Thin plate spline radial basis functions for vibration analysis of clamped laminated composite plates

A meshless method based on thin plate spline radial basis functions and higher-order shear deformation theory are presented to analyze the free vibration of clamped laminated composite plates. The singularity of thin plate spline radial basis functions is eliminated by adding infinitesimal to the zero distance. Convergence characteristics of the present thin plate spline radial basis functions for the vibration analysis of the clamped laminated plates are investigated. The frequencies computed by the present method agree well with the available published results.     

Semi-analytical solution for three-dimensional vibration of thick continuous grading fiber reinforced (CGFR) annular plates on Pasternak elastic foundations with arbitrary boundary conditions on their circular edges

In this paper free vibration of continuous grading fiber reinforced (CGFR) annular plates on an elastic foundation, based on the three-dimensional theory of elasticity, for different boundary conditions at the circular edges is investigated. The foundation is described by the Pasternak or two-parameter model. The CGFR annular plates have an arbitrary variation of fiber volume fraction in the thickness direction. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. Some new results for the natural frequencies of the plate are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. Besides, results for CGFR plate with arbitrary variation of fiber volume fraction in the thickness direction of the plate are compared with discrete laminated composite plate. The main contribution of this work is to present useful results for continuous grading of fiber reinforcement in the thickness direction of a plate on an elastic foundation and comparison with similar discrete laminated composite plate. The interesting and new results show that non-dimensional natural frequency parameters of a functionally graded fiber volume fraction is larger than that of a discrete laminated and close to that of a 2-layer. The new results can be taken as the benchmark solutions for those from numerical methods and future researches.     

局部损伤复合材料层合板的振动分析

本文采用了一种改进方法对局部损伤复合材料层合板进行了振动分析,将复合材料板中的损伤模拟为局部刚度的削减,并取三个损伤因子来刻画损伤的特性．利用高阶摄动法对其自由振动方程进行求解,主要计算了损伤板的自然振动频率和振动模态．相较于一阶摄动展开法,该方法在计算局部较大损伤问题中具有更高的准确度和敏感度．最后对损伤问题进行了参数研究,分析了不同的损伤因子（包括局部损伤程度、方向、面积大小）对板自由振动频率的影响．该方法为二维板局部损伤检测提供了有效精确的理论依据,并为损伤的定量评价提供了一种思路．     

A hybrid energy approach for vibrational modelling of laminated trapezoidal plates with point supports

An investigation on the free flexural vibration of symmetric angle-ply thin trapezoidal plates continuous over arbitrarily distributed point supports is reported. A hybrid energy approach which combines the pb-2 Rayleigh-Ritz method with the Lagrangian multiplier method is proposed for the modelling of the aforementioned plate problem. The pb-2 Rayleigh-Ritz method uses a set of Ritz functions generated from the product of a two-dimensional polynomial and the equations of boundaries each raised to the power of 0, 1 or 2 corresponding to a free, simply-supported or clamped edge, respectively. The geometric boundary conditions associated with the point supports are satisfied through the use of Lagrangian multipliers. In this paper, some new solutions for the natural frequencies of several laminated trapezoidal plates with different stacking sequences and location of point supports are presented. The first known mode shapes by means of contour plots for such laminated plates are also included.     

VIBRATION AND SOUND RADIATION OF AN ASYMMETRIC LAMINATED PLATE IN THERMAL ENVIRONMENTS

Analytical studies on the vibration and sound radiation characteristics of an asymmetric laminated rectangular plate are carried out in this paper. Theoretical formulations, in which the effects of thermal environments are taken into account, are derived for the vibration and sound radiation based on both first-order shear deformation plate theory and Rayleigh integral. It is found that the natural frequencies, the resonant amplitudes of vibration response and the sound pressure level decrease with the temperature rising. The natural frequencies of asymmetric plates are smaller than those of symmetric plates and the velocity responses of asymmetric plates are larger than those of symmetric plates.     

复合材料层合板自由振动分析的无网格自然邻接点Petrov-Galerkin法

基于一阶剪切变形理论,提出了复合材料层合板自由振动分析的无网格自然邻接点Petrov-Galerkin法。计算时在复合材料层合板中面上仅需要布置一系列的离散节点,并利用这些节点构建插值函数。在板中面上的局部多边形子域上,采用加权余量法建立复合材料层合板自由振动分析的离散化控制方程,并且这些子域可由Delaunay三角形方便创建。自然邻接点插值形函数具有Kronecker delta函数性质,因而无需经过特别处理就能准确地施加本质边界条件。对不同边界条件、不同跨厚比、不同材料参数和不同铺设角度的复合材料层合板,由本文提出的无网格自然邻接点Petrov-Galerkin法进行自由振动分析时均可得到满意的结果。数值算例结果表明,本文方法求解复合材料层合板的自由振动问题是行之有效的。     

The effect of support locations on the steady-state response of viscoelastically point-supported rectangular orthotropic plates

The vibration of orthotropic rectangular plates having viscoelastic point supports symmetrically located on its diagonals is analyzed. The plate is under the effect of a sinusoidally varying force at its center. The Lagrange equations are used in the solution process. To apply the Lagrange equations, the trial function denoting the deflection of the plate is expressed in polynomial form. By using the Lagrange equations, the problem is reduced to the solution of a system of algebraic equations. The influence of the mechanical properties, the damping of the supports and the locations of point supports on the steady-state response of the viscoelastically point-supported rectangular plates is investigated numerically for the concentrated load at center for various values of the mechanical properties characterizing the anisotropy of the plate material and for various values of damping and location of the supports for a certain stiffness value of the supports. The results are given for the considered frequency range of the external force. Convergence studies are made. The validity of the obtained results is demonstrated by comparing them with other solutions based on the Kirchhoff–Love plate theory.     

Non-linear analysis of functionally graded fiber reinforced composite laminated plates,Part I: Theory and solutions

This paper deals with the large amplitude vibration, non-linear bending and postbuckling of fiber reinforced composite laminated plates resting on an elastic foundation in hygrothermal environments. Two kinds of fiber reinforced laminated plates, namely, uniformly distributed and functionally graded reinforcements, are considered. The material properties of fiber reinforced laminated plates are estimated through a micromechanical model and are assumed to be temperature-dependent and moisture-dependent. The motion equations are based on a higher order shear deformation plate theory that includes plate-foundation interaction and the hygrothermal effect. A two-step perturbation technique is employed to determine the non-linear to linear frequency ratios of plate vibration, the load-deflection and load-bending moment curves of plate bending, and postbuckling equilibrium paths of laminated plates.     

用高阶位移模式分析复合材料层合板的自由振动

建立一种新的高阶位移模式：分层假设复合材料层合板的位移场，利用各层间应力及位移的连续性条件，导出了作为整体的层合板控制微分方程。最后，采用这一高阶剪切理论来分析复合材料矩形层合板的自由振动。数值计算采用有限差分法，并编制了计算程序，计算得出了六种不同边界条件、不同铺层、不同宽厚比工况下的矩形层合板的自振频率。同经典理论解进行比较，可以看出本文方法简便易行，且精度较高，可以在微机上实用。     

求解不连续中厚板自由振动的微分容积单元法

基于区域叠加原理和微分容积法，发展了一种新型的数值方法——微分容积单元法，用以分析具有不连续几何特征的中厚板的自由振动。根据板的不连续情况将其划分为若干单元，在每个单元内用微分容积法将控制微分方程离散成为一组线性代数方程．在相邻的单元连接处应用位移连续条件和平衡条件，引入边界约束条件后得到一套关于各配点位移的齐次线性代数方程，由此可导出求解系统固有频率的特征方程。本文用子空间迭代法求解特征方程，并以开孔板、混合边界条件板和突变厚度板为例研究了方法的收敛性和计算精度。     

Static and free vibration analysis of graphene platelets reinforced composite truncated conical shell,cylindrical shell,and annular plate using theory of elasticity and DQM

Abstract

In this paper, three-dimensional static and free vibration analysis of functionally graded graphene platelets-reinforced composite (FG-GPLRC) truncated conical shells, cylindrical shells and annular plates with various boundary conditions is carried out within the framework of elasticity theory. The main contribution of the present work is that formulation for free vibration and bending behavior of the FG-GPLRC truncated conical shell based on theory of elasticity has not yet been reported. Additionally, formulation and solution for cylindrical shell and annular plate are derived by changing the semi vertex angle in formulation and solution of FG-GPLRC truncated conical shell. A semi-analytical solution is proposed base on employing differential quadrature method (DQM) together with state-space technique. Validity of current approach is assessed by comparing its numerical results with those available in the literature. An especial attention is drawn to the role of GPLs weight fraction, patterns of GPLs distribution through the thickness direction, geometrical parameters such as semi-vertex angle, length to mid-radius ratio on natural frequencies and bending characteristics. Numerical results reveal that desirable static and free vibration response (such as lower radial deflection and higher natural frequencies) can be achieved by locating more square shaped GPLs near inner and outer surfaces.     

On a method for determining the fixing conditions for a rectangular plate from the natural frequencies

We prove the duality of solutions for the problem of determining the boundary conditions on two opposite sides of a rectangular plate from the frequency spectrum of its bending vibrations. A method for determining the boundary conditions on two opposite sides of a rectangular plate from nine natural frequencies is obtained. The results of numerical experiments justifying the theoretical conclusions of the paper are presented. Rectangular plates are widely used in various technical fields. They serve as printed circuit boards and header plates, bridging plates, aircraft and ship skin, and parts of various mechanical structures [1–4]. If the plate fixing cannot be inspected visually, then one can use the natural bending vibration frequencies to find faults in the plate fixing. For circular and annular plates, methods for testing the plate fixing were found in [5–7], where it was shown that the type of fixing of a circular or annular plate can be determined uniquely from the natural bending vibration frequencies. The following question arises: Is it possible to determine the type of fixing of a rectangular plate on two opposite sides of the plate from the natural bending vibration frequencies if the other two sides are simply supported? Since the opposite sides of the plate are equivalent to each other, a plate with “rigid restraint—free edge” fixing will sound exactly the same as a plate with “free edge—rigid restraint” fixing. Hence we cannot say that the type of fixing of a rectangular plate on two opposite sides can be uniquely determined from its natural bending vibration frequencies. But it turns out that we can speak of duality in the solution of this problem. Here we observe an analogy with the problem of determining the rigidity coefficients of springs for elastic fixing of a string [8]: the rigidity coefficients of the springs are determined by the natural frequencies uniquely up to permutations of the springs.     

Free vibration of rectangular plates with continuously distributed spring-mass

The vibratory characteristics of rectangular plates attached with continuously and uniformly distributed spring-mass in a rectangular region are studied which may represent free vibration of a human–structure system. Firstly, the governing differential equations of a plate with uniformly distributed spring-mass are developed. When the spring-mass fully occupies the plate, the natural frequencies of the coupled system are exactly solved and a relationship between the continuous system and a series of discrete two degrees-of-freedom system is provided. The degree of frequency coupling is defined. Then, the Ritz–Galerkin method is used to derive the approximate solution, when the spring-mass is distributed on a part of the plate, by using the Chebyshev polynomial series to construct the admissible functions. Comparative studies demonstrate the high accuracy and wide applicability of the proposed method. Finally, the frequency and modal characteristics of the plate partially occupied by distributed spring-mass are numerically analysed. It has been observed that both the natural frequencies and the modes appear in pairs. Moreover, a parametric study is performed for rectangular plates with three edges simply supported and one edge free. The effects of occupation size and position of the distributed spring-mass on natural frequencies of the coupled system are studied in detailed. The present investigation provides an improved understanding of human–structure interaction, such as grandstands or floors occupied by a stationary crowd.     

横观各向同性热电材料简支矩形板的自由振动

从耦合的三维压电热弹性理论出发分析了横观各向同性热电材料简支矩板的自由振动,证明其存在两类振动,即解耦的第一类振动和耦合的第二类振动,如果板的结构和上下表面边界条件关于中面对称,则第二类振动又可进一步分解为对称振和反对称振动,给出了热电材料简支矩形板自由振动的三维精确解,采用Ｍｏｎｔｅ Ｃａｒｌｏ法克服了超越方程求复根的困难。对于ＰＺＴ－４矩形板给出了数值结果,分析了耦合效应对振动频率的影响,计算     

A comprehensive analytical solution for free vibration of rectangular plates with classical edge coditions: Experimental verification

The problem of obtaining free vibration frequencies and mode shapes of rectangular plates resting on combinations of classical (i.e., clamped, simply supported, or free) edge supports is one that has been investigated for more than one hundred years. More recently, the superposition method has been developed for obtaining accurate analytical-type solutions for this family of problems. The object of this paper is to report on the results of numerous experimental tests carefully performed in order to verify the superposition method and associated computer software. Experimental and computed results are compared for a wide range of plate configurations. Very good agreement between theory and experiment has been obtained with regard to both plate natural frequencies and mode shapes. It is concluded that this computational procedure constitutes a powerful new tool for analysis of rectangular plate vibration problems.     

Nonlinear vibration analysis of piezo-thermo-electrically actuated functionally graded circular plates

A theoretical model for geometrically nonlinear vibration analysis of thermo-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. The material properties of the FG 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. Dynamic equations and boundary conditions including thermal, elastic and piezoelectric couplings are formulated and solutions are derived. An exact series expansion method combined with perturbation approach is used to model the nonlinear thermo-electro-mechanical vibration behavior of the structure. Control of the FG 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 mixtures of ceramic and metal are presented in dimensionless forms. A parametric study is also undertaken to highlight the effects of the thermal environment, applied actuator voltage and material composition of the FG core plate on the nonlinear vibration characteristics of the composite structure.     

Non-linear analysis of functionally graded fiber reinforced composite laminated plates,Part II: Numerical results

In this Part, the extensive parametric studies performed are reported and numerical results are presented for the non-linear vibration, non-linear bending and compressive postbuckling of uniformly distributed and functionally graded fiber reinforced unsymmetric cross-ply and/or antisymmetric angle-ply laminated plates resting on Pasternak elastic foundations under different hygrothermal environmental conditions. The numerical results show that the functionally graded fiber reinforcement has a significant effect on the postbuckling response and load-bending moment curves of plate bending, whereas this effect is less pronounced on the load-deflection curves of plate bending and the linear and non-linear frequencies of the same plate.     

Erratum to: Vibration Analysis of Perforated Plates Using Time-Average Digital Speckle Pattern Interferometry

Digital Speckle Interferometry is a non invasive full-field coherent optical technique used in mechanical vibration measurement. In this research, it is used for tuning resonant frequencies of vibrating plates in order to investigate the dynamical behavior of perforated plates. The plate was excited to resonant vibration by a sinusoidal acoustic source. Fringe pattern produced during the time-average recording of the vibrating plate, for several resonant frequencies were registered. Results of plates fixed at one edge having internal holes and attached masses are presented. Experimental natural frequencies and modal shapes are compared to those obtained by an analytical approximate solution based on the Rayleigh–Ritz method with the use of orthogonal polynomials as coordinate function. A high degree of correlation between computational analysis and experimental results was observed, proving the potentiality of the optical technique as experimental validation of the numerical simulations.     

Vibration Analysis of Perforated Plates Using Time-Average Digital Speckle Pattern Interferometry

Digital Speckle Interferometry is a non invasive full-field coherent optical technique used in mechanical vibration measurement. In this research, it is used for tuning resonant frequencies of vibrating plates in order to investigate the dynamical behavior of perforated plates. The plate was excited to resonant vibration by a sinusoidal acoustic source. Fringe pattern produced during the time-average recording of the vibrating plate, for several resonant frequencies were registered. Results of plates fixed at one edge having internal holes and attached masses are presented. Experimental natural frequencies and modal shapes are compared to those obtained by an analytical approximate solution based on the Rayleigh–Ritz method with the use of orthogonal polynomials as coordinate function. A high degree of correlation between computational analysis and experimental results was observed, proving the potentiality of the optical technique as experimental validation of the numerical simulations.     

Natural frequencies of composite plates with tailored thermal residual-stresses

Thermal residual-stresses introduced during manufacture and their effect on the natural frequencies and vibration modes of stringer stiffened composite plates is investigated. The principal idea in the work is to include stiffeners on the perimeter of a composite plate in which the laminate design of the stiffeners and plate are different. Such an arrangement yields manufacturing induced thermal residual-stresses; these stresses result from the difference in manufacturing and operating temperatures as well as the difference in thermal expansion coefficients and elastic properties of the plate and the stiffeners. The analysis is based on an enhanced Reissner–Mindlin plate theory and involves two separate calculations. In the first, the thermal residual-stress state is determined for an unconstrained plate. In the second, the free vibration problem is solved; thermal effects from the first calculation are included by way of nonlinear membrane-bending coupling which in turn defines the free vibration reference state. The problem is solved using a 16-node bi-cubic Lagrange element in a finite element formulation. Three different plate-stiffener geometries are used to illustrate the effects of stringer size, stringer placement and temperature difference. Two principal results are obtained: first, it is shown that thermal residual-stresses can have a significant effect on the natural frequencies; secondly, thermal residual-stresses can be tailored to increase natural frequencies. Therefore it is concluded that an evaluation of these stresses and a judicious analysis of their effects must be included in the design of this class of composite structures.