Vibration analysis of circular cylindrical double-shell structures under general coupling and end boundary conditions

In this paper, the free and forced vibration analysis of circular cylindrical double-shell structures under arbitrary boundary conditions is presented. This is achieved by employing the improved Fourier series method based on Hamilton’s principle. In the formulation, each displacement component of the cylindrical shells and annular plates is invariantly expanded as the superposition of a standard Fourier series with several supplementary functions introduced to remove the potential discontinuities of the original displacement and its derives at the boundaries. With the introduction of four sets of boundary springs at the coupling interfaces and end boundaries of the shell–plate combination, both elastic and rigid coupling and end boundary conditions can be easily obtained by assigning the stiffnesses of the artificial springs to certain values. The natural frequencies and mode shapes of the structures as well as frequency responses under forced vibration are obtained with the Rayleigh–Ritz procedure. The convergence of the method is validated by comparing the present results with those obtained by the finite element method. Several numerical results including natural frequencies and mode shapes are presented to demonstrate the excellent accuracy and reliability of the current method. Finally, a number of parameter studies concerning various end and coupling boundary conditions, different dimensions of shells and annular plates are also performed.     

Free vibration of two elastically coupled rectangular plates with uniform elastic boundary restraints

An analytical method is derived for determining the vibrations of two plates which are generally supported along the boundary edges, and elastically coupled together at an arbitrary angle. The interactions of all four wave groups (bending waves, out-of-plane shearing waves, in-plane longitudinal waves, and in-plane shearing waves) have been taken into account at the junction via four types of coupling springs of arbitrary stiffnesses. Each of the transverse and in-plane displacement functions is expressed as the superposition of a two-dimensional (2-D) Fourier cosine series and several supplementary functions which are introduced to ensure and improve the convergence of the series representation by removing the discontinuities that the original displacement and its derivatives will potentially exhibit at the edges when they are periodically expanded onto the entire x-y plane as mathematically implied by a 2-D Fourier series. The unknown expansions coefficients are calculated using the Rayleigh-Ritz procedure which is actually equivalent to solving the governing equation and the boundary and coupling conditions directly when the assumed solutions are sufficiently smooth over the solution domains. Numerical examples are presented for several different coupling configurations. A good comparison is observed between the current results and the FEA models. Although this study is specifically focused on the coupling of two plates, the proposed method can be directly extended to structures consisting of any number of plates.     

A unified approach for predicting sound radiation from baffled rectangular plates with arbitrary boundary conditions

This paper discusses sound radiation from a baffled rectangular plate with each of its edges arbitrarily supported in the form of elastic restraints. The plate displacement function is universally expressed as a 2-D Fourier cosine series supplemented by several 1-D series. The unknown Fourier expansion coefficients are then determined by using the Rayleigh-Ritz procedure. Once the vibration field is solved, the displacement function is further simplified to a single standard 2-D Fourier cosine series in the subsequent acoustic analysis. Thus, the sound radiation from a rectangular plate can always be obtained from the radiation resistance matrix for an invariant set of cosine functions, regardless of its actual dimensions and boundary conditions. Further, this radiation resistance matrix, unlike the traditional ones for modal functions, only needs to be calculated once for all plates with the same aspect ratio. In order to determine the radiation resistance matrix effectively, an analytical formula is derived in the form of a power series of the non-dimensional acoustic wavenumber; the formula is mathematically valid and accurate for any wavenumber. Several numerical examples are presented to validate the formulations and show the effect of the boundary conditions on the radiation behavior of planar sources.     

Geometrically nonlinear dynamic response of stiffened plates with moving boundary conditions

An approach is presented to investigate the nonlinear vibration of stiffened plates. A stiffened plate is divided into one plate and some stiffeners, with the plate considered to be geometrically nonlinear, and the stiffeners taken as Euler beams. Lagrange equation and modal superposition method are used to derive the dynamic equilibrium equations of the stiffened plate according to energy of the system. Besides, the effect caused by boundary movement is transformed into equivalent excitations. The first approximation solution of the non-resonance is obtained by means of the method of multiple scales. The primary parametric resonance and primary resonance of the stiffened plate are studied by using the same method. The accuracy of the method is validated by comparing the results with those of finite element analysis via ANSYS. Numerical examples for different stiffened plates are presented to discuss the steady response of the non-resonance and the amplitude-frequency relationship of the primary parametric resonance and primary resonance. In addition, the analysis on how the damping coefficients and the transverse excitations influence amplitude-frequency curves is also carried out. Some nonlinear vibration characteristics of stiffened plates are obtained, which are useful for engineering design.     

Vibrations of a polygonal plate having orthogonal straight edges by an extended rayleigh-ritz method

An extended Rayleigh-Ritz method is presented for solving vibration problems of a polygonal plate having orthogonal straight edges. The polygonal plate is considered as an assemblage of several rectangular plates. For each element rectangular plate, the transverse displacement is approximated by interpolation functions corresponding to unknown displacements and slopes at the discrete points which are chosen along the edges, and series of trial functions which satisfy homogeneous artificial boundary conditions. By minimizing the energy functional corresponding to the assumed displacement function, the dynamic stiffness matrix of the element rectangular plate, which is similar to that obtained in the finite element method, is derived. The dynamic stiffness matrix of the whole system is obtained by summing up those of the element rectangular plates. Numerical results are presented for the natural frequencies and mode shapes of cantilever L-shaped and T-shaped plates.     

Buckling and free vibration analyses of stiffened plates using the FSDT mesh-free method

This paper presents a mesh-free Galerkin method for the free vibration and stability analyses of stiffened plates via the first-order shear deformable theory (FSDT). The model of a stiffened plate is formed by (1) regarding the plate and the stiffener separately, (2) imposing displacement compatible conditions between the plate and the stiffener so that displacement fields of the stiffener can be expressed in terms of the mid-surface displacement of the plate, and (3) superimposing the strain energy of plate and stiffener. Because there are no meshes used in this method, the stiffeners can be placed anywhere on the plate and need not be placed along the mesh lines. Several numerical examples are computed by this method to show its accuracy and convergence. The present results demonstrate good agreement with the existing solutions given by other researchers and the ANSYS. Influences of support size and order of the complete basis functions on the numerical accuracy are also investigated.     

Scattering by cavities of arbitrary shape in an infinite plate and associated vibration problems

This paper presents a solution for the displacement of a uniform elastic thin plate with an arbitrary cavity, modelled using the biharmonic plate equation. The problem is formulated as a system of boundary integral equations by factorizing the biharmonic equation, with the unknown boundary values expanded in terms of a Fourier series. At the edge of the cavity we consider free-edge, simply-supported and clamped boundary conditions. Methods to suppress ill-conditioning which occurs at certain frequencies are discussed, and the combined boundary integral equation method is implemented to control this problem. A connection is made between the problem of an infinite plate with an arbitrary cavity and the vibration problem of an arbitrarily shaped finite plate, using the jump discontinuity present in single-layer distributions at the boundary. The first few frequencies and modes of displacement are computed for circular and elliptic cavities, which provide a check on our numerics, and results for the displacement of an infinite plate are given for four specific cavity geometries and various boundary conditions.     

Large amplitude flexural vibration of eccentrically stiffened plates

The large amplitude free flexural vibrations of thin, orthotropic, eccentrically and lightly stiffened elastic rectangular plates are investigated. Clamped boundary conditions with movable in-plane edge conditions are assumed. A simple modal form of one-term transverse displacement is used and in-plane displacements are made to satisfy the in-plane equilibrium equations. By using Lagrange's equation, the modal equations for the nonlinear free vibration of stiffened plates are obtained for the cases when the stiffeners are assumed to be smeared out over the entire surface of the plate, and when the stiffeners are located at finite intervals. Numerical results are obtained for various possibilities of stiffening and for different aspect ratios of the plate. By particularizing the problem to different known cases, the results obtained here are compared with available analytical and experimental results, and the agreement is good.     

PARAMETRIC STUDY FOR AN EFFICIENT MESHLESS METHOD IN VIBRATION ANALYSIS

Based on the moving least-squares (MLS) approach, an efficient meshless method is employed to generate the displacement functions for vibration analysis of elastic bodies. The equation of motion is established by following the standard procedure and the boundary conditions are imposed by applying penalty functions. As the displacement functions are expressed in terms of weight functions, the accuracy will depend on the parameters of the weight functions. Therefore, a parametric study is carried out to determine the best values for these parameters. To demonstrate the accuracy, modal analyses of the beams and plates with different boundaries have been carried out. In addition, the responses of these structures under dynamic excitation have been analyzed. The examples include simply supported beams subjected to sudden excitations and simply supported plates subjected to initial displacements.     

Three-dimensional free vibration analysis of isotropic rectangular plates using the B-spline Ritz method

This paper presents three-dimensional free vibration analysis of isotropic rectangular plates with any thicknesses and arbitrary boundary conditions using the B-spline Ritz method based on the theory of elasticity. The proposed method is formulated by the Ritz procedure with a triplicate series of B-spline functions as amplitude displacement components. The geometric boundary conditions are numerically satisfied by the method of artificial spring. To demonstrate the convergence and accuracy of the present method, several examples with various boundary conditions are solved, and the results are compared with other published solutions by exact and other numerical methods based on the theory of elasticity and various plate theories. Rapid, stable convergences as well as high accuracy are obtained by the present method. The effects of geometric parameters on the vibrational behavior of cantilevered rectangular plates are also investigated. The results reported here may serve as benchmark data for finite element solutions and future developments in numerical methods.     

Vibration of electro-elastic versus magneto-elastic circular/annular plates using the Chebyshev–Ritz method

The aim of this paper is to analyze three-dimensional free vibration of magneto-elastic/electro-elastic circular/annular plates with different boundary conditions using the Chebyshev–Ritz method, in which a set of duplicate Chebyshev polynomial series multiplied by the boundary function satisfying the boundary conditions are chosen as the trial functions of the displacement components, the electric potential and the magnetic potential. Convergence of the method is checked using various Chebyshev polynomial terms. The effect of geometrical parameters and material properties of magneto-elastic/electro-elastic circular/annular plates on the eigenfrequencies of free vibration is considered.     

Free vibration of polar-orthotropic sector plates

The free vibration of ring-shaped polar-orthotropic sector plates is analyzed by the Ritz method using a spline function as an admissible function for the deflection of the plates. For this purpose, the transverse deflection of a sector plate is written in a series of the products of the deflection function of a sectorial beam and that of a circular beam satisfying the boundary conditions. The deflection function of the sectorial beam is approximately expressed by a quintic spline function, which satisfies the equation of flexural vibration of the beam at each point dividing the beam into small elements. The frequency equation of the plate is derived by the conditions for a stationary value of the Lagrangian. The present method is applied to ring-shaped polar-orthotropic sector plates with some combination of boundary conditions, and the natural frequencies and the mode shapes are calculated numerically up to higher modes. This method is very effective for the study of vibration problems of variously shaped anisotropic plates including these sector plates.     

加筋板结构声导纳的面接触判据

通过采用有限元方法建立加筋板模型,计算不同边界条件下具有不同加筋形式的薄板在不同加载面尺寸下的导纳差值,进而得到与频率相关的加筋板面接触判据,揭示了加筋板面导纳和点导纳之间的相关关系,得到了“点”和“面”是否能够等效处理的面接触判据。进一步通过考察薄板的不同加筋形式和加载面形状以及加筋板的速度分布,可以发现:在不加筋、单加筋和十字加筋3种情况中不加筋平板的面接触判据最严格,而十字加筋板的面接触判据最宽松;单加筋板的速度差值与导纳差值的计算结果较为吻合;单加筋板的加载面沿垂直加筋方向的边长越长导纳差值越大。      

Study on plate silencer with general boundary conditions

A plate silencer consists of an expansion chamber with two side-branch rigid cavities covered by plates. Previous studies showed that, in a duct, the introduction of simply supported or clamped plates into an air conveying system could achieve broadband quieting from low to medium frequencies. In this study, analytical formulation is extended to the plate silencer with general boundary conditions. A set of static beam functions, which are a combination of sine series and third-order polynomial, is employed as the trial functions of the plate vibration velocity. Green?s function and Kirchhoff–Helmholtz integral are used to solve the sound radiation in the duct and the cavity, and then the vibration velocity of the plate is obtained. Having obtained the vibration velocity, the pressure perturbations induced by the plate oscillation and the transmission loss are found. Optimization is carried out in order to obtain the widest stopband. The transmission loss calculated by the analytical method agrees closely with the result of the finite element method simulation. Further studies with regard to the plate under several different classical boundary conditions based on the validated model show that a clamped-free plate silencer has the worst stopband. Attempts to release the boundary restriction of the plate are also made to study its effect on sound reflection. Results show that a softer end for a clamped–clamped plate silencer helps increase the optimal bandwidth, while the same treatment for simply supported plate silencer will result in performance degradation.     

Non-linear flexural vibrations of orthotropic rectangular plates

This study is an analytical investigation of free flexural large amplitude vibrations of orthotropic rectangular plates with all-clamped and all-simply supported stress-free edges. The dynamic von Karman-type equations of the plate are used in the analysis. A solution satisfying the prescribed boundary conditions is expressed in the form of double series with coefficients being functions of time. The model equations are solved by expanding the time-dependent deflection coefficients into Fourier cosine series. As obtained by taking the first sixteen terms in the double series and the first two terms in the time series, numerical results are presented for non-linear frequencies of various modes of glass-epoxy, boron-epoxy and graphite-epoxy plates. The analysis shows that, for large values of the amplitude, the effect of coupling of vibrating modes on the non-linear frequency of the fundamental mode is significant for orthotropic plates, especially for high-modulus composite plates.     

Free vibration analysis of stiffened plates using finite difference method

Free vibration characteristics of rectangular stiffened plates having a single stiffener have been examined by using the finite difference method. A variational technique has been used to minimize the total energy of the stiffened plate and the derivatives appearing in the energy functional are replaced by finite difference equations. The energy functional is minimized with respect to discretized displacement components and natural frequencies and mode shapes of the stiffened plate have been determined as the solutions of a linear algebraic eigenvalue problem. The analysis takes into consideration inplane deformation of the plate and the stiffener and the effect of inplane inertia on the natural frequencies and mode shapes. The effect of the ratio of stiffener depth to plate thickness on the natural frequencies of the stiffened plate has also been examined.     

高效半解析方法分析周期正交加筋板的振动-声辐射特性

为了对水下无穷大双周期正交加筋板结构模型在简谐面力激励下的振动响应及声辐射特性进行更为合理的理论预测与分析,建立了加筋板结构的数学模型。结合傅里叶变换、泊松迭加公式及空间波数法,将周期加筋板的振动响应及辐射声压表达为关于结构位移谐波分量的函数方程,对加筋板模型提出了高效分析求解方法并进行了谐波分量截断求解。验证了方法的正确性,并分析了结构的振动特性以及加强筋周期间距和扭矩对辐射声压的影响。结果表明,加强筋的扭转作用影响加筋板结构的振动模态频率,对于较高精度要求的工程应用,加强筋的扭转作用不能忽略。通过调节加强筋周期间距及横截面尺寸,可以降低薄板在较低频域区间的远场辐射声压。      

Rayleigh-Ritz vibration analysis of Mindlin plates

The Rayleigh-Ritz method is applied to the prediction of the natural frequencies of flexural vibration of square plates having general boundary conditions. The analysis is based on the use of Mindlin plate theory so that the effects of shear deformation and rotary inertia are included. The spatial variations of the plate deflection and the two rotations over the plate middle surface are assumed to be series of products of appropriate Timoshenko beam functions. Results are presented for a number of types of plate and these demonstrate the manner of convergence of the method as the number of terms in the assumed series increases.     

Vibrations of rectangular plates with elastically restrained edges

The Rayleigh-Ritz method is used to determined natural frequencies in transverse vibration of rectangular plates with elastically restrained edges. By treating an elastically restrained edge as intermediate between an appropriate pair of classical boundary conditions and using the corresponding vibration mode shapes of beams with classical boundary conditions as assumed functions, a relatively small number of functions is required; consequently only a modest quantity of computation is necessary. The good accuracy of the method is demonstrated by solving test problems. The method can be applied to a wide range of elastic restraint conditions, any aspect ratio and for higher modes in addition to the fundamental. The usefulness and accuracy of existing simplified approaches to the problem are assessed. The effect of in-plane forces on the natural frequencies and the determination of critical loads for plates with these restraint conditions are considered also.     

Vibration characteristics of thin rotating cylindrical shells with various boundary conditions

An analysis is presented for the vibration characteristics of thin rotating cylindrical shells with various boundary conditions by use of Fourier series expansion method. Based on Sanders’ shell equations, the governing equations of motion which take into account the effects of centrifugal and Coriolis forces as well as the initial hoop tension due to rotating are derived. The displacement field is expressed as a product of Fourier series expressions which represents the axial modal displacements and trigonometric functions which represents the circumferential modal displacements. Stokes’ transformation is employed to derive the derivatives of the Fourier series expressions. Then, through the process of formula derivation, an explicit expression of the exact frequency equation can be obtained for a thin rotating cylinder with classical boundary conditions of any type. Once the frequency equation has been determined, the frequencies are calculated numerically. To validate the present analysis, comparisons between the results of the present method and previous studies are performed and very good agreement is achieved. Finally, the method is applied to investigate the vibration characteristics of thin rotating cylindrical shells under various boundaries, and the results are presented.