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.     

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.     

The free vibration analysis and optimal design of an adhesively bonded functionally graded single lap joint

In this study, three-dimensional free vibration and stress analyses of an adhesively bonded functionally graded single lap joint were carried. The effects of the adhesive material properties, such as modulus of elasticity, Poisson's ratio and density were found to be negligible on the first ten natural frequencies and mode shapes of the adhesive joint. Both the finite element method and the back-propagation artificial neural network (ANN) method were used to investigate the effects of the geometrical parameters, such as overlap length, plate thickness and adhesive thickness; and the material composition variation through the plate thickness on the natural frequencies, mode shapes and modal strain energy of the adhesive joint. The suitable ANN models were trained successfully using a series of free vibration and stress analyses for various random geometrical parameters and compositional gradient exponents. The ANN models showed that the support length, the plate thickness and the compositional gradient exponent played important role on the natural frequencies, mode shapes and modal strain energies of the adhesive joint whereas the adhesive thickness had a minor effect. In addition, the optimal joint dimensions and compositional gradient exponent were determined using genetic algorithm and ANN models so that the maximum natural frequency and the minimum modal strain energy conditions are satisfied for each natural frequency of the adhesively bonded functionally graded single lap joint.     

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.     

Vibration analysis of orthotropic circular and elliptical nano-plates embedded in elastic medium based on nonlocal Mindlin plate theory and using Galerkin method

In the present study a continuum model based on the nonlocal elasticity theory is developed for free vibration analysis of embedded orthotropic thick circular and elliptical nano-plates rested on an elastic foundation. The elastic foundation is considered to behave like a Pasternak type of foundations. Governing equations for vibrating nano-plate are derived according to the Mindlin plate theory in which the effects of shear deformations of nano-plate are also included. The Galerkin method is then employed to obtain the size dependent natural frequencies of nano-plate. The solution procedure considers the entire nano-plate as a single super-continuum element. Effect of nonlocal parameter, lengths of nano-plate, aspect ratio, mode number, material properties, thickness and foundation on circular frequencies are investigated. It is seen that the nonlocal frequencies of the nano-plate are smaller in comparison to those from the classical theory and this is more pronounced for small lengths and higher vibration modes. It is also found that as the aspect ratio increases or the nanoplate becomes more elliptical, the small scale effect on natural frequencies increases. Further, it is observed that the elastic foundation decreases the influence of nonlocal parameter on the results. Since the effect of shear deformations plays an important role in vibration analysis and design of nano-plates, by predicting smaller values for fundamental frequencies, the study of these nano-structures using thick plate theories such as Mindlin plate theory is essential.     

基于高阶改进方法的层合板固有频率分析

针对一般的计算层合板振动特性的方法过于繁复的问题,采用了一种高阶改进方法来计算复合材料层合板的固有频率。建立了高阶改进方法的层合板动能和势能表达式,利用哈密顿原理构建了层合板的自由振动方程并计算了层合板的固有频率。在此基础上,通过该方法计算了铺设角度、跨厚比和弹性模量比对层合板固有频率的影响。结果表明：该算法可以有效地计算出反对称铺设层合板和正交铺设层合板的固有频率,同时也可以计算任意形式铺设的层合板的固有频率。     

含裂纹板断裂韧度厚度效应的理论与应用

从理论角度对断裂韧度厚度效应进行了研究。在线弹性断裂力学与含裂纹板的二维位移场的基础上,给出分离变量型的具有待定函数的三维位移场的表达式,进而通过几何方程与物理方程获取三维应变场和三维应力场。进一步,通过虚位移原理,使用变分方法建立待定函数所应满足的支配方程与边界条件,从而确定待定函数。基于上述分析,建立一个断裂韧度与试样厚度关系的理论表达式。最后通过两种试验材料LY12CZ(L-T)与TC4(L-T)进行了验证。     

含裂纹板断裂韧度厚度效应的理论研究

从理论角度对断裂韧度厚度效应进行研究。在线弹性断裂力学与含裂纹板的二维位移场的基础上，给出分离变量型的具有待定函数的三维位移场的表达式，进而通过几何方程与物理方程获取三维应变场和三维应力场。进一步，通过虚位移原理，使用变分方法建立待定函数所应满足的支配方程与边界条件，从而确定待定函数。基于上述分析，建立一个断裂韧度与试样厚度关系的理论表达式。最后通过两种试验材料LY12CZ（L-T）与TC4（L-T）进行验证。     

Electro-thermo nonlocal nonlinear vibration in an embedded polymeric piezoelectric micro plate reinforced by DWBNNTs using DQM

In the present paper, electro-thermo nonlinear vibration of a piezo-polymeric rectangular micro plate made from polyvinylidene fluoride (PVDF) reinforced by zigzag double walled boron nitride nanotubes (DWBNNTs) is studied. This plate is embedded in an elastic medium which is simulated by Winkler and Pasternak foundation models. Using nonlinear strain-displacement relations and nonlocal elasticity plate theory as well as considering charge equation for coupling between electrical and mechanical fields, the motion equations are derived based on energy method and Hamilton??s principle. The differential quadrature method (DQM) is employed to computation of nonlinear frequency for different mechanical and free-free electrical boundary conditions. The results indicate that smart composite and consequently the generated G4 improved sensor and actuator applications in several process industries, because it increases the nonlinear vibration frequency. Furthermore, it can be also found that the nonlinear frequency increases as the values of the elastic medium constants, the geometrical aspect ratios and DWBNNTs volume fraction increase but it decreases as nonlocal parameter increases.     

圆盘式MRF离合器的研究

磁流变液离合器是一种利用磁流变液剪切应力来进行离合的一种装置，它传递的力矩随外加磁场的变化迅速变化。本文在理论上给出了盘式磁流变液离合器的设计方法，推出了磁流变液离合器传递力矩的方程，得出了盘式磁流变液离合器中磁流变液体积、厚度等的计算公式。为盘式磁流变液离合器的设计奠定了理论基础。     

A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates

An exact closed-form frequency equation is presented for free vibration analysis of circular and annular moderately thick FG plates based on the Mindlin's first-order shear deformation plate theory. The edges of plate may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. The material properties change continuously through the thickness of the plate, which can vary according to a power-law distribution of the volume fraction of the constituents, whereas Poisson's ratio is set to be constant. The equilibrium equations which govern the dynamic stability of plate and its natural boundary conditions are derived by the Hamilton's principle. Several comparison studies with analytical and numerical techniques reported in literature and the finite element analysis are carried out to establish the high accuracy and superiority of the presented method. Also, these comparisons prove the numerical accuracy of solutions to calculate the in-plane and out-of-plane modes. The influences of the material property, graded index, thickness to outer radius ratios and boundary conditions on the in-plane and out-of-plane frequency parameters are also studied for different functionally graded circular and annular plates.     

Winkler地基上四边自由矩形薄板的亚谐共振与奇异性分析

研究Winkler地基上四边自由矩形薄板的亚谐共振问题.按照弹性力学理论建立Winkler地基上四边自由受简谐激励作用矩形薄板的动力学方程.利用Galerkin方法将其转化为非线性振动方程.应用非线性振动的多尺度法求得系统满足1/3次亚谐共振情况时的一次近似解,并进行数值计算.分析激励、阻尼等对系统响应曲线的影响.应用奇异性理论对1/3次亚谐共振幅频响应方程进行奇异性分析,得到选取不同分岔参数时开折参数平面的转迁集和分岔图.     

Aeroelastic behavior of cantilevered rotating rectangular plates

Aeroelastic behavior of a supersonic rotating rectangular plate in the air medium is studied. For simulating the plate structure, the Mindlin first-order shear deformation plate theory along with Von Karman nonlinear terms is employed. Air dynamic pressure is modeled using first-order piston theory. The plate is placed inside a rigid baffle to remove shock waves generated by plate rotation and provide a uniform flow passing over the plate. Nonlinear dimensionless generalized equations of motion are presented based on the Kane dynamic method. After linearization of the nonlinear equations of motion, effect of different parameters including plate aspect ratio, thickness ratio, hub radius ratio and dimensionless rotation speed on aeroelastic behavior of the system are investigated. Frequency locking and high frequency flutter phenomenon are observed in the numerical results.     

Analysis of Natural Vibrations of Round Bimorph Piezoelectric Ceramic Plates of Arbitrary Dimensions. II. Rigidly Fixed Piezoelectric Plate

The problem of natural vibrations of a round bimorph rigidly fixed supported piezoelectric ceramic plate (piezoplate) of an arbitrary thickness with arbitrary axisymmetric electrodes is solved using the method of finite elements. The natural frequency spectra are analyzed in the resonance and antiresonance regimes. The displacement distributions over the piezoplate surfaces (vibration modes) are also analyzed and the dynamic electromechanical coupling coefficient (DCC) is investigated as a function of the relative plate thickness for various piezoelectric ceramic compositions. Round bimorph piezoplates with partial electrodes are considered; it is shown that the DCC can be significantly (by tens times) raised using partial electrodes. The results obtained make it possible to estimate the limits of applicability of the approximate one-dimensional theory and choose the optimum geometric dimensions of the plate and electrodes that ensure the maximum DCC.     

复杂机械系统多维耦合振动传递矩阵分析

以特性传递矩阵描述复杂机械系统中子结构输入、输出力与速度矢量关系,提出了复杂振源激励、多弹性支承与基础板结构三维耦合隔振系统传递矩阵力学模型。考虑隔振支承多维波动效应,推证了弹性支承耦合振动传递矩阵,为复杂系统功率流特性及隔振器多维驻波效应研究确立了有效途经。数值模拟计算表明,基础板结构的前两阶弯曲共振模态是高频域系统功率流提高的主要原因。振源力矩或垂向激振力诱发的隔振器纵向谐振使系统功率流明显提高:增大基础板厚度,驻波共振峰值提高。     

Free vibrations of thin periodic plates interacting with an elastic periodic foundation

A dynamic behaviour of thin periodic plates interacting with an elastic periodic foundation is analysed. The main aim is to investigate free vibrations and calculate frequencies for these plates. The presented modelling approach to the linear-elastic plates, having periodic structure in planes parallel to the midplane, is adopted from the tolerance averaging method developed by Woźniak and Wierzbicki (Averaging techniques in thermomechanics of Composite Solids, Wydawnictwo Politechniki Cz stochowskiej, Cz stochowa, 2000) for periodic composites. The obtained plate model describes the effect of the periodicity cell size on the overall plate behaviour. In this paper this effect is analysed in the free vibrations problem. Two special cases are taken into account: a square plate with a periodic thickness interacting with a homogeneous foundation and a square plate with a constant thickness interacting with a periodic foundation. A certain justification of the model is also presented.     

磁电弹薄板的非线性主共振

针对磁电弹性薄板,结合大挠度板理论,考虑几何非线性,建立了非线性无阻尼强迫振动的数学模型,应用改进的L—P法,对非线性无阻尼强迫振动问题进行求解,得到薄板稳定状态下的幅频响应方程,数值模拟了不同情况下系统的幅频响应曲线图。通过比较分析得出：板的厚度、外激励力、参数e的不同取值以及压电材料BaTi03、压磁材料CoFe204及磁电弹材料BaTi03／CoFe204与Terfenol—D／PZT均会对系统的幅频响应曲线产生影响,主要表现为对系统主共振区间,弹簧软硬特性,幅值跳跃现象的影响。这些结论在理论上可以更好地指导工程结构的设计。     

Analysis of Natural Vibrations of Round Bimorph Piezoelectric Ceramic Plates of Arbitrary Dimensions. I. Simply Supported Piezoelectric Plate

The problem of natural vibrations of a round bimorph simply supported piezoelectric ceramic plate (piezoplate) of an arbitrary thickness with arbitrary axisymmetric electrodes is solved in the exact formulation using the method of finite elements. The natural frequency spectra are analyzed in the resonance and antiresonance regimes. The displacement distribution over the piezoplate surfaces (vibration modes) is also analyzed and the dynamic electromechanical coupling coefficient (DCC) is investigated as a function of the relative plate thickness for various piezoelectric ceramic compositions. Round bimorph piezoplates with partial electrodes have been also considered. The results obtained make it possible to estimate the limits of applicability of an approximate one-dimensional theory and choose the optimum geometric dimensions of the plate and electrodes that ensure the maximum DCC.     

The refined sinusoidal theory for FGM plates on elastic foundations

Using the refined sinusoidal shear deformation plate theory and including plate-foundation interaction, a thermoelastic bending analysis is presented for a simply supported, rectangular, functionally graded material plate subjected to a transverse uniform load and a temperature field, and resting on a two-parameter (Pasternak model) elastic foundation. The present shear deformation theory is simplified by enforcing traction-free boundary conditions at the plate faces. No transversal shear correction factors are needed because a correct representation of the transversal shearing strain is given. Material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The equilibrium equations of the present plate are given based on various plate theories. A number of examples are solved to illustrate the numerical results concerning bending response of homogeneous and functionally graded rectangular plates resting on two-parameter elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The influences played by transversal shear deformation, plate aspect ratio, side-to-thickness ratio, volume fraction distributions, and elastic foundation parameters are studied.