基于重构核粒子法的带孔薄板模态分析研究

无网格法是一种新颖的工程数值计算方法,与有限元法相比,具有很多独特的优势。重构核粒子法具有变时-频特性和多分辨率特性等优点,在结构动力学中有广泛的应用。本文针对带孔薄板这一不连续问题,根据无网格法和板壳振动的基本理论,采用了可视性准则处理场函数的不连续性,编写了基于重构核粒子法的模态分析程序,得到了不同边界条件下带孔薄板的前五阶固有频率和模态,并和有限元分析结果进行了比较。算例表明该方法收敛性好、精度高,为带孔薄板的模态分析提供了一种新的求解方法。     

均匀化理论在多孔板结构优化中的应用研究

多尺度均匀化理论自从上世纪 70年代产生以来 ,就被认为是确定复合材料当量性能的一种不可替代的方法。本文将多尺度均匀化方法应用于基于有限元分析的多孔板结构优化设计中 ,简述了求解当量板有效弹性常数的均匀化理论 ,在此基础上 ,建立了基于有限元分析的多孔板结构优化流程 ,编写了FORTRAN程序 ,并将其与AN SYS软件相连接 ,完成了一个多孔板结构的优化设计 ,所得结果表明 ,它能够满足工程应用的需要 ,并为多孔板结构设计提供了一种新思路     

Vibrations of point-supported rectangular plates with variable thickness using a set of static tapered beam functions

This paper studies the free vibrations of point-supported rectangular plates with variable thickness using the Rayleigh–Ritz method. The domain of the plate is bounded by x=αa′, a′ (0α<1); y=βb′, b′ (0β<1) in the Cartesian coordinate system. The thickness of the plate varies continuously and is represented by a power function (x/a′)^s(y/b′)^t. Varieties of tapered rectangular plates can be described by giving s and t different values. A set of static tapered beam functions which are the solutions of a tapered beam (a unit width strip taken from the particular plate under consideration in one or the other direction parallel to its edges) under a Taylor series of static loads, are developed as the admissible functions for the vibration analysis of point-supported rectangular plates with variable thickness in one or two directions. The eigenfrequency equation is derived through the Rayleigh–Ritz approach, supplemented by the zero deflection conditions at the point-supports. A very simple program in common use has been compiled. The convergence study shows a small computational cost and the comparison with known solutions for point-supported rectangular plates with uniform thickness demonstrates the accuracy of the present method. Finally, some new numerical results are given, which may serve as the benchmarks for future research on the aforementioned problem.     

Vibration analysis of delaminated composite beams and plates using a higher-order finite element

In order to analyze the vibration response of delaminated composite plates of moderate thickness, a FEM model based on a simple higher-order plate theory, which can satisfy the zero transverse shear strain condition on the top and bottom surfaces of plates, has been proposed in this paper. To set up a C⁰-type FEM model, two artificial variables have been introduced in the displacement field to avoid the higher-order derivatives in the higher-order plate theory. The corresponding constraint conditions from the two artificial variables have been enforced effectively through the penalty function method using the reduced integration scheme within the element area. Furthermore, the implementation of displacement continuity conditions at the delamination front has been described using the present FEM theory. Various examples studied in many previous researches have been employed to verify the justification, accuracy and efficiency of the present FEM model. The influences of delamination on the vibration characteristic of composite laminates have been investigated. Especially the variation of ‘curvature of vibration mode’ (i.e., the second-order differential of deflections in vibration mode) caused by delamination has been studied in detail to provide valuable information for the possible identification of delamination. Furthermore, two approaches have been investigated to detect a delamination in laminates by employing this information.     

Three-dimensional free vibration of variable thickness thick circular and annular isotropic and functionally graded plates on Pasternak foundation

This paper describes a study of three-dimensional free vibration analysis of thick circular and annular isotropic and functionally graded (FG) plates with variable thickness along the radial direction, resting on Pasternak foundation. The formulation is based on the linear, small strain and exact elasticity theory. Plates with different boundary conditions are considered and the material properties of the FG plate are assumed to vary continuously through the thickness according to power law. The kinematic and the potential energy of the plate-foundation system are formulated and the polynomial-Ritz method is used to solve the eigenvalue problem. Convergence and comparison studies are done to demonstrate the correctness and accuracy of the present method. With respect to geometric parameters, elastic coefficients of foundation and different boundary conditions some new results are reported which may be used as benchmark solutions for future researches.     

DSC-element method for free vibration analysis of rectangular Mindlin plates

A novel DSC-element method is proposed to investigate the free vibration of moderately thick plates based on the well-known Mindlin first-order shear deformation plate theory. The development of the present approach not only employs the concept of finite element method, but also implements the discrete singular convolution (DSC) delta type wavelet kernel for the transverse vibration analysis. This numerical algorithm is allowed dividing the domain of Mindlin plates into a number of small discrete rectangular elements. As compared with the global numerical techniques i.e. the DSC-Ritz method, the flexibility is increased to treat complex boundary constraints. For validation, a series of numerical experiments for different meshes of Mindlin plates with assorted combinations of edge supports, plate thickness and aspect ratios is carried out. The established natural frequencies are directly compared and discussed with those reported by using the finite element and other numerical and analytical methods from the open literature.     

3-D vibration analysis of skew thick plates using Chebyshev–Ritz method

The free vibration characteristics of skew thick plates with arbitrary boundary conditions have been studied based on the three-dimensional, linear and small strain elasticity theory. The actual skew plate domain is mapped onto a basic cubic domain and the eigenvalue equation is then derived from the energy functional of the plate by using the Ritz method. A set of triplicate Chebyshev polynomial series multiplied by a boundary function chosen to satisfy the essential geometric boundary conditions of the plate is developed as the trial functions of the displacement components. The vibration modes are divided into antisymmetric and symmetric ones in the thickness direction and can be studied individually. The convergence and comparison studies show that rather accurate results can be obtained by using this approach. Parametric investigations on rhombic plates with fully clamped edges and completely free edges are performed in detail, with respect to the thickness-span ratio and skew angle. Some results known for the first time are reported, which may serve as the benchmark values for future numerical technique research.     

Pseudospectral analysis of buckling and free vibration of rectangular Mindlin plates

A study of buckling and free vibration of rectangular Mindlin plates is presented. The analysis is based on the pseudospectral method, which uses basis functions that satisfy the boundary conditions. The equations of motion are collocated to yield a set of algebraic equations that are solved for the critical buckling load and for the natural frequencies in the presence of the in-plane loads. Numerical examples of rectangular plates with SS-C-SS-C boundary conditions are provided for various aspect ratios and thickness ratios, which show good agreement with those of the classical plate theory when the thickness ratio is very small. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin Jinhee Lee received B.S. and M.S. degrees from Seoul National University and KAIST in 1982 and 1984, respectively. He received his Ph.D. degree from the University of Michigan, Ann Arbor in 1992 and joined the Dept. of Mechanical and Design Engineering of Hongik University in Choongnam, Korea. His research interests include inverse problems, pseudospectral method, vibration and dynamic systems.     

Eshelby-Mori-Tanaka approach for vibrational behavior of functionally graded carbon nanotube-reinforced plate resting on elastic foundation

In this study, based on the three-dimensional theory of elasticity, free vibration characteristics of functionally graded (FG) nanocomposite plates reinforced by randomly-oriented straight single-walled carbon nanotubes (SWCNTs) resting on an elastic foundation are considered. Material properties are graded in the thickness direction of the plate according to the volume fraction power law distribution. An embedded carbon nanotube (CNT) in a polymer matrix and its surrounding inter-phase which is perfectly bonded to surrounding resin is replaced with an equivalent fiber to predict the mechanical properties of the carbon nanotube/polymer composite. The Mori-Tanaka approach is employed to calculate the effective elastic moduli of the plate. The natural frequencies of the plate are obtained by means of the generalized differential quadrature (GDQ) method. Detailed parametric studies have been carried out to investigate the influences of the CNT volume fraction, Winkler foundation modulus, shear elastic foundation modulus and various geometrical parameters on the vibration behavior of the functionally graded carbon nanotube-reinforced (FG-CNTR) plates.     

Natural frequencies of stepped thickness rectangular plates

This paper presents the natural frequencies of stepped thickness square and rectangular plates together with the mode shapes of vibration. The transverse deflection of a stepped thickness plate is written in a series of the products of the deflection functions of beams parallel to the edges satisfying the boundary conditions, and the frequency equation of the plate is derived by the energy method. By use of the frequency equation, the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically in good accuracy for square and rectangular plates with edges simply supported or elastically restrained against rotation, having square, circular or elliptical stepped thickness, from which the effects of the stepped thickness on the vibration are studied.     

Three-dimensional vibration analysis of spherical shell panels subjected to different boundary conditions

This paper presents the elasticity solutions for free vibration analysis of doubly curved shell panels of rectangular planform. The vibratory characteristics of panels subjected to different boundary conditions have been obtained via a three-dimensional displacement-based energy formulation represented in the spherical coordinates. The p-Ritz method is employed for the solution of the problem. To validate the accuracy of the results and the ease of implementation of the present methodology, selected cases have been compared against the results existing in the literature.     

An optimal placement of CLD treatment for vibration suppression of plates

A study on optimal placement of constrained-layer damping (CLD) treatment for vibration suppression of plates is presented. A mathematical model is derived using the energy approach. This model allows both the constraining layer (CL) and the viscoelastic layer (VEM) to be either as thick as the host or very thin. The influence of treatment thickness on damping effects is particularly addressed. From the simulated results, it is found that the best damping performance occurs as the CL thickness is twice that of the VEM thickness. As to the optimization process, an objective function including structural damping ratios, resonant frequencies’ shift, and CLD thickness is defined. In optimization, the structural damping plays the main performance index and the frequencies’ shift and CLD thickness play as penalty functions. A topographical method in conjunction with the complex method is used to search for the global minimum. Examples are illustrated and the effects of weighting factors are discussed. Individual effects of weighting factors can be useful for designers, since the designers can build up a better sense of how CLD treatment improves the vibration suppression.     

Three-dimensional vibration of laminated composite plates and cylindrical panels with arbitrarily located lateral surfaces point supports

On the basis of fully three-dimensional elasticity considerations, this paper presents a free vibration analysis of simply supported, cross-ply laminated plates and cylindrical panels that are subjected to an arbitrary number of lateral surfaces point supports. The analysis is based on a recursive approach suitable for the vibration analysis of corresponding unconstrained structural elements. By means of dynamic equilibrium considerations, the reaction of the point supports is imposed by using the Lagrange multipliers method. This yields the eigendeterminant of a constrained panel by appropriately coupling the response of a suitably large number of natural vibration modes of the corresponding unconstrained structural element.     

The numerical simulation of a rectifying device with a perforated plate

The aim of this research is to verify a perforated plate design technique for generating a certain velocity and turbulence distribution by introducing the connection condition that demonstrates the disturbance attenuation characteristics of the plate. The application range on the downstream side of the perforated plate is studied based on a numerical simulation of the resistance characteristics of perforated plates. The optimum numerical simulations of turbulent flows through perforated plates are then clarified. The numerical simulation results and the experimental data for the disturbance attenuation are highly consistent with the numerical simulation and experimental data for the perforated plates with fixed opening ratios and pore diameters. In addition, the calculation results for the downstream disturbance distribution almost correspond with the experimental results when the opening ratio of the perforated plate is fixed but the pore diameter is variable. Consequently, the disturbance numerical simulation technique developed in this research by utilising the connection conditions for perforated plates clarifies the reliability of the rectifying device design method used to generate the velocity and turbulence distribution of perforated plates.     

Axisymmetric free vibration of thick annular plates

This paper presents a numerical analysis of the axisymmetric free vibration of moderately thick annular plates using the differential quadrature method (DQM). The plates are described by Mindlin’s first-order shear-deformation theory. The first five axisymmetric natural frequencies are presented for uniform annular plates, of various radii and thickness ratios, with nine possible combinations of free, clamped and simply supported boundary conditions at the inner and outer edges of the plates. The accuracy of the method is established by comparing the DQM results with some exact and finite element numerical solutions and, therefore, the present DQM results could serve as a benchmark for future reference. The convergence characteristics of the method for thick plate eigenvalue problems are investigated and the versatility and simplicity of the method is established.     

Vibration of rectangular Mindlin plates resting on non-homogenous elastic foundations

This paper is concerned with the vibration behaviour of rectangular Mindlin plates resting on non-homogenous elastic foundations. A rectangular plate is assumed to rest on a non-homogenous elastic foundation that consists of multi-segment Winkler-type elastic foundations. Two parallel edges of the plate are assumed to be simply supported and the two remaining edges may have any combinations of free, simply supported or clamped conditions. The plate is first divided into subdomains along the interfaces of the multi-segment foundations. The Levy solution approach associated with the state space technique is employed to derive the analytical solutions for each subdomain. The domain decomposition method is used to cater for the continuity and equilibrium conditions at the interfaces of the subdomains. First-known exact solutions for vibration of rectangular Mindlin plates on a non-homogenous elastic foundation are obtained. The vibration of square Mindlin plates partially resting on an elastic foundation is studied in detail. The influence of the foundation stiffness parameter, the foundation length ratio and the plate thickness ratio on the frequency parameters of square Mindlin plates is discussed. The exact vibration solutions presented in this paper may be used as benchmarks for researchers to check their numerical methods for such a plate vibration problem. The results are also important for engineers to design plates supported by multi-segment elastic foundations.     

Nonlinear free vibration of laminated composite plates on elastic foundation with random system properties

The present investigation is concerned with free vibration analysis of laminated composite plates resting on elastic foundation undergoing large amplitude oscillation with random system properties. The lamina material properties and foundation stiffness parameters are modeled as basic random variables for accurate prediction of the system behavior. The basic formulation of the problem is based on higher-order shear displacement theory including rotatory inertia effects and von Karman-type nonlinear strain displacement relations. A C⁰ finite element is used for descretization of the laminate. A direct iterative method in conjunction with first-order Taylor series based perturbation technique procedure is developed to solve random nonlinear generalized eigenvalue problem. The developed probabilistic procedure is successfully used for the nonlinear free vibration problem with a reasonable accuracy. Typical numerical results (second-order statistics) are obtained for the composite plates resting on Winkler and Pasternak elastic foundations with different support conditions, side-to-thickness ratio, aspect ratio, oscillation amplitude ratio, stacking sequences and foundation parameters for symmetric and anti-symmetric cross-ply and angle-ply laminates. The results are validated with existing available results and independent Monte Carlo simulation.     

Vibration of stepped thickness plates

The vibration characteristics of stepped thickness plates of rectangular geometry with simply supported edges are investigated through an exact analysis. To illustrate the effect of sudden variation of thickness in the step form on the dynamic characteristics of plates, numerical calculations have been made for various combinations of thickness step, step-length ratio and side ratio. Natural frequencies for the first eight to nine modes are presented in the form of tables. Nodal lines are also plotted for a few selected cases. Several interesting features such as that of the thickness step splitting the degenerate frequencies of uniform rectangular plates into distinct ones, and the “frequency crossing” of some of the modes are observed.     

Vibration of skew plates by the MLS-Ritz method

This paper presents a study on the vibration of skew plates by a numerical method, the moving least square Ritz (MLS-Ritz) method which was proposed by the authors in a previous study [Zhou L, Zheng WX. A novel numerical method for the vibration analysis of plates. Computational mechanics WCCM VI in conjunction with APCOM’04, Beijing, China, 5-10 September 2004; Zhou L, Zheng WX. MLS-Ritz method for vibration analysis of plates. Journal of Sound and Vibration 2006;290(3-5):968-90]. One of the most challenging numerical difficulties in analysing the vibration of a skew plate with a large skew angle is the slow convergence due to the stress singularities at the obtuse corners of the plate. The MLS-Ritz method is employed in this paper to address such problem. This method utilises the moving least square technique to establish the trial function for the transverse displacement of a skew plate and the Ritz method is applied to derive the governing eigenvalue equation for the skew plate. The boundary conditions of the plate are enforced through a point substitution technique that forces the MLS-Ritz trial function satisfying the essential boundary conditions along the plate edges. Due to the flexibility of the arrangement of the MLS-Ritz grid points, more grid points can be placed around the obtuse corners of a skew plate so as to address the stress singularity problem at the corners. A series of cases for rhombic plates of various edge support conditions are presented to demonstrate the efficiency and accuracy of the MLS-Ritz method.     

Geometrically nonlinear vibration analysis of piezoelectrically actuated FGM plate with an initial large deformation

A theoretical model for geometrically nonlinear vibration analysis of piezoelectrically actuated circular plates made of functionally grade material (FGM) is presented based on Kirchhoff’s-Love hypothesis with von-Karman type geometrical large nonlinear deformations. To determine the initial stress state and pre-vibration deformations of the smart plate a nonlinear static problem is solved followed by adding an incremental dynamic state to the pre-vibration state. The derived governing equations of the structure are solved by exact series expansion method combined with perturbation approach. The material properties of the FGM core plate are assumed to be graded in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents. Control of the FGM plate’s nonlinear deflections and natural frequencies using high control voltages is studied and their nonlinear effects are evaluated. Numerical results for FG plates with various mixture of ceramic and metal are presented in dimensionless forms. In a parametric study the emphasis is placed on investigating the effect of varying the applied actuator voltage as well as gradient index of FGM plate on vibration characteristics of the smart structure. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin Farzad Ebrahimi received his B.S. and M.S. degree in Mechanical Engineering from University of Tehran, Iran. He is currently working on his Ph.D. thesis under the title of “Vibration analysis of smart functionally graded plates” at Smart Materials and Structures Lab in Faculty of Mechanical Engineering of the University of Tehran. His research interests include vibration analysis of plates and shells, smart materials and structures and functionally graded materials.