# Vibration Characteristics Analysis of Composite Laminated Annular/Circular Plate Using High-Order Shear Deformation Theory

Authors: Guodong Jin; Lei Ma; Hong Zhang; Qingshan Wang
Journal: Shock and Vibration
 (2021)

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## Abstract

In this paper, the free vibration behaviors of composite laminated annular and circular plates under complex elastic boundary constraints are investigated. Firstly, Reddy’s high-order shear deformation theory (HSDT) and Jacobi polynomial method are effectively combined to establish the unified vibration analysis model of composite laminated annular and circular plates. Secondly, the simulation of complex elastic boundary and coupling boundary is realized by using artificial virtual spring technology. Then, the energy equation of the composite laminated plate is established by using Rayleigh–Ritz energy technology. Finally, the free vibration solution equation of the laminated plate is obtained through the Hamilton differential principle. The fast and uniform convergence of this method and the accuracy of the calculated results are verified by numerical examples and the model experimental method. On this basis, the parameterization study is conducted, and the effects of material parameters, geometric parameters, spring stiffness values, and lamination scheme on the vibration characteristics of the annular or circular plate are fully discussed, which can provide a theoretical basis for future research.


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## Body

## 1. Introduction

The composite laminated annular or circular plates are regarded as the basic components of complex structures in various engineering and industrial application. Therefore, in-depth investigation of the free vibration characteristics of this kind of structure is quite meaningful for the vibration reduction, noise reduction, and impact resistance of the plate. Many scholars have been trying to find new methods or improve the existing methods to further study the vibration mechanical properties of the composite plate structure and have made fruitful research results. Among these research results, the commonly used theories are two-dimensional equivalent single-layer plate theory, zig-zag theory, layerwise theory, and three-dimensional elastic theory. The equivalent single-layer plate theory mainly includes classical thin plate theory (CPT), first-order shear deformation theory (FSDT), and high-order shear deformation theory (HSDT). Next, these theories are taken as the main line and briefly introduce the existing research results.According to the Ritz method and CPT, Afsharmanesh et al. [1] solved the forced vibration problem of laminated circular plates resting on a Winkler-type foundation. Parametric research is mainly focused on the various classical edge constraints and fiber orientation. Amabili et al. [2] investigated the free vibrations of a bolted (or riveted) circular plate having elastic constraints variable by using the Rayleigh–Ritz method. Arshid and Khorshidvand [3] studied the free vibration analysis of saturated porous functionally graded (FG) circular plates based on the differential quadrature method (DQM) and CPT. Bahrami and Teimourian [4] presented a wave propagation approach to analyze free vibration of nonuniform annular and circular membranes. As the first theory to be put forward, CPT completely ignores the shear deformation of the plate, so it is only suitable for studying the low frequency vibration behavior of thin plates or shells.Compared with CPT, the FSDT simply considers the effect of shear deformation on structural vibration by introducing the shear factor. Therefore, this theory can easily solve the vibration behaviors of the plate or shell with medium thickness. Mercan et al. [5] investigated important parametric results for annular plates and conical panels with isotropic, laminated, and FG materials by using FSDT and discrete singular convolution (DSC) method. On the basis of FSDT, Viswanathan et al. [6] expanded the Bickley-type spline function approximation to study the vibration problem of the symmetric angle-ply laminated annular and circular plate of variable thickness. Powmya and Narasimhan [7] discussed the free vibration analysis of orthotropic laminated circular and annular plates based on FSDT. They found that the transverse shear effects of orthotropic laminates are more prominent than that of isotropic laminates. Draiche et al. [8] presented an analytical model to predict the static analysis of laminated reinforced composite plates subjected to sinusoidal and uniform loads by using a simple first-order shear deformation theory. Although the FSDT further considers the effect of transverse shear deformation by introducing shear factor, this theory cannot reflect the warping deformation of the plate in the thickness direction more comprehensively.In order to overcome the limitations of the CPT and FSDT, the HSDT has been developed, which can better describe the vibration of the composite laminated plate. Amabili et al. [9] investigated forced nonlinear vibrations around the frequency of the fundamental mode by introducing a geometrically nonlinear damping model based on third-order shear deformation theory. In addition, he and his team have carried out a series of studies on the nonlinear mechanics of composite plate and shell structures, based on the third-order shear deformation theory [10–12]. Bisadi et al. [13] employed an exact closed-form solution for freely vibrating annular thick plates under traditional boundary constraints based on Reddy’s HSDT. Asadi and Jam [14] employed Mori–Tanaka method to calculate the buckling property of annular plates reinforced by carbon nanotubes on the basis of HSDT. The effects of CNTs’ volume fractions, orientation angles, boundary conditions, and geometric ratio of the plate had been analyzed. Vinyas et al. [15] studied the free vibration behavior of annular and circular magneto-electro-elastic plates through the combination of finite element method and HSDT. Hosseini-Hashemi et al. [16] proposed an exact analytical solution to discuss the freely vibrating piezoelectric coupled circular/annular thick plate based on Reddy’s HSDT. The boundary constraints of the plate are set to the combinations of free, simply support, or fixed support. Besides, he and his team [17] presented exact closed-form solutions to investigate free vibration behavior of circular thin, moderately thick, and thick plates with classical boundary constraints based on Reddy’s HSDT. Compared with the FSDT, the results obtained by HSDT get closer to the results obtained by 3D elastic theory. Najafizadeh and Heydari [18] considered mechanical buckling of FG circular plate with uniform radial compression based on HSDT. The study found that HSDT accurately predicts the behavior of the FG circular plate, whereas the CPT and FSDT overestimate buckling loads. In addition, some scholars have proposed a simplified higher-order shear deformation theory to further study the natural characteristics of composite plate or shell structures [19–23]. Katariya et al. [24, 25] studied the vibration characteristic of composite structures based on the HSDT and experimental test method. It is not difficult to find that there are still limitations in the study of vibration characteristics of composite laminated plate structures by using HSDT. Especially, for complex elastic boundary conditions of the laminated plate, the research results are relatively few.According to the limitations of the above research results, this paper effectively combines Reddy’s HSDT and Jacobi polynomials method to establish the unified analytical model of the laminated annular plate and circular plate. This analysis model can quickly and accurately obtain the free vibration behaviors of laminated annular and circular plates under complex elastic boundary constraints. Through the parametric study of boundary spring stiffness, the range of elastic boundary conditions is determined. On this basis, several groups of different boundary settings are given. Besides, in order to ensure the continuity when the rotation angle is 360 degrees, the stiffness value of the connection spring is clearly given. Through the parametric study, the effects of geometric parameters, material parameters, ply angle, and other important parameters on the vibration behaviors of the plate structures are effectively revealed. These research results can provide a theoretical basis for future research.


## 2. Theoretical Formulations

### 2.1. Model Description of the Laminated Annular/Circular Plate

The theoretical analysis model can be established because of the correlation between the geometry and elasticity of the annular plate and the circular plate. According to the relationships between geometry and elasticity of the annular plate and circular plate, the theoretical analysis model can be found in Figure1. First of all, as shown in Figure 1(a), it is not difficult to find that the cylindrical coordinate system of the plate is built on the midplane. In this coordinate system, r-axis represents the radius direction, and the corresponding geometric parameters are internal radius R0 and external radius R1, respectively. R represents the difference between R1 and R0. In particular, the circular plate is set when the inner radius is zero. θ-axis represents the rotation direction of the laminated plate, which ranges from 0 to 2π. Z-axis donates the thickness direction of the plate, which ranges from 0 to h. Then, the overall displacement functions U,V,andW of this kind of plate correspond to r-axis, θ-axis, and z-axis, respectively.Figure 1
The unified formulation and boundary conditions for the composite laminated annular plate and circular plate. (a) Coordinate system and unified formulation. (b) Setting of boundary conditions for the circular plate.
(a)(b)In this paper, simulations of various complex elastic boundary constraints are achieved by applying artificial virtual spring technology. All in all, spring components are placed evenly at the boundaries of the annular or circular plate. Among them, the number and type of spring components should be determined according to the displacement function on the middle plane. In this paper, the spring components are shown in Figure1(b). Taking the outer radius of the laminated circular plate as an example, three linear springs (kr1u, kr1v, and kr1w) and four rotary springs (Kr1ψr, Kr1ψθ, Kr1wr, and Kr1wr) form a boundary spring component. When the study model is changed from the annular plate to the circular plate, not only the inner radius is set to 0 but also the stiffness of these boundary springs at this position is set to 0. Through this series of settings, the simulation of different boundaries can be easily realized.


### 2.2. Kinematic Relations and Stress-Strain Relations

In the displacement field, the displacementsU and V can be extended to the cubic functions of the z-axis. Besides, transverse deflection W is assumed to be constant along the thickness direction of the plate. Therefore, the relationship between the global displacement function and the midplane displacement function can be expressed as follows [26, 27]:(1)Ur,θ,z,t=ur,θ,t+zψrr,θ,t+z2ϕrr,θ,t+z3λrr,θ,t,Vr,θ,z,t=vr,θ,t+zψθr,θ,t+z2ϕθr,θ,t+z3λθr,θ,t,Wr,θ,z,t=wr,θ,t,where u,v,andw are the displacements of the midplane. Then, ψr and ψθ denote the rotation foundations of normal to midplane about the r-axis and θ-axis, respectively. Besides, ϕr, ϕθ, λr, and λθ can be solved by using the vanish of transverse shear stresses at the top and bottom of the plate [26, 27]. Next, a simple solution process will be given.According to the three-dimensional elasticity theory of the annular/circular plate, the strain-displacement relations in the cylindrical coordinate system can be obtained:(2)εr=∂U∂r,εθ=∂Vr∂θ+Ur,γrz=∂W∂r+∂U∂z,εrθ=∂V∂r,εθr=∂Ur∂θ−Vr,γθz=∂Wr∂θ+∂V∂z.Based on the simple high-order theory put forward by Reddy [26, 27], ϕr, ϕθ, λr, and λθ can be obtained by setting γrzr,θ,±h/2=γθzr,θ,±h/2=0. So, their expressions can be written as(3)ϕr=0,λr=−43h2∂w∂r+ψr,ϕθ=0,λθ=−43h2∂wr∂θ+ψθ.In this way, combining equations (1) and (3), the displacement function can be rewritten as follows:(4)Ur,θ,z,t=ur,θ,t+gzψrr,θ,t+fz∂wr,θ,t∂r,Vr,θ,z,t=vr,θ,t+gzψθr,θ,t+fz∂wr,θ,tr∂θ,Wr,θ,z,t=wr,θ,t,in which gz=z−4z3/3h2 and fz=−4z3/3h2. It is not difficult to find from equation (4) that the unknown number of displacement functions involved is the same as that of the FSDT. But the present HSDT explains the parabolic distribution of transverse shear strains along the thickness of the composite laminated annular or circular plate.Substituting equation (4) into (2), it can be rewritten as(5)εr=εr0+zkr0+z2kr1+z3kr2,εθ=εθ0+zkθ0+z2kθ1+z3kθ2,εrθ=εrθ0+zkrθ0+z2krθ1+z3krθ2,εθr=εθr0+zkθr0+z2kθr1+z3kθr2,γrz=γrz0+zkrz0+z2krz1+z3krz2,γθz=γθz0+zkθz0+z2kθz1+z3kθz2,where the strain‐displacement relationships on the mid‐plane are given in Table 1.Table 1
The strain‐displacement relationships on the mid‐plane.
iεi0γi0ki0ki1ki2r∂u/∂r∂ψr/∂r0−4/3h2∂ψr/∂r+∂2w/∂r2θ∂v/r∂θ+u/r∂ψθ/r∂θ+ψr/r0−4/3h2∂ψθ/r∂θ+ψr/r+∂2w/r2∂θ2+∂w/r∂rrθ∂v/∂r∂ψθ/∂r0−4/3h2∂ψθ/∂r+∂2w/r∂r∂θθr∂u/r∂θ−v/r∂ψr/r∂θ−ψθ/r0−4/3h2∂ψr/r∂θ+∂2w/r∂r∂θ−ψθ/r−∂w/r2∂θrz∂w/r+ψr0−4/h2ψr+∂w/∂r0θz∂w/r∂θ+ψθ0−4/h2ψθ+∂w/r∂θ0The unidirectional fiber reinforced layers are the basic components of composite laminated structures. Therefore, the stress-strain relationship of the fiber reinforced layer of the laminated plate is focused. It assumes that the proposed plate hasN layers. Then, the relationship of stress-strain for the kth layer can be expressed as(6)σrkσθkτrθkτθrkτrzkτθzk=Q11k¯Q12k¯Q16k¯Q16k¯00Q12k¯Q22k¯Q26k¯Q26k¯00Q16k¯Q26k¯Q66k¯Q66k¯00Q16k¯Q26k¯Q66k¯Q66k¯000000Q55k¯Q45k¯0000Q45k¯Q44k¯εrkεθkεrθkεθrkγrzkγθzk,in which σik (i = r and θ) and τjk (j = rθ, θr, rz, and θz) denote normal stresses and shear stresses, respectively. Besides, the stiffness coefficients Qplk¯ (p and l = 1, 2, 4, 5, and 6) are expressed as(7)Q11k¯=Q11kcos4γk+2Q12k+2Q66kcos2γksin2θk+Q22ksin4γk,Q12k¯=Q11k+Q22k−4Q66kcos2γksin2γk+Q12ksin4γk+cos4γk,Q22k¯=Q11ksin4γk+2Q12k+2Q66kcos2γksin2γk+Q22kcos4γk,Q16k¯=Q11k−Q12k−2Q66kcos3γksinγk+Q12k−Q22k+2Q66kcosγksin3γk,Q26k¯=Q11k−Q12k−2Q66kcosγksin3γk+Q12k−Q22k+2Q66kcos3γksinγk,Q66k¯=Q11k+Q22k−2Q12k−2Q66kcos2γksin2γk+Q66ksin4γk+cos4γk,Q44k¯=Q44kcos2γk+Q55ksin2γk,Q45k¯=Q55k−Q44kcosγksinγk,Q55k¯=Q55kcos2γk+Q44ksin2γk,in which the angle between the fiber direction and the radius direction is defined as γk. Qplk is the material coefficient and is given as(8)Q11k=E1k1−μ12kμ21k,Q12k=μ21kQ11k,Q22k=E2k1−μ12kμ21k,Q44k=G23k,Q55k=G13k,Q66k=G12k,where μ12k and μ21k are Poisson’s ratios. Young’s modulus is expressed as E1k and E2k. The relationship between Poisson’s ratio and Young’s modulus is μ12E2k=μ21E1k. Besides, the shear modulus is denoted as G12k, G13k, and G23k. When E2k=E1k and G12k=G13k=G23k=E1k/21+μ12k, the isotropic plate can be simulated.So far, the relations between generalized force and strain can be written as the matrix form:(9)NMN1M2=ABCDBCDECDEFDEFGεkk1k2,QPQ1P2=AκBκCκDκBκCκDκEκCκDκEκFκDκEκFκGκγzkzkz1kz2,N=NrNθNrθNθrT,M=MrMθMrθMθrT,N1=Nr1Nθ1Nrθ1Nθr1T,M2=Mr2Mθ2Mrθ2Mθr2T,ε=εr0εθ0εrθ0εθr0T,k=kr0kθ0krθ0kθr0T,k1=kr1kθ1krθ1kθr1T,k2=kr2kθ2krθ2kθr2T,Q=QrQθT,Q=PrPθT,Q=Qr1Qθ1T,Q=Pr2Pθ2T,γz=γrz0γθz0T,kz=krz0kθz0T,kz1=krz1kθz1T,kz2=krz2kθz2T,A=A11A12A16A16A12A22A26A26A16A26A66A66A16A26A66A66,B=B11B12B16B16B12B22B26B26B16B26B66B66B16B26B66B66,C=C11C12C16C16C12C22C26C26C16C26C66C66C16C26C66C66,D=D11D12D16D16D12D22D26D26D16D26D66D66D16D26D66D66,E=E11E12E16E16E12E22E26E26E16E26E66E66E16E26E66E66,F=F11F12F16F16F12F22F26F26F16F26F66F66F16F26F66F66,G=G11G12G16G16G12G22G26G26G16G26G66G66G16G26G66G66,Aκ=A55A45A45A44,Bκ=B55B45B45B44,Cκ=C55C45C45C44,Dκ=D55D45D45D44,Eκ=E55E45E45E44,Fκ=F55F45F45F44,Gκ=G55G45G45G44,AplBplCplDplEplFplGpl=∑k=1NQplk¯∫ZkZk+11zz2z3z4z5z6dz,in which Apl, Bpl, Cpl, Dpl, Epl, Fpl, and Gpl are marked as the stiffness coefficients. Besides, Zk is the thickness coordinate value of the under surface of the kth layer plate. Zk+1 is the thickness coordinate value of the upper surface of the kth layer plate. So, the thickness of this layer plate is zk = Zk+1 − Zk.


### 2.3. Energy Equation of Laminated Annular/Circular Plate

In order to decrease the requirement of displacement function selection, a kind of multisegment segmentation technique is extended which can effectively relax the boundary constraints of the laminated annular or circular plate. Specifically, the laminate structure is subdivided intoNr segments along the r-axis direction. Then, a penalty parameter is introduced to ensure the continuity and boundary constraints between segments. Firstly, the general expression of the Lagrange energy equation L for composite laminated annular or circular plates is given, which fully considers the effect of boundary constraints:(10)L=∑qNrTq−Uq−∑qNr−1Vcpq−VBC,where the kinetic energy and potential energy of the qth segment plate are expressed by the symbols Tq and Uq. When the plate is segmented, the coupling potential energy is introduced, which is represented by symbol Vcpq. For the whole plate, the total kinetic energy and total potential energy should be the superposition of Nr segments Tq and Uq, while the total coupling potential energy should be the superposition of (Nr − 1) segment Vcpq. According to the boundary spring simulation technique, the spring potential stored at the plate boundary can be represented by symbol VBC. Next, according to Reddy’s HSDT, their specific expressions are given.For theqth segment plate, the kinetic energy Tq is expressed as(11)Tq=12∫rqrq+1∫02πI0u˙2+v˙2+w˙2+2I1u˙ψ˙r+v˙ψ˙θ+I2ψ˙r2+ψ˙θ2−83h2I3u˙∂w˙∂r+u˙ψ˙r+v˙∂w˙r∂θ+v˙ψ˙θ−83h2I4ψ˙r2+ψ˙θ2+ψ˙r∂w˙∂r+ψ˙θ∂w˙r∂θ+169h4I6∂w˙∂r+ψ˙r2+∂w˙r∂θ+ψ˙θ2rdrdθ,in which(12)I0,I1,I2,I3,I4,I5,I6=∫ZkZk+1ρz1,z,z2,z3,z4,z5,z6dz.The specific expression ofUq stored in the qth segment laminated plate is(13)Uq=12∫rqrq+1∫02πεTAε+Bk+Ck1+Dk2+γzTAκγz+Bκkz+Cκkz1+Dκkz2+kTBε+Ck+Dk1+Ek2+kzTBκγz+Cκkz+Dκkz1+Eκkz2+k1TCε+Dk+Ek1+Fk2+kz1TCκγz+Dκkz+Eκkz1+Fκkz2+k2TDε+Ek+Fk1+Gk2+kz2TDκγz+Eκkz+Fκkz1+Gκkz2rdrdθ.As mentioned above, when the plate structure is segmented along the radius direction, the coupling potential energy between adjacent segments is needed to ensure the continuity of displacements. Similarly, the coupling boundary is simulated by virtual spring technology. In short, the coupling springs (kcu, kcv, kcw, Kcψr, Kcψθ, Kcwr, and Kcwθ) are added between adjacent segments. Then, the reasonable spring stiffness is set, which will be further discussed in the following numerical analysis. For the qth segment and (q + 1)th segment plate, the coupling potential energy Vcpq can be written as(14)Vcpq=12∫02πkcuuq−uq+12+kcvvq−vq+12+kcwwq−wq+12+Kcψrψr,q−ψr,q+12+Kcψθψθ,q−ψθ,q+12+Kcwr∂wq∂r−∂wq+1∂r2+Kcwθ∂wqr∂θ−∂wq+1r∂θ2rdθ.There are a lot of research results on the elastic boundary conditions of plates and shells by using artificial virtual spring technology [28–31]. In Figure 1(b), the setting and types of elastic boundary springs have been given. The expressions of coupling potential energy VBC of the boundary springs are given directly below:(15)VBC=12∫02πR0kr0uu2+kr0vv2+kr0ww2+Kr0ψrψr2+Kr0ψθψθ2+Kr0wr∂w∂r2+Kr0wθ∂wr∂θ2r=R0+R1kr1uu2+kr1vv2+kr1ww2+Kr1ψrψr2+Kr1ψθψθ2+Kr1wr∂w∂r2+Kr1wθ∂wr∂θ2r=R1dθ.So far, the energy equation of the elastic laminate structure has been clearly expressed. Next, we will focus on the setting of the displacement function on the reference midsurface and the specific solving process of the natural frequencies and mode shapes.


### 2.4. Solution Process of Laminated Annular/Circular Plate

In the previous section, the global energy equation of the laminated annular or circular plate has been established, so this section will solve it by using the Hamilton differential principle. Firstly, it is necessary to determine the concrete displacement function expression of the middle plane. This paper applies the multisegment segmentation technique to investigate the free vibration behaviors of composite annular or circular plates, which can reduce the requirements of choosing displacement functions. Because compared with directly building the displacement function of the whole structure, the displacement function only needs the low order polynomials for the segment structure. Here, the Jacobi orthogonal polynomials are applied to construct displacement functions [32–36], which can be expressed as(16)u=∑m=0M∑n=0NUmn,qPmα,βrcosnθ+sinnθejωt,(17)v=∑m=0M∑n=0NVmn,qPmα,βrcosnθ+sinnθejωt,(18)w=∑m=0M∑n=0NWmn,qPmα,βrcosnθ+sinnθejωt,(19)ψr=∑m=0M∑n=0Nψmn,qrPmα,βrcosnθ+sinnθejωt,(20)ψθ=∑m=0M∑n=0Nψmn,qθPmα,βrcosnθ+sinnθejωt,where the Jacobi coefficients are expressed as Umn,q, Vmn,q, Wmn,q, ψmn,qr, and ψmn,qθ for the qth segment plate. The Jacobi polynomial of the mth in the radius direction is set to Pmα,β, and α and β denote the type of polynomials. Jacobi polynomials are orthogonal polynomials, so we can get Legendre polynomials, Chebyshev polynomials, and so on by setting the values of α and β. For example, if the values of α and β are 0, the Legendre polynomial will be obtained. When α = β = −0.5, it is the first kind of Chebyshev polynomial. When α = β = 0.5, it is the second kind of Chebyshev polynomial. This also shows the universality of the Jacobi polynomials. In addition, n represents the number of half wave in the rotation direction. Then, j, ω, and t are pure imaginary number, circular frequency, and time variable.According to the Hamilton differential principle, the partial derivative ofL to the unknown Jacobi coefficient matrix E is zero, which can be written as(21)∂L∂E=∑qNr∂Tq∂E−∂Uq∂E−∑qNr−1∂Vcpq∂E−∂VBC∂E=0,(22)E=Umn,qVmn,qWmn,qψmn,qrψmn,qθT.By introducing equations (11)–(15) into (21), a vibration characteristic solution equation can be obtained:(23)K+Kcp+KBC−ω2ME=0,in which K, Kcp, and KBC are stiffness matrix, coupling matrix between segments, and boundary spring matrix, separately. M denotes the mass matrix. From this equation, it is easy to find that the circular frequencies can be obtained by finding the generalized eigenvector. In addition, the corresponding generalized eigenvectors are introduced in equations (19)–(23), and the modal shapes of the laminated plate could be easily gained.


## 2.1. Model Description of the Laminated Annular/Circular Plate

The theoretical analysis model can be established because of the correlation between the geometry and elasticity of the annular plate and the circular plate. According to the relationships between geometry and elasticity of the annular plate and circular plate, the theoretical analysis model can be found in Figure1. First of all, as shown in Figure 1(a), it is not difficult to find that the cylindrical coordinate system of the plate is built on the midplane. In this coordinate system, r-axis represents the radius direction, and the corresponding geometric parameters are internal radius R0 and external radius R1, respectively. R represents the difference between R1 and R0. In particular, the circular plate is set when the inner radius is zero. θ-axis represents the rotation direction of the laminated plate, which ranges from 0 to 2π. Z-axis donates the thickness direction of the plate, which ranges from 0 to h. Then, the overall displacement functions U,V,andW of this kind of plate correspond to r-axis, θ-axis, and z-axis, respectively.Figure 1
The unified formulation and boundary conditions for the composite laminated annular plate and circular plate. (a) Coordinate system and unified formulation. (b) Setting of boundary conditions for the circular plate.
(a)(b)In this paper, simulations of various complex elastic boundary constraints are achieved by applying artificial virtual spring technology. All in all, spring components are placed evenly at the boundaries of the annular or circular plate. Among them, the number and type of spring components should be determined according to the displacement function on the middle plane. In this paper, the spring components are shown in Figure1(b). Taking the outer radius of the laminated circular plate as an example, three linear springs (kr1u, kr1v, and kr1w) and four rotary springs (Kr1ψr, Kr1ψθ, Kr1wr, and Kr1wr) form a boundary spring component. When the study model is changed from the annular plate to the circular plate, not only the inner radius is set to 0 but also the stiffness of these boundary springs at this position is set to 0. Through this series of settings, the simulation of different boundaries can be easily realized.


## 2.2. Kinematic Relations and Stress-Strain Relations

In the displacement field, the displacementsU and V can be extended to the cubic functions of the z-axis. Besides, transverse deflection W is assumed to be constant along the thickness direction of the plate. Therefore, the relationship between the global displacement function and the midplane displacement function can be expressed as follows [26, 27]:(1)Ur,θ,z,t=ur,θ,t+zψrr,θ,t+z2ϕrr,θ,t+z3λrr,θ,t,Vr,θ,z,t=vr,θ,t+zψθr,θ,t+z2ϕθr,θ,t+z3λθr,θ,t,Wr,θ,z,t=wr,θ,t,where u,v,andw are the displacements of the midplane. Then, ψr and ψθ denote the rotation foundations of normal to midplane about the r-axis and θ-axis, respectively. Besides, ϕr, ϕθ, λr, and λθ can be solved by using the vanish of transverse shear stresses at the top and bottom of the plate [26, 27]. Next, a simple solution process will be given.According to the three-dimensional elasticity theory of the annular/circular plate, the strain-displacement relations in the cylindrical coordinate system can be obtained:(2)εr=∂U∂r,εθ=∂Vr∂θ+Ur,γrz=∂W∂r+∂U∂z,εrθ=∂V∂r,εθr=∂Ur∂θ−Vr,γθz=∂Wr∂θ+∂V∂z.Based on the simple high-order theory put forward by Reddy [26, 27], ϕr, ϕθ, λr, and λθ can be obtained by setting γrzr,θ,±h/2=γθzr,θ,±h/2=0. So, their expressions can be written as(3)ϕr=0,λr=−43h2∂w∂r+ψr,ϕθ=0,λθ=−43h2∂wr∂θ+ψθ.In this way, combining equations (1) and (3), the displacement function can be rewritten as follows:(4)Ur,θ,z,t=ur,θ,t+gzψrr,θ,t+fz∂wr,θ,t∂r,Vr,θ,z,t=vr,θ,t+gzψθr,θ,t+fz∂wr,θ,tr∂θ,Wr,θ,z,t=wr,θ,t,in which gz=z−4z3/3h2 and fz=−4z3/3h2. It is not difficult to find from equation (4) that the unknown number of displacement functions involved is the same as that of the FSDT. But the present HSDT explains the parabolic distribution of transverse shear strains along the thickness of the composite laminated annular or circular plate.Substituting equation (4) into (2), it can be rewritten as(5)εr=εr0+zkr0+z2kr1+z3kr2,εθ=εθ0+zkθ0+z2kθ1+z3kθ2,εrθ=εrθ0+zkrθ0+z2krθ1+z3krθ2,εθr=εθr0+zkθr0+z2kθr1+z3kθr2,γrz=γrz0+zkrz0+z2krz1+z3krz2,γθz=γθz0+zkθz0+z2kθz1+z3kθz2,where the strain‐displacement relationships on the mid‐plane are given in Table 1.Table 1
The strain‐displacement relationships on the mid‐plane.
iεi0γi0ki0ki1ki2r∂u/∂r∂ψr/∂r0−4/3h2∂ψr/∂r+∂2w/∂r2θ∂v/r∂θ+u/r∂ψθ/r∂θ+ψr/r0−4/3h2∂ψθ/r∂θ+ψr/r+∂2w/r2∂θ2+∂w/r∂rrθ∂v/∂r∂ψθ/∂r0−4/3h2∂ψθ/∂r+∂2w/r∂r∂θθr∂u/r∂θ−v/r∂ψr/r∂θ−ψθ/r0−4/3h2∂ψr/r∂θ+∂2w/r∂r∂θ−ψθ/r−∂w/r2∂θrz∂w/r+ψr0−4/h2ψr+∂w/∂r0θz∂w/r∂θ+ψθ0−4/h2ψθ+∂w/r∂θ0The unidirectional fiber reinforced layers are the basic components of composite laminated structures. Therefore, the stress-strain relationship of the fiber reinforced layer of the laminated plate is focused. It assumes that the proposed plate hasN layers. Then, the relationship of stress-strain for the kth layer can be expressed as(6)σrkσθkτrθkτθrkτrzkτθzk=Q11k¯Q12k¯Q16k¯Q16k¯00Q12k¯Q22k¯Q26k¯Q26k¯00Q16k¯Q26k¯Q66k¯Q66k¯00Q16k¯Q26k¯Q66k¯Q66k¯000000Q55k¯Q45k¯0000Q45k¯Q44k¯εrkεθkεrθkεθrkγrzkγθzk,in which σik (i = r and θ) and τjk (j = rθ, θr, rz, and θz) denote normal stresses and shear stresses, respectively. Besides, the stiffness coefficients Qplk¯ (p and l = 1, 2, 4, 5, and 6) are expressed as(7)Q11k¯=Q11kcos4γk+2Q12k+2Q66kcos2γksin2θk+Q22ksin4γk,Q12k¯=Q11k+Q22k−4Q66kcos2γksin2γk+Q12ksin4γk+cos4γk,Q22k¯=Q11ksin4γk+2Q12k+2Q66kcos2γksin2γk+Q22kcos4γk,Q16k¯=Q11k−Q12k−2Q66kcos3γksinγk+Q12k−Q22k+2Q66kcosγksin3γk,Q26k¯=Q11k−Q12k−2Q66kcosγksin3γk+Q12k−Q22k+2Q66kcos3γksinγk,Q66k¯=Q11k+Q22k−2Q12k−2Q66kcos2γksin2γk+Q66ksin4γk+cos4γk,Q44k¯=Q44kcos2γk+Q55ksin2γk,Q45k¯=Q55k−Q44kcosγksinγk,Q55k¯=Q55kcos2γk+Q44ksin2γk,in which the angle between the fiber direction and the radius direction is defined as γk. Qplk is the material coefficient and is given as(8)Q11k=E1k1−μ12kμ21k,Q12k=μ21kQ11k,Q22k=E2k1−μ12kμ21k,Q44k=G23k,Q55k=G13k,Q66k=G12k,where μ12k and μ21k are Poisson’s ratios. Young’s modulus is expressed as E1k and E2k. The relationship between Poisson’s ratio and Young’s modulus is μ12E2k=μ21E1k. Besides, the shear modulus is denoted as G12k, G13k, and G23k. When E2k=E1k and G12k=G13k=G23k=E1k/21+μ12k, the isotropic plate can be simulated.So far, the relations between generalized force and strain can be written as the matrix form:(9)NMN1M2=ABCDBCDECDEFDEFGεkk1k2,QPQ1P2=AκBκCκDκBκCκDκEκCκDκEκFκDκEκFκGκγzkzkz1kz2,N=NrNθNrθNθrT,M=MrMθMrθMθrT,N1=Nr1Nθ1Nrθ1Nθr1T,M2=Mr2Mθ2Mrθ2Mθr2T,ε=εr0εθ0εrθ0εθr0T,k=kr0kθ0krθ0kθr0T,k1=kr1kθ1krθ1kθr1T,k2=kr2kθ2krθ2kθr2T,Q=QrQθT,Q=PrPθT,Q=Qr1Qθ1T,Q=Pr2Pθ2T,γz=γrz0γθz0T,kz=krz0kθz0T,kz1=krz1kθz1T,kz2=krz2kθz2T,A=A11A12A16A16A12A22A26A26A16A26A66A66A16A26A66A66,B=B11B12B16B16B12B22B26B26B16B26B66B66B16B26B66B66,C=C11C12C16C16C12C22C26C26C16C26C66C66C16C26C66C66,D=D11D12D16D16D12D22D26D26D16D26D66D66D16D26D66D66,E=E11E12E16E16E12E22E26E26E16E26E66E66E16E26E66E66,F=F11F12F16F16F12F22F26F26F16F26F66F66F16F26F66F66,G=G11G12G16G16G12G22G26G26G16G26G66G66G16G26G66G66,Aκ=A55A45A45A44,Bκ=B55B45B45B44,Cκ=C55C45C45C44,Dκ=D55D45D45D44,Eκ=E55E45E45E44,Fκ=F55F45F45F44,Gκ=G55G45G45G44,AplBplCplDplEplFplGpl=∑k=1NQplk¯∫ZkZk+11zz2z3z4z5z6dz,in which Apl, Bpl, Cpl, Dpl, Epl, Fpl, and Gpl are marked as the stiffness coefficients. Besides, Zk is the thickness coordinate value of the under surface of the kth layer plate. Zk+1 is the thickness coordinate value of the upper surface of the kth layer plate. So, the thickness of this layer plate is zk = Zk+1 − Zk.


## 2.3. Energy Equation of Laminated Annular/Circular Plate

In order to decrease the requirement of displacement function selection, a kind of multisegment segmentation technique is extended which can effectively relax the boundary constraints of the laminated annular or circular plate. Specifically, the laminate structure is subdivided intoNr segments along the r-axis direction. Then, a penalty parameter is introduced to ensure the continuity and boundary constraints between segments. Firstly, the general expression of the Lagrange energy equation L for composite laminated annular or circular plates is given, which fully considers the effect of boundary constraints:(10)L=∑qNrTq−Uq−∑qNr−1Vcpq−VBC,where the kinetic energy and potential energy of the qth segment plate are expressed by the symbols Tq and Uq. When the plate is segmented, the coupling potential energy is introduced, which is represented by symbol Vcpq. For the whole plate, the total kinetic energy and total potential energy should be the superposition of Nr segments Tq and Uq, while the total coupling potential energy should be the superposition of (Nr − 1) segment Vcpq. According to the boundary spring simulation technique, the spring potential stored at the plate boundary can be represented by symbol VBC. Next, according to Reddy’s HSDT, their specific expressions are given.For theqth segment plate, the kinetic energy Tq is expressed as(11)Tq=12∫rqrq+1∫02πI0u˙2+v˙2+w˙2+2I1u˙ψ˙r+v˙ψ˙θ+I2ψ˙r2+ψ˙θ2−83h2I3u˙∂w˙∂r+u˙ψ˙r+v˙∂w˙r∂θ+v˙ψ˙θ−83h2I4ψ˙r2+ψ˙θ2+ψ˙r∂w˙∂r+ψ˙θ∂w˙r∂θ+169h4I6∂w˙∂r+ψ˙r2+∂w˙r∂θ+ψ˙θ2rdrdθ,in which(12)I0,I1,I2,I3,I4,I5,I6=∫ZkZk+1ρz1,z,z2,z3,z4,z5,z6dz.The specific expression ofUq stored in the qth segment laminated plate is(13)Uq=12∫rqrq+1∫02πεTAε+Bk+Ck1+Dk2+γzTAκγz+Bκkz+Cκkz1+Dκkz2+kTBε+Ck+Dk1+Ek2+kzTBκγz+Cκkz+Dκkz1+Eκkz2+k1TCε+Dk+Ek1+Fk2+kz1TCκγz+Dκkz+Eκkz1+Fκkz2+k2TDε+Ek+Fk1+Gk2+kz2TDκγz+Eκkz+Fκkz1+Gκkz2rdrdθ.As mentioned above, when the plate structure is segmented along the radius direction, the coupling potential energy between adjacent segments is needed to ensure the continuity of displacements. Similarly, the coupling boundary is simulated by virtual spring technology. In short, the coupling springs (kcu, kcv, kcw, Kcψr, Kcψθ, Kcwr, and Kcwθ) are added between adjacent segments. Then, the reasonable spring stiffness is set, which will be further discussed in the following numerical analysis. For the qth segment and (q + 1)th segment plate, the coupling potential energy Vcpq can be written as(14)Vcpq=12∫02πkcuuq−uq+12+kcvvq−vq+12+kcwwq−wq+12+Kcψrψr,q−ψr,q+12+Kcψθψθ,q−ψθ,q+12+Kcwr∂wq∂r−∂wq+1∂r2+Kcwθ∂wqr∂θ−∂wq+1r∂θ2rdθ.There are a lot of research results on the elastic boundary conditions of plates and shells by using artificial virtual spring technology [28–31]. In Figure 1(b), the setting and types of elastic boundary springs have been given. The expressions of coupling potential energy VBC of the boundary springs are given directly below:(15)VBC=12∫02πR0kr0uu2+kr0vv2+kr0ww2+Kr0ψrψr2+Kr0ψθψθ2+Kr0wr∂w∂r2+Kr0wθ∂wr∂θ2r=R0+R1kr1uu2+kr1vv2+kr1ww2+Kr1ψrψr2+Kr1ψθψθ2+Kr1wr∂w∂r2+Kr1wθ∂wr∂θ2r=R1dθ.So far, the energy equation of the elastic laminate structure has been clearly expressed. Next, we will focus on the setting of the displacement function on the reference midsurface and the specific solving process of the natural frequencies and mode shapes.


## 2.4. Solution Process of Laminated Annular/Circular Plate

In the previous section, the global energy equation of the laminated annular or circular plate has been established, so this section will solve it by using the Hamilton differential principle. Firstly, it is necessary to determine the concrete displacement function expression of the middle plane. This paper applies the multisegment segmentation technique to investigate the free vibration behaviors of composite annular or circular plates, which can reduce the requirements of choosing displacement functions. Because compared with directly building the displacement function of the whole structure, the displacement function only needs the low order polynomials for the segment structure. Here, the Jacobi orthogonal polynomials are applied to construct displacement functions [32–36], which can be expressed as(16)u=∑m=0M∑n=0NUmn,qPmα,βrcosnθ+sinnθejωt,(17)v=∑m=0M∑n=0NVmn,qPmα,βrcosnθ+sinnθejωt,(18)w=∑m=0M∑n=0NWmn,qPmα,βrcosnθ+sinnθejωt,(19)ψr=∑m=0M∑n=0Nψmn,qrPmα,βrcosnθ+sinnθejωt,(20)ψθ=∑m=0M∑n=0Nψmn,qθPmα,βrcosnθ+sinnθejωt,where the Jacobi coefficients are expressed as Umn,q, Vmn,q, Wmn,q, ψmn,qr, and ψmn,qθ for the qth segment plate. The Jacobi polynomial of the mth in the radius direction is set to Pmα,β, and α and β denote the type of polynomials. Jacobi polynomials are orthogonal polynomials, so we can get Legendre polynomials, Chebyshev polynomials, and so on by setting the values of α and β. For example, if the values of α and β are 0, the Legendre polynomial will be obtained. When α = β = −0.5, it is the first kind of Chebyshev polynomial. When α = β = 0.5, it is the second kind of Chebyshev polynomial. This also shows the universality of the Jacobi polynomials. In addition, n represents the number of half wave in the rotation direction. Then, j, ω, and t are pure imaginary number, circular frequency, and time variable.According to the Hamilton differential principle, the partial derivative ofL to the unknown Jacobi coefficient matrix E is zero, which can be written as(21)∂L∂E=∑qNr∂Tq∂E−∂Uq∂E−∑qNr−1∂Vcpq∂E−∂VBC∂E=0,(22)E=Umn,qVmn,qWmn,qψmn,qrψmn,qθT.By introducing equations (11)–(15) into (21), a vibration characteristic solution equation can be obtained:(23)K+Kcp+KBC−ω2ME=0,in which K, Kcp, and KBC are stiffness matrix, coupling matrix between segments, and boundary spring matrix, separately. M denotes the mass matrix. From this equation, it is easy to find that the circular frequencies can be obtained by finding the generalized eigenvector. In addition, the corresponding generalized eigenvectors are introduced in equations (19)–(23), and the modal shapes of the laminated plate could be easily gained.


## 3. Numerical Analysis and Discussion

By effectively combining Reddy’s HSDT and Jacobi polynomials, the equations for solving vibration behaviors of laminated annular or circular plates are established. Next, some numerical examples are applied to study the free vibration characteristics of the laminated plate structures. This section mainly studies the convergence and accuracy, new results, and experimental tests. In addition, it should be pointed out that different boundary constraints are represented by a combination of corresponding first characters. Specifically, since there is only one boundary in the direction of the outer radius for a circular plate, there is only one character to represent it. Among them, the character sign corresponding to the fixed support is C, the sign corresponding to the simple support is S, the sign corresponding to the free boundary is F, and the sign corresponding to the elastic support is E. For annular plates with fixed support in the inner radius direction and simply supported in the outer radius direction, the boundary constraint can be expressed as C-S.


### 3.1. Model Validation

Convergence and accuracy analysis are important indexes for evaluating an algorithm. Therefore, this section focuses on the convergence of this method and the accuracy of the analysis model in the study of vibration characteristics. By parameterizing the stiffness of the boundary spring, different boundary constraints are defined. In addition, the parameterization of the stiffness of the coupling spring between segments is examined. From the expression of the displacement function, it is not difficult to find that the truncated valueM of Jacobi polynomial and segment number Nr directly represent the convergence of the proposed method. In Figure 2, the convergence of frequency parameters for the laminated annular or circular plate with fixed boundary constraints is investigated. For the laminated plate, the material parameters are set to E1 = 20E2, E2 = 10 GPa, G12 = G13 = 0.6E2, G23 = 0.5E2, μ12 = 0.25, and ρ = 1600 kg/m3. The lamination scheme is [0°/90°], and the geometric parameters are R1 = 1 m and h = 0.1 m. In addition, the internal radius R0 is 0.3 m. Figure 2 shows the curves of the frequency parameters Ω=ωR2/ρ/h2E2 of the annular plate and the circular plate with the increase of M or Nr. It shows intuitively that the frequency parameters of the two types of plate structures converge when M is 8 or Nr is 2. The proposed method shows good and consistent convergence in solving the natural frequency of this structure. In order to ensure the accuracy of the numerical example, the truncated value in the following examples is defined as M = 10 and Nr = 2.Figure 2
Frequency parameterΩ=ωR2/ρ/h2E2 of the composite laminated annular plate and circular plate under fixed boundary constraints with different M and Nr. (a) Annular plate under fixed boundary constraints. (b) Circular plate under fixed boundary constraints.
(a)(b)(c)(d)As previously described, different types of polynomials could be obtained by setting the values of Jacobi parametersα and β. Table 2 shows error analysis of frequency parameter Ω=ωR1ρh/A11 for the clamped composite laminated annular plate with various Jacobi parameters α and β. The material parameters are as follows: E1 = 15E2, E2 = 10 GPa, G12 = G13 = 0.6E2, G23 = 0.5E2, μ12 = 0.25, and ρ = 1500 kg/m3. The lamination scheme is [0°/90°0°/90°], and the geometric parameters are R0 = 1 m, R1 = 3 m, and hp = 0.1 m. It is not difficult to find out from Table 2 that the Jacobi parameters α and β have little effect on the calculation results of the frequency parameter Ω under the accuracy of five effective digits. In addition, the results obtained in this paper are in excellent agreement with those in [37]. If there is no special description, the Jacobi parameters are chosen as α = β = 0 in other calculation examples in this paper.Table 2
Error analysis of frequency parameterΩ=ωR1ρh/A11 for Jacobi parameters α and β for the annular plate with clamped supported boundary.
Method(α, β)Mode number12345Present(0, 0)0.416180.416570.427190.466410.54884(0, −0.5)0.416180.416570.427190.466410.54884(0, 0.5)0.416180.416570.427190.466410.54884(−0.5, 0)0.416180.416570.427190.466410.54884(−0.5, −0.5)0.416180.416570.427190.466410.54884(−0.5, 0.5)0.416180.416570.427190.466410.54884(0.5, 0)0.416180.416570.427190.466410.54884(0.5, −0.5)0.416180.416570.427190.466410.54884(0.5, 0.5)0.416180.416570.427190.466410.54884(1, 1)0.416180.416570.427190.466410.54884Reference [37]—0.416220.416690.427510.467030.54960As mentioned above, the complex boundary conditions are simulated by artificial virtual spring technology. Next, Figure3 displays the influence of boundary spring values on natural frequency of the laminated annular plate and a circular plate. Through numerical analysis, the variation range of spring stiffness for the elastic boundary can be determined. Except for the boundary constraints, the structural and material parameters of Figure 3 are exactly the same as those used in Table 2. The range of values of the linear boundary spring (defined as k) and rotary boundary spring (defined as K) is both 10−4 to 1016. It can be seen from Figure 3 that the waterfall plots of the frequencies to the change of boundary spring stiffness values have the same trends. In brief, when the linear boundary spring stiffness value (k) and the rotary boundary spring stiffness value (K) are less than 104, the frequency value is 0 corresponding to the free boundary condition. When k and K are higher than 1010, the frequency reaches the peak corresponding to the clamped boundary condition. Besides, the elastic boundary conditions can be determined when k and K appear in the following four combinations: 104<k < 1010, K < 104; 104 < k < 1010, K > 1010; 104 < K < 1010, k < 104; 104 < K < 1010, k > 1010. In light of the conclusions above, the values k and K corresponding to different boundary conditions are given in Table 3.Figure 3
Effect of the boundary spring value on frequency (Hz) of the composite laminated annular plate and circular plate. (a) Annual plate. (b) Circular plate.
(a)(b)(c)(d)Table 3
Stiffness values of the boundary spring corresponding to different boundary conditions.
BCStiffness valueskr0ukr0vkr0wKr0ψrKr0ψθKr0wrKr0wθF0000000C1014101410141014101410141014S1014101410140000E11081081081014101410141014E2108108108108108108108Now, let us focus on the connective spring. Figure4 displays the influence of the connective spring stiffness value on the natural frequency of laminated annular and circular plates. In Figure 4, the annular plate is simply supported and the circular plate is clamped. Other parameters are exactly the same with Figure 3. Stiffness values of the connective spring vary from 10−2 to 1016. The results show that the effect of connective spring stiffness on the natural frequencies is the same whether it is an annular plate under the SS boundary constraint or a circular plate under the C boundary constraint. The frequencies are almost unaffected when the value of the connective spring stiffness is less than 104. Thereafter, the frequencies increase with the increase of the connective spring stiffness until the value of connective spring stiffness is greater than 1010. To ensure continuity, the value of connective spring stiffness is chosen as 1010.Figure 4
Effect of the connective spring value on frequency (Hz) of the composite laminated annular plate and circular plate. (a) Annular plate under the S-S boundary condition. (b) Circular plate under the C boundary condition.
(a)(b)So far, a numerical analysis model for free vibration behaviors of the composite laminated annular or circular plate has been established. To further validate the correctness of the proposed method, Table4 gives the comparisons of the frequency parameter Ω=ωR1ρh/A11 for the laminated annular plate between the results obtained by HSDT and existing research [37]. Obviously, these two methods have a great agreement, which proves the correctness of the present vibration analysis model. In addition, it is not difficult to see from this table that the stronger the boundary constraint, the greater the natural frequency of the plate.Table 4
Comparison of the frequency parameterΩ=ωR1ρh/A11 for the composite laminated annular plate with different boundary conditions.
BCMethodMode number12345F-SPresent0.042450.083020.172030.280550.37333Reference [37]0.042550.084080.173020.281320.37312F-CPresent0.116870.146040.240210.367390.49685Reference [37]0.116710.146850.241530.368350.49659S-SPresent0.216710.218400.238690.301350.40908Reference [37]0.216690.218490.239090.302120.40990S-CPresent0.328320.329810.347400.405470.51367Reference [37]0.328250.329860.347750.406190.51446C-CPresent0.416180.416570.427190.466410.54884Reference [37]0.416220.416690.427510.467030.54960Then, Table5 shows the comparison of the frequency parameter Ω=ωR12/ρ/h2E2 for the isotropic circular plate with different thickness ratio h/R1. The parameters of the clamped circular plate are E1 = E2 = 185 GPa, G12 = G13 = G23 = 7.12 GPa, μ12 = 0.3, ρ = 1600 kg/m3, and R1 = 1 m. The results of FEM in the table are obtained by ABAQUS software. It is not difficult to find that the results obtained by the present method are in good agreement with those obtained by FEM when h/R1 is 0.2. In addition, the material used throughout this numerical example is actually an isotropic material. Therefore, the accuracy and universality of the present method in solving isotropic plate structures are also verified by this example. From this table, it can be found that the natural frequency of the plate decreases with the increase of the ratio of h/R1.Table 5
Comparison of the frequency parameterΩ=ωR12/ρ/h2E2 for the composite laminated circular plate with different plate thicknesses.
h/R1MethodMode number135679110.01FEM3.0906.44610.57512.07815.47718.55921.138Present3.0936.35110.50412.02515.36618.25820.9840.02FEM3.0876.43510.54812.04315.42018.48121.034Present3.0916.33810.47612.00115.30918.17620.8820.05FEM3.0696.36210.36211.81215.03917.96220.355Present3.0746.25810.29411.77514.93817.66020.2210.08FEM3.0376.23410.04411.41814.41217.11819.275Present3.0426.1209.98311.39114.32716.82219.1710.1FEM3.0086.1239.77811.09213.90616.44518.435Present3.0136.0019.72411.07313.83516.15518.3550.15FEM2.9155.7829.00910.15612.52213.67714.644Present2.9215.6248.98110.16412.49214.33914.7070.2FEM2.7985.3938.2029.19010.25811.17511.905Present2.8065.0918.1969.23011.02911.18511.880


### 3.2. Study on Free Vibration Characteristics

In Section3.1, the vibration analysis model for the composite annular and circular plate based on Reddy’s HSDT and Jacobi polynomials method has been established. The correctness and universality of the present method have been validated, too. In this part, we will give some new results and conclusions.Firstly, Table6 shows the frequency parameter Ω=ωR1ρh/A11 for the composite annular plate with various radius ratios and boundary constraints. The material parameters are E1 = 30E2, E2 = 10 GPa, G12 = G13 = 0.6E2, G23 = 0.5E2, μ12 = 0.27, and ρ = 1600 kg/m3. And, the lamination scheme is [60°/−30°/60°]. The invariant geometric parameters are R1 = 1 m and h = 0.15 m. It can be found from Table 6 that radius ratios and boundary constraints have a profound impact on the frequency parameter. On the one hand, the frequency parameter increases when the boundary spring stiffness value increases. On the other hand, the influence of the radius ratio on the frequency parameter is more complex and does not show monotonicity. Then, Table 7 gives the frequency parameter Ω=ωR1ρh/A11 for the laminated annular plate with various boundary constraints and thicknesses. The material parameters involved in Table 7 are E1 = 15E2, E2 = 10 GPa, G12 = G13 = 0.5E2, G23 = 0.2E2, μ12 = 0.25, and ρ = 1500 kg/m3. The lamination scheme is [30°/−30°], and R1 = 1 m. It is noticeable that the frequency parameter increases when the boundary spring stiffness and plate thickness increase.Table 6
The frequency parameterΩ=ωR1ρh/A11 for the composite laminated annular plate with different radius ratios and boundary conditions.
R0/R1BCMode number123456780.1C-E10.24060.30270.64000.64000.73370.73370.75600.8754E1-E10.15810.18470.52360.64000.64000.70130.70130.8754E1-E20.15770.18470.29200.29200.51340.51900.51900.7013E2-E20.15750.18470.27480.27480.51210.51850.51850.70130.3C-E10.32220.83240.84800.84800.98900.98901.06921.1834E1-E10.17250.19030.65010.75150.75150.84800.84800.9890E1-E20.17220.19030.37620.37620.55940.55940.63320.7515E2-E20.17200.19030.29400.29400.52540.52540.61830.75150.5C-E10.49151.28111.28111.35761.35761.47901.47901.6372E1-E10.19730.20740.73850.73850.96241.21611.21611.2811E1-E20.19720.20740.54100.54100.69930.69930.73850.7385E2-E20.19700.20740.32330.32330.56010.56010.73850.73850.8C-E10.49151.28111.28111.35761.35761.47901.47901.6372E1-E10.29900.30130.68890.68891.03581.03581.13161.3101E1-E20.29900.30130.68890.68891.03581.03581.13161.3101E2-E20.29890.30130.45610.45610.68890.68890.75470.7547Table 7
The frequency parameterΩ=ωR1ρh/A11 for the composite laminated annular plate with different boundary conditions and thicknesses.
BCMode numberH0.01 m0.02 m0.04 m0.08 m0.10 m0.14 m0.18 m0.20 mS10.00640.01270.02530.05040.06280.08710.11070.122130.02860.05660.10930.22710.26810.34240.40100.425150.04640.09190.17990.33680.40300.51090.59040.6216E110.01750.03470.06750.11220.11870.11470.10570.101430.04110.08050.15360.18370.16460.13940.12310.116850.06150.12120.23500.26690.28340.33760.39670.4245E210.01760.03470.06730.11060.11660.11240.10370.099530.04090.08010.14760.18340.16420.13890.12260.116350.06170.12080.22870.23350.23200.22230.21410.2108C10.01750.03480.06900.13430.16450.21950.26660.287330.04150.08230.15770.31940.37090.45890.52340.548850.06140.12090.23450.42590.50090.61590.69580.7262Next, Table8 gives the frequency parameter Ω=ωR12/ρ/h2E2 for the laminated annular plate and a circular plate with diverse anisotropy degrees and lamination schemes. The material parameters implicated in Table 8 are E2 = 10 GPa, G12 = G13 = 0.5E2, G23 = 0.6E2, μ12 = 0.25, and ρ = 1500 kg/m3. The geometrical parameters are R1 = 1 m, h = 0.08 m, and R0 = 0.3 m (for the annular plate). The boundary constraints of the annular plate and circular plate are E2-E2 and E2, respectively. It is not difficult to find that the frequency parameter increases with the increase of anisotropy degrees. It is noteworthy that the lamination scheme has an important influence on the frequency parameters. However, due to data limitations, it is difficult to draw general conclusions.Table 8
The frequency parameterΩ=ωR12/ρ/h2E2 for the composite laminated annular/circular plate with different anisotropy degrees and lamination schemes.
Plate typeE1/E2[0/90°][0/90°/0][0/90°/0/90°]135135135Annular53.77594.06295.04523.91014.06825.41083.82864.06435.1982103.85364.07545.22713.96984.07985.57903.91024.07755.4256203.92194.08475.40824.00404.08815.68283.96384.08765.6096303.95494.08945.51084.01744.09225.72853.98524.09255.7084403.97494.09245.58134.02514.09495.75683.99734.09575.7798503.98834.09475.63494.03044.09705.77794.00524.09815.8379603.99814.09655.67824.03454.09865.79514.01104.09995.8884Circular53.74236.22176.86974.29556.20448.06434.03286.22247.4192104.21706.20837.48864.90936.19708.94434.67226.21218.3725204.72026.20518.20095.34316.19879.61225.19496.21059.2159305.00076.20698.64035.51366.20219.95065.41746.21229.6415405.18196.20948.94805.60516.205010.18385.54016.21429.9247505.30876.21169.18045.66276.207510.36535.61826.216010.1405605.40246.21379.36465.70266.209610.51535.67246.217510.3174For the sake of fully revealing the effect of material parameters on vibration behaviors of the laminated annular plate and circular plate, the parametric research is conducted. Figure5 studies the effect of the layer number on natural frequencies of the laminated annular plate and circular plate with fixed boundary constraints. The material parameters are E1 = 5E2, E2 = 10 GPa, G12 = G13 = 0.5E2, G23 = 0.6E2, μ12 = 0.25, and ρ = 1500 kg/m3. The geometrical parameters of the clamped plates are R1 = 1 m, h = 0.08 m, and R0 = 0.3 m (for an annular plate). It should be noted that p represents the layer number. And, [0/−30°]1 represents [0/−30°], while [0/−30°]2 represents [0/−30°/0/−30°], and so on. As can be seen from Figure 5, the frequencies increase when the layer number increases and tend to be stable after the layer number exceeds 4. Moreover, the frequency changes most significantly when the layer number changes from 1 to 2.Figure 5
Effect of the layer number on frequency (Hz) of the fixed supported laminated annular plate and circular plate. (a) Annual plate. (b) Circular plate.
(a)(b)(c)(d)In addition, for the annular plate, its frequency decreases monotonously with the increase of the ply angle, but not for the circular plate. In order to reveal the effect of the ply angle on the vibration behaviors of the laminated annular and circular plate with fixed boundary constraints, Figure6 shows the variation of frequency with various ply angles. The lamination scheme used in Figure 6 is chosen as [0/γ°], and α ranges from 10° to 170°. From Figure 6, it is easy to find that the frequency variation is symmetrical with respect to [0/90°]. On the contrary, when α is less than 90°, the frequency of the annular plate decreases monotonously with the increase of γ. For the circular plate, the frequency is not significantly affected by the change of the ply angle and does not have monotonicity.Figure 6
Effect of the ply angle on frequency (Hz) of the fixed supported laminated annular plate and circular plate. (a) Annular plate. (b) Circular plate.
(a)(b)


### 3.3. Model Experimental Study

In Section3.2, free vibration behaviors of the laminated annular plate and circular plate are explored in depth based on the high-order shear deformation theory proposed by Reddy and Jacobi polynomials method. However, the previous studies only consider numerical validation. There is a lack of experimental test data to support the qualitative conclusion of the correctness of the above analysis model. In order to more fully verify the correctness of this study, this part applies the model experimental method. At present, equipment and principle of the structural model test can give reliable and more accurate experimental data. In addition, the fixed or free boundary conditions are insensitive to the external environment and easy to get realized. So, the model experiments of annular and circular plates are completed under the fixed boundary condition or the combination of free and fixed boundary conditions, which can provide more accurate experimental results. The experimental details on circular and annular plates can be found in [38, 39].In this paper, the force measurement method is used for the model experiment. The measurement method is a single-point vibration pickup method, that is, the position of the acceleration sensor remains unchanged all the time, and the hammer strikes all measuring points in turn. This test method requires that the vibration pick-up point avoid the joint position of the test piece and tries to select the position with large deformation. During the experiment, the 32-channel DH5922N dynamic signal test and analysis system and the corresponding modal analysis software are used, which were produced by Jiangsu Donghua Testing Technology Co., Ltd. In addition, the general piezoelectric acceleration sensor (1A116E) and the charge amplifier (DH5857-1) are also needed. Table9 shows the experimental instruments and their related information used in the model experiment.Table 9
Experimental instruments for the modal test of rotating plate structures.
NameModel numberParameterUseHammerLC-01AHammer head: aluminum hammer head; sensitivity: 4 pC/N; output signal: electric charge typeProvide incentives for plate structuresCharge amplifierDH5857-1Input resistance: >1011 Ω; input electric charge: <105 pC; sensitivity: 0.1, 10 mv/pCAmplify the excitation force and acceleration signalAcceleration sensor1A116EWeight: 5.5 g; sensitivity: 10 mv/m·s−2; measuring range: 500 m·s−2; frequency range: 1∼10000 HzPicking up acceleration signal of plate vibrationDynamic signal test and analysis systemDH5922NFrequency range channel number: 32 channels; maximum power: 100 W; communication interface: USB3.0; power supply: 220 VAC, 12 VDCCollect force signal and acceleration signalIn this experiment, according to the experimental schematic diagram in Figure7(a), the above experimental equipment and workpiece are installed and arranged. Figures 7(b) and 7(c) show the connection diagram of experimental equipment and test object of the annular plate and circular plate. In this experiment, the 30 cm thick foundation frame, 10 cm thick batten, and annular plate or circular plate are connected by 60 bolts with the same diameter 8 mm to simulate the fixed boundary conditions. Table 10 gives specifications of test plates and arrangement. It should be noted that the geometric dimensions given in this table are all effective dimensions. The first test step is to select a point on the plate as the vibration pick-up point and place the acceleration sensor on it. In the second step, the hammer is used to excite each grid intersection on the plate, and the excitation force signal is measured by the force sensor on the hammer. Then, the acceleration sensor at the pick-up point of the plate collects the vibration acceleration signal and amplifies the signal through the charge adapter. Finally, using the force measurement method modal analysis software in the computer, modal parameters identification and frequency response function analysis are carried out on the experimental data. Through frequency response curve fitting, steady-state diagram calculation, and mode calculation, the vibration mode and natural frequency of the plate are finally obtained.Figure 7
Schematic diagram of the model experiment and object of annular and circular plates. (a) Schematic diagram of the model experiment. (b) Annular plate with the F-C boundary condition. (c) Circular plate with the C boundary condition.
(a)(b)(c)Table 10
Specifications of test plates and arrangement.
NameAnnular plateCircular plateInternal radius of plate,R0 (m)0.05—External radius of plate,R1 (m)0.250.25Plate thickness,h (mm)55.2Mass density,ρ (kg/m3)78507850Modulus of elasticity,E (GPa)205.8205.8Poisson’s ratio,μ0.30.3Total number of measurement points8897Figures8 and 9 show the difference of experimental test results, FEM results, and present results. Among them, the maximum deviation of the experimental test results and present results is 7.57% for the natural frequency of the annular plate and 2.57% for the natural frequency of the circular plate. Besides, the maximum deviation of the experimental test results and FEM results is 9.83% for the natural frequency of the annular plate and 1.90% for the natural frequency of the circular plate. For the prediction results of the model shape, the two methods are highly consistent. The deviation of the above experimental results is within the acceptable range, which fully proves the correctness of the analysis model for the vibration behaviors of annular and circular plates. The experimental error is caused by many reasons. First of all, the fixed boundary conditions of the plate cannot be fully simulated by the way of clamping the foundation frame and the battens. Then, the material parameters used in the numerical calculation of the plate deviate from the actual material parameters of the work piece, and the work piece cannot be completely ideal isotropic material. In addition, the accuracy deviation of the force sensor and acceleration sensor and the human error of the experimenter in the process of hammering will cause the error of experimental data.Figure 8
Comparison of experimental results for the annular plate with the F-C boundary condition. (a) Present results. (b) Experimental results. (c) FEM results.
(a)(b)(c)Figure 9
Comparison of experimental results for the circular plate with the fixed boundary condition. (a) Present results. (b) Experimental results. (c) FEM results.
(a)(b)(c)


## 3.1. Model Validation

Convergence and accuracy analysis are important indexes for evaluating an algorithm. Therefore, this section focuses on the convergence of this method and the accuracy of the analysis model in the study of vibration characteristics. By parameterizing the stiffness of the boundary spring, different boundary constraints are defined. In addition, the parameterization of the stiffness of the coupling spring between segments is examined. From the expression of the displacement function, it is not difficult to find that the truncated valueM of Jacobi polynomial and segment number Nr directly represent the convergence of the proposed method. In Figure 2, the convergence of frequency parameters for the laminated annular or circular plate with fixed boundary constraints is investigated. For the laminated plate, the material parameters are set to E1 = 20E2, E2 = 10 GPa, G12 = G13 = 0.6E2, G23 = 0.5E2, μ12 = 0.25, and ρ = 1600 kg/m3. The lamination scheme is [0°/90°], and the geometric parameters are R1 = 1 m and h = 0.1 m. In addition, the internal radius R0 is 0.3 m. Figure 2 shows the curves of the frequency parameters Ω=ωR2/ρ/h2E2 of the annular plate and the circular plate with the increase of M or Nr. It shows intuitively that the frequency parameters of the two types of plate structures converge when M is 8 or Nr is 2. The proposed method shows good and consistent convergence in solving the natural frequency of this structure. In order to ensure the accuracy of the numerical example, the truncated value in the following examples is defined as M = 10 and Nr = 2.Figure 2
Frequency parameterΩ=ωR2/ρ/h2E2 of the composite laminated annular plate and circular plate under fixed boundary constraints with different M and Nr. (a) Annular plate under fixed boundary constraints. (b) Circular plate under fixed boundary constraints.
(a)(b)(c)(d)As previously described, different types of polynomials could be obtained by setting the values of Jacobi parametersα and β. Table 2 shows error analysis of frequency parameter Ω=ωR1ρh/A11 for the clamped composite laminated annular plate with various Jacobi parameters α and β. The material parameters are as follows: E1 = 15E2, E2 = 10 GPa, G12 = G13 = 0.6E2, G23 = 0.5E2, μ12 = 0.25, and ρ = 1500 kg/m3. The lamination scheme is [0°/90°0°/90°], and the geometric parameters are R0 = 1 m, R1 = 3 m, and hp = 0.1 m. It is not difficult to find out from Table 2 that the Jacobi parameters α and β have little effect on the calculation results of the frequency parameter Ω under the accuracy of five effective digits. In addition, the results obtained in this paper are in excellent agreement with those in [37]. If there is no special description, the Jacobi parameters are chosen as α = β = 0 in other calculation examples in this paper.Table 2
Error analysis of frequency parameterΩ=ωR1ρh/A11 for Jacobi parameters α and β for the annular plate with clamped supported boundary.
Method(α, β)Mode number12345Present(0, 0)0.416180.416570.427190.466410.54884(0, −0.5)0.416180.416570.427190.466410.54884(0, 0.5)0.416180.416570.427190.466410.54884(−0.5, 0)0.416180.416570.427190.466410.54884(−0.5, −0.5)0.416180.416570.427190.466410.54884(−0.5, 0.5)0.416180.416570.427190.466410.54884(0.5, 0)0.416180.416570.427190.466410.54884(0.5, −0.5)0.416180.416570.427190.466410.54884(0.5, 0.5)0.416180.416570.427190.466410.54884(1, 1)0.416180.416570.427190.466410.54884Reference [37]—0.416220.416690.427510.467030.54960As mentioned above, the complex boundary conditions are simulated by artificial virtual spring technology. Next, Figure3 displays the influence of boundary spring values on natural frequency of the laminated annular plate and a circular plate. Through numerical analysis, the variation range of spring stiffness for the elastic boundary can be determined. Except for the boundary constraints, the structural and material parameters of Figure 3 are exactly the same as those used in Table 2. The range of values of the linear boundary spring (defined as k) and rotary boundary spring (defined as K) is both 10−4 to 1016. It can be seen from Figure 3 that the waterfall plots of the frequencies to the change of boundary spring stiffness values have the same trends. In brief, when the linear boundary spring stiffness value (k) and the rotary boundary spring stiffness value (K) are less than 104, the frequency value is 0 corresponding to the free boundary condition. When k and K are higher than 1010, the frequency reaches the peak corresponding to the clamped boundary condition. Besides, the elastic boundary conditions can be determined when k and K appear in the following four combinations: 104<k < 1010, K < 104; 104 < k < 1010, K > 1010; 104 < K < 1010, k < 104; 104 < K < 1010, k > 1010. In light of the conclusions above, the values k and K corresponding to different boundary conditions are given in Table 3.Figure 3
Effect of the boundary spring value on frequency (Hz) of the composite laminated annular plate and circular plate. (a) Annual plate. (b) Circular plate.
(a)(b)(c)(d)Table 3
Stiffness values of the boundary spring corresponding to different boundary conditions.
BCStiffness valueskr0ukr0vkr0wKr0ψrKr0ψθKr0wrKr0wθF0000000C1014101410141014101410141014S1014101410140000E11081081081014101410141014E2108108108108108108108Now, let us focus on the connective spring. Figure4 displays the influence of the connective spring stiffness value on the natural frequency of laminated annular and circular plates. In Figure 4, the annular plate is simply supported and the circular plate is clamped. Other parameters are exactly the same with Figure 3. Stiffness values of the connective spring vary from 10−2 to 1016. The results show that the effect of connective spring stiffness on the natural frequencies is the same whether it is an annular plate under the SS boundary constraint or a circular plate under the C boundary constraint. The frequencies are almost unaffected when the value of the connective spring stiffness is less than 104. Thereafter, the frequencies increase with the increase of the connective spring stiffness until the value of connective spring stiffness is greater than 1010. To ensure continuity, the value of connective spring stiffness is chosen as 1010.Figure 4
Effect of the connective spring value on frequency (Hz) of the composite laminated annular plate and circular plate. (a) Annular plate under the S-S boundary condition. (b) Circular plate under the C boundary condition.
(a)(b)So far, a numerical analysis model for free vibration behaviors of the composite laminated annular or circular plate has been established. To further validate the correctness of the proposed method, Table4 gives the comparisons of the frequency parameter Ω=ωR1ρh/A11 for the laminated annular plate between the results obtained by HSDT and existing research [37]. Obviously, these two methods have a great agreement, which proves the correctness of the present vibration analysis model. In addition, it is not difficult to see from this table that the stronger the boundary constraint, the greater the natural frequency of the plate.Table 4
Comparison of the frequency parameterΩ=ωR1ρh/A11 for the composite laminated annular plate with different boundary conditions.
BCMethodMode number12345F-SPresent0.042450.083020.172030.280550.37333Reference [37]0.042550.084080.173020.281320.37312F-CPresent0.116870.146040.240210.367390.49685Reference [37]0.116710.146850.241530.368350.49659S-SPresent0.216710.218400.238690.301350.40908Reference [37]0.216690.218490.239090.302120.40990S-CPresent0.328320.329810.347400.405470.51367Reference [37]0.328250.329860.347750.406190.51446C-CPresent0.416180.416570.427190.466410.54884Reference [37]0.416220.416690.427510.467030.54960Then, Table5 shows the comparison of the frequency parameter Ω=ωR12/ρ/h2E2 for the isotropic circular plate with different thickness ratio h/R1. The parameters of the clamped circular plate are E1 = E2 = 185 GPa, G12 = G13 = G23 = 7.12 GPa, μ12 = 0.3, ρ = 1600 kg/m3, and R1 = 1 m. The results of FEM in the table are obtained by ABAQUS software. It is not difficult to find that the results obtained by the present method are in good agreement with those obtained by FEM when h/R1 is 0.2. In addition, the material used throughout this numerical example is actually an isotropic material. Therefore, the accuracy and universality of the present method in solving isotropic plate structures are also verified by this example. From this table, it can be found that the natural frequency of the plate decreases with the increase of the ratio of h/R1.Table 5
Comparison of the frequency parameterΩ=ωR12/ρ/h2E2 for the composite laminated circular plate with different plate thicknesses.
h/R1MethodMode number135679110.01FEM3.0906.44610.57512.07815.47718.55921.138Present3.0936.35110.50412.02515.36618.25820.9840.02FEM3.0876.43510.54812.04315.42018.48121.034Present3.0916.33810.47612.00115.30918.17620.8820.05FEM3.0696.36210.36211.81215.03917.96220.355Present3.0746.25810.29411.77514.93817.66020.2210.08FEM3.0376.23410.04411.41814.41217.11819.275Present3.0426.1209.98311.39114.32716.82219.1710.1FEM3.0086.1239.77811.09213.90616.44518.435Present3.0136.0019.72411.07313.83516.15518.3550.15FEM2.9155.7829.00910.15612.52213.67714.644Present2.9215.6248.98110.16412.49214.33914.7070.2FEM2.7985.3938.2029.19010.25811.17511.905Present2.8065.0918.1969.23011.02911.18511.880


## 3.2. Study on Free Vibration Characteristics

In Section3.1, the vibration analysis model for the composite annular and circular plate based on Reddy’s HSDT and Jacobi polynomials method has been established. The correctness and universality of the present method have been validated, too. In this part, we will give some new results and conclusions.Firstly, Table6 shows the frequency parameter Ω=ωR1ρh/A11 for the composite annular plate with various radius ratios and boundary constraints. The material parameters are E1 = 30E2, E2 = 10 GPa, G12 = G13 = 0.6E2, G23 = 0.5E2, μ12 = 0.27, and ρ = 1600 kg/m3. And, the lamination scheme is [60°/−30°/60°]. The invariant geometric parameters are R1 = 1 m and h = 0.15 m. It can be found from Table 6 that radius ratios and boundary constraints have a profound impact on the frequency parameter. On the one hand, the frequency parameter increases when the boundary spring stiffness value increases. On the other hand, the influence of the radius ratio on the frequency parameter is more complex and does not show monotonicity. Then, Table 7 gives the frequency parameter Ω=ωR1ρh/A11 for the laminated annular plate with various boundary constraints and thicknesses. The material parameters involved in Table 7 are E1 = 15E2, E2 = 10 GPa, G12 = G13 = 0.5E2, G23 = 0.2E2, μ12 = 0.25, and ρ = 1500 kg/m3. The lamination scheme is [30°/−30°], and R1 = 1 m. It is noticeable that the frequency parameter increases when the boundary spring stiffness and plate thickness increase.Table 6
The frequency parameterΩ=ωR1ρh/A11 for the composite laminated annular plate with different radius ratios and boundary conditions.
R0/R1BCMode number123456780.1C-E10.24060.30270.64000.64000.73370.73370.75600.8754E1-E10.15810.18470.52360.64000.64000.70130.70130.8754E1-E20.15770.18470.29200.29200.51340.51900.51900.7013E2-E20.15750.18470.27480.27480.51210.51850.51850.70130.3C-E10.32220.83240.84800.84800.98900.98901.06921.1834E1-E10.17250.19030.65010.75150.75150.84800.84800.9890E1-E20.17220.19030.37620.37620.55940.55940.63320.7515E2-E20.17200.19030.29400.29400.52540.52540.61830.75150.5C-E10.49151.28111.28111.35761.35761.47901.47901.6372E1-E10.19730.20740.73850.73850.96241.21611.21611.2811E1-E20.19720.20740.54100.54100.69930.69930.73850.7385E2-E20.19700.20740.32330.32330.56010.56010.73850.73850.8C-E10.49151.28111.28111.35761.35761.47901.47901.6372E1-E10.29900.30130.68890.68891.03581.03581.13161.3101E1-E20.29900.30130.68890.68891.03581.03581.13161.3101E2-E20.29890.30130.45610.45610.68890.68890.75470.7547Table 7
The frequency parameterΩ=ωR1ρh/A11 for the composite laminated annular plate with different boundary conditions and thicknesses.
BCMode numberH0.01 m0.02 m0.04 m0.08 m0.10 m0.14 m0.18 m0.20 mS10.00640.01270.02530.05040.06280.08710.11070.122130.02860.05660.10930.22710.26810.34240.40100.425150.04640.09190.17990.33680.40300.51090.59040.6216E110.01750.03470.06750.11220.11870.11470.10570.101430.04110.08050.15360.18370.16460.13940.12310.116850.06150.12120.23500.26690.28340.33760.39670.4245E210.01760.03470.06730.11060.11660.11240.10370.099530.04090.08010.14760.18340.16420.13890.12260.116350.06170.12080.22870.23350.23200.22230.21410.2108C10.01750.03480.06900.13430.16450.21950.26660.287330.04150.08230.15770.31940.37090.45890.52340.548850.06140.12090.23450.42590.50090.61590.69580.7262Next, Table8 gives the frequency parameter Ω=ωR12/ρ/h2E2 for the laminated annular plate and a circular plate with diverse anisotropy degrees and lamination schemes. The material parameters implicated in Table 8 are E2 = 10 GPa, G12 = G13 = 0.5E2, G23 = 0.6E2, μ12 = 0.25, and ρ = 1500 kg/m3. The geometrical parameters are R1 = 1 m, h = 0.08 m, and R0 = 0.3 m (for the annular plate). The boundary constraints of the annular plate and circular plate are E2-E2 and E2, respectively. It is not difficult to find that the frequency parameter increases with the increase of anisotropy degrees. It is noteworthy that the lamination scheme has an important influence on the frequency parameters. However, due to data limitations, it is difficult to draw general conclusions.Table 8
The frequency parameterΩ=ωR12/ρ/h2E2 for the composite laminated annular/circular plate with different anisotropy degrees and lamination schemes.
Plate typeE1/E2[0/90°][0/90°/0][0/90°/0/90°]135135135Annular53.77594.06295.04523.91014.06825.41083.82864.06435.1982103.85364.07545.22713.96984.07985.57903.91024.07755.4256203.92194.08475.40824.00404.08815.68283.96384.08765.6096303.95494.08945.51084.01744.09225.72853.98524.09255.7084403.97494.09245.58134.02514.09495.75683.99734.09575.7798503.98834.09475.63494.03044.09705.77794.00524.09815.8379603.99814.09655.67824.03454.09865.79514.01104.09995.8884Circular53.74236.22176.86974.29556.20448.06434.03286.22247.4192104.21706.20837.48864.90936.19708.94434.67226.21218.3725204.72026.20518.20095.34316.19879.61225.19496.21059.2159305.00076.20698.64035.51366.20219.95065.41746.21229.6415405.18196.20948.94805.60516.205010.18385.54016.21429.9247505.30876.21169.18045.66276.207510.36535.61826.216010.1405605.40246.21379.36465.70266.209610.51535.67246.217510.3174For the sake of fully revealing the effect of material parameters on vibration behaviors of the laminated annular plate and circular plate, the parametric research is conducted. Figure5 studies the effect of the layer number on natural frequencies of the laminated annular plate and circular plate with fixed boundary constraints. The material parameters are E1 = 5E2, E2 = 10 GPa, G12 = G13 = 0.5E2, G23 = 0.6E2, μ12 = 0.25, and ρ = 1500 kg/m3. The geometrical parameters of the clamped plates are R1 = 1 m, h = 0.08 m, and R0 = 0.3 m (for an annular plate). It should be noted that p represents the layer number. And, [0/−30°]1 represents [0/−30°], while [0/−30°]2 represents [0/−30°/0/−30°], and so on. As can be seen from Figure 5, the frequencies increase when the layer number increases and tend to be stable after the layer number exceeds 4. Moreover, the frequency changes most significantly when the layer number changes from 1 to 2.Figure 5
Effect of the layer number on frequency (Hz) of the fixed supported laminated annular plate and circular plate. (a) Annual plate. (b) Circular plate.
(a)(b)(c)(d)In addition, for the annular plate, its frequency decreases monotonously with the increase of the ply angle, but not for the circular plate. In order to reveal the effect of the ply angle on the vibration behaviors of the laminated annular and circular plate with fixed boundary constraints, Figure6 shows the variation of frequency with various ply angles. The lamination scheme used in Figure 6 is chosen as [0/γ°], and α ranges from 10° to 170°. From Figure 6, it is easy to find that the frequency variation is symmetrical with respect to [0/90°]. On the contrary, when α is less than 90°, the frequency of the annular plate decreases monotonously with the increase of γ. For the circular plate, the frequency is not significantly affected by the change of the ply angle and does not have monotonicity.Figure 6
Effect of the ply angle on frequency (Hz) of the fixed supported laminated annular plate and circular plate. (a) Annular plate. (b) Circular plate.
(a)(b)


## 3.3. Model Experimental Study

In Section3.2, free vibration behaviors of the laminated annular plate and circular plate are explored in depth based on the high-order shear deformation theory proposed by Reddy and Jacobi polynomials method. However, the previous studies only consider numerical validation. There is a lack of experimental test data to support the qualitative conclusion of the correctness of the above analysis model. In order to more fully verify the correctness of this study, this part applies the model experimental method. At present, equipment and principle of the structural model test can give reliable and more accurate experimental data. In addition, the fixed or free boundary conditions are insensitive to the external environment and easy to get realized. So, the model experiments of annular and circular plates are completed under the fixed boundary condition or the combination of free and fixed boundary conditions, which can provide more accurate experimental results. The experimental details on circular and annular plates can be found in [38, 39].In this paper, the force measurement method is used for the model experiment. The measurement method is a single-point vibration pickup method, that is, the position of the acceleration sensor remains unchanged all the time, and the hammer strikes all measuring points in turn. This test method requires that the vibration pick-up point avoid the joint position of the test piece and tries to select the position with large deformation. During the experiment, the 32-channel DH5922N dynamic signal test and analysis system and the corresponding modal analysis software are used, which were produced by Jiangsu Donghua Testing Technology Co., Ltd. In addition, the general piezoelectric acceleration sensor (1A116E) and the charge amplifier (DH5857-1) are also needed. Table9 shows the experimental instruments and their related information used in the model experiment.Table 9
Experimental instruments for the modal test of rotating plate structures.
NameModel numberParameterUseHammerLC-01AHammer head: aluminum hammer head; sensitivity: 4 pC/N; output signal: electric charge typeProvide incentives for plate structuresCharge amplifierDH5857-1Input resistance: >1011 Ω; input electric charge: <105 pC; sensitivity: 0.1, 10 mv/pCAmplify the excitation force and acceleration signalAcceleration sensor1A116EWeight: 5.5 g; sensitivity: 10 mv/m·s−2; measuring range: 500 m·s−2; frequency range: 1∼10000 HzPicking up acceleration signal of plate vibrationDynamic signal test and analysis systemDH5922NFrequency range channel number: 32 channels; maximum power: 100 W; communication interface: USB3.0; power supply: 220 VAC, 12 VDCCollect force signal and acceleration signalIn this experiment, according to the experimental schematic diagram in Figure7(a), the above experimental equipment and workpiece are installed and arranged. Figures 7(b) and 7(c) show the connection diagram of experimental equipment and test object of the annular plate and circular plate. In this experiment, the 30 cm thick foundation frame, 10 cm thick batten, and annular plate or circular plate are connected by 60 bolts with the same diameter 8 mm to simulate the fixed boundary conditions. Table 10 gives specifications of test plates and arrangement. It should be noted that the geometric dimensions given in this table are all effective dimensions. The first test step is to select a point on the plate as the vibration pick-up point and place the acceleration sensor on it. In the second step, the hammer is used to excite each grid intersection on the plate, and the excitation force signal is measured by the force sensor on the hammer. Then, the acceleration sensor at the pick-up point of the plate collects the vibration acceleration signal and amplifies the signal through the charge adapter. Finally, using the force measurement method modal analysis software in the computer, modal parameters identification and frequency response function analysis are carried out on the experimental data. Through frequency response curve fitting, steady-state diagram calculation, and mode calculation, the vibration mode and natural frequency of the plate are finally obtained.Figure 7
Schematic diagram of the model experiment and object of annular and circular plates. (a) Schematic diagram of the model experiment. (b) Annular plate with the F-C boundary condition. (c) Circular plate with the C boundary condition.
(a)(b)(c)Table 10
Specifications of test plates and arrangement.
NameAnnular plateCircular plateInternal radius of plate,R0 (m)0.05—External radius of plate,R1 (m)0.250.25Plate thickness,h (mm)55.2Mass density,ρ (kg/m3)78507850Modulus of elasticity,E (GPa)205.8205.8Poisson’s ratio,μ0.30.3Total number of measurement points8897Figures8 and 9 show the difference of experimental test results, FEM results, and present results. Among them, the maximum deviation of the experimental test results and present results is 7.57% for the natural frequency of the annular plate and 2.57% for the natural frequency of the circular plate. Besides, the maximum deviation of the experimental test results and FEM results is 9.83% for the natural frequency of the annular plate and 1.90% for the natural frequency of the circular plate. For the prediction results of the model shape, the two methods are highly consistent. The deviation of the above experimental results is within the acceptable range, which fully proves the correctness of the analysis model for the vibration behaviors of annular and circular plates. The experimental error is caused by many reasons. First of all, the fixed boundary conditions of the plate cannot be fully simulated by the way of clamping the foundation frame and the battens. Then, the material parameters used in the numerical calculation of the plate deviate from the actual material parameters of the work piece, and the work piece cannot be completely ideal isotropic material. In addition, the accuracy deviation of the force sensor and acceleration sensor and the human error of the experimenter in the process of hammering will cause the error of experimental data.Figure 8
Comparison of experimental results for the annular plate with the F-C boundary condition. (a) Present results. (b) Experimental results. (c) FEM results.
(a)(b)(c)Figure 9
Comparison of experimental results for the circular plate with the fixed boundary condition. (a) Present results. (b) Experimental results. (c) FEM results.
(a)(b)(c)


## 4. Conclusions

This paper applies the Jacobi polynomial method to establish a unified analytical model for the vibration characteristics of laminated annular and circular plates under complex elastic boundary constraints based on Reddy’s HSDT. Due to the adoption of multisegment segmentation technique, coupling springs are applied between adjacent segments to ensure continuity. In addition, various elastic boundary constraints are simulated by setting boundary springs at the edges of the plate. The Lagrangian energy equation of composite laminated annular or circular plates under elastic boundary constraints is established. The solution equation of free vibration behavior for the laminated plate is easily obtained by the Hamilton differential principle. On the basis of verifying the fast convergence and great accuracy of this method, further parametric study is carried out. Some important conclusions are revealed:(1)
Through the analysis of the stiffness value of the boundary springs, the value range of the elastic boundary condition is determined which are 104 < k < 1010 and K < 104, 104 < k < 1010 and K > 1010, 104 < K < 1010 and k < 104, and 104 < K < 1010 and k > 1010. When θ = 2π, the value of connective spring stiffness should be chosen as 1010 to ensure continuity.(2)
Material parameters, geometric parameters, and lamination schemes have an important effect on the vibration characteristics of the annular and circular plates. Specifically, the natural frequency will increase with the increase of plate thickness and anisotropy degrees.(3)
The frequency increases with the increase of the layer number and tends to be stable after the layer number exceeds 4. Moreover, the frequency changes most significantly when the layer number changes from 1 to 2.(4)
The deviation of the model experimental results is within the acceptable range, which fully proves the correctness of the unified analysis model for the vibration behaviors of annular and circular plates.



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*Source: 5913860-2021-07-31.xml*