Vibro-acoustic characteristics analysis of the rotary composite plate and conical – cylindrical double cavities coupled system

Purpose – The purpose of the study is to obtain and analyze vibro-acoustic characteristics. Design/methodology/approach – A unified analysis model for the rotary composite laminated plate and conical – cylindrical double cavities coupled system is established. The related parameters of the unified model are determined by isoparametric transformation. The modified Fourier series are applied to construct the admissible displacement function and the sound pressure tolerance function of the coupled systems. The energy functional of the structure domain and acoustic field domain is established, respectively, and the structure – acoustic coupling potential energy is introduced to obtain the energy functional. Rayleigh – Ritz method was used to solve the energy functional. Findings – The displacement and sound pressure response of the coupled systems are acquired by introducing the internal point sound source excitation, and the influence of relevant parameters of the coupled systems is researched. Through research, it is found that the impedance wall can reduce the amplitude of the soundpressureresponseandsuppresstheresonanceofthecoupledsystems.Besides,thecompositelaminatedplatehasagoodnoisereductioneffect. Originality/value – This study can provide the theoretical guidance for vibration and noise reduction.


Introduction
Rotary laminated plate structures based on fiber reinforced composite materials have been widely applied in aerospace, marine and other fields. In engineering practice, it is inevitable to produce the coupled system between rotary composite laminated plate and acoustic cavities.
Taking the submarine as an instance, the two cabins inside the hull of the submarine can be regarded as a rotary acoustic cavity coupled system, and the two cabins are separated by the cabin door composed of composite laminated plate. Noise in each cabin causes the variation of sound pressure, which leads to the vibration of the cabin door. Meanwhile, this vibration also contributes to noise, forming the composite laminated plate and double acoustic cavities coupled system. To compare the vibro-acoustic characteristics of structure-acoustic coupled system containing rotary composite laminated plate and acoustic cavity coupled system, it is important to conduct in-depth research on the vibro-acoustic characteristics of the rotary composite laminated plate and conical-cylindrical double acoustic cavities coupled system and reveal the vibro-acoustic coupling mechanism, which can provide the theoretical basis for structural optimization and low noise design.
The acoustic field characteristics of acoustic cavity coupled system under different conditions have been investigated by many scholars in recent years. Tanaka et al. (2012) derived the eigenpairs, which verified the validity by an experiment of coupled rectangular cavity, and investigated the fundamental properties of the eigenpairs derived. Moores et al. (2018) demonstrated an acoustical analog of a circuit quantum electrodynamics system that leverages acoustic properties to enable strong multimode coupling in the dispersive regime, and the operating frequencies where this emission rate is suppressed are identified. Unnikrishnan Nair et al. (2010) proposed a simplified modeling approach for numerical simulation of a coupled cavity-resonator system, and the influence of damping and resonator volume fraction on the coupled system performance is shown. A high-order doubly asymptotic open boundary for modeling scalar wave propagation in two-dimensional unbounded media was presented by Birk et al. (2016), which can handle domains with arbitrary geometry by using a circular boundary to divide these into near field and far field. Through modal energy analysis (MODENA), Zhang et al. (2016) defined a dimensionless coupling quotient, which is equal to the ratio of the gyroscopic coupling coefficient and the critical coefficient at modal frequencies. Chen et al. (2017) presented a domain decomposition method to predict the acoustic characteristics of an arbitrary enclosure made up of any number of sub-spaces and revealed the effect of coupling parameters between sub-spaces on the natural frequencies and mode shapes of the overall enclosure. Based on the energy principle in combination with a 3D modified Fourier cosine series approach, Shi et al. (2018) studied the modeling and acoustic eigen analysis of coupled spaces with a coupling aperture of variable size. It can be seen from the existing literature that the current research on acoustic cavity coupled system is mostly for rectangular cavity. However, the research on the acoustic characteristic model of the rotary acoustic cavity is rarely involved, let alone the research on the modeling of rotary acoustic cavity coupled system. At the same time, most of the investigations are based on the coupling mechanism analysis between acoustic cavities, while no analysis model has been established specifically.
Unlike the acoustic cavity coupled system, the coupling of rotary composite laminated plate and conical-cylindrical double acoustic cavities belongs to structure-acoustic coupling. Many scholars have researched the coupling mechanism of structure-sound coupled system. On the foundation of these investigations, the models of sectional structure-sound coupled system are established, and the vibro-acoustic characteristics of the coupled system are analyzed. Van Genechten et al. (2011) presented a newly developed hybrid simulation technique for coupled structural-acoustic analysis, which applies a wave-based model for the acoustic cavity and a direct or modally reduced finite element (FE) model for the structural part. By invoking the energy distribution approach, De Rosa et al. (2012) proposed a similitude for the analysis of the dynamic response of acoustic-elastic assemblies and discussed the FE modeling of a flexural plate coupled to an acoustic room and an infinite cylinder containing a fluid. Wang et al. (2015) developed a coupled smoothed finite element method (S-FEM) to deal with the structural-acoustic problems consisting of a shell JIMSE 3,1 configuration interacting with the fluid medium; numerical examples of a cylinder cavity attached to a flexible shell and an automobile passenger compartment were conducted to illustrate the effectiveness and accuracy of the coupled S-FEM for structural-acoustic problems. Shu et al. (2014) proposed a level set-based structural topology optimization method for the optimal design of coupled structural-acoustic system with a focus on interior noise reduction. Some scholars have also studied the vibro-acoustic coupling characteristics of the composite laminated plate-double cavity coupled system and obtained some achievements. Larbi et al. (2012) demonstrated the theoretical formulation and the FE implementation of vibro-acoustic problems with piezoelectric composite structures connected to electric shunt circuits. Sarig€ ul andKarag€ ozl€ u (2014, 2018) presented the results of modal structure-sound coupling analysis for rectangular plates with different composite parameters and compared with the performance of isotropic plate systems. Ferreira et al. (2014), Carrera et al. (2018) and Cinefra et al. (2021) established a theoretical model of rectangular composite laminated plate and constructed a rectangular cavity model based on the FE formula of sound field under standard pressure. In the existing literature, the research object of the composite laminated plate-cavity coupled system is mostly concentrated on the rectangular composite laminated plate-cavity coupled system, while the rotary composite laminated plate-cavity coupled system is rarely studied, much less the rotary composite laminated plate and conical-cylindrical double acoustic cavities coupled system.
In view of the shortcomings of existing research, the vibro-acoustic characteristics analysis of the rotary composite laminated plate and conical-cylindrical double acoustic cavities coupled system are conducted in this paper. First, the expressions of admissible displacement function and sound pressure function of laminated plate are constructed. Second, the energy functional of the structure domain and sound field domain is established, respectively, and the coupling potential energy is added to acquire the energy functional of the whole coupled system. Then, Rayleigh-Ritz method is applied to gain the equation of the vibro-acoustic characteristics of the system. Finally, based on the fast convergence and accuracy of the model, the effect mechanism of relevant parameters on the vibro-acoustic characteristics is analyzed, and the response of the coupled system is investigated by introducing internal point force and point source excitation.
2. Unified analysis model of the coupled systems 2.1 Model description Figure 1 demonstrates the rotating cross-section geometrical parameters and coordinate system of the conical-cylindrical acoustic cavity coupled system and rotary composite laminated plate and conical-cylindrical double cavities coupled system. As shown in Figure 1a, the coordinate of conical-cylindrical acoustic cavity coupled system is composed of two local coordinate systems (o-s 1 , θ 1 , r 1 ) and (o-s 2 , θ 2 , r 2 ). R 1 and R 2 separately represent the short radius and long radius of conical cavity 1, and α and L 1 are the cone-apex angle and the length of generatrix of conical cavity 1. R 2 and R 3 denote the inner radius and outer radius of cylindrical cavity 2, and the height of cylindrical cavity 2 is marked as L 1 . Thickness of the acoustic cavity coupled system is represented as H. Compared with the coupled system in Figure 1a, the rotary composite laminated plate and conical-cylindrical double acoustic cavities coupled system add a composite plate between the two acoustic cavities. It can be found from Figure 1b that the coordinate of rotary composite laminated plate and conicalcylindrical double cavities coupled system is composed of three local coordinate systems (o-s 1 , θ 1 , r 1 ), (o-s 2 , θ 2 , r 2 ) and (o-s 3 , θ 3 , r 3 ). L p is the thickness of composite laminated plate, and other geometric parameters have the same meanings as in Figure 1a. The shapes of rotary acoustic cavity coupled system at different rotation angles are given in Figure 2. The parameter ϑ is introduced here to express the rotation angles of coupled systems, and Vibration and noise reduction whether rotation angles are 2π is related to the number of acoustic walls in the coupled systems. In order to study the vibro-acoustic characteristics of the coupled systems, a monopole point sound source Q is placed.
In Figure 3, the general boundary conditions on the edge of rotary composite laminated plate are represented by introducing three groups of linear springs k u , k v , k w along u, v, w directions and two groups of torsion springs K r and K θ , which are continuously distributed along the boundary. u, v and w denote r, θ and s directions, respectively. k u θ0 , k v θ0 , k w θ0 , K r θ0 and K θ θ0 are introduced to represent the five groups of boundary springs at the boundary θ 5 0, and the boundary springs at θ 5 ϑ, r 5 0 and r 5 H can also be expressed in a similar way. When the rotation angle ϑ 5 2π, the coupling boundary shown in Figure 3b is generated for the rotary composite laminated plate. Three groups of linear coupling springs k uc , k vc and k wc and two groups of torsion springs K rc and K θc are uniformly set on the coupling boundary (θ 5 0 and θ 5 2π). Therefore, the coupling of the rotating composite laminated plate can be realized.

Related parameter of unified model
To ensure that the systems can be coupled during the modeling process, the parallelogram section in conical acoustic cavity is necessary to be transformed into a square section. It can be found from Figure 4 that the plane ros coordinate system is transformed to plane ξoη coordinate system by iso-parametric transformation, where each vertex in plane ξoη coordinate system has been determined as (0, 0), (1, 0), (1, 1) and (0, 1).
Using the four-node coordinate transformation in the FE method, the coordinate transformation equations are as follows: Vibration and noise reduction in which (r (i) , s (i) ) and (ξ (i) , η (i) ) separately denote the coordinate of the ith vertex in the plane ros coordinate system and plane ξoη coordinate system, and N i (ξ, η) is the shape function of the ith vertex of the plane ros coordinate system. The specific coordinate system transformation process is expressed in the matrix form: where jJ j is the determinant of the Jacobian matrix. Table 1 shows the related parameters between the local coordinate systems of conical and cylindrical acoustic cavity and the coordinate system of double curvature cavity element (α, β, z). In addition, the Lame coefficient of the cavity and the value range of the corresponding coordinate axis are also given:

Construction of admissible displacement and sound pressure functions
Based on the first-order shear deformation theory (FSDT) and two-dimensional modified Fourier series theory, the admissible displacement function of rotary composite laminated plate is established. The admissible sound pressure functions of conical and cylindrical cavities are constructed by three-dimensional modified Fourier series. The expression is the superposition of a cosine function and six sines and cosines functions. The specific expressions can be written as follows (Zhang et al., 2019(Zhang et al., , 2020a in which u (r, θ, t), v (r, θ, t) and w (r, θ, t) separately represent the admissible displacement function of the surface in the r, θ and z directions of the rotary composite laminated plate, and f r ðr; θ; tÞ and f θ ðr; θ; tÞ denote the lateral rotation in the r and θ directions, respectively. p n (n 5 1, 2) is the expression of the admissible sound pressure function for the nth cavity in the coupled system. The displacement supplement polynomials of rotary composite laminated plate can be expressed as Φ M and Φ Nq (N q 5 1, 2), and the sound pressure supplement polynomials of the nth cavity are written as P n Ω and P n Θq (Θ q 5 1, 2, . . . ,6). While A mn , B mn , C mn , D mn and E mn represent the unknown two-dimensional Fourier coefficient vectors and unknown three-dimensional Fourier coefficient vectors. These parameters can be expressed as follows: P Ω n ðr n ; θ n ; s n Þ ¼ P Θ 1 n ðr n ;θ n ; s n Þ ¼

Stress-strain and displacement relationship
The normal strain and shear strain at any point on the composite laminated plate can be defined by the changes of strain and curvature in midplane: where ε 0 r , ε 0 θ , γ 0 rθ , γ 0 rz and γ 0 θz are the strain components on the middle surface of the laminated plate, and χ r , χ θ and χ rθ are the curvature variation components on the middle surface. The specific expressions are as follows: According to Hooke's law, the corresponding stress-strain relationship at the k-layer can be obtained as follows: where the stiffness coefficients of laminated plate can be denoted by Q k ij (i, j 5 1, 2, . . ., 6), and they can be acquired from the following equations: In Equation (35), T is the transformation matrix, which is defined as follows, where θ is the included angle between the main direction and the r direction of the layer, namely, the layer laying direction: T ¼ 2 6 6 6 6 4 cos 2 θ sin 2 θ À2 sin θ cos θ sin 2 θ cos 2 θ 2 sin θ cos θ cos θ sin θ Àsin θ cos θ sin θ cos θ Àsin θ cos θ cos 2 θ À sin 2 θ 3 7 7 7 7 5 in which Q k ij (i, j 5 1, 2, . . ., 6) denotes the material coefficient of the k-layer of plate, and its value can be obtained by the engineering constant: The forces and torques applied to the laminated plate are obtained by integrating the stresses in the plane. From one layer of laminated plate to another layer, the thickness is integrated to obtain the following equation: where N r , N θ and N rθ are the resultant forces in the plane, M r , M θ and M rθ represent bending and torsional torque and Q θ , Q r denote the resultant forces of horizontal shear. Shear correction factor which can guarantee the strain energy caused by transverse shear stress is expressed as κ s , and N L represents the number of layers.

Energy equation and solution procedure
The Lagrange equation of t the conical-cylindrical acoustic cavity coupled system and rotary composite laminated plate and conical-cylindrical double cavities coupled system can be expressed as follows: where T P and T Cn (n 5 1, 2) are the total kinetic energy of the rotary composite plate and the nth acoustic cavity in the coupled systems, respectively. U P and U Cn separately denote the total potential energy in the plate and the nth acoustic cavity. U PÀcoupling and U CnÀcoupling represent the coupling potential energy of the composite laminated plate and acoustic cavities when rotation angle ϑ 5 2π. The boundary spring potential energy of plate can be expressed as U SP . W P&C 1 , W P&C 2 W C 1 &P and W C 2 &P are the coupling potential energy generated when the composite plate is coupled with acoustic cavity 1 and acoustic cavity 2, and W P&C 1 ¼ W C 1 &P , W P&C 2 ¼ W C 2 &P . W Cn−wall represents the impedance potential energy caused by the impedance wall of the nth acoustic cavity in the coupled systems. W C 1 &C 2 and W C 2 &C 1 denote the coupling potential energy between acoustic cavity 1 and acoustic cavity 2, in which W C 1 &C 2 ¼ W C 2 &C 1 . W F is the work done by harmonic point force F on the composite laminated plate, and W Qn is the work done by the monopole point sound source in the nth acoustic cavity of the coupled systems.
For the two coupled systems, partial energy equations do not exist completely: (1) the conical-cylindrical acoustic cavity coupled system: (2) the rotary composite laminated plate and conical-cylindrical double cavities coupled system: The total kinetic energy of the rotating composite laminated plate and the nth acoustic cavity represented by T P and T C n can be written as follows: in which N l represents the number of layers of rotary composite laminated plate. Z k is the coordinate value of bottom surface thickness of k-layer, and Z k þ 1 is the coordinate value of upper surface thickness. The material density of k-layer can be expressed as ρ k , and ρ Cn denotes the density of acoustic media in nth acoustic cavity.

Vibration and noise reduction
The specific expressions of the total potential energy U P and U C n are as follows: Vibration and noise reduction In the formula, the total potential energy of rotary composite laminated plate includes tensile potential energy U stretch , bending potential energy U bend , tensile and bending coupling potential energy U sÀb , whose expressions have also been given, and c n is the propagation speed of sound in the acoustic medium of the nth cavity.
As the boundary conditions of the rotary composite laminated plate model established in this paper are determined by the boundary springs, the boundary spring potential energy U SP will be generated, and the expressions are as follows: When 0 < ϑ < 2π, the impedance wall dissipation energy of the nth sound cavity in the coupled systems is as follows: W Cn−wall ¼ W n wall 1 þ W n wall 2 þ W n wall 3 þ W n wall 4 þ W n wall 5 þ W n wall 6 (59) JIMSE 3,1 where j represents a pure imaginary number. S r denotes the area of the rth uncoupled acoustic wall, and the corresponding acoustic wall impedance value is expressed by Z r . When ϑ 5 2π, the coupling potential energy of the middle plate U PÀcoupling and the nth acoustic cavity of the coupled systems U CnÀcoupling can be written as follows: The coupling potential energy between acoustic cavity 1 and acoustic cavity 2 W C 1 &C 2 can be expressed as follows: W P&C 1 and W P&C 2 are the coupling potential energy when the laminated plate is coupled with acoustic cavity 1 and 2, respectively, and the expressions are as follows: Vibration and noise reduction jJ 1 j$ðr 1 þR 2 Þdr 1 dθ 1 (64) The expressions of the work W F done by the harmonic point force F on the composite laminated plate and the work W F done by the monopole point sound source Q in the nth acoustic cavity in the coupled systems are as follows: where f i (i 5 u j , v j , w j ) is the function of external load distribution, and the action position of harmonic point force F is (r 0 , θ 0 ). Q n s represents the distribution function of the point sound source acting on the nth cavity, the amplitude of the point sound source is A (kg/s 2 ) and the specific position of the point sound source Q is (r e , θ e , s e ). δ and δ c are two-dimensional and three-dimensional Dirac functions.
Each energy equation is substituted into Equations (41)-(43). According to Rayleigh-Ritz method, the partial derivatives of the unknown two-dimensional and threedimensional Fourier coefficients are obtained in the Lagrange equation, and the results are equal to zero: vL C 2 vG m t n t l t ¼ vT C 2 vG m t n t l t À vU C 2 vG m t n t l t À vU C 2 Àcoupling vG m t n t l t À vW P&C 2 vG m t n t l t À vW C 2 &C 1 vG m t n t l t þ vW C 2 −wall vG m t n t l t þ vW Q 2 vG m t n t l t ¼ 0 Transform Equation (71)-(73) into matrix form: À in which K p and K cn respectively represent the stiffness matrix of the rotary composite laminated plate and the nth acoustic cavity in the coupled systems, and M p , M cn are the mass matrices. The impedance matrix of the nth acoustic cavity in the coupled systems is denoted by Z cn . C Cn&P is the vibro-acoustic coupling matrix between the nth cavity and the composite laminated plate in coupled systems, and C P&Cn ¼ C Cn&P .
When F and Q n are equal to zero, Equations (75)-(77) can be combined to obtain the natural frequency and mode solving equations of the coupled systems. To facilitate the solution, the equations need to be converted into linear equations for solving: S ¼ 2 6 6 6 6 6 6 4 G ¼ ½ P mn ωP mn F mt nt lt ωF mt nt lt G mt nt lt ωG mt nt lt T Finally, the eigen solution ω is the natural frequency of the coupled systems, and the eigenvector G is the corresponding mode. By substituting harmonic point force and monopole point sound source into Equations (78)-(81), the steady-state response of the coupled systems can be obtained.

Numerical discussion and result analysis
According to the unified analysis model of the conical-cylindrical acoustic cavity coupled system and rotary composite laminated plate and conical-cylindrical double cavities coupled system, numerical discussion and result analysis are carried out to further research the vibroacoustic characteristics. This section mainly verifies the convergence and accuracy of the coupled system model, analyzes the influence factors on natural frequency of the coupled systems under free vibration and studies the steady-state response analysis of the coupled systems under the action of harmonic point force and point sound source excitation. In this section, the boundary conditions of the composite laminated plate in the coupled system are represented by artificial virtual spring boundary technology, which can simulate complex boundary conditions. The spring stiffness settings of each boundary condition are presented in Table 2. Table 3 shows the material parameters of the composite laminated plate in the examples. The acoustic medium of acoustic cavities is air, where the density of air is defined as ρ air 5 1.21 kg/m 3 , and the speed at which sound waves travel through air is defined as c air 5 340 m/s.

Unified analysis model validation
The convergence analysis and accuracy verification of the model of the coupled systems established above will be carried out. Tables 4 and 5 Tables 4 and 5 are as follows: R 0 5 0.5 m, R 1 5 1m, R 2 5 1.5 m, H 5 0.5 m, α 5 π/6, L 2 5 1.5 m, h p 5 0.02 m, ϑ 5 π/2. The material used is [Glass/epoxy j Boron/epoxy], and the layering angle is [0 jπ/2]. It can be seen from Tables 4 and 5 that when M c 3 N c 3 Q c 5 5 3 5 3 5 and M p 3 N p 5 10 3 10, the natural frequencies of each order basically complete convergence. In contrast to the FE simulation results, the maximum error of the natural frequency is less than 0.1%, indicating the accuracy of the proposed method. Therefore, it is reliable to select truncation values M c 3 N c 3 Q c 5 5 3 5 3 5 and M p 3 N p 5 10 3 10 in the numerical calculation of present method. When the rotation angle of the coupled systems ϑ 5 2π, the rotary composite laminated plate and acoustic cavities in the systems will generate additional coupling potential energy, exerting an influence on the vibro-acoustic characteristics of the systems. Therefore, the accuracy of the proposed method is verified again as shown in Table 6. The acoustic walls of cavities in this example are all rigid walls, and the boundary conditions of the rotary plate are set as S-S-S-S. The geometric parameters of the coupled systems in Table 6 are as follows: R 0 5 0.7 m, R 1 5 1.4 m, R 2 5 2.1 m, H 5 0.7 m, α 5 π/6, L 2 5 2.5 m, h p 5 0.03 m. The material used is [Glass/epoxy j Boron/epoxy j Glass/epoxy], and the layering angle is [π/4 j 0 jπ/4]. The maximum error between the numerical calculation results and the FE simulation results in Table 6 is less than 1%, which indicates that present method still has well accuracy when the rotation angle ϑ 5 2π.

Free vibration analysis
In this section, the free vibration of the conical-cylindrical acoustic cavity coupled system and rotary composite laminated plate and conical-cylindrical double cavities coupled system is conducted to investigate the effects of related parameters on the vibro-acoustic characteristics of the coupled systems. Figure 5 shows the variations of natural frequencies of the coupled systems with different rotation angles. The acoustic walls of cavities in this example are all rigid walls, and the boundary conditions of the rotary plate are Material ρ plate (kg/m 3 ) set as S-S-C-C. The determined geometric parameters are as follows: R 0 5 0.6 m, R 1 5 1.2 m, R 2 5 1.7 m, H 5 0.5 m, α 5 π/4, L 2 5 2m, h p 5 0.024 m. The material used is [Glass/epoxy j Boron/epoxy], and the layering angle is [Àπ/4 jπ/4]. As shown in Figure 5, the natural frequency of the same order decrease with the increase of rotation angle ϑ. However, the situation changes when the rotation angle ϑ 5 2π. Taking the rotary composite laminated plate and conical-cylindrical double cavities coupled system as an instance, Table 7 gives the first eight natural frequencies of the coupled system with various rotation angles. It can be found from Table 7 that the natural frequency at ϑ 5 2π is basically the same as that at ϑ 5 π.
In the case of coupling of the two acoustic walls, a closed loop will be formed, and repeated modes will be generated, resulting in continuous expansion of the natural frequencies of each order. These conditions explain why the natural frequency rises again when ϑ 5 2π in Figure 5.  Table 6. Accuracy analysis of the conical-cylindrical acoustic cavity coupled system and rotary composite laminated plate and conicalcylindrical double cavities coupled system at rotation angle of 2π

Vibration and noise reduction
For the conical-cylindrical acoustic cavity coupled system and rotary composite laminated plate and conical-cylindrical double cavities coupled system, it is necessary to analyze the effects of conical cavity cone-apex angle α and cylindrical cavity height L 2 on the natural frequency of the coupled systems. The variations of natural frequencies of the coupled systems with different cone-apex angle α and height L 2 are presented in Figure 6.
The acoustic walls of cavities in this example are all rigid walls, and the boundary conditions of the rotary plate are set as S-S-C-C. The determined geometric parameters are as follows: R 0 5 0.5 m, R 1 5 1m, R 2 5 1.5 m, H 5 0.5 m, ϑ 5 π/2, h p 5 0.018 m. The material used is [Glass/epoxy j Boron/epoxy], and the layering angle is [Àπ/2 jπ/2]. As can be seen from Figure 6, the natural frequency of the coupled system degrades with the decrease of coneapex angle α and the increase of height L 2 .
Compared with the conical-cylindrical acoustic cavity coupled system, the rotary composite laminated plate and conical-cylindrical double cavities coupled system is also affected by the relevant parameters of the composite laminated plate. The first eight natural frequencies of the coupled system under different boundary conditions are shown in Table 8. The acoustic walls of cavities in this example are all rigid walls. The geometric parameters of the coupled system in Table 8 are as follows: R 0 5 0.5 m, R 1 5 1m, R 2 5 1.5 m, H 5 0.5 m, α 5 π/6, L 2 5 1.5 m, h p 5 0.02 m, ϑ 5 π/2. The material used is [Glass/epoxy j Boron/epoxy], and the layering angle is [0 jπ/2]. It can be found from Table 8 that the natural frequency of the coupled system calculated by present method is basically consistent with the FE simulation  The conical-cylindrical acoustic cavity coupled system The rotary composite laminated plate and conical-cylindrical double cavities coupled system results with different boundary conditions, which again proves the accuracy of the proposed method. Meanwhile, the natural frequency of the coupled system increases with the increase of the stiffness value of boundary springs, and the calculation results of elastic boundary 1 (E 1 ) and elastic boundary 2 (E 2 ) also verify the reliability of this conclusion. Besides boundary conditions, the thickness of the plate is also necessary to be studied parameterized. Figure 7 shows the influence trend of laminated plate thickness h p on the natural frequency of the coupled system with different size conditions. The acoustic walls of cavities in this example are all rigid walls, and the boundary conditions of the rotary composite laminated plate are set as C-C-C-C. The determined geometric parameters are as follows: α 5 π/6, L 2 5 1.5 m, ϑ 5 π/2. The material used is [Glass/epoxy j Boron/epoxy], and the layering angle is [0 jπ/2]. As can be seen from Figure 7, the natural frequency of the coupled system shows an upward trend with the increase of h p .

Steady state response analysis
In this section, the displacement response and sound pressure response of the conicalcylindrical acoustic cavity coupled system and rotary composite laminated plate and conicalcylindrical double cavities coupled system under the excitation of point sound source are researched. Figures 8 and 9 Figures 8 and 9 that the response curve obtained by the proposed method is consistent with the results of FE simulation, which proves the accuracy of the displacement and sound pressure response analysis model of the coupled systems under the excitation of point sound source. The sound pressure response of the coupled systems under the excitation of a point sound source with different impedance value of the acoustic walls is presented in Figure 10. All the six acoustic walls in the coupled systems are impedance walls, and the impedance values of the acoustic walls are rigid, Z 1 5 ρ c c 0 (100j) and Z 2 5 ρ c c 0 (30j). The boundary condition, geometric parameters, materials used and layering angle of the coupled systems are consistent with those in Tables 4 and 5. The action position of point sound source is at (0.80 , 0.71, 0.25 m) in cavity 1, observation point 1 is located at (0.85 , 0.42, 0.33 m) in cavity 1 and observation point 2 is located at (1.15 , 0.45, 0.52 m) in cavity 2.
As shown in Figure 10, the amplitude of the sound pressure response is reduced, and the effect of resonance suppression is achieved when the acoustic wall is changed to the impedance wall, whose influence increases with the increase of the impedance value. However, the variation of acoustic walls in the coupled systems does not affect the waveform of the response. The conical-cylindrical acoustic cavity coupled system The rotary composite laminated plate and conical-cylindrical double cavities coupled system

Vibration and noise reduction
In the rotary composite laminated plate and conical-cylindrical double cavities coupled system, if the point source excitation and the observation point are not in the same cavity, the sound pressure response of the coupled system will be affected by the laminated plate between the two cavities. To investigate the influence mechanism on the thickness of plate, Figure 11 gives the sound pressure response curves of the observation points in the upper conical cavity under the excitation of the point sound source in the lower cylindrical cavity with different thicknesses of the laminated plate, where h p 5 0 m denotes that there is no plate between the two acoustic cavities, and the system is cylindrical-conical acoustic cavity coupled system. The acoustic walls, boundary condition, geometric parameters, materials used and layering angle of the coupled systems are consistent with those in Tables 4 and 5. The action position of point sound source is at (1.23 , 0.41, 0.86 m) in cavity 2, observation point 1 is located at (0.77 , 0.69, 0.29 m) in cavity 1 and observation point 2 is located at (0.91 , 0.35, 0.38 m) in cavity 1. It is not difficult to know from Figure 11 that the amplitude of sound pressure response curves at the same observation point decreases with the increase of the plate thickness, especially the difference between the sound pressure response and the sound pressure response of the composite laminated plate is apparent. The results show that the composite laminated plate in the coupled system has an impact on the noise reduction, and the effect increases with the increase of the thickness of composite laminated plate.

Conclusion
A unified analysis model of the rotary composite laminated plate and conical-cylindrical double cavities coupled system is constructed in this investigation. First, the admissible displacement and sound pressure functions of the rotary laminated plate and acoustic cavity are presented. Second, the energy functional of the laminated plate structure domain and cavity sound field domain is proposed. Then, the coupling potential energy between the cavities and the plate-double cavities is introduced to obtain the total energy functional of the coupled systems. Finally, the energy functional is solved by the Rayleigh-Ritz method. The free vibration and steady-state response of the coupled systems are studied by the numerical results of examples, and the following significant conclusions are obtained: (1) The unified analysis model established in this paper has good convergence and accuracy when the truncation values M c 3 N c 3 Q c 5 5 3 5 3 5 and M p 3 N p 5 10 3 10.
(2) Under the condition of free vibration, the natural frequency of the rotary composite laminated plate and conical-cylindrical double cavities coupled system increases with the increase of the rotation angle, the spring stiffness, thickness of the plate and cone-apex angle of the conical acoustic cavity. It decreases with the height of the cylindrical acoustic cavity increasing.
(3) In the conical-cylindrical acoustic cavity coupled system and rotary composite laminated plate and conical-cylindrical double cavities coupled system, the impedance wall can reduce the amplitude of the sound pressure response and suppress the resonance of the coupled systems. The effect of the impedance wall increases with the increase of the impedance value but has no effect on the waveform of the response. In the rotary composite laminated plate and conical-cylindrical double cavities coupled system, the composite laminated plate has a good noise reduction effect and increases with the increase of the plate thickness.