Fundamental frequency and design of the CFCF composite sandwich plate

The design efficiency of sandwich panels is often associated with the value of fundamental frequency. This paper investigates the free vibrations of rectangular sandwich plates having two adjacent edges fully clamped and the remaining two edges free (CFCF). The vibration analysis is performed by applying Hamilton’s principle in conjunction with the first-order shear deformation theory. The analytical solution determining the fundamental frequency of the plate is obtained using the generalised Galerkin method and verified by comparison with the results of finite element modal analysis. The approach developed in the paper and equations obtained are applied to the design of sandwich plates having composite facings and orthotropic core. Design charts representing the effects of the thickness of the facings and core on the mass of composite sandwich panel for a given value of the fundamental frequency are obtained.     

基于剪切耗能假设的黏弹性夹芯梁的振动和阻尼特性

基于一阶剪切变形理论和哈密顿原理建立了三层粘弹性夹芯梁结构的有限元模型并对其振动和阻尼特性进行了研究。建模时认为粘弹材料层不可压缩,振动能量是依靠粘弹性层的剪切变形来耗散的。为验证本模型的正确性,将其与解析解作了对比。同时,为了证明本方法的优越性,将其与常用的“实特征模态”、“近似复特征模态”、“钻石法”和“近似法”四种数值方法做了比较。结果表明本方法的精度在这几种数值方法中是最好的。最后,讨论了粘弹性夹芯梁结构参数变化对系统固有频率和损耗因子的影响,得到了一些有工程实际意义的结论。     

Active constrained layer damping of geometrically nonlinear vibrations of smart laminated composite sandwich plates using 1–3 piezoelectric composites

This paper deals with the analysis of active constrained layer damping (ACLD) of geometrically nonlinear vibrations of sandwich plate with orthotropic laminated composite faces separated by a flexible core. The constraining layer of the ACLD treatment is composed of the vertically/obliquely reinforced 1?C3 piezoelectric composites. The Golla?CHughes?CMcTavish method has been implemented to model the constrained viscoelastic layer of the ACLD treatment in time domain. The first-order shear deformation theory and the Von Kármán type nonlinear strain displacement relations are used for analyzing this coupled electro-elastic problem. A three dimensional finite element model of smart laminated composite sandwich plate integrated with ACLD patches has been developed to investigate the performance of these patches for controlling the geometrically nonlinear vibrations of the plates. The numerical results indicate that the ACLD patches significantly improve the damping characteristics of the sandwich plates with laminated cross-ply and angle-ply facings for suppressing their geometrically nonlinear vibrations. Particular emphasis has been placed on investigating the effect of the variation of piezoelectric fiber orientation angle on the performance of the ACLD treatment.     

Vibration Modeling of Multilayer Composite Structures with Viscoelastic Layers

This work deals with the vibration of orthotropic multilayer sandwich structures with viscoelastic core. A finite element model is derived from a classical zigzag model with shear deformation in the viscoelastic layer. The aim of the present work is to establish numerical models and develop numerical tools to design multilayer composites structures with high damping properties. To fulfill this purpose, a finite element model has been developed for vibration analysis of a sandwich plate (elastic orthotropic)/(viscoelastic orthotropic)/(elastic orthotropic). A numerical study from the variation of the damping properties of the structures was performed according to the faces materials fibers orientation.     

Fundamental frequency of the CCCF composite sandwich plate

The analytical closed-form solution providing the value of fundamental frequency of the CCCF composite sandwich plate is proposed in this article. The variational equations of motion are derived using Hamilton’s principle in conjunction with the first-order shear deformation theory. The solution approach involves reduction of the governing equations of motion to a system of ordinary equations using Kantorovich procedure and subsequent application of the generalised Galerkin method. The analytical solution is verified using the finite element modal analysis. The comparisons of computational results have shown that the fundamental frequency of the CCCF sandwich plates can be calculated with sufficient accuracy using the analytical technique developed in this work.     

A study on material damping of 0° laminated composite sandwich cantilever beams with a viscoelastic layer

The damping behavior of a 0° laminated sandwich composite beam inserted with a viscoelastic layer is investigated. A quantitative analysis of damping in the sandwich laminated composite beam has been conducted through the theoretical method. Results showed that the viscoelastic core thickness in the sandwich beam and the length of the beam have a great effect on the damping loss factor. They also demonstrate the great capability of laminated sandwich composites with embedded viscoelastic layer to considerably enhance structural damping. It is shown that the extended Ni–Adams’ theory can be efficiently used to identify the damping characteristics of the laminated sandwich composite beam.     

SENSITIVITY ANALYSIS AND A MIXED APPROACH TO THE OPTIMIZATION OF SYMMETRIC LAYERED COMPOSITE PLATES

This paper studies the sensitivity analysis of the static structural compliance of symmetric layered composite plates with respect to the fiber orientation in terms of continuous and finite element discretized formulations. From the optimality criterion obtained based on the sensitivity analysis, it is found that the number of different angles of the fibers in an optimal plate is no more than 4 under rather general conditions. Making use of the sensitivity information and optimality criterion approach, we construct a mixed method for adjusting the fiber orientations to minimize the structural compliance of composite plates. Furthermore, the non-convcxity of compliances of composite plates is shown by numerical experiment.     

考虑芯层横向变形的粘弹性复合材料夹层板结构的声振特性分析

夹层板结构具有很高的比强度和比刚度。若芯层采用粘弹性阻尼材料，夹层板结构还具有良好的隔振和隔声特性，因此在工程结构中得到广泛应用。以往的夹层板理论大多忽略了芯层的横向正应变和横向正应力，在分析芯层较厚的夹层板或者夹层结构的高频振动问题时由于不能体现芯层的横向压缩变形，往往显得不够合理。针对这一不足，构造了一个复合材料夹层板单元：夹层板的上下面板采用基于一阶剪切变形理论的Mindlin假定以及层合板理论进行分析；采用文献[6，7]中提出的Timoshenko层合厚梁理论构造了单元每边的转角和剪应变场，消除了Mindlin板单元当板厚变小时的剪切锁死问题；假定芯层的位移沿厚度方向线性变化，并用上下面板的自由度表示，最终形成以上下面板自由度表示的系统总的运动方程。该单元不仅考虑了芯层的横向剪切变形，还考虑了芯层的横向压缩变形。数值计算结果表明：无论对于静力问题、动力问题还是声辐射等问题，考虑芯层的横向压缩变形是合理的，也是有必要的。     

基于Layerwise理论的共固化粘弹阻尼复合材料动特性分析

共固化粘弹性复合材料兼具结构承载和阻尼减振功能。针对传统的混合单元法在应用于粘弹性夹层复合材料结构阻尼性能分析时存在着前处理困难、计算规模大、精度低以及难以考虑正交各向异性铺层自身损耗能力的缺点,推导了一种基于Layerwise离散层理论的四节点四边形复合材料层合板单元,并利用直接复特征值解法建立了共固化粘弹性复合材料结构的阻尼性能分析方法。将该方法应用于不同的阻尼结构,分析结果与文献中已公开结果和混合单元法的计算结果进行了对比验证。结果表明,基于离散层理论的层合板单元具有计算精度高、前处理建模简单和计算规模小的优点,可有效应用于复杂共固化粘弹性复合材料结构的阻尼性能分析和设计。     

Vibration analysis of sandwich plates with lattice truss core

An effective methodology is developed to investigate the vibration of the sandwich plate with pyramidal lattice core. Equation of motion of lattice sandwich plate is established by Hamilton's principle. Displacement fields are expressed with a simple method, and the natural frequencies of the lattice sandwich plate are conveniently calculated. The correctness of the analytical method is verified by comparing the present results with published literatures. The effects of structural and material parameters on the vibration characteristics of lattice sandwich plate are analyzed. The present method will be useful for vibration analysis and design of lattice sandwich plates.     

Torsion of carbon fiber composite pyramidal core sandwich plates

A combined theoretical, experimental and numerical investigation of carbon fiber composite pyramidal core sandwich plates subjected to torsion loading is conducted. Pyramidal core sandwich plates are made from carbon fiber composite material by a hot compression molding method. Based on the Classical Laminate Plate Theory and Shear Deformation Theory, the equivalent mechanical properties of laminated face-sheet are obtained; based on a homogenization concept combined with a mechanical of materials approach, the equivalent in-plane and out-of-plane shear moduli of pyramidal core are obtained. A torsion solution is derived with Prandtl stress function and can be used in the sandwich plate with orthotropic face-sheets and orthotropic core. The influences of material properties and geometrical parameters on the equivalent torsional stiffness are explored. In order to verify the accuracy of the analytical torsion solution, experimental tests of sandwich plate samples with different face-sheet thicknesses are conducted and two types of finite element models are developed. Good agreements among analytical predictions, finite element simulations and experimental evaluations are achieved, which prove the validity of the present derivation and simulation. The proposed method could also be applied in design applications and optimization of the pyramidal core sandwich structures.     

Probabilistic fracture mechanics by Galerkin meshless methods – part I: rates of stress intensity factors

 This is the first in a series of two papers generated from a study on probabilistic meshless analysis of cracks. In this paper (Part I), a Galerkin-based meshless method is presented for predicting first-order derivatives of stress-intensity factors with respect to the crack size in a linear-elastic structure containing a single crack. The method involves meshless discretization of cracked structure, domain integral representation of the fracture integral parameter, and sensitivity analysis in conjunction with a virtual crack extension technique. Unlike existing finite-element methods, the proposed method does not require any second-order variation of the stiffness matrix to predict first-order sensitivities, and is, consequently, simpler than existing methods. The method developed herein can also be extended to obtain higher-order derivatives if desired. Several numerical examples related to mode-I and mixed-mode problems are presented to illustrate the proposed method. The results show that first-order derivatives of stress-intensity factors using the proposed method agree very well with reference solutions obtained from either analytical (mode I) or finite-difference (mixed mode) methods for the structural and crack geometries considered in this study. For mixed-mode problems, the maximum difference between the results of proposed method and finite-difference method is less than 7. Since the rates of stress-intensity factors are calculated analytically, the subsequent fracture reliability analysis can be performed efficiently and accurately. Received 20 February 2001 / Accepted 19 December 2001     

Rates of change of closed-loop eigenvalues and eigenvectors of actively controlled structures

The sensitivity of the closed-loop eigenvalues and eigenvectors of actively controlled flexible structures with distinct eigenvalues is considered. The sensitivity information is essential in the optimization of large space structures with stability/performance robustness, closed-loop eigenvalues or damping ratios treated as objective or constraint functions. The efficiency of the analytical expressions derived for the rates of change of the closed-loop eigenvalues with respect to structural design variables is demonstrated through two numerical examples. The utility of the sensitivity of the closed-loop eigenvalues and eigenvectors in evaluating the derivatives of the stability and performance robustness indices is also indicated and illustrated through numerical examples.     

Bending response of inhomogeneous fiber-reinforced viscoelastic sandwich plates

The static response of an inhomogeneous fiber-reinforced viscoelastic sandwich plate is investigated by using the first-order shear deformation theory. Several types of sandwich plates are considered taking into account the symmetry of the plate and the thickness of each layer. In addition, two cases are considered depending on the viscoelastic material which are included in the core or the faces of the sandwich plates. The method of effective moduli and Illyushin’s approximation method are used to solve the equations governing the bending of simply supported inhomogeneous fiber-reinforced viscoelastic sandwich plates. Numerical computations were carried out to study the effect of the time parameter on deflections and stresses at different values of the aspect ratio, side-to-thickness ratio and constitutive parameter.     

玻璃纤维增强塑料夹层板振动能量流可视化研究

为探索复合材料结构在外部激励作用下的振动能量传递及分布特性,基于结构声强法对玻璃纤维增强塑料夹层板的振动能量可视化技术进行了研究。基于结构声强概念,结合复合材料结构层合理论推导了玻璃纤维增强塑料夹层板的结构声强解析表达式;给出了基于有限元数值方法的结构声强可视化计算实现流程,利用Python和Matlab语言编写了相应计算程序;接着从固有频率、振型及结构声强矢量三个方面验证了所提出可视化程序的有效性;以玻璃纤维增强塑料夹层板为例,利用结构声强技术提供的图形信息实现了激励源定位及振动能量传递特性的可视化分析;最后提取了剪切分量、扭转分量和弯曲分量对结构声强矢量的贡献情况,直观展示了剪切分量在振动能量传递过程中的决定性作用。     

Electroelastic effect of thickness mode langasite resonators

Langasite is a very promising material for resonators due to its good temperature behavior and high piezoelectric coupling, low acoustic loss, and high Q factor. The biasing effect for langasite resonators is crucial for resonator design. In this article, the resonant frequency shift of a thickness-mode langasite resonator is analyzed with respect to a direct current (DC) electric field applied in the thickness direction. The vibration modes of a thin langasite plate fully coated with an electrode are analyzed. The analysis is based on the theory for small fields superposed on a bias in electroelastic bodies and the first-order perturbation integral theory. The electroelastic effect of the resonator is analyzed by both analytical and finite-element methods. The complete set of nonlinear elastic, piezoelectric, dielectric permeability, and electrostrictive constants of langasite is used in the theoretical and numerical analysis. The sensitivity of electroelastic effect to nonlinear material constants is analyzed.     

Free vibration analysis of composite sandwich plates with different truss cores

The free vibration of composite truss core sandwich plates is investigated. The natural frequencies of the sandwich plate are calculated by using the classic laminated plate theory, the first-order shear deformation theory, Reddy's third-order shear deformation theory, and a Zig-Zag theory. The differences between the natural frequencies, obtained from the four theories, are compared. The influences of structural parameters on the natural frequencies and the ratios of natural frequency to equivalent density of the sandwich plates with pyramidal core, tetrahedral core, and 3D-Kagome core are studied with the aid of the Zig-Zag theory.     

Multiscale approach for the design of composite sandwich structures for train application

In the present work a multiscale approach is considered for the design of composite sandwich structures for a roof of railway vehicle. The procedure consists in different steps that start from cost/benefit analysis on materials and their manufacturing process and cycle up to analysis of sub-components and entire structures. Each step is characterized by experimental, theoretical and numerical studies. The design activities herein presented count experimental campaigns able to characterize both the properties of novel sandwich material, manufactured expressly for transportation industry, the sandwich and joint behaviors. Analytical and numerical approaches have been used to validate and optimize the structural layout. Finite element analysis has been also performed on a test article to verify the “new” sandwich roof in regard to structural requirements suggested by European Code.     

粘弹性复合材料夹芯板的稳态响应分析

芯材采用Kelvin粘弹性本构模型,推导了复合材料夹芯板的动力学方程,运用模态正交原理,以Navier完备解形式求解了四边简支正交对称铺层层合板的稳态响应,并给出了固有频率和结构损耗因子的解析解。通过固有频率的有限元解对比验证了数值计算的可靠性。分析了芯材剪切模量和芯材厚度对结构固有频率和损耗因子的影响。探讨了稳态响应的收敛性,并得到结构稳态响应振幅与频率的关系,分析了芯材损耗因子对结构稳态响应的影响。结果表明：芯材剪切模量存在最佳设计值;结构首阶模态特性主导结构的稳态动态响应。     

Conforming boundary elements in plane elasticity for shape design sensitivity

The structural design sensitivity analysis of a two-dimensional continuum using conforming (continuous) boundary elements is investigated. Implicit differentiation of the discretized boundary integral equations is performed to obtain design sensitivities in an efficient manner by avoiding the factorization of the perturbed matrices. A singular formulation of the boundary element method is used. Implicit differentiation of the boundary integral equations produces terms that contain derivatives of the fundamental solutions employed in the analysis. The behaviour of the singularity of these derivatives of the boundary element kernel functions with respect to the design variables is investigated. A rigid body motion technique is presented to obtain the singular terms in the resulting sensitivity matrices, thus avoiding the problems associated with their numerical integration. A formulation for obtaining the design sensitivities of the continua under body forces of the gravitational and centrifugal types is also presented. The design sensitivity results are seen to be of the same order of accuracy as the boundary element analysis results. Numerical data comparing the performance of conforming and non-conforming formulations in the calculation of design sensitivities are also presented. The accuracy of the present results is demonstrated through comparisons with existing analytical results.