# A Semianalytic Method for Vibration Analysis of a Sandwich FGP Doubly Curved Shell with Arbitrary Boundary Conditions

Authors: Zhongyu Zhang; Jiayang Gu; Jianjun Ding; Yanwu Tao
Journal: Shock and Vibration
 (2021)

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## Abstract

Due to the excellent mechanical properties of doubly curved structure and functionally graded porous (FGP) material, the study of their vibration characteristics has attracted wide attention. The main aim of this research is to establish a formulation for free and forced vibration analysis of a new Sandwich FGP doubly curved structure. Four models of Sandwich materials are considered. The potential energy and kinetic energy functions are obtained on the foundation of the first-order shear deformation theory (FSDT). The idea of domain energy decomposition is applied to the theoretical modeling, where the structure is segmented along the generatrix direction. The continuity conditions for the interfaces between adjacent segments are balanced by the weighted parameters. For each segment, the displacement functions are selected as the Jacobi orthogonal polynomials and trigonometric series. The boundary conditions of the structure are obtained by the boundary spring simulation technique. The solution is obtained by the variational operation of the structural functional. The convergence performance and correctness of the theoretical model are examined by several numerical examples. Finally, some novel results are given, where free and forced vibration characteristics of Sandwich FGP doubly curved structures are examined in detail.


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## Body

## 1. Introduction

The doubly curved structure is widely used in the construction industry because of its excellent structural and mechanical properties [1–5]. Due to the appearance of pore, functionally graded porous (FGP) materials have better mechanical properties than traditional functionally graded materials [6–10]. Sandwich structures, which own outstanding mechanical properties, have been extensively applied in aerospace, transportation, and other fields [11–15]. However, the Sandwich structure combined with FGP materials has not been investigated yet. The objective of the paper is to establish a model for free and forced vibration analysis of a new Sandwich FGP doubly curved structure based on the semianalytical method.Zenkour [16, 17] conducted the bending and free vibration analysis of functionally graded (FG) ceramic-metal Sandwich plates by proposing a two-dimensional solution. It was assumed that the faces of structure are with isotropic two-constituent material distribution along the thickness direction, and their elasticity modulus and Poisson’s ratio vary in terms of a power-law distribution, while the core layer is homogeneous and consisted of isotropic ceramic material. Natarajan and Manickam [18] studied the bending and free flexural vibration behavior of Sandwich functionally graded material (FGM) plates by using QUAD-8 shear flexible element. Mahi et al. [19] presented a new hyperbolic shear deformation theory to conduct the bending and free vibration analysis of isotropic, FG, Sandwich, and laminated composite plates. Extending the Moving Kriging-based mesh-free method, Thai et al. [20] surveyed the static, dynamic, and buckling analysis of FG isotropic and Sandwich plates with classical boundary conditions. Xiao et al. [21] applied the finite element method to study the vibration behavior of Sandwich panel with mass density gradient (DG) foam core on the basis of the high-order Sandwich plate theory. Liu et al. [22] investigated the free vibration characteristics of FG material Sandwich plates based on the refined higher order Sandwich panel theory. Zenkour [23] proposed a quasi-3D shear deformation theory to study the bending responses of FG single-layered and Sandwich plates with porosities by using Navier’s technique. Using a layer-wise finite element method, Pandey and Pradyumna [24] studied the free vibration of FG Sandwich shells in the framework of FSDT. Chen et al. [25] conducted vibration investigation of FGM Sandwich doubly curved shallow shell by a novel shear deformation theory and the Navier method. Trinh and Kim [26] proposed a closed-form solution to study the nonlinear characteristics of the FG Sandwich shells with thermomechanical loadings.A retrospect of previous researches shows that most of the investigations were focusing on Sandwich FGM structures, while FGP materials were rarely studied. In addition, the Sandwich rectangular plate or cylindrical shell has been sufficiently examined, while the research on Sandwich doubly curved structure is scarce.This paper aims to construct a vibration analysis model for Sandwich FGP doubly curved shell subjected to arbitrary boundary conditions based on the FSDT, where four Sandwich material models are considered. The domain energy decomposition technique is employed, which can make the selection of displacement functions become rather flexible. The weighted parameters are utilized to deal with the continuity conditions between each neighbored segment. For every segment, the displacement functions are represented by Jacobi polynomials along the generatrix orientation and trigonometric series in the circumferential orientation. Free vibration and forced response can be obtained by standard variational operation. The convergence and validity of the model are shown by several numerical cases.


## 2. Theoretical Formulations

### 2.1. Geometric Description

The orthogonal coordinate system (φ, θ, z) of the Sandwich FGP doubly curved shell is shown in Figure 1, where oz and o′z′ are the spatial coordinate axis and geometric central axis, respectively. By spinning the generatrix with respect to the central axis o′z′, structural surface can be achieved. Rφ and Rθ are primary curvature radius, the centers of which are Oφ and Oθ respectively. Rs represents the offset distance between o′z′ and oz. R0 signifies the sum of horizontal radius and Rs, where R0 = Rθsin φ. In this paper, three kinds of doubly curved shells are studied, and the corresponding generatrix characteristics are shown in Figure 2. The specific geometric equations are expressed as follows [27, 28]:(1)
Elliptical shell (Figure2(a)):(1.a)Rφφ=a2b2a2sin2φ+b2cos2φ3,Rθφ=a2a2sin2φ+b2cos2φ+Rssinφ,wherea indicates the length of the semimajor axes, and b represents that of the semiminor axes.(1.b)b=aLa2−R02−a2−R12(1.c)φ0=arctanbR0aa2−R02,φ1=arctanbR1aa2−R12.(2)
Paraboloidal shell (Figure2(b)):(2.a)Rφφ=k2cos3φ,Rθφ=k2cosφ+Rssinφ,where(2.b)k=R12−R02L(2.c)φ0=arctan2R0k,φ1=arctan2R1k.(3)
Hyperbolic shell (Figure2(c)):(3.a)Rφφ=−a2b2a2sin2φ−b2cos2φ3,Rθφ=a2a2sin2φ−b2cos2φ+Rssinφ.Specially,(3.b)b=ac/R02−a2=aD/R12−a2,(3.c)φ0=arctanbR0aR02−a2,φ1=π−arctanbR1aR12−a2.Figure 1
Geometric and reference system of a Sandwich FGP doubly curved structure.Figure 2
Generatrix diagram of three Sandwich FGP doubly curved structures.


### 2.2. Material Model

For FGP materials, there are three kinds of commonly used mechanical models: symmetric distribution model, nonsymmetric distribution model, and uniform distribution model [29]. In this paper, symmetric distribution model and nonsymmetric distribution model are considered. Four types of Sandwich FGP materials are shown in Figure 3. The material properties along the thickness direction are expressed as follows:(4.a)Ez=E11−e0α,(4.b)Gz=G11−e0α,(4.c)ρz=ρ11−emα.Figure 3
Schematic diagrams of four types of Sandwich FGP materials.The expressions of pore characteristic coefficientse0 and em are(5)e0=1−E2E1=1−G2G1,0≤e0≤1,em=1−ρ2ρ1,0≤em≤1,em=1−1−e0.It is obvious from equation (4) that, for different types of Sandwich FGP materials, only material parameter α can be adjusted. The expressions of material parameters α are as follows:(6)α=0z∈z1,z2cosπz−zizi+1−zi−π2z∈z2,z30z∈z3,z4,(7)α=1z∈z1,z2cosπz−zi2zi+1−ziz∈z2,z30z∈z3,z4,(8)α=cosπz−zizi+1−zi−π2z∈z1,z20z∈z2,z3cosπz−zizi+1−zi−π2z∈z3,z4,(9)α=cosπz−zi2zi+1−ziz∈z1,z20z∈z2,z3,1−cosπz−zi2zi+1−ziz∈z3,z4


### 2.3. Energy Functional of Shell Segment

In this paper, the FSDT is applied in the process of modeling [30, 31]. The displacement fields of Sandwich FGP doubly curved shells are(10.a)U¯φ,θ,z,t=uφ,θ,t+zψφφ,θ,t,(10.b)V¯φ,θ,z,t=vφ,θ,t+zψθφ,θ,t,(10.c)W¯φ,θ,z,t=wφ,θ,t,where u, v and w are displacements in the φ, θ and z directions on the middle surface, and ψφ and ψθ are rotations of the normal with respect to the θ and φ orientations. The linear strain–displacement relationships can be shown as(11.a)εφ0=1A∂u∂φ+vAB∂A∂θ+wRφ,(11.b)εθ0=1B∂v∂θ+uAB∂B∂φ+wRθ,(11.c)γφθ0=AB∂∂θuA+BA∂∂φvB,(11.d)kφ=1A∂ψφ∂φ+ψθAB∂A∂θ,(11.e)kθ=1B∂ψθ∂θ+ψφAB∂B∂φ,(11.f)kφθ=AB∂∂θψφA+BA∂∂φψθB,(11.g)γφz0=1A∂w∂φ−uRφ+ψφ,(11.h)γθz0=1B∂w∂θ−vRθ+ψθ,(11.i)A=Rφ,B=Rθsinφ.The relations between force/moment resultants and strains/curvatures change are(12.a)NφNθNφθMφMθMφθ=A11A120B11B120A12A220B12B22000A6600B66B11B120D11D120B12B220D12D22000B6600D66εφεθγφθkφkθkφθ,(12.b)QφQθ=κshearA6600A66γφzγθz,where the stiffness coefficients are obtained as follows:(13)Aij,Bij,Cij=∫−h/2h/2Qijz1,z,z2dz,Q11z=Ez1−μ2z,Q12z=μzEz1−μ2z,Q66z=Ez21+μz.In this research, the domain energy decomposition technique is exploited. To this end, the structure is divided intoN segments along the axis direction. By introducing penalty parameters, the continuity conditions for the interfaces can be realized. According to the FSDT, the strain energy Ui and kinetic energy Ti of ith segments can be expressed as(14)Ui=12∭VNφεφ0+Nθεθ0+Nφθγφθ0+Mφkφ+Mφkθ+Mφθkφθ+Qφγφz0+Qθγθz0ABdφdθdz,(15)Ti=12∭VρzU¯˙2+V¯˙2+W¯˙21+zRφ1+zRθABdφdθdz=12∫φ0φ1∫02πI0u˙2+v˙2+w˙2+2I1u˙ψ˙φ+v˙ψ˙θ+I2ψ˙φ2+ψ˙θ2ABdφdθ,where(16)I0,I1,I2=∫−h/2h/2ρz1+zRφ1+zRθ1,z,z2dz.As mentioned previously, the boundary spring simulation technique is exploited to simulate the corresponding boundary conditions. In addition, the continuity conditions are simulated by weighted parameters. The corresponding boundary and coupling potential energy can be described as(17)Ub=12∫02π∫−h/2h/2ku,0u2+kv,0v2+kw,0w2+kφ,0ψφ2+kθ,0ψθ2φ0=0Bdθdz+12∫02π∫−h/2h/2ku,1u2+kv,1v2+kw,1w2+kφ,1ψφ2+kθ,1ψθ2φ0=LBdθdz.(18)Uci=12∫02π∫−h/2h/2kuui−ui+12+kvvi−vi+12+kwwi−wi+12+kφψφi−ψφi+12+kθψθi−ψθi+12Bdθdz,where i and i+1 represent the ith and (i+1)th shell segments. When considering forced vibration, the work done by the external loads may be represented as(19)Wi=12∫φ∫θfuiui+fvivi+fwiwi+mφiψφi+mθiψθiRφR0dφdθ,where fui, fvi and fwi signify the related distributed forces along the φ, θ and z orientations, respectively; and the corresponding distributed couples with respect to the reference surface are represented by mφi and mθi.As a result, the Lagrangian energy functions (L) can be displayed as(20)L=∑iNTi−Ui+Wi−∑i=1N−1Uc,i−Ub.


### 2.4. Solution Methodology

With regard to the trial function method, the key issue is constructing a displacement admissible function, which satisfies the boundary conditions. In this paper, by introducing the domain energy decomposition technique, the structure is segmented, and weighted parameters are brought in to simulate the continuity conditions for the interfaces between each neighbored segment. This can make the choices of displacement admissible function become quite flexible. Since all segments are free boundaries, only the displacement functions are required to be continuous and orthogonal. Herein, the Jacobian orthogonal polynomials [32–34] are utilized for the expansion of the displacement. The displacement functions are exhibited as follows:(21.a)u=∑m=0M∑n=0NUmn,iPmα,βφcosnθ+sinnθeiωt,(21.b)v=∑m=0M∑n=0NVmn,iPmα,βφsinnθ+cosnθeiωt,(21.c)w=∑m=0M∑n=0NWmn,iPmα,βφcosnθ+sinnθeiωt,(21.d)ψφ=∑m=0M∑n=0Nψφmn,iPmα,βφcosnθ+sinnθeiωt,(21.e)ϕφ=∑m=0M∑n=0Nϕφmn,iPmα,βφcosnθ+sinnθeiωt,where Umn,i, Vmn,i, Wmn,i, ψφmn,i and ϕφmn,i are the Jacobi expanded coefficients; Pmα,βφ is the Jacobi polynomial of mth order, representing the displacement components along the generatrix orientation. In practice, different types of orthogonal polynomials can be obtained by resetting Jacobian parameters α and β. It must also be pointed out here that the relative errors between the calculated results under different Jacobian parameters are very small. In this paper, the parameters of Jacobi are selected as α = β = 0 (Legendre polynomials).Substituting equations (21) into equation (20) and expressing it in the form of partial differential equation, the discretized equation of motion is obtained:(22)∂L∂q=0,q=Umn,i,Vmn,i,Wmn,i,ψxmn,i,ϕxmn,i.Then, the forced vibration characteristic equation may be achieved:(23)Mq¨+K+KC+KBq=F,where M denotes the generalized mass matrix; K indicates the disjoint generalized stiffness matrix; KC and KB are the stiffness matrices related with weighted and boundary parameters, respectively. By assuming harmonic motionsq=q˜eiωt, the governing equations can be derived from equation (23):(24)det−ω2M+K−K¯λ−K¯γ−K∧B=0.Through the solution of equation (24), the frequencies as well as the mode shapes can be obtained.


## 2.1. Geometric Description

The orthogonal coordinate system (φ, θ, z) of the Sandwich FGP doubly curved shell is shown in Figure 1, where oz and o′z′ are the spatial coordinate axis and geometric central axis, respectively. By spinning the generatrix with respect to the central axis o′z′, structural surface can be achieved. Rφ and Rθ are primary curvature radius, the centers of which are Oφ and Oθ respectively. Rs represents the offset distance between o′z′ and oz. R0 signifies the sum of horizontal radius and Rs, where R0 = Rθsin φ. In this paper, three kinds of doubly curved shells are studied, and the corresponding generatrix characteristics are shown in Figure 2. The specific geometric equations are expressed as follows [27, 28]:(1)
Elliptical shell (Figure2(a)):(1.a)Rφφ=a2b2a2sin2φ+b2cos2φ3,Rθφ=a2a2sin2φ+b2cos2φ+Rssinφ,wherea indicates the length of the semimajor axes, and b represents that of the semiminor axes.(1.b)b=aLa2−R02−a2−R12(1.c)φ0=arctanbR0aa2−R02,φ1=arctanbR1aa2−R12.(2)
Paraboloidal shell (Figure2(b)):(2.a)Rφφ=k2cos3φ,Rθφ=k2cosφ+Rssinφ,where(2.b)k=R12−R02L(2.c)φ0=arctan2R0k,φ1=arctan2R1k.(3)
Hyperbolic shell (Figure2(c)):(3.a)Rφφ=−a2b2a2sin2φ−b2cos2φ3,Rθφ=a2a2sin2φ−b2cos2φ+Rssinφ.Specially,(3.b)b=ac/R02−a2=aD/R12−a2,(3.c)φ0=arctanbR0aR02−a2,φ1=π−arctanbR1aR12−a2.Figure 1
Geometric and reference system of a Sandwich FGP doubly curved structure.Figure 2
Generatrix diagram of three Sandwich FGP doubly curved structures.


## 2.2. Material Model

For FGP materials, there are three kinds of commonly used mechanical models: symmetric distribution model, nonsymmetric distribution model, and uniform distribution model [29]. In this paper, symmetric distribution model and nonsymmetric distribution model are considered. Four types of Sandwich FGP materials are shown in Figure 3. The material properties along the thickness direction are expressed as follows:(4.a)Ez=E11−e0α,(4.b)Gz=G11−e0α,(4.c)ρz=ρ11−emα.Figure 3
Schematic diagrams of four types of Sandwich FGP materials.The expressions of pore characteristic coefficientse0 and em are(5)e0=1−E2E1=1−G2G1,0≤e0≤1,em=1−ρ2ρ1,0≤em≤1,em=1−1−e0.It is obvious from equation (4) that, for different types of Sandwich FGP materials, only material parameter α can be adjusted. The expressions of material parameters α are as follows:(6)α=0z∈z1,z2cosπz−zizi+1−zi−π2z∈z2,z30z∈z3,z4,(7)α=1z∈z1,z2cosπz−zi2zi+1−ziz∈z2,z30z∈z3,z4,(8)α=cosπz−zizi+1−zi−π2z∈z1,z20z∈z2,z3cosπz−zizi+1−zi−π2z∈z3,z4,(9)α=cosπz−zi2zi+1−ziz∈z1,z20z∈z2,z3,1−cosπz−zi2zi+1−ziz∈z3,z4


## 2.3. Energy Functional of Shell Segment

In this paper, the FSDT is applied in the process of modeling [30, 31]. The displacement fields of Sandwich FGP doubly curved shells are(10.a)U¯φ,θ,z,t=uφ,θ,t+zψφφ,θ,t,(10.b)V¯φ,θ,z,t=vφ,θ,t+zψθφ,θ,t,(10.c)W¯φ,θ,z,t=wφ,θ,t,where u, v and w are displacements in the φ, θ and z directions on the middle surface, and ψφ and ψθ are rotations of the normal with respect to the θ and φ orientations. The linear strain–displacement relationships can be shown as(11.a)εφ0=1A∂u∂φ+vAB∂A∂θ+wRφ,(11.b)εθ0=1B∂v∂θ+uAB∂B∂φ+wRθ,(11.c)γφθ0=AB∂∂θuA+BA∂∂φvB,(11.d)kφ=1A∂ψφ∂φ+ψθAB∂A∂θ,(11.e)kθ=1B∂ψθ∂θ+ψφAB∂B∂φ,(11.f)kφθ=AB∂∂θψφA+BA∂∂φψθB,(11.g)γφz0=1A∂w∂φ−uRφ+ψφ,(11.h)γθz0=1B∂w∂θ−vRθ+ψθ,(11.i)A=Rφ,B=Rθsinφ.The relations between force/moment resultants and strains/curvatures change are(12.a)NφNθNφθMφMθMφθ=A11A120B11B120A12A220B12B22000A6600B66B11B120D11D120B12B220D12D22000B6600D66εφεθγφθkφkθkφθ,(12.b)QφQθ=κshearA6600A66γφzγθz,where the stiffness coefficients are obtained as follows:(13)Aij,Bij,Cij=∫−h/2h/2Qijz1,z,z2dz,Q11z=Ez1−μ2z,Q12z=μzEz1−μ2z,Q66z=Ez21+μz.In this research, the domain energy decomposition technique is exploited. To this end, the structure is divided intoN segments along the axis direction. By introducing penalty parameters, the continuity conditions for the interfaces can be realized. According to the FSDT, the strain energy Ui and kinetic energy Ti of ith segments can be expressed as(14)Ui=12∭VNφεφ0+Nθεθ0+Nφθγφθ0+Mφkφ+Mφkθ+Mφθkφθ+Qφγφz0+Qθγθz0ABdφdθdz,(15)Ti=12∭VρzU¯˙2+V¯˙2+W¯˙21+zRφ1+zRθABdφdθdz=12∫φ0φ1∫02πI0u˙2+v˙2+w˙2+2I1u˙ψ˙φ+v˙ψ˙θ+I2ψ˙φ2+ψ˙θ2ABdφdθ,where(16)I0,I1,I2=∫−h/2h/2ρz1+zRφ1+zRθ1,z,z2dz.As mentioned previously, the boundary spring simulation technique is exploited to simulate the corresponding boundary conditions. In addition, the continuity conditions are simulated by weighted parameters. The corresponding boundary and coupling potential energy can be described as(17)Ub=12∫02π∫−h/2h/2ku,0u2+kv,0v2+kw,0w2+kφ,0ψφ2+kθ,0ψθ2φ0=0Bdθdz+12∫02π∫−h/2h/2ku,1u2+kv,1v2+kw,1w2+kφ,1ψφ2+kθ,1ψθ2φ0=LBdθdz.(18)Uci=12∫02π∫−h/2h/2kuui−ui+12+kvvi−vi+12+kwwi−wi+12+kφψφi−ψφi+12+kθψθi−ψθi+12Bdθdz,where i and i+1 represent the ith and (i+1)th shell segments. When considering forced vibration, the work done by the external loads may be represented as(19)Wi=12∫φ∫θfuiui+fvivi+fwiwi+mφiψφi+mθiψθiRφR0dφdθ,where fui, fvi and fwi signify the related distributed forces along the φ, θ and z orientations, respectively; and the corresponding distributed couples with respect to the reference surface are represented by mφi and mθi.As a result, the Lagrangian energy functions (L) can be displayed as(20)L=∑iNTi−Ui+Wi−∑i=1N−1Uc,i−Ub.


## 2.4. Solution Methodology

With regard to the trial function method, the key issue is constructing a displacement admissible function, which satisfies the boundary conditions. In this paper, by introducing the domain energy decomposition technique, the structure is segmented, and weighted parameters are brought in to simulate the continuity conditions for the interfaces between each neighbored segment. This can make the choices of displacement admissible function become quite flexible. Since all segments are free boundaries, only the displacement functions are required to be continuous and orthogonal. Herein, the Jacobian orthogonal polynomials [32–34] are utilized for the expansion of the displacement. The displacement functions are exhibited as follows:(21.a)u=∑m=0M∑n=0NUmn,iPmα,βφcosnθ+sinnθeiωt,(21.b)v=∑m=0M∑n=0NVmn,iPmα,βφsinnθ+cosnθeiωt,(21.c)w=∑m=0M∑n=0NWmn,iPmα,βφcosnθ+sinnθeiωt,(21.d)ψφ=∑m=0M∑n=0Nψφmn,iPmα,βφcosnθ+sinnθeiωt,(21.e)ϕφ=∑m=0M∑n=0Nϕφmn,iPmα,βφcosnθ+sinnθeiωt,where Umn,i, Vmn,i, Wmn,i, ψφmn,i and ϕφmn,i are the Jacobi expanded coefficients; Pmα,βφ is the Jacobi polynomial of mth order, representing the displacement components along the generatrix orientation. In practice, different types of orthogonal polynomials can be obtained by resetting Jacobian parameters α and β. It must also be pointed out here that the relative errors between the calculated results under different Jacobian parameters are very small. In this paper, the parameters of Jacobi are selected as α = β = 0 (Legendre polynomials).Substituting equations (21) into equation (20) and expressing it in the form of partial differential equation, the discretized equation of motion is obtained:(22)∂L∂q=0,q=Umn,i,Vmn,i,Wmn,i,ψxmn,i,ϕxmn,i.Then, the forced vibration characteristic equation may be achieved:(23)Mq¨+K+KC+KBq=F,where M denotes the generalized mass matrix; K indicates the disjoint generalized stiffness matrix; KC and KB are the stiffness matrices related with weighted and boundary parameters, respectively. By assuming harmonic motionsq=q˜eiωt, the governing equations can be derived from equation (23):(24)det−ω2M+K−K¯λ−K¯γ−K∧B=0.Through the solution of equation (24), the frequencies as well as the mode shapes can be obtained.


## 3. Convergence Studies and Numerical Verification

The accuracy of the whole theory should be closely related to the expansion coefficients of Jacobian orthogonal polynomials, the number of segments, and the weighted parameters between segments. Therefore, it is essential to study the convergence and correctness of the numerical results. In order to simplify the description of edge conditions, the F, C, and Ei (i = 1,2,3) represent the free, clamped, and elastic edge conditions, respectively.Table1 shows the convergence analysis of the number of segments for Sandwich FGP doubly curved structure. The material parameters are defined as follows: E1 = 70 GPa, μ = 0.3, ρ = 2702 kg/m3, e0 = 0.2. Sandwich material type is 2–1. If the material parameters are not specified later, the material parameters will remain unchanged. The thickness ratio coefficient of Sandwich structure is defined as 1 : 3:1. The geometric parameters are defined as follows: paraboloidal shell: R0 = 2 m, R1 = 4 m, Rs = 1 m, k = 1, h = 0.1 m; hyperbolic shell: R1 = 2 m, Rs = 1 m, c = 1 m, D = 4 m, h = 0.1 m; elliptical shell: a1 = 1 m, b1 = 2 m, Rs = 1 m, h = 0.1 m, φ0 = π/6, φ0 = 5π/6. As shown in Table 1, it is obvious that when the number of segments exceeds 3, the computed results tend to be stable. Therefore, in all subsequent examples, the number of segments is chosen to be N = 4. Table 2 shows the convergence analysis of truncation terms for Sandwich FGP doubly curved structure with C-F boundary conditions. The geometric dimensions and material parameters are in accordance with those in Table 1.Table 1
Convergence analysis of number of segments for Sandwich FGP doubly curved structure.
NParaboloidal shellHyperbolic shellElliptical shell12341234123429.469.8015.9423.3042.9147.4956.5874.9445.5760.60128.43167.7539.459.8015.9123.3042.8347.4756.4174.9345.5660.59128.40167.7549.459.8015.9023.3042.7947.4756.3074.9345.5660.59128.40167.7559.459.8015.9023.3042.7847.4756.2574.9445.5660.59128.40167.7569.459.8015.9023.3042.7747.4756.2374.9445.5660.59128.40167.7589.459.8015.9023.3042.7747.4756.2174.9445.5660.59128.40167.75109.459.8015.9023.3042.7747.4756.2174.9445.5660.59128.40167.75Table 2
Convergence analysis of truncation terms for Sandwich FGP doubly curved structure.
mParaboloidal shellHyperbolic shellElliptical shell12341234123429.469.8015.9423.3042.9147.4956.5874.9445.5760.60128.43167.7539.459.8015.9123.3042.8347.4756.4174.9345.5660.59128.40167.7549.459.8015.9023.3042.7947.4756.3074.9345.5660.59128.40167.7559.459.8015.9023.3042.7847.4756.2574.9445.5660.59128.40167.7569.459.8015.9023.3042.7747.4756.2374.9445.5660.59128.40167.7589.459.8015.9023.3042.7747.4756.2174.9445.5660.59128.40167.75109.459.8015.9023.3042.7747.4756.2174.9445.5660.59128.40167.75From Table2, it is also found that when the Jacobian orthogonal polynomial is expanded to six terms, the calculated results can be regarded as exact ones. Therefore, in Table 1 and all the following examples, Jacobian orthogonal polynomials are uniformly expanded into eight terms. The results of the convergence study of weighted parameters are shown in Table 3. From Table 3, it can be found that when the weighted parameter is small, the result is unstable. With the increase of weighted parameters, the calculated results converge rapidly. When the weighted parameter is more than 1020, the results diverge. Therefore, it is indicated that when the weighted parameters are set between 1014 and 1018, stable converged results can be obtained. In all subsequent examples, the weighted parameters are set as 1014.Table 3
Convergence analysis of weighted parameters for Sandwich FGP doubly curved structure.
ΛParaboloidal shellHyperbolic shellElliptical shell1234123412341083.845.978.4614.239.0314.4922.3423.1617.9538.6939.4139.8510109.019.2715.8621.5955.4160.9869.3182.1345.2055.59126.38154.6810129.459.7915.9023.2842.8647.5956.3375.0745.5660.53128.38167.6110149.459.8015.9023.3042.7947.4756.3074.9345.5660.59128.40167.7510169.459.8015.9023.3042.7947.4756.3074.9345.5660.59128.40167.7510189.459.8015.9023.3042.7947.4756.3074.9345.5660.59128.39167.7510209.459.8315.8823.3042.7147.4656.2974.9345.6860.55128.48167.7610229.8321.0927.2835.3257.0464.6866.6577.9546.3271.05170.45174.491024131.67134.72160.47169.09140.82181.57200.96210.62369.36445.81481.35566.60Based on the study of convergence, the correctness of the present method is further studied. Table4 shows the numerical comparison of Sandwich FGP doubly curved structure with various boundary conditions. The geometric parameters are as follows: elliptical Shell: a1 = 1 m, b1 = 2 m, Rs = 1 m, h = 0.1 m, φ0 = π/6, φ0 = 5π/6; hyperbolic shell: R1 = 2 m, Rs = 0 m, c = 1 m, D = 4 m, h = 0.1 m. The dimensionless frequencies are as follows: Ω=ωRs/hρ1/E1(elliptical Shell) and Ω=ωD2ρ1/E1(hyperbolic shell). Sandwich material type is 1–2, and the thickness coefficient of Sandwich structure is defined as 0-1-0. Table 4 shows that the results obtained in this paper are very close to those obtained by Zhao et al. [35] using Ritz method. Table 5 gives the comparative data of Sandwich FGP doubly curved elliptical shell, which are obtained from finite element method (FEM) through ABAQUS. The boundary condition is defined as F-C. Geometric parameters are defined as follows: a1 = 1 m, b1 = 2 m, Rs = 0 m, h = 0.1 m, φ0 = 0, φ0 = π/2. The Sandwich material type and thickness coefficient are consistent with those in Table 1. Table 5 shows that the present method is appropriate for calculating free vibration features of Sandwich FGP doubly curved shells.Table 4
Numerical comparison of frequencies for Sandwich FGP doubly curved structure with various boundary conditions.
TypeModeElliptical shellHyperbolic shellC-CC-FC-CC-FPresentReference [35]PresentReference [35]PresentReference [35]PresentReference [35]Type1-113.013.0090.5740.5742.6872.681.8391.83823.0123.0110.7380.7383.0253.0132.2182.21733.143.141.6121.6124.1824.1512.6332.62743.1473.1462.0282.0294.4044.3862.9842.974Type1-213.0033.0040.5620.5632.6712.6651.8141.81423.0053.0060.7350.7352.9692.9592.2172.21733.1293.1291.5811.5824.0874.0582.6132.60743.1383.1392.0282.034.3334.312.9252.916Table 5
Numerical comparison of frequencies for Sandwich FGP elliptical shell accompanied with F-C boundary condition.
Mode12345678910Present272.39420.47466.94550.27591.89595.34701.03712.70757.62789.95FEM273.15420.2466.42553.54592.22595.2703.33712.35757.27790.86Next, the correctness of this method for forced response prediction is further verified. Due to the lack of relevant literature results, the finite element software ABAQUS is used. The geometric dimensions and material parameters are identical with those in Table5. Figure 4 shows three loads: point load, line load, and surface load. In the steady-state response analysis, the load form is point load, and the position of action and observation point are Aφ,θ=0,0,0,0 and Oφ,θ=π/4,π/4,0,0, respectively. It should be noted that the first item in parentheses represents generatrix direction φ and the second one represents circumferential direction θ. The point load is as follows: fw=f¯wδφ−φAθ−θA, where f¯w=−1N, and δ is the Dirac delta function. Frequency ranges from 500 Hz to 1200 Hz, with a total of 501 sweeps. Figure 5 shows four specific forms of impact load function: (a) rectangular pulse signal; (b) triangular pulse signal; (c) half-sine pulse signal; and (d) exponential pulse signal, respectively [36]. In transient response analysis, the shock load function is chosen as rectangular wave. The loading time is 10 ms, and the interpolation point is 501. The loading position is consistent with that of the steady-state response, and the coordinates of observation points are defined as (π/3, 0). Figure 6 shows the comparison of steady-state response and transient response, respectively. It is obvious from Figure 6 that the proposed method has excellent accuracy for the prediction of forced response of Sandwich FGP doubly curved structure.Figure 4
Schematic diagram of three categories of loads. (a) Point force. (b) Line force. (c) Surface force.
(a)(b)(c)Figure 5
The diagram of load signal. (a) Rectangular pulse signal. (b) Triangular pulse signal. (c) Half-sine pulse signal. (d) Exponential pulse signal.
(a)(b)(c)(d)Figure 6
Verification of the correctness of responses: (a) steady-state response; (b) transient response.
(a)(b)


## 4. Numerical Results and Discussion

The third part has made a detailed comparative study on the correctness of the theoretical modeling, which verifies that the proposed model has excellent convergence speed and computational accuracy. In this part, numerical discussions will be further carried out.Tables6 and 7 show the first four frequencies of four Sandwich FGP materials, respectively. The geometric dimensions are as follows: paraboloidal shell: Case 1:R0 = 2 m, R1 = 4 m, Rs = 1 m, k = 1, h = 0.1 m; Case 2:R0 = 2 m, R1 = 4 m, Rs = 1 m, k = 2, h = 0.1 m; Case 3:R0 = 2 m, R1 = 4 m, Rs = 1 m, k = 3, h = 0.1 m; hyperbolic shell: Case 1:R1 = 2 m, Rs = 1 m, c = 1 m, D = 4 m, h = 0.1 m; Case 2:R1 = 2 m, Rs = 1 m, c = 2 m, D = 4 m, h = 0.1 m; Case 3:R1 = 2 m, Rs = 1 m, c = 3 m, D = 4 m, h = 0.1 m; Elliptical shell: Case 1: a1 = 1 m, b1 = 1 m, Rs = 1 m, h = 0.1 m, φ0 = π/6, φ0 = 5π/6; Case 2: a1 = 1 m, b1 = 2 m, Rs = 1 m, h = 0.1 m, φ0 = π/6, φ0 = 5π/6; Case 3: a1 = 1 m, b1 = 3 m, Rs = 1 m, h = 0.1 m, φ0 = π/6, φ0 = 5π/6. For parabolic shells, the frequency parameters of shells increase with the increase of characteristic coefficient k. For hyperbolic shells, when the structure coefficient c increases, the frequency parameters of the shells decrease. For elliptical shells, when the structural coefficient b increases, the frequency parameters decrease accordingly. The above results indicate that the change of parameters directly affects the stiffness matrix of the Sandwich FGP doubly curved structure. Besides, the frequency parameters of Type 1--1 Sandwich FGP materials are the largest, while those of Type 1-2 Sandwich materials are the smallest. For the study of free vibration, modal modes can provide more information. Therefore, modal modes under some elastic boundary conditions are given in Figures 7, 8, 9.Table 6
First four frequencies for Type 1-1 Sandwich FGP doubly curved structure with diverse boundary conditions.
TypeModeParabolic shellHyperbolic shellElliptical shellCCSSCFCCSSCFCCSSCFCase 1166.0563.4015.27107.90105.2443.61361.55276.5871.18267.3964.7218.55112.79109.1847.32389.58315.40130.70372.2869.6923.88135.21132.5358.47461.71407.96146.15477.9075.5427.64139.24133.6874.45551.69515.53220.14Case 21118.03109.8423.74100.6498.6241.48247.30219.5847.232120.89112.1025.83110.98107.2442.22247.48247.1060.643121.30114.1331.06120.62118.7358.03258.01247.46132.534131.65125.8741.13139.31133.3164.42258.59257.41166.64Case 31162.09143.7529.2697.2195.5539.66135.46135.3235.272163.01144.2131.77108.38105.4739.99137.59137.3143.473165.72148.3036.00109.28107.0857.33153.59153.53111.174168.36149.7645.25115.59112.4357.87157.70157.48120.46Table 7
First four frequencies for Type 1-2 Sandwich FGP doubly curved structure with diverse boundary conditions.
TypeModeParabolic shellHyperbolic shellElliptical shellCCSSCFCCSSCFCCSSCFCase 1164.5261.7914.92105.94102.8842.55352.10267.8569.38266.2263.4817.88109.70105.7746.70379.98307.29126.54370.1967.5423.57133.47128.7156.51451.42400.48143.67476.8374.4326.56134.57130.3473.61539.98507.61216.17Case 21115.63107.9922.9898.7496.4440.40243.99213.5145.662118.02109.7125.32107.84103.7841.64244.17243.9059.703119.20112.6229.90119.08116.8156.05254.14243.95128.404129.66124.5040.56134.59128.3163.70254.95254.14164.83Case 31158.57142.0928.3295.3493.4238.88133.21133.0834.732159.15143.0231.05106.03101.9139.10135.84135.6041.983162.41145.9934.67107.00105.3755.87149.92149.87109.984164.02148.8844.51113.32109.8456.69155.88155.69117.50Figure 7
Several typical mode shapes for Sandwich FGP paraboloidal shell. (a) Paraboloidal shell (k = 1) with E1-E1 boundary conditions. (b) Paraboloidal shell (k = 2) with E2-E2 boundary conditions. (c) Paraboloidal shell (k = 3) with E3-E3 boundary conditions.
(a)(b)(c)Figure 8
Several typical mode shapes for Sandwich FGP hyperbolic shell. (a) Hyperbolic shell (c = 1) with E1-E1 boundary conditions. (b) Hyperbolic shell (c = 2) with E2-E2 boundary conditions. (c) Hyperbolic shell (c = 3) with E3-E3 boundary conditions.
(a)(b)(c)Figure 9
Several typical mode shapes for Sandwich FGP elliptical shell. (a) Elliptical shell (b = 1) with E1-E1 boundary conditions. (b) Elliptical shell (b = 2) with E1-E1 boundary conditions. (c) Elliptical shell (b = 3) with E1-E1 boundary conditions.
(a)(b)(c)Next, the parameters of the forced response are examined. The geometric parameters are consistent with those of Case 1 in Tables6, 7, 8, and 9. In the steady-state response, the load type is linear. The boundary is defined as C-C, and the Sandwich FGP material is selected as Types 1-2. For different doubly curved structures, the location of load action is defined as follows: paraboloidal shell: Aφ,θ=π/3,2π/3,0,π; hyperbolic shell: Aφ,θ=π/2,π/2,0,π; and elliptical shell: Aφ,θ=π/2,π/2,0,π. The locations of observation point are defined as follows: paraboloidal shell: Oφ,θ=π/3,π/3,0,0; hyperbolic shell: Oφ,θ=π/4,π/4,0,0; and elliptical shell: Oφ,θ=π/4,π/4,0,0. In order to simplify the transient study, the shock load function is chosen as rectangular wave. The position of the impact load is consistent with that of the steady-state response. Similarly, observation points are defined as follows: paraboloidal shell: Oφ,θ=3π/10,3π/10,0,0; hyperbolic shell: Oφ,θ=π/2,π/2,0,0; and elliptical shell: Oφ,θ=π/4,π/4,0,0.Table 8
First four frequencies for Type 2-1 Sandwich FGP doubly curved structure with diverse boundary conditions.
TypeModeParabolic shellHyperbolic shellElliptical shellCCSSCFCCSSCFCCSSCFCase 1164.9462.4415.01107.20104.7042.79354.72273.6269.75267.0364.5217.74110.00106.6447.47383.30313.01126.40370.2367.7923.95134.10128.9356.30456.33406.35145.66478.0475.8326.29135.71133.1774.93546.42513.89218.24Case 21116.71109.1422.9299.8697.9540.55248.11219.3345.562118.70110.5025.56108.03104.5342.31248.57248.0760.593120.66114.0829.62121.11119.3055.79257.72248.33128.404131.52126.2540.85134.06128.5064.85259.37257.72167.75Case 31159.78142.4928.2396.3994.8038.96135.12134.9935.262160.05143.6331.25106.05102.5339.72138.25138.0141.873163.93146.1634.37108.83107.5755.60151.02150.98111.924164.60149.6645.07114.48111.5457.71158.70158.51118.26Table 9
First four frequencies for Type 2-2 Sandwich FGP doubly curved structure with diverse boundary conditions.
TypeModeParabolic shellHyperbolic shellElliptical shellCCSSCFCCSSCFCCSSCFCase 1165.0762.5315.04107.44104.8542.87355.24273.6169.86267.1964.6417.76110.18106.7347.59383.91313.29126.55370.3467.8624.01134.25128.9856.36457.13407.18145.98478.2376.0026.31136.07133.4275.14547.42515.20218.49Case 21116.95109.4122.95100.0898.1040.62248.72219.3545.612118.92110.7325.62108.20104.6142.41249.20248.6860.733120.94114.4029.65121.42119.5355.85258.30248.95128.554131.84126.6340.88134.21128.5465.03259.95258.30168.22Case 31160.10143.0028.2696.5994.9539.02135.43135.3035.342160.35144.1931.32106.20102.5839.82138.61138.3741.913164.28146.6334.40109.11107.7855.66151.29151.24112.234164.89150.2845.18114.72111.7057.86159.12158.94118.45Figures10 and 11, respectively, study the steady and transient responses of the structure under different elastic boundary conditions. It can be found that the boundary condition parameters have a significant effect on the forced response of the Sandwich FGP doubly curved structure. Figure 12 and 13, respectively, present the steady and transient responses of the Sandwich FGP doubly curved structure under different porous coefficient. For steady-state response analysis, with the same frequency range, the number of resonance peaks decreases with the increase of pore coefficient. For transient response analysis, the increase of porous coefficient results in the increase of response amplitude to a certain extent. The main reason for this is that the increase of porous coefficient can reduce the quality of the structure. Figures 14 and 15 show the effect of the thickness of FGP materials on the forced response of Sandwich FGP doubly curved structure. From these two graphs, it is observed that the thickness coefficient of FGP layer is not very sensitive to the forced response of the Sandwich FGP doubly curved structure. For the steady-state response, the resonance peak occurs at larger frequencies as the thickness increases.Figure 10
Study on the influence of elastic boundary conditions on the steady-state response of Sandwich FGP doubly curved structure: (a) Sandwich FGP paraboloidal shell; (b) Sandwich FGP hyperbolic shell; (c) Sandwich FGP elliptical shell.
(a)(b)(c)Figure 11
Study on the influence of elastic boundary conditions on the transient response of Sandwich FGP doubly curved structure: (a) Sandwich FGP paraboloidal shell; (b) Sandwich FGP hyperbolic shell; (c) Sandwich FGP elliptical shell.
(a)(b)(c)Figure 12
Study on the effect of porous coefficient on the steady-state response of Sandwich FGP doubly curved structure: (a) Sandwich FGP paraboloidal shell; (b) Sandwich FGP hyperbolic shell; (c) Sandwich FGP elliptical shell.
(a)(b)(c)Figure 13
Study on the effect of porous coefficient on the transient response of Sandwich FGP doubly curved structure: (a) Sandwich FGP paraboloidal shell; (b) Sandwich FGP hyperbolic shell; (c) Sandwich FGP elliptical shell.
(a)(b)(c)Figure 14
Study on the effect of thickness on the steady-state response of Sandwich FGP doubly curved structure: (a) Sandwich paraboloidal honeycomb shell; (b) Sandwich hyperbolic honeycomb shell; (c) Sandwich elliptical honeycomb shell.
(a)(b)(c)Figure 15
Study on the effect of thickness on the transient response of Sandwich FGP doubly curved structure: (a) Sandwich paraboloidal honeycomb shell; (b) Sandwich hyperbolic honeycomb shell; (c) Sandwich elliptical honeycomb shell.
(a)(b)(c)


## 5. Conclusions

In this research, free and force vibration analyses of a novel Sandwich FGP doubly curved shell subjected to arbitrary boundary conditions are conducted in the framework of the FSDT. The theoretical modeling is based on the domain energy decomposition and boundary spring simulation techniques. For every segment, the displacement functions are represented by Jacobi polynomials along the generatrix orientation and trigonometric series in the circumferential orientation. Free vibration and forced response of Sandwich structures can be obtained by standard variational operation of unknown coefficients of displacement functions. The convergence and validity of the established analysis model are given through several numerical cases. Some new results are given, which may be served as reference data. In addition, parameterized study of forced response is also carried out for pore parameters, boundary parameters, and thickness parameters.



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