{(k)}) = 0, \end{array}

\begin{array}{l} - m (F_{12} + F_{66}) r_{k}^{- 1} U_{, x}^{(k)} - m (F_{22} + F_{66}) r_{k}^{- 2} \sin λ U^{(k)} + F_{66} V_{, x x}^{(k)} + F_{66} r_{k}^{- 1} \sin λ V_{, x}^{(k)} \\ + [Y_{44} r_{k}^{- 1} \cos λ - (m^{2} F_{22} + F_{66} \sin^{2} λ) r_{k}^{- 2}] V^{(k)} + m (Y_{44} r_{k}^{- 1} - F_{22} r_{k}^{- 2} \cos λ) W^{(k)} \\ - m (H_{12} + H_{66}) r_{k}^{- 1} X_{, x}^{(k)} - m (H_{22} + H_{66}) r_{k}^{- 2} \sin λ X^{(k)} + H_{66} Θ_{, x x}^{(k)} + H_{66} r_{k}^{- 1} \sin λ Θ_{, x}^{(k)} \\ - [Y_{44} + (m^{2} H_{22} + H_{66} \sin^{2} λ) r_{k}^{- 2}] Θ^{(k)} - m (T_{12} + T_{66}) r_{k}^{- 1} Ψ_{, x}^{(k)} - m (T_{22} + T_{66}) r_{k}^{- 2} \sin λ Ψ^{(k)} \\ + T_{66} Φ_{, x x}^{(k)} + T_{66} r_{k}^{- 1} \sin λ Φ_{, x}^{(k)} - [Z_{44} + (m^{2} T_{22} + T_{66} \sin^{2} λ) r_{k}^{- 2}] Φ^{(k)} + ω^{2} (I_{3} V^{(k)} + I_{4} Θ^{(k)} + I_{5} Φ^{(k)}) = 0, \end{array}

(32)

where the subscripts ,x and ,xx indicate the first-order and the second-order derivatives with respect to x_k, respectively.

By inserting equation (23) into equations (27)–(29), performing some simplifications for the simply supported condition, and applying the solution presented in equation (31), the relations below are obtained as the boundary conditions:

Clamped:

\begin{array}{l} U^{(k)} = 0, & V^{(k)} = 0, & W^{(k)} = 0, & X^{(k)} = 0, & Θ^{(k)} = 0, & Ψ^{(k)} = 0, & Φ^{(k)} = 0 . \end{array}

(33)

Simply supported:

\begin{array}{l} \begin{array}{l} V^{(k)} = 0, & W^{(k)} = 0, & Θ^{(k)} = 0, & Φ^{(k)} = 0, \end{array} \\ A_{11} U_{, x}^{(k)} + A_{12} r_{k}^{- 1} \sin λ U^{(k)} + B_{11} X_{, x}^{(k)} + B_{12} r_{k}^{- 1} \sin λ X^{(k)} + F_{11} Ψ_{, x}^{(k)} + F_{12} r_{k}^{- 1} \sin λ Ψ^{(k)} = 0, \\ B_{11} U_{, x}^{(k)} + B_{12} r_{k}^{- 1} \sin λ U^{(k)} + D_{11} X_{, x}^{(k)} + D_{12} r_{k}^{- 1} \sin λ X^{(k)} + H_{11} Ψ_{, x}^{(k)} + H_{12} r_{k}^{- 1} \sin λ Ψ^{(k)} = 0, \end{array} F_{11} U_{, x}^{(k)} + F_{12} r_{k}^{- 1} \sin λ U^{(k)} + H_{11} X_{, x}^{(k)} + H_{12} r_{k}^{- 1} \sin λ X^{(k)} + T_{11} Ψ_{, x}^{(k)} + T_{12} r_{k}^{- 1} \sin λ Ψ^{(k)} = 0 .

(34)

Free:

\begin{array}{l} A_{11} U_{, x}^{(k)} + A_{12} r_{k}^{- 1} \sin λ U^{(k)} + m A_{12} r_{k}^{- 1} V^{(k)} + A_{12} r_{k}^{- 1} \cos λ W^{(k)} + B_{11} X_{, x}^{(k)} + B_{12} r_{k}^{- 1} \sin λ X^{(k)} \\ + m B_{12} r_{k}^{- 1} Θ^{(k)} + F_{11} Ψ_{, x}^{(k)} + F_{12} r_{k}^{- 1} \sin λ Ψ^{(k)} + m F_{12} r_{k}^{- 1} Φ^{(k)} = 0, \\ - m A_{66} r_{k}^{- 1} U^{(k)} + A_{66} V_{, x}^{(k)} - A_{66} r_{k}^{- 1} \sin λ V^{(k)} - m B_{66} r_{k}^{- 1} X^{(k)} + B_{66} Θ_{, x}^{(k)} - B_{66} r_{k}^{- 1} \sin λ Θ^{(k)} \\ - m F_{66} r_{k}^{- 1} Ψ^{(k)} + F_{66} Φ_{, x}^{(k)} - F_{66} r_{k}^{- 1} \sin λ Φ^{(k)} = 0, \\ A_{55} W_{, x}^{(k)} + A_{55} X^{(k)} + Y_{55} Ψ^{(k)} = 0, \\ B_{11} U_{, x}^{(k)} + B_{12} r_{k}^{- 1} \sin λ U^{(k)} + m B_{12} r_{k}^{- 1} V^{(k)} + B_{12} r_{k}^{- 1} \cos λ W^{(k)} + D_{11} X_{, x}^{(k)} + D_{12} r_{k}^{- 1} \sin λ X^{(k)} \\ + m D_{12} r_{k}^{- 1} Θ^{(k)} + H_{11} Ψ_{, x}^{(k)} + H_{12} r_{k}^{- 1} \sin λ Ψ^{(k)} + m H_{12} r_{k}^{- 1} Φ^{(k)} = 0, \\ - m B_{66} r_{k}^{- 1} U^{(k)} + B_{66} V_{, x}^{(k)} - B_{66} r_{k}^{- 1} \sin λ V^{(k)} - m D_{66} r_{k}^{- 1} X^{(k)} + D_{66} Θ_{, x}^{(k)} - D_{66} r_{k}^{- 1} \sin λ Θ^{(k)} \\ - m H_{66} r_{k}^{- 1} Ψ^{(k)} + H_{66} Φ_{, x}^{(k)} - H_{66} r_{k}^{- 1} \sin λ Φ^{(k)} = 0, \\ F_{11} U_{, x}^{(k)} + F_{12} r_{k}^{- 1} \sin λ U^{(k)} + m F_{12} r_{k}^{- 1} V^{(k)} + F_{12} r_{k}^{- 1} \cos λ W^{(k)} + H_{11} X_{, x}^{(k)} + H_{12} r_{k}^{- 1} \sin λ X^{(k)} \\ + m H_{12} r_{k}^{- 1} Θ^{(k)} + T_{11} Ψ_{, x}^{(k)} + T_{12} r_{k}^{- 1} \sin λ Ψ^{(k)} + m T_{12} r_{k}^{- 1} Φ^{(k)} = 0, \\ - m F_{66} r_{k}^{- 1} U^{(k)} + F_{66} V_{, x}^{(k)} - F_{66} r_{k}^{- 1} \sin λ V^{(k)} - m H_{66} r_{k}^{- 1} X^{(k)} + H_{66} Θ_{, x}^{(k)} - H_{66} r_{k}^{- 1} \sin λ Θ^{(k)} \\ - m T_{66} r_{k}^{- 1} Ψ^{(k)} + T_{66} Φ_{, x}^{(k)} - T_{66} r_{k}^{- 1} \sin λ Φ^{(k)} = 0 . \end{array}

(35)

By inserting equations (23) into (30), performing some simplifications, and applying the solution presented in equation (31), the following relations can be attained as the compatibility conditions between two parts of the shell at the location of the ring:

\begin{array}{l} U^{(1)} = U^{(2)}, & V^{(1)} = V^{(2)}, & W^{(1)} = 0, & W^{(2)} = 0, & X^{(1)} = X^{(2)}, & Θ^{(1)} = Θ^{(2)}, & Ψ^{(1)} = Ψ^{(2)}, & Φ^{(1)} = Φ^{(2)}, \\ U_{, x}^{(1)} = U_{, x}^{(2)}, & V_{, x}^{(1)} = V_{, x}^{(2)}, & X_{, x}^{(1)} = X_{, x}^{(2)}, & Θ_{, x}^{(1)} = Θ_{, x}^{(2)}, & Ψ_{, x}^{(1)} = Ψ_{, x}^{(2)}, & Φ_{, x}^{(1)} = Φ_{, x}^{(2)}, \end{array}

(36)

Solution in the meridional direction

It seems that no analytical solution can be found for the governing equation (32) under the compatibility conditions (36) and any combinations of the standard boundary conditions (33)–(35). Thus, an approximate solution is presented in this section via the DQM.

Based on the main idea of the DQM, first, the solution domain is discretized into a pre-selected set of grid points. Then, derivatives of each unknown function at each of the points are approximated in terms of values of the function at all points, Finally, the set of the differential equations are converted into a set of algebraic equations.

According to the Lagrange method, the function F(x) can be fit on a set of N discrete points (x₁,F₁), (x₂,F₂),…, (x_N,F_N) as follows:

F (x) = L_{1} (x) F_{1} + L_{2} (x) F_{2} + \dots + L_{N} (x) F_{N} = \sum_{j = 1}^{N} L_{j} (x) F_{j},

(37)

in which L_j(x) are defined to satisfy the relation below:

L_{j} (x) = {\begin{cases} 1 & x = x_{j} \\ 0 & x \neq x_{j} \end{cases} .

(38)

A simple selection for L_j(x) can be considered as follows:

L_{j} (x) = \frac{(x - x_{1}) (x - x_{2}) \dots (x - x_{j - 1}) (x - x_{j + 1}) \dots (x - x_{N})}{(x_{j} - x_{1}) (x_{j} - x_{2}) \dots (x_{j} - x_{j - 1}) (x_{j} - x_{j + 1}) \dots (x_{j} - x_{N})} = \frac{\prod_{\begin{array}{l} k = 1 \\ k \neq j \end{array}}^{N} (x - x_{k})}{\prod_{\begin{array}{l} k = 1 \\ k \neq j \end{array}}^{N} (x_{j} - x_{k})} .

(39)

A sample derivative of the function F(s) can be presented as follows:

\begin{array}{l} {\frac{d^{r} F}{d x^{r}} |}_{x = x_{i}} = \sum_{j = 1}^{N} \frac{d^{r} L_{j} (x_{i})}{d x^{r}} F_{j}, & i = 1, 2, 3, \dots, N \end{array}

(40)

which results in the following relation:

{\frac{d^{r} F}{d x^{r}}}_{N \times 1} = {[D^{(r)}]}_{N \times N} {F}_{N \times 1},

(41)

where [D^(r)] is the weighting coefficient matrix associated with the rth-order derivative. For the first-order derivative, this square is presented as³³:

D_{i j}^{(1)} = {\begin{cases} \begin{array}{l} \sum_{\begin{array}{l} q = 1 \\ q \neq i \end{array}}^{P} {(x_{i} - x_{q})}^{- 1} & i = j \\ \frac{\prod_{\begin{array}{l} q = 1 \\ q \neq i, j \end{array}}^{P} (x_{i} - x_{q})}{\prod_{\begin{array}{l} q = 1 \\ q \neq j \end{array}}^{P} (x_{j} - x_{q})} & i \neq j \end{array} & i, j = 1, 2, 3, \dots, N \end{cases},

(42)

and the following relation can be used to estimate higher-order derivatives:

\begin{array}{l} [D^{(r)}] = [D^{(1)}] [D^{(r - 1)}], & r = 2, 3, 4, \dots \end{array}

(43)

for the distribution of the grid points in the interval 0 ≤ x ≤ L, the following relation is utilized in this paper³³:

x_{i} = \frac{L}{2} [1 - \cos (\frac{i - 1}{N - 1} π)], i = 1, 2, 3, . . ., N .

(44)

Which is known as the Chebyshev-Gauss–Lobatto distribution pattern.

Using equation (41), the governing equation (32) are presented in the following algebraic form:

[K] {y} = ω^{2} [M] {y},

(45)

where

{y}

is the total displacement vector, and [K] and [M] stiffness and mass matrices, respectively.

Using equation (41), the compatibility conditions (36), and any combinations of the boundary conditions (33)–(35) can be presented in the algebraic form below:

[Λ] {y} = {0},

(46)

in which, the matrix [Λ] is the matrix associated with the compatibility and boundary conditions.

Simultaneous solutions of the set of algebraic equations (45) and (46) result in an inconsistency in the numbers of algebraic equations and unknown variables.³⁴ To overcome this challenge, the set of grid points should be divided into two sets: the points at the edges of the shell (s₁ and s_N) known as the boundary points, and the other interior ones (s₂-s_N-1) called the domain points.³⁵ If one neglects satisfying the algebraic form of the governing equations at the boundary points, this equation can be presented as follows:

[K_{tr}] {y} = ω^{2} [M_{tr}] {y},

(47)

in which the subscript tr shows corresponding non-square truncated stiffness and mass matrices. By separating the columns of each matrix associated with the boundary (bo) and domain (do) points in the non-square truncated stiffness and mass matrices (equation (47)) and the matrix associated with the compatibility and boundary conditions (equation (46)), these algebraic equations can be represented as follows³⁴:

{[K_{t r}]}_{b o} {y}_{b o} + {[K_{t r}]}_{d o} {y}_{d o} = ω^{2} ({[M_{t r}]}_{b o} {y}_{b o} + {[M_{t r}]}_{d o} {y}_{d o}),

(48)

{[Λ]}_{b o} {y}_{b o} + {[Λ]}_{d o} {y}_{d o} = {0},

(49)

Inserting equations (49) into (48) results in the following eigenvalue equation:

[K_{f}] {y}_{do} = ω^{2} [M_{f}] {y}_{do},

(50)

where

\begin{array}{l} [K_{f}] = {[K_{0}]}_{do} - {[K_{0}]}_{bo} {[Λ]}_{bo}^{- 1} {[Λ]}_{do}, & [M_{f}] = {[M_{0}]}_{do} - {[M_{0}]}_{bo} {[Λ]}_{bo}^{- 1} {[Λ]}_{do} . \end{array}

(51)

The natural frequencies and the vibrational mode shapes can be attained through the solution of the eigenvalue equation (50).

Numerical results

The numerical examples are provided and presented in this section. The boundary conditions are indicated via two capital letters which show conditions at x₁ = 0 and x₂ = L-L_r, respectively. Except for the cases which are described otherwise, the numerical results are presented for an FC conical shell of R₁ = 0.25 m, λ = 45°, L/R₁ = 3, and h/R₁ = 0.02. The sandwich FG (p = 1) shell is fabricated from the composition of Aluminum (E_me = 70 GPa, ν_me = 0.35, and ρ_me = 2700 kg/m³) as the metal and Alumina (E_ce = 380 GPa, ν_ce = 0.22, and ρ_ce = 3950 kg/m³) as the ceramic according to the power-law function (P-FGM). The thickness of the honeycomb core is considered as h_c/h = 0.8, and the geometric parameters of its cells are selected as β = 45°, η₂ = 0.02, and $η_{1}^{H} = 1$ ( $η_{1}^{R} = 4.2815$ ). The shell is enriched with an intermediate ring support located at L_r/L = 0.5. Results are presented in dimensionless forms in which the dimensionless frequency parameter is defined as follows:

Ω_{mn} = ω_{mn} R_{1} \sqrt{\frac{ρ_{me}}{E_{me}}},

(52)

where the first subscript (m = 0,1,2,…) is the circumferential wave number defined in equation (31), and the second subscript (n = 1,2,3,…) is utilized to indicate the sequence of vibrational modes in the meridional direction.

For the described case of study, the average values of the elastic and shear moduli and the density of the FG honeycomb core and FG face sheets are presented in Table 1. As observed, the FG honeycomb core benefits from lower density, and the FG face sheets benefit from higher elastic and shear moduli.

Table 1.

The average values of the elastic and shear moduli and the density of the FG honeycomb core and FG face sheets.

FG bottom layer	FG honeycomb core	FG top layer
E = 85.50 GPa	E₁₁ = 0.7119 MPa	E = 364.50 GPa
G = 31.83 GPa	E₂₂ = 18.129 MPa	G = 148.61 GPa
ρ = 2762.5 kg/m³	G₁₂ = 0.0617 MPa	ρ = 3887.5 kg/m³
	G₁₃ = 349.16 MPa
	G₂₃ = 1388.5 MPa
	ρ = 82.64 kg/m³

Convergence analysis and verification

By considering the same values for the numbers of grid points in two parts of the shell (N₁ = N₂ = N), the dimensionless frequency parameters are tabulated in Table 2 for m = 3, n = 1,2,3,4, and several values of the number of grid points utilized. As observed, the numerical solution presented in the meridional direction converges rapidly. In what follows, the numerical results and examples are presented for N = 21.

Table 2.

The convergence analysis of the approximate numerical solution presented in the meridional direction via the DQM.

		N = 9	N = 11	N = 13	N = 15	N = 17	N = 19	N = 21	N = 23
n = 1	Hexagonal core	0.2027	0.2027	0.2025	0.2024	0.2023	0.2022	0.2020	0.2020
	Re-entrant core	0.2013	0.2010	0.2008	0.2006	0.2005	0.2003	0.2001	0.2001
n = 2	Hexagonal core	0.3197	0.3195	0.3194	0.3194	0.3194	0.3194	0.3194	0.3194
	Re-entrant core	0.3179	0.3175	0.3174	0.3174	0.3174	0.3174	0.3174	0.3174
n = 3	Hexagonal core	0.4722	0.4718	0.4717	0.4716	0.4716	0.4716	0.4716	0.4716
	Re-entrant core	0.4645	0.4638	0.4636	0.4636	0.4635	0.4636	0.4636	0.4636
n = 4	Hexagonal core	0.4926	0.4924	0.4923	0.4922	0.4922	0.4922	0.4923	0.4923
	Re-entrant core	0.4908	0.4906	0.4905	0.4905	0.4905	0.4905	0.4906	0.4906

Three numerical examples are provided in the current section to check the accuracy of the present model. As the first verification example, an SS single-layer P-FGM cylindrical shell (λ = 0) of R₁ = R₂ = R = 1 m, h/R = 0.002, and L/R = 20 with no ring support is considered. The mechanical properties of the shell vary from the mechanical properties of nickel (E₁ = 205.098 GPa, ν₁ = 0.31, ρ₁ = 8900 kg/m³) at the inner surface to the mechanical properties of stainless steel (E₂ = 207.788 GPa, ν₂ = 0.317756, ρ₂ = 8166 kg/m³) at the outer surface. For several values of the material index (p = 0,0.7,2,15), the natural frequencies associated with m = 1,2,…,10 and n = 1 can be found in Table 3 (in Hz) along with the corresponding ones reported by Loy et al.³⁶ As this table shows, there is high agreement between the results which confirms the accuracy of the present model.

Table 3.

Natural frequencies (in Hz) of an SS P-FGM cylindrical shell with no ring support.

	p = 0		p = 0.7		p = 2		p = 15
	Present	Ref. 36	Present	Ref. 36	Present	Ref. 36	Present	Ref. 36
m = 1	13.548	13.548	13.269	13.269	13.103	13.103	12.933	12.933
m = 2	4.5921	4.5920	4.4995	4.4994	4.4436	4.4435	4.3837	4.3834
m = 3	4.2646	4.2633	4.1761	4.1749	4.1247	4.1235	4.0667	4.0653
m = 4	7.2279	7.2250	7.0718	7.0691	6.9847	6.9820	6.8885	6.8856
m = 5	11.547	11.542	11.295	11.290	11.155	11.151	11.003	10.999
m = 6	16.904	16.897	16.533	16.527	16.330	16.323	16.108	16.101
m = 7	23.253	23.244	22.743	22.735	22.462	22.454	22.158	22.148
m = 8	30.584	30.573	29.913	29.903	29.544	29.533	29.144	29.132
m = 9	38.895	38.881	38.041	38.028	37.572	37.559	37.063	37.048
m = 10	48.185	48.168	47.126	47.122	46.545	46.529	45.915	45.897

As the second verification example, consider an SS single-layer isotropic homogenous conical shell of λ = 60°, Lsinλ/R₂ = 0.25, and h/R₂ = 0.01 without ring support in which its density, elastic modulus, and Poisson’s ratio are shown with ρ, E, and ν = 0.3, respectively. Dimensionless frequency parameters defined as χ_mn = ω_mnR₂[ρ (1-ν²)/E]^0.5 are presented in Table 4 For m = 1,2,..,9 and n = 1 along with the corresponding ones reported by Dai et al.³⁷ As observed, the results are in excellent agreement which confirms the precision of the present model.

Table 4.

Dimensionless frequency parameters (χ_mn) of an SS single-layer isotropic homogenous conical shell without ring support.

	m = 1	m = 2	m = 3	m = 4	m = 5	m = 6	m = 7	m = 8	m = 9
Present	0.4749	0.5716	0.5994	0.6045	0.6066	0.6146	0.6326	0.6629	0.7058
Ref. 37	0.4754	0.5721	0.6001	0.6053	0.6075	0.6156	0.6340	0.6646	0.7080

As the third verification example, consider a single-layer isotropic homogenous (ν = 0.3) conical shell of λ = 30°, L/R₁ = 10, h/R₁ = 0.05 reinforced with an intermediate ring support located at L_r/L = 0.5. For various vibrational modes associated with m = 1,2,3,4 and n = 1,2,3,4 the dimensionless frequency parameters (χ_mn = ω_mnR₁[ρ (1−ν²)/E]^0.5) are presented for two boundary conditions in Table 5 with those reported by Bagheri et al.³⁸ As observed, the results are in high agreement which confirms the accuracy of the present model.

Table 5.

Dimensionless frequency parameters (χ_mn) of a single-layer isotropic homogenous conical shell stiffened with an intermediate ring support.

		CS				FC
		m = 1	m = 2	m = 3	m = 4	m = 1	m = 2	m = 3	m = 4
n = 1	Present	0.1034	0.0860	0.0746	0.0620	0.1138	0.0621	0.0617	0.0692
	Ref. 38	0.1034	0.0860	0.0746	0.0621	0.1138	0.0621	0.0617	0.0692
n = 2	Present	0.1318	0.1173	0.0844	0.0751	0.1518	0.1061	0.0850	0.0734
	Ref. 38	0.1318	0.1173	0.0844	0.0751	0.1518	0.1061	0.0850	0.0734
n = 3	Present	0.1615	0.1467	0.1324	0.1192	0.1634	0.1489	0.1218	0.1223
	Ref. 38	0.1615	0.1467	0.1324	0.1192	0.1634	0.1489	0.1219	0.1224
n = 4	Present	0.1783	0.1709	0.1598	0.1448	0.1838	0.1641	0.1398	0.1327
	Ref. 38	0.1783	0.1709	0.1598	0.1449	0.1838	0.1641	0.1398	0.1327

Parametric study

In this section, the influences of several factors on the natural frequencies are examined. Figure 5 shows the variations of the natural frequencies versus the variation of the circumferential wave number. This figure shows that the natural frequencies experience initial irregular variations followed by a steadily increase as the circumferential wave number increases from zero to higher values. In other words, there are some small values of the circumferential wave number which are associated with the lowest natural frequencies. For this case of study, the lowest four natural frequencies of the sandwich shell with either hexagonal or re-entrant honeycomb cores are associated with (m,n)=(4,1), (m,n)=(3,1), (m,n)=(5,1), and (m,n)=(7,1). The corresponding vibrational modes are illustrated in Figures 6 and 7. As these figures show, the type of the honeycomb core has no remarkable influence on the vibrational modes of the shell.

Figure 5.

The variations of the natural frequencies versus the variation of the circumferential wave number.

Figure 6.

The vibrational modes shapes of the FC conical sandwich shell with a hexagonal core associated with the lowest four natural frequencies.

Figure 7.

The vibrational modes shapes of the FC conical sandwich shell with a re-entrant core associated with the lowest four natural frequencies.

For some selected boundary conditions, the dimensionless frequency parameters are presented in Table 6 for the vibrational modes associated with m = 0,1,2,…,9 and n = 1. As this table shows, the highest natural frequencies belong to the CC shell, and depending on the vibrational modes, the lowest natural frequencies belong to the SS or CF shells. This table also shows that for all vibrational modes, the natural frequency of the SC shell is higher than the natural frequency of the CS shell. Thus, it can be concluded that utilizing the conditions with a lower degree of freedom especially at the large edge of the shell leads to higher natural frequencies. These variations can be explained by the larger perimeter of the shell at the large radius compared to the perimeter of the shell at the small radius. Table 6 reveals that by adjusting the aspect ratios of the hexagonal and re-entrant cells to achieve the same density of the honeycomb core, the type of the honeycomb which results in higher natural frequencies depends on the boundary conditions and vibrational mode number.

Table 6.

Dimensionless frequency parameters for some selected boundary conditions.

		CC	SC	CS	SS	FC	CF
Ω ₀₁	Hexagonal core	0.4445	0.4436	0.3492	0.1748	0.4436	0.2288
	Re-entrant core	0.4454	0.4450	0.3526	0.1695	0.4450	0.2288
Ω ₁₁	Hexagonal core	0.4150	0.4148	0.3605	0.3236	0.4068	0.1690
	Re-entrant core	0.4157	0.4154	0.3641	0.3246	0.4081	0.1660
Ω ₂₁	Hexagonal core	0.3557	0.3535	0.3466	0.3446	0.2975	0.1332
	Re-entrant core	0.3544	0.3519	0.3466	0.3444	0.2960	0.1302
Ω ₃₁	Hexagonal core	0.3100	0.3060	0.2967	0.2923	0.2021	0.1016
	Re-entrant core	0.3067	0.3025	0.2929	0.2882	0.2002	0.0997
Ω ₄₁	Hexagonal core	0.2802	0.2760	0.2631	0.2592	0.1814	0.0788
	Re-entrant core	0.2763	0.2716	0.2591	0.2547	0.1799	0.0778
Ω ₅₁	Hexagonal core	0.2572	0.2572	0.2369	0.2367	0.212	0.069
	Re-entrant core	0.2540	0.2539	0.2341	0.2337	0.2108	0.0683
Ω ₆₁	Hexagonal core	0.2394	0.2392	0.2158	0.2157	0.2381	0.0712
	Re-entrant core	0.2363	0.2362	0.2136	0.2136	0.2355	0.0707
Ω ₇₁	Hexagonal core	0.2318	0.2317	0.2056	0.2056	0.2317	0.0821
	Re-entrant core	0.2283	0.2283	0.2037	0.2037	0.2282	0.0818
Ω ₈₁	Hexagonal core	0.2342	0.2342	0.2071	0.2071	0.2341	0.0989
	Re-entrant core	0.2302	0.2302	0.2052	0.2052	0.2302	0.0986
Ω ₉₁	Hexagonal core	0.2455	0.2455	0.2189	0.2189	0.2455	0.1197
	Re-entrant core	0.241	0.241	0.2168	0.2168	0.241	0.1195

In what follows, the effects of various parameters on the natural frequencies are studied in the vibrational modes associated with the four lowest natural frequencies described in Figures 6 and 7. Figure 8 shows the effects of the core-to-shell thickness ratio on the natural frequencies. As shown in Table 1, in comparison with the FG face layers, the FG honeycomb core has lower elastic and shear moduli and benefits from lower density. Thus, by considering a specific value for the thickness of the shell, as the thickness of the FG honeycomb core increases, both the stiffness and mass of the shell decrease. Consequently, as the thickness of the honeycomb core increases, the natural frequencies experience initial increases followed by reductions. It can be concluded that for each vibrational mode, there is an optimal core-to-shell thickness ratio which leads to the highest natural frequency. Figure 8 shows that the optimal core-to-shell thickness ratio is weakly affected by the type of the honeycomb core. However, it may be strongly affected by the variations of other parameters.

Figure 8.

The influences of the thickness of the honeycomb core on the natural frequencies.

Figure 9 is devoted to examining the influences of the wall thickness of the cells in the honeycomb core on the natural frequencies. According to equations (6) and (7), an increase in the wall thickness of the cells in the honeycomb core provides higher elastic and shear moduli and increases the density of the honeycomb core. Since the most percentage of the stiffness of the shell comes from its face layers, an increase in the wall thickness of the cells in the honeycomb has a stronger effect on the mass of the shell rather than its stiffness. Thus, as Figure 9 shows, the natural frequencies diminish as the wall thickness of the cells in the honeycomb core increases.

Figure 9.

The influences of the wall thickness of the cells on the natural frequencies.

The effects of the material index on the natural frequencies of P-FGM, S-FGM, and E-FGM shells are examined in Figures 10–12, respectively. According to Figure 3, by increasing the material index the volume fraction of ceramic decreases in the P-FGM and E-FGM shells, and experiences either increase or decrease in the S-FGM. It results in a small reduction in the mass of the shell (ρ_ce/ρ_me≈1.46) and a significant reduction in the stiffness of the shell (E_ce/E_me≈5.43). As a result, as Figures 10–12 show, the natural frequencies decrease as the material index increases. These figures show that by considering the same material index, the natural frequencies of P-FGM and E-FGM shells are approximately the same which is expected according to Figure 3.

Figure 10.

The influences of the material index on the natural frequencies of a P-FGM shell.

Figure 11.

The influences of the material index on the natural frequencies of an S-FGM shell.

Figure 12.

The influences of the material index on the natural frequencies of an E-FGM shell.

Figure 13 shows the effects of the inclined angle on the natural frequencies. The authors’ investigations on equations (6) and (7) reveal that by increasing the inclined angle from zero to 90°, the elastic constants experience different trends and the density experiences an initial reduction followed by an increase. Thus, as shown in Figure 13, as the inclined angle increases, the natural frequencies experience an initial smooth increase followed by a significant reduction. In other words, for each vibrational mode, there is an optimal value of the inclined angle which brings about the highest natural frequency. As observed, the optimal value of the inclined angle is easily influenced by the type of the honeycomb core.

Figure 13.

The influences of the inclined angle on the natural frequencies.

Figure 14 is presented to study the influences of the location of the intermediate ring support on the natural frequencies. As this figure shows, as the intermediate ring support moves from the small radius of the shell toward the large radius of the shell, the natural frequencies experience initial increases followed by a decrease. In other words, there is an optimal location for the intermediate ring support which brings about the highest natural frequencies. To explain the reason behind this trend, it should be noted that when the intermediate ring support moves, the length and consequently the stiffness of the shell parts on two sides of the intermediate ring support experience opposite trends. Figure 14 shows that the optimal location of the intermediate ring support is not the same for all vibrational modes and can be affected by the vibrational mode number. As observed, the optimal location of the intermediate ring support is not dependent on the type of the honeycomb core. However, it might be influenced by the variations of other parameters.

Figure 14.

The influences of the location of the intermediate ring support on the natural frequencies.

Figure 14 shows abnormal changes in the curvature of the curves at the optimal point. To find the reason behind these unusual changes, the vibrational mode shapes of a sandwich shell with an FG re-entrant core are presented in Figure 15 for m = 4, n = 1,2, and three different locations of the intermediate ring support. The selected locations are L_r/L = 0.33, L_r/L = 0.3482 (the optimal location), and L_r/L = 0.36. As this figure shows, the vibrational mode switching may happen due to the variations in the location of the intermediate ring support.

Figure 15.

The influences of the location of the intermediate ring support on the sequence of vibrational modes.

Conclusions

In the present paper, the free vibration analysis was investigated for ring-stiffened FGM conical sandwich shells with honeycomb cores. The sandwich shell was modeled based on Murakami’s zig-zag theory and the intermediate ring support was modeled as a rigid ring. The main findings of the work are described as follows:

• By adjusting the aspect ratios of the hexagonal and re-entrant cells to attain the same density of the honeycomb core, the type of the honeycomb which brings about higher natural frequencies depends on the boundary conditions and vibrational mode number.

• By considering the same material index, the natural frequencies of P-FGM and E-FGM shells are approximately the same.

• An optimal value can be found for the core-to-shell thickness ratio which results in the highest natural frequency.

• The natural frequencies decrease by increasing the wall thickness of the cells in the honeycomb core.

• For each vibrational mode, an optimal value can be found for the inclined angle which leads to the highest natural frequency.

• For each vibrational mode, there is an optimal location for the intermediate ring support which results in the highest natural frequency.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Hassan Afshari

Appendix

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