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A theoretical solution method for the in-plane vibration characteristics of the plate with cutouts is proposed in this paper. The energy principle in conjunction with the Rayleigh–Ritz solution technique is adopted for the theoretical modeling of the in-plane vibration of the structure. The energy functions for the plate with cutouts are established by subtracting the energies of the cutout domains from the total energies of the whole plate. To ensure continuity over the entire solution domain, the in-plane displacements are composed of two-dimensional standard Fourier series and supplementary functions. The in-plane eigenmodes of the plates with different square cutouts are compared with the results obtained from the finite element method (FEM), with good agreements. The influences of the cutouts on the in-plane vibration characteristics of the plates with cutouts are investigated by varying the number, size, and position of the cutouts.

This paper presents a three-dimensional formulation for the free vibrations of thick rectangular plates with general boundary conditions and resting on elastic foundations. The general boundary conditions are imposed by means of penalty function method. The displacements of the plates are expressed by a three-dimensional cosine series and some simple polynomial functions which introduced to ensure and accelerate the convergence of the series representation. All the unknown coefficients can be obtained by using the Rayleigh–Ritz method. Comparisons of the present results with those in available literature demonstrate the accuracy and reliability of the present formulation. Furthermore, parametric investigations are presented including the effects of boundary conditions, geometrical parameters, and elastic foundation.

This paper is concerned with the modeling and acoustic eigenanalysis of coupled spaces with a coupling aperture of variable size. A modeling method for this problem is developed based on the energy principle in combination with a 3D modified Fourier cosine series approach. Under this theoretical framework, the energy exchange property and acoustically transparent characteristics of the opening are taken into account via the inflow and outflow sound powers through the opening without any assumptions. The sound pressure in the subrooms is constructed in the form of the three-dimensional modified Fourier series with several auxiliary functions introduced to ensure the uniform convergence of the solution over the entire solution domain. The accuracy of the natural frequencies and mode shapes of three exemplary coupled rooms systems is verified against numerical data obtained by finite element method, with good agreement achieved. The present method offers a unified procedure for a variety of cases because the modification of any parameter from one case to another, such as the size and location of the coupling aperture, is as simple as modifying the material properties, requiring no changes to the solution procedures.

This paper is concerned with the modeling and solution method of a three-dimensional (3D) coupled acoustic system comprising a partially opened cavity coupled with a flexible plate and an exterior field of semi-infinite size, which is ubiquitously encountered in architectural acoustics and is a reasonable representation of many engineering occasions. A general solution method is presented to predict the dynamic behaviors of the three-dimensional (3D) acoustic coupled system, in which the displacement of the plate and the sound pressure in the cavity are respectively constructed in the form of the two-dimensional and three-dimensional modified Fourier series with several auxiliary functions introduced to ensure the uniform convergence of the solution over the entire solution domain. The effect of the opening is taken into account via the work done by the sound pressure acting at the coupling aperture that is contributed from the vibration of particles on the acoustic coupling interface and on the structural–acoustic coupling interface. Both the acoustic coupling between finite cavity and exterior field and the structural–acoustic coupling between flexible plate and interior acoustic field are considered in the vibro-acoustic modeling of the three-dimensional acoustic coupled acoustic system. The dynamic responses of the coupled structural–acoustic system are obtained using the Rayleigh-Ritz procedure based on the energy expressions for the coupled system. The accuracy and effectiveness of the proposed method are validated through numerical examples and comparison with results obtained by the boundary element analysis. Furthermore, the influence of the opening and the cavity volume on the acoustic behaviors of opened cavity system is studied.

The eigenmodes of vibro-acoustic coupled spaces with a mixed interface are predicted by the energy principle, in which the mixed interface is composed of an acoustic aperture and the structure with a cutout. Both the vibro-acoustic coupling and the acoustic coupling have been considered in the acoustic vibration analysis owing to the coexistence of sound transmission paths for the structure and acoustic aperture in the coupled system. The transverse displacement of the structure and the sound pressure inside the coupled spaces are, respectively, expressed as two and three dimensional improved Fourier series to ensure the uniform convergence of the solutions over the entire solution domain. The vibro-acoustic coupling energy is composed of the work produced by the interaction of between the coupled spaces and the structure on the rectangular regions decomposed by the aperture position in the establishment of the energy functions for the coupled spaces and the structure with a cutout. The accuracy of the eigenmodes for the vibro-acoustic coupled systems having variable cutouts within the mixed interface is verified against numerical results obtained by finite element method. The influence of the acoustic aperture within the mixed interface on the modal characteristics of the vibro-acoustic coupled system is investigated by the comparisons between the modal results of the coupled system possessing different cutout sizes and cutout positions.

A modeling method is proposed for the acoustic analysis of a three-dimensional (3D) rectangular opened enclosure coupled with a semi-infinite exterior field by a rectangular opening of arbitrary size, and with general wall impedance. In contrast to existing modeling methods that solve the differential equations, the energy principle in combination with a 3D modified Fourier cosine series is employed in the present method for the modeling of this system. Under this theoretical framework, the effect of an opening in the wall of a rectangular enclosure is taken into account via the work done by the sound pressure acting on the opening between the finite enclosure and exterior domain. The sound pressure inside the opened enclosure is expressed as the combination of a 3D trigonometric cosine series and one supplementary 2D expansion introduced to ensure uniform convergence of the solution over the entire solution domain including opening boundary. The acoustic responses of the opened enclosure are obtained based on the energy expressions for the enclosure system. The effectiveness and reliability of the current method are checked against the results obtained by the boundary element method and experimental results, and excellent agreement is achieved. The effects of sizes and positions of the opening and wall impedance on the acoustic behaviors of opened enclosure system are investigated.

A unified solution for coupled cylindrical shell and annular plate systems with general boundary and coupling conditions is presented in the study by using a modified Fourier-Ritz method. Under the framework, regardless of the boundary and continuity conditions, each displacement for the cylindrical shell and the annular plate is invariantly expressed as the modified Fourier series composed of the standard Fourier series and auxiliary functions. The introduction of the auxiliary functions can not only remove the potential discontinuities at the junction and the extremes of the combination but also accelerate the convergence of the series expansion. All the expansion coefficients are determined by the Rayleigh-Ritz method as the generalized coordinates. The arbitrary axial position of the annular plate coupling with the cylindrical shell considered in the theoretical formulation makes the present method more general. The theoretical model established by present method can be conveniently applied to cylindrical shell-circular plate combinations just by varying the inner radius of the annular plate. The convergence and accuracy of present method are tested and validated by a number of numerical examples for coupled annular plate-cylindrical shell structures with various boundary restraints and general elastic coupling conditions. The effects of the axial position of the annular plate and elastic coupling conditions on the vibration behavior of the coupled system are also investigated. The power of present method compared to conventional finite element method is demonstrated with less computation cost. Some new results are presented to provide useful information for future researchers.

This paper presents a modified Fourier–Ritz approach for free vibration analysis of laminated functionally graded shallow shells with general boundary conditions in the framework of first-order shear deformation shallow shell theory. The displacement and rotation components of the shells are represented by the modified Fourier series consisted of standard Fourier cosine series and several closed-form auxiliary functions introduced to ensure and accelerate the convergence of the series representation. The material properties are assumed to vary continuously through the thickness according to power-law distribution. Four common types of sandwich functionally graded shallow shells are studied. The bi-layered and single-layered functionally graded shallow shells are obtained as special cases of sandwich shells. By setting groups of boundary springs and assigning corresponding stiffness constants to the springs, different boundary conditions including free, clamped, simply supported, elastic boundaries and their combinations are considered. A comprehensive investigation concerning the free vibration of the laminated functionally graded shallow shells with completely free and simply-supported edges is given. The results show that the present method enables rapid convergence, high reliability and accuracy. Numerous new vibration results for shallow shells with different material distributions, lamination schemes and elastic restraints are provided. Some mode shapes of the shallow shells are depicted. Parameter studies illustrate that changes of boundary conditions, material types, thickness schemes and power-law exponents will affect obviously the free vibration characteristic of the shallow shells. In contrast to most existing techniques, the current method can be universally applicable to a variety of boundary conditions including all the classical cases, elastic restraints and their combinations without the need of making any change to the solution procedure.

Free vibration analysis of truncated conical shells with general elastic boundary conditions is presented in this paper. An accurate modified Fourier series solution is developed, in which, regardless of the boundary conditions, each displacement of the conical shell is invariantly expressed as a new form of improved series expansions composed of a standard Fourier series and closed-form auxiliary functions introduced to ensure and accelerate the convergence of the series expansion. All the expansion coefficients are treated as the generalized coordinates and determined using the Rayleigh–Ritz method. By using the present method, conical shells with arbitrary boundary conditions including all classical and elastic end restraints can be solved in a unified form. The accuracy and convergence of the current approach are validated by numerical examples and comparison with FEM results and those from the literature, and excellent accuracy is demonstrated. Comprehensive studies on the effects of elastic restraint parameters, semi-vertex angle and the ratio of length to radius are also reported. Some new results are presented for cases with elastic boundary restraints which may serve as benchmark solution for future researches.

A unified method based on the three-dimensional theory of elasticity is developed for the free vibration analysis of thick cylindrical shells with general end conditions and resting on elastic foundations. Each shell displacements, regardless of boundary conditions, is expanded as a standard Fourier cosine series supplemented with four auxiliary functions introduced to eliminate any possible discontinuities of the original displacement and its derivatives throughout the entire shell space including the boundaries and then to effectively enhance the convergence of the results. Mathematically, such series expansions are capable of representing any functions (including the exact displacement solutions). Since the displacement field is constructed adequately smooth throughout the whole solution domain, an accurate solution can obtained by using Rayleigh–Ritz procedure based on the energy functions of the shell. The current method can be universally apply to a variety of end conditions including all the classical cases and their combinations and arbitrary elastic foundations. The excellent accuracy and reliability of current solutions are demonstrated by numerical examples and comparisons with the results available in the literature. Effects of the boundary restraining parameters and foundation coefficients on frequency parameters are investigated as well. New results for thick cylindrical shells with various end conditions and resting on elastic foundations are presented, which may serve as benchmark solutions for validating new computational techniques in future.

A unified accurate solution procedure for free vibration analysis of arbitrary functionally graded spherical shell segments with general end restraints is presented. The material properties of the spherical shells are assumed to change continuously in the thickness direction and two different four-parameter power-law distributions are considered. The proposed method is formulated by the Ritz procedure on the basis of the first-order shear deformation shell theory. Each of admissible functions, regardless of boundary conditions, is composed of a standard Fourier cosine series and several auxiliary functions introduced to ensure and accelerate the convergence of series representations. The accuracy and reliability of the current solution are validated by comparing the results with existing results and those generated from the finite element analyses, and numerous new results for functionally graded spherical shells subjected to elastic restraints are presented, which can serve as the benchmark solutions for other computational techniques in the future research. The effects of the boundary conditions, power-law exponents, and shell segments on the free vibrations of the spherical shells are also investigated, and some interesting insights into the parameter effects on frequency behaviors are illustrated.

This paper presents a unified formulation for the free vibration analysis of open shells subjected to arbitrary boundary conditions and various geometric parameters such as subtended angle, conicity. Under the current framework, a general classical shell theory in conjunction with Chebyshev polynomials and Rayleigh–Ritz procedure is developed. Each displacement components of the open shell, regardless of shell types and boundary conditions, is expanded as Chebyshev polynomials of first kind in both directions. All the Chebyshev expanded coefficients are treated equally and independently as the generalized coordinates and solved directly by using the Rayleigh–Ritz procedure. The convergence, accuracy and reliability of the current formulation are validated by comparisons with existing experimental and numerical results published in the literature, with excellent agreements achieved. A considerable number of new vibration results for open cylindrical, conical and spherical shells with various geometric parameters and boundary conditions are presented, which may be used for benchmarking of researchers in the field. The effects of boundary stiffness, subtended angle and conicity on the frequency parameters of different open shells are also discussed in detail. The presented formulation is general compared to the existing literature. Different boundary conditions and geometric dimensions (shallow or deep), different types of shells (cylindrical, conical or spherical), can be easily accommodated in this formulation. It also offers a unified operation for the entire restraining conditions and the change of boundary conditions from one case to another is as easy as changing structural parameters without the need of making any change to the solution procedure. In addition, it can be readily applied to open shells with more complex boundaries such as point supports, non-uniform elastic restraints, partial supports and their combinations.

In this paper, a unified solution method for free vibration analysis of functionally graded cylindrical, conical shells and annular plates with general boundary conditions is presented by using the first-order shear deformation theory and Rayleigh–Ritz procedure. The material properties of the structures are assumed to change continuously in the thickness direction according to the general four-parameter power-law distributions in terms of volume fractions of constituents. Each of displacements and rotations of those structures, regardless of boundary conditions, is expressed as a modified Fourier series, which is constructed as the linear superposition of a standard Fourier cosine series supplemented with auxiliary polynomial functions introduced to eliminate all the relevant discontinuities with the displacement and its derivatives at the edges and accelerate the convergence of series representations. The excellent accuracy and reliability of the current solutions are confirmed by comparing the present results with those available in the literatures, and numerous new results for functionally graded cylindrical, conical shells and annular plates with elastic boundary conditions are presented. The effects of boundary conditions and the material power-law distribution are also illustrated.

In this paper, a simple yet accurate solution procedure based on the Haar wavelet discretization method (HWDM) is applied to the free vibration analysis of composite laminated cylindrical shells subjected to various boundary conditions. The Reissner–Naghdi’s shell theory is adopted to formulate the theoretical model. The initial partial differential equations (PDE) are first converted into system of ordinal differential equations by the separation of variables. Then the discretizations of governing equations and corresponding boundary conditions are implemented by means of the HWDM, which leads to a standard linear eigenvalue problem. Accuracy and reliability of the current solutions are validated by comparing the results with those available in the literature. The effects of several significant aspects including boundary conditions, length to radius ratios, lamination schemes and elastic modulus ratios on natural frequencies are discussed. The advantages of the current solutions consist in its simplicity, fast convergence, low computational cost and high accuracy.

An analytical study on the vibro-acoustic behaviors of a double-panel structure with an acoustic cavity is presented. Unlike the existing studies, a structural–acoustic coupling model of an elastically restrained double-panel structure with an acoustic cavity having arbitrary impedance on sidewalls around the cavity is developed in which the two dimensional (2D) and three dimensional (3D) modified Fourier series are used to represent the displacement of the panels and the sound pressure inside the cavity, respectively. The unknown expansions coefficients are treated as the generalized coordinates and the Rayleigh–Ritz method is employed to determine displacement and sound pressure solutions based on the energy expressions for the coupled structural–acoustic system. The effectiveness and accuracy of the present model is validated by numerical example and comparison with finite element method (FEM) and existing analytical method, with good agreement achieved. The influence of key parameters on the vibro-acoustic behaviors and sound transmission of the double-panel structure is investigated, including: cavity thickness, boundary conditions, sidewall impedance, and the acoustic medium in the cavity.

A new three-dimensional exact solution for the free vibrations of arbitrarily thick functionally graded rectangular plates with general boundary conditions is presented. The three-dimensional elasticity theory is employed to formulate the theoretical model. According to a power law distribution of the volume of the constituents, the material properties change continuously through the thickness of the functionally graded plates. Each of displacements of the plates, regardless of boundary conditions, is expanded as a three-dimensional (3-D) Fourier cosine series supplemented with closed-form auxiliary functions introduced to eliminate all the relevant discontinuities with the displacements and its derivatives at the edges. Since the displacement fields are constructed adequately smooth throughout the entire solution domain, an exact solution is obtained based on Rayleigh–Ritz procedure by the energy functions of the plate. The excellent accuracy and reliability of the current solutions are demonstrated by numerical examples and comparison of the present results with those available in the literature, and numerous new results for thick FG plates with elastic boundary conditions are presented. The effects of gradient indexes are also illustrated.

This study presents a forced acoustic analysis and investigation of the energy distribution of a theoretical model of coupled rooms with a transparent coupling aperture. The energy principle is employed in combination with a 3D modified Fourier cosine series approach to formulate the theoretical model. The sound pressures inside individual subrooms are expanded as 3D Fourier cosine series supplemented with auxiliary functions, which are introduced to ensure the uniform convergence of the solution over the entire solution domain. Both the energy transmission through the coupling interface and the work done by the external load are considered as the external work in the energy expressions for the coupled system. The theoretical results are verified using the experimental results obtained for the room with a transparent opening. The influences of the coupling aperture and boundary conditions on the acoustic behaviors and energy distribution inside the coupled room system are investigated, including: opening positions, opening size, and impedance boundaries.

This paper presents elasticity solutions for the vibration analysis of isotropic and orthotropic open shells and plates with arbitrary boundary conditions, including spherical and cylindrical shells and rectangular plates. Vibration characteristics of the shells and plates have been obtained via a unified three-dimensional displacement-based energy formulation represented in the general shell coordinates, in which the displacement in each direction is expanded as a triplicate product of the cosine Fourier series with the addition of certain supplementary terms introduced to eliminate any possible jumps with the original displacement function and its relevant derivatives at the boundaries. All the expansion coefficients are then treated equally as independent generalized coordinates and determined by the Rayleigh-Ritz procedure. To validate the accuracy of the present method and the corresponding theoretical formulations, numerical cases have been compared against the results in the literature and those of 3D FE analysis, with excellent agreements obtained. The effects of boundary conditions, material parameters, and geometric dimensions on the frequencies are discussed as well. Finally, several 3D vibration results of isotropic and orthotropic open spherical and cylindrical shells and plates with different geometry dimensions are presented for various boundary conditions, which may be served as benchmark solutions for future researchers as well as structure designers in this field.

A general modeling method is developed for the vibroacoustic analysis of an arbitrarily restrained rectangular plate backed by a cavity with general wall impedance. The present method provides a uniform way to obtain the solution of the coupled structure-cavity system, making changes of both boundary conditions of the plate and impedance of the cavity walls as simple as the modifications of geometrical or material parameters without requiring any altering of the whole solution procedure. With the displacement of the plate and acoustic pressure in the cavity expanded as double and triple Chebyshev polynomial series, respectively, a simple yet efficient solution to the problem of the modal and vibroacoustic behavior of the coupled system is obtained under the Rayleigh–Ritz frame. The current method can be applied to handle strong structural-acoustic coupling cases and this is illustrated explicitly by considering one case with a shallow cavity and very thin plate while the other with a water-filled cavity. The spatial matching of velocity at the interface is checked by numerical examples. The excellent orthogonal and complete properties of the Chebyshev series representations enable excellent accuracy and numerical stability. An experiment is conducted to validate the present method. In addition, the accuracy and reliability of the current method are also extensively validated by numerical examples and comparisons with theoretical solutions, finite element results, and results available in the literature. The effects of several key parameters are analyzed, including structural boundary conditions, plate thickness, cavity depth, and wall impedance.

This paper presents an analytical and experimental study on active control of structurally radiated sound from an elastic cylindrical shell. An analytical model is developed for the active structural acoustic control (ASAC) of the cylindrical shell. The structural response of a cylindrical shell to the harmonic point force is derived by applying the modal supposition method. Expressions for the far‐field radiated sound pressure and radiated sound power are derived by using the Kirchhoff–Helmholtz integral and the appropriate Green’s function. Both global and local control strategies are considered. The optimal control forces corresponding to each control strategy are obtained by using the linear quadratic optimal control theory. Numerical simulations are performed to examine and analyze the control performance under different control strategies. Results show that using point force as the control input of the ASAC system can achieve the global sound attenuation of the cylindrical shell at resonance frequen...

An analytical study of active control of sound transmission into a cabin enclosure is presented. A cabin enclosure with four acoustically rigid walls and two flexible plates is considered as the analytical model. Two types of actuators are used, i.e., acoustic actuators and distributed lead zirconate titanate piezoelectric (PZT) actuators instead of point force actuators. Using the modal acoustic transfer impedance‐mobility matrices, the excitation and interaction in the coupled sound transmission system can be described with clear physical significance. With the control system designed to globally reduce the sound field, different control configurations are considered, including the structural actuator on the incident panel, actuator on the receiving panel, acoustic actuator on the cavity, and their combinations. The effectiveness and performance of the control system corresponding to each configuration are compared and discussed. The roles and attenuation mechanisms of each type of actuator are of parti...