# Modeling and Vibration Analysis of a Porous Rotational Shell Based on Biot Theory

Authors: Jing Lu; Yu Xiang; Sha Chen; Hao Jin; ZhengXuan Chen
Journal: Shock and Vibration
 (2021)

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## Abstract

Combining the Biot theory and classical elastic theory for thin shells, a new dynamic model of a thin fluid-saturated porous rotational shell is proposed. First-order ordinary differential control equations of the porous rotational shell are derived in the frequency domain. These equations are then solved by using the precise element method. The accuracy of this model has been verified by comparing with a vibration experiment. Moreover, the comparisons between the present model and two equivalent property models are carried out. Because the present approach considers the fluid-solid coupling effect and makes no assumptions for the fluid displacements, it is more accurate in the high-frequency range. Lastly, the dynamic characteristics of porous rotational shells are demonstrated by the proposed method.


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## Body

## 1. Introduction

Porous materials are widely used for passive absorption and noise control in many fields, such as the vehicle, aerospace, geophysics, and civil engineering. Many researchers have studied the vibration and acoustic characteristic of poroelastic structures since the 1950s. In 1956, Biot [1, 2] established the three-dimensional theory of wave propagation in fluid-saturated porous elastic solids to describe the linear acoustic behaviours of these porous media. Several representative formulations are presented by Zienkiewicz et al. [3, 4], such as u-U-p, u-U, u-p, and u-w forms. Recently, Cieszko and Kubik [5] put forward the analysis of equations to describe wave propagation in fluid-saturated porous materials. By applying some assumptions, the Biot theory has been applied to the research on porous plates and shells. For instance, neglecting the relative fluid/solid displacements and defining the in-plane solid displacements as functions of the transverse solid displacement, Manuel et al. derived a mixed displacement-pressure formulation for the bending vibration of porous plates by combining the thin plate theory and Biot theory [6]. This formulation is useful for porous plates saturated by air. In order to evaluate the energy dissipation due to the viscous friction, Leclaire and Horoshenkov proposed an analytical model of a fluid-saturated porous plate by considering the fluid loading terms and introducing relative transverse fluid/solid displacements [7]. However, it was pointed out that the model was only valid for low frequencies since the thickness of the plate was assumed to be smaller than the acoustic wavelength and the variation of the amplitude of the transverse fluid displacement was also assumed to be small. Xiang et al. [8] proposed a precise dynamic model of a thin rectangular porous plate based on the classical theory of thin plate. Theodorakopoulos and Beskos [9] assumed that the in-plane fluid flow relative to the motion of the solid was negligible compared to the transverse fluid flow, and the governing equation written in terms of the three solid displacements and fluid transverse displacement was derived to demonstrate the flexural plate dynamic characteristic in the frequency domain. Based on a similar hypothesis, research on porous cylindrical shells has been carried out. Ahmed Shah et al. obtained a frequency domain equation for the axially symmetric vibration of an infinite poroelastic circular cylindrical shell with different boundary conditions [10]. Employing a first-order shear deformation theory and the 3D Biot theory, Julien et al. derived an analytical model of an infinite sandwich cylindrical shell to demonstrate the influence of the structural damping [11]. However, these infinite cylindrical shell models are quite different from the shells used in practical applications. Due to the difficulty of analytical modeling of this problem, many numerical methods were adopted [12, 13], and the finite element method (FEM) combined with equivalent models is frequently implemented instead of the analytical method [14, 15]. Boily and Charron created a finite element model of a sandwich cylindrical shell [16]. In the model, the poroelastic core was simplified to an equivalent fluid model, and the modal reduction method was used to reduce the size of the system. The finite element method is convenient and widely used, but the FEM model of porous structure is usually large, complex, and time-consuming, especially at high frequencies. Therefore, this method can only be practically applied in low-frequency cases. On the other hand, rotational shells are widely used in various fields, such as aerospace, ships, and civil machinery. Porous is usually treated on these rotational shells to reduce the noise and isolate the vibration. However, the existing research studies mostly focus on porous beams, plates, and cylindrical shells, and vibration analyses of rotational porous shells are not available in the literature.In this article, accounting for the coupling interaction between the fluid and solid phases, the dynamic governing equations of the porous rotational shell based on classical thin shell theory and Biot theory are derived. The equation is written in a first-order ordinary differential matrix form and solved using a precise integration method. Subsequently, corresponding experiment results are used to verify the accuracy of the present method. Finally, some additional numerical examples are demonstrated.


## 2. Governing Equations

### 2.1. Constitutive Equations of the Porous Rotational Shell

The stress-strain relations of a homogeneous, isotropic poroelastic material are [1, 2](1)σij=2Neij+Ae+Qεδij,i,j=1,2,3,−ϕp=Qe+Rε.The first set of formulas in equation (1) describes the constitutive relations of the solid phase, and the second set of formulas describes the constitutive relations of the fluid phase. The elastic coefficients of the porous material A and N have a similar meaning with the Lamé coefficients for an elastic material. N is the shear modulus of the skeleton material. R is a pressure coefficient forcing a certain volume of the fluid into a constant aggregate. Q represents the coupling coefficient between the volume change of the solid and liquid. δij is the Kronecker delta. p is the fluid pressure. ϕ is the porosity. σij and eijare the stress and strain tensor of the solid skeleton, respectively. e and ε are the dilations of solid and fluid, respectively, which are given by(2)e=ekk=uk,k,ε=εkk=Uk,k,k=1,2,3,where u is the solid displacement tensor and U is the fluid displacement tensor. Note that the repetition of the index k implies summation; commas indicate differentiation with respect to space variables; and εij denotes the fluid strain tensor. In a spherical coordinate system, k is equal to φ,θ, or z.The middle plane of the thin porous rotational shell coincides with the natural coordinates, as shown in Figure1, in which, φ, θ, and z are meridian, circumferential, and normal coordinates, respectively, with relevant curvature radii of Rφ and Rθ, respectively. The midplane displacements in the meridian, circumferential, and normal directions are u,v,andw, respectively. The midplane Lamé coefficients in meridian and circumferential directions are Rφ and R0, respectively, and are both the functions of φ.Figure 1
The porous rotational shell.Applying the classical elastic theory for a thin shell, the normal line (z-direction) of the middle plane is assumed to be straight. Additionally, the angles between the normal line and the vertical lines of the meridian and circumferential direction are constants. Furthermore, the total normal stress can be neglected. Applying these assumptions, one obtains(3)eφz=0,eθz=0,σz−ϕpη=0.The fluid dilation can be evaluated from the second set of formulas in equation (1) as(4)ε=−ϕp−QeR.Substituting equation (4) into the first set of formulas of equation (1) and omitting the two shear stress components according to equation (3) yield(5)σφ=2Neφ+A−Q2Re−QRϕp,σθ=2Neθ+A−Q2Re−QRϕp,σz=2Nez+A−Q2Re−QRϕp,τφθ=Neφθ.By eliminatingez from the third formula of equation (5) with the third formula of equation (3), the solid dilation is obtained:(6)e=1−B¯eϕ+eθ+A¯ϕp,where A¯ and B¯ are constants, with A¯=Q+R/2NR+AR−Q2,B¯=AR−Q2/2NR+AR−Q2.Combing equations (5) and (6), the solid constitutive equations of a porous rotational shell take the following form:(7)σφ=N¯¯eφ+A¯¯eθ+B¯¯ϕp,σθ=N¯¯eθ+A¯¯eφ+B¯¯ϕp,τφθ=Neφθ,where A¯¯=A−Q2/R1−B¯,B¯¯=A−Q2/RA¯−Q2/R, N¯¯=2N+A−Q2/R1−B¯.Substituting equation (6) into equation (4), the fluid constitutive equations of a porous rotational shell are obtained:(8)ε=A¯¯′ϕp+B¯¯′eφ+eθ,where A¯¯′=−1+QA¯/R,B¯¯′=−Q/R1−B¯.


### 2.2. Governing Equations of the Solid Frame

Applying the Reissner model with classical thin shell theory, the solid displacements can be written as [17](9)u^s=us+zθφ,v^s=vs+zR0vssinφ−∂ws∂θ,w^s=ws,where u^s=u^1, v^s=u^2, w^s=u^3 are the solid displacements at an arbitrary point of the shell. The displacements us, vs, ws correspond to the midplane displacements. The midplane rotation θφ is(10)θφ=1Rφus−∂ws∂φ.The corresponding solid strains can be represented using the midplane solid displacements as follows [17]:(11)eφ=1Rφ∂us∂φ+1Rφws+zRφ∂θφ∂φ,eθ=cosφR0us+1R0∂vs∂θ+sinφR0ws+zR0∂∂θsinφR0vs−1R0∂ws∂θ+zcosφR0θφ,eφθ=1R0∂us∂θ+1Rφ∂vs∂φ−cosφR0vs+2zR0∂θφ∂θ.By assuming that the dynamic external excitation is harmonic and expanding each variable functionfφ,θ,t with the Fourier series along the circumferential direction, one obtains(12)fφ,θ,t=∑nf˜nφejnθejωt,where f˜n is the amplitude of the nth order component of f in the frequency domain, j=−1, and ω is the circular excitation frequency. In the following sections, the factors ejωt and ejnθ, and the superscript n are omitted for convenience.By combing equations (7) and (11) and applying a Fourier series expansion, the internal force-displacement relations of the solid frame are(13)N˜θ=∫−h/2h/2σ˜θdz=A¯¯Rφhdu˜sdφ+N¯¯njR0hv˜s+N¯¯cosφR0hu˜s+N¯¯sinφR0+A¯¯Rφhw˜s+B¯¯hP˜,N˜φθ=N˜θφ=∫−h/2h/2τ˜φθdz=NhnjR0u˜s+NhRφdv˜sdφ−NhcosφR0v˜s,M˜φ=∫−h/2h/2σ˜φzdz=h312N¯¯Rφdθ˜φdφ+h312A¯¯njsinφR02v˜s+h312A¯¯n2R02w˜s+h312A¯¯cosφR0θ˜φ+h312B¯¯M˜p,M˜θ=∫−h/2h/2σ˜θzdz=h312N¯¯njsinφR02v˜s+h312N¯¯n2R02w˜s+h312A¯¯Rφdθ˜φdφ+h312N¯¯cosφR0θ˜φ+h312B¯¯M˜p,M˜φθ=Mθφ=∫−h/2h/2τ˜φθzdz=h3122NnjR0θ˜φ.In equation (13), the following midplane variables are introduced: u˜s=1/h∫−h/2h/2u˜sdz, v˜s=1/h∫−h/2h/2v˜sdz, w˜s=1/h∫−h/2h/2w˜sdz, p˜=1/h∫−h/2h/2ϕp˜dz, M˜p=12/h3∫−h/2h/2zϕp˜dz, where h is the thickness of the shell; N˜φ and N˜θ are the component amplitudes of the axial force per unit length in the φ and θ directions, respectively; N˜φθ is the amplitude of the shear force per unit length; M˜φ and M˜θ are the amplitudes of the bending moment per unit length in the φ and θ directions, respectively; and M˜φθ is the amplitude of the torque per unit length.According to the Biot theory, the dynamic equations of the solid frame are [1, 2](14)σij,j−bu˙i−U˙i+Fi=ρ11u¨i+ρ12U¨i,i,j=1,2,3,where the dots indicate differentiation with respect to time. Fii=1,2,3 are the body forces per unit volume. Ui is the fluid midplane displacement. ρ11=1−ϕρs+ϕρfα∞−1 and ρ12=ϕρfα∞−1 are the mass density coefficients related to the frame and fluid mass densities, respectively. ρs and ρf are the frame mass density and fluid mass density, respectively. α∞ is the tortuosity. b=υϕ2/q is a coefficient related to the permeability q, and υ is the fluid viscosity.Based on thin shell theory and considering equation (14), the solid dynamic equations of the fluid-saturated porous rotational shell can be derived in the frequency domain as(15)RφcosφN˜φ+R0∂N˜φ∂φ+Rφ∂N˜θφ∂θ−RφcosφN˜θ+R0Q˜φ+RφR0q˜φ−RφR0hΩ11u˜s−Ω12u˜f=0,Rφ∂N˜θ∂θ+2RφcosφN˜φθ+R0∂N˜φθ∂φ+RφR0Q˜θRθ+RφR0q˜θ−RφR0hΩ11v˜s−Ω12v˜f=0,RφcosφQ˜φ+R0∂Q˜φ∂φ+Rφ∂Q˜θ∂θ−RφR0N˜φRφ−RφR0N˜θRθ+RφR0q˜z−RφR0hΩ11w˜s−Ω12w˜f=0,Rφ∂M˜θ∂θ+RφcosφM˜φθ+R0∂M˜φθ∂φ+RφcosφM˜θφ−RφR0Q˜θ=0,RφcosφM˜φ+R0∂M˜φ∂φ+Rφ∂M˜βφ∂θ−RφcosφM˜θ−RφR0Q˜φ=0,where Ω11=bωj−ρ11ω2and Ω12=ρ12ω2u˜f+bωj are frequency parameters. u˜f=U˜1, v˜f=U˜2 and W˜f=U˜3 are the amplitudes of the fluid midplane displacements. q˜ii=φ,θ,z is the amplitudes of external loads. Qφand Qθ are the amplitudes of the shear forces per unit length in the φ and θ directions, respectively. These forces can be represented by Kelvin–Kirchhoff equivalent shear forces as follows:(16)S˜φ=N˜φθ+sinφR0M˜φθ,V˜φ=Q˜φ+njR0M˜φθ.


### 2.3. Governing Equations of the Fluid

By adopting the Biot theory, the dynamic equations of the fluid are [1, 2](17)−ϕp,i+bu˙i−U˙i+Xi=ρ21u¨i+ρ22U¨i,where Xi is the body force vector per unit volume. ρ22=ϕρf+ϕρfα∞−1 and ρ21=ρ12 are the mass density coefficients related to the frame and fluid mass densities, respectively.Thus, based on thin shell theory, the fluid dynamic equations of the fluid-saturated porous rotational shell are written in the frequency domain as(18)−1Rφdϕp˜dφ=−Ω21u^˜s+Ω22u^˜f,−njR0ϕp˜=−Ω21v^˜s+Ω22v^˜f,−dϕp˜dz=−Ω21w^˜s+Ω22w^˜f,where u^˜, v^˜f, and w^˜f are the amplitudes of the fluid displacements at arbitrary point of the porous shell. As well, Ω21=bωj+ρ21ω2 and Ω22=bωj−ρ22ω2 are the frequency parameters.The amplitudes of the fluid displacements can be determined from equation (18) as(19)u^˜f=Ω21Ω22u^˜s−1RφΩ22dϕp˜dφ,v^˜f=Ω21Ω22v^˜s−njR0Ω22ϕp˜,w^˜f=Ω21Ω22w^˜s−1Ω22dϕp˜dz.The fluid strain-displacement relations of the fluid-saturated porous rotational shell are(20)ε˜φ=1Rφ∂u^˜f∂φ+w^˜f,ε˜θ=1R0u^˜fcosφ+njv^˜f+sinφw^˜f,ε˜z=∂w^˜f∂z≈0.Inserting the third formula of equation (19) into the third formula of equation (20), it can be shown that d2ϕp˜/dz2=0 since the derivative dw^˜s/dz is nearly equal to zero for a thin shell. Therefore, it is reasonable to assume that the fluid pressure, p˜, within a thin shell, is linearly distributed along z direction.Combining equations (8) and (20), the fluid constitutive equations of a porous rotational shell are derived in the frequency domain as(21)1Rφdu^˜fdφ+njR0v^˜f+cosφR0u^˜f+sinφR0+1Rφw^˜f=A¯¯′ϕp+B¯¯′eφ+eθ.Substituting equation (11) into equation (21), integrating over the shell thickness with the aid of equation (9), and assuming the fluid pressure in linearly distributed along the z-direction, one can achieve(22)−RφΩ21−Ω22B¯¯′du˜sdφ+RφcosφR0−Rφ′RφdPdφ+d2Pdφ2=Rφ2cosφR0Ω21−Ω22B¯¯′u˜s+Rφ2njR0Ω21−Ω22B¯¯′v˜s+Rφ2sinφR0+1RφΩ21−Ω22B¯¯′w˜s−Rφ2n2R02+Ω22A¯¯′P˜−sinφR0+1RφRφ2hϕp˜1−p˜2,where Rφ′=∂Rφ/∂φ, and p˜1 and p˜2 are the amplitudes of the pressure on the surface of the shell.Similarly, multiplying equation (21) by z, and integrating over the thickness of the shell with the aid of equation (9), one can obtain(23)−1Rφ2d2M˜Pdφ2−n2R02+Ω22A¯¯′MP+Rφ′Rφ3dM˜Pdφ+Ω21−Ω22B¯¯′njsinφR02v˜s−Ω21−Ω22B¯¯′n2R02w˜s+Ω21−Ω22B¯¯′1Rφdθφdφ+Ω21−Ω22B¯¯′cosφR0θφ=0.Equations (22) and (23) are the governing equations of the fluid in the fluid-statured porous rotational shell. It should be emphasized that, in the above derivation, there are not assumptions for the displacements. Meanwhile, the coupling interaction between the phases are considered. The absence of displacement simplifications and consideration of fluid-solid coupling is an advantage of the present method.


### 2.4. The Integrated Governing Equations of a Porous Rotational Shell Saturated in Fluids

By introducing dimensionless variables, the physical variables can be expressed as(24)u˜s=Lu¯s,v˜s=Lv¯s,w˜=Lw¯s,θ˜φ=θ¯φ,N˜φ=hN¯¯N¯φ,S˜φ=hN¯¯S¯φ,V˜φ=hN¯¯V¯φ,M˜φ=N¯¯h312LM¯φ,P˜=N¯¯B¯¯P¯,M˜p=N¯¯B¯¯LM¯p,q˜φ=hN¯¯Lq¯φ,q˜θ=hN¯¯Lq¯θ,q˜z=hN¯¯Lq¯z,where L is a length for nondimensionalization.Combining the solid governing equations (10), (13), (15), and (16), and the fluid governing equations (22) and (23), the integrated governing equations of a porous rotational shell can be obtained after the elimination of some dependent variables. After implementing the dimensionless procedure using equation (24), the integrated governing equations can be expressed as the following first-order differential matrix equation:(25)AdZdφ=BZ+F,where Z=u¯sv¯sw¯sθ¯φN¯φS¯φV¯φM¯φP¯dP¯/dφM¯pdM¯p/dφT is the integrated state vector, A and B are coefficient matrixes, and F is the integrated load vector. The nonzero components are listed in Appendix.Equation (25) can be solved effectively by the extended homogeneous capacity precision integration method [17]. The relationship between the start point and end point is written as(26)Ze=TZs,where Zeand Zsare the state vector of the start point and end point, respectively, and T is the transfer matrix.In general, there are six known boundary conditions and six unknown boundary conditions at each boundary. Taking the fixed end as an example, its known boundary conditions areu¯s,v¯s,w¯s,θ¯φ,P¯,M¯p, denoted as Zk, while its unknown boundary conditions can be denoted as Zu. Applying the boundary conditions to equation (26), the unknown boundary conditions Zu are determined with some rearrangement of transfer matrix T. Combining the 12 boundary conditions of the starting point and precision integration method, the integrated state vector at any point of the shell is obtained based on equation (25).


## 2.1. Constitutive Equations of the Porous Rotational Shell

The stress-strain relations of a homogeneous, isotropic poroelastic material are [1, 2](1)σij=2Neij+Ae+Qεδij,i,j=1,2,3,−ϕp=Qe+Rε.The first set of formulas in equation (1) describes the constitutive relations of the solid phase, and the second set of formulas describes the constitutive relations of the fluid phase. The elastic coefficients of the porous material A and N have a similar meaning with the Lamé coefficients for an elastic material. N is the shear modulus of the skeleton material. R is a pressure coefficient forcing a certain volume of the fluid into a constant aggregate. Q represents the coupling coefficient between the volume change of the solid and liquid. δij is the Kronecker delta. p is the fluid pressure. ϕ is the porosity. σij and eijare the stress and strain tensor of the solid skeleton, respectively. e and ε are the dilations of solid and fluid, respectively, which are given by(2)e=ekk=uk,k,ε=εkk=Uk,k,k=1,2,3,where u is the solid displacement tensor and U is the fluid displacement tensor. Note that the repetition of the index k implies summation; commas indicate differentiation with respect to space variables; and εij denotes the fluid strain tensor. In a spherical coordinate system, k is equal to φ,θ, or z.The middle plane of the thin porous rotational shell coincides with the natural coordinates, as shown in Figure1, in which, φ, θ, and z are meridian, circumferential, and normal coordinates, respectively, with relevant curvature radii of Rφ and Rθ, respectively. The midplane displacements in the meridian, circumferential, and normal directions are u,v,andw, respectively. The midplane Lamé coefficients in meridian and circumferential directions are Rφ and R0, respectively, and are both the functions of φ.Figure 1
The porous rotational shell.Applying the classical elastic theory for a thin shell, the normal line (z-direction) of the middle plane is assumed to be straight. Additionally, the angles between the normal line and the vertical lines of the meridian and circumferential direction are constants. Furthermore, the total normal stress can be neglected. Applying these assumptions, one obtains(3)eφz=0,eθz=0,σz−ϕpη=0.The fluid dilation can be evaluated from the second set of formulas in equation (1) as(4)ε=−ϕp−QeR.Substituting equation (4) into the first set of formulas of equation (1) and omitting the two shear stress components according to equation (3) yield(5)σφ=2Neφ+A−Q2Re−QRϕp,σθ=2Neθ+A−Q2Re−QRϕp,σz=2Nez+A−Q2Re−QRϕp,τφθ=Neφθ.By eliminatingez from the third formula of equation (5) with the third formula of equation (3), the solid dilation is obtained:(6)e=1−B¯eϕ+eθ+A¯ϕp,where A¯ and B¯ are constants, with A¯=Q+R/2NR+AR−Q2,B¯=AR−Q2/2NR+AR−Q2.Combing equations (5) and (6), the solid constitutive equations of a porous rotational shell take the following form:(7)σφ=N¯¯eφ+A¯¯eθ+B¯¯ϕp,σθ=N¯¯eθ+A¯¯eφ+B¯¯ϕp,τφθ=Neφθ,where A¯¯=A−Q2/R1−B¯,B¯¯=A−Q2/RA¯−Q2/R, N¯¯=2N+A−Q2/R1−B¯.Substituting equation (6) into equation (4), the fluid constitutive equations of a porous rotational shell are obtained:(8)ε=A¯¯′ϕp+B¯¯′eφ+eθ,where A¯¯′=−1+QA¯/R,B¯¯′=−Q/R1−B¯.


## 2.2. Governing Equations of the Solid Frame

Applying the Reissner model with classical thin shell theory, the solid displacements can be written as [17](9)u^s=us+zθφ,v^s=vs+zR0vssinφ−∂ws∂θ,w^s=ws,where u^s=u^1, v^s=u^2, w^s=u^3 are the solid displacements at an arbitrary point of the shell. The displacements us, vs, ws correspond to the midplane displacements. The midplane rotation θφ is(10)θφ=1Rφus−∂ws∂φ.The corresponding solid strains can be represented using the midplane solid displacements as follows [17]:(11)eφ=1Rφ∂us∂φ+1Rφws+zRφ∂θφ∂φ,eθ=cosφR0us+1R0∂vs∂θ+sinφR0ws+zR0∂∂θsinφR0vs−1R0∂ws∂θ+zcosφR0θφ,eφθ=1R0∂us∂θ+1Rφ∂vs∂φ−cosφR0vs+2zR0∂θφ∂θ.By assuming that the dynamic external excitation is harmonic and expanding each variable functionfφ,θ,t with the Fourier series along the circumferential direction, one obtains(12)fφ,θ,t=∑nf˜nφejnθejωt,where f˜n is the amplitude of the nth order component of f in the frequency domain, j=−1, and ω is the circular excitation frequency. In the following sections, the factors ejωt and ejnθ, and the superscript n are omitted for convenience.By combing equations (7) and (11) and applying a Fourier series expansion, the internal force-displacement relations of the solid frame are(13)N˜θ=∫−h/2h/2σ˜θdz=A¯¯Rφhdu˜sdφ+N¯¯njR0hv˜s+N¯¯cosφR0hu˜s+N¯¯sinφR0+A¯¯Rφhw˜s+B¯¯hP˜,N˜φθ=N˜θφ=∫−h/2h/2τ˜φθdz=NhnjR0u˜s+NhRφdv˜sdφ−NhcosφR0v˜s,M˜φ=∫−h/2h/2σ˜φzdz=h312N¯¯Rφdθ˜φdφ+h312A¯¯njsinφR02v˜s+h312A¯¯n2R02w˜s+h312A¯¯cosφR0θ˜φ+h312B¯¯M˜p,M˜θ=∫−h/2h/2σ˜θzdz=h312N¯¯njsinφR02v˜s+h312N¯¯n2R02w˜s+h312A¯¯Rφdθ˜φdφ+h312N¯¯cosφR0θ˜φ+h312B¯¯M˜p,M˜φθ=Mθφ=∫−h/2h/2τ˜φθzdz=h3122NnjR0θ˜φ.In equation (13), the following midplane variables are introduced: u˜s=1/h∫−h/2h/2u˜sdz, v˜s=1/h∫−h/2h/2v˜sdz, w˜s=1/h∫−h/2h/2w˜sdz, p˜=1/h∫−h/2h/2ϕp˜dz, M˜p=12/h3∫−h/2h/2zϕp˜dz, where h is the thickness of the shell; N˜φ and N˜θ are the component amplitudes of the axial force per unit length in the φ and θ directions, respectively; N˜φθ is the amplitude of the shear force per unit length; M˜φ and M˜θ are the amplitudes of the bending moment per unit length in the φ and θ directions, respectively; and M˜φθ is the amplitude of the torque per unit length.According to the Biot theory, the dynamic equations of the solid frame are [1, 2](14)σij,j−bu˙i−U˙i+Fi=ρ11u¨i+ρ12U¨i,i,j=1,2,3,where the dots indicate differentiation with respect to time. Fii=1,2,3 are the body forces per unit volume. Ui is the fluid midplane displacement. ρ11=1−ϕρs+ϕρfα∞−1 and ρ12=ϕρfα∞−1 are the mass density coefficients related to the frame and fluid mass densities, respectively. ρs and ρf are the frame mass density and fluid mass density, respectively. α∞ is the tortuosity. b=υϕ2/q is a coefficient related to the permeability q, and υ is the fluid viscosity.Based on thin shell theory and considering equation (14), the solid dynamic equations of the fluid-saturated porous rotational shell can be derived in the frequency domain as(15)RφcosφN˜φ+R0∂N˜φ∂φ+Rφ∂N˜θφ∂θ−RφcosφN˜θ+R0Q˜φ+RφR0q˜φ−RφR0hΩ11u˜s−Ω12u˜f=0,Rφ∂N˜θ∂θ+2RφcosφN˜φθ+R0∂N˜φθ∂φ+RφR0Q˜θRθ+RφR0q˜θ−RφR0hΩ11v˜s−Ω12v˜f=0,RφcosφQ˜φ+R0∂Q˜φ∂φ+Rφ∂Q˜θ∂θ−RφR0N˜φRφ−RφR0N˜θRθ+RφR0q˜z−RφR0hΩ11w˜s−Ω12w˜f=0,Rφ∂M˜θ∂θ+RφcosφM˜φθ+R0∂M˜φθ∂φ+RφcosφM˜θφ−RφR0Q˜θ=0,RφcosφM˜φ+R0∂M˜φ∂φ+Rφ∂M˜βφ∂θ−RφcosφM˜θ−RφR0Q˜φ=0,where Ω11=bωj−ρ11ω2and Ω12=ρ12ω2u˜f+bωj are frequency parameters. u˜f=U˜1, v˜f=U˜2 and W˜f=U˜3 are the amplitudes of the fluid midplane displacements. q˜ii=φ,θ,z is the amplitudes of external loads. Qφand Qθ are the amplitudes of the shear forces per unit length in the φ and θ directions, respectively. These forces can be represented by Kelvin–Kirchhoff equivalent shear forces as follows:(16)S˜φ=N˜φθ+sinφR0M˜φθ,V˜φ=Q˜φ+njR0M˜φθ.


## 2.3. Governing Equations of the Fluid

By adopting the Biot theory, the dynamic equations of the fluid are [1, 2](17)−ϕp,i+bu˙i−U˙i+Xi=ρ21u¨i+ρ22U¨i,where Xi is the body force vector per unit volume. ρ22=ϕρf+ϕρfα∞−1 and ρ21=ρ12 are the mass density coefficients related to the frame and fluid mass densities, respectively.Thus, based on thin shell theory, the fluid dynamic equations of the fluid-saturated porous rotational shell are written in the frequency domain as(18)−1Rφdϕp˜dφ=−Ω21u^˜s+Ω22u^˜f,−njR0ϕp˜=−Ω21v^˜s+Ω22v^˜f,−dϕp˜dz=−Ω21w^˜s+Ω22w^˜f,where u^˜, v^˜f, and w^˜f are the amplitudes of the fluid displacements at arbitrary point of the porous shell. As well, Ω21=bωj+ρ21ω2 and Ω22=bωj−ρ22ω2 are the frequency parameters.The amplitudes of the fluid displacements can be determined from equation (18) as(19)u^˜f=Ω21Ω22u^˜s−1RφΩ22dϕp˜dφ,v^˜f=Ω21Ω22v^˜s−njR0Ω22ϕp˜,w^˜f=Ω21Ω22w^˜s−1Ω22dϕp˜dz.The fluid strain-displacement relations of the fluid-saturated porous rotational shell are(20)ε˜φ=1Rφ∂u^˜f∂φ+w^˜f,ε˜θ=1R0u^˜fcosφ+njv^˜f+sinφw^˜f,ε˜z=∂w^˜f∂z≈0.Inserting the third formula of equation (19) into the third formula of equation (20), it can be shown that d2ϕp˜/dz2=0 since the derivative dw^˜s/dz is nearly equal to zero for a thin shell. Therefore, it is reasonable to assume that the fluid pressure, p˜, within a thin shell, is linearly distributed along z direction.Combining equations (8) and (20), the fluid constitutive equations of a porous rotational shell are derived in the frequency domain as(21)1Rφdu^˜fdφ+njR0v^˜f+cosφR0u^˜f+sinφR0+1Rφw^˜f=A¯¯′ϕp+B¯¯′eφ+eθ.Substituting equation (11) into equation (21), integrating over the shell thickness with the aid of equation (9), and assuming the fluid pressure in linearly distributed along the z-direction, one can achieve(22)−RφΩ21−Ω22B¯¯′du˜sdφ+RφcosφR0−Rφ′RφdPdφ+d2Pdφ2=Rφ2cosφR0Ω21−Ω22B¯¯′u˜s+Rφ2njR0Ω21−Ω22B¯¯′v˜s+Rφ2sinφR0+1RφΩ21−Ω22B¯¯′w˜s−Rφ2n2R02+Ω22A¯¯′P˜−sinφR0+1RφRφ2hϕp˜1−p˜2,where Rφ′=∂Rφ/∂φ, and p˜1 and p˜2 are the amplitudes of the pressure on the surface of the shell.Similarly, multiplying equation (21) by z, and integrating over the thickness of the shell with the aid of equation (9), one can obtain(23)−1Rφ2d2M˜Pdφ2−n2R02+Ω22A¯¯′MP+Rφ′Rφ3dM˜Pdφ+Ω21−Ω22B¯¯′njsinφR02v˜s−Ω21−Ω22B¯¯′n2R02w˜s+Ω21−Ω22B¯¯′1Rφdθφdφ+Ω21−Ω22B¯¯′cosφR0θφ=0.Equations (22) and (23) are the governing equations of the fluid in the fluid-statured porous rotational shell. It should be emphasized that, in the above derivation, there are not assumptions for the displacements. Meanwhile, the coupling interaction between the phases are considered. The absence of displacement simplifications and consideration of fluid-solid coupling is an advantage of the present method.


## 2.4. The Integrated Governing Equations of a Porous Rotational Shell Saturated in Fluids

By introducing dimensionless variables, the physical variables can be expressed as(24)u˜s=Lu¯s,v˜s=Lv¯s,w˜=Lw¯s,θ˜φ=θ¯φ,N˜φ=hN¯¯N¯φ,S˜φ=hN¯¯S¯φ,V˜φ=hN¯¯V¯φ,M˜φ=N¯¯h312LM¯φ,P˜=N¯¯B¯¯P¯,M˜p=N¯¯B¯¯LM¯p,q˜φ=hN¯¯Lq¯φ,q˜θ=hN¯¯Lq¯θ,q˜z=hN¯¯Lq¯z,where L is a length for nondimensionalization.Combining the solid governing equations (10), (13), (15), and (16), and the fluid governing equations (22) and (23), the integrated governing equations of a porous rotational shell can be obtained after the elimination of some dependent variables. After implementing the dimensionless procedure using equation (24), the integrated governing equations can be expressed as the following first-order differential matrix equation:(25)AdZdφ=BZ+F,where Z=u¯sv¯sw¯sθ¯φN¯φS¯φV¯φM¯φP¯dP¯/dφM¯pdM¯p/dφT is the integrated state vector, A and B are coefficient matrixes, and F is the integrated load vector. The nonzero components are listed in Appendix.Equation (25) can be solved effectively by the extended homogeneous capacity precision integration method [17]. The relationship between the start point and end point is written as(26)Ze=TZs,where Zeand Zsare the state vector of the start point and end point, respectively, and T is the transfer matrix.In general, there are six known boundary conditions and six unknown boundary conditions at each boundary. Taking the fixed end as an example, its known boundary conditions areu¯s,v¯s,w¯s,θ¯φ,P¯,M¯p, denoted as Zk, while its unknown boundary conditions can be denoted as Zu. Applying the boundary conditions to equation (26), the unknown boundary conditions Zu are determined with some rearrangement of transfer matrix T. Combining the 12 boundary conditions of the starting point and precision integration method, the integrated state vector at any point of the shell is obtained based on equation (25).


## 3. Results and Discussion

### 3.1. Comparison with the Experiment Result

To verify the present model, a vibration experiment of a cantilevered cylindrical shell is performed. The experiment is degenerated from the porous rotational shell whenRφ⟶∞,R0=R,φ=π/2. The shell is made of the foamed aluminium, and the geometric and physical parameters are listed in Table 1. The experimental setup is shown in Figure 2. In this case, a unit harmonic excitation is loaded at the bottom of the free end θ=0. The deflections of the bottom and top points at the free end are both measured. The frequency response function is then given by FRF=log10w¯, in which w¯ is the dimensionless displacement amplitude of the free end. The experimental results compared with the numerical ones obtained by the present method are shown in Figure 3. The results given by the two method are very similar in most frequency range, especially the values of the natural frequencies. The relative errors of the first- and second-order natural frequencies are both less than 3%. It should be noted that, due to the singularity of the theoretical model and insufficient consideration of the damping in the experiment, the vibration amplitude at the natural frequency is far from the experimental value.Table 1
Parameters of the porous cylinder shell.
PropertiesDescriptionValueLLength500 mmRRadius100 mmHThickness10 mmρsSolid density2700 kg/m3ρfFluid density1.213 kg/m3μPoisson’s ratio0.3EYoung’s modulus0.48 GPaϕPorosity0.74α∞Tortuosity1.2qPermeability2.7 × 10−10 m2Figure 2
Vibration experiment of a porous cylinder shell.Figure 3
Comparison between experimental and present result.


### 3.2. Comparison with Equivalent Models

Among all the porous material models with saturated fluid, equivalent models are the most widely used, such as the effective medium method and the equivalent density fluid method. The former method is applied to the porous material in case of a rigid solid frame. It assumes that the microelements of the porous materials can be seen as an effective elastic element [18]. The latter method is used for porous material in the case of a limp solid frame. Here, it is assumed that the frame is almost as soft as a fluid, and the microelements of the porous material can be treated as an effective limp element. Both models describe the porous mediums’ mechanical behaviours by using an equivalent mass density and ignore the solid-fluid coupling.In this section, two cases are considered for two different kinds of rotational shells using the effective medium method and the equivalent density fluid method. The numerical results are compared with the ones obtained by the present method.The effective medium method combined with FEM software is used to analyze a foamed aluminium cantilever shell and compared with the proposed method. For the calculation, the shell has a major axisa=21mm, minor axis b=105mm, starting angle φ=π/6, end angle φ=5π/6, and layer thicknesses h=210mm. The material parameters are listed in Table 2. Applying the equivalent elastic modulus and the equivalent density in FEM software, the comparison of the results for the natural frequencies is listed in Table 3 with m denoting the half wave numbers of the meridian direction and n denoting that of the circumferential direction. Table 3 indicates that the proposed method and the equivalent medium method have good agreement with each other.Table 2
Parameters of the rigid framed rotational shell.
ρsρfμEs (GPa)ϕqα∞2700 kg/m31.213 kg/m30.36.90.32.7×10−10 m21.2Table 3
Comparison of natural frequency.
Order (m, n)Present method (Hz)FEM (Hz)(1, 1)76.3774.72(1, 2)181.93180.92(1, 3)233.78234.92(1, 4)369.00375.60(1, 5)610.50621.37For the limp framed porous shell, the effective density fluid model is used. In this case, the solid-fluid coupling is negligible (u=U,ρ12=0) [19], and the porous material is described by an effective bulk modulus Keff and an effective density ρeff:(27)Keff=1−ϕKs+ϕKf−1,ρeff=ρρ˜−ρf2ρ˜+ρ−2ρf,where ρ˜=α∞ρf/ϕ+jFυ/qω and ρ=ϕρf+1−ϕρs, and Ksand Kf are the bulk modulus of the solid and the pore fluid, respectively. υ is the dynamic viscosity coefficient of the fluid, and F is also a coefficient of the fluid. For this calculation, a thin Y foam rotational shell with major axis a=260mm, minor axis b=105mm, starting angle (free end) φ=π/6, and end angle (fixed end) φ=π/2 is considered and surrounded by water. The shell is excited by a unit harmonic excitation Fz=1ejωtN at the free end (φ=π/6,θ=0), and Ks=6.9Gpa and Kf=2.395Gpa. The parameters of the shell are listed in Table 4. Figure 4 shows the frequency response curves given by the present method and the equivalent density fluid method.Table 4
Parameters of the limp frame porous rotational shell.
ρsρfμE (GPa)ϕqα∞ΥF1136 kg/m31.293 kg/m30.350.210.691×10−8 m21.030.00105 kg/m·s1.0Figure 4
Comparison between the effective density fluid method and the present method.By comparing the curves in Figure4, we note that the two methods give very similar results at low frequencies, but differences in results arise at higher frequencies. This may be due to the fact that the present method has fully considered the interactions between the fluid and solid phases, while the equivalent density fluid method has ignored this effect, so this simplification is only valid at low frequencies. As the frequency increases, the interactions between the fluid and solid phases gradually increase and it can no longer be neglected at higher frequencies. Therefore, compared to the equivalent fluid density method, the present model can be applied over a wider frequency range.


### 3.3. The Effects of Parameters

In this section, the influence of material and structural parameters, for instance, porosity, thickness, and ellipticity on structural vibration have been further studied. Letting the end angleφ=π/2, a porous rotational shell, the same as the first case in Section 3.2, is considered with a unit harmonic excitation Fz=1ejωtN excited at the free end. Figure 5 gives the amplitude frequency response curves with porosities of 0.3, 0.4, and 0.5. We note from the figure that higher porosity results in higher resonant frequencies. However, higher porosity does not always damp better, especially in the low-frequency range. This observation may be explained by the fact that the increasing porosity leads to the reduction of equivalent mass and equivalent stiffness. The former will reduce the vibration, and the latter will increase it. The descent velocities of the two factors is different in different frequency ranges, so the variation of resonance peak with porosity is not uniform.Figure 5
FRF curve for various porosities.To demonstrate the effect of the thickness, different thicknesses (10 mm, 12 mm, and 14 mm) are examined with the porosity of 30%. The response curves are illustrated in Figure6. It can be concluded from the figure that increasing the shell thickness leads to an increase in the resonant frequencies. However, increasing the thickness does not necessarily achieve better damping effect, e.g., the vibration at the first resonance peak. The interpretation of this behaviour is that although thicker structure can increase energy loss, it also can increase the vibration for the increases of the inertia.Figure 6
FRF curve for various thicknesses.For a thickness of 10 mm, the influence of ellipticity on the vibrational performance of the shell is discussed. In the meridian plane, the geometric equation of the porous rotational shell isz/a2+r/b2=1. The effects of the ellipticity on the frequency response curves are shown in Figure 7 with various values of ellipticity (a/b = 2.0, 2.5, and 3.0). It is seen from the figure that the resonant frequencies decrease as the ellipticity is increased. As well, this phenomenon is more obvious in the low-frequency range.Figure 7
FRF curve for various ellipticities.


## 3.1. Comparison with the Experiment Result

To verify the present model, a vibration experiment of a cantilevered cylindrical shell is performed. The experiment is degenerated from the porous rotational shell whenRφ⟶∞,R0=R,φ=π/2. The shell is made of the foamed aluminium, and the geometric and physical parameters are listed in Table 1. The experimental setup is shown in Figure 2. In this case, a unit harmonic excitation is loaded at the bottom of the free end θ=0. The deflections of the bottom and top points at the free end are both measured. The frequency response function is then given by FRF=log10w¯, in which w¯ is the dimensionless displacement amplitude of the free end. The experimental results compared with the numerical ones obtained by the present method are shown in Figure 3. The results given by the two method are very similar in most frequency range, especially the values of the natural frequencies. The relative errors of the first- and second-order natural frequencies are both less than 3%. It should be noted that, due to the singularity of the theoretical model and insufficient consideration of the damping in the experiment, the vibration amplitude at the natural frequency is far from the experimental value.Table 1
Parameters of the porous cylinder shell.
PropertiesDescriptionValueLLength500 mmRRadius100 mmHThickness10 mmρsSolid density2700 kg/m3ρfFluid density1.213 kg/m3μPoisson’s ratio0.3EYoung’s modulus0.48 GPaϕPorosity0.74α∞Tortuosity1.2qPermeability2.7 × 10−10 m2Figure 2
Vibration experiment of a porous cylinder shell.Figure 3
Comparison between experimental and present result.


## 3.2. Comparison with Equivalent Models

Among all the porous material models with saturated fluid, equivalent models are the most widely used, such as the effective medium method and the equivalent density fluid method. The former method is applied to the porous material in case of a rigid solid frame. It assumes that the microelements of the porous materials can be seen as an effective elastic element [18]. The latter method is used for porous material in the case of a limp solid frame. Here, it is assumed that the frame is almost as soft as a fluid, and the microelements of the porous material can be treated as an effective limp element. Both models describe the porous mediums’ mechanical behaviours by using an equivalent mass density and ignore the solid-fluid coupling.In this section, two cases are considered for two different kinds of rotational shells using the effective medium method and the equivalent density fluid method. The numerical results are compared with the ones obtained by the present method.The effective medium method combined with FEM software is used to analyze a foamed aluminium cantilever shell and compared with the proposed method. For the calculation, the shell has a major axisa=21mm, minor axis b=105mm, starting angle φ=π/6, end angle φ=5π/6, and layer thicknesses h=210mm. The material parameters are listed in Table 2. Applying the equivalent elastic modulus and the equivalent density in FEM software, the comparison of the results for the natural frequencies is listed in Table 3 with m denoting the half wave numbers of the meridian direction and n denoting that of the circumferential direction. Table 3 indicates that the proposed method and the equivalent medium method have good agreement with each other.Table 2
Parameters of the rigid framed rotational shell.
ρsρfμEs (GPa)ϕqα∞2700 kg/m31.213 kg/m30.36.90.32.7×10−10 m21.2Table 3
Comparison of natural frequency.
Order (m, n)Present method (Hz)FEM (Hz)(1, 1)76.3774.72(1, 2)181.93180.92(1, 3)233.78234.92(1, 4)369.00375.60(1, 5)610.50621.37For the limp framed porous shell, the effective density fluid model is used. In this case, the solid-fluid coupling is negligible (u=U,ρ12=0) [19], and the porous material is described by an effective bulk modulus Keff and an effective density ρeff:(27)Keff=1−ϕKs+ϕKf−1,ρeff=ρρ˜−ρf2ρ˜+ρ−2ρf,where ρ˜=α∞ρf/ϕ+jFυ/qω and ρ=ϕρf+1−ϕρs, and Ksand Kf are the bulk modulus of the solid and the pore fluid, respectively. υ is the dynamic viscosity coefficient of the fluid, and F is also a coefficient of the fluid. For this calculation, a thin Y foam rotational shell with major axis a=260mm, minor axis b=105mm, starting angle (free end) φ=π/6, and end angle (fixed end) φ=π/2 is considered and surrounded by water. The shell is excited by a unit harmonic excitation Fz=1ejωtN at the free end (φ=π/6,θ=0), and Ks=6.9Gpa and Kf=2.395Gpa. The parameters of the shell are listed in Table 4. Figure 4 shows the frequency response curves given by the present method and the equivalent density fluid method.Table 4
Parameters of the limp frame porous rotational shell.
ρsρfμE (GPa)ϕqα∞ΥF1136 kg/m31.293 kg/m30.350.210.691×10−8 m21.030.00105 kg/m·s1.0Figure 4
Comparison between the effective density fluid method and the present method.By comparing the curves in Figure4, we note that the two methods give very similar results at low frequencies, but differences in results arise at higher frequencies. This may be due to the fact that the present method has fully considered the interactions between the fluid and solid phases, while the equivalent density fluid method has ignored this effect, so this simplification is only valid at low frequencies. As the frequency increases, the interactions between the fluid and solid phases gradually increase and it can no longer be neglected at higher frequencies. Therefore, compared to the equivalent fluid density method, the present model can be applied over a wider frequency range.


## 3.3. The Effects of Parameters

In this section, the influence of material and structural parameters, for instance, porosity, thickness, and ellipticity on structural vibration have been further studied. Letting the end angleφ=π/2, a porous rotational shell, the same as the first case in Section 3.2, is considered with a unit harmonic excitation Fz=1ejωtN excited at the free end. Figure 5 gives the amplitude frequency response curves with porosities of 0.3, 0.4, and 0.5. We note from the figure that higher porosity results in higher resonant frequencies. However, higher porosity does not always damp better, especially in the low-frequency range. This observation may be explained by the fact that the increasing porosity leads to the reduction of equivalent mass and equivalent stiffness. The former will reduce the vibration, and the latter will increase it. The descent velocities of the two factors is different in different frequency ranges, so the variation of resonance peak with porosity is not uniform.Figure 5
FRF curve for various porosities.To demonstrate the effect of the thickness, different thicknesses (10 mm, 12 mm, and 14 mm) are examined with the porosity of 30%. The response curves are illustrated in Figure6. It can be concluded from the figure that increasing the shell thickness leads to an increase in the resonant frequencies. However, increasing the thickness does not necessarily achieve better damping effect, e.g., the vibration at the first resonance peak. The interpretation of this behaviour is that although thicker structure can increase energy loss, it also can increase the vibration for the increases of the inertia.Figure 6
FRF curve for various thicknesses.For a thickness of 10 mm, the influence of ellipticity on the vibrational performance of the shell is discussed. In the meridian plane, the geometric equation of the porous rotational shell isz/a2+r/b2=1. The effects of the ellipticity on the frequency response curves are shown in Figure 7 with various values of ellipticity (a/b = 2.0, 2.5, and 3.0). It is seen from the figure that the resonant frequencies decrease as the ellipticity is increased. As well, this phenomenon is more obvious in the low-frequency range.Figure 7
FRF curve for various ellipticities.


## 4. Conclusions

In this paper, a new theoretical model of thin porous rotational shells has been established based on the Biot theory and the classical elastic thin shell theory. The accuracy of the present model has been verified by comparing to experimental results and FEM results combined with effective medium method. In the low-frequency range, the proposed model agrees well with the above methods. However, in medium- and high-frequency ranges, the results do not agree well. This is certainly a consequence of the fact that the present model does not make any assumptions for the displacements and considers the interactions between the fluid and solid phases. These factors, usually neglected in the finite element method combined with the equivalent model, are more significant in medium and high frequencies. It indicates that the present model can be applied over a wide frequency range. The present investigation of the parameters of the porous rotational shell leads to the following conclusions: (i) increasing porosity and the thickness of the shell can increase in the resonant frequencies. (ii) The influence of porosity and thickness on the damping effect varies with the frequency. (iii) Increasing the ellipticity reduces the resonance amplitudes and frequencies. The effect of ellipticity in low frequency is more pronounced than that at medium and high frequencies.



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*Source: 6682451-2021-01-13.xml*